Q U A N T U M T H EO RY
O F M A N Y -PA R T IC LE
SY ST EM S
A LEXA N D ER L. FEU ER
Associate Professorof Physics
Stanford U niversity
JO H N D IRK W A LECKA
Professorof Physics
Stanford University
M cG raw-llill,Inc.
New York St.Louis San Francisco Auckland Bogotl
Caracas Lisbon London Madrid M exico City M ilan
M ontrecl New Delhi San Juan SingaN re
Sydney Tokyo Toronto
C O N T EN TS
PREFACE
PART ON EQ INTRO DU CTIO N
1
CHA PTER 1 SECO ND G UA NTIZATIO N
1 THE SCHRODINGER EQUATION IN FlRST AND SECOND
GUANTIM TION
4
B'osons 7
M any-particle Hilbertspace and creation and destruction operators
Fermions 15
FlELDS
19
EXAM PLE:DEGENERATEELECTRON GAS
CHA PTER 2 STATISTICALM ECHANICS
REVIEW OFTHERM ODYNAM ICS AND STATISTICALM ECHANICS
IDEAL GAS 36
Bosons 38
Fermions 45
33
34
CONTENTS
PART TW O ClGROUND-STATE (ZERO-TEM PERATURE)
FO RM A LISM
CHAPTER 3 G REEN'S FUNCTIONS AND FIELD THEORY (FERM IONS)
53
6 PICTURES 53
Scbrödingerpicture 53
lnteraction picture 54
Heisenberg picture 68
Adiabatic ''switching on'' 59
Gell-M ann and Low theorem on the ground state in quantum field
theoœ
61
7 GREEN'S FUNCTIONS 64
Definition
64
Relation to observables 66
Example:free ferm ions 70
The Lehmann representation 72
Physicalinterpretation ofthe Green'sfunclion 79
8 W îCK-S THEOREM
83
9 DIAGRAM MATIC ANALYSIS OF PERTURBATION THEORY
Feynman diagrams in coordinate space 92
Feynman diagrams in momentum space 1G0
Dyson'sequations 1O5
Goldstone'stheorem
111
CHAPTER 4 FERM ISYSTEM S
120
10 HARTREE-FOCK APPROXIMATION
11 IM PERFECT FERM IGAS 128
Scalering from ahard sphere 128
Scattering theorv in momentum space 130
Ladderdiagrams and the Bethe-salpeterequation
Galilskii'sintegralequations 139
The properself-energy 142
Physicalquantities 146
Justification ofterms pelained 149
DEGENERATE ELZCTRON GAS 151
Ground-state energy and the dielectric constant 151
Ring diagrams 154
'
Evaluation ofFID 158
Correlation energy 163
Effective interaction 166
CHAPTER 5 LINEAR RESPO NSE AND COLLECTIV E M ODES
13 GENERAL THEORY OF LINEAR RESPONSE TO AN EXTERNAL
PERTURBATION 172
14 SCREENING IN AN ELECTRON GAS
175
15 PLASM A OSCILLATIONS IN AN fLECTRON GAS 180
16 ZERO SOUND IN AN IM PERFECT FERM IGAS
183
17 INELASTIC ELECTRON SCATTERING 188
171
CHA PTER 6 BO SE SYSTEM S
198
18 FO RM ULATION OFTHE PROBLEM
19 GREEN'S FUNCTIONS 203
199
xi
CONTENTS
20 PERTURBATION THEOBY AND FEYNMAN RULES 207
lnteraction picture 207
Feynmanrulesincoordinatespace 208
Feynmanrulesinm omentum space 209
Dyson's equations 211
Lehmann representation 214
21 W EAKLY INTERACTING BOSE GAS 215
22 DILUTE BOSE GAS W ITH 9EPULSIVECO9ES 218
PART THREEL FINITE-TEM PERATU RE FO RM A LISM
226
CHAPTER 7 FIELD THEORY AT FINITE TEM PERATURE
227
23 TEM PERATURE GREEN'S FUNCTIONS
Definition 228
227
Relattontoobservables 229
Example :noninteracting system
232
PERTURBATION THEORY AND W ICK'S THEOREM FOR FINITE
TEM PERATURES 234
Interaction picture 234
Periodicityof@ 236
ProofofW ick'stheorem
237
25 DIAGRAM MATIC ANALYSIS 241
Feynman rulesin coordinatespace 242
Feynman rules in momentum space 244
Evaluatjonoffrequencysums 248
26 DYSON'S ECUATIONS 250
CHAPTER 8 PHYSICALSYSTEM S AT FINITE TEM PERATURE
265
27 HARTREE-FOCKAPPROXIM ATION 255
28 IM PERFECT BOSE GAS NEAR Tc 259
29 SPECIFIC HEAT OFAN IM PERFECT FERM IGAS AT LOW
TEM PERATURE 261
Low-temperalureexpansion of@
262
Hartree-Fock approximation 262
Evaluation ofthe entropy 265
30 ELECTRON GAS 267
Approximate properself-energy 268
Summation ofring diagrams 271
Approximatethermodynamicpotential 273
ClassicalIim it 275
Zero-temperature limit 281
CHAPTER 9 REAL-TIM E G REEN'S FUNCTIO NS AND LINEAR RESPONSE
31 GENERALIZED LEHM ANN 8EPRESENTATION
292
Definitionofö
292
Retarded and advanced functions 294
Temperature Green's functions and analytic continuation 297
32 LINEAR RESPONSE AT FINITETEM PERATURE 298
Generallheory 298
Density correlationfunction 300
33 SCREENING IN AN ELECTRON GAS 303
34 PLASM A OSCILLATIONS IN AN ELECTRON GAS 3Q7
291
xii
CONTENTS
PART FO UR L CA NO NICAL TRA NSFO RM ATD NS
311
CHAPTER 14 CANONICALTRANSFORM ATIONS
313
35 INTERACTING BOSE GAS 314
36 COOPER PAIRS 320
37 INTERACTING FERMIGAS 326
PA RT FIV EQ A PPLICATIONS TO PHYSICAL SYSTEM S
339
CHAPTER 11 N UCLEAR M AU ER
341
38 NUCLEAR FORCES:A REVIEW
341
39 NUCLEAB M ATTER 348
Nucjearradiiand chargedistributions 348
The semiempiricalmassfonnu#a 349
40 INDEPENDENT-PARTICLE (FERMI-GAS)MODEL 352
41 INDEPENDENT-PAIRAPPROXIMATION (BRUECKNER'STHEORY) 357
Self-consistentBethe-Goldstone equation 358
Solution fora nonsingularsquare-wellpotential 360
So
lutionfœ apurehard-coreqotential 363
Properties ofnuclearmatterwrth a ''realistic''potential
366
42 RELATION T0 GREEN'S FUNCTIONS AND BETHE-SALPETEB
EQUATION 377
43 THEENERGY GAP IN NUCLEAR M AU ER 383
CH APTER 12 PHO NONS AND ELECTRO NS
389
44 THENONINTERACTING PHONON SYSTEM
390
Lagrangian andhamiltonian 391
Debyetbeoryoftbespecificheat 393
45 THE ELECTRON-PHONON INTERACTION 396
46 THECOUPLED-FIELD THEORY 399
Feynman rulesfor'r= 0 399
The equivalentelectroo-electroninte?actîon 401.
Vertex parts :nd Dyson's equations 402
47 MIGDAL'S THEOREM 406
CHAPTER 13 SUPERCOND UCTIVITY
48 FUNDAM ENTAL PROPERTIES OFSUPERCONDUCTORS 414
Basic experimentalfacts 414
The/modynamic relations 417
49 LONDON-PIPPARD PHENOMENOLOGICALTHEORY 420
Derivation ofLondon equations 42Q
Solutionforhalfspaceand slab 421
Conservationandquantization offluxold 423
Pippard'sgeneralizedequation 425
50 GINZBURG-LANDAU PHENOM ENOLOGICALTHEORY 430
Expansion ofthe frae energy 430
Solution in sim ple cases 432
Flux quantization 435
Sudaceenergy 436
413
CONTENTS
51 MICROSCOPIC (BCS)THEORY
Generalformulation 439
Solution foruniform medium
439
444
Delerminali
on ofthe gap functionZ(F)
447
Thermodyoam ic functions 449
52 LINEAR RESPONSETO A W EAK MAGNETIC FIELD
Derivationofthegeneralkernel 455
M eissnereffect 459
454
Penetrationdepth i
n Pippard (nonlocal)Iimit 461
Nonlocalintegralrelation 463
53 M ICROSCOPIC DERIVATION OF GINZBURG-LANDAU
EQUATIONS 466
CHAPTER 14 SUPERFLUID H ELIU M
54 FUNDAM ENTAL PROPERTIES OF He 11 481
Basic experimentalfacts 481
Landau's quasiparticle model 484
55 W EAKLY INTERACTING BOSE GAS 488
Generalformulation 489
Uniform condensate 492
Nonuniform condensate 495
479
CHAPTER 15 APPLICATIO NS TO FINITE SYSTEM S :THE ATOM IC NU CLEU S
56 GENERALCANONICAL TRANSFORM ATION TO PARTICLES AND
HOLES 504
57 TH E SINGLE-PARTICLE SHELL M ODEL 508
Approximate Hartree-Fock wave functions and Ievelorderings in a
centralpotentiaf 508
Spin-orbitsplitîing
511
Single-padicle matrix elements 512
58 M ANY PARTICLES IN A SHELL 515
503
Two val
ence particles;generalinteraction and8(x)force
Severalparticles:normalcoupling 519
The pairing-force problem
523
The boson approximation 526
The Bogoliubov transformation 527
59 EXCITED STATES :LlNEARIZATION OF TH E EQUATIONS OF
MOTION
538
Tamm-Dancoffapproximaîion (TDA) 538
Random-phaseapproximation (RPA) 54O
Reduction ofthe basis
543
Soluti
onforthe ElBl-dimensionalsupermul
tipletwi
th a8(x)force
An application to nuclei:O16 555
60 EXCITED STATES :G REEN'S FUNCTION M ETHO DS
The polarization propagator 558
Random -phase approximation 564
Tamm -Dancoff approximation 565
Construction of1R(r
,))inthe RPA
547
558
566
61 REALISTIC NUCLEAR FORCES 567
Tw o nucleons outside closed shells: the independent-pair approximation 567
Bethe-Goldstone equation 568
Harmonic-oscillatorapproximation 570
Pauliprinciple correction 574
Extensionsandcalculationsofotherquantities 574
xiv
CO NTENTS
A PPENDIXES
A
B
579
DEFINITEINTEGRALS 579
REVIEW OFTFIETHEOHY OFANGULAR MOMENTUM
58
Basiccommutation relations 581
1
Coupl
i
ng
of
t
wo
angk
l
l
ar
moment
a
:
Cl
ebs
ch
-Gor
C
dancoefficients 582
ing ofthree angularmomenta:îhe6lrroupl
/
c
oe
f
fi
cients 585
ble tensoroperatorsandtbe Wigner-Eckar
T educi
ttheorem
586
ensoroperatorsincoupledschemes 587
INDEX
589
N U M ER ICA L VA LU ES O F SO M E
PH YS ICA L Q U A NTIT IES
h- hlz,
z
= l.055 x 10-27erg sec
c
- 2.998 x 1010cm sec-l
ks
- l.381x 10-16erg(OK)-t
me
m#
- 9.110 x 10-28g
= 1.673 x 10-24g
mplme
= 1836
t,
Avogadro'snum ber
= 4.803 x 10-10esu
= 6.022 x 1023m ole-t
ellhc= a
p.a= ehlzmec
y,s= ehllmpc
lRy= eljlan= :4melzhl
= (137.0)-1
= 9.274x 10-21erggauss-l
= 5.051 x 10-2*erg gauss-l
= 13.61eV
tu = hnlmeel
= 5.292 x 10-9cm
Ae=hlm.c
Ap= hlmpc
= 3.862 x 10-1'cm
= 2.103 x 10-14cm
mpc2
= 938.3M eV
(
a = hclle
= 2.068 x 10-7gausscm2
hc
1ev
= 197.3 x 10-:3M eV cm
= 1.602 x 10-12erg
Source: This table is compiled from B.N.Taylor,W .H.Parker.and
D.N.LangenG rg.Ret'.M od.Phys..41:375(1969).
part one
Introd uctio n
1
Second Q uantization
The physicalworld consists ofinteracting m any-particle system s. A n accurate
description ofsuch system s requires the inclusion of the interparticle potentials
in the m any-particle Schrödinger equation. and this problem form s the basic
subjectofthe presentbook. ln principle,theN-body wavefunction in conhguration space contains all possible inform ation,but a direct solution of the
Schrödingerequation isim practical. ltis therefore necessary'to resortto other
techniques, and we shall rely on second quantization, quantum -tield theory,
and the use ofG reen'sfunctions. ln a relativistic theory,the conceptofsecond
quantization is essentialto describe the creation and destruction of particles.i
Even in a nonrelativistic theory.however,second quantization greatly sim plifies
the discussion of m any identical interacting particles.z This approach m erely
!P.A.M ,Dirac,Proc.Rtly'.Soc.çl-ondon),114A :243 (1927).
2P.Jordan and 0.Klein.Z.Physik.45:751 (I927);P,Jordan and E.P.W'igner,Z.Phvsik,
47:63l(1928);V.Fock,Z.Physik,75:622(1932).Although diflkrentin detail,the approach
presented herefollowsthespiritofthislastpaper.
3
lN-rqoouczloN
rerorm ulates the original Schrödinger equation. N evertheless, ithas several
distinct advantages:the second-quantized operators incorporate the statistics
(BoseorFermi)ateachstep,whichcontrastswiththemorecumbersomeapproach
of using sym m etrized or antisymm etrized produets of-single-particle wave
functions. The m ethods ofquantum seld theory also allow usto concentrate
on thefew m atrix elem entsofinterest,thusavoidingtheneed fordealingdirectly
wîth the m any-partiele wave function and the coordinates of a1lthe rem aining
particles. Finally,the G reen'sfunetionscontain the m ostim portantphysical
informationsuch astl4eground-stateenergyandotherthermodynamicfbnctions,
the energy and lifetim e of excited states,and the linear response to external
pcrturbations.
U no rtunately.theexactG reen'sfunctionsarenoeasierto determ inethan
the origillalwave function-and we therefore m ake use ofperturbation theory,
which isherepresented intheconciseand systematiclanguageofFeynm an rules
and diagram s.: These rules allow us to evaluate physical quantities to any
orderin perturbationtheory. W eshallalsoshow,asfirstobserved by Feynm an.
that the disconnected diagram s cancel exactly. This cancellation Ieads to
linked-cluster expansions and m akes explicit the volume dependence of aI!
physical quantities. It is possible to form ulate a set of integral equations
(oyson-s equations) NNhose iteratîons yicld the Feynman-Dyson perturbation
theory to any arbitrary order in the perturbation param eter and which are
indepcndeatofthe originalperturbation series.z since the proa rties ofm anyparticle system s freqttently involve expressions that are nonanalytic in the
couplïng constant, the possibility of nonperturbative approxïm ations is very
im portant.
In addition.itisfkequently possibleto m akephysicalapproxim ationsthat
reduce the second-quantized harniltonian to a quadratic tbrm . The resulting
problem is then exactly solvable either by m aking a canonicaltransform ation or
by exam ining thelinearequationsofm otion.
IITHE SCHRöDINGER EQUATION IN FIRST
A ND sEco No O UA NTIZATIO N
w-e shallstartourdiscussion by m erely reform ulating the Schrödingerequation
in the language of second quantization. ln alm ost all eases of interest,the
ham iltonian takesthe form
N
N
H = .N
.
k-k)+ J O
N- Vlxk,rI)
.'
. Tl
#=1
k=u1=1
where r isthe kinetic energy and Vristhe potentialenergy ot-interaction between
the particles. The quantity xk denotes the coordinates of the é-th particle,
1 R.P . Feynman,Phys.Ret'..76:749(1949);76:769(1949).
' F. J.Dyson-#/?
y'
î.Ret'..75:486(1949),
.
*7521736(1949).
SECO ND QUANTIZATfON
5
including the spatial coordinate xk and any discrete variables such as the z
com ponent of spin for a system of-ferm ions or the z com ponent of isotopic
spin for a system ofnucleons. The potential-energy term represents the interaction between every pair of particles, counîed tlp?t-tx. which accounts for the
factorof1.and thedoublesum runsovt
lrtheindiccsk and/separately.excluding
the valtte k equalto J. svith thishamiltonian-thetim e-dependentSchrödinger
equation is given bq
togetherw ith anappropriatesetofboundaryconditionsforthewavefunctionT .
W e start by expanding the m any-particle wave function t1'in a com plete
set of tim e-independent single-particle wave functions that incorporate the
boundary conditions. For exam ple,ifwe have a large hom ogeneous system itisnaturaltoexpandinasetofplanewavesinaIargeboxwith periodicboundary
eonditions:alternatively,ifwehaveasystem ofinteracting electronsin an atom ,
a com plete set of single-particle coulom b wave functions is com m only used,
.
finally,ifw'
ehave particlesm oving in acrystallattice.aconvenientchoiceisthe
com pletesetofBloch wavefunctionsintheappropriateperiodicpotential. W e
shaliusethegeneralnotation forthe single-particlewavefunctîon
m
1.sk(xk)
.
where Ek represents a com plete set of single-particle quantum num bers. For
exam ple,Ekdenotesp fora system ofspinlessbosonsin a box,orE,J,and -$f
forasetofspinlessparticlesin acentralt'
ield.orp.s'zforahom ogeneoussystem
offerm ions.and so on. It isconvenient to im agine thatthisinl
inite set of
single-particle quantum numbers is ordered (1.2.3.. .. -r-s.t, .. ..u:)and
that Eg runs over this set efeigenvalues. W 'e can now expand the m any-body
w ave function asfollows:
As
--'
'X,
v.;1= yy:,...s.- ,jrN,tyy
-(jj
x1'.
.
Thisexpression iscom pletely generaland issim ply theexpansion ofthem any-
particle w'
ave function in a complete set of states. Since the 4s(.
v)are time
independent,a1lofthe time dependence of the wave function appears in the
coemcientst7(f'1 .. .Es..r).
Letusnow insertEq.(1.3)into theSchrödingerequationand thenm ultiply
b3,
'the expression 4:,(xh)A ''''/isvtxxll,which istheproductoftheadjoint
wave functionscorresponding to aJxpl sett?/-qualltunlr/lfn/ycr.
îEj...Es
J.
IntegrateoveralIthe appropriatecoordinates(thismay include asum overspln
coordinatesifthe particles have spin). On the left-hand side.thisprocedure
.
INTRODUCTION
projectsoutthecoeëcientC correspondingtothegivensetofquantum numbers
fl ' ..fx,andwethereforearriveattheequation
N
ih&d C(Ej...fs.t)= .-,
)
()
(Jdxk/s.txJlF(xk)/p,txJ
p,
. . .
Ek-qpz.+,...Es,t)
x C(E1 '.'f:-IW'fk.
yl
f1-lWz
''Et+l
(1.4)
Sincethe kinetic energy isa single-particle operatorinvolving the coordinates
oftheparticlesoneata tim e,itcan changeonly one ofthesingle-particlewave
functions. The orthonorm ality of the single-particle wave functions ensures
thatallbutthekth particlem usthave theoriginalgiven quantum num bers,but
thewavefbnction ofthe1th particlecanstillrun overtheinsnitesetofquantum
num bers. To be very explicit,wehavedenoted thisvariableindex by l
#'in the
aboveequation. A sim ilarresultholdsforthepotentialenergy. Thesituation
isa little m orecom plicated,however,because the potentialenergyinvolvesthe
coordinatesoftwo particles.Atm ost,itcan changethewavefunctionsoftwo
particles,the kth and /th particles for exam ple,while aIlthe other quantum
numbers mustbe the same as those wehaveprojected out. The quantum
num bersofthekth particlestillrun overan infnitesetofvalues,denoted by H?'
,
andthequantum numbersofthe/th particlestillrunoveraninfnitesetofvalues,
denoted by Hz
''. Each given setofquantum num bersEj '' 'Es leads to a
diflkrentequation,yielding an inhnite setofcoupled diflkrentialequationsfor
thetim e-dependentcoeë cientsofthem any-particlewavefunction.
W enow incorporatethestatisticsoftheparticles. Them any-particlewave
function isassum ed to havethefollowing property
(1.5)
where,asdiscussed above,thecoordinatex:includesthespin foran assem bly of
fermions. Equation(1.5)showsthatthewavefunctionmustbeeithersymmetric
orantisym m etricundertheinterchange ofthecoordinatesofany two particles.
A necessaryandsuëcientconditionforEq.(1.5)isthattheexpansioncoeëcients
them selvesbe eithersym m etric or antisymm etric underthe interchange ofthe
corresponding quantum num bers
(y(...g;j...l;y...,f)..:u(y(...1?; ...jgj...,j)
(j.6j
Thesuëciency ofEq.(1.6)iseasily seenbysrstcarrying outtheparticleinterchangeon thew ave function and then carrying outtheappropriateinterchange
sccoNo OUANTIZATION
ofdummysummation variables. Thenecessityisshownby projectingagiven
coem cientoutofthewavefunction with theorthonorm alityofthesingle-particle
wavefunctionsand thenusingproperty(1.5)ofthetotalwavefunction. Thus
wecan putallthe symmetry ofthe wavefunction into the setofcoeëcientsC.
Boso Ns
Particles thatrequire the plus sign arecalled bosons,and we tem porarily concentrate on such systems. The symmetry ofthecoeë cientsunderinterchange
of quantum num bers allows us to regroup the quantum num bers appearing in
anycoemcient. Outofthegivensetofquantum numbersE l ' ..fx,suppose
thatthestate1occursrljtim es,thestate2occursnztim es,andsoon,forexam ple,
c(121324 -.-,tj= C(=1l
l '-- 2
22 .w=
= . v' =
a!
Nz
A1lofthoseterm sin theexpansion ofthewavefunction withnlpartfclesin the
state1,n2particlesinthestate.
2,andsoon,havethesamecoey cientinthewave
function. ltisconvenientto give thiscoemcienta new nam e
C(nlnz ...n.,/)> C(111 -.. 222 --fl
(1.7)
ng
Consider the normalization of the many-particle wave function. The
normalization condition can berepresented symbolically as
J1'Fl2(#'
r)= 1
(1.8)
which means:takethewave function,multiply itby itsadjoint,and integrate
overa11the appropriate coordinates. The resulting normalization guarantees
thatthetotalprobabilityofsnding thesystem som ewherein consguration space
isunity. Theorthonormality ofthesingle-particlewavefunctionsimmediately
yieldsa correspondingcondition on ourexpansion coeëcientsC
Z
Ej ' ''Es
(C(fl
f.
v,t)12= l
(1.9)
W enow m akeuseofthe equality ofal1coeë cientscontaining the sam enum ber
ofparticlesin thesamestatesto rewritethiscondition in termsofthecoemcientC
J(
1= 1
(l.l0)
(nE,ka......EJ.)
Here thesum issplitinto two pieces:first,sum overa1lvaluesofthe quantum
num bers ELFz ' ' ' fx consistent with the given set of occupation num bers
(n,nzna ...n.),and then sum overallsetsofoccupation numbers. It is
clear that this procedure is merely a way of regrouping a1l the term sin the
sum . The problem of sum m ing over allsetsofquantum num bers consistent
with a given setofoccupation num bersisequivalentto the problem ofputting
8
INTRODUCTION
N objectsintoboxeswithr/jobjectsinthesrstbox,nzobjectsinthesecondbox,
and so on,which can be done in N !/z71!nz!. ''n.!ways. W e thusobtain
them odised norm alization statem ent
X
2-.ni= N
i=l
(1.12)
Bydelinition,however,0!isequalto l,andEq.(1.l1)iswelldehnedaswritten.
O ur resultscan be expressed more elegantly if we desne stillanother coeë cient
-
fx,?)'
)s,(xl)-. .
psslx,
s')
x * ,1az . , . n.(x1xz .
wherewehavedetined
'
/Js,(xl)'''/sx(x,
v)
E ...E
(nl,
1a...;.)
Equation (1.15)isanimportantresult,anditsimplysaysthatatotallysymmetric
wave function can be expanded in term s of a complete orthonorm al basis of
com pletely sym m etrized wave functions t
llu,
a.(
xI .''vN). Furthermore,
sEcoNo aUANTIZATION
9
thecoeëcientsin thisexpansion arejustthesetof/'s. Note the following
Propertiesofthe* 's
ljj(...xj...xy...jc
::
u
a(j)(...xy ...xj...j
j-dxj.--#xsm.,,n,,...a.-(xl.-.xs)%*n,np . . .,,.(xl.-.xs)
,
=
3n:-n,
. . -
3,
,.,.. (1.18)
The srst result follows im m ediately from interchanging particle coordinates,
and then,correspondingly,dummy variablesin the desning equation (1.16);
the second resultfollow sfrom the orthonormality of the single-particle wave
functions. As an explicitexam ple,we shallwrite outthe wave function for
three spinless bosons,two of which occupy the ground state (denoted by the
subscript1)and oneofwhich occupiesthe firstexcited state(denoted bythe
subscript2):
*21:...(
)(.
Yi.
Y2.
X3)
(/1(1)/1(2)/2(3)+/1(1)/X2)/i(3)+/2(1)/1(2)/1(3)2
=Y
x/q
W ereturnto theanalysisofEq.(1.4),whereitisimportantto remember
thatE j . . ' fzv isagi
vensetofquantum numbersin thisexpression. Consider
irst the kinetic-energy term ,which can be rewritten in an obviousshorthand
N
N
Z Z Icfkfrlr#')C(fl'''Ek-jI#'Q ..I'''EN,t)=k=1
ZZ
((fktrrrP')
:=1 <
<
(1.19)
H ere,the right side m akes explicitthe observation thatthe quantum num ber
Ek occurs one lesstim e and the quantum num ber Hzroccurs one m ore tim e.
N ow every tim eEktakesthesam evaluein the sum m ation overk,1etussay E
(thisoccursnztimes),itmakesthesamecontributionto thesum. Therefore,
and thisisthe crucialpointin the wholetreatm ent,instead ofperform ing a sum
overk from 1to N ,wecan equally wellsum overE and write
Z
nz
E
Thatis,asum overparticlesisequivalentto asum overstates,and Eq.(1.19)
becom es
N
Z Z f.Ek1T!< )Ctfl '''Ek-t1.
#f,+l '''Es,?)
#=l G
=
)
()((Fjr(rf')ns
E +
x C(nInz-.-ns- 1 ---nw.+ 1 .'-n.,t)
INTROOUCTION
The sum on E is now insnite.running overa1lofthe single-particle quantum
num bers,butm ostof the A7s are zero since those statesare notoccupied in the
original given set of quantum num bers E l . ' . E ,s.. Finally, it is convenient
to sim plify the notation,which yields
VN N',
.
.l;i: xEk.
IF'1
4/z
')C(Ej...E<-,I
ZI
Z'
F:-,...E.
s.,t)= jié-N
'ni/i'
Erpj'
.
k.
J-
h((:%(,7j?7a . . .??i- 1 . ..11,-h1 .../7aa,t) (l.(!1)
Exactly the sam e m anipulationsapply to the potentialenergy term :
V
1r N' N'N'''EkE li;z'i
IjJ,
'4,
6,
/'-'
.
)
kcptI=1 W' H'
E 1- l U/'E l+l
%'
=
Jl
N' N N'.xEk E I'p'!
.jf.'j,
jz?.
- . .-.
R= l=1 $$' 1i''
' 'l:Ek- 1 - - - 11wr-u.1
l.
lEI - 1 - - -
,7p-'+ 1 ' ..n.,t) (l.22)
As in the preceding discussion.the statesEkand flare each occupied one less
tim e,and the states Wzrand 11,
''are eac'
h occupied one m ore tim e in thissum .
Again.every timeEktakesthe sam evalue,say f (thisoccursnEtimes),and L
takesthesamevalue,sayE'(thisoceursns,times).itmakesthesameeontribution
to the sum . There is only one slight further com plication here.owing to the
restriction k # l in the double sum . Thus it is necessary to use the fbllowing
counting forthe num berofterm sappearing in thissum
1.??s??s,ifE % E'
!ws(?7s- l)ifE = f'
because the restriction k ,
#nldoes not afl
kct the counting ifE ynF ',while the
eigenvalueE iscounted onelesstim ein thesecondsum ifE = F '. Thepotential
energy now becom esa doubly intinite sum ,butm ostofthe factors?7sand ny'
arezero. Thtls.justasbefore.wecanwriteEq.(1.22)as
N
1- V
.,
u'
.5
=-1, L.
'Ekfjrp'(4,
f.
,rf.
,''
y
,
-
krh.l-1 #fkp''
. .
.
Ek-lrJ.
'
ék-.l . . . E I- l jf,z'E I.+l .
. .
.
tlX,t)
(1.23)
SECOND QUANTIZATION
lfthecoeëcientsf defined in Eq.(1.13)aresubstituted into Eqs.(1.21)
and (1.23),wearriveatthefollowinginsnitesetofcoupled diFerentialequations
. . .a.,t)
. . nj- .
! . . . ng.j.j .
' ' '
X
(ni 2)!'''(nk>Fl)!'''(nlM-l)!'''1
.
N!
(1.24)
where(çetc.''standsfortherem aining 13 possibleenum erationsoftheequalities
and inequalitiesbetween the indicesi.j,k,and /. M ultiplication by the factor
(N l/nj!nz!...n.!)1on both sidesoftheequation finally yieldsthecoupled
setofequations
ih & .é(,
7,
nx,?)
n.,t)
xJtaj '''ni--2 '''nk+ 1 -''nt+ 1 '.'n.,?)+ ete. (1.25)
There is such an equation for each setofvalues ofthe occupation num bers
nlnz . . .n.;in thisform ,the equations are very com plicated. A sshow n in
the following discussion,however,itispossibleto recastthese equationsin an
extrem elycom pactand eiegantform .
INTRO DUCTION
M A NY-PARTICLE HILBERT SPACE AN D CREATION
A ND D ESTRUGTION OPERATO RS
W e shalltem porarily forgetthe jlzoviousanalysisand instead seek a com pletely
dipkrent quantum -m echanical basis that describes the num ber of particles
occupyingeach statein a com pletesetofsingle-particlestates. Forthisreason
weintroducethetim e-independentabstractstate vectors
.. .yjx)
wherethenotationm eansthattherearenjparticlesintheeigenstate1,nzparticles
in the eigenstate 2,etc. W e wantthisbasisto be com plete and orthonormal,
which requiresthatthese statessatisfy the conditions
.ttX
'!
nz .
.nj
orthogonality
(1.26)
com pleteness
N ote thatthe com pletenesssum ig overallpossible occupation num bers,with
no restriction. To m ake this basis m ore concrete,introduce tilne-independent
operatorsbk,b%
sfy thecom m utation rules
k thatsati
(/u,?(')=bkk. bosons
p b,
k'kJ- pl
k5:k
1,2- 0
(1.27)
Thesearejustthecommutationrulesforthecreaticnand destructionoperators
of the harmonic oscillator. AIIof theproperties of these operatots,follow
directly from thecom m utation rules,forexam plel
bk
%bk(
pk)= akfakh nk= 0,1,2,...,:s
'kjnk)-=(ak)1)Ak- 1)
bk
ttn.)= (n.+ l)1(nk+ 1)
(1.28)
The num ber operator>l
genvaluesthatincludesa11the
k bk has a spectrum of ei
positive integers and zero. bk is a destruction operator that decreases the
occupation numberby1and multipliesthestatebyn1;b%
kisacreationoperator
thatincreasestheeigenvaluebyoneand multipliesthestateby(nk+ 1)+. The
proofofthese relationsappearsin any standard book on quantum m echanics;
since itiscrucialthatallofthese resultsfollow directly from the com m utation
rules,wehereincludeaproofofEqs.(1.28).
1Compare L.1.Schi/,ttouantum M echanics,''3d ed-,pp.182-183.M cGraw-HillBook
Company,New York,1968. '
SECON D QUANTIZATION
Theoperatorb'b isaherm itian operatorand thereforehasrealeigenvalues,'
callthisoperatorthenum beroperator. Theeigenvaluesofthe num beroperator
are greater than orequalto zero.as seen from the relation
/?,= Nk
-(.n.b%.?A)
/1= '
x11,'J)Tb '
m b ,7 2 ..- ()
Now considerthecommutation relation,which followsfrom Eq.(1.27).
(/7ï'b.>),
.
-.
-.
b
W ith Eq.(1.30).itiseasy to seethatthe operatorb acting on an eigenstate NNith
eigenvalue ?; produces :1 ncNs eigenstate of the num beroperatorbutu.ith eigenvalue reduced by one unit. This result is proNbed ur
ith the follou'ing relations
b%b(b'?? )= blb'
bd?)'??'-,-g/lfb ?
7)11
= (
n- l)(b.
1
,3(
')
Repeated applications of b to any eigenstate m ust eventually give zero. since
otherwise Eq.(1.31)could produce a state with a negative eigens'alue ofthe
number operator.in contradiction to Eq.(l.29). Hence zero is one possible
eigenvalue of the number operator. In exactly thc same way. the adjoint
com m utation rule
(l'#b.d?1')= b%
showsthatb%isthecreation operatorand increasesthet.l. . ..alue ofthe num bc'r
operatorby one unit. The hrstofEqs.(l.28)isthusproved. Furthermore.a
combination ofEqs.(l.29)and (1.3l)l'ields the second of Eqs.(1.28).apart
from an overallphase which can be chosen to be unity with no lossofgenerality.
FinallythelastofEqs.(1.28)isproved in exactlysimilarfashion.l
The preceding discussion has been restricted to a single m ode.
however.readily verised thatthe num beroperatorsfordifTerentm odescom m ute.
which m eansthattheeigenstatesofthetotalsystem can besim ultaneouseigen-
statesofthe set(??k
'
t>t.
)-(ê&). ln particular,ourdesired occupation-number
basis states are sim ply the direct product ofeigenstates ofthe number operator
foreach m ode
1?7l?7: ' .- z7as''=-1??j
Considernow the qtlestion.can we rewrite the Schrödingereqtlation in
term softhesem ore abstractstatevectors? Form thefollow ing state
1
'
1
1.
*(t)'
N=
'
V
11In2 ' ' ' Ra;
f(r?lnz '.'nx,t)!nInz''
14
INTBODUCTION
wherethef'saretakentobethesetofexpansioncoeëcientsofEq.(1.15)and
satisfy the coupled partialdiserentialequations(1.25). Thisstate vector in
the abstractHilbertspacesatissesthediflkrentialequation
ih ê
ê
jj(Y'(J))= a)nz X
ihj-jf(a!nz .-'ax.t)ialnz '''a.)
' ' ' n.
(1.35)
Since thebasisstatevectorsareassumed to be timeindependent, theentiretim e
dependen'ceoftheequation iscontained inthecoeëcientsJL Asanexample,
lookatthesecondkinetic-energyterm inEqs.(1.25).
(1.36)
The dum m y indices in this sum m ation m ay be relabeled with the substitution
ni- 1Y n; nl+ 1> nJ
'
,
N
n
t' t= h
N-??/
nke ny
'
(k.+ iorj)
(1.37)
Furtherm ore,itispossibleto sum theprim ed occupation num bersoverexactly
the sam e values as the originalunprimed occupation num bers, because the
coeëcient(nî)1(?n + 1)+vanishesfornJ
'= 0and forz?;= -1. ThusEq.(1.36)
m ay berewrittenas
?
ihj/1F(/))- .-.+ n 'n ')() n ' i)
g(iIF1/)/('#
l .l
' '. @ $
z7
/
.
,---,tj
.nl
.
((n3'=s)
x(n;+ l(
)1(nJ(
)11...n'
j+ l ...nj- 1 .
(1.38)
N ow observethatthestatevectorwiththevalueofa'
fraised by oneand thevalue '
nj.lowered byone,togetherwiththemultiplicativestatisticalweightfactor,can
be sim ply rewritten in term softhe creation and destruction operatorsacting
onthestatevectorwitha;andnJ
'
(n;+ 1)1(nJ)1In(.-.n;+ 1 -..ng
'- 1 -.
(1.39)
The only dependence on the occupation numberleftin this expression is con-
tained in thecoeëcientsf and in thestatevector;hencethesummationcanbe
carried outand givesouroriginalabstractstatevectorlkle(/))desned by Eq.
(1.34). Thusthisterm intheenergyreducesto thefollowingexpression
P
.
ihjj1
:lP4/))- --.+ )( ,
Lilrl/)bi
#:/1*1/))
.+ --f# J
.
.
(1.40)
SECOND Q IJANTIZATION
1g
Theothertermsin the hamiltonian can be treated in exactly the same fashion;
asa consequence,thisabstract state vector 1
N'41)) satishesthe Schrödinger
equation
f/ij-j1,1,(r))- 4 I,.
I'(f))
(1.41)
where the circumcex denotes an operator in the abstractoccupation-number
Hilbertspace(exceptwherethisisobvious,asin thecreation and destruction
operators)and thehamiltonianX isgivenbytheexpression
X=Z
b%
i(i1Fl/)bi+'
itZ
b%
i4''
J(#lFl
k')bkbk
CJ
-lkl
.
(1.42)
It is im portant to distinguish between the optrators and c num bers in
'
Eq.(1.42). ThusW isan operatorin thisabstractoccupation-numberspace
becauseitdependson the creation and destruction operators. In contrast,the
matrix elementsofthe kinetic energy and the potentialenergy taken between
the single-particle eigenstatesofthe Schrödinger equation in srstquantization
aremerely complexnumbersmultiplyingtheoperators. Equations(1.41)and
(1.42)togetherrestatetheSchrödingerequationinsecondquantization,and al1
ofthestatisticsand operatorpropertiesarecontained in thecreationand destruc-
tion operators !/ and b. The physicalproblem is clearly unchanged by the
new formulation. Inparticularthecoeëcients/specifytheconnectionbetween
Erstand second quantization.
Forevery solution to theoriginaltime-dependentmany-particleSchrödinger
equationthereexistsasetofexpansioncoescientsf Giventhissetofexpansion
coes cientsh itispossibletoconstructasolutiontotheproblem fasecondquantizatl
bn,asshownabove. Conversely t
f theproblem f.
çsolvedI'
nsecondquandzation,
wecandetermineasetofexpansioncoes cientsh whichthenyield asolution to
theoriginaltime-dependentmannparticleSchrödingerequation.
FERM Io Ns
Ifthenegativesign inEq.(1.6)isused,theparticlesarecaliedfermions. The
sam e general analysis applies,but the details are a little m ore com plicated
because ofthe minus signs involved in the antisymmetry ofthe coeëcient C.
C(...Fj...E.
j...,t)= -C(...Eg...Ji
e
t...jf)
(1..43)
The C%s are antisymmetric in the interchange ofany two quantum numbers.
which impliesthatthequantum numberEkmustbe diflkrentfrom the quantum
numberEJ or the coeëcientvanishes. Thisresultshowsthat the occupation
numbernkmustbeeitherzeroorone,whichisthestatementofthePauliexclusion
principle. Any coeëcientsthathavethe same statesoccupied areequalwitlkin
aminussign,and itispossibleto desneacoeëcientC
C(a,nz.''n.,t)* C('''Ej< EJ< Ek ''
(1.44)
16
INTRODUCTION
wherewesrstarrangeallthequantum numbersinthecoeëcientCinanincreasing
sequence. Exactlyasbefore,themany-particlewavefunctionT canbeexpanded
as
l
T'(xj...xs,t4= nl ' ')(
2 flnk .--n.,t)*n,.....(xj ---xxj
'nqo=(1
(1.45)
wherethebasiswavefunctions* aregiven by a normalized determinant
'
t
r
4,.,n(xl) .
nj!
*.: . . ...
(xl.--xsj=
kzzjotxxlj
r
a. !+1
(
j
(
$
'
x!
j
j (1.46)
1/z.
x('(xI) -'-/ss(,
(x.
v)!
The single-particle quantum num bersofthe occupied statesare now assum ed
to be ordered Eç
l< fz
0...< fj. These functions form a complete setof
orthonorm alantisym m etrictim e-independentm any-particlewavefunctionsand
areusuallyreferred to astheSlaterdeterminants.t
ltisonce m oreconvenientto introduce the abstractoccupation-number
spaceand desne
î
Xl'(f))= Sl5z Z
fçnjn2'''n.'t)1n1nz -''n.)
' * '?I*
(1.47)
Here thecoeëcientsfobey equationswllich diflkrfrom (1.25)only byphase
factors(seeEq.(1.57))andtherestrictionsthatnt= 0,I. Thisrestrictionswhich
reqects the particle statistics,must be incorporated into the operators in the
abstract occupation-num ber space. As a convenien'
t procedure, we shall
follow them ethod ofJordan and W ignerzand work withanticom mutation rules
llrslf
sl= 3rs
fermions
(1.48)
lJ.,J,)=(t0,J1)=0
wheretheanticom m utatorisde:ned bythefollowingrelation
(#,.
B)- !-.
1,.
B)+% ,
.
4.
B+ .
9,4
(1.49)
Itiseasilyseen thatthisdiFerentsetofcomm utation rulesproducesthecorrect
statistics:
l*a.l=JJ
12=0*
'therefore d f010)=0, which
occupyingthesam estate.
'J.C.Slater,Phys. Rev.,34:1293(1929).
2P.Jordan and E.P. W igner.loc.cit.
prevents two particles from
SECOND QUANTIZATION
17
2 a%a= 1- aa% (where we take the operatorsreferring to the same mode);
therefore
(fzfas2- 1- laa%+ aa%alf= 1- 2/2 + a(l- J2)a%= 1- aa%= a%a
or
a1d l- a%* - 0
(1.50)
ThislastrelationimpliesthatthenumberoperatorforthexthmodeH,=u1J,
has the eigenvalueszero and one.asrequired. Furtherm ore,itis straight-
forward to prove that the commtltator lâr.r
lsl vanishes,even though the
individual creation and destruction operators anticommute. This result
permitsthesimultaneousdiagonalizationoftheset(Hr),inagreementwiththe
desnition oftheoccupation-numberstate vectors.
3.Theanticom m utation rulesthem selves.alongwith an overallchoiceofphase,
therefore yield the following relationsfortheraising and lowering operators
c'
fpO)- I1))
af'j
h
.I
, = 0
The anticom m utation rules slightly com plicate the direct-product state
1nlnz...r?.),because it becomes essentialto keep track ofsigns. With
thedeflnition
'
1n,nz .' 'n.x = (t7!
t)n!tut
la...(u'
t
j
2lr
r)n.jg.
.
(j.j;)
,
wecan com pute directly the esectofthe destruction operatorason thisstate;
ifns= 1,we 5nd
as':
nb?71
, . .
&.
x))= (-j).
VLak
jjn1...Las/s
Y)...(a.
Y)n1
oj()j
(j.j3j
wherethephasefactorSsisdetined by
Ss= n!+ nz.
k..' '+ n,-!
(1.54)
N ote thatifns= 0,the operatorascan be m oved a1lthe way to the vacuum ,
yielding zero. Ifthereisone particle in the statens,itisconvenientsrstto use
1= l - a%
the anticom m utation relation asas
,a, and then to take the as of the
second term to the vacuum where it gives zero. Thus we finally arrive at the
threerelations
ifns= 1
otherwise
ifns= 0 (l.55)
otherwise
a#
..
& asj
@.. .ns '
18
INTRODUCTION
The third relation istbe m ost useful,forit shows thatthe number operator
introducesno extra phases. Thereason forwriting thesrsttwo equationswith
thefactor(n,)+and (a,+ 1)1instead ofthemorefamiliarnotationusingnsand
1- nsisthattheynow assumeexactlythesameform asbeforesoryanish,except
fortheextraphasefactor(-1)*.
As an example ofthe role played by thisfactor in the case offermions,
considerthekinetic-energyterm inEq.(1.4)
i
!
s
p
z
'
l
k
'
ihjjC(E1.''fs,tj= k=l
jg1)
;EkJFI1
zP')
G
xC(El--.Ek-lWz
fkml-'-Ex.
,t)+ --- (1.56)
whereEj '. .Es isagiven setofquantum num bers. Thesequantum num bers
can bereordered simultaneously on 80thsidesofthisequation into theproper
sequence. O ne problem rem ains,however,because the quantum num ber I
'
F
appearson theright,whereEkappearson theleft. IfI'
P ism oved to itsproper
place in the ordered form ,an extra phase factor
(-1)npr+l+ap'+z+'.'+nEk-l F < Ek
(- 1)az:k+I+afk+a+ ''''
bnw-l j?
f'
r> Ek
isneeded,representing the num berofinterchangesto putF 'in itsproperplace.
Justasbefore,wenow go overtothe/coeëcientsandagain changevariables.
For exam ple,consider partofthe kinetic-energy term
P .
1h.
jj7
%'(t))= ...+ nj'aa'.
N
.
I
,
.
S, (irFIj)
N
'''nco' i< j
.
/'(.. .t'
t'
j .. .t);...,(j(n'
jy jj1(nJj1jay.jJay,j
x(-1)'I-+I+ai'+z+'''+nJ'-l(...ni
'uI
-'
1 ..-'
t
n*J
'.
-*
1 ...%
)+ ... (l.58)
.
x
.
,
N ote that the phase factor appearing in this expression is equivalent to
(-l)SJ-S'-nï'.furthermore,thisterm contributesonlyifn;isequalto0and n)is
equalto 1,sothatthisphasefactorisjust(-1)&-:f. Equations(1.55)now allow
us to rewrite the relevantfactoras
bni-()3,)'I(??;+ l)1'(n,
')1(-1ls.
l-siJ...ni
'+ 1 .--nJ
'- 1 '.-)
- J!
lajjn1...n.
') (l.59)
which demonstratesthatthe creation and destruction operatorsdefned wff/l
.
the anlicomm tl:ation rules indeed have the required properties. ln this way
tiieSchrödingerequation again assumesthefollowing form in second quantization
0 . -
,zgt 1
,.
1,(?)p- z?'1
.1.(?)>
H'
*= j)ar
tyxr(
ty$
ju
vx
z
.f7s+ j x u'
uj
c
.
s
s!
$
uy
auaf
rt
s'
x.y
(p'
1t
,.,
,
rs
.
rst;l
,
sEcoNo ûUANTIZATION
19
N ote particularly the ordering of the snal two destruction om rators in the
ham iltonian,which isoppositethatofthelasttwo single-particlewavefunctions
inthematrixelementsofthepotential(thereaderisurgedtoverifythisindetail
forhimself). Inthecaseofbosons,ofcourse,thisorderingisirrelevantGcause
the Enaltwo destruction operatorscom m ute with each other,butforferm ions
the order asects the overall sign. Exactly as before, Eq.(1.60)is wholly
equivalent to the Schrödinger equation in Erst quantization,but the phases
arising from theantisym m etfy oftheexpansion coeE cientshavebeen incorporated'into theham iltonian and thedirect-productstatevectors.
Z/FIELDS
Itisoftenconvenienttoform thelinearcom binationsofthecreationanddestruc-
tionoperators(denotedc'andcforgenerality)
#(x)- Z /2xlck
k
C(x)-Xk /2x)fcl
.(2.j)
'
where the coeE cientsarethesingle-particle wavefunctionsand the sum isover
the com plete setofsingle-particle quantum num bers. ln particular,the index
kforspin-èfermionsmaydenotethesetofquantum numbers(k,Jz)orLE.L,LM ;
and thecorresponding wave functionshave two components
x)I
jz
(ao
/2x)= /k
/k(
(x)2 O Sklxi tz= ,
Thesequantities'
#,and '
(darecalled/el#operators. Theyareoperatorsinthis
.
abstractoccupation-num berH ilbertspacebecause they depend on thecreation
and destruction operators. The Eeld operatorssatisfy sim ple com m utation or
anticom m utationrelationsdepending on thestatistics
(fk(x),'
/'
;(x'))v-I
k/k(x)./.(x');-3.j:tx-x')
(2.3J)
(fk(x),#j(x,))v- E#.
t(x),##
1.(x'))v- ()
(z.a,)
wheretheupper(lower)signreferstobosons(fermions). Herethesrstequality
in Eq.(2.3J)followsfrom thecommutation oranticommutation relationsfor
the creation and destruction operators.and the second followsfrom the com pletenessofthesingle-particlewavefunctions.
Theham iltonian operatorcan berewrittenin term softheseseld operators
asfollows:
4 =J#3x##(x)r(x)#(x)+ .
l.JJd'xdqx'##(x)##(x')F(x,x')Wx')Wxl
.
(2.4)
Thisexpression isreadily verised sincethe integration overspatialcoordinates
producesthesingle-particle m atrix elem entsofthe kinetic energy and potential
2:
INTRODUCTION
energy,leaving a sum ofthese m atrix elem entsm ultiplied by the appropriate
creation and destruction operators. An additional matrix element in spin
spaceisimpliediftheparticleshavespin-è. Notecarefullytheorderingofthe
lasttwo held operatorsin thepotentialenergy,which agreeswith ourprevious
remarksandensuresthat# ishermitian. lnthisform,thehamiltoniansuggests
thenam esecondquantization,fortheaboveexpression looksliketheexpectation
va
1lueofthehamiltoniantakenbetweenwavefunctions. Thequantities# and
# arenotwavefunctions,however,butfleldoperators;thusinsecondquantization the heldsaretheoperatorsand the potentialand kinetic energy arejust
complex coeëcients.
Theextensiontoanyotheroperatorisnow clearfrom theforegoinganalysis.
Forexample,considerageneralone-body operator
N
J= f1
2J(xj)
x.zl
(2.5)
written in ârst-quantized form. Thecorresponding second-quantized operator
isgiven by
7=72
trIJlJ)4 cs
'J
=
Jd% J
2#,(x)#J(x)#,(x)cv
%cs
FJ
=
Jd3x#A(x)J(x)#(x)
(2.6)
where the last form is especially useful. In particular,the number-density
operator
N
n(x)==l1
4 3(x--xt)
-1
(2.7)
becomes
H(x)- )
(/.(x)#/a(x)c#
rca- 4'(x)4(x)
r'
(2.8)
whilethetotal-num beroperatorassum esthesim pleform
X =J#3x:(x)=72cr
1c,= I Av
=
J:3x4#(x)lx)
.jg,.9j
because of the orthonorm ality of the single-particle wave functions. The
numberoperatorcommuteswiththehamiltonianofEq.(2.4),ascanbeverifed
by usingeitherthecom m utation rulesforthecreation and destruction operators
or those of the seld operators. This result is physically obvious since the
ordinarySchrödingerham iltonian doesnotchangethetotalnum berofparticles.
W e infer thatX is a constant of the motion and can be diagonalized simultaneously with the ham iltonian. Thus the problem in the abstract H ilbert
sEcosD aUANTIZATION
21
spaceseparatesinto asequenceofproblemsin the subspacescorresponding to a
fixed totalnumberofparticles. Nevertheless,theabstractHilbertspacecontains
stateswithanynumberofparticles.
K EXAM PLE : DEG ENERATE ELECTRON GAS
To illustratetheutility ofsecond quantization,weconsidera simple modelthat
provi'
desa srstapproximation to a metalora plasma. Thissystem isan interacting electrongasplacedin auniformlydistributed positivebackground chosen
to ensurethatthetotalsystem isneutral. ln arealm etalorplasm a,ofcourse,
thepositivechargeislocalizedin theioniccores,whosedynam icalm otion m ust
also beincluded in the calculation. These positiveionsare m uch heavierthan
theelectrons,however,and itisperm issible to neglecttheionicm otion entirely.
In contrast,the assumption ofa uniform background ismore drastic;forthis
reason,thepresentm odelcan provideonly aqualitativeaccountofrealm etals.
W e are interested in the properties ofthe bulk medium, Ittherefore is
eonvenientto endose the system in a large cubicalbox with sidesoflength L ;
thelim itL -* :c;willbetaken attheend ofthecalculation. Inauniform inhnite
m edium ,a1lphysicalproperties m ust be invariant under spatial translation;
thisobservation suggeststhe useofperiodicboundary conditionson thesingleparticlew avefunctions,which then becom eplane-wavestates
/kA(x)= F-lezfk'xTà
(3.1)
Here P-(>Z3)isthevolumeofthebox,and '
qàarethetwo spinfunctionsfor
spin-up and spin-down along achosenzaxis,
1
Tt= 0
0
T(= 1
Theperiodicboundary conditionsdeterminetheallowed wavenumbersas
ki= lvni
ni= 0,+1,+2,
L
Thetotalhamiltonian can be writtenasthesum ofthreeterm s
H = H et+ H v+ H el-v
where
H
N
e2=
z 1 N e-Hj
ry-yjj
PJ2
2 + -e
-
i= 1 - 2
i*:J
rrf- ryI
i
istheham iltonian fortheelectrons,
N(X)D(X')E'-BIX-X't
Ix x,l
Hb=1'
C2JJd'X#3X'
-
(3.4)
22
INTRODUCTION
istheenergyofthepositivebackgroundwhoseparticledensityisnlxj,and
H
N
D(X)'-VlX-'''
et-b=-E'
2fI
J
#3
x
- ,
(3.6)
a.l
i
X- rjI
is the interaction energy between the electrons and the positive background.
W e have inserted an exponentialconvergence factor to dehne the integrals,
and Jzwilleventually be allowed to vanish. Because ofthelong-rangenature
ofthecoulomb interaction,thtthreetermsin Eq.(3.3)individually divergein
the e4therm odynamic lim it'' N ->.cc, F -->cc, but n = N/P'constant. The
entire system is neutral,however. and tht sum of these terms must remain
m eaningfulin this lim it. The presence of the convergence factor Jz ensures
thatthe expressions are mathematically welldesned atevery step and allows
ustom akethiscancellationexplicit. Sinceweareinterested in thebulkproper-
tiesoftheneutralmedium,forexample E(N (which dependsonly on n),our
limiting procedure willbe firstL -+.co and then p.--+0. Equivalently.we can
assum ep,-i<tL ateach stcp inourcalculation;thisallowsusto shifttheorigin
of integration atwill,apartfrom surface corrections,which are negligible in
thislim it.
InEq.(3.3)theonlydynamicalvariablesarethosereferringtotheelectrons,
because the positivebackground isinert. ThusH bis a purec num berand is
readilyevaluated fora unifbrm distribution n(x)= NIP':
Hb-.
p2(v
X)2JJd'xd'x'C
I
x
-Y'
Xx
X'!
'
p2(X
p
v)2jy'.
xjdqzeZ'z
- .
M 24,
r'
=
J,2'.'
p''H
M
H erethe translationalinvariancehasbeen used to shiftthc origin ofintegration
in the second line. Thequantity N -1.Jf,diverges in the limitJ.
t-+.0,because
the long range of the coulomb potentialallows every element of charge to
interactwith every otherone.
In principle,H et- b isaonc-particle operatorsinceitactson each electron.
Forthe presentsystem ,however, we m ay again usethetranslationalinvariance
to write
N N
t'-9lX-T6t
H el-1
)= -e2 - d3x
-
d.1/'
N
lx--rf!
1
e-PZ
= -e2f -P d?z
-l
z
N 24.
= - e2
+ 0
#
(3.8)
SECOND QUANTIZATION
23
showing thatH et- b isinfacta cnum ber. Thetotalham iltonian thusreducesto
S = -Je2NlP'-14,
ap.-2+ Sel
and a11ofthe interesting physicaleflkctsare contained in H et. W e shallnow
rewrite Eq.(3.9) in second quantization. The kinetic-energy term requires
the m atrix elem ent
hlk2
jA z j
-d3xel(kz-kI).x
m P' : 2'
=l
A21.2.
=
'
-'
lm-'2JAjAajklka
(3.10)
wherethe usualdesnition oftheK roneckerdelta
J-d5xgf(ka-.
kl).x= prjk1ka
hasbeen used. Thekinetic-energy operatorbecom es
j'=
h2k2tz1 a
k/
2m
k/ kA
which can be interpreted as the kinetic energy ofeach m ode m ultiplied by the
corresponding num ber operator. For the potential energy it is necessary to
evaluate thefollowing m orecom plicated m atrix elem ent
(3.l3)
W ith the substitution x = xz and y = xl- xc,this txpression reduces to
e2
.
)kl21k2AzrlZl
k3A3k4A4)= Fj J3xe-ffklmka-kJ-k4l*x
&
=
e2
4.
,:
.
T 3AIza8za,14îklykz,ka+k.(kj- ky)2+ y.2 13-14)
wheretheK roneckerdeltasinthespinindicesarisefrom theorthogonality ofthe
spin wavefunctions. O nceagain,wehaveshifted theorigin ofintegration, and
24
INTRO DUCTION
thefinalKroneckerdeltarepresentstheconservationofmomentum ina uniform
system . Thetotalhamiltoniancan now bewritten as
#
1e2N14=.
= -'
i #' --j+
p
:2
h2k2
e2
t/kztuj+.
jp
2X Y
=
J) J( J) 3,
41Ay3:2A4êkImka,kyyk.
k:AlkzAa kzA:sk4A4
X '- ---
471.
#
'
t
(kly-k3)
-i-+--pzJk!AIJkzâzJk4AAak,:
13
,
(3.15)
The electricalneutrality ofour system m akes itpossible to elim inate p,
from theham iltonian,aswe shallnow show. The conservation ofm om entum
really limitsthesummation over(kf)to threeindependentvariablesinstead of
four. Thechangeofvariables
kj= k + q
k)= k
kz= p - q
k4= p
guaranteesthatk:+ kc= ks+ kl,and furthermoreidentihesâtkl- k3)= âqas
the m omentum transferred in the two-particle interaction. W ith thest new
variables,thelastterm ofEq.(3.15)becomes
el
2P'
4,
r t
#
q2+.jz'iJk4.q,A)Jp-q,AzJpAa t'kp
à1
(3.16)
kpq A:Az
wheretwoofthespinsum m atiollshavebeenevaluatedwiththeK roneckerdeltas.
ItisconvenienttoseparateEq.(3.16)intotwoterms,referringtoq# 0andq= 0,
respectively,
e2 ,
4.
r. t
t
c Jk+q.A:Jp-q,AzJpAz tzkA
2P'kx AlAaq2 + p.
:
/2
+ -2F
4,
r & &
---jJkzjJpzzJpA,JkAl
kp âl/z >
wheretheprimeon thefirstsummation means:omittheterm q= 0. Thesecond
sum m ation m ay berewritten with theanticom m utation relation as
el 4,
v
el 4,
v
Jk
fA,akAl(ap
#A:Jpz, - 3kp3A:zz)= jys-.
j(X2- Sj
p. kAl pzz
Jt
2#'--i
whereEq.(2.9)hasbeen used to identify 4. Sinceweshallalwaysdealwith
statesofhxed N,the operatorX may be replaced by itseigenvalue Ar,ihereby
yieldinga T-numbercontribution tothehamiltonian
,2N l4v ,2N 4o
YT-O
8 -Y PO
M
(3.18)
sEcoND GUANTIZATIO N
ltisclearthatthefirstterm ofEq.(3.l8)cancelsthefirstterm ofEq.(3.15).
Thesecondterm ofEq.(3.18)representsanenergy -J4=c2(Ug2)-1perparticle
and vanishesin the properphysicallim itdiscussed previously:firstL --+.:
yzand
then Jz.
-+.0 (alwayskeepingp.-l<<
:A). Thustheexplicitp.-2divergencecancels
identically in the therm odynam ic lim it,w hich resects the electricalneutrality
ofthe totalsystem ;furtherm ore,itis now perm issible to setpt= 0 in the srst
term ofEq.(3.17),sincetheresultingexpressioniswelldehned. W etherefore
obtain thesnalham iltonian fora bulk electron gasin a uniform positivebackground
''
h2k2 &
02 , 4n #
t
H =
-2m
- JkAJkA-hl I
-uq,AlJp-q,AatlpyhaflkAI
''
qjJkk/
kpq 21/2
where it is understood that the limit N --+.cc,
N(P'= n= constant is
im plicitly assumed.
Itis now convenientto introduce dim ensionlessvariables. W e define a
length roin termsofthevolumeperparticle:
P'H Gr
a r)N
rt
lisessentially the interparticle spacing. The coulom b interaction providesa
second length,given bytheBohrradius
tzfl=
h2
2
andthe(dimensionless)ratiobetweenthesetwoquantities
r& H -Q
an
evidently characterizesthe density ofthe system . W ithrf
lasthe unitoflength,
wedesnethefollowingquantities
F = rà-3P'
Q = rvk
j= rop
I = roq
andthusobtainthefollowingdimensionlessform ofEq.(3.19)
e2
1f2
r, , 4.
t
-q
un.tuj- q,A,cp-q
,A
.
J'
.
?= c j alkAtuz+ afp
auyzitzkz,
tu r, kA
kya a,az
(3.24)
This is an im portant result,foritshows thatthe potentialenergy becom es a
smallperturbation asrs->.0,corresponding to the high-density limit(ra-.
>.0).
Thus the leading term in the interaction energy of a high-density electron gas
can beobtained with frst-orderperturbation theory,even though the potential
isneitherweak norshortrange. O nem ightexpectthattheground-stateenergy
hasa power-seriesexpansion in the sm allparam eterrs,but,in fact,thesecond-
26
INTRODUCTIO N
orderterm divergeslogarithmically(seeProbs.1.4and 1.5). lnstead,theseries
takestheform
N e2
E = anrsz(a+ brs+ cr,
2lnrs+ #r,
2+ ...)
wherea,:,c,d .. .are num ericalconstants. W e shallnow evaluatea and b.
whilec m aybeinferred from thecalculation in Prob.1.5. Thepropercalculation ofc and highercoeë cientsisvery dië cult, however,and requiresthem ore
elaboratetechniquesdeveloped in Chaps.3 to 5.
In the high-density lim it,the preceding discussion enablesus to separate
theoriginaldimensionalform ofthehamiltonian (Eq.(3.19))into two parts:
X0= h2
zk2(/kzakz
(3.254)
k2
- e2 ,
H L=
2 P'
kx AjAz
4,
p. .
'
-ilj
.
j
q k+q,AlJp-q,AaJpAaJkAl
(3.25:)
where Xnisthe unperturbed hamiltonian, representing a noninteracting Ferm i
system ,and X 1isthe(small)perturbation. Correspondingly,theground-state
energy E m ay be written asE f0)+ E(1)+ .. . where E t01is the ground-state
energy ofa free Fermigas.while Eçl$isthesrst-orderenergy shift. Since the
Pauliexclusionprincipleallowsonlytwo ferm ionsin each m om entum eigenstate,
onewith spin-up and onewith spin-down,thenormalizedground-state IF)is
obtained by slling the m om entum states up to a m axim um value,the Ferm i
m om entum pF= hkF. In the lim it that the volum e of the system becom es
infnite,wecanreplacesum soverstatesbyintegralswith the following fam iliar
relation
Z/
k
lX(k)-nxn
ZyaaZ
AX(
!
$
2L
71
u/L.
+
.
rJjjdnxdnydnxI
A)/Aj(M)
=
p'(2,
rr)-3jAjjdnkyA(k)
(3.26)
Here,Eq.(3.2)hasbeen used to convertthesum overmomentaintoa sum over
theintegerscharacterizingthewavenumbers. ForverylargeZ,thefunctionf
varies slowly when the integerschange by unity so thatnx,nyand nz m ay be
considered continuousvariables. Finally,Eq.(3.2)again allowsusto replace
thevariables(?
1âJby(kiJ,leavinganintegraloverwavenumbers.
Them axim um w avenum berkrisdeterm ined bycom putingtheexpectation
valueofthenumberoperatorinthegroundstate IF)
N = X'FIX 1F)- )((FIHkAlF)= Z %kF- k4
kâ
=
kâ
7(2,r)-3J2(fd5kptks- k)= (3*2)-îFkl.= N
(3.27)
SECOND GUANTIZATION
27
whtre#(x)denotesthestepfunction
8@ )= 1 x > 0
(3.28)
0
x< 0
This im portant relation between the Ferm iwavenum ber ke and the particle
densityn- h'
/P-willbeused repeatedly in subsequentwork;analternativeform
k
3=2N i '9zr 1
p=
s
=
(j-rc
-1.
'
4C1.
92rn
-1
showsthatkv
-tiscomparable with the interparticle spacing. The expectation
valueofXtmaybeevaiuatedinthesamefashion
e'ïg'
h2
=kF(
/%(
F)=jyé
/
c2(F(
éuzt
F)
k/
h2
-jsy
kz#(/
c,-k)
k;
h2
=jyUAé742z4-3jd5kk2#(/
cs-kj
=
3X2k/ el N:$ 9* '
à' cl 2..2,)
J lm N = 2a: Fi
. N rj
sà % = 25:
s
In afreeFerm igas,theground-stateenergy perparticleEf0)N is.
j.oftheFermi
e
N may beexpressed as2.21r,-2rytrydbnergy ef= hlkljlm ;alternativelyEt0)/.
erg),where 1 ry = elllaz;
k:13.6 ev isthe ground-state binding energy ofa
hydrogen atom .
W eshallnow computethefirst-orderenergy shift
s (l)= ;sjs-l!
Isx
z;
.
eî
-
2pz
,
- -
kx â:Az
4n
t/kvq,AIup,q.gaan>aakz:ys)
qi(sl
The m atrix elem entis readily analyzed asfollows;the states lAcand kAîmust
z
be occupied in the ground state IF '
z,, s
ince the destruction fperatorsacting to
the rightwould otherwise givezero. Sim ilarly the statesk +.q A,andp - q,Aa
,
FA
mtlStalso be occupied in jj
g, sincetheoperatorsa%actiag to the Ieftwould
otherwisegivezero. Finally, thesamestate appearson each sideofthe matrix
elem ent,which requiresthetwo creation operatorsto tillup the holesm ade by
the two destruction operators. The operators must therefore be paired o&
,
and thereare onlytwo possibilities'
.
k + q,A1= kAj
k + q,Al= pAc
p - q,Aa - pA2
p - q,A2= kAI
28
INTRODUCTION
The srstpairing ishereforbidden because the term q= 0 isexcluded from the
sum ,and thematrix elementbecomes
8k+q,p3A,
Az1FlJ1+q.A,J'
lzlJk-bq,AlflkA!1F1
ôk+qa3AlzzfF1?
%+q,A,?
lkz,lF)
= - 8k+q ôA,zz%k
F- lk+ ql)%kF- O (3.33)
.p
A combinationofEqs.(3.31)and(3.33)yields
- -
e2 x.xv -,4,
p,
S(1)= - rsy g y $#s- j
k+q1)olky.-#)
A! kq
e2 4=7
=--j.(2=)62J#3kd3qq-2ptks- lk+q(
)#tks-k)
(3.34)
where the factor2 arisesfrom thespin sum,and the restriction q # 0 may now
be om itted sinceitalectstheintegrand atonlya single point. Itisconvenient
tochangevariablesfrom k toP = k + iq,whichreducesEq.(3.34)tothesymm etricalform
Etl'=-4=e2F(2=)-6fd%q-2j#3.
ppt/cs- (P+jqj)Sks- jp-jqjl
Theregion ofintegration overP isshownin Fig.3.1. Both particleslie inside
the Fermisea,so that IP + èqland IP - !qlmustb0th be smallerthan kF.
P+àq
P-à4
P
V
k
r
P+l'
ql<k'
Region 1
lP-!ql<ày
Fig. a.1 Integration region in momentum
spaceforFt''.
The evaluation ofthisvolume isa simple problem in geometry,with theresult
f#3#ptks - IP+h 1)ùtks- IP -iqj)=4
=k)(1-l'
-y.
x+l.
x3)P(1-x)
x > q (33j)
lkr
.
Therem aining calculation iselem entary,and wefm d
Eçtt=-4=e2F42,
v)-6jgr
kl2ksj0
'dx4741-lx+èx3)
= -
,2 N 9.
n.1 3
2u,-rs -i- i7 =
-
e2 0.916
2a,N rs
(3.36)
SECOND LUANTIZATION
29
Thus the ground-state energy per particle in the high-density lim it is given
approxim ately as
E
el 2.21 0.916
X r,..02az rz
r,
N otethattheenergy perparticleishnite,which showsthatthetotalenergy isan
extensivequantity. Thefirstterm in Eq.(3.37)issimplythekineticenergyof
the Ferm igas ofelectrons;itbecom esthedom inantterm asr,.
->.0,thatis,in
the lim it ofvery high densities. The second term is known as the exchange
energy and isnegative. ltarisesbecausetheevaluation ofthe m atrix elem entin
Eq.(3.31)involvestwo termsEEq.(3.32)1,directand exchange,owing to the
antisym m etry ofthe wave functions. A swe have seen,the directterm arises
from the q = 0 part ofthe interaction and servesto cancelH b+ H ei- v. This
E/-V
O. 10t,ê/2a()
Exactresultasr,-...0
2
-
0 l0e1I2ao
-
I
4
6
:
rs= 4.83
1c 5
- - +
.
GW ignersolid
' EIN = - 0.095egIjav
Fig. 3-2 Approximate ground-state energy' (first tw'
o
termsinEq.(3.37))ofanel
ectron gasin auniform positive
background.
cancellation leads to the restriction q # 0 and reiects the electricalneutrality of
thesystem. Al1thatremainsisthe(negative)exchangeenergy. Theremaining
termsinthisseries(indicatedbydots)arecalledthecorrelationenergy;lweshall
returntothisproblem inSecs.12and 30,wheretheleadingterm in thecorrelation
energy willbe evaluated explicitly.
For the presentshowever,it is interesting to consider the first tw o term s
ofEq.(3.37)asafunction ofrs(Fig.3.2). Theattractivesign oftheexchange
energy ensures that the curve has a m inim um occurring for negative values of
theenergy;thesystem isthereforebound. Asrs-+.0(thehigh-densitylimit),
Eq.(3.37)represents the exactsolution to the problem. For larger valtlesof
r,,our solution isonly approxim ate,butwe can now use the fam iliar RayleighRitz variationalprinciple,w hich assertsthattheexactground state ofa quantum m echanicalsystem always has a lowerenergy than thatevaluated by taking the
expectation value ofthe totalham iltonian in any norm alized state. The conditions ofthisprinciple are clearly satissed,since we have m erely com puted the
expectation value ofthe hamiltonian X in the state 1F='. ltfollowsthatthe
!E,P.W igner.Phys.Rer.,46:1002(1934).
3c
INTRODUCTION
exactsolution to our m odelproblem m ustalso representa bound system with
energy lying below thecurvein Fig.3.2. Theminimum ofEq.(3.37)occurs
atthevalues
(rxlmin= 4.83
E
y
e2
=-0.0952az
m ln
(3.38)
Although thereisno reason to expectthatoursolutioniscorrectin thisregion,
itisinteresting to observe thatthese values
E
rs= 4.83
W = -1.29ev
atminimum
(3.39)
com pare favorably with the experim ental values for m etallic sodium under
laboratory conditionsl
Na(experiment)
(3.40)
where the binding energy is the heatofvaporization ofthe m etal. Thus this
very simple m odelis able to explain the largestpartofthe binding energy of
m etals. In realm etals,one m ustfurther localize the positive background of
chargeon thecrystallatticesites,assrstdiscussed by W ignerand Seitz.2,3
ltisalsointerestingtouseEq.(3.37)toevaluatethethermodynamicpropertiesoftheelectron gas. Thepressureisgiven by
p=
-
'E
.,
v.zu.
= -
(JP'x
6IEdrs= Nelrs 2(2.21)- 0.9.16
drsdV 2J03P' rl
r,
t
(3.41)
The pressurevanishesatthepointrs= 4.83,w herethesystcm isin equilibrium .
Furtherm ore,thebulk m odulus
B-
-
e.
p
.
xèz 2 5(2.21) 2(0.916)
''tap.l
.
v-zao9p(rs-
r-1
(3.
42)
vanishesatthehighervalue r,= 6.03,where the system ceasesto be m etastable
in thisapproxim ation.
Inthelow-densitylimit(r,-+.cc))W igner4hasshown thatonecan obtain
a lower energy ofthe system by allowing the electrons to Sicrystallize''in a
tfW igner solid.'' This situation occurs because the zero-point kinetic energy
associated with localizing the electrons eventually becom es negligible in com parison to the electrostatic energy ofa classicallattice. W ignerhas shown that
1See,forexample,C.Kittel.àsouantum Theory ofSolids,''p.115,John W iley and Sons.Inc.,
New Y ork,1963.
2E.P.W ignerandF.Seitz.Phys.Rev.,43:804(1933);46:509(1934).
3A modern accountofthe relevantcorrectionsmay befound in C.Kittel,op.cl'r.,pp.93-94.
115.
4E.P.W igner.Trans.Farad.Soc.,34:678(1938);W .J.Carr,Jr.,Phys.At'&.,122:1437(1961).
SECOND QUANTIM TION
31
theenergy perparticlein thissolid isgiven asym ptotically by
E
e2
1.79 2.66
+ j. + ...
rs
=
9 u.
.. zao rs
:x
wignersolid,.
(3.43)
and itisclearthatthisexpression givesalowervalue oftheenergy than thatof
Eq.(3.37). Thelow-densitylimit(Eq.(3.43))issketched asthedotted linein
Fig.3.2. ThevariationalprincipleguaranteesthatthisW ignersolid represents
a betterwavefunction asr,-.
>.x),becauseithasalowerenergy.
PRO BLEM S
1.1. Provethatthenumberoperator# = J#*(x)#(x)J3x commuteswith the
hamiltoniansofEqs.(1.42)and(1.60).
1.2. Given a homogeneoussystem ofspin-.
iparticlesinteracting through a
potential F
(J)show thatthe expectation valueofthehamiltonian in the noninteracting
ground state is
kr h2k2
k :
s(o+s(l)-2yk a. +!k
yàkj
'
A'ftkzk,z,jzl
àAu,y,j
-
whereA isthezcomponentofthespin.
(kàk'à'jp'Ik'z'kl;4
(b) Assume F is centraland spin independent. If F(Ixl- xal
)< 0 for al1
Ixl- xcIandJlF(x)1#3x< cc,provethatthesystem willcollapse(Hint:stal
from (f(0)+ Eçtt)lN asafunction ofdensity).
1.3. Given a homogeneoussystem of spin-zero particles interacting through
a potentialF
(J)show thattheexpectation value ofthehamiltonian in the noninteracting
groundstateisECLSIN = (# - 1)F(0)/2F = èaF(0),where
P'(q)= Jd3xF(x)e-tq*x
and
F(0)meansF(q=0)
(b) RepeatProb.1.2â.
(c)t Show that the second-order contribution to the ground-stateenergy is
E(2)=
N
-
N - 1 d% lF(q)12 n #% jF(q)j2
2F (2z43h2qa(m = -i (2=)ahaqzlm
1Usestandardv ond-orderx rturbationtheory:IfH = Ho+ Htandtheunm rtur-.
deigen-
vKtorsI/)satisfy/51.
/)= FgI/),then
Etz)=7)
l(0
EI
z
fA-1
E
/J
)11=t0IKFaPH.HLI0)
J# e
wherel
0)istheground-stateeigenvKtorofHowithenern Sq,andP = 1- 1
0)(01isapro- tion
o> ratorontheexcited Mates.
:2
INTRODUCTION
1.4.1 Show thatthesecond-ordercontributiontotheground-stateenergyofan
electrongasisgivenbyEt21= (Nelj2av)(kL+ 61),where
fl= - 3 d?q r
z,g r
d,pp(1- k):(1-r)
g.
p,5 q4 jjk-yqj>j- :j
plqjwj . q2+.q.(k+.p)
3
dsq
.- ,
-
9(l- k)p(l-p4
e'- 16,5 qz ,
k+q1>,d-kjlp+qI>l#-#(q+k+pli-lq
-i+q-tk+p)j
1.5. Theexchangeterm d inProb.1.4ishnite,wllilethedirectterm e:diverges.
(* Considerthefunctionflq)dehned as
/(
p(1- k)d(1-p4
ç)=Jl
kiql>L
d3kJl
p+ql>ld3#q2.j
-qptk+p)
Show thatflq);
k)(4=/3)2:-2asq->.:
x)and flq);
kJi'
(2=)2(1- ln2)q asq-+.0.
Hence conclude that eq= -(3/8=5)Jdhflqlq-k diverges logarithmically for
smallq.
(:)Thepolarizabilityoftheinterveningmedium modifestheefl
kctiveinteraction
between two electronsatlongwavelength,whereitbehavesas
l'
X#leff;
Q$4=82:2+ L4rs(=jk((4(9=)%j-t
forq-.
>.0. EseeEq.(12.65).) Inthelimitrs.
-+.0,usethisresulttodemonstrate
that
e:= 2=-2(1- ln2)lnrs+ const= 0.0622lnrs+ const
(c) Howdoese:ofpartbaflkcttheequationofstate? Findthedensityatwhich
thecom pressibility becom esnegative.
1.6. Consider a polarized electron gas in which N x denotes the num ber of
electronswithspin-up(-downl.l
(a)Find theground-stateenergy to Erstorderin theinteraction potentialasa
functionofN = N++ N-andthepolarization (= (N+- N-jIN.
(:)Provethattheferromagneticstate((= 1)representsalowerenergythanthe
unmagnetized state ((= 0)ifr,> (2=/5)49*/4)1(21+ 1)= 5.45. Explain why
thisisso.
(c)Show thatè%EJN)(D(2j(.(
)becomesnegativeforr,>(3=2/2/ =6.03.
(#)Discussthephysicalsignihcanceofthetwocriticaldensities. Whathappens
for5.45< r,< 6.03?
1.7. RepeatProb.1.6forapotentialFtlx-yI)= g3t3'(x- y). Show thatthe
system ispartially magnetized for20/9<gNIVT< (5/3)21= 2.64,where T is
the mean kinetic energy per particlein the unmagnetized state,and N(V isthe
corresponding particledensity. W hathappensoutsideoftheselimits?
tSeefootnoteonp.31.
lF.Bloch,Z.Physik,57:545(1929).
2
StatisticalM echanics
Before fbrm ulating the quantum -m echanical description of many-particle
assem bliessitisusefultoreview sometherm odynam icrelations. Theelem entary
discussionsusually considerassem bliescontaining a hxed num berofparticles,
but such a description is too restricted for the present purposes. W e m ust
thereforegeneralize the treatmentto include thepossibility ofvariable num ber
ofparticlesN . Thisapproach ism ostsim ply expressed in thegrand canonical
ensemble,which isgenerally more tractable than the canonicalensemble (h'
sxed). In addition,there are physicalsystems where the variable number of
particles is an essentialfeaturt,rather than a m athem aticalconvenience;for
exam ple.them acroscopiccondensateinsuperiuidhelium andinsuperconductors
acts as a particle bath that can exchange particles w ith the rem ainder ofthe
system . Indeed,thesesystem sare bestdescribed with m odelham iltoniansthat
do noteven conserve N,and them oregeneraldescriptionm ustbeused.
:u
INTRODUCTION
K REVIEW O F THERM ODYNAM ICS
A ND STATISTICAL M ECHANICS
Although it is possible to treat system s containing severaldiserent kinds of
particles,the added generality isnotneeded formostphysicalapplications,and
we shallconsider only single-componentsystem s. The fundamentaltherm odynam ic identity
dE = TdS - P dv + p.dN
specihes the change in the internalenergy E arising from sm all independent
changesintheentropyS,thevolum eP',and thenum berofparticlesN . Equation
(4.1)showsthattheinternalenergyisa thermodynamicfunctionofthesethree
variables,E = EIS,F,.
N'),and thatthe temperatureF,thepressureP,and the
chem icalpotentialp,arerelated to thepartialderivativesofE :
OE
T = DS
vs
Jf
-# = .
x
-=- sx
dP
OE
g = ON sz
'
In the particularcase ofa quantum -m echanicalsystem in itsground state,the
entropy vanishes,and thechem icalpotentialreducesto
M (y
-.)F
-
whereE istheground-stateenergy. Moregenerally,Eqs.(4.1)and (4.2)may
be interpreted asdefningthechemicalpotential.
Theinternalenergyisusefulforstudying isentropicprocesses;in practice.
however,experim entsare usually perform ed atsxed F.and itisconvenientto
make a Legendre transformation to the variables(r,V,N)or(F,#,N). The
resultingfunctionsare known astheHelmholtzfreeenergy FIT,P)zV)and the
Gibbsfreeenergy G(r,#,.
N),definedby
F = E - TS G = E - TS+ PV
(4.4)
Thediserentialofthesetwoequationsmaybecombined with Eq.(4.1)toyield
dF= -SdT- PdV+ y.dN dG = -SdT+ VdP + JzdN
(4.5)
which dem onstratesthat F and G are indeed thermodynamic functions of the
sx ciâed variables. In particular, the chemical potentialmay be desned as
.-
(lx
')TF-(
yW
'M
a
G
vJ
àTP
(4.
6)
Furthermore,itisoften importantto considerthe setofindependentvariables
(F,F,p),whichisappropriateforvariableN. A furtherLegendretransformation
leadstothethermodynamicpotential
Q(F,F,Jz)= F- gN = E - TS- mN
(4.7)
36
STATISTICAL M ECHANICS
with thecorresponding diFerential
dkt= -SdT- PdV- A #rz
(4.8)
Thecoeëcientsareim m ediatelygiven by
s- -
(%
,,.
)-- ,--(a
?-;
?,)-- x--(J
?s)-
(4.
9)
which willbe particularly usefulin subsequentapplications.
Although elF,G,and flrepresentformally equivalentwaysofdescribing
thesamesystem ,theirnaturalindependentvariablesdiflkrin oneimportantway.
lnparticular,theset(u
% P'
,N)consistsentirelyofextensivevariables.proportional
to the actualam ountofm atterpresent. The transformation to F and then to
G or D may be interpreted as reducing the number of extensive variables in
favor ofintensive ones that are independent of the totalamount of matter.
Thisdistinction between extensiveand intensivevariablesleadsto an im portant
result. Considerascalechangeinwhich allextensivequantitles(includingE,
F,G,and f1)aremultiplied byafactorA. Fordesniteness,weshallstudythe
internalenergy.which becom es
hE = E(hS,hV,hNj
Diflkrentiatewith respectto A and setA= l:
s-s(kf)FN+p'(j
pf
r
s)Sb.+xta
îE,
)S'
F-rs-,p-+,,N
whereEq.(4.2)hasbeenused. Equation(4.10),whichherearisesfrom physical
arguments,is a specialcase of Euler's theorem on hom ogeneous functions.
Theremainingthermodynamicfunctionsareimmediatelyfound as
F= -PV + yh'
G = y'
N
D = -PI.
'
(4.ll)
which showsthatthechemicalpotentialin aone-componentsystem istheGibbs
free energy perparticle /.
t= N -1G(r,#,N)and that P = - P'-1D(F,P'
,/z). This
lastresultisalso an obviousconsequence ofEq.(4.9),becausefland P'are
extensive,whereasF and J,
tareintensive.
To this point. we have used only m acroscopic thermodynam ics,which
m erely correlates bulk properties of the system . The m icroscopic content of
thetheory m ustbeadded separately through statistiealm echanics,which relates
thetherm odynam icfunctionsto theham iltonian ofthe m any-particleassem bly.
In thegrand canonicalensem bleatchem icalpotentialp,and tem perature
T
-
1
ks#
(4.12)
x
INTRODUCTION
where ks is Boltzmann's constant,ks= 1.381x 10-16 erg/degree,the grand
partition function Zoisdeâned as
Zo- ;
J(
2e-CCEJ-I'N'
N J
=
=
2
1Z (Njl
d-/6X-B*'lA7)
N J
Tr(e-b(R-HRtj
(4.13)
where.jdenotesthesetofa11statesforasxed numberofparticlesN,and thesum
implied inthetraceisoverboth.jand N. A fundamentalresultfrom statistical
mechanicsthen assertsthat
Dtr,F,p,
)=-ksFlnzs
(4.14)
which allows us to compute a1lthe macroscopic equilibrium thermodynamics
from thegrand partitionfunction.
ThestatisticaloperatorlocorrespondingtoEq.(4.13)isgiven by
)s=z(;!e-#(#-pA)
W iththeaidofEq.(4.14),)(;mayberewrittencompactlyas
)s= e#(t'
)-8+#l#)
Foranyoperatorô,theensembleaverage(O)isobtainedwiththeprescription
(ö)- Trtloö)
= Trt
elto-f+Mxld)
Tr
(e=
Tr
el(
jW(pB
.N
s'
pö)
;
.
(4.17)
The utility ofthese expressions willbe illustrated in Sec.5,which reviewsthe
thermodynamicbehaviorofidealBose and Fermigases.
SEIDEA L GAS.
W enow apply theseresultsby reviewing thepropertiesofnoninteracting Bose
and Fermigases. Throughoutourdiscussion we usethenotationalsimpliscation
#= (kBF)-1
'The argumentsin this section are contained in any good book on statisticalmechanics,for
txampl
e.L.D.I-andau and E.M .Lifshitz,''StatisticalPhysics.''chap.V,Pergamon Press,
London,1958.
STATISTICAL M ECHANICS
37
IfEq.(4.13)iswrittenoutindetailwiththecompletesetofstatesintheabstract
occupation-num berHilbertspace,wehave
ZG= Tr(e--ltJle-BA))
))jal ...n.j
Jg ;nL...n.1e#(MA-#(
=
#1: ' ' 'Flx)
= e- #f)(j
(T,F,s)
(5.2)
sincethesestatesareeigenstatesofthehamiltonian.
4:andthenumberoperator
#,both operatorscan bereplaced bytheireigenvalues
'n.1exp j p,)()ni- J2e.n, 1a,-'.a.)
Theexponentialisnow acnum berand isequivalenttoaproductofexponentials;
hence the sum over expectation values factors into a product of traces,one
referringtoeach m ode,
za= (
j)(njIe#tBnl-f.a''1n1) ''.12(?utc#tB>*-'*a*'I
n.)
aj
(5.4)
5*
which m aybewritten com pactlyas
*
zG= yjTrje-lfel-/
zlât
(5.5)
fa.1
For bosons the occupation numbers are unrestricted so that we must
sum ntovera1lintegersinEq.(5.5)
X
*
*
zo= (
-! pl(c/(>-el))a= 1
-I(1- e#(8-fl))-1
l-1 a=0
l=1
(5.6)
Thelogarithm ofEq.(5.6)yieldsthethermodynamicpotential
*
fhntr,F,p)= -k.T ln 17 (1- eftB-eI')-1
f-l
*
Datr,F,p,
)= ksF jzln(1- eltB-ft')
t=l
Bose
(5.8)
Themean numberofparticlesisobtained from t'lnbydiflkrentiatingwith respect
tothechemicalpotential,asin Eq.(4.9),keeping Fand F (equivalentlytheej)
Nxed:
*
IN?$= lI
f=
.ln9
*
1
1(ff-H)- 1
e
f=l
Bose
wheren2isthemeanoccupation numberinthefthstate.
(5.9)
3g
INTRODUCTCON
For fermions,the occupation numbers areeither 0 or 1,and the sum in
Eq.(5.5)isrestrictedtothesevalues
*
l
eo
Zc= H ; (e#t8-fl)X = FI(1+ e#ïB-fl')
1-1 n-(1
(5.10)
1-l
Taking thelogarithm ofboth sides,wehave
*
f1a(7lF,/z)= -kaF Z ln(1+ eltB-fl))
Fermi
(5.11)
l-1
while thenumG rofparticlesG comes
*
*
1
r#)- 11
2n2- I e#(ej-p)+ j
..l
Fermi
(5.12)
-1
Although bosons and fermions diflkr only by the sign in the denominator in
Eqs.(5.9)and (5.12),this sign leadsto ratherremarkable diFerencesin the
G havioroftheseassemblies.
BosoNs
W e shallErstconsidera collection ofnoninteracting bosons,where the energy
sm ctrum isgivenby
p2 :2k2
fP= 2- = 2-
W eassumethattheassemblyiscontained inalargevolume F andapplyperiodic
boundaryconditionsonthesingle-particlewavefunctions. JustasinEq.(3.26),
sumsoversingle-particlelevelscan bereplaced byanintegraloverwavenumbers
according to
72-.
>.gJd3n=gF(2=)-3Jd3k
(5.14)
whereg isthedegeneracyofeach single-particlemomentum state. Forexample,
g= 1forspinlessparticles. W ith Eq.(5.13),thedensityofstatesin Eq.(5.14)
can lyerewritten as
'V 4
(293.1adk-(.
p
F.
,(2
h>
al
j12
ee
d+
n=4
g=
V2jV
zzjlede
(j.jj)
andtâethermodynamicpotentialEq.(5.8)f- anidealBosegasGcomes
t'lo = P V .
-kBr kBT 4
A'
*F2(2
#
zi
j
-9j0
*derljntj-e#(s-e)
-
A simple partialintegration then yields
pp'- 4
#4F
a
(l
ym)1j
2j:
*wejjrj).
-j
(5.16)
STATISTICAL M EcHANlcs
w
Alternatively,acombination ofEqs.(4.9),(4.10),and (5.8)allowsustowrite
gF /2v 1 *
el
E=z/nîfI=ozj,a (j#ee#(.-.)-j
(5.18)
showingthattheequationofstateofanidealBosegasisgvenby
(5.19)
PV= JS
In a similarway,thenum- rofparticlesG comes
N
g 2- 1 *
e*
(5.20)
P = 4=a y
:peej(..j,j j
Although Eqs.(5.17)and (5.20)determinethethermodynamicvariablesofan
idealBosegasasfunctionsofF,F,and/z,Eq.(5.20)caninprindplel)einverted
.
to obtain thechemicalpotentialasafunction ofthenum berofparticles. Sub-
stitution into Eq.(5.17)thenyieldsthethermodynamicvariablesasafunction
ofr,F,and N.
Equation(5.20)ismeaningfulonlyif
e- Jz> 0
Otherwisethemeanoccupationnum bera0would% lessthanzeroforsomevalues
ofe. ln particular,ecan vanish so thatthechemicalpotentialofan idealBose
gasmustsatisfy thecondition
p,< 0
(5.21)
To understand thisrelation,werecalltheclassicallimitofthechemicalpotential
forfixed N 1t
kXr * '-*
s
r .- >.a)
(5.22)
In thislimitweseethatBoseand Fermigasesgivethesameexpression
N -.
sr el(!
,-et)
(5.23)
whichisjustthefamiliarBoltzmanndistribution,and
f1:= -Pv= -ksT Jlge'çln-q6b= -Nk.T
T --+*
(5.24)
whichistheequation ofstateofanidealclassicalgas. Thesum m ay% evaluated
approxim ately asan integral
7l2ebqt=g(2.)-$FJd'ke-O:2/2-4T
'-
m kzT 1
= gv 2 a
=h
tL.D.ïxndauandE.M.Lifshitz,op.cit..= .45.
(5.25)
40
INTRODUCTION
andEq.(5.23)thenyieldstheclassicalexpression y.cforthechemicalpotential
/2e
N l=h2 1
k = ln
gv vk'sF
sF
intermsofr,F,N. NotethatEq.(5.22)isindeedsatissedasF->.(
x). Furthermore,itisclearthatFerm i,Bose,and Boltzm ann statikticsnow coincide,since
tho particles are distributed over m any states. Thus the m ean occupation
num ber ofany one stateism uch lessthan one,and quantum restrictionsplay
no role.
TheclassicalchemicalpotentialofEq.(5.26)issketchedin Fig.5.1. As
thetemperature isreduced atsxed density,y'
clkBl-passesthrough zero and
$
l N 2=*2 7
nr
#
t
k.T
$y
.
Myrrml
XNv ..''-.x k:r
N
N
N.
x
pp'-'h.'h'
i
XBO9C
x
x
F
w.'.h. xx
'h.
.h.'h.Nx
WMX
-k
sl-
Bc
I
a#
Fig.5.1 The chemical potential of ideal
classical,Fermi,and Bose gases for hxed N
and F.
becom es positive,diverging to +:s atF = 0. Since this behaviorviolatesEq.
(5.21),thechemicalpotentialforan idealBose gasmustliebelow theclassical
value,staying negative orzero. LetTobe thetem peraturewherethechem ical
potentialofan ideal Bose gas vanishes. This criticaltem perature is readily
determined with Eq.(5.20)
N
.
g' lm @' *
el
V = 4=2 hl t)Jeef/v T0- 1
whichmayberewrittenwiththenew variablexEB6//fsFaas
N
V
=
jê =
g 2m ks.T'
..
4=2
h2
#X
x1
x
a e -1
(5.28)
Theintegralisevaluated in Appendix A
N - g zz/lksL ê
P 4nz ,2
(('
1)1>0)
andEq.(5.29)maybeinvertedtogive
h2
r-'--y-à/
.
g-
4x2
!' N '
i 3.31 hl N '#
(,,r(.
s((#$)(w)--g.-,-(v)
(5.30)
STATISTICAL M ECHANICS
41
asthe tem peratureatwhich thechemicalpotentialofan idealBosegasreaches
zero. Thisvalue hasthesimplephysicalinterpretation thatthethermalenergy
ksrcis comparablewith the only other intensive energy fora perfectgas,the
zero-pointenergy(h2/m)(N/P')1associatedwithIocalizingaparticleinavolume
VIN.
W hathappensasweIowerthetemperaturebelow To? ltisclearphysically
thatmany bosonswillstartto occupy the lowestavailable single-particle state,
namely the ground state. For p.= 0 and r < Fn, however, the integral in
Eq.(5.20)islessthan NlP'becausetheseconditionsincreasethe denominator
relative to itsvalue at L . Thusthe theory appears to break down because
Eq.(5.20)willnotreproducethefulldensityNlF. Thisdimcultycan betraced
to the replacementofthe sum by an integralin Eq.(5.14),and wetherefore
exam inetheoriginalsum
N = )f((eltel-?2)- 1)-1
Asp,-.
+.0.allofthetermsexceptthehrstapproach aEnitelimit;thesum ofthese
hnitetermsisjustthatgivenbytheintegralevaluatedabove.l Incontrast,the
:rstterm hasbeen lostin passing to the integralbecause the E1 in the density
of states vanishes at e= 0. W e see, however, that this srst term becomes
arbitrarily large asp.-+.0 and can therefore make up the restofthe particles.
Thisbehaviorreqectsthe macroscopic occupation ofthe single quantum state
e = 0.
FortemperaturesF< L .weconcludethatthechemicalpotentialp,must
beinsnitesimally smalland negative
p.= 0-
forF< To
(5.31)
In thistemperaturerange,the density ofparticleswith energies e> 0 becomes
dN
g 2m 1 e+JE
V S = 402 h2 e#e- j
(5.32)
with theintegrated value
Ne>0.
g l2M/fsT11 *yx '
Y* =NiT 1
P' 4=2!. h2 ) a e - 1 X yF:
(5.33)
Theremaining particlesarethen in thelowestenergystatewith e= 0
N --,- x j-
p' p (w
ro).
(5.34)
tStrictly speaking,theoccupationnumG rofthelow-lyingexcited statesisoforderN k.which
lv omes negligible only in the thermodynamic limit. A rigorous dixussion of the BoseEinstein condensation maylx foundin R.H.Fowlerand H .Jones,Proc.C'
alnàrflgePhil.Soc..
M :573(1938).
42
INTFyooucTloN
Inthedegenerateregion(F< Tz)wherethechemicalpotentialisgiven by
Eq.(5.31),theenergyoftheBosegasarisesentirelyfrom thoseparticlesnotin
thecondensate
E = g zvksFï1a :. u xl
# 4.2 :2 ) AaFJoGxex-1
Thisintegralisagain treated in AppendixA,and we5nd
E - g 2?nksF 1
4=2
P (Aa)k.r(0)P0)
(5.35)
whichmayberewrittenihtermsofL from Eq.(5.29)as
E- 10)lx lxks
((
1)r(l) rtjj'-tl
.77oxksrt.
jll rvz.
,
(5.36)
Theconstant-volume heatcapacity then becom es
5o
cv-j /xxks(s
r0)1 p<rc
.
which variesasF1 and vanishesatF = 0. Equation(5.
35)also can beusedto
rewritetheequation ofstate:
# 2E 22V1
mhk.T)%g
- jy,- j 4.c((.
1.
)ln(1) yq
-
0.0851v1(ksT4%h-?g
T< ra
Thepressurevanishesatzero tem peraturebecause a1loftheparticlesarein the
zero-momentum stateand thereforeexertnoforceon thewallsofthecontainer.
Furthermore,thepressureisindependentofthedensityN/U,depending only
onthetem perature r < ra.
W e have seen thattheidealBose gashasa criticaltemperature Fowhere
thechemicalpotentialchangesitsanalyticform. SinceJz(F,V,N)isrelated to
thefreeenergybyEq.(4.6),itisnaturaltoexpectsimilardiscontinuitiesin other
therm odynam ic functions,and we now show thattheheatcapacity atconstant
volume Ck hasa discontinuousslope atF(). The behaviorforF < Fcis given
in Eq.(5.37);thecorrespondingquantity forF> rocan befound asfollows:
Defnethe(tktitious)numberofparticlescomputed forp,= 0 and F> F(jby
P' 2m i' 'o
e+
(
)dee,tuw. . j
N'
:(r)H 4=z hz
.
Thisexpression clearly implies
#0(F)= #0(F)
#c(Fc)
x
=
F1
(y1 F>rc
STATISTICAL M ECHANICS
43
Equation(5.20),which determinestheactual/ztr,V,NjforF> r;,can now be
rewritten
gv lm 1 * u
N -.
N;(F)= 4*z hg
1
.
1
/ks,
r..j- eqfk.r.
a aee- eç'-P)
j
Thedominantcontribution totllisintegralarisesfrom smallvaluesofebecause
ptksr is smalland negative for0< F- Fc< Fn. Thus we shallexpu d the
integrand to give
N-&(F)=4
g,,
vz(2
j,
i,j1g,
gswo
'
sas+(,+
1j
sj
)
;
k;-4
g.
pz
'(2
sp,
j1swsvoj
sj
1
where we have set F = Fo to leading order in F - Fo. A combination with
Eq.(5.29)leadsto
Jz= = -
(0)P(1)2ksro Xn(F)- 1 2
=
N
(0)F0)2
,v.
F @' 2
ksz'
;(y
J-1 FkFa
-
NotethatJzvanishesquadraticallyasT-+TI sothatJz(F,V.N)hasadiscontinuoussecondderivativeatro(seeFig.5.1).
Theremainingcalculationcanbecarried outbydiflkrentiatingtheequation
ofstate(5.19)
(>
''
E
;
-)r'
--i
3ê(e
.
PsP
.')r'
--1x
wherethelastequalityfollowsfrom Eqs.(4.9)and(4.1l). Thisresultallowsus
to:ndthechangein energyarisingfrom asmallchangein p.atconstantrand P-.
IfEIT,F)istheenergyforzerochemicalpotential(Eq.(5.35)),then theactual
energyisgiven approxim atelyas
E = E(F,P')
F < Fn
)f'
(r,P'
)+j,
N. T>Fo
W enow changevariablesto E P',and N usingtheexpression obtained abovefor
p,
tr,V,N). ThejumpintheslopeofCp,isthengivenbyl
'Cv
â ?r r,-
'
-
tNk.zo t0)PQ)2 02 r 1
=
l?rz(rL)-12)w,
)r(!.)2Nk.
= -2
y7 tf!.
wo = -3.66Nk.
=
Fp
(5.39)
1F.London,issuperfluids,''vol.lI,sec.7,Dover,New York,1964;w,eherefollow theapproach
ofL.D.Landau and E.M .Lifsbitz,op.cit.,p.170.
44
INTRODUCTION
The fullcurveissketched in Fig.5.2. Such discontinuitiesim ply thatan ideal
Bosegasexhibitsaphasetransition ata tem peratureF;. Thistem peraturehas
a physicalinterpretation asthe pointwherea hnitefraction ofa11theparticles
beginsto occupy thezero-m om entum state. Below F(
)the occupation num ber
aîofthelowestsingle-particlestateisoforderN,ratherthan oforderl. As
emphasized by F.London,lthe assembly is ordered in momentum space and
notin coordinate space;thisphenomenon iscalled Bose-Einstein condensation.
Cg
Nka
?l
?
1
7 Fig.5.2 Constant-volumeheatcapacity C'vofan
75
T idealBosegas.
To estim ate the m agnitude ofthe quantitiesinvolved,we recallthatthe
densityofliquid H e4at1ow tem peratureis
P4= 0.l45gcm -3
InsertingthisquantityintoEq.(5.30),we5ndthevalue
L = 3.140K
(5.40)
asthetransitiontem peratureofanidealBosegaswiththeparam etersappropriate
to liquid helium . Below this tem perature,the foregoing discussion indicates
that the assem bly consistsoftwo diferentcom ponents,one corresponding to
theparticlesthatoccupythezero-m om entum stateandthereforehaveno energy,
and theothercorresponding to theparticlesin theexcited states. Indeed,itis
anexperimentalfactthatliquidHe4hasatransitionat2.20K (theApoint)between
the two phasesH eIand H elI. Below thistem peratureH e4actslikea m ixture
ofa superiuid and a norm alfluid,and the superquid hasno heatcapacity or
viscosity. Itisalso truethatthefraction ofnorm alcom ponentvanishesasthe
tem perature goes to zero. The Bose-Einstein condensation ofthe idealBose
gasthereforeprovidesa qualitative description ofactualH e4. ln detail,however,the idtalBose gas is an ovtrsim plised m odel. For exam ple,the actual
specischeatvariesasr3atlow tem peratureand becomeslogarithm icallyinsnite
attheA pointforliquid He4. Inaddition,itisincorrecttoidentifythesuperiuid
com ponent ofH e 11 with the particles in the zero-m om entum state. Indeed,
theexcitation spectrum oftheidealBosegasprecludessuperCuidityatany linite
velocity. Thesequestionsarediscussed in detailin Chaps.6,10,and 14,where
we show thatthe interparticle interactionsplay a crucialrole in understanding
thepropertiesofquantum iuidssuch asliquid H e4.
1F.London,op.cit..pp.39,143.
sTATlsTlcAu M EcHANlcs
*s
FERM IO NS
w enow discussEqs.(5.11)and (5.12)referring to an assembly offermions,
which servesasa m odelform any physicalsystem s. The bàsicequation isthe
mean occupatiop num ber
J= (e#(ef-/
z)+ 1)-1
(5.41)
w ith thenonrelativisticenergy spectrum (Eq.(5.13)),thesameanalysisasfor
bosonsgives
2 #F 2m 1 =
el
Pv= j.
F - j4=a hz
()#eej(r-s)j.j
N
e1
g lm 1 *
# - 4= 2 V
.
cdeebç'-l,'+ 1
(5.42)
(5.43)
whereg isthe degeneracy factor(g= 2 fora spin-è Fermigas). As noted
previously,theonly diflkrencebetween bosonsand fermionsistheminusorplus
signinthedenominatorsofEqs.(5.42)and(5.43).
Consider the distribntion function a0. Equation (5.41)showsthatthe
condition
n0 < 1
isguaranteedforallvaluesofp.andF. ItisinterestingtoinvertEq.(5.43)and
determ inethechem icalpotentialforsxed N ;thisfunction issketchedin Fig.5.1.
ln thehigh-temperatureorclassicallimit,weagain End
n0 = ebllà-'b
which isjust the familiar Boltzmann distribution. Unlike the situation for
bosons,however,thereisnothingtopreventthechem icalpotentialfrom becom ing
positiveasthetem peratureisreduced;in particular,wehave
when e= p,
In the zero-tem peraturelim it,the Ferm idistribution reducesto a step function
1
0
:-#I)/ks'r+ j r--..c j
e(
e > Jz
e < jz
=
9(/z- e)
(5.44)
Thisbehaviorisreadilyunderstood,becausethelowestenergystateofthesystem
isobtained by fllingtheenergy levelsupto
atF = 0
Henccthechemicalpotentialofan idealFermigasatzero temperatureisa hnite
positive number, equal to the Fermi energy. The equilibrium distribution
numbersin threerepresentativecasesaresketched in Fig.5.3.
*e
INTRODUCTION
W eshallfrstevaluatethtprom rtiesofan idealFermigasatr= 0. From
Eqs.(5.43)and(5.44),thedensityisgivenby
N
g 2- 1 #
#= 0 2V
dq:+
G cause the distribution numberisthen a step function. Thisintegraliseasily
evaluated as
N = ..c 2- 12
# 4.2 j-2
- à#
I
1
(5.45)
T= 0
1
l
F > ()
Jt > 0
l
1
i
'
T -'+ x
:1-+ -=
e M &sr
#'
- <F
.(F)
Fig.B.3 Ahematic distribution functions a(e) for an idealFermigas at vari
ous
temN aturo.
which may% inverttd to *nd theFermienergy
o a:xa N 1 R2k;
e,- AT = 0)= -# S i 'P = 2m
ortheFermiwavenuma r
k
6.2N 1
F=
gF
Similarly,theenergyisobtained from
E g 2- 1 #
g 2- 12
'P= 4=2 V
;dqel= 4=2 V
A combinationwith Eq.(5.45)yields
(5.47)
k
ày.
E
N = js= :es
-
Finally,theequationofstate(5.42)G comes
PV= JF = qNœF
P=à
l(-2
6g
*j1i
h;
2ijx
Vj1
(5.46)
(5.48)
(5.494)
(5.49:)
STATISTICAL M ECHANICS
47
which show sthata Fermigasexertsasnitepressureatzero tem perature. This
resultarisesbecause the Pauliprinciple requiresthatthe m om entum statesbe
slled up to the Ferm imomentum,and these higherm omentum statesexerta
pressureon thewallsofany container.
W enow turn to smallbutsnitetem perature,wherethedimcultpartisthe
inversion of-Eq.(5.43)to determinethechemicalpotentialin termsofthe total
number ofparticles. Desne the variable x> (e- Jzl/ksr. Equation (5.42)
m ay then berewrittenas
Pp'-J
24
K.
n
V
.
j(2
â?
jê(ksr)1j2
'
ov'
k.rdxZ-6
e
-x
6+
%q
1.
T
-)
.%
(5.
50)
Itis also convenientto introduce = H Jz//csr;since ptisliniteas F -.
>.0,weare
interested in thelim it= .-+.:v.. Considertheintegral
x
(x+ a)1
o
(=+ .
x)'
l
x (z.
#.x)1
f(œ)-J-.dxex.
j
-j-J-.dxex--i
-+Jodx-exs.1
-
.
The change of variable x --*-x in the firstintegraland use of the identity
(e-x+ 1)-1H 1- (ex+.1)-1yield
Il
a
x
(a+ x)1- (a- x)'
l
xj=jodx(a-x)!'+jf
jdx
x (a- -v)1
ex.
-4
-.
j +Jzdxt
ax-o-j-.
Thelastterm isexponentiallysm allin thelim itoflarge(
x,and wecanapproxim ate
thenumeratorin thesecond integralas
A straightforward calculation (see Appendix A)therefore givesthe asymptotic
expansion
(5.52)
Not'
e that this resultgives #P'(r,1'./2),B'hich are the proper thermodq'nam ic
variablesforthetherm odynamicpotential. Thecorrectiontermsinthiseqtlation
(indicated bydots)areofhigherorderin 7-2and thusnegligibleto thepresent
order.
The num ber of particles can be determ ined im m ediately from thisexpresx=
jP(P
#/z
F')jwz=4
t
%
*Vj'
z
Vzz
ljlJ
2gsêo.(z.
sw)cj=
/
.
,
l
1.
+..
4:
INTRODUCTION
IfN/P'isrewritten in termsoftheFermienergy fs using Eq.(5.46)wehave
*2Ikul'jl
-1
Jz=es1+yj'
s)+''.
(5.55)
which m ay besolved forJzasa powerseriesin 7-2
,
p.2 k w z
#' = t' 1- i
BF + .. .
-j s
(5.56)
Theentropycanbedeterminedfrom Eq.(5.53)bydiflkrentiatingat/xe#Vandg
s(r,p,p.
)-(0(a
PwP'
)jzs-4
#.
n
Pa
'(2
s?
jèJ
2j24
=2:jws++...j
(5.
j7)
SinceS is a therm odynam ic function,itm ay be expressed in any variables'
,in
particular,substitutionofEq.(5.56)yields
ksr *2
&(r,v.N)= Nks fF -j-
(5.58)
.
to lowestorderin F. W ecan thuscompute the heatcapacity from therelation
'
es
,.2
k r
cv-rtajss,-j.yk.2s
(5.59)
which gives
j.s
2 N i pr!.
Cr,= S= m
-/?a 6; VF x
-
(5.60)
..
N ote thatthe heatcapacity fora Ferm igasatlow tem perature islinearin the
tem perature. In contrast, at high tem perature. where Boltzm ann statistics
apply,theheatcapacityofaperfect(BoseorFermi)gasis
C)-.
->.jNku T-->.:t)
(5.61)
and the heatcapacity ofan idealFerm igasatalltem peraturesisindicated in
Fig.5.4. N ote thata Ferm igashas no discontinuities in the therm odynam ic
variablesatany tem perature.
Fig.6.4 Constant-volume heatcapacity ofan
idealFermigas.
srATlsTlcAu M EcHANlcs
4:
The noninteracting Ferm igas form sa usefulsrstapproximation in the
theory ofm etals,inthetheory ofliquid H e3,in studiesofnuclearstructure,and
even forunderstanding such diverse phenomena asthe structureofwhite-dwarf
and neutron stars. Fora detailed understandingofthebehaviorofthesem anybody assem blies,however,wem ustincludetheinteractionsbetweentheparticles,
which formsthecentralproblem ofthisbook.
PRO BLEM S
2.1. Provethattheentropy ofan idealquantum gasisgiven by
S= -ks)i(ln2lnn?7F:(1+ n3)ln(1:!uay))
wheretheupper(lower)signsrefertobosons(fermions). Findthecorresponding
expression forBoltzmann statistics. Provethatthe internalenergy is given by
E = jlEfnîforallthreecases.
f
2.2. Given the energy spectrum ep= R#c)2+ znàc4)1-->.pc (p --.
>.cc)),prove
thatan ultrarelativistic idealgassatissestheequation ofstatePV = E/? where
E isthetotalenergy. tcomparewithEqs.(5.19)and (5.42).)
2.3. Show thatthereisno Bose-Einsteincondensation atanysnitetemperature
fora two-dim ensionalidealBose gas.
2.4. Consideranidealgasinacubicalbox(P'= L3)withtheboundarycondition
thatthesingle-particlewavefunction vanish atthewalls.
(J) Find the density ofstates. In the thermodynamic Iimit,show thatthe
thermodynamicfunctionsforboth bosonsand fermionsreducetothoseobtained
in Sec.5.
(:) Discuss the onsetofBose condensation and compute the propertiesfor
F<L .
2.5. W hen a metalisheated to a suë ciently high tem perature,electrons are
emitted from the m etalsurface and can be collected as thermionic current.
A ssum ingtheelectronsform anoninteracting Ferm igas,derivetheR ichardson-
Dushman equationl for the currenti= (4=emkàTl(h2)e-W'
'
kBT where !#'is
the work function for the metal (i.e., the energy necessary to rem ove an
electron).
2.6. ProvethattheparamagneticspinsusceptibilityofafreeFermigasofspin-!
particles at F= 0 is given by y(F= 0)= j.
L2m(h2k;)y,
jNJV where y,o is the
magnetic m oment of one of the particles. Derive the corresponding high-
temperatureresulty(F-+.cc))= SXiN/ICBTV.
'S.Dushman,Rev..
M'
tl#.Phys..2:381(1930).
so
INTRODUCTION
2.7. For a srst approxim ation to atom ic nuclei,consider the nucleus as a
degeneratenoninteracting Ferm igasofneutronsand protons.
(t8 W hatisthedegeneracyfactorforeachlevel?
(:)Iftheradiusofa nucleuswith ,4 nucleonsisgiven by R = rcWlwith rc=
1.2 x 10-13cm ,whatareksand er? H ow do theyvarywith W ?
(c)W hatisthepressureexertedbythisFermigas?
(#)lfeachnucleonisconsideredto bemovinginaconstantpotentialofdepth
Prc,how largemustL be?
(e)Atwhattemperaturewillthenucleusactlikeacollectionofparticlesdescribed
by Boltzm ann statistics?
2.8. A s a m odel of a white-dwarf star,consider an electrically neutralgas
composedoffullyionizedHe(aparticles)anddegenerateelectrons.
(c)Writetheequation ofIocalhydrostaticequilibrium inthelow-density (nonrelativisticelectrongas)andhigh-density(relativisticelectrongas)limitsassuming
an idealFerm isystem .
(b)Hence :nd expressions for the density p(r)and the relation M = M (Rj
between thetotalmassM and theradiusR ofthestar.
(c)Show thereexistsamaximum massM maxcomparablewith thesolarmass
M e. Explain thephysicsofwhythisisso.
(#)Check the initialmodel using the typicalparameters of a white dwarf
p q$107 g/cm 3;
k;107pe,M ;
k:1033 g;
k;M escentraltemperature;k;107OK $
1 rz.
N otethefollowingresultsobtained by num ericalintegration:l
)1
'
ik
#
t(yad
..
#
J
(j--ygimplies'fy,
t
l
j
);=j,
ys.a
t jaaoy
T'(0)-0;/(1)-0
a -
P
1W
d(ycù
df().-y3)impliesfy,
(
ol6.
89,
7
tl'
--2.018
f,(0)-0,
'.
/.(1)-01
lL.D.Landau and E.M .Lifshitz,op.cit.,seec.106.
part 1w o
G rou nd -state
(Zero-tem perature)
Fo rm alism
3
G reen's Functions and Field T heory
(Ferm ions)
In m ost cases ofinterest,the Nrstfew orders of perturbation theory cannot
provide an adequate description of an interacting m any-particle sàstem . For
thisreason,itbecom esessentialto develop system atic m ethodsforsolving the
Schrödingerequation to a11ordersin perturbation theory.
6LPICTURES
Asa preliminary step,weshallintroducethreeimportan:pictures(Schrödinger,
interaction,and Heisenberg)thatare usefulin analyzingthe second-quantized
form oftheSchrödingerequation(Eqs.(1.41)and(1.60)J.
SCHRöDINGER PICTURE
Theusualelementary description ofquantum m echanicsassum esthatthe state
vectorsaretim edependent.whereastheoperatorsare tim eindependentand are
53
54
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
constructed by the fam iliar rules from the corresponding classicalquantities.
TheSchrödingerequation thereforetakestheform
? ,
ihjjI'I-s(?))- W'$
:Fs(/))
(6.1)
where.
f.
?'isassumed to haveno explicittimedependence. SinceEq.(6.1)isa
hrst-orderdiflkrentialequation,theinitialstate attodeterm inesthe subsequent
behavior,and aform alsolution isreadilyobtained by writing
1'1's(r))- e-iWf'-t:'/9l'l's(fp)
(6.2)
Here the exponential of an operator is defined in term s of its power-series
expansion. Furthermore,X ishermitian sothattheexponentialrepresentsa
unitary operator. G iven the solution to the Schrödingerequation atthe tim e
to,theunitarytransformationinEq.(6.2)generatesthesolution attimet.
INTERACTIO N PICTURE
A ssum e,asisusuallythecase,thattheham iltonian istim eindependentand can
beexpressed asthe sum oftwo term s
X = Xt
l+ X1
(6.3)
whereXtactinyaloneyieldsa solubleproblem. How can wenow includeaIl
the eflkcts of S l? Deine the interaction state vector in the follow ing way
(
'
t1,
's(?'))M einz'/9t'
tl7s(J))
(6.4)
whichism erelyaunitarytransform ation carried outatthetim ez. Theequation
ofmotion ofthisstatevectoriseasilyfound by carryingoutthe timederivative
ih I''
I-;(?))=-.
f% e a,ysjv x(/))+ ://?;,/,m jP-jjv sttl)
? .
js
=ekûztlhL-lh +#(j+#j)e-t/lef/AjNpI(,))
and we thereforeobtain thefollowingsetofequationsin theinteraction picture
ê ,
ih 1.1.,(?))- .
??j(r)1
'.
Ir,(,))'
(6.5)
z;l(?)- eiBztl'i1e-fnee/.
Ingeneral,/% doesnotcommutewithX1.so thattheproperorderofthese
operators isvery im portant. An arbitrary matrix elemcntin the Schrödinger
picture may be written as
('l''s
'(?)1:sI'1'
-s(r))- ('l'J(/)Iei4:l'.dse-f/el''I'I'z(f))
(6.6)
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
s5
which suggeststhefollowing desnition ofan operatorin theinteraction picture
oz(,). eknotthose-fnat/,
(6.p
Equations (6.4) and (6.7) show thatthe operatorsOï(t)and the state
vectors1
kF'z(/))bothdepend ontimein theinteraction picture. Theimportant
pointhere isthatthe tim e dependence ofthe operatorsisparticularly sim ple.
DiflkrentiateEq.(6.7)withrespecttotime.
ihP :,(?)- etn(,!/,(osJ.
?.
o- J'
èoos):-(J?:f/,
.
--
(t),(r),z?c1
(6.8)
H ere the tim e independence ofthe Schrödinger operatorhasbeen used along
with theobservationthatanyfunctionofanoperatorcom m uteswiththeoperator
itself. Considerarepresentation in which Xoisdiagonal.
J% - jkghœkclck
Thetim edependenceofthecreationanddestructionoperatorsintheinteraction
picturecan bedeterm ined from thediflkrentialequation
?
J.
ih& ckz(?)- el?(',ysEcks,p()je-tavt/s. yj.:cgglt)
which iseasilysolved to yield
Ckf(f)= Cke-'t
l?kî
alongwithitsadjoint
ck
ll(/)=4eîa'.f
(6.10J)
(6.10à)
Thus the tim e occurs only in a com plex phase factor,which m eans that the
operatorpropertiesofhlt)and c1(r)arejustthesameasintheSchrödinger
picture. ln particular,thecommutation relationsofckand claresimply the
canonicalonesfrom Chap.1. Furtherm ore,any operatorin the Schrödinger
picturemay beexpressedintermsofthecompletesetckand c1.andthecorresponding operatorin the interaction picture is obtained with the substitution
ck-+.ckglt),cl--+.ti(?). Thislastresultfollowsfrom theidentity
1= :- fR(
)tlhekR(
)tlh
which m ay beinserted between each operatorintheSchrödingerpicture.
w eshallnow trytosolvetheequationsofm otion in theinteractionpicture.
D esne a unitary operator 1.
'
1(/,/:)thatdeterminesthe state vector attime tin
.
term softhe state vectoratthe tim e tv.
I'l'z(?)'
)- tJ'(?,J(
)!
'Fz(?o))
56
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
Evidently,('
)mustsatisfy therelation
0(/0./0)= l
(6.12)
Forhnitetimes L'
/ltstz)can beconstructed explicitly byusingtheSchrödinger
picture:
1
:1C,(/))- e'J?'
ef/'I'l's(!))- e'*n'/'e-'9t'-'n'/'1
T ,(f:))
= ef/1:1/hé'-t/?(t-t())/âe-4Rot(
)/ht
'
tj.
pJ(/();j
which thereforeidentises
pLt.0
t)zczoin.
ot/,e-fn(t-t(
)),
/:e-iJ)tlto)pj tjinjtetjmes)
(6.13)
SinceX andXodonotcommutewitheachother,theorderoftheoperatorsmust
be carefully maintained. Equation (6.13)immediately yieldsseveralgeneral
propertiesofP
t'
)'
t(?,/.
())f;(/.
,/0)- L'
lltnto)6-1(/,/0)- 1
which impliesthatL'
/isunitary:
t'
)t(?,/o)- r7-1(r,/(
))
P(/1,/c)Cltz,tss- P(?1'/3)
whichshowsthatP hasthegroupproperty,and
0(/,/0)Cltçtntj= 1
(6.14)
(6.15)
which im pliesthat
P(fa,r)= Il'
jt,tzj
(6.16)
Although Eq.(6.13)istheformalsolution to the problem posed by Eq.
(6.1l),itisnotveryusefulforcomputationalpurposes. Insteadweshallconstruct
an integralequation for0,which can thenbesolved by iteration. Itisclear
from Eqs.(6.5)and (6.11)that0 satissesadiferentialequation
ih& I')(r,?a)= P1(/)I'
/ltttè
(6.17)
lntegratethisequation from toto t
Clt,tf
i 'dt'Xj(/'')t/lt,,G)
)l- I'
/ltovtoj= -j
re
Thisresult,combined with the boundary condition (6.12),yieldsan integral
equation
r7(?,ra)- 1- i
j tdt'4j(/')t'
)'
(/',to4
rc
(6.18)
GREEN'S FUNGTIONS AND FIELD THEORY (FERM IONS)
67
lfP wereac-numberfunction,Eq.(6.18)wouldbeaVolterraintegralequation,
because the independentvariable tappears asthe upperlimit ofthe integral.
Under very broad conditions Volterra equations may be solved by iteration,
and the solution isguaranteed to converge,no matter how large the kernel.'
Thereisno assurancethatthepresentoperatorequation hasthesameproperties;
neverthelessweshallattemptto solveEq.(6.l8)by iteration,alwaysmaintaining
theproperorderingofthe operators. Thesolution thustakestheform
Considerthethird term in thisexpansion. ltmay be rewritten as
t
J'ejs,jto
'dt,pi(,,
)sj(/,)
-'
l
'jt
'odt'jo
''#?'
-tîkç
;
t'
)I'
èjlt')+!.j,
t
at
s'
:
/'
-.
f,
'dt'.
41(/.
'
).
?l(r,
'
) (6.
20)
since the lastterm on therightisjustobtained by reversing theorderofthe
integrations,asillustrated in Fig.6.1. W enow changedum m yvariablesin this
Fig.6.1 lntegration regionsforsecond-order
term in Olt.fc).
secondterm,interchangingthelabelsf'andf'.andthesecondterm ofEq.(6.20)
thereforebecom es
Thesetwo term sm aynow berecom bined to give
t
t'
,,
j#/
jyt
jdt(y(jdtjjj(t,).
#jjj,).jjyt
.
,s
jy
tj#y,
,
.
.
x (.
/.
/l(t').
f'
?I(r')0(t'- t')+ Ièklt').
/.
31(/')0(tn- tz))
lSee,for example.F.Smithits.tilntegralEquations,''p.31,Cambridge University Press,
Cambridge,1962.
'
s:
GHOUND-STATE (ZERO-TEM PERATURE) FORMALISM
wherethestepfunction(Eq.(3.28))isessentialbecausetheoperatorsX:donot
necessarilycommuteatdiflkrenttimes. Equation(6.21)hasthecharacteristic
feature thatthe operatorcontaining the latest tim e stands farthestto the left.
Wecallthisatime-orderedproductofoperators.denotedbythesymbolr. Thus
Eq.(6.21)uanberewrittenas
'
t
r
*t
(-#?, (-#Jv jyk(?t)j.,j,(?t)-.
!.#
, , r p j.j t,
jy jp
,jj (6.jgj
.'l(
).
't(
)
.c
l'J,
(
,dtrE.(?) !(
,.
.
,
Thisresultisreadilygeneralized,andtheresultingexpansionforC'becomes
X
L..J
' -.
-
(fa/())=
n=0
.n
'.
i l r
-h
-' 1-1''
! t(
)f/f) '
.
(6.23)
wherethe??= 0term isjusttheunitoperator.' Theproofot
-Eq.(6.23)isas
follows. Considerthenthterm inthisseries. Therearen!possibletim eorderings
ofthe Iabels tI . ' ' tn. Pick a particularone,say,Jl '
> rc > ?a . ' .> ?,,. A ny
othertimeorderinggivesthesamecontribution to P. Thisresultiseasily seen
bJ'relabeling the dum m y integration variables ti to agree with the previous
ordering.and then using the sym metry ofthe r productunderinterchange of
itsarguments:
lIjltjl
. .
.
.
(6.24)
Eqtlation (6.24) follows from the deénition of-the F product.w'
hich puts the
operatoratthe latesttim e farthestto the left,the operatoratthe nextlatesttime
next.and so on,since theprescription holdsequally w'
ellforboth sidesofEq.
(6.24). Inthisway,Eq.(6.23)reproducestheiteratedseriesofEq.(6.19).
HEISENBERG PICTU RE
The statevectorin the Heisenberg pictureisdesned as
1kI.-H(t)'EëEeiJ?''91
51-$(/).
(6.25)
ltstimederivativemaybecombinedB'ith theSchrödingerequation(6.l)toyield
.
P .
ih& TVI-s(/).= 0
(6.26)
which show'sthat !
Vl.*s '.istim e independent. Since an arbitrary m atrix element
in the Schrödingerpicture can be w ritten as
v)jastjyjjQs,
!l.
j*
a.
u-y4p
kj.
lny,
s)j,j.m..
l.
fg
hei/1tjhpsyy-iJ)th(
.
1Equation(6.23)issometimeswritten asa formaltime-orderedexponential
(
/6/,
?:)-Ft
f
expg-l
'
â-'.(,
'
0dt'.
4,(?'
)
j/
l
sincethepower-seriesexpansion reproducesEq.(6.23)term byterm.
*
$
>
QK= m
b -cX
A=.T Y :
(6.gyj
GREEN'S FUNCTIONS AND FIELD THEORY (FERM IONS)
59
a generaloperatorin the H eisenberg picture isgiven by
ozf(/)useipt:osp-f/?fh
(j.gg)
NotethatOH(t)isacomplicatedobjectsinceX andOsingeneraldonotcommute.
w e see that the H eisenberg picture ascribes a11 the tim e/dependence to the
operators, whereas the corresponding state vectors are time independent. ln
contrast,the operatort)s in the Schrödinger picture is time independent,and
thetimederivativeofEq.(6.28)yields
P
ih)
-Ovqt)= eis,/,gos,yyje-iatfh = jywt/jssj
r
Thisim portantresultdeterm inestheequation ofm otion ofany operatorin the
Heisenbergpicture. Inparticular.ift'
hscommuteswithX,therightsidevanishes
identically.and Ou isa constantofthemotion.
Equation (6.28)can be rewritten in termsofinteraction-picture operators
(Eq.(6.7)q
OJf(/1= Pi/?t/:t'-iJ?ot/hO1(t)(
yfz
'
?(
):/âC?-iJ1tt
'h
(6'3(j)
and theformalsolution forthe operatorC'gEq.(6.13)1yields
ôH(?)- I;r((),?)O,(t4t)
-/(?,0)
ln addition.thevariousdehnitionsshow that
1àI.'H',= I
.
Y1-S(0)'.= 1
.
V1'
*f(0)z
.
''
Os- :s(0)- ö;(0)
so thatallthree picturescoincideatthetime t= 0. Thestationary solutionsto
the Schrödinger equation have a definite energy,and the corresponding state
vectors in the H eisenberg picture satisfy the tim e-independent form of the
Schrödingerequation
J'
?1
à'
1-s - E 'i-n-'
.
.
These state vectors are therefore the exact eigenstates of the system and are
naturally very complicated foran interacting system. Equation (6.32)and the
desnitionoftheoperatorC'togetherleadtotherelation
I
'
l.
j*H'
xc:
ml
j.
q1(()ju= S;(0'y(;j;
Vj.>j(y(;j'
y
(6.y4j
;t
.
w hich allowsusto construetthese exacteigenstates from the interaction state
vectorsatthetimetowiththeunitaryoperatorCL
ADIA BATIC ''SW ITCHING ON''
Thenotionofsw itchingontheinteractionadiabaticallyrepresentsam athem atical
devicethatgeneratesexacteigenstatesoftheinteracting system from thoseofthe
e0
GROUND-STATE (ZERO-TEMPEBATURE) FORMALISM
noninteracting system . Sincewepresumablyknow allaboutthenoninteracting
system ,forexample,theground state,theexcited states,etc.,thisprocedurelets
us follow the developmentofeach eigenstate as the interaction between the
particles is switched on. Speciscally, we introduce a new tim e-dependent
ham iltonian
X = Pn+ e-fI'lP l
(6.35)
where 6isa smallpositive quantity. Atvery largetimes,both in thepastand
inthefuture,thehamiltonianreducestoJ%,whichpresentsasolubleproblem.
Atthetime /= 0,X becomesthefullhamiltonian oftheinteracting system.
Ifetendstozero attheend ofthecalculation,theperturbation isturned onand
ofl-insnitely slowly,or adiabatically,and any meaningfulresult must be independentofthequantity e.
Thehamiltonian (6.35)presentsa time-dependentproblem tha.
tdepends
on theparam eterE.
,and weshallseek asolutionin theinteraction picture. ltis
readilyverifedthatEqs.(6.17)and (6.23)remain correcteven when Plistime
dependentinthe Schrödingerpicture,and weim m ediatelyobtain
('l'z(/))- L.
'E(f.G)t'l'l(G))
(6.36)
where the time-developmentoperatordependsexplicitly on e and is given by
,.o
- r
,/.
()), .
&,(
,
i i'
ïn 1
t dtk.-. , dt
n
,
,
.
j
)
n
!
tQ
,(
n=0
j
-
.
x e-f(ltll+.''+Ir,I)Fg/l'
jLtj) ...J)j(/j)j
Now letthetimetoapproach-co;Eq.(6.35)showsthat# thenapproachesJ%.
In thislimit,theSchrödinger-picturestatevectorreducesto
l
T s(G))= e-'fn'Q/âI*(
))
(6.38)
where (*0)issometime-independentstationaryeigenstateoftheunperturbed
hamiltonian 4:
XaI*0)- .
Q I*(
))
(6.39)
and thecorrespondinginteraction-picturestatevectorbecom es
1
T z(G))- efNnt0/'1
kI7's(G))= 1*0)
(6.40)
Thusl'lPz(/())becomestimeindependentas to-+.-cs,
'alternatively,the same
conclusion followsfrom theequation
ê
ih.
y1
k1Cz(?))= e-fl
'lXl(/)1klPz(J))-->.0
(6.41)
lfthere wereno perturbation,theseeigenstatesin theinteraction picturewould
rem ainconstantin tim e,being thestationary-statesolutionsto the unperturbed
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
61
Schrödinger equation. A s t increases from -cc, however, the interaction is
turned on,and Eq.(6.36)determ ineshow the statevectordevelopsin time,a1l
the way to the tim e t= 0,when theinteraction isatfullstrength. Forsnitetim es
If1< e-',al1ofourpreviousresultsremain valid,in particularEqs.(6.32)and
(6.34). W ethusobtainthebasicre
-la
-tion
ltloh= lT-z(0))= tk(0,-oa)t*()l
(6.42)
whichexgressesanexacteigenstateoftheinteractingsystem intermsofaneigenstate ofH o.
W e m ust now ask what happens in the lim it e -->.0. D o we get tinite
m eaningfulresults? This question is answered by the G ell-M ann and Low
theorem,which isproved in thenextsection.
GELL-M ANN AND LOW THEOREM O N THE G RO UND STATE
IN QUANTU M FIELD THEO RYI
The G ell-M ann and Low theorem is easily stated :If the follow ing quantity
existsto allordersin perturbation theory,
1im b-'f(0'-:
x))'
1(l)a) EE 1
k1--0'b
r.o.
:t*:j.r
.
,/6(0,-:
----'c)j
1*(
)., umolà!-pkc,.
..)
-- - .
(6.43)
then itisan eigenstate ofH-,
- .j
tl.
*()'
.
s
H'
,
,
,
=
r.*01
%1,0)
E
j
'
kl.
'pq
.
z
a.
(*:1
'1os
(6.44)
Thisprescription generatesthe eigenstate thatdevelopsadiabatically from p4)o.,
astheinteractionisturnedon. lfkmo)isthegroundstateofthenoninteracting
system,thecorrespondingeigenstateofX isusuallytheinteractinggroundstate,
but this is by no m eans necessary. For exam ple, the ground-state energy of
som e system s does not have a perturbation series in the coupling constant.
(ForanotherexampleseeProb.7.5.) M ultiplyEq.(6.44)from theleftby the
statefmai;sinceXop(1)c)= Eo1
,f1)()2),weconclude
tQ)(
)iH-1(
kl*'
)
.
E - E ()= o,
$j
s
.
! 0)
.*0
(6.45)
A n essentialpointofthe theorem is thatthe numeràtorand the denom inator
ofEq.(6.43)donotseparatell'é'.
x'l
'
JrasE'-->.0. An equivalentstatementisthat
Eq.(6.42)becomesmeaninglessinthelimite--+0,
'indeed,itsphasedivergesIike
e-lin thislimit. The denominatorin Eq.(6.43)servesprecisely to cancelthis
inhnitephase gsee Eq.(6.51)and subsequentdiscussionl. The theorem thus
assertsthatiftheratio in Eq.(6.43)exists,theeigenstateiswelldesnedandhas
theeigenvaluegiveninEq.(6.45). W eproceed totheproofgiven byGell-Nlann
and Low.
lM .Gell-M ann and F.Low'
,Phys.Rtu'.,84:350(1951).
62
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
Considertheexpression
(P0-.
G )I'l%(e))-(Xt-fo)t%(0,-*)1*0)- (Pn,1.
1(0,-*))1*0) (6.46)
W e shallexplicitly evaluatethecom m utatorappearing on therightside. C on-
sidertheathterm inEq.(6.37)fortheoperatorC,andpickanarbitrarytime
ordering of the n tim e indices. The associated com m utator can be written
identicallyas
I)
f'
.
?'
.
(,,z?'
l(?f).
f?'
l(?,) .-..
J?1(&)1- (z?t-z?'
1(?.
y)(
I.
4,(/.
,) .--Iltltk)
+ 4I(?f)(z?'
e,.
41(/,))--.41(r.)+ --+ Iltlti).
41()
/,) --.1)
.
4(,,z?'
1(&))
Furthermore,Eq.(6.8)allowsustowrite
h84,(/)-
i p?
Ez%,z3,(J)1
(6.47)
Inconsequence,each ofthecommutatorswith Xtjyieldsa timederivativeofthe
interaction ham iltonian,
L.
lk
ïçt-l.
.
îtltillî'
jlto.
l...z?'1(&)1
h ?
P
ê
= I 0fj+ 't2+ --'+ 0la Xj(/f)XI(/J) ***.
f?j(1:)
forallpossibletinleorderings. Equation(6.46)thusbecomes
* z j n-l j 0
(
)
(z?t-Eè!
'l%(e))'--n (h âi -.dtk. -.dtn
=
l
ê
xen(1l+''.+ln)
i= 1
01i rg.
4jLtj) ...II'L(/a)jjtp4
)l (6.4s)
In deriving Eq.(6.48),allthe time derivatives have been taken outside ofthe
tim e-ordering sym bol. The validity ofthisstep can be seen from the identity
' a
$= l
P/f oltp- tq)p(/v- tr)''-bltu- tv)e 0
'
wherep,q,r, . .. ,u,risany perm utation ofthe indicesl,2, ..
diFerentiation is m osteasily evaluated with the representation
0(t)-j'dt'ô(?'
)
The
O
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
which immediately yields deltjldt= 8(f)= &(-t). n us the integrand of Eq.
(6.48)may% rewrittenas
r
g
(y
l
.
'
la
:
,,
)#(,
l
)..z?
.
l
(&)
j-(7
d
)1a
'
,)rE
#.
('l
) J?l
(
f
a
)l
Allthetime-derivativetermsinEq.(6.48)makethesamecontribution to
theintegral,asshown by changing dummyvariables;wethereforeretainjust
one,sayWêfj,andmultiplybyafactora. Integratebypartswithreslxcttott:
n isprocedureleadstotwo terms,oneofwhichissimplytheintegrand evaluated
attheend points,andtheotherarisesfrom thederivativeoftheadiabaticfactor.
W ethereforeobtain
(#c- Fc)l'l%(e))- -#!l'l%(e))+ eihgt-1
:F,(M)
g
(6.
49)
where #jisassumed proportionalto acouplingconstantg in orderto whte
-
j a-1
-#
j
a -/ aj
(n- 1)!#n= ihgbg -#' h-ign
Bythismeans,weobtainaseriesthatreproducesthestatevectorl
7 :(M)again.
Equation(6.49)isreadilyrewrittenas
(z?- ek)I'l%(M)- ihegj-I'l%(M)
(6.50)
g
Multiply thisequationontheleftby ((*:l1l%(e))1-1(*e1;since(a/êg)(*aI= 0.
we:nd
(*(
)lXllV%(e))= ihegvê ln(*c1l%(e))- E - Ez= LE
(*
:l
1l%(e))
dg
(6.51)
Ifewereallowed to vanish atthispoint,itwould % tempting to concludethat
LE = 0,whichisclearlyabsurd. Infact,theamplitude(*ôl
N%(M)mustacquire
an in:nitephascproportionalto ie-'so thate1n(tN l
q%(4)remains ânite as
e->.0.1 Equation(6.50)maybemanipulatedto$ve
(z;-s,-fAegj:
j)jol
i
(
,1
v
etf
nl
(
)
e))-r*1
:
(
F
,I
2
'
l%0()
e))(fAdgj
lI
n(*cI
.I
',(e))j
andacombinationwithEq.(6.51)snallyyields
(# - E) 11l%(M) = iyegyê oI'l%(
j
vM)
(
4:*(,1
'1%(e))
#'4: (h :v))
(6.jz)
-
p'
e are aow in a posîtion to lete go to zero. By assumption,the quantity in
bracketson the rightside ofEq.(6.52)issniteto allordersin m rturbation
t> .forexample.J.Hubbard,Proc.Roy.Soc.(& -z#na),m
:539(195D.
64
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
theory,thatis,in g,and the derivatiNe with respect to g cannotchange this
property. Sincetherightside ism ultiplied bl e,itvanishesasftendsto zero,
which provesthebasictheorem
(# E4lim !:1,
*,
0(-e)7
: -0
e..o(*(
)t1-(
)(.'))
-
Thisproofappliesequally wellto thequantity
t%(
0,+œ)
))
t*o
IG(
0,+cl
s*o
)I*o
(6.54)
where
tk(0,+co)- P1(-t.x.0)
Herethesystem ''comesback''from ?= +.
x,,where theeigenstateis /*0),. If
thestatethatdevelopsoutof)*a)isnondegenerate,thenthesetwodesnitions
m ustbe thesam e. They could difl
krby a phase factor,butthecom m on norm alization condition
(*n
l
à1%)
1(*o1
kl'())'- 1
(6.55)
precludes even this possibility. Thus,for a nondegenerate eigenstate
lim 0e(-0,+cc))*(
)J-
= lim
C(.0.-'
x))l(1)()X
?
.
e-yo(mof&r(O,+cc,)f*o) e-.o'
tmof&r(O,-cc)(*0)
(6.56)
A snotedbefore,thestateobtained from theadiabaticswitching procedureneed
notbe the true ground state,even if *t)'
xisthe noninteracting ground state.
The Gell-M ann and Low theorem merely asserts that itis an eigenstate; in
addition,ifitisa nondegenerate eigenstate,then both waysofconstructing it
IEq.(6.56)Jmustyieldthesameresult.
7QGREEN'S FUNCTIONS
Thissection introducesthe conceptofa Green'sfunctionl(orpropagator,asit
issometimescalled),whichplaysafundamentalroleinourtreatmentofmanyparticleassem blies.
DEFINITION
The single-particle Green'sfunction isdesned bytheequation
iG (
xbxl,x't')=
('Pn1FE#,,a(x/)'
l
;Lb(x'/,))1
*1*0)
(v oj
v ())
lV.M .Galitskiiand A.B.M igdal,Sov.Phys.
-JETP,7:96 (1958);A.Klein and R.Prange.
Phys.Ret'
.,112:994 (1958);P.C.M artin and J.Schwinger,Phys.Aer-,115:1342(1959).
GREEN'S FUNcTloNs AND FIELD THEORY (FERMIONS)
66
where 1
Y%) istheHeisenbergground stateoftheinteracting system satisfying
X1
'F:)= E 1'F())
(7.2)
andguqlxt)isaHeisenbergoperatorwiththetimedependence
;tHx(x/)= eiBtlh,
#
/)a(x)e-ikltlh
'
(7.3)
Heretheindicesa and# labelthecomponentsof'thefeld operators;xand j
cantaketwovaluesforspin-!fermions,whereastherearenoindicesforspin-zero
bosons,because such a system is described by a one-component seld. The
F producthererepresentsageneralization ofthatin Eq.(6.22):
Fl
iuulxt)'
Jlfjtx'/')
#zratx/l'
fàjtx'r'))- +f1/x,t,)'iuxlxt)
'
'
(7.4)
wheretheupper(Iower)sign refersto bosons(fermions). Moregenerally,the
r productofseveraloperatorsordersthem from rightto leftin ascending tim e
orderandaddsafactor(-1)#,whereP isthenumberofinterchangesoîfermion
operators from the originalgiven order. Thisdehnition agreeswith thatin
Eq.(6.22),becauseX jalwayscontainsanevennumberoffermionselds. Equation(7.1)maynow bewrittenexplicitlyas
'
t'I%l#,,atxrl'A blx'/.)1
:F(,)
iG (
xbxt,x'/')=
r'l%1'I%)
v ; (x,t,)ysz
+'
C o1f,#
tx/)jkl't))
t'FaI'Po)
(7.5)
The G reen'sfunction isan expectation value offeld operators;assuch,
itis simply a function of the coordinate variables xtand x't'. If X is time
independent,then G dependsonly on the tim e diflbrence f- t',which follows
immediatelyfrom Eqs.(7.2)and(7.3):
iG j(x/,x't')
eçt-tetlh('
tl'01
4œ(x)e-tWtf-'''/''
(7(
)1
,
tlr0)
ei
b x'
=
f'FoI'Pn)
r
,
.
l
.
'
I
4#
(
x
'
f-r')/.4x(x)I'l%)
+e ir(l-le)?: 0 # )ef/(
f'l%l'l%)
(7.6)
-
HerethefactorexpfzjziE(?- t'jlh?ismerelya complex cnumberand may be
taken outofthematrixelement;in contrast,theoperatorP between theEeld
operatorsm ustremain aswritten.
66
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
RELATION T0 OBSERVABLES
ThereareseveralreasonsforstudyingtheGreen'sfunctions. First,theFeynman
rulesforsnding the contribution ofath orderperturbation theory are sim pler
for G than forothercom binations of5eld operators. Thisresultis discussed
in detail in Sec.9. Second,although the ground-state expectation value in
Eq.(7.1)impliesthelossofmuch detailed information abouttheground state,
the single-particle Green's function stillcontains the observable properties of
greatestinterest:
l.Theexpectation valueofany single-particle operatorin the ground state of
tbesystem
2. Theground-stateenergyofthesystem
3.Theexcitation spectrum ofthesystem
Thehrsttwo points are demonstrated below,while the third followsfrom the
Lehm ann representation,which isdiscussed laterin thissection.
Considerthesingle-particleoperator
â=f:3xy(x)
where #(x)isthe second-quantized density forthehrst-quantized operator
Jbxlxt:
#(x)-a)#(#;txlljztxl'
(ktxl
Theground-stateexpectation valueoftheoperatordensityisgiven by
(#(x))-('l%1,/(x)1
:1'n)
v jvo)
(.1'oIv-,
f(x'),
y
1a(x)1:Po)
-
m.x)
2.
& .(x)
'
Xli
up
(:F(,1:1%)
+.flim lim JJ./j.txlG.jtx/,x'/')
-
=
1,..1+x'-xa#
= Lkil
im 1im trV(x)G(x/,x't'))
t'-yf+ x'ex
HeretheoperatorJjatx)mustactbeforethelimitx'-->.xisperformedbecause
J m aycontain spatialderivatives,asin them om entum operator. Furtherm ore,
thesymboltsdenotesa timeinhnitesimally laterthan t,which ensuresthatthe
:eld operatorsin thethird lineoccurin theproperorder(compare Eq.(7.5)).
Finally,the sum overspin indices m ay be recognized asa trace ofthe m atrix
productJG ,which is here denoted by tr. For exam ple,the num ber density
(?1(x)),the spin density (ê(x)),and the totalkinetic energy (f') are readily
found tobe
(H(x))= zkitrGlxtnxt+)
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
(ê(X))= iitr(*G(Xl,Xl+)l
(/)=zkiJ#3xl
m.x
Ki
'-
h2/2
- 2m trGtx/,x'I+)
*7
l
wgsgj
(7.10)
The interesting question now arises:Is italso possible to constructthe
potentialenergy
fP) iI fd'x#3x'('I%If4(x)#/(x')p-(x,x')..,,,#,4,.(x')4.,(x)Ivo)
(.t%1
v ,)
- '
,
pq'
(7.11)
andtherebydeterminethetotalground-stateenergy? SinceEq.(7.11)involves
fourfield operators,wemightexpectto need thetwo-particleGreen'sfunction.
The Schrödingerequation itselfcontainsthe potentialenergy,however,which
allowsusto5nd(P)intermsofthesingle-particleGreen'sfunction. Consider
theHeisenbergseld operator#sz(x?),withthehamiltonian
J?- 72f:3x#1(x)r(x)4ztxl
.
+ !.xx
)('Jd3x#3x'#)(x)#j
ftx')F(x,x')..,,#j,#j.(x')#ae(x) (7.12)
)#'
The identity of the particles in the assembly requires thatthe interaction be
unchanged underparticle interchange
F(x,x').a-,pb,= F(x',x)j#,..z(7.13)
(Moreformally,such aterm istheonly kind thatgivesanonvanishing contribution inEq.(7.12).) TheHeisenbergequation ofmotion (Eq.(6.29))relates
thetimederivativeofftothecommutatorofç,with#.
ih P#s.(x/)= elsf/s(jx(x),#je-fntlh
,
(7.j4)
where
(#.(x),#)-)
#:j':3z(#.(x),4)(z)r(z)4,(z))+!.#
z#fdqzy'z'
yyJ
x(#.(x),f#z)#y
#(z')F(z,z')jj,,yy,fy-(z')#j-(z)) (7.15)
W enow usethe very im portantidentity
(,4,1C')= ABC - BCA = ABC - BAC + BAC- BCA
(.
x,#)C - B(C,W)
=
(z4,1)C - .
(7.I6)
#(C,z4)
which allowsusto expressEq.(7.15)in termsofeithercommutatorsoranticomm utators. Fordesniteness,consider the ferm ion case,since this is m ore
68
GROUND-STATE (ZERO-TEMPERATURE) FORMALISK!
complicated. W ith the canonicalanticommutation relations (Eq.(2.3))the
com m utatorisreadily evaluated,and we5nd
E#.(x),#!- F(x)4atx)- !. I fd'z,
g(z)p'(z,x)jj.,.y,4y,(x)4j,(z)
l#'y'
.
+!.b'Iyy'J#3z'#;(z')F(x,z').j,,yy,#y,(z')'
t
Jj.txl
Inthe:rstpotential-energyterm.changethedummyvariables# -+.y,j'-+.y',
y'-+.$',x-+z'. The symmetry of the potential (Eq.(7.13)1and the anticommutativityoftheNelds?;thenyield
E'X(x),P1= F(x)1;alx)+#')y(?'J#3z'#;(z?)P'(x,z').j,,yy,#y'(z')'
t
/j,txl
(7.18)
whiletheseldequation(7.14)becomes
ihy - F(x) #satxr)
-
Z Jd'z'#iytz'?)#'(x,z')aj.,y.'#,,y.tz'/),
;,,;'(x/) (7.19)
b'?y'
Equations(7.18)and (7.19)arealsocorrectforbosons.
M ultiplyEq.(7.19)by'
(1.(x't'4ontheleft,andthentaketheground-state
expectation value
ih P
satx'/'
)#sa(x?)1'l%)
#az,
& r(x) t'Fthl##
I
J
,
(:F(,1
'l%)
#,yy,
r'
klrolfzsatx'r')#s
'ytz't4F(x,z')a,',,,y'#sy'(z'tj'Jsj'(x?)!'1%)
x
(:1%1:F0)
..
(,
y.ao
-
In thelim itx'-,.x,t'-+.t+,theleftsideisequalto
ê
,,
ui
ufll'imyt+ x'
limyx ihs
- F(x) Gxxlxt,x t)
w,w
..
(7.21)
-
W enow sum over= and integrate overx,which snafly yields(compare Eq.
(7.11)j
P
,,
(P)= utrl.
f.f#3x l'li
m
1
i
m
i
h
j
j
r(
x
)
Ga
a
t
x
/
,
x
t)
(7.22)
-+f+ x'-yx
A combination ofEqs.(7.10)and (7.22)then expressesthetotalground-state
energysoleiy in term softhesingle-particleG reen'sfunction.
E- (J-+ P)- (Iî)
?
,,
= +!.ïJ#3x limyr+ 1
im ihx
+ F(x) trG(x/,x /)
x' jx
%'T.
'
...
-
P hlV2
j
lm trGlxt,x't'
= +!.fJ#3x 1im
lim ih.
-.t+ x' ,x
Ac
'-.
-
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
69
Theseexpressionsassume a simplerform fora homogeneoussystem in a
largeboxofvolum eF,wherethesingle-particleGreen'sfunctionmaybewritten
as(compareEq.(3.1))1
Ga/xr,x't')=Fk
* dul
.
.
gxjzefk*tf-f'e-fœtr-r'G.#(k,(t4
-X
(7.24)
InthelimitP'.
-.
>.cc,thesum overwavevectorsreducestoanintegral(Eq.(3.26))
G.b(x?,x'J'
)=(2*)-4Jd3kj*
-.dt
>efk*tx-x'e-l
L
uç'
-t'Gxb(k,t
.)
j
andacombinationofEqs.(7.8),(7.23),and(7.25)gives
N-.
fd3xt?
i(x))-+,
'
(zj)4,
l
.
i
ntfdqkj--,Jr
xlet-ntrctk,
f
z,
)
F
co
(7.
26)
h2k2
E = +!.
ïta l4qlim
d?kJ..#(
x
)el
ul'
lam +hattrG(k,
(
s)
o,. J
-
H eretheconNergenctfactor
jim :ia?(l'-t). lim eiu''l
l'..+f+
'r)-+0+
desnesthe appropriate contourin the complex utplane;henceforth,the limit
T -+.0+willbeim plicitwheneversuch afactorappears.
For some purposes.itwould be more convenientto have the diflkrence
hœ - hlk2j2m appearing in Eq.(7.27). Thisresultcan be achieved with the
following trick,apparently due to Pauliand since rtdiscovered m any times.z
Theham iltonianiswritten with a variablecouplingconstantA as
z?(A)- J?o+ AJ?l
then
4(1)= J? and 4(0)-/%
and we attempt to solve the time-independent Schrödinger equation for an
arbitraryvalueof/
à:
4(A)î
'1%(A))- f(A)î'1%(A))
(7.28)
where thestatevectorisassum ed norm alized
('Fc(A)i'1'a(A)'
)- 1
'Fora proofthatG dependsoùly on thecoordinatediferencex - x'ina uniform system.see
thediscussion precedingEq.(7.53).
2See,forexample,D .Pi
nes,l6-l-heM any-BodyProblem ,
''p.43,W .A.Benjamin,Inc.,New
York,1961;T.D.Schultz,*souantum FieldTheoryand the M any-BodyProblem,''p.18,
Gordon andBreach,SciencePublishers.New York,1964.
7:
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
ThescalarproductofEq.(7.28)with(Ta(1)1immediatelyyields
f(l)- 4'l%(4l#(A)I'l%(z))
and itsderivativewith respectto theparameterAreducesto
a#z
-zr(A)--4td*F
:)(A)jy?(A)4
n?c(z))+4nr:(A)(
y?(A)d'P
:)(l);)
-
AF(Tc(A)lJJ#?j(?)l,po(A))
--1:(z)#-tvo(z)I
vc(A))+t'rc(z)lz?lInra(z))
-
dz
--
(vo(A)I#lI
v n(A))
(7.29)
where the normalization condition has been used in obtaining the last line.
IntegrateEq.(7.29)with respecttoàfrom zerotooneand notethatE(0)= Eo
andf(l)= E
E-Ez-
1dh
Q
w ('l%(A)lAJ%l'l%(A))
(7.30)
Theshiftintheground-stateenergyishereexpressed solelyinterm softhematrix
elementoftheinteractionAXI. Unfortunately,thismatrixelementisrequired
foralIvaluesofthecoupling constant0< A< 1. ln the usualsituation,where
4 1representsthe potentialenergy (Eq.(7.12)),a combination ofEqs.(7.22)
and(7.30)gives
1dt
j
d
f - Eo= H f 0 -j- #3x t1êim
l
i
m
i
h
z
-j- F(x) trGA(x/,x't')
yg+ xz-yx
..
with thecorresponding expression fora uniform system
#' l#A
c
o
hlkl
E-Ev=i-à.
f(2*)4 (
)y #3kj-c
odt
.
oetû'
ghul- 2m trGA(k,a)(7.
32)
ExAM etE: FREE FERM IONS
A s an exam ple ofthe above form alism ,consider the G reen's function for a
noninteracting homogeneous system of ferm ions. It is hrst convenient to
m rform a canonicaltransformation to particles and holes. In the deânition
oftheield (compareEqs.(2.1)and(3.1))
Wx)= Z
/kA(X)CkA
kA
weredehnetheferm ion operatorckaas1
ckA
k > kF particles
fk3= b!kA
.
k< kr
holes
C'M I
'n eabsenceofaparticlewithmomentum +.
k from tbehlledFermisea impli
esthatthesystem
pos= ses a momentum -k. For a prom r interpretation ofthe spin of the hole state,see
Sec.56.
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
n
which is a canonicaltransformation thatpreservesthe anticommutation rules
fJ.,J1-1-f*.:1,1= 3.,.
(7.35)
and therefore leavesthe physicsunchanged. Here theJ'sand b'sclearly anticommute with each otherbecause they referto diferentmodes. 'Fhe a%and a
om a'
atorscreate and destroy particlesabove the Fermisea,while the b%and b
om ratorscreate and destroy holesinside the Fermisea,asisevidentfrom Eq.
(7.34). n efeldsmay now be rewritten in termsofthese new om ratorsas
#s(X)= kAZ
/kA(X)DkAV kâ<11kF /k2X)Y kA
>kF
(7.36)
#z(Xf)= kà>k
Z w /kz(X)Wl*&îWA+ kA<Xkr /kA(X)d-f**îY kA
(7.3D
wherethehrstequation isin the Schrödingerpictureand thesecond equationis
in theinteraction picture. Equations(7.36)and (7.37)diFeronly in thatthe
interactionpicturecontainsacomplextime-dem ndentphase. Correspondingly,
theham iltonian becom es
Xc= )
(hœkclAckA
kA
=
J( hutkJ1AJkz-kâX
haikldkz5kz+ k/L
âttlk
< kr
< kr
kâ> kr
(partlclex)
(âplez)
(7.38)
(fllled'Yrml:e.)
In the absence ofparticles orholes,the energy isthatofthe 5lled Ferm isea.
Creating a holelowerstheenergy,whereascreating a particle raisestheenergy.
lfthetotalnumberoffermionsisExed,however,particlesand holtsnecessarily
occurin pairs. Each particle-hole pairthen hasa netpositiveenergy,showing
thattheilled Ferm isea representstheground state.
By desnition,the noninteracting fermion Green's function is given by
fGî,(xr,x'r')- (*oIrE#z.(x/)41/x'r')!I*e)
(7.39)
where the noninteracting ground state vector is assumed normalized,and the
superscriptzero indicates thatthis isa Green's function with no interactions.
W enow observethattheparticleand holedestruction om ratorsb0th annihilate
theground state
àkzI*o)= Jkàl*0)= 0
(7.K )
sincetherearenoparticlesaboveorholesY low theFermiseainthestate 1
*:2).
Equation(7.40)showstheusefulnessoftheparticle-holenotation. Theremainingterm foreach timeordering iseasilycomputed,and we:nd
iGkblxt.xzt;)= jzj)z'
-1jkefk.lx-x')e-ltthlf-1')
k
x (p(?- t')0(k- kz.
)- #(J'- t)94kz-- k)) (7.41)
72
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
where the factor 3a, arises btcause the sum over spin states iscomplete. In
the lim itofan infnite volum e,the sum m ation overk becomesan integration
iGQ
aj(x/,x't')= 3aj(2z8-3j-d3kpik*tx-x')d-iahff-r')
x I)
û(/- t')ù(k- ks)- 9(/'- ?)bqkF- A)) (7.42)
ltisnow usefulto introduceanintegralrepresentationforthestepfunction
8(/- t')= -
(
:
ci dœ e-it
,,
l(t-t')
.o)+ I.T
2=,
-.
(7.43)
Equation (7.43)isreadilyveritied asfollows:lft> t',then thecontourmustbe
closed in the lower-half u)plane,including the sim ple pole at (
.
v= -iT with
residue-1. lfl< /',then thecontourm ustbeclosed in theupper-halftzlplane
andgiveszero,becausetheintegrand hasnosingularitiesforlm f.
s > 0. Equation
(7.43)maybecombinedwith Eq.(7.42)togive
(k-kp) y ft/cs-Fcj
x3,jg(
,bc
o,+,
j .-o,,-,
.
,
yj(,
y.
44)
-
which im m ediatelyyields
G1j(k,œ)= îaj #(k- ks) + 0(kF- k)
(
.o- o)k+ i'
q f
.e- œk- i,
rl
ltisinstructivetoverifyexplicitlythatEq.(7.44)indeedreproducesEq.(7.42),
and also thatEq.(7.45)givesthecorrectvaluefor(i
9?
$(Eq.(7.26))andE H Efj
(Eq.(7.27)). Equation (7.45)also can be derived directly by evaluating the
Fouriertransform ofEq.(7.42),inwhichcasetheiiytermsarerequiredtorender
thetim eintegralsconvergent.
TH E LEHM AN N REPRESENTATIO N l
Certain features of the single-particle G reen's function follow directly from
fundam entalquantum -m echanicalprinciples and are therefore independentof
the specifk form ofthe interaction. This seetion is devoted to such general
properties. Although oursnalexpressions areformally applicable to both
bosons and ferm ions,the existence of Bose condensation atT = 0 introduces
additionalcomplications(seeChap.6),and weshallconsideronlyfermionsin
thenexttwo subsections. TheexactG reen'sfunction isgiven by
iGxblxt,x't')= (%%)rlvMsxtx/)vef
sjtx't'))1
k1%)
(7.46)
1H.Lehmann,Nuovo Cimento, 11:342(1954). OurtreatmentfollowsthatofV.M .Galitskii
and A.B.Migdal,Ioc.cit.and A.A.Abrikosov.L.P.Gorkov,and 1.E.Dzyaloshinskii,
çtM ethodsofQuantum Field Theory in Statisti
calPhysics,''sec.7,Prentice-llall,lnc.,Englewood Clifs.N .J.,1963.
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
73
wheretheground stateisassumed normalized f(kP()I
kI'%)= l). Ingeneral,the
Heisenberg held operators and state vectors in thisexpression are very comPlicated. Nevertheless, it is possible to derive some interesting and general
results. lnserta com plete setofHeisenbergstatesbetween theheld operators;
these states are eigenstates of the fullhamiltonian, and include allpossible
numbersofparticles. TherightsideofEq.(7.46)becomes
dGzjtxr,x't')= )(;(p(?- t')ttl'
-0f#s.(x?)('.
Fn)ttl''at#ijtx'/')(t1%)
0(t'-t)tkl'
ccll
/1#(x't')I
T,)t'l'al#s.(x?)lkI%)J (7.47)
-
Each Heisenbergoperatormay berewritten in thefbrm
0H(/)= diW'/â0Se-iW'/â
which allowsusto m ake explicitthetim e dependenceofthese matrix elements
l
'Gaj(xt,x'/')= )((#(/- t')e-içEn-E't'-'''/'(kl.
'()11
/a(x)1kra)(51.
7
-nr
#jf(x')41F'
())
a
- 0(t,- /)ejtz.
a-sl(f-t')/,(Aj.
c(
)jyt
#tx')jkl.
pn)(kj.
aajyu(x)4T*(j)j (7.4j)
Asapreliminarystep,weshow thatthestatesJT,,)containN + 1particles
ifthe state !
H7'(j) containsN particles. The number operator has the form
#-)
(fd3x#J(x)4.(x)
a
anditscommutatorwiththeseldoperatoriseasilyevaluated(fbrb0th bosons
andfermions)as
(#,'
;/z)!- -#j(z)
or,equivalently,
4'
;j(z)- ,
J#(z)(4 - 1)
Applythislastoperatorrelationtothestate 1
T1):
X(';'
j(z)1'l''(,))- CN - 1)(#j(z)l'1%)1
(7.49)
where we havenoted thatl
kFc) isan eigenstateofthe numberoperatorwith
eigenvalueN. Thustheseld9 actingonthestate1T'0)yieidsaneigenstateof
thenumberoperatorwithonelessparticle. Similarly,theoperator#1'increases
thenumberofparticlesbyone. Equation(7.48)thuscontainsonenew feature
thatdoes not occur in the ordinary Schrödinger equation,for we mustnow
considerassem blieswith diserentnum bersofparticles.
U ntilthispoint,thediscussion hasbeen com pletely general,assum ingonly
thatX istimeindependent. Although itispossibleto continue thisanalysis
withoutfurtherrestriction.we shallnow consideronlythesim plercaseoftranslationalinvariance. This implies thatthe momentum operator,which is the
generatorofspatialdisplacements,commuteswith 4. Itisnaturalto usethe
74
GROUND-STATE (ZERO-TEMPEBATURE) FORMALISM
plane-wavebasisofEq.(3.1)forsuchasystem,and themomentum operatoris
given by
êw )?(J#3x(û(x)(-fâV)fk(x)= I
âkclzckA
kA
(7.50)
Thecommutatorofê withtheseld operator9 (forbothbosonsandfermions)
iseasilyevaluated as
-
/âVfz(x)= I#.(x),ê)
Which can also berewritten in integralform ;
y1(x)= e-it.xlhyX((pvie.xlh
(7*5g)
Sincef:isaconstantofthe motion,thecomplete setofstatesalso can be taken
as eigenstates ofmomentum. W e therefore extractthe x dependence ofthe
matrixelementsinEq.(7.48):
'Gab(x/,x'f')= )(
i
)(#(t- t')e-ftfk-f'f'-J''''efPp.tx-x')/5
x(.1.
e(
)lfkt0)1T,n)('1-nl##'
f(0)1.1-o)- 0(t'- t)eiç%-E'('-t''''e-''%.(x-x''/.
x('lP:6'
;1
/0)t'F'
n'
)('l'n(
fa(0)(
'lCc)1
(7.53)
where wehaveobserved thatêikl'o')= 0. Equation(7.53)makesexplicitthat
G dependsonlyonthevariablesx- x'and t- f/.:k ThtcorrespondingFourier
transform is
Gxb(k,(,>)= f#3(x- x')j-#lf- t')ta-ik*tx-x')eiwst-t''Gxj(xr>x't')
Cl%1#a(0)i
'F.,
'
)t'L lf1
/0)1V1'
e0)
Iz')
a(dk,pa/, t5 ,-,(s.- y)+ j,y
.
-
+FZ3k.-w/,s-r:F0
.
;)(0h)-l'
1
-a
'I''
a2
1kk(
)
.1%)' (7.54)
-1
j'
l
k)
zt
e)
-0
,
.
,21
n
* +
where thezki'
rlisagain necessary to ensure the convergence ofthe integralover
lnthesrst(second)term,thecontributionvanishesunlessthemomentum
ofthe state l9'a)correspondsto a w'
avenumberk(-k),which can be used to
t - t'.
restricttheinterm ediatestates2
G. /k,fa))= P' 4'l%l
#a(0)l
'?k)(akI'3(
0)
I'l%)'
ts â-j(sa.s).
j.t
y
-
+r:1%1,/
1'
)(0)(a,-ik)(n,s)
-kt
v
1.
(l0)l
%'a)
-'
ix
'
-
(
.
o+ h (E'
n-
(7.55)
tFormanyprobiemsitismoreconvenienttoassumethattheinteractingparticlu moverelative
toasxedframeofreference. Forexample,intheproblem ofinteractingelectronsincrystalline
solidsand atoms,thecrystalline lattice and heavy atomic nucleusprovide such sxed frames.
ln tbiscasetheê oftheinteractingparticlesnolongercommuteswith 4,and theGrœn's
function may dex nd explicitly on x and x'. Thismorecomplicated situation isdiscussed in
Chap.l5.
GREEN'S FQINCTîONS AND FIELD THEORY (FERMIONS)
75
Thusthegeneralprinciplesofquantum mechanicsenable usto exhibitthefrequency dependence ofthe Green'sfunction,G causeutnow appearsonly lh the
denominatorofthissum.
It is helpful to examine these denominators in a little more detail. In
the Erst sum the intermediate state has N + lparticles,and the denominator
m ay be written as
(.
o- h-3LEnIN + 1)- S(1))= t,y- h-LLX LN + 1)- E(N + l))
A-i(F(# + 1)- ElNjj (7.56)
-
Now E(N + 1)- E(Njisthechangeinground-stateenergyasoneextraparticle
is added to the system . Since the volume ofthe system iskeptconstant,this
'
changeinenergyisjustthechemicalpotential(compareEq.(4.3)1. Furthermore.
the quantity En(N + 1)- E(N + 1)> E&(# + 1)isthe excitation energy ofthe
N + 1particlesystem ;bydefnition,%(# + 1)isgreaterthan orequaltozero.
Sim ilarly,thedenom inatorofthesecond term can bewritten
(
.
o+ â-lLEnIN - 1)- F(#))= o)- â-1(f(#)- E(N - l))
+ â-1LEnIN - lj- E(N - 1))
= ts- â-1Jz+ â-!e.(N - 1)
(7.57)
sinceE(N)- E(N - 1)isagainthechemicalpotentialrz,apartfrom corrections
oforderN-t. Indeed,theverydeEnitionofthethermodynamiclimit(N -.
+ cc,
F ->.cc.butN1Fconstant)implies
p.
IN + 1)= b4N)+ 0(#-1)
(7.58)
A lthough we shallnot attem pt to prove this relation in general.it is readily
dem onstrated forafreeFerm igasatzerotem perature,wherethePauliprinciple
furtherensuresthatp,= 4. Equations(7.56)and(7.57)can now becombined
withEq.(7.55)togivetheLehmannrepresentation
G
('l%)#.(0))nk)rak)#;(0))tl%)
.,(k,
tu)=AFY(uw.g-egvfyoj)ojp
+6:1%1#/0)1a,-k)(n,-kl#.(0)I'l%) yjo
hut-;+e-.-.(N-1)-i,
q 1(.
'
ltis possible to simplify the matrix structure ofG in the sm cialcase of
spin-l.. SinceG isa2x 2matrix,itcanbeexpanded in thecompletesetconsisting ofthe unitm atrix and the three Paulispin m atrices n. lfthere is no
preferreddirectionintheproblem,then G mustbeascalarunderspatialrotations.
Sincek istheonly vectoravailable to combine with e,G necessarily takesthe
form
G(k,ts)= JI+ :l.k
76
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
where the invariance underrotations impliesthata and b arefunctionsofk2
and œ. 1f,in addition,the hamiltonian isinvariantunder spatialreqections,
then G mustalso have thisproperty;but l.k is a pseudoscalar under spatial
reqections,so thatthe coeë cientb m ustvanish. Thussifthe ham iltonian and
ground state are invariantunder spatialrotationsand reiections,the G reen's
function hasthefollowingm atrixstructure
Ga#tk,t,al= 3.#G(k,f,
t))= 8=bG(!k1,(z>)
proportionalto theunitm atrix.
ItisinstructivetouseEq.(7.59)toreproduceourpreviousexpressionfor
G0(k,fs)(Eq.(7.45))in afreeFermisystem. Forthefirstterm ofEq.(7.59),
theadded particlem ust1ieabove the Ferm isea,and them atrixelem entsofthe
feld operatorsbecom e
f'l'014a(0)i?'k)(nkl#)(0)l
'Fn)-*P'-'3œj9(k-:,)
(7.61)
In thedenom inatorofthisterm ,theexcitation energy isthediflkrencebetween
theactualenergy oftheadditionalparticle and theenergy thatitwould haveat
theFerm isurface. Thustheenergydiflkrenceisgivenby
f
tN + 1)- E(X + 1)M ek(X + 1)--'
>'ek- ek=
k
â2(k2- k/)
lm
Thesecond term ofEq.(7.59)clearly correspondsto a holebelow the Fermi
surface,and them atrixelementsoftheEeld operatorsbecom e
('FoI'
;')(0)ln,-k)(n,-kl#z(0)lk1'o)-+.#'-'3zjgtks-k4
The ground state ofthe N - 1 particle system is reached by letting a particle
from theFermisurfacecomedownandflllupthehole;hencetheenergydifrerence
in thesecond denom inatorisgiven by
Ek
(N - 1)- f(N - 1)- e-k(X - 1)->'ek- f0k= hzlk;
2m- k2)
-
Sincep,= ek-foranoninteractingsystem,weobtainEq.(7.45).
Asnoted above,Eq.(7.59)exhibitsthedependenceoftheexactGreen's
function on thefrequency u),and itisinterestingto considertheanalyticproper-
tiesofthisfunction. Thecrucialobservation isthatthefunction G(k,(o)isa
meromorphicfunction of âa),with simplepolesattheexactexcitation energies
of the interacting system corresponding to a momentum âk. For frequencies
below glh,thesesingularitieslieslightlyabovetherealaxis,andforfrequencies
abovelxlh,thesesingularitieslieslightlybelow therealaxis(compareFig.7.l).
ln this way.the singularities of the Green's function immediately yield the
energiesofthoseexcited statesforwhich the numeratordoesnotvanish. For
an interacting system ,the itld operator conneets the ground state with very
m any excited states of the system containing N :
i:1 particles. For the non-
GREEN'S FUNCTIONS AND FIELD THEORY (FERM IONS)
77
interacting system ,however,the seld operator connects only one state to the
ground state,so thatG0(k,(s)hasonly asingle pole,slightly below the realaxis
athut= h2k2(2m ifk > ày and slightly above therealaxisatthe samevalue of
hulifk < kF.
Itisclear from thisdiscussion thatthe G reen'sfunction G isanalytic in
neitherthe upper northe lower (s plane. Forcontourintegrations,however,
itisusefulto considerfunctionsthatareanalytic in onehalfplaneortheother.
Aozplane
Fz-œn.-k(X-1)+iT e(k)t'
Nointeractions)
XXXX XXXXXXX AX X M
XXXXXXXXX X
/*+ 'nk(X+1)-l
h
Fig. 7.1 Singularities of G(k,(
z9 in the
complexhl.oplane.
W e therefore deine a new pair offunctions,known asretarded and advanced
G reen'sfunctions
I
'G'
ljtx/,x't')= ((
tl%r(#s.(x/),#).
Ijtx't'))(1
F0)0(t- t')
(7.62)
ftojtxr,x't')= -(Yeol
(1
Jsa(xr),'
f'
sjtx't'))411)0(t'- r)
where the braces denote an anticom m utator. The analysis of these functions
proceedstxactly asforthetim e-ordered G reen'sfunction. In a hom ogeneous
system ,we fnd the following Lehm ann representation oftheir Fouriertransform s:
GR
A(k(s)= h1z- '
IVI
1
/x(0)q
Nk)tDk1f)(0)1'Fo',
x)
Jjo1
,
N
t,
o- y - raktx + 1)s iv
tkl'01.
t
;j
f(0)1n,-k>Ln,-k(f,
z(0)$
kl'o)
hol- p,+ %
,- k(
N - 1)+ iy
NotethattheFouriertransformsGR(k,(s)and GA(k,(s)areagainmeromorphic
.
+
functions ofts. A1lthe poles of GR(k,ts)1ie in the lowerhalf plane,so that
GR(k,(o)isanalyticforlmf,
t
a>0,
'incontrast,a1lthepolesofGA(k,œ)liein the
upper half plane.so thatfP(k,t
x?)isanalytic for Im t
,
e< 0. Forrealop,these
funetionsaresim ply related by
EG1k(k,t,3.)*- G/.(k,u9
(7.64)
wheretheasterisk denotèscomplex conjugation. Theretardedand advanced
Green'sfunctionsdiflkrfrom each otherandfrom thetime-orderedGreen'sfunc-
tiononlyintheconvergencefactorsië,whichareimportantnearthesingularities.
If(.
o isrealand greaterthan â-1Jz,then the insnitesim alim aginary parts +f4
7:
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
inthesecondterm ofEqs.(7.59)and(7.63)playnorole. Wethereforeconclude
thatin thislimited dom ain ofthecom plex u;plane
Gl#(k,œ)= G.#(k,œ)
hœ real,>rz
(7.65/)
hœ real,< Jz
(7.655)
Similarly,
tQ,(k.œ)= G.,(k,tt0
Asnoted previously,G.j is usually diagonalin the spin indices:G.,= G3aj.
W ith the same assum ptions,the retarded and advanced Green'sfunctions are
also diagonal,and wemaysolveforG asG = (2x+ 1)-1EaGux- (2J+ 1)-1G.a
with theconvention thatrepeated indicesareto besummed.
lfthespacingbetweenadjacentenergylevelsischaracterizedbyatypical
valueàe,the discretelevelstructurecan be resolved only overtimescaleslong
compared withhjLe. Conversely,ifan observation laàtsfora typicaltimem,
then thecorresponding energy resolution isoforderhlr. Since àebecomes
vanishingly smallfora macroscopic sample,itgenerally satissestherestriction
Ae< hlr.andwethereforedetectonlytheIeveldensity,averagedoveranenergy
intervalhlr. In thethermodynamic limitofa bulk system ,itfollowsthatthe
discrete variable a can be replaced by a continuous one. Iftfa denotes the
numa roflevelsin asmallenergy intervale< eak< e+ dE,then thesumm ations
inEqs.(7.59)and(7.63)canberewrittenas
(2J.+ 1)-1F )(2!(akI
#1(0)lN%)12..'
œ (2J+ 1)-1FJdnI(akl#1(0)1
'1'a)12''.
=
#a
(2a+ 1)-1FfffeI(ak(#1(0)I'F())12w ...
M â-1J#eW(k,eâ-').
(7.66/)
and
(N + 1)-1FJJI@,.-kIfk(0)lt1%)12 ...- â-1.
fde1(k,eâ-1)
(7.66:)
which dtinethepositive-deNniteweightfunctionsadtk,e/âland #(k,e/â). The
corresponding Fouriertransform ofthesingle-particleG reen'sfunction becomes
Glk,
a(k,tx)')
:1,4
.
,:.)
t
,
al=jodt
z'o,-,ls-u),+i
.
o+(
.
,
)-,.ls+
u,,-y
.
,
(7.
67)
which now has a branch cutin thecomplex o)planealong the whole realaxis.
Thustheinfnite-volumelimitcompletelyalterstheanalyticstructureofG(k,tM,
becausethediscretepoleshave merged to form a branch line. Thesameresult
descria sasnitesystem whenevertheindividuallevelscannotberesolved.
GREEN'S FUNCTIONS AND FIELD THEORY (FERM IONS)
79
A sim ilaranalysisfortheretarded and advanced Green'sfunctionsyields
GR'
.
W(k,(,a')
,
#(k,(s')
X(k,t
z)=jodf
z
l(
.
o-:-ls-f
s,siy
l+(
s-:-1s+oà':
jui
xl (7.
68)
which showsthata1lthree Green'sfunctionscan be constructed if,4 and B are
known. ln addition,the sym bolicidentityvalid forreal(.
o
1 =
(.
o+ i.
rl
1
@:Fi'
rrblu))
(t)
.
(7.69)
showsthatGRand GA satisfy dispersion relations
R
eGR.?(k,oa)= EF.
t
#
* dœ'lm GR.A(k (
.
v'j
-
X C
* - œ
,' -
(7.70)
where,
' denotesa Cauchy principalvalue. Thisequation also holdsforsnite
system s,whereIm G isasum ofdeltafunctions.
These Green'sfunctions allhave a simple asymptotic behaviorfor large
Considerthe ground-state expectation value ofthe anticomm utator
'
t
:'l'
%If'
ik(A2),'
;l(X')1lV0)-îajî(X-X')
AnanalysissimilartoEq.(7.53)showsthat
3(x- x')= (2J+ 1)-1j)(efPn*tx-f'l/lttk1'a;
1J1(0)i
kFo)i
,2
.
j
.g-fP?!@(x-x?)/hj)1.
j.
*aj))
.
,x((;)kj%O
j
and itsFouriertransform with respectto x - x'yields
1= (2J+ 1)-1P-)((ttnk11
/1(0)1
'1'a)12+ i(a,.-.
k!
,1Ja(0)1
àI%)l2!
-
j0
=#ts(
:(k,
(
s)+a(k,(
s)J
wherethelastlinefollowsfrom Eq.(7.66). For1f
.
z)1-+.cc,Eqs.(7.67)and(7.68)
yield
c(k.(
s)-c/(k,
oa)=G,
.(k,
(
z,
)-f
1
z
aj0
*t
ït
s'gz
4(
k,(
s'
)+s(k,(
s')j
which rem ainscorrectforan arbitraryinteracting system .
PHYSICAL INTERPRETATION OF THE G REEN'S FUNCTION
Tounderstandthephysicalinterpretation ofthesingle-particleGreen'sfunction,
considertheinteraction-picture state 1
T z(/')),and add a particleatthepoint
(x't'):f4#(x'?')lH'
*J(?')). Althoughthisstateisnotingeneralaneigenstateof
80
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
thehamiltonian,itstillpropagatesintimeaccordingtop(t,/')41gx'f')ItF,(/')).
lc-ort> t'whatistheoverlapofthisstatewiththestate';1a(x?)I
.
kl'z(?))?
('l',(r)i,;za(x/)C'(t,t'4'Jljtx'?')i'F,(/'))
- (
*()1t'
)(vc,?)EP(?,0)'
;sz(x?)t')(0,J))I'
/ltnt'j
x Er
.
'
)(?',0);,
.
Lblx'?')t'
)(0,/')1Cl
'(t',-:
s)1
*0)
t'1'
-olf,latxrl#1,/x'?/)1
:1-c)
where we have used the resultsofSec.6. Thisquantity isjustthe Green's
function for t> /', which therefore characterizes the propagation of a state
containing an additionalparticle. In a similarway,ift< t',the seld operator
srstcreatesaholeattim et,and thesystem thenpropagatesaccordingtothefull
hamiltonian. These holes can be interpreted as particles going backward in
tim e,asdiscussedinthefam ouspapersofFeynm an.l Theprobabilityam plitude
ata latertim e t'forfnding a single hole in the ground state ofthe interacting
system isagainjusttheGreen'sfunctionfort< t'.
W e shallnow study how this propagation in tfm e is related to the function
G(k,(s),and,fordefniteness,weshallconsideronly theusualcasewherethe
time scale istoo shortto resolvethe separateenergy 1evels.2 By the deEnition
ofthe Fouriertransform ,the tim e dependence isgiven by
G(k,
az dol
?)= x ;
;-c-fœlG(k,(s)
=TF
-
'
Ift> 0,theintegralmay beevaluated by deforming thecontourinto thelower
halfo.
kplane. Since G(k,co)hasa rathercomplicated analytic structure,itis
convenienttoseparateEq.(7.73)intotwoparts:
G(k,l)=
$11hJl.
s
* dul
-2= e-ft
i''G(k,œ)+ lh- e-iLO:Gtk,tz)l
. -a7
/2 2=
(7.74)
In thesrstterm ((srealand < h-1p,
),G(k,f
z))coincideswiththeadvanced Green's
functionGA(k,t
r)
.(Eq.(7.65:)1,andtheintegralthusbecomes
lAIhe,
u'
llhlts
- ...
2* e-i(
.,tG(k'ts)=
.
alrGA(ky(
x))
2=.e-i
Now GA(k,(o)isanalyticinthelowerhalfplane,andthecontourcanbedeformed
from C'1to CL
'(Fig.7.2J). Equation (7.72)showsthatGA(and GR)behaves
!R.P.Feynman,Phys.Rev.,76:749(1949);76:769(1949).
2OurargumentfollowsthatofV.M .GalitskiiandA.B.M igdal,Ioc.cit.andofA.A.Abrikosov,
L.P.Gorkov,and1.E.Dzyaloshinskii,loc.cit.
GREEN'S FUNCTIONS AND F'ELD THEORY (FERMIONS)
81
liketr-ifbr14
.
,
ai-->.az;Jordan'slemmalthusensuresthatthecontributionfrom
thearcatinûnityvanishes,and Eq.(7.75)reducesto
Hlht/*
-co
.-
2-o,rG(k,f
x))=
.n.e f
v/
'h
Jco
à--e-izotGz
ttk,(s)
v/:-f. zrr
(7.76)
Thesecond term ofEq.(7.74)can betreated similarly,because G(k,(o)coincides
with GR(k,f
.
t))for real(
z?> Jzh. Thereisone importantnew feature,however,
Fig.7.2 Contoursused inevaluating G(k,t)fclrt> 0.
becauseGR(k,(s)isnotanalytic in thelowerhalf(.
oplane butinstead hassingularities. Fordefinitenesswe m ake a l)pr)'eIem elk/arq'm odelof the interacting
assem blyand assumethatGB(k.co)hasa sinl
plepoleclose ;t?the realaxisin the
lower halfplane at (
.
o= lt-lEk- iyk with residue a.where E'k> p,and 6k- /.
t>
hykl
.
p0. (lfGR hasseveralpoles,thesameanalysisappliesto each one separately.) The contour (72can be deformed to Ca
'(Fig.7.2:4,and the large arc
atinfinity again makesno contribution;the second term ofEq.(7.74)then
becom es
. ty*
s?&
r
-..
,'i-ix d
'
20.e-f
c
'
rG'
(k,(
,)-j j*c-ïa'
rGA(k,
(
o)-iat
?-f6
k'
7'e-k'
k'
. vfs
,n.
A combination ofEqs.(7.76)and(7.77)yields
Gtk,t)=
v,
'h
lxl9-ifn
l(
s
'
jot'g6'
X(k'to)- G&(k,ct))J- iae-iEkf/
'
9e-k'
k:
zre-i
Ifrisneithertoo largenortoo small,theintegralin Eq.(7.78)isnegligible,
and thestate containing oneadditionalparticle propagateslikean approxim ate
eigenstate with a frequency ekf
/h and dam ping constant vk. M ore precisely,
we shallnow show thatif
1tllek- rz)à>h
it1yk;
:l
tSee,for example.E.G .Phillips,tçll-unctions ofa Complex Variable.'gp.122,lnterscience
Publishers,Inc.,New York.1958.
GROUND-STATE (ZERO-TEMPERATURE) FORMALjSM
:2
thtnl
'G'
tk#/)Q;-iae-fe:f/âe-Mkt
(7.79)
Notethatthecondition 6k- Jz> hykisassumed implicitly,so thatthepolemust
lievery closeto therealaxis. Inthiscase,Eq.(7.79)showsthattherealand
imaginary partsofthe polesoftheanalyticcontinuation ofGR(k,to)into the
lower halt-plane determine the frequency and lifetime of the excited states
obtained by addinga particletoan interactinggroundstate. Equation(7.79)
isreadilyprovedbynotingthattheintegrandinEq.(7.78)isexponentiallysmall
as Im f
.
s becom es large and negative,so thatthe dom inantcontributionscom e
from theregion nearthe realaxis. O n therealaxis,in thevicinity ofthe pole
we have
GR(k,tt));
ksto- rka
//j+ iyg
GA(k,u))= (GR(k,(s))*;
J
k;
(
.o- çyjh- iyz
whertthesecond relationfollowsfrom Eq.(7.64). Theserelationsallow usto
analytically continueGR(k,(s)and GA(k,(s)into the complexf.
splane,and the
integralin Eq.(7.78)canthereforebewrittenas
v:h
#4s
2 t?-foafgcAtk,ts,
l- GR(k,ts)J
BIh-ico 0.
;
k;liyka
#1h
#(s
=((z)- h j6k)a+--yk
2
#t/h-tx)2.
-
'
?ka e- iytlh co
= -
e-if
.
t
)f
=
du z
e-l
zr
.j
y
(
) y.+ (â (p.- q)- syz
;
k;-4=1)-1y:ahls - ex)-2e-iHtih.
t- iae-*ktlhe-tkt (780)
where the third line isobtained with thesubstitution u= fl(,)- â-ip). The
hnalform follows by usipg assumptions 1 and 2,along with the condition
y'k.
<4h-1(q - /z). Notethatthelastine4ualityinEq.(7.80)failsiftistoolarge
or too small. ln a wholly analogous fashion,the poles ofthe analytic con-
tinuation ofGzttk,f
.
olinto the upperhalfoaplanedeterminethefrequencyand
lifetime ofthestateobtained by creating a hole(destroying a particle)in the
interacting ground state.
'The apparentexponentialdecay isslightly misleading because condition 2 restrictsusto the
region wheree-'';kJ1- yt.
83
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
8DW ICK'S THEO REM '
Thepreceding section defined thesingle-particleG reen'sfunction and exhibited
itsrelation to observableproperties. Thisanalysisin no way solvesthefundam entalm any-bodyproblem ,however,and wem uststillcalculateG fornontrivial
physical system s. A s our general m ethod of attack,we shallevaluate the
Green's function with perturbation theory. This procedure is most easily
carried outintheinteraction picture,wherethevarioustermscan beenumerated
with a theorem ofW ick,derived in thissection. Therem ainderofthischapter
(Sec.9)isdevoted to thediagram maticanalysisoftheperturbation series.
TheG reen'sfunction consistsofam atrixelem entofH eisenberg operators
intheexactinteracting ground state. Thisform isinconvenientforperturbation
theory,and we now prove a basic theorem thatrelatesthe m atrix elem entofa
HeisenbergoperatortX(/)tothematrixelementofthecorrespondinginteraction
optratorOg(t):
('F'(
)I:,,(J)l
'l%).
l
*
(:
1'
-01
'
1P(
) .
t*cl
,
9i
*oljooj*j-jf
-jv21juta, J--dtv
.-
v-o
Xe-etItJ/+ . +I,pl)r(.
J.
?.j(/y)
Heretheoperator.
Lisdefinedby
.
??!(?v)t),(/)31*c) (8.1)
S= t'
Vc
s,-aa)
(8.2)
Theproofisasfollows:TheG ell-M ann and Low theorem expressestheground
state ofthe interactingsystem in theinteraction picture
t1l%)
CO,+cc'l(*0)
fê*cl'l'0) ((p:(t%(0.uiu:x?)((l':).
.
=
The denominator on the leftside ofEq.(8.1)can be calculated by writing
I.
k(0,-cc)1*0,
N
)ontherightandt%(0,:
c)t*o))ontheleft
(11,
-()1
àF0) (*cIG(0,cs)fG(0,-cc)1*n)
tf*c1
,
'l'e)t2
tt*o1
'1'o)l2
C*()rG('
x,,0)f$(0,-:c)1*0)
'
=
ttèmelklp
.ol'ta(*01.
91*0)
==((*
z
ot
'l'0)l
(8.3)
wherebothEqs.(6.15)and(6.16)havebeenused. Inasimilarway,thenumeratorontheleftsideofEq.(8.1)becomes,withtheaidofEq.(6.31),
t*(
,It%(cc,0)t'
)e(O,?)dz(?)t%(r,0)t%(:,-'
x))1*0)
lf*cl
''
I'(,)12
-
4*01I
cc,J
)öz(?)Lhlt.-cc)1*0) (s.4)
.%(
lo js-;jc
.
:
ro o
lG .C.w i
ck,Phys.#et,..&);268(1950).
>
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
Thecommon denominatorsofEqs.(8.3)and(8.4)cancelin formingtheratio,
and we5nd
f'l%I:,,(/)l
'Fc)- (*o1t%(=,r)ör(f)G(f,-cc)I*c)
(g.5)
- -
.:.1'c1.1%)
(*aI,9I*(,)
The rem aining problem isto rewrite the num erator ofthe rightside of
Eq.(8.5),containingtheoperator
t%(*,f)0z(f)t'Vf,-*)
* - j a j xj
h -- dfj .. .
n! t
n=0
*
djae-g(j.j1+...+jtaj)pg.
/'
.
/'
G
l(jlj....
j'
g.
e
l(jajj
==
l
*
x 0z(/)
-
im 1
r
t
h m ! -. dtl..' -.dtm
x e-e(Itl1+'''+lt-I)TgJ)l(tj)....
J'
.
J
'j(ty
pj)) (8.6)
whereEq.(6.37)hasbeenused. Thetheorem willnow beprovedbydemonstratingthattheoperatorin thenumeratorontherightsideofEq.(8.1)isequalto
Eq.(8.6). lnthevthterm ofthesum inEq.(8.1),dividetheintegrationvariables
into nfactorswith ti> tand m factorswith tt< t,wherem + n = v. There are
v!/v !n!waystomakethispartition,andasummationovera1lvaluesofm anda
consistentw ith the restriction p= m + n com pletely enum erates the regions of
integrationinthisp-foldmultipleintegral. TheoperatorinEq.(8.1)therefore
becom es
(x)
p.0
i v -1 * * 31.m+n '
y'y. x dtl h lz.
'n=0 m =0 ' m .
!n .
' l
a)
-
- -
!
dtne-f(If.1+ '''+1taI
)
xF(4j
(fI
)....
4l(K)(
1:,(/)J'dtt---J'dt
m
.
xe-e(I
tlI+'.'+I1-1)r(# k(fj) ...J/jtfa))
(8.7)
TheKroneckerdeltahereensuresthatm + n= p,butitalsocanbeusedtoperform
thesummationoverp,whichprovesthetheorem becauseEq.(8.7)thenreduces
toEq.(8.6).
In a sim ilar m anner,the expectation value of time-ordered H eisenberg
operatorsm aybewritten as
('l
%l
F(0'
z
I(
?)
0
(f'
)11
'1C0)-yoj,j1joa (o(,jy)(
i>;
1
t
4
%
1
:
1%s)
v)
j
,
x
..,
j*
-*:/,...J
f*
-*#f.e-e(I
,!l
+...+1
,
.1
)
x F(Xl(ll)---#I(f.)öz(l)öz(/'))I*c) (8.8)
GREEN'S FUNCTIONS AND FIELD THEORY (FERMiONS)
85
Thisresultdependson the observation
tqmorl
9ttcc,-ol(r7e(0,?)ô;(t46f,(/.
-0)1(P,(0,?-)O,.
(t')t'
.
)t(?',0))f-,
.(O,-x)!
* ()'
p
-'
:tp(
)It7
.
rr(*,?)t'
),(?)Pt(?,J')öz(J')C''(t',-cr
:.)r*())
.
.
.
and we m ust therefore partition the integration variables into three distinct
groups. Otherwise,the proofis identicalwith thatofEq.(8.1). Since Eqs.
(8.1)and (8.8)both consistofratios,thedivergentphasefactorscancel,and it
is perm issible to take the lim it e'-.
+0. ln this last form ,these theorem sare
am ongthe m ostusefulresultsofquantum field theory.
A san interesting exam ple,the exactG reen'sfunction m ay be written as
X z
p j cc
.
>x)
l
.
caj(x,.
,,
)- L-hip
-!-.dt,...j #?p=0
. -.
f(l)o!F(X j(/I) ...X I(tp))
;a(x)'
ly14y))'(l)()zz
y '
'
-- -
- -
tmolS l*())'
where the notation x Eë(x,r())- (x,Jx)has been introduced. Here and hence'
forth, the subscript 1 wil) be om itted.since we shall consistently work in the
interaction picture. lt is also convenient to rewrite the interparticle potential
in X jas
which allows us to write the integrations sym m etrically.l For exam ple, the
numeratorofEq.(8.9),which wewilldenoteby I'
C,becomes
.
j,
)
ic'
.;(x,y)-I
'
GI
Ojtx,
y).
+.(-/
j
.) j dnx,J4x/(
v.
(xj,
.
v()ha,
.
ss,
2/$
ppt'
r'#t w *Y ' '2' ? 'A
'q'F k
xt
*o1
.t
yA(A1)'
t
/,
'
s(xl)'
/?s,(xl)'
t
/?A-(A'l)v'''
a(x)'
z
'j(A'
)1I(l)az'-F
-'' '
.Fk
.
WherefG0
aé
y(x,J.)= t't1)()rll
/'
x)9-.
1(-I')J.*() refersto the noninteracting system.
a(
?'
3
. .
.
Thisexpression showsthatwe m ustev'aluatetheexpectation v'alue in the noninteracting ground state ofF productsofcreation and destruction operators of
theform
/f1)01F(y,,1 . . .#Gk'
ha(x),
t
/
1j'4-3.
'
)qI
1)0)
.(
.
(8.l2)
'TheGreen'sfunction now assumesacovariantappearanceand,indeed,isjustthatobtained
in relativistic quantum electrodynamics.where the interaction ofEq.(8,10)is mediated b5'
the exchange ofvirtual photons of the electromagnetic field. The only dipkrence is that
quantum electrodynam ics involves the retarded electromagnetic interaction, whereas the
present theory involves a static instantaneous potentialproportional to a delta function
&(t3- /a). ltshould be emphasized.however.thattheforl'lalislttdeN'
eloped here applies
equally wellto relativisticquantum field theory.which isespeciallyevidentin Chap. 12.where
weconsideranonrelativistic retarded interaction arisingfrom phononexchange.
86
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
lt is clear thatthe creation and destruction operators must be paired or the
expectation valuevanishes;eveninthislowest-orderterm ,however,thestraightforward approach ofclassifying al1possiblecontributionsby directapplication
ofthecom mutation oranticommutation relationsisvery lengthy. Instead,we
shallrelyon W ick'stheorem,which providesageneralprocedureforevaluating
such m atrix elem ents.
Theessentialideaisto movealldestruction operatorsto theright,where
they annihilate the noninteracting ground state. In so doing, we generate
additionalterm s,proportionalto thecom m utatorsoranticom m utatorsofthe
operatorsinvolved in the interchanges ofpositions. Formost purposes,itis
more convenientto use the seld operators directly ratherthan the operators
(ck)referringtoasinglemode. Inmostsystemsofinterest,#(x)canbeuniquely
separated into a destruction part#t+'(x) thatannihilatesthe noninteracting
groundstateandacreationpart'
/-'(x).t
#(x)= #t+'(x)+ '
/ -'(x)
(8.13)
47+'(x)1*0)= 0
(8.14)
'
Correspondingly,theadjointoperatorbecomes
f7f(x)= #t+)'(x)+ #(-)f(x)
(8.15)
where
'
4
JC-'è(x)(
*0)= 0
(8.16)
Thus<t+'(x)and 1Jf-'1(x)areb0th destruction parts,while<t-'(x)and W+)è(x)
areboth creation parts. Thenotation isa vestige ofthe originalapplication of
Wick's theorem to relativisticquantum Eeld theory,where(+)and (-)signs
referto a Lorentz-invariantdecom position into positiveand negativefrequency
parts. Forourpurposes,however,theycanbeconsidered superscriptsdenoting
destruction and creation parts. A san explicitexam ple ofthisdecom position,
consider the free fermion seld,rewritten with the canonicaltransformation of
Eq.(7.34).
'
,
j(xl= J( F-1edtk*x-t/kr)TàakA+ jl F-+eltk.x-uhfl'
flày!-kâ
kâ>kF
c1
Jt+'(x)+ '
/-)(x)
k/<kF
(8.17)
In thiscase,thesymbols(+)and (-)maybeinterpreted asthesignofthefrequenciesoftheseld com ponentsm easured withrespecttotheFerm ienergy.
To presentW ick'stheorem in aconciseand usefulmanner,itisnecessary
to introduce somenew deGnitions.
l. Tproduct:The F productofa collection offield operatorshasalready
been deGned (Eq.(7.4)). ItorderstheNeld operatorswith thelatesttime on
1ForadiscussionofthespecialproblemsinherentinthetreatmentofcondensedBox systems,
seeChap.6.
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
87
the leftand includesan additionalfactorof-1foreach interchangeofferm ion
operators. By desnition
Tlviâê.b ...)- (-1)ertc,i'x/ ...)
(8.18)
whereP isthenum berofperm utationsofferm ion operatorsneeded to rearrange
theproductasgivenontheleftsideofEq.(8.18)to agreewiththeorderonthe
rightside. Itisclearlyperm issibleto treatthe boson fieldsasifthey com m ute
and the ferm ion seldsas if they anticom m ute when reordering felds within a
T product.
2.N orm alordering:Thisterm representsadiflkrentorderingofa product
ofseld operators,in which alltheannihilation operatorsareplaced to theright
ofal1thecreation operators,againincluding afactorof-1forevery interchange
offerm ion operators. Bydesnition
Nlziâcb ...)- (-l)P.
N'
(dWXé ..')
(8.19)
so thattheseldswithin a norm al-ordered productcanagain betreated asifthey
commute(bosons)oranticommute(fermions). Forexample,ifwedealwith
ferm ion helds,
N(#t+'(x)#t-)(y))= -1
Jt-'(y)4t+'(x)
.
(8.20)
N'
(1
/f+)(x)1Jt+)f(y))= -1
Jf+)1(y)'
?
;t+'(x)
.
Inboth casesthecreation partofthefield iswrittentotheleft,and thefactor- 1
reqects the single interchange of ferm ion operators. The reader is urged to
writeoutseveralexam plesofeach defnition.
A normal-ordered product of seld operators is especially convenient
because its expectation value in the unperturbed ground state 1*:) vanishes
identically (see Eqs.(8.14)and (8.16)J. Thisresultremainstrue even ifthe
productconsistsentirely ofcreation parts,asisclearfrom the adjointofthe
equations desning the destruction parts. Thus the ground-state expectation
value ofa F productofoperators gforexample (8.12)Jmay beevaluated by
reducing it to the corresponding a
N product;the fundamentalproblem is the
enum eration oftheadditionalterm sintroduced in thereduction. Thisprocess
issim plised by notingthatboth the Tproductand the N productaredistributive.
Forexam ple,
N LIX + é)(C + X) ''')= NIA-C .
Itisthereforesufl
icientto provethe theorem sseparately forcreationordestruction parts.
3.Contractions:The contraction oftwo operatorsCJand P isdenoted
1?.P'and isequalto thediserencebetween ther productand theaV product.
L1.P'> r(PP)- A(PP)
(s.21)
88
GROLIND-STATE (ZERO-TEMPCRATURE) FORMALISM
It represents the additional term introduced by rearranging a tim e-ordered
productinto a norm al-ordered productand is therefore diflkrentfordifl
krent
tim eorderingsoftheoperators. Asanexam ple,aIIofthefollowingcontractions
vanish
j(+)'j(-).= y(+)j'.y(-)'
f.= ,
j(+)'
f.yt-).= .
j(+).y(-).
f.= ()
.
(y.,
);)
becausetheT productoftheseoperatorsisidenticalwiththeN productofthe
same operators. To be m ore specifc,consider'
the frstpair of operators in
Eq.(8.22). TheirFproductisgivenby
F(#t+'(x)'
#t-)(J')1H f7(+
1(x)#C-1(y)
+#-)(y,)yy+)(x)
.
.
tx> ty
ty> tx
(8-23)
where the + in the second line refers to bosons or fermions. But the seld
operator'
($isalinearcombinationofinteraction-pictureoperatorsoftheform
cke-ia'
k'(compare Eq.(6.10f8). Thus,foreitherstatistics,Eq.(8.23)may be
rewritten as
F!#t+J(x)1Ji-h(.
p))= +fC-'(y)1;t+)(x)
because #(-)and '
?
/t+)commute oranticommuteatany time. Notethatthis
resultistrue only in the interaction picture,wherethe operator properties are
the same asin theSchrödingerpicture. By the desnition ofa norm al-ordered
product,we have
Aë'/t+'(x)#t-'(>'
)1M :i:#f-'(.
$'
Jt+'(x)
(8.25)
and theircontraction thereforevanishes
(8.26)
TheothercontractionsinEq.(8.22)also vanish becauseallofthepaired interaction-pictureoperatorscom m uteoranticom m utewith each other.
Equation (8.22)showsthatmostcontractionsarezero. In particular,a
contractionoftwocreation partsortwo destructionpartsvanishes,and theonly
nonzerocontractionsaregiven by
y(+)(x)'y(+)t(y).= iGhxsyj tx> tv
y(-)(x)'y(-)#(y,)'=
0
ty> tx
()
tx7-ty
fGXx,
. .
$
ty> tx
Forferm ions,thisresultisderived withthecanonicalanticom m utation relations
ofthecreation and destructionoperators(Eq.(1.48))andthedehnitionofthe
free Green's function given in Eq,(7.41). A similar derivation applies fbr
noncondensed bosons(see,forexample,Chap.12). Notethatthecontractions
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
89
arecnumbersin theoccupation-num berH ilbertspace,notoperators. Equation
(8.27)ismore-sim ply derived with theobservation
fJ'P).(tpa,= P 'P'
ttporr(PP)1m:')- (*(
)!P'P'l*(
)z +.tx*o;
.N (
.
s
(8.28)
since(*o1N(PP)1*o'vanishesbydefinition. Thedistributivepropertiesthen
yield thecontraction oftheNeld operatorsthem selves
,
?
;a(x)'.
,
/$j
1(3,
)'= iG'
çtj(x,è,)
a
.
(8.29)
4. z'
f convention : W e introduce a further sign convention. Norm alordered productsoffleld operatorswitb m orethan onecontraction w illhavethe
contractions denoted by pairs ofsuperscripts with single dots,double dots,etc.
Two factors that are contracted m ust be brought together by rearranging the
order of the operators within the normalproduct,always keeping the standard
sign convention for interchange of operators. The contracted operators are
then to be replaced by the value ofthe contraction given by Eq.(8.27). Since
thiscontractionisnowjustafunctionofthecoordinatevariables.itcanbetaken
outsideofthe norm al-ordered product.
Nl.,
i''écq'b ...)- +At,g
t
-'cn./z')..,)-zbzzi.tl''Nlâb ...)
(8.30)
Finally,note that
(.
/.P'= :i:p*C'
(8.31)
wbich follows from Eq.(S.2l) and the desnition of r product and normalordered product. ltisnow possible to state
5. W ick'stheorem :
Thebasicidea ofthetheorem isasfollows:Consideragiven tim eordering,and
startm ovingthecreation partsto theleftw'ithin thisproductoffield operators.
Each tim e a creation part fails to com m ute or anticom m ute,itgenerates an
additionalterm,which isjustthecontraction. Itispermissibleto includeaI1
possible contractions, since the contraction vanishes if the creation part is
already to the leftofthe destruction part(rememberthatmos:contractionsare
zero);hencethetheorem clearly enumeratesalltheextra termsthatoccurin
reordering a F productinto a norm al-ordered product.
90
cqouqn-s'
rA'rE (ZERO-TEMPERATURE) FoqMALlsu
To prove the theorem,we shallfollow W ick'sderivation and srstprove
thefollowing.
6.BasiclentmalIfAIt)P '..k?jisa normal-ordered productand ;
isafactorlabeled wts
tha timeearlierthanthetimesfor0,P ' ..X,#'
,then
AIPP ...k?)2 = AIPP ...E?.;')+ AIPP .'.E'#2*)
+ N(I/'P ...k fT,)+ AIf.
)P ..'k?24 (8.33)
'
. .
.
Thusifa norm al-ordered productism ultiplied on therightwith any operator
atan earliertim e,w eobtain a sum ofnorm al-ordered productscontaining the
extra operatorcontracted in turn with allthe operatorsstanding in the original
product,along with a term where the extra operator is included within the
norm al-ordered product. To prove the lem m a, note the following points:
(J)If2 is a destruction operator,then all the contractions vanish since
F(X2)= N(vi24. Thus,only thelastterm in Eq.(8.33)contributesand the
lem m aisproved.
(:)Theoperatorproduct0f''''kf'canbeassumedto benormalordered,
since otherwise the operatorscan be reordered on b0th sides ofthe equation.
Oursign conventions ensure the sam e signature factoroccursin each term of
Eq.(8.33)and thereforecancelsidentically.
(c) Wecan furtherassumethat2 isacreation operator,and P ''' ? areaIl
destruction operators. lfthelemm a isproved in thisform ,creation operators
m ay be included by m ultiplying on the left;the additionalcontractions so
introduced vanish identically and can therefore be added to the right side of
Eq.(8.33)withoutchangingtheresult.
Hence itissumcientto prove Eq.(8.33)fork a creation operatorand
P ''' ? destruction operators. Theprooffollowsbyinduction. Equation
(8.33)isevidentlytruefortwooperatorsbydesnition (Eq.(8.21))
32 = F(?2)- ?.2.+ N(fT)
(8.34)
W e now assume itistrue for n operatorsand provt itforn+ l ogerators.
M ultiplythelemma(8.33)ontheleftbyanotherdestructionoperatorD having
atimelaterthanthatof2.
.
X1Y ) (8.35)
Since P,P .''X, ? are alldestruction operatorsand thecontraction of2'
withanydestructionoperatorisacnumber,b hasbeentakeninsidethenormal
orderingexceptfortheveryIastterm in Eq.(8.35),where: isstillanoperator.
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
Considerthislastterm ,which,w eassert, can be written as
DNII/b> ...z
L'?k)- N(D'('
)P ...##:')+ Nlbob'
k...k?2)
(8.36)
Thisequation isreadily verihed.
DNIL'
/Z ...z%?î4
= (
-1),D2I)b7 ...k?
- (
-1)1r(>:)PP '..#f'
= (
-1)'X'2.PP ...vkLr+ (-1),,+*xlzb)tJ/P ...#fr
= (
(-1)e)2b.(')P ...kfq.+ g(-1)r+t?)2Nlbl?b'
y...k?kj
= ,(
.
f).PP ...k?z'l+ slbfkb
s ...##:)
Inthesecondline2 ismoved to theleftwithinthenormal-ordered product,introducingasignaturefactor(-1)P. Thefactorsnow'appearinnormalorder,and
the N productcan be rem oved. Furtherm ore, theproductbz isalready time
orderedbyassum ption. Thefourthlinefollowsfrom thedehnition ofa contrac-
tion,with a factor(-1)0 arising from the interchange ofX and 2. The last
term inthefourthlineisinnormalorder,becausePP .'.Xk'arealldestruction
operators. The sign conventions then allow us to reorder the operators to
obtainthesnalform,whichprovesthebasiclemma(8.33).
The rtsultcan be generalized to norm al-ordered products already containing contractions ofEeld operators. Multipl
y both uidesofEq.(8.33)by
the contraction oftwo m ore operators,.l'
t''S'',say,and then interchange the
operatorson both sides. Each term has the sam e overallsign change which
cancelsidentically. Thuswecan rewritethebasiclemma(8.33)as
N(PP--.-.#'-?42 = sr(t'
)P''...#..17
-.2.)+ ...
+ N(0'P**...k.'Lr
2.j+ x(I?'f-''...k ''92) (s.38)
7.Proofof Wick'stheorem :Againthetheorem willbeprovedbyinduction.
ltisobviously truefortwo operators, by thedefnition ofacontraction
z'
(t7f')= x(t7P)+ t'
)'P.
(8.39)
Assum e itistruefornfactors, andmul
tiplyon therightbyanoperator(1with
atim e earlierthan thatofany otherfactor.
FIPPT .'.X#2)t'
1
- rl
r/fzlfz...8?2th4
- N(
I?b
7çk'...Eî2)fl+ x(r
J?'P')l'...f'::)j1+ ..
- xt
przl,
fz ...##'
zû)
+A'
-tsum overalIpossiblepairsofcontractions) (8.40)
92
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
The operatorL can beincluded in the F productbecause itisata timeearlier
than any ofthosealready in theF product. On therightside,weuseourbasic
lemma (8.33)to introduce the operatorLh into thenormal-ordered products.
The restriction on the time ofthe operator(1 can now be removed by sim ultaneously reorderingtheoperatorsin each term ofEq.(8.40). Againthesign
conventions give the sam e overallsign on both sides of the equation,w hich
therefore rem ains correct. W ick's theorem has now been proved under the
assum ption thatthe operators are either creation or destruction parts of the
fleld, The T product and the norm al-ordered product are both distributive,
however,and W ick'stheorem thusappliesto theheldsthemselves.
ltm ustbe em phasized thatW ick'stheorem is an operator identity that
rem ains true for an arbitrary m atrix elem ent. Its realuse,however,isfor a
ground-state average (*al''. !
*0'
),where al1uncontracted normal-ordered
productsvanish. In particular,theexactGreen'sfunction (Eq.(8.9)Jconsists
ofallpossiblefully contrad ed term s.
9UDIAGRAM M ATIC ANALYSIS OF PERTURBATIO N THEORY
Wick'stheorem allows usto evaluate the exactGreen'sfunction (8.9) asa
perturbation expansion involving only wholly contracted field operatorsin the
interaction pl'
i''lre T'
,.esecontractionsarejustthefree-seld Green'sfunctions
G%(Eq.(8.29)),and G isthereby expressed in a seriescontaining U and G%.
Thisexpansion can beanalyzed directlyincoordinatespace,or(forauniform
system) in momentum space. As noted previously, the zero-temperature
theoryforcondensedbosonsrequiresaspecialtreatment(Chap.6),andweshall
consideronlyferm ionsin thissection.
FEYNM A N DIAGRA M S IN coo RolNATE spAcE
As an exam pleofthe utility ofW ick's theorem ,weshallcalculate the srst-order
contributionsin Eq.(8.11). Theexpectation valueofallthetermscontaining
norm al-ordered products of operators vanishes in the noninteracting ground
statet*;),leavingonlythefullycontracted productsoffeldoperators. Wick's
theorem thenrequiresusto sum overallpossiblecontractions,and Eq.(8.29)
showsthatthe only nonvanishing contraction is between a seld ;'
x and an
adjointheld#:
:. Inthisway,thesrst-orderterm ofEq.(8.11)becomes
jJ(
f1
xl
p)(x,.v)= hl
-
d4xld4xl
z(ytxjjxj
/lag,,ss,
AA'bzy'
(I'
G.
0j(x,y)(jG0
Fz-H(xl
z,x;)jG0
a,ztxj,xj)- iG;
nt.A(xl
',xj)1G0
z-v(xl,x;))
fAl
(s)
+ k'
Go
zztx,xj)gj
'
Go
z,jjtxj,xj
?)ïGs
0'j(xj
',y)- fGo
z,j(xl,y)/Gs
0,stxj
',xl
'))
(c)
fnt
+ 1G.
0H(x,xj
')(;
-Gs
0,Atxj
',xl)iGç
).#(xl,y)- iG;
0
.,j(xj
',y)j'
Gz
0-A(xI,xj)))
(E)
(F)
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
93
Thereaderisurged to obtain Eq.(9.l)directly from Eq.(8.11)by enumerating
allnonvanishing contributionsforal1possible tim eorderings. Thisprocedure
isverycom plicated,even in thefrstorder,and W ick'stheorem clearlyprovides
a very powerfuland sim ple tool.
X
R'
X
X'l à
.
j
A
A'
A1
z'
X1
A.
Jz
#,
#t .
.X'1
y.
'
X1
8
Y!# e
.
#
(zd)
(#)
X
,
A-1
Xl
à
à
(C)
A'
#E
rz
lt,
p.
Jz'
xî
xî
A
A,
'
X1
A
A
X'1 .
.
#
b'
A
(f))
(E)
.
A
#
(F)
Fig.9.1 First-ordercontributionstoCaplxsy).
W ecan now associateapicturewitheach oftheterm sappearing in expres-
sion (9.1),as illustrated in Fig.9.1. The Green'sfunction G0isdenoted by a
straightline with an arrow running from thesecond argum entto the first. while
the interaction potentialisdenoted bya wavy line. Thesediagram sappearing
in theperturbation analysisofG form aconvenientwayofclassifyingtheterm s
obtained with W ick'stheorem . They are know n asFeynmandiagramsbecause
tht firstdiagram m aticexpansion ofthisform w asdeveloped by Feynm an inhis
94
GROUND-STATE (ZERO-TEMPERATURE) FonMAusu
work on quantum electrodynam ics.l The precise relation with quantum seld
theory wassrstdemonstrated byDyson.z
The analytic expression in Eq.(9.1) and the corresponding diagrams
(Fig.9.1)haveseveralinterestingfeatures.
1.Theterms,1,B,D,andFcontainaGreen'sfunctionwithbotharguments
atthe sam e time,which is indicated by a solid line closed on itself. By the
desnition(Eq.(7.41)),theexpressionfGjj(x,x)isambiguous,anditisnecessary
todecidehow tointerpretit. Thisquantityrepresentsacontractionof# and
fh1',but the time-ordered productis undehned at equal times. Such a term,
however,arisesfrom acontractionoftwoseldswithintheinteractionham iltonian
Xl.wheretheyappearintheform '
($(x)'
(ktxlwiththeadjointseldalways
occurringtotheleftoftheseld. Inconsequence,theGreen'sfunctionatequal
tim esm ustbeinterpreted as
ït4;(x,x)-tl'i
mf6 4*oIF(#.(x?)#;(x/'))1*0)
->
(*ol#;(x)#a(x)I*o)
-
-
-
-
(2.
8+ 1)-1b ja0(x)
&xbN
= -(2
unform system
(9.2)
J+ 1)P'
fora system ofspin-xfermions. Herea0(x)istheparticledensity inthe unperturbedground state(compareEq.(7.8))and need notbeidenticalwith a(x)
in the interacting system because theinteraction m ay redistribute the particles.
Forauniform system,however,n0= n= NIV,becausetheinteractiondoesnot
change the totalnum ber ofparticles. The term s D and F thus representthe
lowest-order direct interaction with a11the particles that make up the non-
interactinggroundstate(hlled Fermisea).whilethetermsC and E providethe
corresponding lowest-orderexchangeinteraction. Heretheterm sS'directl'and
ç'exchange''arise from the originalantisym m etrized Slater determ inants,as
discussed below Eq.(3.37).
2.The term s.
,
d and B aredisconnecteddiagram s,containingsubunitsthat
arenotconnectedtotherestofthediagram byanylines. Equation(9.1)shows
thatsuch term stypically have G reen'sfunctions and interactions whose argum entsclose on them selves. A saresult.thecontributionofthissubunitcan be
factored out of the expression for C. Thus,in the terms ,
4 and # above,
fGa
0j(x,y)representsonefactorandtheintegralrepresentsanotherfactor. To
hrstorderintheinteraction.weassertthatEq.(8.11)canberewrittenasshown
in Fig.9.2. Each diagram in thishguredenotesawell-desned integral,givenin
Eq.(9.1). The validity ofFig.9.2 isreadily verihed by expanding theproduct
and retainingonlythehrst-orderterms.whicharejustthoseinFig.9.l. The
1R.P.Feynman.Ioc.cit.
2F.J.Dyson.Phys.Rev.,75:486 (1949);75:1736 (1949).
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
Ië4(x.y)-
+
+
+
+
+'..x 1+
9:
+
+ .-.
Fig.9.2 FactoriyAtionofflrst-ordercontributionstoCG x,#).
additionaltermsofsecond orderintheinteractionarehereunimportantlwr>use
thepresentcalculationisconsistentonlyLojrstorderintheinteraction.
Thedenominator(*:1JI*n)= (*aI0(*,-*)I*:) in Eq.(8.9)hasY n
ignored to thispoint,and weshallnow evaluateitto Erstorderin theinteraction
potential. The operator 04*,-*)isthe same asthatin the numerator of
Eq
.(8.9),exceptthattheomrators#.(x)#J(y)must% dele-d. Timsthe
denominator can also lx evaluated with W ick's theorem,and only the fully
contracted term s contribute. The resulting calculation evidently yields the
termsshown in Fig.9.3,where each diagram again stands for a well-deâned
FIg. 9.3 Disconnœted diagrams in the
denomi
natorofG#(x,y).
A
<*(j1J1% > - l+
+
+ ...
integral. These integralsare preciselythesame asthose apm aring in the terms
A andB ofEq.(9.1). Wethereforeconcludethatthecontributionofthedenomfaa/orin Eq.(8.9)exactly rlaeellthecontribution of thedisconneeted dl
Wgrin thenumerator. Thisimportantresulthasso far% en verihed only to lowest
orderintheinteraction,butweshallnow proveittoallorders.l
A disconnected diagram closeson itself;conm uenuy,itscontribution to
G.j(x,y)factors. n usthevth-orderterm ofthenumeratorofEq.(8.9)r>n
bewritten as
.'
#'
fJt
(x,y)-
* * -i >+1 >.
! *
*
dt.
-A- 3...+--!a,m ! ..dtl - -.
@-0 PI-Q
v
x4:*(
,1F(41(/1)---#l(f-)#.(x)#)(z)!I*:).....,x
j**df,
.+1...J
(*
-*dt.(*cl
r(#l(l.+,
)-..#1(f.)1I
*,)
(9.3)
which cà
'n be seen by applying W ick'stheorem on both sidesofthisexpression.
n e second factor,containing n interations,in generalconsists of many dis-
connected parts. n efactorv!/a!m !representsthenum-rofwaysthatthev
'Herewefollow the proofgivenby A.A.Abrikosov,L.P.Gorkov,and 1.E.Dzyal- hiMkii,
op.cit..= .8.
:6
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
operatorsPl(/j)can bepartitioned into twogroups,and,asnoted before,Xj
canbem ovedinsidetheF productwith noadditionalchangesofsign. Equation
(9.3)mustnow be summed overallp,which is trivially performed with the
Kroneckerdelta,andthenumeratorofEq.(8.9)becomes
'<'
I'
f.#(x,y)m -0
fm 1
-
=
*
h- ,n!
- -. dtL.-' -. dtm
x'
:molrEzhltr,l --.Iljltmt1/
1.(x)v'#
ftzlllmolcennecteu
*
-
x
n
i 1
n=0
*
co
co
dtj .. .
-X
#/n
-X
x ((1)()JF(H'
-j(tj)--.H-I(t,))p*())
(9.4)
Thesrstfactoristhesum ofal1connecteddiagram s,whilethesecond isidentical
with the denominator (*f
)IXi*o). We therefore obtain the fundamental
formula
*
-
iG(
xj(x,y)=
m=0
im 1
h m!
co
c
o
dtj''' -. dtm
OX
x(*(,IrE.
f?.(?.)-..4l(/mlfztxl4ltzllrmnlconnecteu (9.5)
which expressesthefactorization ofdisconnecteddiagram s. A related ççlinked-
cluster''expansionfortheground-stateenergywasfirstconjectured byBrueckner,lwho verihed the expansion to fourth order in the interaction potential;
the proofto allorders was then given by G oldstonez with the techniques of
quantum seld theory. Equation (9.5) isimportantbecause itallows us to
ignoreal1diagram sthatcontain partsnotconnected to theferm ion linerunning
from y to x.
The expansion ofG.j(x,y)into connected diagramsiswhollyequivalent
to theoriginalperturbation series. ThesearethecelebratedFeynmandiagrams,
and we shallnow derivethe preciserulesthatrelatethediagram sto the term s
ofthe perturbation series. Itmustbe em phasized,however,thatthe detailed
structure ofthe Feynm an rulesdependson the form ofthe interaction ham il-
tonianX1,andthepresentderivationappliesonlytoasystem ofidenticalparticles
interacting through atw o-body potential.
3.For any given diagram .there is an identical contribution from all
similardiagram sthatdiSer m erely in the perm utation ofthe labels 1 ' '.m
in theinteractionhamiltonianIîj. Forexample,thetwodiagramsin Fig.9.4
havethesam e numericalvalue because they difl'
erm erely in the labeling ofthe
dum m y integration variables. In addition,they have the sam e sign because
1K .A . Bruœ kner,Phys.#0 .,1* :36(1955).
2J.Goldstone,Proc.Roy.Soc.tf,
()ndt/al,A239:267 (1957).
97
GREEN'
S FUNCTIONS AND FIELD THEORY (FERMIONS)
lz
Xz
x1
x'
1
m
X!
x2
xz'
Fig.9.4 Typicalpermutation ofIh in connected diagram s.
Iîjcontainsan evennumberoffermion seldsand maythereforebemovedat
willwithin the F product. ln vth orderthereare m !possibleinterchangesof
thistype corresponding to the m !waysofchoosing theinteraction ham iltonian
4jinapplyingWick'stheorem. AIIofthesetermsmakethesamecontribution
totheGreen'sfunction,sothatwecancounteachdiagram jusfonceandcancel
thefactor(rn!)-1inEq.(9.5). Notethatthisresultistrueonlyfortheconnected
diagrams,wherethe externalpointsx and z are sxed. lncontrast,the disconnected diagram shown in Fig.9.5 representsonly a single term . Thisresultis
Fig.9.6 Typicaldisconnected diagram.
Xz
xa
Xl
Xl
,
easilyseen byexpanding(*01X 1
*0)withW ick'stheorem. There isonly one
way to contracta1loftheselds,and the diagram obtained by the interchange
x:xl
'= xzxa'doesnotcorrespond to a new and diferentanalytic term. This
distinction between connected and disconnected diagrams is one of the basic
reasons for studying the G reen's function;the sxed external points greatly
sim plifythecounting ofdiagram sin perturbationtheory.
W etherefore 5nd the following ruleforthe ath-ordercontribution to the
single-particleGreen'sfunctionG./x,y):
(J) Draw alltopologically distinctconnected diagramswith n interaction lines
U and 2n+ ldirected G reen'sfunctionsG0.
Thisprocedure can besimpliEed with the observation thatafermion line
eitherclosesonitselfoçrunscontlhuously from z tox. Eachofthesediagrams
9:
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
representsallthea!diFerentpossibilitiesofchoosing among thesetofvariables
(xjxJ) '''(.
L x;). lfthereisa questionasto theprecise meaning oftopologically distinct diagrams,W ick's theorem can always be used to verify the
enum eration.
4.In oursrst-orderexample (Eq.(9.1))wenotethatthe termsC and E
areequal,asarethetermsD and F;theydiferonlyin thatxandx'(and the
corresponding matrix indices)areinterchanged,whereasthepotentialissymmetric underthissubstitution (Eq.(7.13)1. Itistherefore suëcientto retain
justonediagram ofeach type.simultaneously omittingthefactor!.in frontof
Eq.(9.1$ which reqectsthefactor!.in the interaction potential(Eq.(2.4)).1
x
l
z'
/t
y
p'
Fig.9.6 Matrixindicesfor&(x,y'
)zz....,.
W ethereforeobtain theadditionalrules:
(â) Labeleachvertexwithafour-dimensionalspace-timepointxi.
(c) Each solidlinerepresentsaGreen'sfunction(4j(x,y)runningfrom y tox.
(#)Eachwavylinerepresentsaninteraction
&(x,>'
),A-,,
a.'- F(x,y)AA,,,.-bltx- ty)
wheretheassociation ofm atrixindicesisshown in Fig.9.6.
(e)Integrateal1internalvariablesoverspaceandtime.
5.W e notethatthe summationsappearingin thesubscriptindiceson the
Green'sfunctionsandinteractionpotentialsinEq.(9.1)arepreciselyintheform
ofa matrix productthatrunsalong the fermion line. Thuswe state the rule:
(/)Thereisaspinmatrixproductalongeachcontinuousfermionline,including
thepotentialsateâch vertex.
6.Theoverallsign ofthevariouscontributionsappearinginEq.(9.1)or
thediagramsappearinginFig.9.1isdeterminedasfollows. Everytimeafermion
lineclosesonitself,theterm acquiresan extraminussign. Thisisseen bynoting
that the ields contracted into a closed loop can lhe arranged in the order
(:11(1)*
1/(1)'')f##(2)'*
/42)*'')''.I y1(A)'Jwith no change in sign. An odd
number ofinterchange's offerm ion operatorsis now needed to move the last
seld om ratorover to itsproper position atthe left. Thusweobtain the rule:
(g)Aëxasignfactor(-1)Ftoeachterm,whereFisthenumberofclosedfermion
loopsin thediagram.
'Note thatthisresultagain appliesonlytoconnected diagrams,asisevidentfrom Fig.9.12
andB.
GAEEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
D
7.Theath-ordtrterm ofEq.(9.5)hasanexplicitnumericalfactor(-ilh?,
whjle the 2n+ 1contractionsofheld om ratorscontribute an additionalfactor
f>+l(=eEq.(8.29)1. W ethereforeobtaintherule:
fh)To compute G(x,y)assign a factor(-f)(-#â)*(f)2a+l= (ilh? to each athorderterm.
Finally,theearlierdixussionofEq.(9.2)yieldstherule:
(f)A Green'sfunctionwithequaltimevariablesmustlx interpretedas
Gî/XJ.X't'
/)
X
Xl
à
/
Xl
l
!:
A,
#
x;
K1
.
Fig.9.7 A1le st-orderFeynmandiarams
forG #(x,.
y).
,
#
#
@)
(:)
#
The foregoing arguments provide a unique prescription for drawing a11
Feynman diagrams that contribute to G(x,z) in coordinate space. Each
diagram corresponds to an analytic expression thatcan now be written down
explicitly with the Feynman rules. The calculation ofG thusG comesa relatively autom aticprocess.
Asan exampleofthe Feynman rules,weshallnow writeoutthe complete
srst-ordercontributiontoG./x,z),showninFig.9.7,
G$(x,y)=fâ-lfd*xkJd*xl((-1)G2z(x,xl)U(xI,xJ)zz,,pg.Z .#xI,z)
xG1'/xJ,xJ)+ Gîz@.xl)U@l,x1)AA',p,#'G0
à./xl,x1)G:'/x1,XJ (9.6)
Hereand henceforth.an implicitsummation isto% carriedoutoveralIrem ated
spinindices. Thecorresponding second-ordercontribution GtRx,y)requires
morework.and we merely assertthatthere are 10 e ond-order Feynman dia-
grams(Fig.9.0. Thereadtrisurgtd toconvincehimselfthatthesediagrams
exhausttheclassofe ond-ordertopologically distinctconnected diagrams,and
to writedown the analyticexpression associated with each term .
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
100
)) )
V)# @)
(J)
fb)
(c)
(d)
4S1
'g
l
(pr)
(#)
(h4
Fig.9.8 Allsx ond-orderFeynmandiavamsforG.#@ ,y).
(e)
(J)
FEYN M A N DIAGRAM S IN M OM ENTU M SPACE
In principle,the Feynman rules enable us to write down the exact Green's
function to arbitrary order,butthe actualevaluation ofthe termscan lead to
formidable problems because each noninteracting Green's function G0(x,y)
consistsoftwo disjoint pieces. Thuseven the srst-ordercontribution (Eq.
(9.6))mustbesplitintomanyseparatepiecesaccordingtotherelativevaluesof
thetimevariables. lncontrast,theFouriertransform G0(x,y,œ)with respect
to tim ehasa sim ple form ,and itisconvenientto incorporate thisinto the calculations. Although it is possible to consider a m ixed representation
G./x,x',a4,which would applyto spatially inhomogeneoussystemswith a
tim e-independentham iltonian,we shallnow restrictthe discussion to uniform
and isotropic system s, w here the exact G reen's function takes the form
3.#G(x- y). The spatialand temporalinvariancethen allowsafullFourier
representation,and wewrite
Gxb(x,>'
)- (2*-4I#4ke'''f>-''G,
x,(k)
t4/x,z)- (2*-4Jd*kekk'çx-v'(4#k)
(9.7J)
(9.7:)
GqEEN.s FUNCTIONS AND FIELD THEORY (FERMIONS)
1Q1
where the lim it F -->.* has already been taken. Here a convenient fourdim ensionalnotation hasbeen introduced
d*k- d3kdœ
k.x> k.x- œt
(9.8)
In addition,we assum e that the interaction depends only on the coordinate
diFerence
&(x,x')= F(x- x')ô(f- t')
(9.9)
Itmay then bewritten as
&(x,x').a'-,j-=(2,4-4J#4kekk-çx-x'bUlkjœ?,,#j,
(2r4-3J#3kedk*fl-x''F(k)??y
.jj.3(l-t')
(9.10)
where
Ulkjxx-vbp'- F(k)..-.j,'
= Jd5xe-fkexF(x) '#j,
(9.l1tz)
(9.11:)
isthespatialFouriertransform oftheinterparticlepotential.
As an exam ple ofthe transform ation to m om entum space,consider the
diagram shown in Fig.9.7b
G2è'(x,y)= fâ-iJ#4x1dkxL(2,
0-16Jd*kd4p#Yjd*q
XG0
aA(k)U(ç)Az'.pp'Gz
% (X fC'/#1)
X efk'(.x-x1)etq.(xj-x1')elp'(xl-x1')elpl'(.x1'-#)
=
=
fâ-l(2r8-8Jd*k#4##4#jd*qGjztk)U(g)zA.,ps,
xGj,
#(p)G0
M,/pI)ekk'xe-fp:.'êf4)(,+q-kj3(4)(pj-q-pj
(2z4-4J#4kelk'lx-Db(fâ-1GJA(k)(2,4-4Jd*p
x&tk-79,,,,..'cî,r,(79G:,,(k))
(9.12)
wherethefour-dimensionalD iracdeltafunctionhastheusualintegralrepresentation
3f4'477)= (2,
0 -4Jdkxelp'x
Note thatEq.(9.12)indeed hastheexpected form,and comparison with Eq.
(9.7c)identisesthequantityinsquarebracketsasthecorrespondingcontribution
toGajlk)H Gz/kstsl.
Thisapproach isre#dilygeneralized. Considerthetypicalinternalvertex
show n in Fig.9.9. ln accordance with our defnitionsofFouriertransform s
in Eqs.(9.10)and (9.7*,wecanalsoassignaconventionaldirection x'-.
>.x to
theinteraction Utx- x'). (Thisconventioncannotaltertheproblem sincethe
potentialis symmetric Utx- x')AA,,vv.= Ulx'- xjvv..AA..) The coordinate x
14'
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
qq:x
-
e
e/40.7
e/4p*J:
FI:.9.9 Typicalintenœ vertexina Feynman diagram .
now apm nm only in the plane-waveexponential,and thereisa factore+'4'xfor
each incoming line and e-tq'xforeach outgoing line. The integration over x
thereforeyields
jd4xef(4-4'+4D'x=(2.)43(4)(ç-q'+q')
(9.13)
whichconsea esenergy andmomentum atele/linternalvertex. Theonlyremaining question istheend points,wherethe typicalstructure isshown in Fig.9.1û.
x
e
IqYx
q##- q#
/d;J.,,
.-.
y
Fiq.9.1Q TypH lstrx tlzreofF
di-
sforG.a@ - y).
Thetranslaiionalinvarianceensuresthatq'= q',asseenexplicitlyinEq.(9.12);
theremainingfactore*''(x-'tisjustthatneededinthedehnitionoftheFourier
transform ofG.b(q').
w e can now state the Feynman rules for the ath-ordercontribution to
G.l(k,œ)> G./k):
1.Draw a1ltopologically distinctconnected diagtamswith n interaction lines
andln + 1directed Green'sfunctions.
2.A ssign a direction to each interaction line;associateadirected four-m om entum with each lineand conservefour-momentum ateachvertex.
3. Each Green'sfunction correspondsto a factor
Gyj(Kts)= 8.jG0(k,ts)= &xô L9
kpy.q + (8
k,.
IkI
)
0o(lkI
k+.
s(ou.
-y
.
(9.14)
4.Each interaction correspondsto afactor U(ç)AA,,vv.= F(q)AA,,p,s.wherethe
matrixindicesare associated with thefermion linesasin Fig.9.11.
5.Perform a spin summation along each continuousparticle line including the
potentialateach vertex.
6.Integrateoverthenindem ndentinternalfour-momenta.
GREEN'S FUNCTIONS AND FIELD THEORY (FERM IONS)
1*
7.Amx a factor(f/âP(2v)-4*(-1)F where F isthe numerofclomM fermion
loops.
8.Any single-particle linethatformsaclosed loop asin Fk.9.11/ orthatis
linked by the same interaction line as in Fig. 9.11: is interpxted as
ele G'
.#(k,œ),whereg)-*0+attheendofthee>lculation.
k
k
#
à
!&
z' () >
kl kl
k-k,
k
k
R g.9.11 M lM t-order Feynnx diagrams
forG./k).
#
(J)
#
(â)
Asanexampleofthe Feynman rulesin momentum space,wecomputethe
srst-ordercontributionf41
/(k,œ),showninFig.9.11. Althoughthetopological
structureisidenticalwith thecorrespondingdiagramsincoordinatespace(Fig.
9.7),thelabelingandinterpretationarenaturallyquitediFerent. In Fig.9.11J,
the four-vector associated with the interaction vanishes % ux of the conservation requirementateach end.A straightforward identiscation yields
GL)(k)-fâ-1(-l)(2,
/4-4Jd*kb(4z(k)&(0)AA,.vp.Gl,/k)fC,s(kI)elu't'
+fâ-l(2=)-4J#4kjGkz(k)U(k-kjlzv,gp,
XGl's(kl)G:'#(k)el'
*'1'l
=
fl-lG0(k)((2=)-4J#4kj(-U(0)aj,sj,G0(k1)etœt'
+ &(k- k1).4,.!,,GQ(kl)e'''''?!)&'
*(k'
)
(9.15)
where the spin summation has Y n simplised with the Kronx ker delta for
each factorG*. Here,and subsequently,weuse the conventionsthat
U40)= &(k= 0)
(9.1* )
F(0)- F(k= 0)
(9.16â)
To makefurtherprogress,we shallconsiderspin/ particleswith two distinct
possibilitiesfortheinteraction potential.
1M
GROUND-STATE (ZERO-TEM PERATU8E) FO RMALISM
l. Ifthe interaction is spin independent,then ithasthe form 1(1)1(2)in spin
space,namely,the unitspin m atrixwith respectto both particles:
&(t
?).j,zp= Ulqjôaj3Apz
(9.17)
The m atrix elem entsthen becom e
Uub.p'pz= 2673,j
Uxv,sj= &3zj
(9.l8)
2.Iftheinteractionisspindependentoftheform l(l).c(2),then
Ulqjxb,àv= U(ç)@(1)zj*c(2)As
and the relevantquantities are
lzjwlps= 0
Gxv-làzj= ((c)2Jaj= 33aj
(9.20)
These results have been obtained with the observations trl = 0 and tr1= 2.
Forinteractionsofthe form
Eqs.(9.15)to (9.20)show thatGfl)isindeed diagonalin the matrix indices:
(yx
(1
b)= gaj(y(l).
TheexactGreen'sfunction can alwaysbewritten in theform
Glkj= Gè(k)+ G0(k))2(1)G0(/()
(9.22)
which desnestheself-energyZ(:). Thehrstterm isjustthezero-ordercontribution,and the structure of the second term follows from that of Fig. 9.l0.
The same structure occurs in Eq.(9.l5). which thus identihes the first-order
kèjxself-energy as
..
â.
Zf1)(/,
:)= 1
*(2=)-4fdnkjg-2P%40)v'l
Gtk -kl)
+ 3lz-jtk- kl))GQ(kI)eicotn
Thefrequency integralcan now be performed explicitlywith Eq. (9.l4)
* dœ j j.,q
2= e
% 1kl1- kF4
u)1 - tzlkj +
#(/cs- ikl1)
i'
r
l+ (t'j- ttlkl - i'
b
gLk
= i s jkj1
,)
-
where the convergencefactorrequiresusto close the contourin the upper-half
plane. The momentum integralin the firstterm ofEq.(9.23)then gives the
particledensityn = N(P'(compare Eq.(3.27)j,and we5nd
âXt1'(/t
-)MâY(1)(k)
= nFL(0)- (2rr)-3fdqk'(I
Gtk- k')+ 3Ujtk - k')Jblky- k')
(9.24)
GgEEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
1QB
Note thattheflrst-orderself-energy isfrequency independent. The two terms
appearing in Eq.(9.24)have the following physicalinterpretation. Thesrst
term representstheBornapproximationforforwardscatteringfrom theparticles
inthemedium (Fig.9.I1J),and thesecond representstheexchangescattering
withtheparticlesinthemedium,againin Bornapproximation(Fig.9.1148.
ovsoN's EQUATIONSi
W eshallnowclassifythevariouscontributionsinan arbitraryFeynmandiagram .
This procedure yields Dyson's equations, which summarize the FeynmanDyson perturbation theoryin a particularlycompactform.
l.Self-energy insertion:Ourgraphicalanalysismakesclearthattheexact
Green'sfunction consistsoftheunperturbed G reen'sfunction plusa1lconnected
x
x
x
A'l
=
Self-enelgy X
+
x;
Fig.9.12 GeneralstructureofXéx,yj. .p
y
y
terms with a free Green's function ateach end. This structure is shown in
Fig.9.12.where the heavy line denotesG and the lightline denotes G0. The
corresponding analyticexpression isgiven by
G'a;(.
A-,A'
)=Gîj(x,.
O +fd*xbJ#*xJG.
0A@,xl)X(xi,xl
')zsfcjtxt
',y) (9.25)
which desnestheself-energy Z@1,xj
')As. A self-energy insertionisdehned as
any partofadiagram thatisconnected totherestofthediagram bytwo particle
lines(onein and oneout).
W enextintroducetheconceptofaproperself-energy insertion,which isa
self-energy insertion thatcannotbe separated into two piecesby cuttinga single
particleline. Forexam ple,Figs.9.84,9.8:,9.8c,and 9.8#allcontain im proper
self-energy insertions,while the remaining termsofFig.9.8containonlyproper
self-energy insertions. By desnition,the proper self-energy is the sum of aIl
properself-energyinsertions,andwillbedenotedX*(xj,m').#. ltfollowsfrom
1F.J.Dyson,/oc.ct
'
t. (Adiscussionofthevertexpartandthecompl
etesetofDyson'sequations
ispresented inChap.12ofthisbook.)
1x
GROUND-STATE ('ERO-TEM PERATUHE) FOY ALISM
thesedesnitionsthattheself-energy cœ sistsofa sum ofallx ssiblerem titions
oftheproperself-energy.
Z(xI,xl
')=X*(xl,x;)+Jd%xzd*xz
'X*(xl,.
n)G0(xz,x1)X*(xJ.xl
')
+ fd*xzdkxz
'j'd*xsd*xs
'X*(xlvxz)G'
'lxzqxl)
x X*(xa',x3)fP(x,,xJ)Z*(xa',x;)+ '' (9.26)
H erc each quantity denotesa m atrix in the spinorindices,and the indicesare
thereforesuppressed. ThestructureofEq.(9.26)isshowninFig.9.13. CorreX1
Pro- rself-eneqy
z*
Al
Self-ene
tr-gy
X1
-
x'
l
%1
.
+
xf
xf
f1
.
Fig.9.1: Relationbetween self-energy
1:and properself-energyE*.
spondingly,thesingle-particleGreen'sfunction (Eq.(9.25))becomes(Fig.9.14)
G(.
x,y)= G0(x,y)+ jJ4xjdnxj
'G0(x,xj)X*(xj,xJ)G0(xj
',y,
)
+ f#4x1#4xJfdfxgdhxz
'GotxsxjjX*(xjsxj
')
x G0(xj
',xa)X*(xz,xz')G0(xzz,.
p)+ ... (9.27)
Proxrjelf-energy
Y
œ
Fig.9.14 Dyson'sequation forGaplx,y).
whichcan besummed formallytoyield anintegralequation(Dyson'sequation)
fortheexactG.
G.
x/x,.
;'
)=t4/x,.
$ +.
fdkxb#4xJt4z(A'.xl)Z*(.
:':,l'I)A'zt2.,(fI.F) (9.28)
The validity ofEq.(9.28)can be verised by iterating therightside,which reproducesEq.(9.27)term by term .
GREEN'S FtlNc'
noNs ANo FIEL; THEORY (FERMIONS)
1e7
Dyson'sequation naturally becomesmuch simjler ifthe interaction is
invariantundertranslations and the system isspatially uniform . In thiscase
thequantitiesappearingin Eq.(9.27)dependonlyonthecoordinatediserences,
and it is possible to introduce four-dimensional Fourier transforms in these
diFerences. W ith thedesnition
Y*(x,y)zj=(29-4Jd% efk'ïx-''X*(k).#
(9.29)
and Eq.(9.7).the space-timeintegrationsin Eq.(9.28)are readily evaluated,
and we ;nd an algebraicequation in momentum space (compare Eq.(9.22))
G./k)= (4#k)+ (4a(k)X*(OA,G/z#k)
(9-30)
Intheusualcase.G,G0,andX*areaIldiagonalinthematrixindices,and Dyson's
equationcan then besolved explicitlyas
G(k)=
1
(Gf)(k)1-j sw(k;
(9.31)
-
The inverseofG%isgiven by
IG0(k))-!- IG0(k,o3)-l=(
.o- (z)kM ttl-â-lek
(9.32)
becausethezkiylinEq.(9.14)isnow irrelevant,and we5nd
G
1
.
/k)> Gz#(k,tt))= œ - ,-j4 - xwjk,og %ô
(9.33)
In the generalcase,thisexpression mustbe replactd by an inversem atrix that
solvesthematrixequation (9.30). Asshown inSec.7,thesingularitiesofthe
exactGreen'sfunctionG(k,tM,consideredasafunctionoft,),determineboththe
excitation energies ek ofthe system and theirdamping yk. Furthermore,the
Ixhm ann representationensuresthatforrealo)
Im X*(k,tM > 0
(
.
o< p,
/â
(9.34)
Im X*(k,o4< 0 o)> +lh
sothatthechemicalpotentialcanbedeterminedasthepointwhereImX*(k,o))
changessign.
Asan example ofthe presentanalysis,We shallconsideral1the irst-and
second-orderdiagrams,shownin Figs.9.7and 9.8. Itisevidentthatb0th Erstorder terms representproper self-energy insertions;as a result the Nrst-order
properself-energy E?k)isgiven bythediagramsin Fig.9.15. Herethesmall
arrowsatthe endssm cify how the Green'sfunctionsare to be connected,and
the diagramscan be inttrpreted either in coordinate space or in momentum
space. The situation is considerably m ore complicated in second order. ln
particular,the diagrams in Fig.9.8/ to d represent a1lpossible second-order
iterationsofX!l)and thereforecorrespond to impromrself-energyinsertions.
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
1e8
1.
z*
(l)=
t
+
t
t
(J)
Fig.9.15 First-orderproperself-energy X4*1,.
(b)
On theotherhand,theremainingterms(Fig.9.8,tojja1lcontain properself-
energyinsertions,andwenow exhibitalIcontributionsto)2!z)inFig.9.16.
A particularly simple approximation isto write X*(k,f.
t));
k;Xh)(k,(z
'
))>
Xh)(k)(seeEq.(9.24))inthesolutionofDyson'sequation(9.33). Thisapproximationcorrespondstosumminganinhniteclassofdiagramscontainingarbitrary
t
*
1
Z t2)-
1
t
'
t (c)
1
t
1
t
t
(e)
(f)
Fig.9.16 Second-orde.
rproperself-energyE(1).
iterationsofZ?k)(Fig.9.17). The polesofthe approximateGreen'sfunction
occuratthe energy
ekl?= d + âxh)(k)
h2k2
= 2m + aF(
)(0)- (2=)-3J#3k'lFn(k- k')+ 3Fl(k- k'))#tks- k')
(9.35)
which determinestheenergy ek'ofastate with momentum âk containing an
additionalparticle. Herethe term nPV0)isaconstantenergy shift;itarises
from the 'ttadpole''diagram Fig.9.l5a and represents the forward scattering
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
1Q9
)')'
)'
)'
)
Fig.9.17 ApproximateGobtainedwiththesubstitutionX*= X1).
ofl-aIlthe other particles. The integralterm depends on k and arises from
Fig.9.13b. Inthepresent(srst-order)approximation,theproperself-energyis
rec/,andthesystem propagatesforeverwithoutdam ping. Thisexam pleclearly
dem onstrates the power ofD yson's equation,because any approxim ation for
X* generates an inhnite-order approximate series for the Green's function.
Dyson'sequation thusenablesusto sum an insnite classofperturbation term s
in acom pactform .
TheexplicitsolutionforG (Eq.(9.33))allowsustorewritetheground-state
energyofauniform system (Eqs.(7.27)and(7.32))inaparticularlysimpleform.
ConsiderEq.(7.27)forspin-.
çfermionswith Y*and G diagonalin thematrix
indices. A combinationwithEq.(9.33)yields
s--,
.p(c,s1)jjjk
).-,--.L,.-);-+,)t(k,
-j
= = -
fl'(2x+ 1)(20-4jd*keit
apntlE.k+ JâX*(k,œ))G(k,op)+ !.â)
fF(2J+ 1)(20-4f#4kcia'9(4 +.
RX*(k,a)))G(k,(z>)
(9.36)
wherethelastlineisobtained with thelim iting procedure
.
oeku''l- lim lim * e-E1a''efa'g#f
1im * dt
.
u.
-.t)
n-,o+ -x 2rr
n-yo+ :-+9+ -.
2*
l e
li
m
1-i
m()+n'
-Ta+ E'a
n
.
e'b e
.
=0
(9.37)
ltisreadilyveriEedthatthisisthecorrectlimitingprocessbyapplyingEq.(9.36)
toanoninteractingFermisystem. Inthesameway,Eq.(7.32)canberewritten
=
as
E-Ez--!.
fp'(2,+1)J0
'J
jA
-J(f
z
,
'
,
'
k
I
'
Re'-nhgho--eA
k
;
7
-ât
ek
*A(k,(
s)1
=-!.
fF(2J+1)(20-4j0
'JAA-'J#4/
t
-E>
it
'9âX*A(k,
tg)G1(k,oa)
(9.38)
whereb0th Z*Aand GAmustbeevaluated forallAbetween0 and 1.
41:
GROUND-STATE (ZERO-TEMPEM TUBE) FORMALISM
2.Polarization insertionkA similar analysis can be carried out for the
interaction G tween two particles,which always consists of tht lowest-order
interaction plus a series ofconneded diagramswith lowest-ordtrinteractions
cominginandout(Fig.9.18). W ecanevidentlywriteanintegralequationfor
.
p œ
x
T'
p .
x
N +
z
'r
lz
à
e
:1
:
A
N
r
Polxn'ution n
FI:.9.18 GeneralstructtlreoftheeFe ivelteraction U...p..
theexactinteraction;thisequationagaina comessimplerfora uniform system,
whereitispossibleto work in momentum space. IfUlqjxb,p.and &;(q).#,p.
denotethtexactandlowest-orderinteractions,thecorresponding equationtakes
theform
Ulqtxp.
p.= Ua(ç).#,
pm+ L%(4).#,s.lR.v,,?A(ç)t&(ç),
?z,pm
(9.39)
which defnes the polarization insertion 1Rsv,,
?z(ç). It is also convenient to
introducethe conceptofa prom rpolarization 1R*,which isapolarization part
thatcannotbeseparatedintotwopolarization partsbycuttingasingleinteraction
lint(Fig.9.19). Equation(9.39)canthen bertwrittenasan equation Gtwetn
Impm-
Paw
Fig.9.19 Typicalimproperandproperpolarizationinsertions.
the exactihteraction and the promr polarization (Fig.9.20). For a homogeneoussystem thisequation becomesan algebraicequation
Uçqsup,p.= t&(:).#,pm+ Ua(ç).,.
s.1Rs
5,,?A@)U(ç),?z,pr
(9.40)
In general,Eqs.(9.39)and (9.40)have a complicated matrix structure,
and weshallusually consideronly spin-independentpotentials
Uéqj.p.p.= Ua(ç)3./&pr
.
#
,p %N
N œ
e #
(9.41)
#.
N + z
m #
y
Pm- N larization n*
FIg.9.2: W son'sm x tionforU.p.p..
à
P
,
m
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
111
Itthen followsim m ediately thattheexactinteraction hasthesam estructure
Ulqjxb,p.= &(ç)ôz#h.
(9.42)
wherethefunction U(q)isdeterminedbythesimplerequations
&(ç)- &()(ç)+ &o(ç)11(ç)L%(ç)
Uçqb- Uéqb+ (&fç)1'
1*(ç)Uçqs
(9.43*
(9.43:)
Herewehaveintroduced theabbreviations
I1(ç)U l1ax.AA(ç)
l1*(ç)O 11X.AA(t
?)
andadirectsolutionofEq.(9.43:)yields
&tV1= 1 17*
Uo
(ç)Uulq)
(ç)
-
Thisresultcanbeusedtodesneageneralizeddielectricfunctionv(ç)
U(q)= Unlqj
Klqj
(9.44*
(9.44:)
69*451
(9.46)
which characterizes the modihcation of the lowest-order interaction by the
polarizationofthemedium. ComparisonofEqs.(9.45)and (9.46)yields
Klqs= 1- UzlqjI1*(ç)
Go kDs-ro NE's THEOREM
Theapplication ofquantum fieldtheorytothemany-bodyproblem wasinitiated
byG oldstonein 1957.1 H eproved thecancellationofthedisconnecteddiagram s
to allorders,and derived the following expression for the energy shiftofthe
ground state
* z
E - En- '
:4)01
'Jf
-l
a=0
1
n
* - Jf
-1 14)oz
N'connecteu
Eo- H o
(9.48)
where/% and .
4:arethetime-independentoperatorsintheSchrödingerrepresentation. Thisresultcan beinterpreted by insertinga com pletesetofeigen-
statesoflîobetweeneachinteractionXj. ThePointhedenominatorcanthen
be replaced by the corresponding eigenvalue. A l1 m atrix elements of the
operatorin Eq.(9.48)thatstartfrom theground state 1
*0)and end with the
groundstate1*a)aretobeincluded. Wecanvisualizethesematrixelementsin
thefollowingway:theoperatorlî'
jactingonthestateI*g)createstwoparticles
and two holes. Thisstatethenpropagateswith (.
G - Sc)-1,and thenextXj
'J.Goldstone,loc.ctt.
112
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
canthencreatemoreparticlesandholesorscattertheexistingqarticlesorholes.
Theresultingintermediatestateagain propagateswith (f%- Ho)-t,and so on.
ThesnalXjmustthen returnthesystem to theground state1*c). A typical
processm aybepictured asshown in Fig.9.21,wherean arrow running upward
l% >
lèj
1
Eo-lh
W!
1
z?1
Eo-%
1
HtEo-lh
1
*07.
Fig.9.21 TypicalGoldstone diagram in theexpansion
ofE - Eo.
represents the presence of a particle,an arrow running downward represents
thegresenceofa hole,and ahorizontalwavylinerepresentstheapplicationof
an H L. Thus the sequence ofeventsstarts atthe bottom ofthe diagram and
proceeds upward. These diagram s are known as Goldstone diagram s and
m erely keep track ofallthe m atrix elem ents thatcontribute in evaluating Eq.
(9.48). The subscriptçiconnected''meansthatonly thosediagramsthatare
connected to the l
inalinteraction are to be included. In particular,the state
*()),which hasno particlesorholespresent,canneveroccurasanintermediate
statein Eq.(9.48),fortheresultingmatrixelementwould necessarilyconsistof
disconnected parts.
Goldstone'stheorem (9.48)isan exactrestatement(to a11orders)ofthe
fam iliartim e-independentperturbation expression forthe ground-state energy.
Thisequivalenceisreadilyverised in the srstfew termsby inserting a com plete
setofeigenstatesofXabetweeneachinteractionXj.
t*ol4l1*a
E )r*nI#lI*o)'+
E - Ev- ,
tm01
4 ,1mc)+
,#0
o- En
.
(9.49)
ThecorrespondingGoldstonediagramsforahomogeneousmedium (seeProb.
3.13)areshownin Fig.9.22. Thefirsttwo diagramsrepresenttheusualdirect
andexchangecontributionsin(*:141t*0).
In applying Goldstone'stheorem to a uniform system ,we observethatthe
m omentum willbeconserved ateveryinteraction becausethem atrixelem entsin
/% involvean integration overallspace. Furthermore,theparticlesin the
intermediatestatestuvephysicalunperturbedenergiesezrelatedtotheirmomentum q,and thevirtualnatureoftheinterm ediatestateissum m arizedintheenergy
denom inators. In contrast,the Feynm an-D yson perturbation theory forthe
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
113
G reen'sfunction,from which wecanalso com puteE - Eo,conservesbothenergy
andmomentum atevery vertex,buttheintermediateparticlescan propagatewith
any frequency f.
t),independent of q. For this reason. the Feynm an-D yson
approach hasthe advantage ofbeing m anifestly covariant,which isessentialin
any relativistic theory. Neyertheless,the /u'
t?approaches m erely represent /wt?
dterentw'
cysofgroupingandl'
nterpretingthetermsd??//
?(,perlurbationexpansion,
andaI1physicalresultsmustbe l
dentical.
CM 'Q'V 'V '
Fig.9.22 AIlsrst-and second-orderGoldstonediagramsfor
E - Eoin auniform system .
We now proveGoldstone'stheorem (Eq.(9.48)2. lftheground stateof
theinteracting system isobtained adiabatically from thatofthe noninteracting
system,theGell-Mann and Low theorem (Eq.(6.45))expressestheenergyshift
oftheground stateas
E Eo= (tD0!Xl0(0,-:
r:)1*0)
.
(*01xt.,
7(0,-:f)l*o)
-
(9.50)
Thenumeratorcan beevaluated by writing
'
t*n1.
41P(0,-'
x))l*()h-
* -j rj
p=0
h -j
r
0
dt,
-.
0
J-.t
u-
x IT()IFEXI.
41(/,) -''Xl(Jv))1*0) (9.51)
HerethefactorX1appearingonthelefthasbeenincorporatedin theF product
since X1- Xj(0)correspondsto a latertime than a11the otherfactorsin the
.
integrand. U se W ick'stheorem to evaluateaIlthecontractionsthatcontribute
to thematrixelementin Eq.(9.51). ThefactorX1(0)providesafixed external
point that enables us to distinguish between connected and disconneeted
diagrams;aconnecteddiagram isonethatiscontractedinto X140). Thedistinction is illustrated in Fig. 9.
23. Suppose thatthereare n connected 4l's
#,(c)
.
Fig. 9.23 Typical connected apd disconnected Goldstonediagrams.
Connœte
.
x
!'
'
Disconnxte
114
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
and m disconnected Wl'swherep= n+ m ;thispartition can beperformed in
pl/n!m !ways. Thesummation overvin Eq.(9.51)can thereforeberewritten
-
j n+m v! 1 0
â
dtj
n!m !7.
0
-.
dtn
-.
x(*(
)f
rEl?,#l(r!
)---4l(/n)1$
*n)cJ0*dtn.,---J0
-œdtn..
m
x (*olrE4l(K+l) ---4l(fa+,211*o)
(9.52)
justasin Eq.(9.4). (For simplicity,we now use a subscript C to indicate
connected.) Thesummationoverm reproducesthedenominatorofEq.(9.50),
and wethusobtain
E - f'
t
l=
* -f & j p
t)
- -j.
dtL .' '
dt.
pj n. -.
-.
n=0
x'
C
:*oIr(4lP14/1)...H'-l(/,)1I*clc (9.53)
which demonstrates the cancellation of the disconnected diagrams in this
expression.
wenow proceedtocarry out//7:timel
htegratîonsth Eq.(9.53)explicitly.
Considertheath-ordercontribution and inserttherelation between4 14/)and
41from Eq.(6.5)
.
'.'-
.n
(E- E()('''= h/
0
.
-
dtl
.
tI
1..-,
dtz --.
-.
-.
dtneftîi+fz+ '''+'.'
'
'
*1efR()t,/h/
x(*:1/.
f
j
,le-f#otl/'efR()
tz/hJjle--R(
)fz/h..v
gjIe-l/lota-l/:elR'
ntplh.
Jjle.
-.tRq%lhjm()
jc
.
Here wehave observed thatalln!possibletimeorderingsmakeidenticalcontri-
butions(see Sec.6)and thereforework with onedehnitetimeordering ofthe
operatorsin thism atrix element. The adiabatic dam ping factorhasalso been
explicitly restored. Changevariablesto relativetim es
X j= tl
xz= fz- tl
':)= tj- tz
.
tj= x j
x = t- t I
tn= xs+ xa-l+ '
,
and use
/:1.
'
N )-fol% )
tz= xc+ xl
tj= xa+ xz+ xj
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
115
Thistransformation yields
-
ia
e
I.
F-.
f'
c)t.)= -J
j t*a1
.
/.
hJ-c
oen'x,t
l
it
ll
o-ro'xt/Af
dxlX1
xJ--eç
n-i)exaei
f/
?(
,
-zz)xalh:x
0zI'
kj...
gfxaeilRq-rn)xa/:dy
J-gjj
(j
ljko
tl
t,
Theintegrationscan now al1becarried outexplicitly,and we5nd
1
1
'
.f
?'l
.
W1
if - .
G )tn'= (*01/.
/,
.
Eo- Xo+ Ieaâ Ev- Xu+ ieln- 1)h
4i
1
--
lîjtmolc
Ef
t- ,
/.
% +ieh
This result immediately yields Goldstone's theorem Eq. (9.48) because the
limitationto connected diagramsensuresthat1*0)cannotappearasan intermediatestate,andthestate1*0)isnondegenerate. ItfollowsthatEz- Xo+ ieh
cannevervanish,so thattheconvergencefactor+iebecomesirrelevant,and we
canusethepropagator(.
G - Xo1-1,asinEq.(9.48).
This formalproofcan be m ade m ore concrete by explicitly considering
a1lath-orderFeynman diagramsthatcontributeto Eq.(9.53). Each diagram
consists of unperturbed Green's functions G0,which evidently contain both
particleand hole propagation (compare Eq.(7.41)). These diagramscan be
groupedintosetscontainingn!equivalentdiagram sthatdifl
kronlybyperm uting
thetim evariables. Thesym m etryoftheintegrand againallowsthereplacem ent
tn-I
0
0
tl
1 0
dtn=
#/j
dtz '
dtn
-5
dtl
n . -x
-x
ex
-x
W iththechoiceofadesnitetimeordering,eachofthen!diagramsnow represents
a distinctprocess. The integraloverrelativetimes(0> xi> -cc)then yields
n!distinctGoldstone diagramscorresponding to the n!possible timeorderings
ofthe originalFeynm an diagram . Thus the setofal1possible tim e-ordered
connected Feynman diagramsgives the complete set ofconnected Goldstone
diagram s. The Feynm an-D yson and G oldstone approaches are clearly
equivalentto everyorderinperturbationtheory,buttheFeynm an-Dysonanalysis
Flc.
ç thefundamentaladvantage of combining many terms of time-independent
perturbation theory frl/o asingleFeynman diagram . W em aynotethata sim ilar
analysis applies to any ground-state expectation value of H eisenberg feld
operators,forexample,the single-particleGreen'sfunctionG(x,y,(s),which is
theFouriertransform ofEq.(9.5). Iftheintegration overa1ltimesiscarried
outexplicitly,the resulting perturbation expansion m ay be classised according
to the intermediate states,justasin Fig.9.21. In thisway wecan obtain a
uniquecorrespondencebetweenagiven Feynm an diagram and asetofGoldstone
diagrams(ordiagramsoftime-independentperturbationtheory).
116
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
Goldstone's theorem (9.48) was originally stimulated by Brueckner's
theory of strongly interacting Fermi systems. This Brueckner-Goldstone
approach hasformed thebasisforextensivework ontheground-stateproperties
ofnuclearm atter,l14e3,2and atom s.3
PROBLEM S
3.1. Show thatwhent< G theintegralequationfor0lt,tzlcanbewrittenas
àf
* l()
L'
/lt,tè = 1+ j
dt'#j(f')Cl'
lt',toj
Henceshow that
* j nj
lq
.
t7(/,fp -
s o
.
le
dtL'. .
dtn/(z?'
j(rj)...z?'l(f,)J
r
D=() '* '''- t
where? denotestheanti-time-ordering(latesttimesto theright). Derivethis
resultfrom Eqs.(6.16)and(6.23).
3.2. One ofthemostusefulrelationsin quantum seld theory is
f2
j3
el'
%t)e-i:= O + i(J,J)+ j-j(u
f,(xf,d))+ j-j(,9,(u
f,(J,J)))+ ..
Verify this resultto the orderindicated. Evaluate the commutatorsexplicitly
andre-sum thescriestoderiveEqs.(6.10)from Eqs.(6.7)and(6.9).
3.3. Desnethetwo-particleGreen'sfunction by
Gajiy/ôtxlt1,xatz;x;tk
',xc
'/1)
=
(H%IFEfk(xlf1)#jt(x:a
tz4'
fkxz
'tz'j'
f
Jitxif;))1
:F(,)
1%1:F
(-f)2
0)
Provethattheexpectation valueofthetwo-bodyinteraction in theexactground
stateisgiven by
(P)=-.
!.Jd?xJ#3x'F(x,x')j
zA,,SAGAA,;ss,(x'/,x/ix't+,x/+)
iK.A.Brueckner, C.A.Levinson,and H.M .Mahmoud,Phys.#e?
J.,95:217(1954);H .A.
Bethe,Phys.Ret,.,103:1353(1956);K .A.BruecknerandJ.L.Gammel,Phys.Re!
?.,1* :1023
(1958);K.A.Brueckner,TheoryofNuclearStructure,inC.DeW itt(ed.),*1TheM any-Body
Probl
em,''p.47,John W iley and Sons,Inc..New York,1959;H.A.Bethe,B.H.Brandow,
andA.G.Petschek,Phys.Rev..129:225(1963);seealsoChap.11.
2K.A.Bruecknerand J. L.Gammel,Phys..
R0 .,1* :1(M0(1958);T.W .Burkhardt,Ann.Phys.
(#.K ),47:516(1968);E.Ostgaard,Phys..
Re&.,17::257(1968).
3See,for exampley H.P.Kelly,Correlation Structure in Atoms,in K.A.Brueckner(ed.),
SeAdvances in TheoreticalPhysicsj''vol.2,p.75,Academic Presslnc.,New York,1968. A
review ofthistopic isalso given in Correlation Efectsin Atomsand M olecules,R . Lefebvre
and C.Moser(eds.
),AiAdvances in ChemicalPhysics,''vol.XIV,Interscience Publishers,
New York,1969.
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
3.4. Consider a many-body system in the presence ofan externalpotential
&(x)with a spin-independentinteraction potentialP'tx- x'). Show thatthe
exactone-particleG reen'sfunctionobeystheequation ofm otion
wheretheupper(lower)sign refersto bosons(fermions)and thetwo-particle
G reen'sfunctionisdefned in Prob.3.3.
3.5. UseEqs.(3.29)and(3.30)toverifyEq.(7.58)foranidealFermigas,and
show thatrz= 4.
3.6. Considerthefunction
2 ''
Fa@)= =x
a=0
1
n2+ z/ac
and discussitsanalyticstructurein thecom plexzplane.
(t8 Show thattheseriescanbesummedtogiveFatz)= z-lcothtrrzl//a)+ (a/=z),
which hasthe sam e analytic structure.
(:) Examinethelim ittx.
-..0 and comparewith thediscussion ofEq.(7.67).
3.7. ((z) Iffx
-xdx!p(x)f< cc,show that/(z)- Jx
-xJxptxl(z- x)-1isbounded
and analytic forlm z# 0. Provethatf(z)isdiscontinuousacrosstherealaxis
wheneverpta'
l+ 0,andthus/tz)hasabranchcutinthisregion.
(:)Assume the following simple form p(x)= y(y2-.
v2)-i. Ekaluate /-IJ)
.
explicitly forlm z > 0 and hnd itsanalyticcontinuatlonto lm z< 0.
(c)Repeatpart(b)forlmz< 0. Compareanddiscuss.
3.8. Derive the Lehmann representation for D(k,(
s), which is the Fourier
transform of
'
.'
IFg:u(-x)lbub
'p))'Y1'
-?
ilhx.ytEE
SExV1e().
-.
yvj s-. s'-'
.
-
a:
..
(jz
withthedensityfuctuationoperatordehnedby
1
(X)
'
t
;
'
?
i(xJH 'J)(x)'Ja(x)- (Y-()l1J'
Yv
-,
y
j!(
cX.
-)2i
-Ff
ft o. o'
Show thatD(k,f.
t))isameromorphicfunctionwithpolesinthesecondandfourth
quadrantofthe com plex (
.
v plane. lntroduce the corresponding retarded and
advanced functions,and constructa Lehm ann representation fortheirFourier
transform s. D iscussthe analytic propertiesand derive the dispersion relations
analogoustoEq.(7.70).
118
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
3.9. M ake the canonicaltransformation to particles and holes forfermions
ckz= 0(k- kslckA+ %kF- k)X kA. By applying W ick's theorem,prove the
relation
clelcxc3-ytclclcxczl+elkr-kz4(3zxytclcz)-:a:#(e1e.)!
+hkF-kî)(81:1(c1c4)- 3l4#(c1c:)l
+ %kF-k1)%kr- ka)(3lzîa4- :l4:z:)
wherethenorm al-orderedproductson therightsidenow referto thenew particle
andholeoperators,andthesubscriptsindicatethequantum numbersG.A).
3.10. Verifythecancellationofdisconnecteddiagrams(Eqs.(8.11),(9.3),and
(9.4)Jexplicitlytosecondorderintheinteractionpotential.
3.11. Considerasystem ofnoninteractingspin-èfermionsinanexternalstatic
potentialwithahamiltonianXex= J#3xyJ(x)F.#(x)#j(x).
(t8 UseW ick'stheorem to5ndtheFeynmanrulesforthesingle-particleGreen's
functionin thepresenceoftheextcrnalpotential.
(b)Show thatDyson'sequationbecomes
Ge
zqtx,y)-(4gx-y)+:-'fd5z(4A(x-z)LA,(z)c7#(z,A)
(c) Expresstheground-stateenergyinaform analogoustoEqs.(7.23)and(7.31).
W hathappensiftheparticlesalso interact?
3.12. Considera uniform system ofspin-è fermionswith spin-independent
interactions.
(J)Use theFeynman rulesin momentum spaceto writeoutthesecond-order
contributionsto theproperself-energy;evaluatethefrequencyintegrals(some
ofthem willvanish).
(b4 Henceshow thatthesecond-ordercontribution to theground-stateenergy
can bewritten
e't-a)
-#
-
2mh-2f...f(2z8-9dqkJ3p#3/#3a3(3)(k + p- l- n)
.
.
x (2F41- k)2- P'41- k)Ftp- l))#(ks-pj#tks- k)
x p(a- ks)0(1- ks)(#2+ k2- 12- a2+ j,
?;l-l
(c)SpecializetoanelectrongasandrederivetheresultsofProb.1.4.
3.13. D erive the expression for E(23given in Prob.3.12 from G oldstone's
theorem (9.48). From thisresult.givetherulesforevaluatingthoseG oldstone
diagram sshown in Fig.9.22.
GREEN'S FUNCTIONS AND FIELD THEORY (FERMIONS)
519
3.14. UseEq.(9.33)toshow thattheenergye:anddamping lyk!oflong-lived
single-particleexcitationsaregiven by
ek =
d + RehT*(k,%!h)
a = 1- dReX*
o(k,œ)
t.o
:w,
-1Im X*(k,q/â)
3.1B. Considerauniform system ofspin-èfermionswith thespin-dependent
interaction potentialofEq.(9.21),and assumethatI1*
r,e,xA(:)maybeapproximatedby1H0(t
?)3v.8Ay,.
(a)SolveEq.(9.40)tofind
P'o(t
?)ôa;3pm P',@)nxb-np.
&(*.,,p.- j p(ç)u(
)(,
?)+ I- p,jtt
yluotql
0
-
(bjCombineEqs.(9.39)and (9.40)to obtain Dyson'sequationforH in terms
of 1R* and Uo. Solve this equation with the above approximation for 1R*,
and provethat
?A(#)= !'
I10(ç)8v,)3As+ .
i.
fl0(ç)Ulqjvg,nnàI10(ç)
I1.p,,
whereUfqjistakenfrom (J).
4
Ferm iSystem s
ln principle,the perturbation theory and Feynman. diagram s developed in
Chap.3enableustoevaluatetheG reen'sfunctionG toa1lordersintheinteraction
potential. Such aprocedureisimpractical,however,andwemustinstead resort
to approxim ation schemes. Forexam ple,thesim plestapproxim ation consists
in retainingonlythehrst-ordercontributionsX)k)to theproperself-energy,as
discussed in Eqs.(9.24) and (9.35). Unfortunately,this approximation is
inadequate for m ost system sofinterest,and it becom esnecessary to include
certain classes of higher-order term s. Two approaches have been especially.
successful;both include insnite orders in perturbation theory, but they are
otherwisequitedistinct. Inthesrst,a sm allsetofproperself-energy insertions
is reinterpreted,so thatthe particle linesrepresent exactG reen'sfunctions G
instead of noninteracting G reen's functions G0. These approximations are
therefore sef-consistent,because G both determinesand isdetermined by the
12c
FERM ISYSTEM S
121
proper self-energy X*. In contrast, the second approach retains a selected
(insnite)classofproperself-energyinsertions,expressed in termsofG0. The
detailsofthemany-particlehamiltonian determine which procedureissuitable,
and we shallconsiderthreeexamplesthatfllustratehow thischoicecan bemade:
The(self-consistent)Hartree-Fock approximation (Sec.10),thesummation of
ladderdiagrams,appropriate forrepulsivehard-core potentials(Sec.11),and
thesum m ation ofring diagram s,appropriateforlong-rangecoulom b potentials
(Sec.12).
IX HARTREE-FOCK APPROXIM ATION
The starting pointforourdiscussion ofinteracting quantum m echanicalassem -
blieshasbeenthestate )*a),which istheground stateofthehamiltonianXo
Ih =Jk(hu).4c.
In 1*02),each oftheN particlesoccupiesadesnitesingle-particlestate,so that
itsm otion isindependentofthepresenceoftheotherparticles. Thissituation
w illbe clearly m odised by the interactionsbetween the particles;nevertheless,
itisan experim entalfactthata single-particle description form sa surprisingly
good approxim ation in m any diflkrentsystem s,forexam ple,m etals,atoms,and
nuclei. H ence a naturalapproach isto retain the single-particle picture and
assumethateachparticlerzltllptz.
sinasingle-particlepo/entialthatcomesfrom its
average interaction with aIIof the otherparticles. The single-particle energy
should then bethe unperturbed energy plusthe potentialenergy ofinteraction
averaged overthestatesoccupied by alIoftheotherparticles. Thisistheresult
obtained in Eq.(9.35),
.thusasafirstapproximationwecan keepjusttheErst-
ordercontributiontotheproperself-energyY!1). ThecorrespondingFeynman
diagram s are shown in Fig. 10.1. This calculation is not fully consistent,
1.
t
,t
*
= j2(1J=
Fig.10.1 Lowest-orderproperself-energy.
t
+
Y
however,sincethebackgroundparticlescontributingtoZh)aretreatedasnoninteracting, ln reality, of course, these particles also m ove in an average
potentialcom ing from thepresence ofa11the otherparticles. Thusinstead of
justthetwoself-energyttrmsshowninFig.10.1w'eshouldincludea11thegraphs
shown in Fig. 10.2. The shaded circles again denote the proper self-energy,
whichisthequantitywearetryingtocom pute. SincetheexactG reen'sfunction
canbeexpressedasaseriescontainingtheproperself-energy(Eq.(9.27))allthe
122
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
t 1
#
=
è
Y
-
1
+
+
'
t
+ ...
è
Y
+
Y
'' +
i
$ + +.
Y
1'
Fig. 10.2 Series for proper self-energy in Hartree-Fx k
approximation.
term sofFig.10.2 can be summed in thepair ofdiagramsshown in Fig.10.3,
where the heavy line denotesthe exactG,which isitselfdeterm ined from the
proper self-energy as indicated in Fig. 10.4. W e proceed to examine these
equationsin detail.
t
1
t
#t..y
. .
1
=
Fig.1Q.3 Self-consistent proper self-energy in Hartree-Fock
approximation.
+
Fig.10.4 Dyson'sequationforG.
W e shallconsidera system in a static spin-independentexternalpotential
&(x),which destroysthespatialuniformity- forexample,electronsin ametal
oran atom . Thetotalham iltonian thenbecom es
P
:272
0- I#3x1
J1(x) - Jwy + U(x) '
fvtxl
II'L-!.fJlx#lx'j4(x)fj
ftx')P'tx- x')#j(x3'
(ktxl
(10.1X
(10.1:)
whereforsim plicitytbeinterparticlepotentialhasbeenassumedspinindtpendent
P'(x,x')AA,,ss.= Ftx- x')3zz'3jzs,
(10.2)
In the presentapproximation,Dyson'sequation takesthe form shown in Fig.
10.5 where thelightline denotesG0(the noninteracting Green'sfunction corre-
FERM I SYSTEM S
123
spondingtoIhq,andtheheavylinedenotestheself-consistentG. Thecorrespondinganalyticequation isgiven by
G(x,y)= G0(x,y)+ Jd4xLd4xj
'G0(x,xI)X*(xl,x()G(xJ,y)
wherethe K roneckerdelta inthem atrix indiceshasbeen factored out. Exactly
as in Chap.3,the Feynman rules yield an explicitexpression forthe proper
self-energy
âX*(xI,xJ)=.
-iblt,-tLj(î(xI-xJ)(2J+ 1)Jd3x2Glxz/a,x2t(j
x Ftxj- x2)- Ftxl- x;)GtxlJj,xlt))) (10.4)
which isvalid forspin-lfermions. Notethatthefrstterm hasan extra factor
(2s+ 1)relativeto thesecond term ;thisarisesfrom the spin sums (compare
Eq.(9.18)).
= +
+
Fig.50.6 Dyson'sequation forG inHartree-Fockapproximation.
In thepresentexample,both P and W'
oaretimeindependent,and itis
thereforeconvenientto usea Fourierrepresentation
Glxt,x'/')=(2.
*-1Jdcoe-iazfl-f''G(x,x',to)
(10.5c)
G9(x/,x't'j=(274-1Jdutc-fœtt-f''G0(x9x'$(s)
(10.5:)
X*(xf,x't'j=IL*(x,x')3(/- t'j=(2r8-1Jdœe-ia'tt-f''t*(xx') (10.5c)
Asin theErst-orderapproximation (Eq.(9.24)),theproperself-energyishere
independentoffrequency. The time integrationsin Eq.(10.3)can now be
performed explicitly,and we:nd
G(x,y,œ)=G0(x,y,tz4+Jd3xL#3.xj
'G0(x,xj,a4X*(xj,xj
')G(xj',y,a4 (10.6)
Correspondingly,Eq.(10.4)reducesto
âE*(xl,xJ)=-ï(2J'+ 1)ôtxl- xJ)f#3xzF(xl-xz)(2,
0-.Jdœeft'g
x G(xa,xz,tt))+ fF(xl- x;)(2r8-1Jdt.
oel'
a''lG(xl,x(,(z)) (10.7)
Itisconvenienttointroducethecom pletesetoforthonorm aleigenfunctions
ofH f
t:
h2:72
Hog(x)- - 2m + (7(x)yJ(x)- 4p,
9(x)
(10.8)
124
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
These single-particle statesform a naturalbasisforthe seld operators'
h lxtj
in theinteractionpicture, and the noninteracting G reen'sfunctionthen becom es
l
-Golxt, x't?)= T
hht)
5-œt
''
xh
l
t'
x'
b('
p'
%
.Nx''
à*ex'
n- ie
-o
--t'
-
x (#(/- t')tllpfjl
m JJ(*(j)- 0(t'- tjtkmljlt4cgI*a))
Z
2(X)el(X')*eXp ieqlht- tJ)
J 7.
.-
=
x(p(/- t')pteî- e;)-0(t'-t)p(ek.- 4)J (10.9)
where e;istheenergy ofthe lastslled state. The Fouriertransform can be
computedexactlyasinEq.(7.44),whichyields
Go(x,x,,(s)- 1)g',?(x),f(x')+ #(e
--CD
'
'tey-4)
J
œ- s-jsg
o.gq+.-s-:s,
g.,.
x)
J
(10.10)
Weeannow evaluatetheparticledensitya0(x)intheunperturbedground state
1*0)
a0(x)=-j(21'+ 1)(2.
/4-1Jltselu''lG0(x,x,ts)
=
(2J'+ 1)Z
l@Xx)12#(6î.- 4)
p
(10.11)
whilethetotalnum berofparticlesisgiven by
N0=J#3xn9(x)=(2,
g+ 1)j)û(eî.- 6k)
(10.12)
.
because thesingle-particlewavefunctiqnsareassumed normalized.
Equations(10.6)to(10.8)andEq.(10.10)deEneasetofcoupledequations
fortheself-consistentGreen'sfunctionG. SinceX*isindependentoffrequency,
itisnaturalto seek asolution forG in the sam eform asG0:
G(x,x',
eley- es)
pte.
s- ep
(s)= JJ(p/x)+J(x')* f.'A- h-1ef+ fn + t,l ,-jcy. j.y)
.
(10.13)
.
where (+/x)Jdenotes a complete set of single-particle wave functionswith
energieseJ,and es istheenergy ofthe lasthlled state. The associated particle
densityintheinteractingsystem becomesgcompareEq.(10.ll)J
a(x)- (2J'+ 1))J(2l@.
f(x)12#(e,.- e.
,)
(10.14)
w hile
N = NQ= (2,ç+ 1)(
j
2#(es- %)
J
(10.15)
becausetheperturbationXlconservesthetotalnumberofparticles. Thusthe
interaction merely shiftsthesingle-particlelevels,which arestillslled according
FERM I SYSTEM S
125
to theirenergy up to the Ferm ilevelcs. The frequency integralin Eq. (10.7)
can now be evaluated directly,and we 5nd
-
Ztxl- xl
')X fpJ(xl)+j(xJ)*dtes- 6'
J) (10.16)
Notethat1:*dependson +j'
,acombinationofEqs.(10.6)and (10.16)thenyields
anonlinearintegralequationfor%yintermsof(theassumedknown)pq.
Thisequation m ay besim plihed with thedipkrentialoperator
h2V!
LI= hœ + 2m '- &(xI)= ht.
o- Ho
IfLjisappliedtoG0(xj,xl
',r
.
z?)weobtain
-
h)((
r)îtxll+T(xi
')*
=
hblxl- x;)
wherethelastlinefollowsfrom theassumedcompletenessoftheset((
J?9
,) Thus
â-1f.1istheinverseoperator(G0)-1,and application ofLjto Eq (10 64ylelds
f-1Gtxj,x!
',t,?)= â3(xl- xj
')+ .f#3x2âZ*(xl,x2)G(x2,x;,(s)
.
.
Itisusefulto inserttheexplicitform sofG and l- j:
ât hlV)
t?+ --jz?7 - b'(xI)
,+ 0(Ey- ej-)
heti- ej)
(p,(xj)%.y(xI) aj-y.
.-/n.- (------j--f.. j
)-.-.
e,v
..-
J
o-
.
ej. . .
q
M ultiplybywktx()and integrateoverxj
'. Theorthogonalityofl(
py)leadstoa
simple Schrödinger-likeequation for+,(xl)
-
h
2T
-2
. 1
2m > LllxI) %ytxj)-p.(#3x2âY*(xj.x2)y#(xc)= ej(
/).
/txI)
-
.
.
where the properself-energy hs* acts asa static noltlocalpotential. Sincel1*
is hermitian and independentof/-the usualproofoforthogonality remains
126
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
unchangedstherebyjustifying ourinitialassumption. Equation(10.16)shows
thatX*consistsoftwo terms:alocal(direct)term proportionalto theparticle
densityand anonlocal(exchange)term. TheseequationsarejusttheHartreeFock equationslfamiliarfrom atomictheory.
ltisapparentthattheseequationsconstitutea very com plicated problem :
A n initialsetofsingle-particle wave functionsand energies isassum ed known,
andthecorrespondingX*iscalculatedfrom Eq.(10.16). Equation(10.17)then
becomes a one-body eigenvalue equation thatdeterm ines a new setofeigenfunctionsand eigenvalues,which are used to recompute X*. Thisprocessis
continued untila self-consistentsolution isobtained forb0th @ ,)and (e,).
Theground-stateenergycanthenbeevaluatedwithEq.(7.23).suitablygeneralizedtoincludetheexternalpotential&(xl)
E=
-
dol
h2Vl
!.
1.(2J+ l)f#3xI 2, eizo'lxl
mpx, hœ - 2- + &(x1) Gtxj,xj
p,(sl
li
z'
dœ z
= -f(2&+ l).
f#3xjjy)p/x1)# a,preu,,?!.yp,(xj)+.(s,pjxjj
-
-
- 3 >E*(x,,xz)+,(xa)) 0tE'- 6Z
Pt''-- 'J1
l.Jd xa
. - ,-1,,+.iy +
. - ,-,e,- y
'
.
)
(2,+ 1)z
e,gtes- e,)-.
1.(2.
,+ 1)JJ3xlJ3xcz
e,(x,)*
J
j
x âX*(x,,xa)pxxa)0(eF- 0) (10.18)
where the second Iine hasbeen obtained with Eq.(10.17)and the Iastwith Eq.
(9.37)and acontourintegration. Thehrstterm ofthisexpressionhasasimple
interpretation as the sum of the energies ofaIIoccupied states. Each singleparticle state incorporatesthee/ectofthe otherparticlesthrough the nonlocal
self-consistentpotentialâY*. In com puting the ground-stateenergy,however,
thesrstterm ofEq.(10.18)by itselfincludes the interaction energy twice;this
doublecounting isthen com pensated by the second term . A com bination of
Eqs.(10.16)and(10.18)yields
f = (2J+ 1)Z e,Ses- eJ4- .
!(2J+ 1))é8(6s- e,)#(es- ek)j'#3x1
J
Jk
x Jd'xzP'txl- xa)(42J+ l)I+/xl)I2Içy(xz)l2
-
e.
/(xl)*çulxlleklxal*çutxzll (10.19)
which istheusualH artree-Fock result.
Thetwo termsin bracketsin Eq.(10.19)yield the directand exchange
energies,respectively. Forashort-range interparticlepotential(asin nuclear
physics),the direct and exchange terms are comparable in magnitude,for a
'D.R.Hartree.Proc.CambridgePhil..5bc..M :89;llI(1928);J.C.Slater.Phys.Rev..35:210
(l930);V.Fock,Z.Physik.61:126(l930).
FERM f sYs'rEv s
1a7
long-range interparticle potential(asin atomic physics),theexchange contribution isusually much smallerthan thedirectone. Indeed,theexchange term
isoccasionally neglected entirely indeterminingtheself-consistentenero levels
of atoms,and the corresponding equations are known as the self-consistent
Hartree equations. The physicalbasis for the distinction G tween short-and
long-range potentials is the following. The exclusion principle prevents two
particlesofthe samespin from occupying thesame single-particle state. Asa
result,the two-particle density correlation function forparallelspins vanishes
throughouta region comparable with thc interparticlespacing. Iftherangeof
the potentialis less than the interparticle spacing,then thisexclusion hole is
crucial in determining the ground-state energy. In contrast, a long-range
potentialextends far beyond the interparticle spacing,and the exclusion hole
then playsonly a minor role (compare Probs.4.1and 5.10,which exhibitthis
distinctionexplicitly).
Fora generalexternalpotential&(x),the Hartree-Fock equations are
very diëcult to solve,because the single-particle wave functionsexx) and
energieseJm ustboth bedetermined self-consistently. TheseequationsV ome
much simplerfora uniform system,where U(x)vanishesand theprom rxlfenergy takes the form Z*(x- x'). Itis readily verised thata plane wave
@k(x)= F-ktpik*xsatissestheself-consistencyrequirements,sinœ itisasolution
ofEq.(10.17). Thecorrespondingself-consistentsingle-particleenergy* m>
ek= e2+âX*(k)
(10.20)
where
âX*(k)=J#3(x-x')e-dk*tx-x')âX*(x-x')
=
(2J+ 1)F(0)(2=)-3J#3k'0(kF-k'j
(2.)-3Jdsk'F(k-k')%kp-k')
= nF(0)- (2r4-3f#3k'F(k- k')gtks- k')
(10.21)
-
Theground-stateenergy(10.18)reducestothesimpleform
E - (2J+ 1)P'(2rr)-3J#3#(ek- !WX*(k))0(kF- k)
(2J+ 1)P'(2x)-3fdbk(4 + !3X*(k))#tks- k)
(10.22)
which showshow the self-energy modisesthe pound-state energy ofthe noninteracting system. Itis interesting thatthese self-consistentexpressionsfora
untform medium are identical with the contributions evaluated in irst-order
perturbationtheory (Eqs.(9.35)and(9.36)1. Thisequalityarisesonlybecause
theunperturbed (plane-wave)eigenfunctionsin a uniform system arealso the
self-consistent ones; for a nonuniform system, however, the self-consistent
calculation clearlygoesfarbeyond thesrst-orderexpression.
128
GROZND-STATE (ZERO-TEMPERATURE) FORMALISM
IIQIM PERFECT FERM I GA S
W eshallnow considerindetailadiluteFerm igaswith strong short-rangerepul-
sivepotentials(çthardcores''l.l Thissystem isofconsiderableintrinsicinterest,
and italso form sthebasisforstudiesofnuclearm atterand H e3,asinitiated by
Brueckner.z The fundam ental observation is the following. Although the
potentialm ay be strong and singular,the scattering amplitude can be sm allfor
such interactions. For desniteness, the potential will be taken as purely
repulsive with a strong short-range core,thereby neglecting any possibility of a
self-bound liquid. A ny realistic potentialm ust clearly have such a repulsive
core;otherwise there would be no equilibrium density and the system would
collapse. (See Prob.1.2.) In particular,the nucleon-nucleon potentialhasa
repulsivecorearising from thestrongly interacting m eson cloud,and theH e3-H e3
potentialhasa repulsivecore arising from the interaction between the electrons.
SCAR ERING Fno M A HARD SPHERE
To illustratethese rem arks,considerthe scattering oftwo particlesinteracting
through astrongrepulsivepotentialofstrength Utj> 0and rangea(Fig.11.1).
An insnite hard core clearly corresponds to the limit F% .
-.
>.a:. The spatial
Fouriertransform ofthepotentialisjusttheBornapproximationforthescattering amplitude;itisproportionalto Ft and therefore divergesfor a hard core.
P-(q)= je-fqexU(x)J3x.
-->.:c
N
(1l.1)
In fact,the true scattering am plitude isgiven by the partial-waveexpansion3
Since the Schrödinger equation is trivially soluble in the region outside the
potential,each phaseshiftcan be obtained explicitly with theboundary condition
Fig.11.1 Repulsivesquare-wellpotential.
'w efollow theanalysisofV.M .Galitskii,Sot'.Phys.-JETP,7:104(1958).
2K . A.Brueckner,TheoryofNuclearStructure,inC.DeW itt(ed.),i.-rhtM anyBodyProblemv''
p.47,John W ileyand Sons,Inc.,New York,1959.
3Forthe basicelementsofscatteringtheoryused in thissection,thereaderisreferred to any
standard textbook on quantum mechanicssforexample,L.1.Schifr '%ouantum M echani
cs,'q
3d ed.,sec.19.M cGraw-l-lillBook Company,New York,1968.
FERMISYSTEMS
12;
long-rangeinterparticle potential(asin atomic physics),the exchangecontribution isusually m uch sm allerthan the directone. Indeed,the exchangeterm
isoccasionally neglected entirely in determining theself-consistentenern levels
of atom s, and the corresponding equations are known as the self-consistent
Hartree equations. The physicalbasisfor the distinction G tween short-and
long-range potentials is the following. The exclusion principle prevents two
particlesofthe same spin from occupying the samesingle-particle state. Asa
result,the two-particle density correlation function for parallelspinsvanishes
throughouta region comparablewith the interparticlespacing. Ifthe range of
the potentialis less than the interparticle spacing,then this exclusion hole is
crucial in determ ining the ground-state energy. In contrast, a long-range
potentialextends far beyond the interparticle spacing,and the exclusion hole
then playsonly a m inorrole (compare Probs.4.1and 5.10,which exhibitthis
distinctionexplicitly).
Fora generalexternalpotentialU(x),the Hartree-Fock equations are
very dimcult to solve,because the single-particle wave functionsp/x) and
energieselmustboth bedetermined self-consistently. TheseequationsA ome
much simplerfora uniform system,where U(x)vanishesand theprom rRlfenergy takes the form E*(x- x'). Itis readily verised thata plane wave
@k(x)= F-kt'dk*xsatissestheself-consistencyrequirements,sinœ itisaKlution
ofEq.(10.17). Thecorrespondingself-consistentsingle-particleenergy% m%
ek= 62+ âX*(k)
(10.20)
where
âE*(k)=j#3(x-x')e-ïk*tx-x')âX*(x-x')
= (
2.
v+ 1)P'(0)(2,4-3Jd'k'8(#z.-k')
(2=)-3J#3k'F(k-k')%kF-k')
=aP'(0)- (2x)-3J#3k,F(k-k')ptks-k')
(10.21)
-
Theground-stateenergy(10.18)reducestothesimpleform
E-(2&+ 1)P'(2=)-3J#3k(ek-.
R X*(k))0(kF-k)
- (
2J+ 1)F(2=)-3fdbk(4 + !3X*(k))#tks- k)
(10.22)
which showshow the self-energy modisesthe ground-state energy ofthe noninteracting system. ltisinteresting thattheseself-consistentexpressionsfora
unéorm medium are identicalwith the contributionsevaluated in srst-order
perturbationtheory (Eqs.(9.35)and(9.36)1. Thisequalityarisesonlybecause
the unperturbed (plane-wave)eigenfunctionsina uniform system are also the
self-consistent ones; for a nonuniform system, however, the self-consistent
calculationclearlygoesfarbeyond theârst-orderexpression.
128
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
IILIM PERFECT FERM I GAS
W eshallnow considerindetailadiluteFerm igaswith strongshort-rangerepul-
sivepotentials(tthardcores''l.l Thissystem isofconsiderableintrinsicinterest,
and italso form sthebasisforstudiesofnuclearm atterand H e3,asinitiated by
Brueckner.z The fundam ental observation is the following. A lthough the
potentialm ay be strong and singular,the scattering amplitude can be sm allfor
such interactions. For desniteness, the potential will be taken as purely
repulsive with a strong short-range core,thereby neglecting any possibility ofa
self-bound liquid. A ny realistic potentialm ustclearly have such a repulsive
core;otherwise there would be no equilibrium density and the system would
collapse. (See Prob.1.2.) ln particular,the nucleon-nucleon potentialhasa
repulsivecore arisingfrom the strongly interactingm eson cloud,and the He3-He3
potentialhasarepulsivecorearising from theinteraction between theelectrons.
SCATTERING FRO M A HARD SPHERE
To illustrate these remarks,considerthe scattering of two particles interacting
through astrongrepulsivepotentialofstrength J'o>0 and rangea(Fig.11.1).
A n insnite hard core clearly corresponds to the lim it P'o--,
w a:. The spatial
Fouriertransform ofthepotentialisjusttheBornapproximationforthescattering am plitude;itisproportionalto Prtjand therefore divergesfora hard core.
F(q)- Jc-ïq-xP'(x)#3.r-.
+.co
%
In fact,the true scattering am plitude isgiven by the partial-wave expansion3
X
/(k#)- 21+ l is,knj p (cosj)
, l= 0 i-.
.e s , ,
Since the Schrödinger equation is trivially soluble in the region outside the
potential,each phaseshiftcan be obtained explicitly with the boundary condition
Fig.11.1 Repulsivesquare-wellpotential.
'Wefollow theanalysisofV.M .Galitskii,Sot'.Phys.-JETP,7:104 (1958).
2K.A.Brueckner.TheoryofNuclearStructure.in C.DeW itt(ed.),ls-l-heM anyBodyProblem,''
p.47,John W iyeyandSons,Inc..New York,1959.
3Forthe basicelementsofscattering theory used in thissection,the readerisreferred to any
standard textbook on quantum mechanics,forexampl
e.L.1.Schih,'souantum Meehanics,''
3ded.,sec.19,M cciraw-l-lillBook Com pany,New York,1968.
ggaMl SYSTEM S
129
thatthe wave function vanish atr= a,and we:nd the well-known expression
3
(/fJ)2l*1
14
/c)= -(2/+ 1)!!(2l
ka-->.0,I> 0
1)!!
(jj.3)
3a(k)= - ka ka.
-+.0
where(2/+ 1)!!= 1'3'5 ''.(2/- 1)'(2/+ 1). Hencethescatteringamplitude
-
vanishesin the lim itofvanishing hard-corerange,asisphysically obvious. In
contrast,theFouriertransform ofthe potential(Eq.(11.1))isintinite foraIl
valuesofthehard-corerange.
Thisconclusion can beverifed in anotherway. ConsidertheSchrödinger
equation for two particles of m ass m interacting with a potential Pr. The
Schrödingerequation in the center-of-m asscoordinate system isgiven by
(V2+ k2)4(x)= y(x)/(x)
(11.4)
where x isthe separation ofthetwo particles,
uzP-(x)- &lU
(x)
!,(x)> z
-vreâ
h'
z
and znreu= à.
?'
rlis the reduced mass. In scattering problems,itis generally
convenienttorewriteEq.(11.4)asanintegralequation,usingtheoutgoing-wave
G reen'sfunction
3
.
fp@(x-y)
! eLklx-yl
Gt+)(X - y)= (d
2=p)3pce- kz- y = k-é jx- yj
(11.6)
Thefunction G(+)satissesthedifïerentialequation
(V2
X+ k1)Gt+)(x- y)= -3(x- y)
and G reen'stheorem then yields the following equation forthe scattering wave
functionkJk+)(x)representinganincidentplanewavewithwavevectorkplusan
outgoing scattered wave:
l
Jk+)(x)= efk*x- j#3yG(+)(x- y)p(y)/;+)(y)
Theasymptoticform of/k+1(x)isequalto
ikx
/k+'(x)...efk-x+ f(kt,k)ex
x.
-.
>.(zn
.
which desnes the scattering am plitude for a transition from an incidentwave
vectork to a snalwave vcctork'
/(k',k)= -44z4-1.f#3yc-ik'*y?
p(y)4k+)(y)
Equation (11.9)iscorrectforanyfinite-rangepotentialr(x),and wecan now
examine itsbehaviorfora hard core. In thislimit/k*'vanisheswhereverv
becom es insnite, so that the scattering am plitude rem ains snite. Thus the
lx
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
potentialdrastically altersthe wave function from its unperturbed form ,and
this eflkct is entirely absentin the Born approximation (Eq.(11.1)). Any
perturbation expansion ofEq.(1l.9)for such singularpotentialswillatbest
convergeslowly,and itisthereforeessentialto solve the integralequationexactly
if the m odiscations ofthe wave function are to be properly included. This
exactsolution evidentlycontainsa11ordersin perturbation theory.
SCATTERING TH EO RY IN M O M ENTUM SPACE
The preceding discussion hasbeen consned to the coordinate representation,
butitism oreusefulforthepresentanalysisto expressallquantitiesin m om entum
space. Sincethe resulting Schrödingerequation islessfam iliar,weshallderive
theexpressionsin som edetail. W ith thedesnitions
/k(p)= J#3xd-iP*X/U'(x)
r(p)= Jd3xe-ipex!
7(x)
(11.10X
(11.10:)
theSchrödingerequation(11.8)mayberewritteninmomentum space
/k(p)-(2=)33(p-k)-p
-,-kl
u-i
yJ(2
4.
'
.
')
j!
?(q)kzktp-q)
(1l.11)
whereEq.(11.6)hasbeen used ontherightside. Furthermore,itisusefulto
introduceam odihed scatteringam plitudewritten in m om entum space
yRk',k)- -4=/(k',k)= (2zr)-3Jd3q&(q)hktk'- q)
(11.12)
andEq.(1l.11)thenbecomes
k'k(P)= (2=)33(p- k)+ k2/Rp,
ak)
.
-
(1l.13)
jy
p .y.j.
Multiply Eq.(11.13) by r(q- p) and integfate (2771-3j'#3/7;an elementary
substitution then yields
- d?q t,(#- q)/(q.,k)
/
'(P,
k)=r(P-k)+1(zgv
j;-/
cc-qc.r,
.
n
.
.
(11.14)
whichisanintegralequationfbr/intermsofl.. Asnotedbefore,thescattering
amplitude/iswelldefinedevenforasingularpotential(r->.cc). IfEq.(1l.14)
were expanded in a perturbation series,each term would separately diverge;
nevertheless.the sum ot-a1lthe term s necessarily rem ains snite. N ote thatthe
solution ofEq.(1l.14)requiresthefunction.
/(q,k)foraIlq2 0,notjustfor
q2=k2:thisisexpressedbysayingthat/isneededûsofl-theenergyshell''aswell
as tton the energy shell.''
Ifthe potentialhasno bound states,as willbeassum ed throughoutthis
section,then the exactscattering solutionswith a given boundary condition form
acom pletesetofstatesand satisfy the relation
(2x)-3j#3:4k1)(x)/k4)(x')*= 3(x- x')
FERM ISYSTEM S
131
Thenum ericalfactorsherem aybechecked bynotingthattheexactwavefunction
obeys the sam e com pleteness relation as the unperturbed wave function elk*K
(compare the hrstterm in Eq.(11.8)). A combination ofEqs.(11.10c)and
(11.15)yieldsthe corresponding completeness relation in momentum space:
(2.
*-3J:3k/k(p)/k(F')*=(2,
03Xp- p')
(11.16)
Multiply Eq.(11.l2)by /k(P')*and integrateoverk;theabovecompleteness
relation leadsto
(2*-3J#3k.
A p,k)4k(p')*=!)tp-p')
whichmerelyrepresentsacomplicatedwayofwriting!). Thecomplexconjugate
ofEq.(11.13)maynow besubstitutedintothisrelation:
d3k x
.
1
L'
(P-P')=/(P,P')+ (2=)3J(P,k)J(p',k)*ka- p,c u
.
Butthepotentialishermitian and thereforesatisses
rtp- p')*= ??(p'- p)
which hnallyyields
JRp,
p'
)-.
/
Rp',
p)*-J(d
Jt
)J(p,k)/(p',
k)*(,c-sla.
j.y-kz.y),
z.yj
(1l.l7)
Ifthem agnitude ofpisequalto them agnitudeofp',theprincipalpartsin
Eq.(11.17)vanish,andweobtain
JRp,p')-yRp'.p)*--2=f(2zr)-3J#3/c.
Ap,k)JRp',k)*&(p2-k2)
Ipl- Ip'l (11.18)
where theradialk integraliseasily evaluated. If,in addition,the potentialis
spherically sym m etric,then the scattering am plitude./i
safunctiononlyofpl
and 9.j',and the leftside of this relation becomes 2fIm .
/*. The resulting
expression
Im Jtp,p')= 16Pz
-
zr
JD
k)./RP',k)*
k/RP.
.
'k1= 1P'
=)1P'!
(jxI
t'= ?J
(11.19)
isa generalization oftheordinary opticaltheorem forthescattering amplitude.
LA DDER DIAGRA M S AN D THE BETHE-SA LPETER EQUATION
Thepreviousdiscussion hasbeen restricted to the scattering oftwo particlesin
freespace,and wenow turn to the m uch m orecom plicated problem ofa dilute
m any-particle assem bly interacting with singular repulsive potentials. The
hard core clearly precludesany straightforward perturbation expansion in the
strength oftheinterparticlepotential. Instead,itisessentialfrstto incorporate
132
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
the efrectofthe repulsive potentialon the wave function oftwo particles in the
m edium '
.only then can we considerhow the m any-particle background aflkcts
the interacting pair.
A lthough the potentialissingular.a diluteFerm igasstillcontainsone sm all
param eter,namelykpa.wherekpistheFerm iw'
avenum ber,and a isthe scattering
length (equalto the particle diameter for a hard-sphere gas). W e therefore
expectthe ground-state energy to have a series expansion ofthe form
E
hlk2
h = ..2-,,7Fj,
gf+ BL.ya-pC(kya)l+ '.'j
(I1.20)
which should bemeaningfuleitherforsmallscattering length (a -.>.0)orforlow
density (ky-,.0). In thlssection wecalculate the t
irstthree coemcientsin this
series. In addition, the techniques developed here can be used as a basis for
realistic theoriesof Ferm isystem s atphysicaldensities.
A trtlly inGnite repulsive core introduces certain artihcialcom plications,
because every term ofany perturbation expansion diverges. W e shall instead
considera strong butf
inite potentialjFig.ll.1.w ith lz
':< :cJand pass to the
limit IG -->.:
x.only atthe end ofthe calculation. As shown in the fo1lowing
calculations,this procedure yields hnite answers that are independent of U(
,.
Forsuch a ûnite potential,the two-particlescatteringequations(l1.8)and (II.9)
in freespacecan beexpanded as
(Jk
çb)(x)r
su
ré'ik*x- )'(.
/3)'Gç-F)(x.
-y)ljy)é'ikey
+.fd?y#3zGt+J(x- y)!'(y)Gt*1(y- z)l.
(z)faik*z+ ... (11.21tz)
/(k',k)= --(4=)
)-lj'#3r't
rz-ik'erI.
,
(y)/k*'J(y)
.
-
The term sin the wave function have asim pleinterpretation asthe unperturbed
solution plus propagation with one or m ore repeated interactions. W e may
evidently represent the terms in the perturbation series (Born series) for
4=/(k'.k)diagrammaticallyasindicatedin Fig.11.2(therulesforconstructing
these diagram sfollow by inspection). These are notFeynman diagrams but
-
-
4.r'
.f
lk'
,k)
h)
Fig.11.2 Perturbation expansion fortwo-bodyscatteringamplitudein freespace.
FERM I SYSTEM S
13z
areagainjusta wayofkeepingtrack ofthetermsthatcontributeto thetimeindependentperturbation seriesforthescattering am plitude. A weak repulsive
Potentialcan beadequatelydescribedwiththesrstfew termsofEq.(11.21),but
zsfrongrepulsivepotentialrequiresalltheterms. Asisevidentfrom Eq.(11.21c)
the higher-orderterm s representthe m odiscation ofthe wave function by the
Potential,and the sum oftheseriesgivestheexactwavefunction.
Inasimilarway.thehrst-orderproperself-energyhttkt(Fig.10.1)istotally
inadequateforastrongrepulsivepotential,and wem ustretaina selected classof
higher-order Feynman diagram s. From the presentdiscussion ïtisquiteclear
which diagram s m ustbe kept;every tim e the interaction appears.itm ust be
allowed to actrepeatedly so asto include the eflkctofthe potentialon the wave
+
w.
+
+
>
+ ''' +
+
<-
+ '''
.>
Lowestorder
Sœond order
Ladderdiagrams
Fig.11.3 Sum ofladderFeynman diagramsforproperself-energy.
function.therebyyieldingawell-definedproductè'/. lnotherwords.therelevantquantity in a two-particle collision is the two-body scattering am plitude in
the presence of the m edium .which rem ainswelldesned even for a singular
two-body potential. W e therefore retain allthe ladder diagrams indicated in
Fig.l1.3.
'thisehoiceclearlyrepresentsa generalization ofthe abovediscussion
becausetheterm sin Fig.11.3denoteFeynm an diagram sandhencecontain b0th
holeand particlepropagation. ln particular,we sum only theladdersbetween
G reen-s functions with arrows running in the sam e direction. Sincethetwoparticleinteraction isinstantaneous,thissetofdiagram sincludesasa subseta1l
thoseprocesseswhere both interm ediate fkrm ionsareparticlesabove the Ferm i
sea atevery step. Such particle-particle contributions come from the particle
partofthe Feynm an propagator,which propagatesforward in time.
A sshown below,thediagram sin Fig.ll.3 suë ceto obtain the srstthree
termsin Eq.(11.20)forahard-sphereFermigas. Thisresulthasadirectphysical
interpretation:the firstterm in theexpansion isthe energy ofa noninteracting
Ferm i system . The second term , which is linear in the scattering length.
represents the forward scattering (both direct and exchange) from the other
particles in the m edium . This identiscation follows because the 1ow density'
(/t
's-->.0)allowsustoconsideronlylow-energycollisions.wherethefree-particle
scattering am plitude reducestoa constant
f(k,k')--.
>..
-a
.
k= k'-->0
(11.22)
1%
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
AlthoughEq.(11.22)maybederiveddirectlyfrom Eqs.(11.2)and(11.3)inthe
caseofhard spheres,italso servesasageneraldesnition oftheJ-wavescattering
length a. In thepresence ofthemedium,however,the actualscatteringam pli-
tude disers from f Y ause the other particles limit the intermediate states
available to the interacting pair. The Pauliprinciple restriction srst appears
when a particleisexcited abovethe Fermisea and isthen de-excited;ifthe sum
ofladderdiagrams in Fig.ll.3 is reexpressed in termsofthefree scattering
length,thisesectgivesacorrection oforder(kralltotheground-stateenergy.
Any otherprocessthatcontributesto the ground-state energy involvesat
Ieastthreedistinctcollisionsand thusyieldsa contribution oforder(#sJ)3to
Eq.(11.20). In particular,considertheFeynmandiagramsshowninFig.11.4,
#
)
+ ....
Fiq.11.4 A class of additionalcontributions to Z*.
neglM ed in the ladderapproximation.
where the shaded box denotes the sum ofladder diagrams. These processes
clearly include the scattering ofan intermediate particleand hole,which really
representsthe transferofan additionalparticleinside the Ferm isea,slling the
originalhole and leaving a new one in itsplace. Thusa collision between a
particleandaholealwaysinvolvesanextraparticleand introducesanextrapower
ofkpa. Itisevidentfrom Eqs.(11.2)and (11.3)thattwo-bodycollisionsin
relativepstatesalsoleadtocorrectionsoforder(kFaj3,whichisagainnegligible
in ourapproximation. Weshalljustify ourchoiceofdiagramsin moredetail
attheend ofthissection.
Beforeproceedingwith thedetailed analysisofthesum ofladderdiagram s,
wenow show thattheforegoingdiscussionimmediatelyyieldsthesrsttwoterms
intheseries(11.20). Considerauniform system ofspin-èfermionsinteracting
through a spin-indem ndent nonsingular potential. Then the lowest-order
ground-stateenergyisobtainedfrom Eqs.(10.21)and(10.22)
E 3âak/x l kr :3k krd'k'
P=Jlm#'
612J (2,
*3J (2,
036
2ZP)-Z(k-k'
)1 (ll.
23)
13:
FERM I SYSTEM S
Foranonsingularpotential,however,Eqs.(1l.9)and(11.22)show that
4=h2
F(0)> .
f#3xF(x)= -
.&(k,k)-*
4*:2
(1l.24)
Js
where the subscript B indicates the Born approximation obtained with the
substitution4k+1(x)-.
->.eik*x. Itisclearthatourdescription ofscatteringfrom
theparticlesinthemedium can beimproved bythesimplereplacement
as-+.a
(1l.25)
Here a t
's ?/?t2actualscattering Iengthforfree two-particle scattering,which
remains welldesned even for singular potentials. ln the low-density lim it
wherekr-+.0.wecanfurthermoreapproximate
F(k- k');
k;F(0)
'
#
t,
#
k+ q
k- p
+
k
p
q
k
k
+
p+ q
k- q
p- q
k- q -p
#
p
p
-q
p
Fig.11.5 Firsttwo ordersin ladderapproximation to X*.
underthe integral,and Eq.(11.23)thusbecomes
E = 3hlk; 1N4,42aN
J-iQ N + i'P m ï
The standard relation (3.29)between the density and the Fermiwavenumber
NIV= :)/3w.
2thereforegives
E :2:./.3 2
= -jyj
tj+gksa+''
(1l.
26)
An equivalentresultw asderived by Lenzlfrom the relation between theindex
ofrefraction and theforward-scatteringam plitude.
W ith these rem arks in m ind.we turn to ourbasic approxim ation.which
is to retain only the Feynm an diagram sofFig. l1.3 in evaluating the proper
self-energy E*. To clarify thevariousfactors,we shallinitially concentrate on
thesrst-and second-ordercontributionsshown in Fig.l1.5 in momentum space.
'w .Lenz,z.Physik. 56:778(1929).
1a6
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
ltisstraightforward to calculateZ* using the Feynm an rulesofSec. 9,and the
frst-ordercontributionisgivenby(compareEq.(9.23))
âY1)(#)=-2ïL%(0)(2rr)-4Jd*kG0(k)eiknT
+ 1*()
2,r)-4j#QG0(/c)Uclk-p)eikzvl (11.27)
where a spin-independent interaction has been assum ed. The second-order
contribution introducesthefollowingadditionalelem ents:
1.Two extra factorsG0
2. O ne extra interaction line Uo
3.Oneextraindependentfour-momentum (compareFig.11.5)
4.Oneextrafactor(i(h)(2.
n.)-4
J't
p.
j
ht:*4J,)=
/v
.+
p
k
p
Fig.11.6 Properself-energyinladderapproximation.
Thesefactors can be com bined with the Feynm an rulesto yield the second-order
contribution
hXn&jlpl= 2h-142,
77
1-8(d*kG'0(#)(dnq(z
rotq)G0(p-q)GQ(k-yq)Uoq-qj
h-l(2'
;r)-Bfd*kG0(/f)fdkqU()(ty)G0(p +.q'
)G0(# -q)&/
r0(/(-q- p)
-
It is clear that this procedure can be carried out to alI orders and the general
form ofY*(r)willbe(seeFig.11.6)
hY*(p)= -2/(2.
,)-4f(/4/fG'0(/c)I-tp/fipk)+ j(2rr)-4fd4kG0(*)Lykpipk)
(1l.28)
wherew'
e have desned an efective two-particleinteraction r(pipz'
,p)p4)that
m ay be interpreted as a generalized scattering am plitude in the m edium . ln
J1
N
Jl
F(:1#2,#3p4$ = N.x .
pz
q
z'
#1
Z
w + /1- q
#2+:
r pj- p? -N'/4
pj
A ;7)- q- p?
#3
#4
Fig. 11.7 Series expansion for efective interaction in ladder
approximation.
FERM I SYSTEM S
1a7
thisparticularexample,F'includesthe sum ofrepeated (ladder)interactions
(Fig.11.7)and hastheform
jnlpbpzipjpè- Uépk-pjt+iâ-'(2,r)-4fdkq&(,(f
?)G0(pl-q4
x G*(pa+ç)&e(#,- q-#3)+ '.' (11.29)
Thissum ofladderdiagram snow can berewritten asan integralequation for r
thatautomaticallyincludesaIIordersgFig.Il.8andEq.(1l.30)j
ïjptpzipsn)- Uolp,-p3)+ m-l(2'
rr)-4fd% f.
&(ç)
xGolpb- qsG't
'(J,c+q4r(:l- qspz+qipsp.t (11.30)
Inanalogywithsimilarequationsinrelativisticseldtheory,Eq.(11.30)isknown
astheBethe-salpeterequationi(moreprecisely,theIadderapproximationtothe
pj
N
pï
N
#1- q
#2
Z
#1
N
:3
A
pj
N #4
q
#z
z'
72+ q
72
A
+
74
A
p?
N #4
Fig.11.8 Bethe-salpeterequation forefrective interaction.
Bethe-salpeterequation). lfthisequation isexpanded in perturbation theory,
assuming &cissmall,weobtainthesum ofallladderdiagrams(Fig.l1.7). The
srsttwo termsprecisely reproducethose ofEq.(1l.29),which ensuresthatthe
signsand numericalfactorsarealso correct;inparticular,thefactorilhisjust
thatassociated with the extra orderin the perturbation Ufj.
The calculation ofl1* isnow reduced to thatoffnding the solution F to
Eq.(11.30). Although l-'is related to a two-partiele Green's function,it is
m ore usefulto exploitthe sim ilarity between r and the scattering am plitude.?
in freespace. Indeed,to lowestorderin thepotentiai,P isjustequalto the
Fourier transform Uv. W e shallnow pursue this analogy and introduce an
eflkctivewavefunctionQ fortwoparticlesinthemedium (compareEqs.(1l.12)
and (1l.13)J:
1E.E. Salpeterand H.A,Bethe,Phys.ReL'
..84'1232 (1951).
13g
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
Thisrepresentation forP agreeswithEq.(11.30)onlyifQ satishestheintegral
equation
Qçptpzipqp.j= (2,44ôt41(,l-#3)+ ih-,G'
0(#j)c0(pa)(2=)-4
x fd4
q&(,(ç)Qlpj- qnpz+q;p,p.4 (11.32)
.
The labeling and ordering ofthe momentum variables requires considerable
care,and thereaderisurged to verifytheseequationsin detail.
Itisconvenientto introducethetotalcenterofmasswavevector
P = /)1+ fz= pq+ p6
where thelastequality follows from theconservation oftotalfour-m om entum
in a hom ogeneoussystem ,and therelativewavevectors
# = V:1-#2) p'= V#3-Jh)
lnaddition.theinstantaneousinteractionmeansthat(70(4)= &c(q)islhdependent
offrequencye,and wecanperform thefrequencyintegralin Eq.(11.32)fora
hxed center of m ass four-m om entum hp of the interacting pair. Integrate
Eq.(11.32)overtherelativefrequency(2,
r8-1j'dpoanddefnethequantity
r*
z(p,p',#)H(2=)-1jPdpzQ(1'
#+p,I'
.
#-p;'
è'.
#+#','
L'
P-p'
4
-
(2=)-1(p#lloQlptJ'2,
.,3,.)
A sim plerearrangem entthen yieldsan integralequation fory
y(p,p'.#)- (2=)3(
5tp--p')+ m-1(2rr)-1fdpzG0(!.
# + pjG0(!# -pj(2.
,)-3
s fd3
q&a(q)à'
lp- q-p',#) (1l.34)
which,asshown in Eq.(1l.39),isvery similarto thescatteringequation in free
space (Eq.(1l.l1)). lfthisequation isiterated asan expansion in Uo,each
term dependsexplicitlyonthevariables(p,p',#),thusjustifyingournotationin
Eq.(11.33).
Itisnow necessaryto evaluatethecoemcientof-thelastterm in Eq.(11.34).
Sinceeach G0hastwo terms(Eq.(9.14)J,theintegrand hasfourtermsin all.
.
Two ofthese term shave b0th poleson thesame sideofthe realpoaxis;in this
case,weclose thecontourin the opposite halfplane,showing thatthese term s
make no contribution. ln contrast,each ofthe rem aining two term s has one
pole above and one polebelow the realaxis. These contributionsare readily
evaluated with a contourintegral,and weEnd
i * ocn(!
j2,, ,+,)ce(!.
,-,)-'
f1
'P
hh+p
-le
-d
t;
'+
rp
)#fe
l
'
#
lP
p-p+
p.i
l
h
-kè
.
-
#(l-sh1
)-0-Lk
Fp--pI
JP
.
p0IJ
-P
-e+
$.ep+
'p
eî
-.
-iT-$) tjj'yj;
-
FERM I SYSTEM S
4z9
This expression has the physical interpretation that the pair of interacting
particlesin the interm ediate statesofFig.11.3 can propagateeitherasa pairof
particlesabovetheFermiseafthesrstterm inEq.(1l.35))orasapairofholes
below theFermisea(thesecond term inEq.(11.35)). SinceFeynmandiagrams
contain allpossible timeorderings,both modesofpropagation areiMcluded in
thediagram ofFig.ll.8.
Theform ofEq.(11.35)canbesimplisedby introducingthetotalenergy
h2/2
E - hpo- 4m
(1I.36)
oftheinteractingpairin thecenterofmassfram eandthefunction
X(P,P)e l-?10
1p+p-ntp-p
(1l.37)
wherenk=ptks-p)istheoccupationnumberintheunperturbedgroundstate.
ThusS(P,p)= 1ifb0th statesI.
P + p are outsidetheFermisea,A(P,p)= -1
ifboth areinside,and A'(P,p)= 0 otherwise. Withthis notation,Eq.(11.35)
assum esthecom pactform
' * ù e e+
i
ja.c(.
# pbc%!.
#-p)-s-scpZ
Xjm
tP.
j
'
Pj
',x(p,p)
(11.38)
andEq.(1l.34)reducesto
y(p,p',#)= (2=)33tp- p')+ s
X(P,P)
.
d%
J(;x)3
jjzpaym .
j.uxtpyp;
X Uo(q)ZP- q,p',#) (1l.39)
Correspondingly,P may bereexpressed in termsofcenterof-massand relative
wavevectorsusingEqs.(1l.31)and(11.33):
1Xp.p'.#)H f'
(i'
# +p,i' -p;!'#A-p','
è'
# -#')
-
(2,,)-3jJ3t?r
srgq)y(p- q,p',#)
(11.40)
WehavealreadynotedthatEq.(1l.39)issimilarto thescatteringequationfor
thewavefunction/p,(p)oftwoparticlesinfreespace(Eq.(11.11)). Thepresent
equation ismorecom plicated,however,because theexclusion principlerestricts
the available interm ediate states through the factor N , and also because the
function l-'mustbeevaluated forallvaluesofthefrequencyPoEcompare Eq.
(11.28)).
GA LITSKII'S INTEG RAL EQ UATIONS
U ntilthispoint,the equations have been written in term softhe interparticle
potentialUo. Such an approach can neverdescribe an insnite repulsive core,
and wenow follow G alitskiiby rewriting Eqs.(1l.39)and (1I.40)intermsofthe
scatteringamplitude/fortwoparticlesinfreespace. Indeed,thelowestapproxi-
140
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
m ation to l-'isproportionalLo.?andthehigher-ordercorrectionsari
seexplicitly
from the many-particle background. It is convenient to desne the reduced
variables
v> m f70h-2 t.JBm Eh-2
(l).41)
and to consider a simplihed version of Eq,(1I.39)obtained by replacing the
exclusion-principle factor N by l. N ote that this substitution becomes exact
as ks -. 0 and thus describes the low-density 1im it. lf à'
()denotes the corre-
sponding solution to Eq. (l1.39) w'
ith h'-=Is m ultiplication by e- p'- l
h
yieldstheequatioll
Thisrepresentatiol
liseasilyveriéed by substituting Eq.(1l.44)i1)to Eq,(11.42),
and using Eq.(l).43)and the completenessrelatiol
)tEq.(11.164). A combinationwiththecomplexconjugateofEq.(1l.13)thenyields
and itfollowsim m ediately thatl-'
llhasthe illtegralrepresentation
whereEqs.(11.12)and(11.41)h.avebeenused, ThisexpressionhastheimportantfeattlrethatitcontainsonlytheG'
ee-particlescattering anlplittldesalld thus
rem ains m eaningfuleven fora singldar repulsive potential.
1*1
FERM I SYSTEMS
Tbefunction lnadependson theparam etere. lfwesete= p'2,then
rrll''lâ-2= .J(p,p?
)
(11.46)
which isjustthefreescatteringamplitude. MoregenerallysEq.(11.45)determ inesraforscattering ofl-theenergy shell,nam ely,forvaluesofp'1# k. N ote
thatwe also need the quantity./of
l-theenergyshell,forallvaluesofitstwo
m om entum argum ents. ln a free scattering experim ent,however,the scattering
am plitude ./ is measured onl
y for equalmagnitudes ofthe two momentum
arguments jp)= 2p'!(we here consider only elastic scattering). Thus the
simplicity of Eq.(1l.45) is slightly deceptive,for the evaluation of./
-'ofl-the
energy shellrequiresa detailed model.such asa potentialP'(x). Nevertheless,
Eq.(11.45)isvery useful.
W e now try to solve for the fullscattering function z in the m edium.
W ith the reduced variablesofEq.(1l.41),theexactEq.(11.39)becomes
pj- .-z
ytp'
p'.
E:-p.Y+J-putc,
??x)tP,
pjj(2
d.3
y
1t
,
(q)y(p-q,p,,p)-(a.)33tp-p,
)
. .
and a slightrearrangem entyields
xtp.
pz,
p,
4-6-p1
j-g-p
z?j(2
#,r
3
ç
,3r(q)k(p-q,p,p)
=(2,
.)3,
!tp-p,
)+(6.
-.p)+(
-P
yx
!
)
?
N)(p,p;-,
y-g1
z+j,
yjp
D1Ia(p,p,.
p)(jj.
4,
y)
Comparisonwith Eq.(11.42)dividedby6-.p2+ i.
r
)showsthattheoperatoron
theleftsideofEq.(1I.47)isjusttheinverseof;o,which meansthaty can be
expressed in term sofxoasfollows:
d3k
:(p,p',#)= :0(p,#,P1+ (2 3:0(#X,P)
=)
x
N(
k)
1
m
,
ï P,
E - i'+ ï
n#(P,k)- e- kz+ i.
rljyjrtk,p.Pj
Thisequation can beveril
ied bycarrying outtheoperation indicated on the left
sideofEq.(11.47)and byusingEq.(11.42). W enow taketheconvolutionwith
t'EsteEq.(11.40j3,which yieldsourfinalequation forthescattering amplitude
in themedium
Intp.
p',
#)-rn(p.
p',
#)+j(c
d=
'k
)aro(p,
k,
#)
x(p,k)
'le-kz
+,
'?.
N(P,k)e-kl
z+iyjj
m'
is(k,
p,w)(jj.
z
s)
-
Since Eqs.(11.45) and (11.48) are expressed in terms ofthefreescattering
amplitudes,wemaypasstothelimitofanininitehard-corepotential(#%--+(
:
s).
14a
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
Asnotedbefore,however,/(p.p')mustbeknownforaIlvaluesofpandp'(q#theenergyshell),whileitcan bemeasured onlyon theenergy shell. Roughly
sm aking.the quantity ln()(Eq.(11.45))containsthe e/ects ofscattering off
theenergy shell,while Eq.(1l.48),which isstillan integralequation for I',
incorporates the exclusion principle in intermediate states. W e shallrefer to
these equations as Galitskii's integral equations.1 (These equations were
derived jointly with Beliaevqzwho studied similar problemsin an imperfect
Bosegas-)
THE P:OPER SELF-ENEBGY
Fora low-density Fermigas.Eq.(11.48)can besolved iteratively asapower
series in kFa <t 1. This expansion is possible because the integrand vanishes
when thevectorsI.
P + k both lieoutsidetheFermisea;sinceweareinterested
inenergiesoftheorderes,thelastterm ofEq.(11.48)canbeestimated (apart
from numericalfactors)asï'okpmvhz. Thustht orderofmagnitude ofthe
correction willbe given by (r- rp/r;4Jkrmlnqlhl;z:krîxfI-stz<:
t1,where
.
Eqs.(11.46)and(11.22)havebeen used.
Before attempting a carefulexpansion ofEq.(11.48),we notice thatthe
fullsetofvariables(p,p',#)isneverneeded in anycalculation. Indeed,a11that
isrequiredistheproperself-energy(Eq.(1l.28)J:
hX*(p)=-2f(2zr)-4Jd*kG0(#)Vpkipk)+ï(2zr)-4Jd*kG0(1)Imlkpipkj
(11.49)
Dehnetherelativeand totalwavevectorsand frequency
V#- k)= q
P+ k = P
pz+ Vc=Pz
(1l.50)
Since l''dependsonlyon thevariablesshown in Eq.(11.40),the properselfenergy can berewritten as
:x*(,)--2f(2.)-.Jd.knnlk)r(q-q,e)+f(2=)-4JdnkGzlktr(-q,q.#)
(1l.5l)
W enow expand 1''to second orderin kra. By thedesnition ofthew-wave
scatttring length a.the J-wave phase shift has the long-wavelength expansion
3:= -ka+ O((*w)3J
Furthermore,theIeadingterm ofthefthpartialwaveisgivenbyd,= O((kJ)2'+1J.
:V.M .Galitskii,loc.cit.
zS.T.N liaev,Sov.Phys.-J.
ETP,7:299(1958).
143
FERM ISYSTEM S
Thusthescatteringamplitudehasthelimiting value
* 21+ I
f(k,
k')=F
g E'f'
llsin3,Fltcos0)
l=0
k-h(1+ à3o)30+ (412a5)
= -J+ ika2+ O(k2c3) j
k$= 1k'!-+.0
=
It is rem arkable that this Iong-wavelength lim it depends only on the s'
-w ave
scatteringlength;forahard-spheregas.aisjustthediameterofthesphere,but
Eq.(11.52) isclearly more general. ln particular,the scattering amplitude
always reduces to a constant as k and k'vanish.w hich enables us to take the
leadingcontributionto/tksk')ofl-theenergyshellinthelong-wavelength limit.
The second term in Eq.(1l.52) is pure imaginary,and ensures thatthe generalized unitarity equation (1l.19)issatissed to orderal. To this order,the
correspondingamplitude/lEq.(1l.12))becomes
J'f
kf4=a- 4=ia2k
JkJ= Jk'p.
-+.0
(l1.53)
A snoted atthebeginning ofthissection.the expansion param eterforan
imperfectFermigasiskya,whichissmallforeitherIow density(kF-'0)orsmall
scattering length (tz->0). In this Iimit,Galitskii's equations can be used to
evaluate the scattering amplitude in the medium In(q,q,#) to order (kpa42.
Equations(11.45)and(1l.53)togethergive
-iro(q,q-#)
h
=
d3k'i
1
l
'
j+...
4na - 4=iqa2-i
-(4.
;ru)2 (ymi
-.
)j,s .. .js.
j-m j.
tj-r-y(;a.
-c
j2u-jy
y
=
#3/
c'
,,
l
,
?/
4=ë+ (4=w)2 (2 j
,
-j- .-+ -,j. c + .'.
rr) E- k + ln k - q
where.
@ denotes the principalvalue and the im aginary partcancels the term
-
4=iqa2. Thisexpansion to ordera2 now can be combined with Eq.(11.48)
to obtain
m r(
d?k'
zV(P,k')
.
'
#
.
-j.+ I.
kpj-yggz t
D q,q,#)= 4=a+ (4=*2 (2=)) 6--k,
T-y;(sy-z + ...
,
.
(ll.54)
sinct the termscontaining (E- k'2+ f,
j)-1in theintegrand cancelidentically.
NotethatEq.(11.54)isanexplicitrepresentationforF(q,q,#)intermsofknown
quantities'
,furthermore,itiseasilyseen that
F'
tqsqs/')= F(-qsq,#)
(1l.55)
144
GnOOND-STATE (ZERO-TEMPERATURE) FORMALISM
tothisorder. Thusthesecondterm ofEq.(11.51)cancelsone-halfofthehrst,
andweobtain
ht*(p)=-f(2=)-4jd4kG0(k)l>(q,q,#)
(11.56)
which is valid through order a2. This cancellation typifesthe generalresult
noted in Sec.10thatthedirectand exchangetermsarecomparablein magnitude
forashort-rangepotential.
Theexpansion ofl-'inEq.(11.54)leadsto acorresponding expansionof
the self-energyin powersofa
âX*(;)-âYh)(p)+httLblpt+ ''.
where
d*k
4=ah2
,
âZ?'
k3
(;'
)--ij(
-jp)
-4Goçksm ,.
'09
d%k (
) i,nn'lnralzâ
htt
qlçp)=.
-,
ij(z.)4G(k)e m2j(
d
zs
'k'
3
-
x
k')
@ a
,#(
a P,
, -> ,a.
e- k + iTN(P,k) k -q
and the subscriptshere denote powersofa,notordersin perturbation theory.
Thefrstterm iseasily evaluated byclosing thecontourin the upperhalfplane
Itheconvergencefactorcikn'
n'playstbesameroleasinEq.(11.27))
âi!
d3k 4nah2 dk yy
oy;,
hk- /
) ptks- kj
(#)=-ij(s)
jm-JV-e ka
-o,,f+s.
'
pj+ko-g
o,-i
y
4,- 52 #3k
J(2,43oçkr-O
)
-
-
m
hlk;2/csJ
(1l.57)
m 3.
)
7.
Thesecond term isconsiderably m orecom plicated.becausethefrequency
= - .- -
kzappears in the denominator through the combination e= mlvh- iP2=
mpzlh+ mkvlh- .
JP2. It is precisely this dependence that necessitated the
solution for r os the energy shell. The evaluation ofthe ko integralisvery
similartothatofEq.(11.35),and wehnd
âX)2)(p,P0)
:2
z z #3kdjk' p
tj-s- kj$j
.
jP+k'1-ks)%IJP-k'(- ks)
- .
=ul6.
=aj.
js-y-(- t
ngvj
y.-jy
)z.
j.ga.
-g,
z.
j.yy
.
+ hk-k'yyj-jks.
S JP+k'1)%kF- t.
iP-k'1) .
@0(kF- k
.l
- - - --
tytgnjs- jg-c+ qz- k,c- j,y
1
- qz- k,2
(1l.58)
FERM I SYSTEM S
145
whichhasbeen simplihed bywriting gcompareEq.(1l50))
hkl hP2 hC2 hL2
2- k% m àrn
Equations(1l.57)and(1l.58)togetherexpressX*asaseries, including allterm s
oforder(kpall.
.
-
-
The single-particle G reen-s function for a dilute Ferm i gas now takes
theapproxim ateform
of
(l1.60)
hpl
J'o- 2'
,-Olkeaqj
m (1sve can therefore setJ
pc;4:hp2,
'2lz?in the lastterm ofEq. (ll.601,u'
hich isalread-j'
oforder(kya)2:thisconstitutesa majorsim pliscation. and the cxplicitsolution
then becom es
=i2p2 h2k2j(!
*t/3k#3i-'
g
js
.
j..-gy
6jjyky(
Z-'
-16zr2(kya)2j
-jyx
-l
j
-.
9(
f)47(IP - k' - l)û('IP - k''- 1)
. 1- /
0(k - 1)0(l- i
. P - k')0
(1- 1
. P - k')
qc .-- k .
@.
- t.
,
y
.
-yt
k4
y
- -7$1
qj---r
s - ()(k).a3)/
,.
-
P = p- k.q= J(p- k) (11.62)
w here:heintegralhasbeenrendered dim ensionlessb)'expressinga1lu
i
r
avevectors
n term sofkr.
This equation resem bles a second-order expansion obtained with ti
independent perturbation theory for an interparticle potential 4
me:4 2tz'
.
,?2 in
m om entum space that is chosen to reproduce the correct -ç-wave scattering
:*6
GROLJNO-STATE (ZERO-TEMPERAIURE) FORMALISM
am plitude. The quantity in braces is proportionalto the proper self-energy
and has thephysiealinterpretation shown schem atically in Fig.1l.9. (These
are now m eantto be diagram s oftim e-independentperturbation theory, the
analog of the Goldstone diagrams. An arrow running upward denotes a
particle,whilean arrow runningdownward denotesahole.) Theterm oforder
kya in Eq.(1l.62)and in Fig.ll.9J representstheforward scattering from the
otherparticlesinthemedium withaneFectivepotentialnnhlalm. Thesecondordercorrectionsin Eq.(11.62)incorporatetheeflkctofthemedium on the
intermediate states. Oftheselatterterms,thefirsttwo representtheprocesses
indicated in Fig.11.9: and c,whilethelastterm Lt
f
/hql-#'2)-1jmustbçsubtracted explicitly because the realpartofthe exactscattering amplitudew ould
4n'h2a
#m
p.
4zrâ1a
p
221:.+ k'
t
p
lP+k'
+
y1P- k.
k
4. 2#
k
1pk'
)
'
.
p) l
p
4.,r,?a
)m
P
4x:2a
m
(c)
Fig.11.9 Schematicexpansion oîproperself-energy.
be given by -a to this orderifthere were no slled Ferm fsea asa background.
The term sin Fig.11.9: and cdenotethe following physicalprocesses. ln the
firstcase,theincidentparticlecollfdcswith a particlein them edium ,exciting it
to som e stateabove theFerm isea,thereby leaving the hole in the m edium ;the
sam e two particles then collide a second tim e,bringing the system back to its
initialstate ofa Ferm isea and the incident particle. The second case is an
exchange process. Tw'
o particlesin the medium interact'
.they are b0th excited
above the Fermisea,thereby leaving two holesin the m edium . The incident
particle collidesw'
ith one oftheexcited partfcles.and thesetwo partfelesthen fill
the two holes. The system thusreturnsto its initialstate,the only alteration
beingtheexchangeoftheinitialincidentparticlewithoneoftheexcitedparticles.
PHYSICA L G UANTITIES
W enow examinetheimplicationsoîEq.(1l.62).
l.Lifetimeofsingle-particleexcitations:Itisevidentthattherealpartep
containsa shiftin the single-particle energiesforparticles with wave vectorp,
FERM I SYSTEM S
147
whereastheimaginary partypleadsto a snitelifetime (compareEq.(7.79)).
W hen theindicated integration in Eq.(11.62)iscarried out,wel
indl
â
h2/
c/2
z pcs- p l
yp-zm.
-(krJ)(k,)Sgn(kF-p)
which isvalid fbr 1/ -/t'sl.
,
.
:kF. ln accordance with the generalLehmann
representation (Fig.7.l) the pole lies below (above)the realaxis forp > Vs
(p < /fs). Since ys vanishes like (p - kr)l, the lifetime becomes insnite as
p.
->.kr, and the condition ep- J.
t%>hyp is satissed (compare the discussion
leading to Eq.(7.79)J.2 These long-lived single-particle excitations are often
known asquasiparticles. Note thatyv isproportionalto (ksJ)2,because the
.
snitelifetimereqectsthepossibility'ofrealtransitionsandthusinvolvesIf 12v:a2.
2. Single-particle excitation JpEc/rur?
l:The present quasiparticle approxim ation
G(p,p());4:(p0- k'
ph-'- iyp'
)-1
(1..64)
impliesthattheground stateremainsa Fermiseafilled uptowavenumbe ks,t
but with a diflkrent dispersion relation Ep. Since yv changek sign at ky,the
Lehm ann representation showsthatthe chem icalpotentialisgiven by y = càs.
Itistherefore necessary to evaluate ep atthe Ferm isurface. A lengthy integra-
tionwith Eq.(11.62)gives
(11.65)
which wasérstobtained by G alitskii.l
Close to the Ferm i surface. the ellergy spectrum can bc expanded in a
Taylor series
1V . M .Galitskii,loc.cl
't.
2Thisdetailedresulttypisesageneraltheorem ofJ.M .LuttingerLPl;)'s.Rcl'
..121:942(1961))
thatIm E* vanishes like(t,
p- /
.
zâ)2nearthe Fermisurface.u'
hich holds to allorders ofperturbation theory.
tA moredetailedevaluation based on Eq.(11,59)show'
sthatdistribution function npisslightly
altered.butthis does not afrectoursubsequentresults (V.A.Bell'akov,Sot'.#/?y.
j
,-Jf-F#.
13:850(1961)).
148
GROUND-STATE (ZERO-TEMPE9ATURE) FORMALISM
which dehnestheeFectivem ass
m+-nzk,jp
De
pp!
j
ks)-'
(11.
6,
7)
in term softheslopeof6patthe Ferm isurface. A detailedcalculationwith Eq.
(11.62)yields
m ...- .
8
l+ 15.nz(7ln2- 1)(/csJ)2
m
(11.68)
correcttoorder(ksX7. NotethefollowingfeaturesofEq.(11.68):
(J)m*hasnotermslinearinkpa,which reqectstheconstantvalueofE)kjin
Eq.(11.57).
(/8 m*determinestheheatcapacity ofthe system in thezero-tem peraturelimit
becausetheheatcapacity dependson thedensity ofstatesnearthe Ferrhisurface
and thus on the eflkctive mass. A s is shown in Sec.29,the precise relation is
givenby(compareEq.(5.60)J
C!
(gzrkjTm*k4(jj6:)
-P
?hl
and Eq.(1l.68)showsthattheinteractionsenhaneeCv. Althoughthepresent
-
-
.
m odelapplies only forkFa-:
tl1,itisinteresting thatexperim entslon pure H e3
suggest(m*jmjvvs;
k:2.9 in qualitativeagreementwith Eq.(1l.68). Unfbrtunately,the large num ericalvalue precludesasim ple perturbationexpansion, and
a m oresophisticated approach isrequired.z
3.Ground-state energy:The ground-state energy can be readily obtained
w ith therm odynam ic identities,as noted by G alitskii. Itcould,ofcourse, be
calculated directly from Z*(p,p0)with Eq.(9.36),butthe following approach is
m uch sim pler. By definition,the chem icalpotentialatS = 0 isrelated to the
exactground-stateenergy E by theequation (4.3)
Pf
#.= IW
.
S= 0
lntegrateEq.(11.70)atconstantP'(and S = 0)
b'
E-(,#x'p.(s-0,p-,x') constp',s-()
Since N appearsin Eq.(11.65)only through kr= (?=2N(V)k,the integralis
easily evaluated with the relation
N
jv#A',LkF(N,
)1A-3#3.ykts
.
lAn excellentsurvey maybefound in J.W ilks,%h-l-he PropertiesofLiquid and Solid Helium q''
chap.17,Oxford University Press,Oxford,1967.
2Seeforexample, L.D .Landau,Sov.Phys.-JETP.3:920 (1957).
149
FERM I SYSTEM S
and wefind
The firstterm isthe familiarkinetic energy ofa free Fermigas gEq.(3.30)J,
whereasthe second term wasdiscussed following Eq.(11.26). The third term
arisesfrom the m odifcation ofthe interm ediate statesby the exclusion principle
and wasfirstobtained by H uang and Y ang.l The fnalterm ,which we have not
discussed here,was obtained by DeD om inicis and M artin.z Itrequires a study
of three-particle correlations, and also depends on the precise shape of the
potential through the y-wave eflkctive range and p-wave scattering length. A s
written hereaEq.(1l.7l)describesa hard-sphere Fermigaswith two degreesof
freedom ; the corresponding expression for nuclear matter (four degrees of
freedom,neutron and proton,spin-up and spin-down)is
E
Ji2kj.
hg
m
na
uc
tt
!ea
rr jjjg
3
2
-
slà,rk,a+.3152,,2(11 z,lnaltksulz+.O.
J8(ksu)3-h...
j(11.72)
+
.- .
JUSTIFICATIO N O F TERM S RETAINED
Weshallnow furtherjustifyourbasicapproximationofretainingonlytheselfenergy of Fig. l1.3, in which two particles or two holes interact repeatedly.
One ofthese term s isshown in Fig.11.10J,along with a typicalom itted one
F
i
g
.
1.
1
0C
o
m
p
a
r
i
s
o
n
o
f
d
i
a
g
r
a
m
s
(
J
)
r
e
t
a
i
n
e
d> >
and (:)om itted jn ladderapproximation.
1K .Huangand C.N.Yang,Phys.Ret'
.,105:767(1957);T.D ,LeeandC.N.Yang,Phys.Rev.,
105:1119 (1957).
2Strictlyspeaking, C.DeDominicisandP.C.M artinLphys.Ret'.,105:1417(1957))obtained the(/cz.c)3correction fornuclearmatter(Eq.(11.72)1,and the generalexpression wassubsequentlydtrivedbyV.N.Esmovand M .Ya.Amusya,Sov.Phys.-JETP,20:388(1965).
1B0
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
(Fig.ll.10:). Thebasicpointisthatonlyonelinein Fig.l1.l0Jrunsin the
reverse direction,whereas lw'
t?linesin Fig.1l.10: run in the reversedirection.
To show precisely how the direction ofthe lines asects the term , we consider
the following two piecesofthe respective graphsshow n in Fig. l1.1l(
compare
the discussion following Eq.(11.27)). Thecorrespondingintegrationsoverq
aregiven,respectively,by
fâ-l(2'
?r)-4Jd% U(,44)G0(pj+q4G0(pz-q)
fâ-1(2'
r)-4Jd% U(
)(4)Ghpt+qjG0(pz+qj
#1t
t91
#lt
(11.73/)
(11.73:)
4#2
q
#1+ q
pl- q :1+ q
Fz+ q
(è)
Fig.11.11 Pieces ofgraphs (t8 retained and (b)
om itted in ladderapproximation.
Incase(/),thetwo Green'sfunctionscontainq withoppositesigns,whereasin
(d8.theyhavethesamesign. Thisdiflkrencehasacrucialesectonthefrequency
integrals,asisreadily verised by carrying outtheqointegration. In particular,
Eq.(11.73(M contains two terms,with the factors (1- n0
p,+q)(l- nT
02-'
''
à and
n0
()
pl+q%0,-q,whi
lethecorrespondingtermsinEq.(11.73:)contain(1- wj+qlnp
zl.q
and nj,+q(1- w9,+q)(comparethecalculation leadingto Eq.(11.35)). Forthe
presentlow-density system,momentum integrations inside the Fermisea have
very restricted phasespacebecausekF cc(#/F)1issmall,whilethoseoutside
areessentially unbounded. Thusthepresenceofa factora0(ahole)reduces
the term relative to one containing only particles. This calculation explicitly
illustratesthe distinction m ade between particlesand holesin the discussion ofFigs.11.3and l1.4.
Itisinteresting to ask how thepresenttheorycanbeim proved. Them ost
obviousflaw isthelack ofself-consistency,because X*isevaluated withJree
G reen's functions G0,while X* determ ines the fully interacting G. The cal-
culation can bemadeself-consistent(in the senseofSec.10)by changing al1
factorsofG'0into G in Eqs.(11.28)and(11.30);ineflkct,thischangesthefree-
particleenergiese2=hlkl(2m appearing in Eq.(11.58)into interactingenergies
ekand introducesadditionalfrequency dem ndence. In thisapproach,X* thus
depends on the exactsingle-particle energies,which,in turn,depend on Z*.
A lthough diflkrentin detail,thism odifled theory isvtry sim ilarto Brueckner's
theory of-nuclearmatterand He3.1 Thesequestionsarediscussed furtherin
Chap.l1.1
1K.A.Brueckner,Ioc.dt.
.SeealsoA.L.FetterandK.M .W atson,n eOpticalM odel,- .V andW ,inK.A.Brueckner
(ed,),''Advancesin TheoreticalPhysics,''vol.1,p.115,AcademicPress,Inc.,New York. 1965.
151
FERM I SYSTEM S
IZUDEGENERATE ELECTRO N GAS
Forthefnalexamplein thischapter,wereturn to thedegenerate(high-density)
electrongastreated in Sec.3. The moststraightforward approach isto analyze
thehigher-ordertermsintheproperself-energy;justasinSec.-1l,thedominant
contribution arises from a partieular classofdiagram s that m ay be sum m ed
explicitly. Thisprocedure isstudied in detailin Sec.30,w'here we considerthe
electron gasatsnite tem perature. Forvariety,we here describe an alternative
form ulation, in w'hich the ground-state energy is expressed in term s of the
polarization insertions11and generalized dielectricconstant.
GROUN D-STATE ENERGY AND THE DIELECTRIC CONSTANT
To sim plify ourtreatm ent,the presentsection isrestricted to a spatiallyhomogeneoussystem ofparticleswith a spin-independentstatic potential
(12.lJ)
U()(x,x')AA,,/z:-- &0(x- x')3AA-t
sss'
- I
z'tx - xz)&(t- t?)3aa-jss,
(l2.lb)
TheinteractionenergyforbothbosonsandfernnionsgEq.(7.1l))thenreducesto
(f')= !.j-#3xJ3x'F(x- x')('
f)(x)fj
#(x')l;jtx'lfa(x))
=
!.jJ3xJ3x'p'tx-x')((?i(x)H(x'))- t
itx-x')((?
'
i(x))1
,
(12.2)
where the second line has been rewritten with the canonicalcommutation or
anticommutation relations gEq.(2.3)),and the angularbracketsdenote the
ground-state expectation value. lt is convenient to introduce the deviation
operator
H(x)H H(x)- (H(x))
(12.3)
inwhich caseEq.(12.2)becomes
(P)- !.j'J3x#3x'U(x- x')((H(x)H(x'))+ (?
i(x))(?i(x'))
3(x- x')(?
i(x)))
-
Thisequation describesan arbitrary interacting system and is therefore quite
com plicated. lts real usefulness, however, is for a uniform system , where
(z
i(x))isaconstantequalton = N/F. Thelasttwo termsofEq.(12.4)arethen
trivial,andwemayconcentrateonthedensitycorrelationfunction (J(x)J(x')),
which containsal1oftheinteresting physicaleflkcts.
To m ake use of the diagram m atic analysis of-Chap. 3, we introduce a
tim e-ordered correlation function
.
,
'
C'P#t!F(/'s(a'
)Fiy(x')1I
.To)
-
lDlx.x )= ---... 'Cpj-()jtjo)
.
152
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
which is clearly sym m etric in its argum ents:
D(x,x')= D(x',x)
(12.6)
For reasons thatare m ade clear in the subsequentdiscussion,D is frequently
called thepolarizationpropagator. ln the usualcaseofa uniform m edium with
time-independentX.D dependsonly on x- x'. Theinteraction energy (Eq.
(12.4)1cannow berewrittenas
('P)= !.J#3x#3x'Utx- x')(ïD(x't,x;)+ n2- 3(x- x')n)
(12.7)
w'here the sym m etry of D enables us to set /= t'directly. If D Q denotes the
corresponding correlation function fora noninteracting system
iD$
?(x',x4= (*ogFg?
iz(x''
)Hz(x)J1*0),
then the interaction energy can be separated into a hrst-ordercontribution and
allthe higher-ordercontributions
(f')= !.f#3x#3x'Utx - x')gïDotx'/,x/)+ nl- 3(x - x')r?j
+ !.fJ3x#3x'P'tx- x')gïDlx't,xt)- iDQ(x't,xr)J (12.9J)
(0)= (t1):IPjmo'
l.
c-.
è.f#3.
x#3x/P-tx- x?)gïDtx't,x/)
iD0(x't,x?)) (12.9:)
-
Thisseparation isvery convenient,forwe havealready evaluated thefrstterm
in Sec.3. Note thatEq.(12.9:)appliesonly to a hom ogeneoussystem,where
t(H(x)))= a
V/P'is independent of the interaction between particles. In an inhomogeneous system ,the interparticle potentialalters the density,and (H(x))
therefore contains contributions from all orders in perturbation theory. A
sim ple exam ple are the electrons in an atom .where the coulom b repulsion
m odisesthe unperturbed hydrogenic orbitals.
Equation(12.9:)can becom bined with Eq.(7.30)toyield thetotalgroundstate energy and the correlation energy defined in Sec.3
-
rmalz3:*a)+ Fcorr
where the integralover the variable coupling constant A has been evaluated
explicitly in thesrst-orderterm . Forthe presentuniform system ,thisexpression
becom esm uch sim pler in m om entum space,where we 5nd
Ecorr-!.p'(2=)-4(0'JzA-,.
flkzpetqlgfoztq,co)-footq-tslq
.
Here P'(q)denotestheFouriertransform ofP'(x),and
DA(x,x')= DA(x - x'5t- t')
zu:z (
gnj-4j
-y4ggi
q.(x-x')g-it
zl(t-t')pZtq5(
x))
.
(12.11)
(j;.jgj
FERM I SYSTEM S
163
Thesymmetry ofD(x,x')gEq.(12.6)jallowsusto writeEq.(12.11)withouta
convergencefactorel
tio)'t which thusdiflkrsfrom Eq.(7.32).
W e have now expressed the ground-state energy of an interacting system
in term s ofthe tim e-ordered density correlation function D '
ifor an arbitrary
value ofthecoupling constanl. A san introduction to thisfunction,itisuseful
to evaluate D0(x,x')whichdescribesanoninteractingsystem
fo0(x,x')-k
z
.(I)(,qz'y,
J(x),;alx),
J)(x'),
t
;,
j(x')qi(l)())
-
*()l'
(7
')(x)'
?
;alA'
l1
.*0-''
,*0l'
t
Jj
Xx')f/,(x'):
,t1)u)
t'.
Thisexpression iseasily evaluated with W iek'stheorem
iD0(x,x')- iGo
=x(x,x+)iGo
g,b(x',x'+)- iG%
)iG?
x',x)- r
x)>
'?
i(x')))
a,;
3(
-v,x'
pa(
-H(
.-.ç
(2,
s+ 1)G0(x,x')G0(x',x)
Fig.12.1 Lowest-ordercontribution DQto density correla-
tion function (J)in coordinatespace,(b)in momentum space.
12.lJ and is typical of a polarization insertion. lndeed, the lowest-order
contributionsto U(x,x')areshown in Fig.12.2,and maybewritten as
Wethereforeidentify(compare(9.44))
D0(x9x')= -ff4j(x,x')G0
ja(x',x)
.
=
âH0(x,x')
ltis easily verifed thatthisstructure persiststo al1orders,so thatD(x,x')ish
tim esthe totalpolarization insertion
D(x.x'4= âI-I(x,x')= âH(x',x)
(12.14)
1:4
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
Furthermore,Dyson'sequation allowsustorewriteJd4xL(&(x,xj)H(xl,x')as
J#4xl&(x,xI)l1*(xl,x'),Where I1* is the proper polarization. The correspondingrelationinmomentum spaceisgivenby&()
4t
?)114ç)= &(t
?)H*(ç),and
thecorrelationenergy(Eq.(12.1l)Jbecomes
ECor-+!.
fp4(2=)-4j0
bJzz-l.
fd.q(gA(ç)rI*A(ç)-zt&(:)IIo(ç))(12.15)
N ote that
I10(ç)M I1A)(ç)
where11$,(t
?)isthelowest-orderproperpolarizationpropagator.
X1
A'
X
.
>
'= .G
X
(12.16)
X
x'+
X
X1
Fig.12.2 Expansion ofeFectiveinteraction.
Equation(12.15)can alsobeexpressedin termsofthegentralizeddielectric
constantv(q,a?)= KLqj,desned byEqs.(9.46)and (9.47). Thustheintegrand
ofEq.(12.15)mayberewrittenwiththerelation
Ul
Uéqb1-1*(:)
q)H*(t?)- l Uolq)H.(q4
q4= 1 - 1
= 1- x(
Klq) K(#)
-
w hich yields
s--,-+à.
frm(2=)-.j0
'yzz-lj.#k(E
sA(ç)!-1-l-,
jvolq)n0(ç))(12.18)
RIN G D lAG RA M S
Equation (12.15)applies to any uniform system,and we now specialize to a
degenerateelectron gas,described by the hamiltonian in Eq.(3.19). Asshown
in Sec.3,the uniform positive background precisely cancels the q = 0 term in
the potential, so that U(0) vanishes identically. This reiects the physical
observation thatthere is no forward scattering from a neutral medium . In
consequence,a1lçttadpole''diagrams(Figs.9.7a,9.8/,c,#,etc.)disappearfrom
thetheory.which sim plifestheperturbation analysisconsiderably.
FERM I SYSTEM S
155
To understand the structure ofEq.(12.15),we shalltemporarilv expand
&H* ina perturbation series,asfollows:
& 11*= U011*+ &011*&0H *+ ...
= U01
7(
*0)+ &011*
(l)+ U0H*
(0)UOfl(
*0)+ '' '
where1-llcjisgivenbyEq.(12.16),and1-1)
t)isthesrst-orderproperpolarization
withthecontributionsshow'
n inFig.l2.3. ThelirstandIastterms(Fig.12.3/
(JJ
(:)
(c)
(d4
(e)
Fig.12.3 A llhrst-ordercontributionsto properpolarization.
ande)vanishinthepresentexamplegP'(0)= 0),andw'eareleftw'iththemiddle
three. Correspondingly,thecorrelationenergyhastheexpansion
fcorr= fJ+ f)+ Ef+ EC-t''''
(12.19)
wherethe varioussecond-ordercontributionsaregiven by
EJ=JïZ/i
(2'
z
r)-4j0
îdhX-1jd*qg
2L%(g)Z0(t
?)12
(12.20)
e-t.c'd=.
jjp4(2=)-4j0(.
/AA-ljd4qgA&/(
)t(
y)I'
llj)j,,c,cjq)j
(12.21)
-1
Herellh)t,,lllllc,and l1ljl,denotetheproperpolarizationsinFig.12.3: to d.
ltiseasily shown thatthecontributionsin (12.21)are linite(Prob.4.13).
In contrast,FJ diverges logarithmically (w'
e explicitly exhibit this divergence
later in the discussion;see also Prob.1.5).and thepresentexpansion through
second order isclearly insum cient. The source ofthisdivergence isthe singular
behaviorofthe coulomb potential&()(t?)= U(q)= 4rre2y?q2atlong wavelengths;
in particular.ELhastwo factorsofL'
%(t?),leadingto a(q)-4behavior. A similar
behavior occurs in all orders.because there is always a single nth-order term
withtheintegrand(67t
)(42)l10(t
y))n. Fortunately,thesesingulartermsarereadily
includedtoa)lordersinperturbationtheorybyintroducingtheeyeclit'
ez'
a/tart
zt-//t?a
Ihlq)(compareEq.(9.45))
Urlq)= b'
çjlq)i-U()(t
?)170(t?)b'zlq)+ '''
.
t.
t- 1-l
0)
(k(fjj
thlql
.
(12.22)
166
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
*
'*
+
&r = 6/0 + '''
Fig.12.4 Ring approximationforeFectiveinteraction.
w hich is an approxim ation to the true interact
,i
on in themedium Lr(t
?)obtained
by retaining only thezero-orderproperpolarization flA)> H0 in Eq.(9.45).
Equation (12.22)containsthe diagramsshown in Fig.12.4 and is known asthe
sum ofr?
'
?7g diagrams. Forhistoricalreasons,itisalso known asthe random phase approximation,although thisname is notespecially illum inating here. l
Thisselectedclassofhigher-orderringdiagram sm akesthefollowingcontribution
to the ground-state energy:
X
Er= a=2
)(Ea
r
-
!.
/p'
?j
(2=)-4jo
b#Az-l.
fdnq,
t
4a
.:Ar
.
&((
?)I.
Io(t
?))-
-ljp,
/jt2.
(ç)
rrl4j-l#AA-ljd4q IAUO@I
,170u(
)()-2)
-
.
.
-
0
.
1- 'jbvlq) q
j?
'p'
â(7=)'*j0
z(
CAA-l*
fd4qA&o(ç)H0(t
?)&)(t
y)H0(ç)
- .
(12.
23)
Thephysicalinterpretation ofErisclearfrom thelastline,because one ofthe
K#bare''interactions(,
'zlq)inEq.(12.20)hasbeenreplaced bythe(lesssingular)
efl
kctiveinteraetion Urlqq. Although thefirstterm ofE,isformallyofsecond
orderin the potential,we see in the following calculation thatthe sum hasa
whollydipkrentanalytic structurethatcannotbeobtained in any hnite orderof
perturbation theory.
Theeflkctiveinteraction &r(ç)gEq.(12.22))canberewrittenintermsofa
dielectricconstantKrlq)bytherelation gcompareEqs.(9.46)and(9.47)1
:)
xrtç)- l- Uolqb1R0(ç)= &0
&(
(ç)
(12.24)
r
where Krlqqmay be considered the ring-diagram approximation to the exact
dielectricconstant. Theenergy Erthen becom es
Er =
l
-
lfdnq Lluott?lZ0(t?)12
i.
/Fâ(2=)-4jv#àA .
.
KAt(
yj r
(12.
25)
'D.Bohm and D.Pines. Phys.Ret'.,92:609(1953);D.Pines.Phys.Rev.,92:626(1953).
157
FERM I SYSTEM S
In the present approxim ation. the correlation energy of a degenerate
electron gasreduces to'
Ecorr= Er- E2-t-Eî+ Ed
2- .
(l2.26)
Ineach term gcompare Eqs.(12.20)and (12.21)).the tw'o endsofa polarization
insertion arejoined u'ith a bareinteraction L'(
)(ty).and the contributions to the
energy are drawn in term s of the equivalent Feynm an diagram s in Fig. 12.5.
Fig.12.6 Leadingcontributionsto correlation energy'.
It m ust be em phasized that these disconnected diagram s cannot be obtained
directly from the Feynm an rulesofChap.3,becausethecounting ofindependent
contributions diflkrs from that of the connected parts as is evidentin Fig. 12.5
(f'îand.
E'4arethesameFeynmandiagram). Althoughitispossibletointroduce
a diagram m atic analysis of Fcorr.we prefer to study only quantities with tixed
externalpoints.such asf1,Z.G.and soforth.sinceasinglesetofFeynm anrules
then applies to a11 cases. This restriction causes no dië culty, because E is
readilyexpressedintermsofâ:(Sec.11)or17(Sec.12).
'This contribution wasfirstevaluated by M .Gell-Nlann and K .A .Brueckner.Phvs.Rc!'.,
106:364 (19574. W'e here follow the approach ofJ.Hubbard.Proc.Roy.Soc.(Londonq.
AM 3:336(1957).
'seeal
soT.D.Schultz.'-ouantum FleldTheoryandtheMany-BodyProblem .*'
secs.lIl.H to III.J.Gordon and Breach.ScienctPublishers,New York,1964.
1sg
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
EVALUATIO N O F H0
Forfurther detailed analysis,we m ustevaluate the lowest-order polarization
insertion H0,given in Eq.(12.13).1 Thisquantityisindependentoftheinterparticle potentialand isthereforedeterm ined solely by the propertiesot
-a non-
interacting Fermisystem. Thesum overa and j yieldsa factor2forspin-!
ferm ionsso that
rI0(x,x')= -2I'
â-lG0(x,x')G0(x',x)
(12.27)
Thisexpression ism ostsim ply evaluated in m om entum space,and the Fourier
transform H04t?)- H0(q,ç:)isgivenby
110(ç)=-2fâ-l(2=)-4Jd*kG0(k)G0(k+q)
(j2.2s)
ascanbeverisedeitherbyanexplicitcalculationwithEq.(12.27)orbyusingthe
Feynm an rulesofSec.9 with Fig.12.1:.
A sa srststep,itisconvenientto perform the frequency integralin Eq.
(12.28);theintegrandcontainsfourterms,ofwhichtwohavetheirpolesonthe
sam e side ofthe realaxis. In these term s.the contour can be closed in the
oppositehalfplane,and the contribution vanishes. The othertwo term shave
polesonoppositesidesoftherealaxis,anda straightforwardcontourintegration
yields
(j,,j:3'c
-1
)
#(c
I
q-+k
-l
--kF4
-#tk+
sf
-,k)-0(k
.+
F-l
q-+-k-I
)
-0(k--.ks)1(12.29'
nn(ç)- zn
where,as before,o,k= â-1e2= hkz/zm. The second term can be rewritten
with thechangeofvariablesk'= .
-.
k - q;thistransfbrm ation leadsto
l'ln
2
d3k
(q.çn)-jJ(2=)jP(I
q+kl-k,'
)hkr-ks
l
1
x * V f'Ok - f'Oq-bk-f'ill- W -1-f'tlq+k - f'Ok- i'fl (12.30)
'
wherethesuperquousprim ehasnow been om itted. Byinspection,theintegrand
isaneven function ofqn,and weconcludethat
l10(q.ç()--'0(qo
-2)
Thissym m etry allowsusto studyonlypositiveqz.
Ifthefrequencydiflkrencein thedenom inatorsisdenoted
h
z
c
h
a
tzaqkH fzaqsk- tz:k= gm Rk+ q) - k 1= - (q*k+ '
W)
1J.Lindhard,Kgl.DanskeVidenskab.Selskab.M at.-Fys.M edd.,28,no.8(1954).
(12.31)
FEBM ISYSTEM S
159
thenthesymbolicidentity(7.69)immediatelyyields
2.
7 ( d3k -
n-.
.
,-.n.,
,
2ttuk
RerI0(q.ç:)= y jtjx);L1-VbR'F-Ik+qlllUbKF-tlgj.40
-qklz
(12.32)
wherethesrststep function hasbeen rewritten with therelation
#(x)= 1- #(-x)
(12.33)
Thesecond term ofEq.(12.32)vanishesidentically,becausetheproductofstep
functions is even under the interchange k m k + q,while œqk is odd;consequently,Rel-Ioreducesto
2..
, d'k
Renetq.
t
?(
)-,J(a.)a#(k,.-k,
l
'
x(f
s-Ji
p
,,-ltq-k+l.
ç2-)-qo+ljpa-ltq-k +.p z)1(12.34)
W enow introduce thedimensionlessfrequency variable
(12.35)
v= hqoy yF
and measureallwavevectorsin termsofkF. W iththesedimensionlessvariables,
Eq.(12.34)becomes
zmk.,
ReH0(q,v)=hzpj(2
:=
3k
)yyj-k)
I
l
x(v-qkcose-lv,- v+qgcoso.jqz)
..
Thisintegraliselem entary and yields
Reuntq,v)- lmkF
tz(-l+jk
lgI-(t
--!)2jlntl
;
j+
-f
(v
'/:-o
'
i'
t
yp
ll
-j
k
lj,-(k
v+!)2jInjl
j+
-(v
p/
yç
:+j
'
iç)t
j(lc.
36)
,, y,
TheimaginarypartofH0canalsobeevaluatedwithEqs.(12.30)and(7.69)
Im I10(q,çp = -â-l(2x)-2fd3kptlq+ kl- kF)dtks- k)
X (3(#0- œqk)+ hh + œqkll (12.37)
ltisagain sumcientto consideronlyqf
t> 0,and thepairofstep functionsensures
thattsqkisalso positive. W enotethatIml10(q.e)hasadirectphysicalinterpretation,for itis proportionalto the absorption probability for transferring
3:o
GeoklNo-saxre (zEqo--rEMF'ER>akJqE.
) yoquAt-lsM
four-momentum (q,f
?c)to a freeFermigas;theprocessmovesa particlefrom
insidetheFermisea(k < kF)to outside(!
k +ql>ks),whilethedeltafunction
guarantees thatenergy is conserved. This application of I10 is discussed at
length in Sec.l7.
W ith the same dimensionless variables as in Eq.(12.36), the identity
3(Jx)= !
Ji-13(x)
(12.38)
canbeusedto rewritetheonlyrelevantdeltafunctioninEq.(12.37)as
3(ço- Yqk)= hmk;3(1
z- q@k- '
W z)
and wetherefore need to evaluate
Im l70(q,c>0)= -v/cs(2oW)-2j#3/c9()q+k)- 1)#(1- k)
x 3(v- q.k - .
!.
t?2)
The integration isrestricted to the interioroftheFermisphere(k < 1),whilethe
vectork + q m ustsim ultaneously lie outside the Ferm isphere. Furtherm ore,
the conservation ofenergy requiresthat
L'
it?+ I'k = -
'
which defines a plane in the three-dim ensionalk space. The integral in Eq.
(12.39)representsthe area ofintersection ofthisplanewith the allowed portion
oftheFerm isphere,asshowninFig.l2.6. Therearethreedistinctpossibilities:
l. q> 2
jq2+q> ry.jq2-q
(12,40)
lfq > 2,then the two Ferm ispheresin Fig.12.6 do notintersect, and we need
onlytheareaofintersectionoftheplaneand theuppersphere. Thisareaclearly
vanishesiftheenergy transferwistoo large ortoo sm all,and thecondition that
k
l + ''
%.k = qq
q+ k
-- - - q
Fig.12.6 Integration region for
Im H0forq> 2. (The Fermisphere:
areofunitradius.)
FEgM ,SYSTEM S
1e1
theyintersectisjustthesecond condition in Eq.(12.40). Theintegrationcan
beperformed with thesubstitution t= cos0
kt'
Imrln(q.v)- -4m.
n.hi2,Jv
'
/4-l:kzJ/
cJ'ldtq
)&(q
k-i
lk
Y-t)
z
and an elem entary calculation yields
m kF l
v 1 2
Im I10(q,$ = - hz 4 1- q
-- jq
,
a,
:
(12.41)
undertherestrictionsonthevariablessetin Eq.(12.40). IfvIiesoutsidethe
regionsdesned in Eq.(12.40),theintegraliszero.
2. q< 2 q+ jq2> Tz> q- l.
ç2
(12.42)
Ifé< 2,thenthespheresdesnedbytheconditionsk< land I
q+ kl> 1intersect,
with thetypicalconfiguration shown in Fig.12.7. Theplanew illnotintersect
k
q+ k
iq+ '
q.k= v/q
'
4
1
!-l- à
aq
Fig.12.7 Integrationregionsforlm 119
forq < 2.
the uppersphere if-theenergy transferv istoo large,and Im H0vanishesin this
case. As v decreases,the intersection is a circle untily'becom es suë ciently
smallthattheplanebeginstointersectttwforbiddenFermisphereatthebottom.
Thislimiteddomaininwhichtheintersectionremainscircularisjustthatdesned
in Eq.(12.42),
.the integration isperformed exactly asbefore with the result
Im no(q.v)- - m
yk:4.
1
(l2.43)
q
-
g1-(#
'
?-j
lq)2j
os:vxq- !.
ç2
(12.44)
ln thiscase.the intersecting plane passesthrough the forbidden Fermisphere
atthe bottom ,and the allowed region ofintersection becomesan annulus,as
indicated in Fig. l2.7. The area of this annulus can be evaluated with the
geom etric relations in Fig.12.8,which show thatthe m inim um value ofk is
given by
â'lin= (,
1.
t?- 1
zç-')2+ El- (!4 + :/t?-1)2)= l- 2v
162
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
@#
kmm
K
.
'
!
,
#/
'
àq
(l- ('
iq+ œ
q)2(
) Fig.12.8 Geometry in momentum space
usedinobtainingEq.(12.
45).
whilethemaximum valueistheFermimomentum (/cmax= 1inthepresentsystem
ofunits). Theintegralcannow beevaluateddirectly:
Im notq.v)- -mk,F '2 z=
, ,.
= -
= -
j(
'
1-2v)
1'kq
t
''
cJ
(-'1t
,sï
(
b
-v
q
kï-2
.
1k
#-t)
m kF 1
hz 4=q (1- (1- 2$)
rrlks 1
hz j'n.q2:,
(12.45)
Equations(12.41).(12.43),and (12.45)determine ImFl0(q,v)forallq and p.
W esketch -(4=â2/v#s)ImH0(q,v)forfxed tqkin twocasesofinterestin Figs.
12.9/ and 12.9:.
Form any applicationsitisusefulto listsom elim iting form softhe zeroorderpolarization part:
Fixthem om entum qand Iettheenergytransferp'approachzero:
Im H0(t
h0)= 0
(12.46/)
RerIo(ç,
o)-mhk
,F2.
1cg-1+q
1(1-tqzll
nù
1
j+
-j
b'
qj
!
(12.
46:)
2.Fix the energy transfer vand 1etthe m omentum transferq approach zero:
Im lR0(0,v)= 0
2
Re1R0(ç,v);
k;mkF
âz 2lz2
jya
(12.474)
(12.47:)
..
.n
v
3. Finally,fix the ratio ofenergy transferto momentum transfer v(qH x,and
letthe m om entum transferq approach zero:
Im I10(
ç,:.
x)=
-
mkF x
2 i-é
h
0
R
mkF 1
q-->'0,0< .
Y< l
(12.48J)
q->.0,x > l
rl+ xl
erI0(g,çx)=-yzj
.
p(2-xln;j.xj
) q-+0
(12.48:)
16:
FERMI SYSYEM S
-1
2- q
q tt
4 p'
l1 q+ lq
-
2
i4 -q
'
2
> $'
ùq + q
Fig.12.9 Sketeho.-(n=hzlmkè Im H Q(q,v)fortypicalvaluesofq.
TheexpressionsforH0(ç,v)can now beused to 5nd thecorresponding
dielectric constantK'
r(ç.$ = 1- &o(ç)I10(ç.$ in thering approximation. The
zero-order polarization has been expressed as ??7ks/âz times a dimensionless
function;consequently,the dimensionlessquantity t%110hasthetypicalvalue
(-ks/â2)(4rre2/#/.)=4=(#sJo)-1= 4gr(4/9=)1G (compare Eqs.(3.20) to (3.22)
and(3.29)3. lnfact,weshallrtquireonlythtthreelimitingcastsjustdiscussed,
and we5nd
1.Fixq,letv->.0:
lars 1-
1(j
,r
(t
?.0)-1+n.
qz q
uz)InI
(
1l
+j'
qj
-
-
(12.49)
2.Fix v,1etq -.
c
,.():
$x%
s,(0,p)= l- 3'rrpc
(12.50)
3.Fixv/qx x> 0,1etq.->.0:
4œr.
x 11+ xl
lixr-x
G((
A(
B )= l+ n'q; l- jln(
t+ qL #(l- ,
x)
;1 - xl
(12.51)
wbereq and pareboth dim ensionlessand xisa numericalconstantt
4 1
x > j-g-j
(12.52)
CORRELATION ENERGY
W enow returnto theevaluationofthecorrelationenergyofadegenerateelectron
gas. Although itispossibleto evaluate alltheterms in Eq.(12.26),such a
cakulation would largely duplicate thatofSec.30. Henct we here consider
onlyEv,whichcontainsapropertreatmentofthelogarl m' 'ergenceappeartWefollow thishistoricalbutunfortunatenotation;xisnotthehhe-structureconstan'inthis
problem.
:64
GROUNn-
STATE (ZERO-TEMPERATURE) FIRMALISM
inginF1andthereforegivesthedominantcontributiontothecorrelationenergy
Theintegration overA can becarried outexplicitlyin E
Eq.
q.(12.23)and yields(see
(12.24))
.
E--!.
fp4(2z.
)-.fd.qj0
l;
/zzkt.
&(t
?)uo(t
?)Jcil-At&(ç)I.
Ic(t
?))-!
=-.
jjp'
âtz,
rr
j*jd*qll
og(1-UéqjI10(g))+Uolq)Jl0(t
y))
jiFâ(2J
r)-4jd*
qfkog(xr
(t
?)J+1-x,((
?)J
(I2.53)
lt is again helpfulto introduce the dimensionless
-
= -
q'= zlks. Theringenergythen becomes- with theva
abl
qoji
hk; and
ar
idi
ofes
Eqps= nî
(3
(3.21),(3.22),
.
29),and(12.52)-
.
Er=
where
Nel
er
2Jc
(l2.54)
(12.55)
Theenergy ermustbereal, and we shallconsideronly therealpartofEq
(
55). Inaddition,Krisanevenfunctionofy', which aliowsusto simplify the.
li12.
mitsofintegration(wenow omittheprimeonqj:
g
. 2d co
s-'L(1,V
-j. - &,a(g,p)
L%az/.:2 0 q 0dv tan-l .
vrl(ç
(12.56)
,v
)
wherethedielectricfunction hasbeenseparated intoitsrealandim aginary parts
Kr= Krl+ l'
gra
and wehaveused therelatian
logtvrl+ I.
'rz)=llntsr
2k+ Kh)+ ftan-j&'
&'
r
2
r1
A s noted previously, the singular behavior of the electron gas a
sm allwav
rises at
evectors(q.
<.
:1),and weshallthereforedividetheq integration into
tw oparts,q < qcand q > qc
a
qc
co
s ,)
qîdq fop tan-k rz(4.
..
,- - x
2zrrxarz
a a
()
xr)(t?,p) rzfqnv)
a
. 2dq .dv tan-' lrz(f,-v) - v
r2 azracrc
xrl(ty,v) ,2(4,r)
: 4c
0
eg1=
.
.
- -
Thisseparation isolatesthe divergence, which occursonly in the firstterm e
forthisreason, erzissniteand can beexpanded in powersofr
rI;
her
m
or
e,
,. Furt
ifqcischosentobem uch lessthan l,then<rmaybtapproxim ated byitslimiting
form (Eq.(l2.51)Jincvaluating e,l,thereby giving a tractable integral. The
FERMISYSTEMS
166
imaginarypartofx,isproportionalto -F(ç)1m110(:,v),which ispositive or
zero tsee Eq.(12.39)2. Thustan-l(<r2JKr2jvariesfrom 0 to =. ln particular,
tan-l(xrc/xrj)= 0 ifKrz= 0 and K,,> 0,whiletan-'(n2/Krj)= = ifxyc= 0and
n j<::0. In theregion0 < q < qc < l,theapproxim ateexpression in Eq.
(12.51)
maybeused,and w'
etind
3
- -
er.-j.'
r
tav).
(q
,cq3dq(a
'dxé
ll
f
T
1
-j.
Y)
Yt
#
z'
))
jtan-1q
zrp
j%yy
sy
x
)-2%rs.
y(z1-.
.
6
o
j(
q
)-q3dqtj0
'dxtan-l(q,.
j
ly
xytx))-2
j(
.
- 4.
4.
,
+jl
=p.
xvrof-ql-(Ay
(x)y)
12
H 1L+ lz
where
57)
.
f(x) 2 '
!l+ xg
>gjI-jx1n(
1
j..
x,
and
(12.58)
A- 2=rs
(12.59)
Thetwo termsin Eq.(12.57)arisefrom the regionsoftheqvplanewhere Krz> 0
and Kr1= 0,respectively.
lfwe now expand er asa powerseriesin thecoupling constantA x el
l
, the
eading term mustreproducethesecond-orderterm (2av;Ne2)EL. In particular
tbesingularityforsmallq hasbeenisolated in Ik,and wetind
,
lj;k:-6ï.a z
uc
àx :2 /'(x)
7TzsJl0 q3dq 0ldx .a--.-S
,--+...-A-xz
qq
q
=-'Q
6Lq)cd.
'b('
qqJj
)Y*YVCXI*0621
r J(
The srst term of this perturbation expansion diverges lögarithm ically at the
originbecauseoftheq-2behaviorofthecoulombpotentialV(q). Thuswesee
explicitly thelogarithmicdivergence ofEJmentioned atthe beginning ofthis
section. Theexactintegral1t. however,containstermsofaIlordersin 2 and
i
,
ts inttgrand is snite as q -->.û. M ore preciselysthe q-* dependence of the
integrand is cutoffforq2< )s/'(x). In consequence,1,can be evaluated with
Iogarithm icaccuracb'asfollows:
I
6 qc#t
?'
*1
j;
k;-g
-
jyj.j(
;dxxf(x)
,
(lnLq
n
'
j
c)('(
sx(l-!.
xIni
j+))
.
2
.,
0
A
=p (1-ln2)lnO
(1260)
.
1ee
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
which means
ecorr= 2=-241- ln2)lnr,+ const
rs-*0
(l2.61)
Thisresultisoriginallydueto M acke.i Notethatthisexpression isnonanalytic
in r'and hasno powerseriesaround r,= 0.
The constantterm in thecorrelation energy requiresthe evaluation of-a11
theremainingtermsinEq.(12.26). Inparticular,itisessentialtoprovethatthe
arbitrary wavenumberqcdropsoutofthe tinalanswerforkr. Thiscalculation
isvery similarto thatinSec.30 and willnotberepeated here. Furthermore,it
iseasytoseethatELand E1vanishidentically,while
Ne2
E2
b= 2Ja eb
2
isjustthesecond-orderexchangeenergystudiedinProb.l.4. The5nalexpression can only be obtained numerically,and the correlation energy becom es
E
--.
e2 2
N
sFrr= 2J0 =-j(j- In2,
)lnrs-(4.+)4+ ofrsjnr,)
2
= ja
-n (0.0622lnrs- 0.094+ Otr,lnrsl)
(12.62)
correctthroughorderlnr:and4.1 Byanextensionoftheargumentspresented
here,DuBois2showsthatthesum ofthenextmostdivergenttermsin each order
inperturbationtheory(thosetermswithonelesspowerofL%(t
/))givesacorrec'
tionorOtrslnr,)to Eq.(12.62).
EFFECTIVE INTERACTIO N
W ehavealready m entioned thatthe perturbation expansionfailsbecause ofthe
singular(q)-2behaviorof&n(q). In contrast,the ring approximation to the
eflkctiveinteraction Url%sqzjhasaverydiflkrentbehavioratlong wavelengths.
Forsimplicity,weshallconsideronlythestaticIimit(qv= 0),andacombination
ofEqs.(12.24)and(12.49)yields
4.
n.e2
&r(q.0)-q
-u-+(jxrs/.)kkg
ygjks)
(l2.63)
1W .M acke. Z.Naturforsch..5a:192(I950).
1Thelogarithmicterm wassrstobtainedbyW .Macke,Ioc.cit.,andthecompl
eteexpression
was then derived by M .Gell-M ann and K.A.Brueckner,Ioc.cit. Seealso L.Onsager, L.
Mittagaand M .J.Stephen.Ann.Physik,1*:7l(1966).
2D.F.DuBois.Ann.Phys.(N.F.),7:174,appendix C,(1959). DuBois'calculation was
repeated and corrected byW .J.Carr,Jr..and A.A.M aradudin.Ph-vs.Rel'.,133:A37I(1964),
whohndû.ûi8rslnr,asthenextcorrection to ecorr.
167
FERM ISYSTEM S
where
l 1
z 1-j.l
.
x
g(x)- j- s (l- 1.*)lnI
j jx
I
..
(12.64)
Thusthemedium composedoftheelectronsandthepositivebackground modifes
Coulomb'slaw. ltisclearfrom Eq.(12.63)thatthismodifcationisimportant
onlyforwavelengths(q/kpjz< r,;inthehigh-densitylimitwherers-+.0,wecan
thereforeapproximateg(Wks)byg(0)= l,sothat
4=e2
Ur(q.0)ra
;
k
;o.
j
-,
qc+ (4.,
,/.):.
(l2.1)
Hencetheeflkctivepotentialiscutofl-forql;
EGk/andishniteatq=0,which
consrmsthe assertionsbelow Eq.(12.23). Thisbehaviorprovidesa physical
basisforthecutofused to5ndkcorrinEq.(12.60)andinProb.l.5.
Although Eq.(12.65)isonlyanapproximationtotheexact&r(q,0)given
inEq.(12.63),itisveryeasytotaketheFouriertransform ofthisapproximate
expression,which gives a Yukawa potential. W e thereby obtain a qualitative
pictureoftheefectiveinteraction in coordinatespace
P;(x)= e2e-qrrxx-1
(12.66)
Hencethesimpleellx Coulomb'slaw betweentwochargesisttshielded'gwith the
Thomas-Fermilscreeninglengthqv
-ldehnedby
qIr=
4ars
Tr
4 4 1
à'/= s rsk;= 0.66r,*/
Tr
(12.67)
In fact.the nonanalyticstructureof(12.63)complicatestheactualexpression
forFrtx)considerably,asisdiscussedindetailinSec.14.
lnthepresentsection,Urlqsça)hasbeenusedonlytoevaluatethecorrelation
energy,which isanequilibrium property. Asshown in theprecedingparagraph,
however,Urcontainsm uch additionalphysicalinform ation becauseitdeterm ines.
theesective static and dynamicinterparticle potential. Thisbehaviorisreally
a particular example of the response to an external perturbation. For this
reason,we shallhrstdevelop the generaltheory oflinear response (Chap.5)
and then return to the nonequilibrium prom rtiesofthedegenerateelectrongas.
PRO BLEM S
4.1. A uniform spin-.
s Fermi system has a spin-indem ndent interaction
PotentialF(x)= L x-le-x/q
(c)Evaluatetheproperself-energyintheHartree-Fockapproximation. Hence
hnd theexcitation spectrum ekand theFerm ienergy ep= y..
.The Thomas-Fermitheory isdescri- d in Sec.14.
16a
GROUND-STATE (ZERO-TEM PERATURE) FonMAulsu
(!8 Show thatthe exchangecontribution to Es isnegligiblefora long-range
interaction(kya> 1)butthatthedirectandexchangetermsarecomparablefor
ashort-rangeinteraction(kpa.c
41).
(c)Inthisapproximationprovethattheesectivemassm*isdetermined solely
by the exchange contribution. Com pute m *, and discuss the lim iting cases
kFa> 1and kra< 1.
(d)W hatistherelationbetweenthelimita--+.,
x)ofthismodelandtheelectron
gasin a uniform positivebackground?
4.2. UseEq.(11.70)todeterminethefirst-ordershiftintheground-stateenergy
fowthe system considered in Prob.4.1. Comparethiscalculation with a direct
approach.
4.3. Using 15'coulomb wave functions as approximate Hartree-Fock wave
functions,com putethe ionization energiesofatom ic He,and com pare with the
experimentalvaluesHe-.
->He'
b+ e-(24.48eV)andHe-+.He**+ 2c-(78.88eV).
Show thatthisapproach isactually a variationalcalculationand use this observation to im prove your results. H ow would you further im prove these
calculations?
4.4. TheequationofProb.3.4canbeconsidered thesrstofaninsnitehierarchy
ofequationsin which then particleG iscoupled to then - 1and n + 1particle
G's. A com m on calculational schem e is to Kfdecouple''these equations by
approximatingthen particleG intermsoflower-order(inn)Green'sfunctions.
Forexam ple,use W ick'stheorem to show thatthe noninteracting 2 particle G
satishes
Gxbiys(xjt1,x2t2,
-xl
't1
',x2't2
')
G3?(x1t1,x1
-fk
')Gjôtxc?z,xa'?a'
)+ Guô(xktl,xz
'/2
')Gjytxat:
L,x,
't,
')
A pproximate the interacting 2 particle G w ith the sam e expression and verify
thatthe resulting self-consistentapproximation forthe lparticle G reproduces
theH artree-Fock approxim ation.
*.5. How are the Hartree-Fock equationsforspin-è particlesmodihed for
spin-dependentpotentialsoftheform giveninEq.(9.21)?
4.6. A uniform spin-l Fermigasinteractsonly through ar-wavehard-core
potentialofrangeasothat31->.-(kJ)3/3forka-*0.
(J)Show from Galitskii'sequationthattheproperself-energyisgivento order
(kz.X3by
,u*(k)- hl2i'6'*3 3
,-
-
k2
j,+(u)j
FERM I SYSTEM S
1e9
(b)Show thatthe:rsttwotermsintheexpansionoftheground-stateenergyas
apowerseriesin G a are
E - hl
3
à
lmk; 3
à+ V (k,c)3
(c)Show thatthespectrum isstrictlyquadraticwithan efectivemassgivenby
mslm = l- (1yJ)3/rr,correcttoorder(kra)3.
4.7. Given a uniform Fermigas with a degeneracy factor ot
-g,show that
(J)the ground-state energy expansion for a hard-core potentialofrange a
becom es
E=
9 hlm
k/('
3
!.
y.Lg-j)j2k
3=
sa.
4
.35
4=z(jj-zjno(ksulj.
j.gg
tksul3jj
(b) theresultinProb.4.6becomes
E - h2k/ 3
(ksu)3
W am jj
-(g+.))snj
4.8. VerifyEqs.(11.63).(11.65),and (11.68).
4 9. Foradegenerateelectrongasshow thatY*(q)isgiventosrstorderinthe
interaction by
â)2)l/q)--s
e2(/
c/qq2lnt
(
J
:
fs+qj
(+ggFj
Sketch the resulting single-particle spectrum . D iscuss the eflkctive m ass
m*lqjdehned bym*lq)- (hlqllDkqjèql-b.
4.10. Apply Prob.1.7to an imperfectspin-! Fermigas and show thatthe
ground statebecomespartiallymagnetized forkFa > */2.
4.11. Verify Eq.(12.36).
4.12. A system ofspinu ferm ionsinteractsthrough a spin-independentstatic
potentialP'(q).
(c)AnalyzetheFeynman diagramsfortheproperpolarization,and show that
ZZ.
uUz(U = Ilg
*(#)(2#+ 1)-'ôs8as+(1R*(ç)- l1e
*W)1(2J+ 1)-23a,&u (see Eq.
(9. :)).
(!8 SolveDyson'sequation forl-I.j,As(t
?)(compare Prob.3.l5).
(c) Show thatD(q)(Eq.(12.12))isequaltoâH.a,Az(ç),andhencerederivethe
expressionD(q)= âI1*(ç)(1- P'(q)11*(ç)1-l.
4.13. Considerthediagramsin Fig.12.3 foran arbitrary potentialF(q),and
show that only 1R*
4kp contributes to Ez in Eq.(12.21). Use Eq. (12.
21) to
evaluatef1,and show thatitagreeswiththatinProb.1.4.
jx
GROUND-STATE (ZERO-TZM PERATURE) FORMALISM
4,14. (c) Evaluateer,inEq.(12.57)byfirstperformingtheqintegralandthen
.
expandingin powersofrs.
'
(5) Evaluateerzdehnedbelow Eq.(12.56)byexpandinginpowersofrs directly.
(c)Show thate,j+ %zisindependentofqc,and compareyourexpression for
theconstantterm in ecorrwith thatobtained byK . Sawada,Phys.Ret'., 106:372
(1957)and by K.Sawada,K.A.Brueckner,N.Fukuda,and R. Brout,Phys.
Ret'.,10*:507(1957).
i
I
I
I
!
l
l
i
k
l
1
!
l
l
l
t
1
I
t
l
5
Linear R espo nse and
C ollective M odes
The preceding chapter concentrated on the equilibrium properties of a Ferm i
system atzero tem perature,alongwith thespectrum ofsingle-particleexcitations
following the addition or rem ovalofoneparticle. These ferm ion excitations
can be directly observed through such processes as positron annihilation in
m etals,nuclear reactions,etc. In addition,mostphysical system s also have
long-lived excited states thatdo not change the num ber of particles. These
excitations(phonons,spin waves,etc.)havea bosoncharacterand arefrequently
known ascollectivem odes. Theycan bedetected withexperim entalprobesthat
couple directly to theparticledensity,spin density,orotherparticle-conserving
operators. Typical experim ents scatter electrom agnetic waves or electrons
from m etalsandnuclei,orneutronsfrom crystalsand liquid He4. Theseprobes
a1linteractweaklywiththesystem ofinterestand thereforecanbetreatedinBorn
approxim ation. To provide a generalbackground,we shallsrst discuss the
theory oflinearresponseto aweak externalperturbation.
171
172
GROUND.STATE (ZERO.TEM PfRATURE) FORMALCSM
I3LGENERAL THEORY O F LINEAR RESPONSE
TO AN EXTERNA L PERTURBATION
Consider an interacting many-particle system with a time-independenthamiltonianX. TheexaqtstatevectorintheSchrödingerpicture/
k1-s(J)z,satissesthe
Schrödingerequation
ihPl'1'
.s(;))
est(?))= 4 $
%1.,
with theexplicitsolution
(
:.
1,
-(r))= tp-fz?r/ntl,(0))
Supposethatthe system isperturbed att= tsbyturning on an additionaltime-
dependent ham iltonian X extf). The new Schrödinger state vector ,tl--stf))
satissesthemodihedequation(?> ?0)
Ps(J))- (4 .#.z'
ih8)'
?.ex(tlj;,.
ps(t)y
,
ot
,
and we shallseek a solution in the form
I'PN(?))- c-f#''',f(?)1
'.
I-$(0)),
.
,
wheretheoperatorz.
i'
(J)obey'
sthecausalboundarycondition
X(J)= 1
A combination ofEqs.(13.3)and(13.4)yieldstheoperatorequationforvijt):
oziltl fs,/,g
,
*
jj. e
..t c
a
u
zjy
.,Lt)e-jp,/,gtyj
-
#g(?)vjjtl
(13.6)
whereX74/)isintheusualHeisenbergpicturethatmakesuseof-thefullinteractingX.
Equation(13.6)maybesolvediterativelyfor?> tf
j
f(t)=l-ih-lJl
'
adt'X'
7(t')+--'
z.
where the causal boundary condition (Eq. (13.5)) is automatically satissed
because4extr)= ()ift< to. Thecorrespondingstatevectorisgivenby
f
1
Ps(?))=e-iW'7
'(
kI
's(0))-ih-te-f
J
?'/'Jê
'
odt'P7(r'
)t
'
Fs(0)'
)+''.
.
LINEAR RESPONSE AND COLLECTIVE M ODES
173
A 1lphysicalinform ation ofinterestiscontainedinm atrixelem entsofSchrödinger
pictureoperatorsös(J)(whichmaydependexplicitlyontime)
tJ(?))ex> tTx'(?)l:.s.
(?)l'Ps(?))
-
-
f'1p
s
'
(0)l(1+,
â-'J'dt-47(/3+..1ei
*t
/'osl
'
tsc-i
/r
/.
x
(l-ih-1jf
'
:dt'Xjxtf'
)+-..
1l
'
Ps(0))
(kl's
'(0)I
0,(r)I
'1',(û))+xïâ('rkl
'z
'
,(0)IJl
'cdt
'
z?7(?'),:s(?)!f'.
l'
-s(0))+ .. (13.9)
OnlythelineartermsinXe'havebeenretained,andthesubscriptH denotesthe
Heisenbergpieturewith respecttothetime-independenthamiltonianX (compare
Eqs.(6.28)and(6.32)). Thefrst-orderchangeinamatrixelementarisingfrom
an external perturbation is here expressed in term s of the exact H eisenberg
operators ofthe interacting butunperturbed system. In particnlar,if 1
àFs)
and I
tP'
H)b0thdenotethenormalizedgroundstate (
11P0,
)h,thelinearresponse of
the ground-stateexpectation value ofan operatorisgiven by
8rö(?))- tötrllex- (:(?))
=
m-lJC
'
Qdt'(.
I.
'
(
)!
gz?7(?'
).
Js(r))J
'.
1%)
(13.10)
A sa speciscexam ple,considera system with charge e perparticle in the
presence ofan externalscalar potential+f'(x/),which is turned on at t= G.
Thecorresponding externalperturbation isequalto
*bxltj- J#3xgstx/ltyeztx/)
(13.11)
wheregs istheexactparticle density operatorin the unperturbed system . The
linearresponsem aybecharacterized bythechange in thedensity
b'
Lhlxtl?
b=/
â-lJ!
'
:#?'.
f#3x't
yextx't'
)(YI
%Jys(x't'),
Hs(x/)))
Y
1Pa)
=1
4-1Jt
'
o#?'j
i
-#3x'E
'
+*'(X'J'
)(V'0l(
Hl
f(x't'),Hzf(X?)1l
Y'
%) (13.l2)
wherewehavenow introducedthedeviationoperatorsl
qt
jlxtj= ?in(xJ)- Itàktlxtjf
(compare Eq.(12.3)). (Note thatthe c numbers always commute.) If the
retarded densitycorrelation functionisdefined in analogy with Eq.(12.5)
1'
*g((?
1 (x),?
1zf(x')1(.1-q)
a
,
, (:
iD (x,x)=0(t-/) -.-#(-5-wo.-.1,
.
-oj--Y.
...- ...
..
thenEq.(13.12)mayberewrittenas
3(?
Xx/))=â-tj*dt'J#3x'DRlxt,x't'
jt
ve'tx'l'
)
(13.14)
174
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
whertthtcausalbehaviorisenforced bythe retarded natureofD R,and wthave
used thefactthat@exvanishesfort'< ?(). Equation (13.l4)typisesageneral
result that the linear response ofan operator to an externalperturbation is
expressibleasthespace-tim eintegralofasuitableretarded correlation function.
Ifthesystem isspatiallyhomogeneous,thenDR(x,x')= DR(x- x'),andit
isusefulto introduceFouriertransforms
+extk,a))= J#3xJdte-ik*xeico'+extx/)
3(H(k.to))xJd5xfdtE'-ik*xeizo'3(H(x?))
XR(k,(z))>j#3xjWe-ikeltzftufDR(xt)
Equation(13.14)immediatelyreducesto
3(H(k,t
x)))= â-1DR(k,f
.
t))etpextk,tsl
(13.15)
(tl3.l6)
(13.17)
(l3.18)
which showsthatthesystem respondsatthesam ewavevectorand frequency as
the perturbation. This relation is som etim es used to dehne a generalized
susceptibility
J
walkrtzalN
3(?i(k,ts)) -îoatk
rxk (
.
,) = h
'(.
,)
et ( ,
(j?.j9)
Such relations art especially useful in studying transport coeë cients, which
representcertain long-wavelength and low-frequency lim itsofthe generalized
susceptibilities(compareProb.9.7).
The foregoing analysis shows that the linear response is most simply
expressed in term soîretarded correlation functionsofexactHeisenberg operators. Unfortunately, such functions cannot be calculated directly with the
Feynm an-D yson perturbation series becatlse W ick-s theorem appliesonly to a
tim e-ordered produetofoperators. Consequently.itisgenerallyeonvenientto
detine an associated time-ordered correlation function ofthe same operators,
which necessarily hastheform ofEq.(8.8). W ick'stheorem can now be used
to evaluate thetime-ordered correlation function in perturbation theory. The
remaining problem ofrelating the time-ordered and retarded functionscan be
solved with the Lehm ann representation. A specific exam ple has been given
in Sec.7,whereG(k,te)and GR(k.(
.
t
p)wereshown to satisfy Eqs.(7.67)and (7.68).
Thtm ethod isclearlyverygeneral,and westateherethecorrespondingrelations
forthedensitycorrelation fknctions(Prob.3.8)
Rep(q,o4 = ReDR(q,f
.
t))
lm D(q,o4sgnf
.
tl= Im .
DR(q,oa)
(13.20)
which are valid for real(,
t). (In thisexpression.sgnts> f
.u/'jf
.
z
al.l Equations
(13.20)areveryimportant,becauseans'approximationforD(q,(z?)immediately
yieldsan approximate DR(q,u))and hence theassociated linearresponse. ltis
also clear from the Lehmann representation for D(q,to)that the poles ofthis
LINEAR RESPO NSE AND COLLECTtVE M ODES
175
function occurat the exactexcitation energies of those states ofthe interacting
assem bly thatare coupled to the ground state through tl
ne density operator.
IK SC REEN ING IN A N ELECTRO N G A S
A s our firstexam ple,w e consider the response ofa dcgenerate electron gasto a
static im purity with positive charge Ze,w here the externalpotentlalis given by
(reftxl)= Zex-1
(I4.l)
and
tJ'
*X(q,tz?)- brrlzeq-23((zp)
N ote that we have here let tv ...
+ - cc. This point charge alters the electron
distribution in its vicinity,and Eqs.(l3.l6)and (l3.lSJtogetherdeturmine thtt
induced particledensity to be (forelectrons.the i1teraction is-.e(
;ex)
tir
'?
i(X))'= -42=)-3j'J3t?é'fq*XDR(q.0)4n.
zezlhq2)''1
Equation (12.14)showsthatthe time-ordered densitl'correlation function D i$
equalto JkI'Iswhere1-1isthetim e-ordered polarization part. If17Risdefined as
the corresponding retarded polarization. then Eq. (l4.3) assum es the sim p1e
form
3r.
'?
i(x),
'= -(27r)-3(J3gé'IQ*XflR(q>0)4.
c.
276:2tj-2
= .--(
2,
77.)-3fJ3t
yvf
q*x11R(q,0)Z (.'
t)(q)
= - (
2,
c'
)-3z fJbqépiq*xgH *(q,O)f
.,
.
'(q.O)jR
= ,-(.
2:
7.)-3Z (dJt:
/cfq-'
itlK.R(q.0)1'-t- 1)
(l4.4)
wherethethird linehasbeen obtained with Dl'son-sequatlon (seeEqs.(9.43)and
(14.5)),and thefourth with the retardedversionofEq.(l2.1'
;). ThepreNtous
perturbation analysis(Sec.l2)allowsusto calculatethetime-ordered functions
Fl and <,and theLehmann representation then l'ieldsgcompare Eq.(1.
3.20)for
o,
.
,
.
s/jI-I1)
11R(q.co'
)> (Re-c.1sgnu?lm)l1(q,(o)
= Re1
1(q.(o)+ 1sgntslm F1(q.t
.
z?)
xR(
'q.a?)= Rea'
(q,(z?)-z.l'sgnu)lm vtqs(,?l
A com bination ofEqs.(14.4)and (l4.6)then proNidesan exacldescriptio1)of
.
the screening abouta pointcharge.
ln the approximation of retaining 0111/.ring diagrams.s,(q.O)is ptrrell'
realgEq.(12.49)1,andtheretarded functionbecomes
xr
'(q,
0)-,
o(q.
0)-1+4ar,F
cs
zt=f
?2)-igjj.j
176
GROUND-STATE (ZERO-TEMPFRATURE) FORMALISM
Herethefunctiong(x)(Eq.(12.64))isgivenby
r(x)=1j-jlk(1-'
$xa)jn I
1
j+
-è
jxt
1
,
(j
4.g)
and hasthefollow inglim itingbehavior
:(x);
k;1+ O(x2)
g(x);=!.+ .
t(x- 2)lnR1x- 2l)
Ix- 21< 1
(14.9c)
(14.9:)
.
x> 1
(14.9c)
g(x)e-jx-2
Equation(14.7)may besubstituted into Eq.(14.4);theinduced chargedensity
then reducesto
3t)(x))r- -p3(?1(x))r
- -
4ars=-1glqlkè
dTq iq-x
Zej(a.)3e (wks)2+yxrs=-lglqlkp
(l4.10)
Thisexpression hasseveralinteresting features:
l.Thetotalinduced charge iseasilydetermined as
bQr= j#3x(
5()(x)),
nars=-ig(ç//cs)
- zefd?q 3(q)(t
?/k,.)z+ 4ars=-j.g'(t#kF)
(14.11)
which showsthatthe screening iscom pleteatlargedistances.
2.Theintegrand ofEq.(14.10)isboundedforal11t
?Iandvanisheslikeq-4as
q-->.:
;
o (compareEq.(14.9c)1. Hencethe induced chargedensity iseverywherefniteincluding theorigin because
1(3?(x))) < 143/(0))r!
-
Ze
vlsq
zurs,r-lgtwlsl
J(
-2
-,
0
-3(f
pks)2+4ar,,r-1gtq/ks)<*
(14.12)
H erethesrstinequalityarisesfrom theoscillatoryexponentialwhich reduces
thechargedensityforx# 0.
3.Thesingularq-ldependenceforsmallqliscutofl-at
çmin=
4ars 1
4kF 1
'
rr
'
n'
aç
t
kr=
=
6=ne2 +
Ek
M qrF
(14.13)
(seeEqs.(3.20to3.22),(3.29),and(12.67)),whereq1.FistheThomas-Fermil
IL.H.Thomas,Proc.CambridgePhil..
O c.,23:542 (1927);E .Fermi.Z.Physik,48:73(1928).
An elementary accountofitsapplication to metalsmay befound in J.M .Ziman,keprinciples
oftheTheoryofSolids.''secs.5.1to 5.3,CambridgeUniversityPress,Cambridge,1964.
LINEAR RESPONSE AN D COLLECTIVE M ODES
wavenum ber.
wherebpwstx)isthatobtained in theThomas-ll
krmiapproxlmation.
2 h2
P= 5
-2
- -(.
3v:2)'
in!
zA?
(14.16)
lfweputachargeZeintoauniform electrongaS(imposedonaunifbrm.positive
sxed background of charge density (v?a that m akes the unperturbed sl'
stem
neutral).then the condition oflocalhydrostatic cquilibrium requiresthat the
forceson a small(unit)Nolumeelementmustbalance
N'Fj= 0 = -V# - ta/?tj
7
where& istheresulting electricfleld. Poisson-sequation becom es
where (J-isthe electrostatic potential. W e can now write
n - llv= èn
V??= V 3z?
2-k2
l
--(
3,
;r2).
i.- vjp:= rvv
3 2p?
n%
(l4.20)
Since the leftside isalready linearin sm allquantities,wecan use Eq.(3.29)to
m'rite
2
lt2k2 I
. - - z-32
v j??= ev t
j--
z?7 ??f)
178
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
orequivalently
(V2-f
?1.
,.
)3p(x)- zeqlw3(x)
(14.22)
wheretheThomas-Fermiwavenumberisdesned in Eq.(14.13). Thesolution
tothisequationisthatquotedinEq.(14.15)
V rstx)= -ZeqIw(4=x4-te-qrrx
(14.23)
Theapproximateresultin Eq.(14.15)isincorrect,however,becauseglx)
has a singularity at x = 2, where its srst derivative becom es infnite. The
presenceofthissingularityintherangeofintegration(0< q< co)gives3()(x))r
analgebraicasym ptoticdependenceon x in contrastto theapparentexponential
behavior arising from the approxim ate sim ple pole at q= il
'
(
?rs. W e m ay
extractthe correct asym ptotic behavior of3t.)(x))rin thefollowing manner:
firstrewritethelogarithm appearinging((
#ks)as(seeEq.(14.8))
/2+yl
ln;t+ Lk.:.-I
jInj
ni
.m
+o.
(q+uj.sju-y-'
n.
-
--
2/fs;
fq- 2/:
...
-
Sinceg isan even function ofitsargument,theintegralin Eq.(14,14)can be
written as
z
-c
o
2
j.
';(x).r-.j.
-.
r
(.
l
;
.
xj..q#t
?ek
qxq,+ql
t
?
rg
-l
q
.
-k/
v'
l-1
(14.
24)
-Fhe integrand is now an analytic function ofq with the singularity structure
show'n in Fig.14.1,and thebranch cutsofthelogarithmshavebeen chosen so
thatthe logarithm isrealalong the realaxis. Thecontourcan bedeformed as
indicated,and the pole atq;4;iqvy givesthe eontribution ofEq.(14.15),which
vanishesexponentially forIargex. Incontrast,thecutsextend down to (within
n)therealaxis. Theintegralsalongthetwobranchcutsdependonthediference
ofthe function acrossthe cut.
'thisdiPkrence arises solely from the phase ofthe
1ogarithm .and u'ith the branch cutsasshown w'e have
'
llog(q-2
k
;
2
.
r
2
î2
-..
-.
.)
2.-.
.)
2 .
=j
..
tfy- 2l-s) ?-î;
)-=
onCj
onC2
l a
F
t
i
g
on
.
1
o
4
f
.
1
8
(
/
C
(
x
o
)
n
r
t
.
o
urf
ora
s
y
mpt
o
t
i
ce
v
al
u
-
LINEAR RESPONSE AND COLLECTIVE M ODES
179
where u
'
ï indicates the value ofthe phase on the rightside of the cut m inusthe
value of the phase on the left. Because of the decreasing exponential in the
integrand,allthe slowly varying functions in the integrand can then be replaced
by theirvaluesatthe startofthe branch cut. Thus we have
q2
ïH 2k
j;z
r
..
s
which was first deri:ed by Langer and N'osko.1 The expression (14.2647ô i<
qualitatively differentfrom thatpredicted in L-q. (l4.1f)and eNhibits1ong-rangc
oscillationsN.
Nith a radialwavelcngth =,Ik'y and an enN'elope proportionaito .v''.
It is clear that 3 .p
'k(x),is an im proN'
em(!11t oN't
?r èf.
lz.
f(x).s1!
-1cc t11e forrrle1'i11corporates the distribution ftlnction of the interacting mediun) i11com pLlti11g
theresponseto theexternaltield.
From a phl'sicalpointofs'ieu . the long-rallge ost?illations in the scrccl)iI)g
chargearisefrom thesharpFerrriiSurface. because ltis Ilotpossible to construct
a sm ooth functio1 outo1
-the restricted setot
-$5a'
ke vetzt0rsq -'
-ky. T hisetlkct
was t
irstsuggested by Friedeljzand such Frie(l('Ioscillatiotls have been obser:etl
asa broadening ofnuclearm agnetic resonancelinesin dilute alloys.3 A sinlilal
eflkct also occurs i11dilute m agnetic a1loj$,'the conduction electrons lnduce :
).il
indirectinteraction between magnetic in'
lpuritiesoftheform xl/cos(7.
1/..vj,).
where xi.l is the separation of the im purities.4 .
A t lou butfinite telnpcrattl1'es.
the Ferm isurface issm eared overa thickness/
.'sF in energy. and itturns otlttllat
iJ,S.Langerand S. H .Vosko,J.Ph-vs.(-/?t????.Solit1î.12:196(l960).
2J.Friedel,Phil.At
ftzg..43:153(l952):N'llot'oC'
I
'
?'
>l
é'??/(
?,7:287,St
lppl.2(l958).
3N .Bloem bergen and T. J.Rowland,Actu -Wtz?.,1;731(1953);T.J.Rowland.Phb'
s.Ret'
..
119:900(I960);%V.Kohnand S. H .Vosko.#/lp.
î.Rek'..119:912(1960):scealso.J.N.
1.Zlnpan.
op.cit.,secs,5.4.qnd 5.5.
4M .A.Rudermanand C.Kittel, Pll.b'
s.Ret'
.,96:99 (1954).
18c
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
Eq.(14.26J) must be multiplied by the factor expt-zzrn?/ksrx,
//j2:'s). The
im portance ofa sharp Fermisurface isconsrmed by the behavior in a super-
conductor,where the Fermisurface issmeared overan energy width -.
î<.tE'k.
evenatF = 0(seeChap.13). ln thiscase,theasymptotic form ofthescreelling
density isproportionalto x-3coslz/csxlexpt-/
fsxl/ey),completelyanalogous
to thatfora norm alm etalatsnitetem perature.l
ISZPLASM A OSCILLATIONS IN AN ELECTRON GAS
lt hasalready been pointed outthat11(q,cg)has polesatthe exact excitation
energy ofthosecollective statesofthe interacting system thatareconnected to
theground statethroughthedensityoperator. Recalling Eqs.(9.435)and (9.46)
-.y)--.-l.= j.
j-g()tqlI-I(q,(s)
V&(
..
tq) Klq,f
-t)i
o
we observe that v(q,(z?)vanishesatthese same energies. ln the ring approximation,Eq.(12.50)showsthatthedielectric constantK.
rhasone obviouszero,
occurring forfxed energy transfer y'and long wavelengthsq .
-+.0
4ar
Krlqstz?l= 1- 1-.:-,X2
77
.
Thisquantity vanishes at
v2
J'
j= 4=rs(3=)-1
Rewriting thisexpression in dimensionalunits (see Eqs.(3.20)to (3.22),(12.35),
and (12.52))we hnd a collective excitation atthe classicalplasma frequencyz
given by
j)2
#j=
4=nel
--
(I5.4)
W e shallinvestigate these plasm a oscillations in m ore detailby considering
the linear response of a degenerate electron gas to an im pulsive perturbation
extxf)= pïq-x(
)?a3(:)
I
whose Fouriertransform isgiven by
%*'(k,f
.
o)- +0(27r)33(q - k)
The corresponding induced density perturbation becomes
3(l
i(xp))= -t>(2=)-4j-#3/fdulcik*xe-il
x'tI4R(k(sl(
?cextk,(z?l
- -e+()piq-xtzvl-l.(dole-iu'fIIR(q'(,,)
= -e(;)c:iq*x(2=)-1.
fdcop-ia,!&0(q)-1tgxR(q,(.
s)j-l- j)
tA . L.Fetter.Phys.Reï'.,140:,
*1921(1965).
2Theclassicaltheoryofplasmaoscillationsisdiscussed attheend ofthissection.
-
W
'
LINEAR RESPONSE AN D COLLECTIVE M ODES
181
which showsthatthesingularitiesof1lRin the con)plex (
.
o planealso determ ine
the resonantfrequencies ofthe system .
A Ithough Eq.(I5.7)isexact,weshallconsideronIy the approxim ation or
retaining the ring diagrams. In thiscase K.v
R(q.r
.
sliqgiven by L'qs.(l2.24)and
(14.6)as
K
&(q,t
r
o)= l--k'tq)1'10R(q,u,j
where gcompareEqs.(12.29)and Eqs.(I4.f)j
110R(q,r
.
s).
.Re110(q.u,)-.1sgn(
.
z)Im 1IC'tq-(
.
r?)
2 - cl5l'
(1--??C
k'q)??k0
llkî
b-q(1-.n%
k)
h (277.)3 (s.-cok --l
-oq k -- i1t' f.o -.f-tlk- r-zpk q .
.
.
.,
- -
.
)'
. t.3'n'
y
).
h
.
l'b';
jk()-q . 'y-':k()
'y
'
((2.
,
.
)3c
o-tt
okq-.upk)-v/'?/
--
--
-
whereso
k= f?(/fs--k). Thus110RdiPkrsfrom llt
2on1y in the lntinitesim als zix).
.
The frequency and Iifetim tlof the coIlectlve modesare determ lned by the poles
ofthe integrand in Eq.(15.7).J These occuratthe values!
.jq- iyqthatsatiSfy
the equation
1-- k'(q)1-10R(q.f..
1q --iyq)
.
In general, this equation can be soll
ked on1). u.ith 1
)um er!caI a:
1a1jsls: it
-the
dampingissmail('
yqw'
ktjq).houever.thentherealand imaginar),'partsseparate,
and w e lind
l=-Iz'kq)R el-l0Rlqsf.1q)= l'tq)Rel-1C1(q-f.jq)
-.' =
z:
(if.l1)
? R ellOR(q.t.
??) -i
lm l1OR(q-f.1t) . . ..
?
J
#
PRe1l0(q.o,) -l
= sgn.
t2qIm l-l0(q-t2#) .
.
do
th
Equation (l5.1l)determines the disperslon relation (1# ofthe collectise mode,
while Eq.(l5.l2)then yields an explicit form ula for the dam ping constant.
This approxim ate separation of real and im aginary parts u illbe àhow'n to be
valid atlong wavelengths.and we now consider the expansion oflI0R forq -'.0.
A1though itispossib1e to expand Eq.tl2.36)forsm a1lf/.&&e insteadwork
directly with Eq.(l5.9). A sim ple change ofN'
ariablesin the firstterm ofthis
lln general, fl&also hasacutin thecompl
ex (splaneJustbelow'therealaxis. with adiscontinuity proportionalto Im 11R(q.u?)(see,forexanlple, Fig.12.9). As?--'cc,how'ever,thiscut
makes a negligible contribution to Eq.(l5,7);hence the donlinantlong-tlnle behaviorhrre
arisesfrom thecollectivem ode,whlch isundam ped in thepresentapproxlnlation.
182
GROUND-GTATE (ZERO-TEMPERATURC) FORMALSSM
expression reducestheintegralto
no'tq,(
sl-2Jjjk
j
)s,,
kgo,-(.,-1
.k-q
).
,.i
,-.
--(.k+ql
-.,)..
j
,
.
j
2v2 dqk
l
Jta.p''
kt&--:q.k/,,+i
ylz-(hqï/zm'bz (15.l3'
-
m
itisclearthatlmH0R= 0if1(z
J1>hkrqlm + hqllzm;inthisregionoftheq- (.
o
plane,ReI1oA= Rel40may beevaluated asan ascendingseriesinq. To order
4 we have
Re11os(q,ts)- 2y2
syt,-2
,
#3k
2âk.q
âk.q 2
J(ayjjnîl+-m
-(
.
o+3(s;
t
s)+.
3 â/çsg 2
-'
j4
--1m(
s
-il+3(m(
,)+'''
lJtd
z,
'
r
k
/Nk-p
x-3
k
=;
'
i
...
k). y2
..
(15.14)
(l5.l5)
and themean value ofklforthe Fermidistribution is.
j.
#/. The dispersion
relation(Eq.(15.11)Jnow becomes
1 4,
s.
11c2
= yyy
-jjj
yj
3 'hk q l
l+jjyyj
c
lt
'
y +''-
(15.16)
which can be solved iteratively to yield
jlq=+:
Dpll+j9
jtt
y
q
sj
,
j
,2+...
where
j-j
$=ne2 1
pj= -..
-?
?
.1
...
-
(j5.jg)
istheplasmafrequenc),andqvr=(6=ne2(e).
)kistheThomas-Fermiwavenumber
introduced in Sec.l4. SinceIml-l0R(f/,j1q)vanishesifpflqr'
>hqkpt
lm + hqllzm,
these collective m odes are undam ped atlong wavelengths. Thisresultarises
f'
rom the approxim ations used in the presentcalculations;when higher-order
correctionsareincluded,theplasm aoscillationsaredam pedata1lwavelengths.l
Theresonantfrequencyatzerowavelengthistheclassicalplasm afrequency
and is therefore independentofh. To clarify the physics ot-these collective
m odes,weshallreview theclassicalderivationofplasmaoscillations.z C onsider
a uniform electron gas;the equilibrium particle density nnm ustequalthatof
the positivebackground nbto ensure thatthesystem iselectrically neutral. lf
'D.F.DuBois.Ann.Phys.(N.F.).7:174(1959);8:24(1959).
7L.Tonksand 1.Langmuir,Phys.Rev.,33:195(1929).
LINE'AR RESPONSE AND COLLECTIVE M ODES
183
theelectron density isslightly perturbed to
nfxt)= no+ bnlxt)
(15.l9)
the resulting uncom pensated chargegivesriseto an electricheldJ*thatsatisfes
Poisson'sequation
V.d:(x?)= -4rreù(x/)- n,)= .
-Anebnlxt)
(l5.20)
Newton's second 1aw determines the force on the electrons in a small(unit)
volum e elem ent
m#
dln?
fl-mgpt
?
n/
v)+.(y.v)(?y)
j-.
-eni
pv
mno œ .
-ennéb
whiletheequation ofcontinuity may bewritten as
Dn
83n
Jè+.V.(rlY);
4:-t.t
.- ?7oV .v.
-0
(15.22)
Both Eqs.(15.21)and (15.22)havebeen Iinearized in thesmallquantitiesbn and
v. ThetimederivativeofEq.(15.22)maybecombinedwith Poisson'sequation
and thedivergence ofEq.(15.21)to yield
q.
::2
0230n(x?)= -,7f
n-?V .d'(x?)= -4
)yêV .&'tx?)= e
---=n --ènlxt)
....
t2
m
m
or
82bnlx:)=
ètl
- --
-.
j)2
tx/)
>'jgy
--
Thus the perturbed charge density executes sim ple harm onic m otion with a
frequency Dpl. Note thatEq.(15.23)doesnotcontain spatialderivatives,so
that there is no m ass transport. This result agrees with the specisc form of
Eq.(15.17),because thegroup velocity oL'
jqt
'
?t.
pvanishesatlong wavelengths.
16LZERO SO UND IN A N IM PERFECT FERM I GAS
Section 15 shows thata charged system can supportdensity oscillations with the
long-wavelength dispersion relation (.
o;4;f'
lpj. Thisrepresentsa true collective
m odebecausetherestoringforceon thedisplaced particlesarisesfrom theselfconsistentelectric seld generated by the localexcesscharges. ltisinteresting
to ask whether a sim ilar collective m ode occurs in a neutral Ferm i system at
r = 0. A s shown in the subsequent discussion, a repulsive short-range interparticle potentialis suë cientto guarantee such a m ode,atleastin a sim ple
m odel. The resulting density oscillation turns out to have a Iinear dispersion
:84
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
relation (
.o= coq forq --.
>.0 and is known as zero sound.l Nevertheless,zero
sound isphysically very diflkrentfrom ordinary frstsound,despitethe sim ilar
dispersion relation (.
o= cjq. The distinction between the two density oscillations depends on the role of collisions:ordinary sound can propagate only if
the system is in localtherm odynam ic equilibrium ;this condition requires that
them eaninterparticlecollisiontim e'
rbeshortcom pared to theperiod ofoscilla-
tion2.
6 u)(thatis,u)r<:
4l). Incontrast,zerosoundisacollectivemodesustained
by the coherent self-consistentinteraction arising from neighboring particles;
zero sound thusoccursonlyin acollisionlessregimewhereoor> l. Thecrucial
observation is that the Pauliprinciple greatly lim its the possible interparticle
collisionsatlow temperature,and,indeed,rbecomesinhnitelike F-2asF -70.t
Ata sxed frequency,there isa criticaltem perature below which ordinary sound
isstrongly attenuated,while zero sound propagates freely. At T = 0,ordinary
sound ceases to propagate at any frequency, and only zero sound can occur.
ln an electron gas,the plasma oscillationsappeared asa resonantresponse
to an im pulsiveperturbation. A very sim ilaranalysisappliesto a neutralFerm i
system ,where the perturbing ham iltonian m ay be written quite generally as
4extrl= jdsxtyxt)&ex(x?)
(16.1)
Here U'slxt)isan externaltime-dependentpotentialthatcouplesto thedensity.
Thesubsequentanalysisisidenticalwith thatofSec.13,and thelinearresponse
isgiven by
3(t?
i(xr).
)= (2*-4Jd3kJ(
oeik*xF-ït
z
:tHR(k1(sl&ex(k,f.
t))
(16.2)
Forthe specialcase ofan im pulsive perturbation
Utxlxt)= &7)
Keiq-xb(t)
(16.3)
a sim plecalculationyields
:(ri(xf))- t/7eiq-xlzrrl-lfdcot?-fc'
af&a(q)-l((xR(q,(
,a))-l- 1)
(16.4)
in completeanalogy with Eq.(15.7). Theresonantf
-requency forwavevectorq
isagaindetermined bythepolesoftheintegrand,which occuratthezerosofthe
retardedgeneralizeddielectricfunction&R(q,(,?).
Thesimplestapproximation to vtq,(slconsistsinretaining onlythezeroorder proper polarization partl10;in this case.the pole occurs at the value
f.q- iyqdeterm ined by
l= P'tqll-I0A(q,flq- lkq)
Weassumethatfk exhibitsaphonondispersionrelation
fk = cvq
(16.5)
(16.6)
lL.D.Landau.Sov.Phys.4ETP,3:920(1957);5:101(1957).
1Thisresultdependsonlyontheavailablephasespaceandwasfirstnotedbyl.Ia.Pomeranchuk,
Zh.Eksp.Teor.Fl
'
z.,20:9l9(1950).
UINEAR RESPONSE AND COLLECTIVE M ODES
185
sothatthe ratio L'
jq/'
qrenlainsfixed asç -.
+ 0. Therelevantlim itof110hasalready
been calculated in Eq.(12.48),and the associated retarded function may be
writtenas(q--.
>.0)
mk
l
ix ->.lq
i'
rr
170R(t?,(
s)- =ahCj ixln x--i - l- '1'xp(l- :
.x:
)
wherex= mœt
lhkFq. Thefactorx= 1x1sgnx intheimaginarypartretlectsthe
change from the time-ordered to the retarded function. An undam ped m ode
ispossible only if '
:.
x.
1> l. ln thiscase,the long-wavelength dispersion relation
isgiven by
= 2:2
'x + l
q-0tnksj
;
y
((
yj=l.
x1njx.j1
,-1=*(x)
lim
where
o >1
x = lim mLj%= mc?- -
:-+0hkFq #t-s t,F
and L's isthe Ferm ivelocity. W e see thatzero sound ispossible only ifc'o > è's.
The function on the right side of Eq.(l6.8)willbe denoted (.
1)(.
v),
'it is
sketched in Fig.16.1. The mostinteresting feature isthelogarithm icsingularity
atx = 1. lfwe assume that U(t
?)approachesa sniteconstant P,(0)asq -.
-+0,
then the speed ofzero sound isdetermined by the intersection of*(.
v)w ith the
horizontalline rr2â2/p?/t
.sP-(0), lt is clear that there is no intersection unless
U(0)7w0,w'
hich implies a repulsive potential because U'(0)= ft/3x I
z'(x). In
thiscase,the explicitsolution is readily found in the weak- and strong-coupling
lim its:
W eak coupling:
2=2hl
c'
oQ:t'y 14.2exp -mk,p'(@ - 2
hl
P'(0).
,
tmV.
.
(16.l0)
186
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
Strong coupling:
co ;k;rs
F(0) 1 k)F40)1
3=2(ha/
gmks) ;k; -jn.
2m
;4J(nP'(0)rn-1)1
h2
P'(0)> mkF
Equations (16.10)and (16.11) show thatct
lis no'nanalytic in the interparticle
potehtialand thuscannotbe obtained w ith perturbation theory. Indeed,the
present approximation of retaining only the lowest-order proper polarization
cannotbejustihedonthebasisofperturbationtheoryforashort-rangepotential.
lnstead,weexpectthatthelogarithmicsingularity of*(x)forx ,
.
> lwould also
occurin morerealisticapproxim ations,
'aninlprovedcalculationwouldtherefbre
renormalize the numericalvalue ofco/t'y butnotalterthe qualitative physical
phenom enon in the weak-coupling lim it. This assum ption is borne out by'
Prob.5.8.w hereaselected classofhigher-orderpolarization insertiol
lsisincluded.
ltisinterestingto rewrite Eq.(l6.ll)as
(I6.l2)
whichshowstheimportanceofashort-langepotential. Ifjz
'(4
2)wereunbounded
asq -->.0,the characterofthe dispersion relation would bequalitatively diflkrent;
inthespecialcaseofacoulombpotential(l
.
'(ç)= 4=e2/q2à,Eq.(16.12)reproduces
the plasm a frequency found in Sec.15. From this view point,zero sound and
plasm a oscillationsare physically very similar;they diflkr only in the detailed
form ofthelong-wavelength dispersion relation,which issxed by the behavior
ofql1
z'(ç)asq->.0.
Forcom parison,we shallbrieiy review the classicaltheory ofsound waves
in a gas,in which the equilibrium m ass density po= m nois slightly perturbed
Xxt)= Po+ 5P(X.
!)
(16.13)
Therestoring forcearisesfrom thepressuregradient,and N ewton'ssecond law
becom es
d(P5')=0V!)+(v.V)(p&);
k:po0
Y= -V #
ot
...
dt
.
ê/
(16.l4)
Correspondingly,theequation ofcontinuity reducesto (compare Eq.(15.22))
obe -
'
tt
-
v .(pv)x .--pov .v
whereboth Eqs.(16.14)and (16.15)have beenlinearized in the smallquantities.
LINEAR RESPONSE AND COLLECTIVE M ODES
187
A com bination oftheseequationsyields
:2&f= v2p
d/1
(16.16)
The system hasan equation ofstate#(p,5')relating thepressure to the density
and entropy. lf-thisequation isexpanded to srstorderin thedensityperturbationatconstantentropy,we5nd
v,p-v2z'(p,,s)-?
-(P
j
#
pjs3p-...;
z'
P
à#
-)srcap
p
(16.17)
Hence3p obeysawaveequation
82àp,
a
3tl ,
..(yyv2jp.
where thespeed ofsound isgiven by
OP
ct= P
pm s
and the subscriptm now explicitly denotes the m assdensity. H ere the restriction to constantentropy m eans thatthe process is adiabatic and thatno heatis
transfkrred whilethecom pressionalwave propagates through the system .For
aperfectFermigasinitsground state(5'= 0).Eq.(14.16)gives
Cj=j
(PPj1 hks uv
opyjj = sjyj= x.zZ
j
A comparisonofEqs.(16.10)and (16.20)showsthat
f'o= v'jcl
(16.21)
in thew'
eak-coupling limit. A m oregeneralanalysisbased on Landau'sFerm i
liquidtheorylshowsthatt-oliesbetweenthespeed ofl
irstsound and x'1times
the speed ofsrstsound for a1Icoupling strengthsand thatthe two speedsare
approxim ately equalfor strong coupling. There isnow desnite evidence for
zero sound in liquid H e3,which is a strongly interacting system . Experim ents
indicate thatz(c(
)- t-jlycl;z0.03,in good agreement with the theoreticalestim ates. Landau's theory also allowsa detailed study ofthe attenuation ofzero
sound and firstsound,
'experim entsfullyconsrm these predictions.
lL.D .Landau,Ioc.cit.;A.A.Abrikosov and 1.M .Khalatnikov.Rep.Prop.Phys.,22:329
(1959),J.W ilks,S'ThePropertiesofLiquid and Soli
d Helium,''chap.18,Oxford University
Press.Oxford,1967.
2B.E.Keen,P.W .M atthews,and J.W ilks. Proc.Roy.Soc.(London).A2&f:125(1965);W .R.
Abel,A.C.Anderson,and J.C.W heatley.Phvs.#el..Letters.17:74 (1966).
188
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
I7JINELASTIC ELECTRO N SCAU ERINGI
W e nextconsiderinelastic electron scattering from system s such asnucleiand
m etals. For sim plicity we retain only the coulom b interaction between the
electronsand thecharged particlesin the target
#ex= - p2
le1(X')l(X)d'Xd3X'
lx- x'j
...-- .
Throughoutthis section,the charge density operatorfor the targetisdenoted
éj(x),since,in principle,)(x)can difl
krfrom theparticledensityoperatorH(x)
(forexample,inheavynucleiwithalargeneutron excess). Thesmallvalueof
the 5ne structure constant(ezjhc ;
k'1/137)allows us to analyzethescattering
processin Born approxim ation. The m atrix elem entforthe electron to scatter
from an initialplane-wave state lkâ'h(ydenotesthespin projection)to a tinai
plane-wavestate !k's'bisjusttheoverlapoftheinitialand finalelectronwave
functions2
l
f
tk's'')eJ(x')k.
s
-)= :fe.x,us
t'lk,)u,(k)
.
where the u'sare D irac wave functions forthe electrons, t1isthenorm alization
volum e,and we have introduced the three-m om entum transferred from the
electron
âq> âtk- k')
(17.3)
ln an inelastic electron scattering experim ent.thethree-m om entum transferand
the(positive)energyloss
hf.
oEB(V- k')hc> 0
(17.4)
m aybevaried independently,theonlyrestriction beingthatthefour-m om entum
transferbe positive
V.
2% (k- k')2- (k- k')l= 4kk'sin2(.
j9)> 0
q2 - (
.
,,2c-2> ()
where we assume ultrarelativistic electrons with f= hkc and b isthe electron
scattering angle. W ith therelations
fefq-x 1 )(x)dsxdsx,= 4=
jx x'1
qz â(-.q)
,
-
(17.6)
/(-t0K fPCQ*X)(x)#3x
1Foradetailed accountofelectron scatteringfrom nuclei.seeT.deForestand J. D .W alecka,
Electron Scatteringand NuclearStructure,inAdvan.Phys.'15:1(1966).
2See,forexample,L.1.Schië Asouantum M echanics,''3ded.,chap.13,McGraw-HillBook
Company,New York,1968.
LINEAR RESPONSE AN D COLLECTIVE M O DES
189
the electron scattering crosssection for an unpolarized targetm ay be written as
#2
20-1 --q
a= j-j/,
s
.
s'
n
f2)c1?k'
bg/jts-(En-f'
oljk'
.t.
I-,,:;(-q):
tF(),
)r2-jyyjj.
4.
n.2 c,2 2
s
/c. -l
(q?1(j2)I
u1'
(k'
)?
.,
(k)2tjj)
w hich follou'
sfrom Ferm i's -eciolden R ule--along u'
ith theincidentelectron flux
c/L1. ln Eq.(l7.8)thestates :1-t) and '1'
11)
'aretheexactHeisenbergeigenstates
ofthe targetparticles. The spi11sum sare evaluated in the ultrarelativistic 11m 1t
w'
ith the relationl
lj',
1p
'
s(..qjtt'
(; 2jgho.
y- (E4
y- 1-()jj
('',. -
g.
2 24).'2
j
(p: 2 oos2((
'
è(
2)
:-tt?c)
(,c)q-cesz(?)-, (,c)4kzslI
'-.-
Q .,.,,, --
t'7.'C))
.
Forthe rem ainder ofthissectio11$5.e.shallconsideronIy illclastic scattering
(thatis-(.
,?n.0):in thiscasetheoperatorp
'hin Eq.(l7.9)may be replaced by the
Quctuation density
without changing the result.
side of Eq.
(17.9)as
V1-,,?(-q)V1.-() 2bg/kf
.
zp- (En- 1())j
x
(l7.12tz)
Im .N. xV
1oI; (-q)I
H'n . n; --q : y
= n
(
so. (u - y j. jly
.1-'
tj*
F(-q.)
'
tj
@
t
'
jl.
j
= j)m
n'x
n-?
.g;'
-!
'
--.)
-j
.*
.
.(-q.
-ao
.
.
n. N
âr
.
o - (En- Eo)- 11)
= - -
.,- -
.
-.
-Xl'*(
?5
.
t.
t..
q).'
.
V.
l'
,
y-.-1
1
-X
.
,#(q)Vl-q-.j ()-;.)2,cj
jJ
t'
s--
.
.
-
(En-z;0):r
s1,
3 )
190
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
Equation(17.12c)isequivalentto(17.12:)sincethelastterm inbracketshasno
imaginary partif(.
o> 0. Thetwo numeratorsinEq.(17.12c)areequalifthe
ground state isinvariantunderrotations,forthen the sum over n atsxed En
mustyield justafunctionofq2. W eshallhenceforthassumethistobethecase.
Notethatthe rightside ofEq.(17.12)vanishes at(.
o= 0 and thusexplicitly
excludesthe elasticcontribution.
ln thisway we obtain the im portantresultl
1 dla
1
- -
O.
M#f1,dz, =
- -
l #2c =
cM dL1't&'
- -
-
rrIm rltq,q;(sl
(l7.13J)
1
sl
= Im l1R(q,q;(
(17.13:)
wherewehavedesned a generalpolarization propagatorforthetargetz
ïâI1(.
x,>')- (:'l'
-ohrE?,/(x)?'
s(.
p)11%'
-o)
(17.1* )
mI1(x'y)- (
t?. dt
.
v
.
-d%
1,,)3 (#3
))
2.
*3 '
jzre'q-xe-foaflx-fy)c-iq.yfârltq'q';o.
(17.14:)
appropriateto both uniform and nonuniform systems(e.g.,hnitenuclei),and
corresponding retarded function
fâI1R(x,>'
)- bltx- /J ('l'o1(#,z(x).J,,(>')!!kF(
))
(17.15)
Equation(17.13J)isimmediatelyverised byinvertingtheFouriertransform in
(17.14:)andthensettingq= q',whichgives
I1tq,
q;t
z
')-)grklcol
pft-qll
f'
n'
l('l
'nI
#(-q)I
'l'o)
':(ht
.
o-(
'
Alfc)+i
.
r
l-âf
,
a+lEnlsc)-i.
q)g(l,.j6)
Equation(17.13:)followsbecauseHR(q,q;f
.
t))diflkrsfrom Eq.(17.16)only by
having a+iy inthedenom inatorofthelastterm . Them om entum conservation
in a uniform system simplihes these results,
'comparing Eq.(17.12) and the
equivalentofEq.(7.55),wefindHtq,q';f
.
t))= V3qq,I1tqstt)l,wherefI(q,(s)isthe
Fouriertransform in thecoordinate diflkrence x - y and P'isthe volum eofthe
target. Therefore
1 d2J
F
,?)
cu #fl,deo= -=Im Il(q,(
uniform system
= -
F
Tr
x
Im H (q,o
'W .Czyàand K.Gottfried,Ann.Phys.(N.F.
),21:47(1963).
2weconsistentlysuppressthenormalizationfactor((1Fc!H%),
)-tinthissedion.
(17.17)
LINEAR RESPONSE AND COLLECTIVE M ODES
191
Notethatitisthecrosssection N runitvolume(orN rtargetparticle)thatis
themeaningfulquantity foran extended system .
Inelasticelectron scattering therefore measuresthe imaginary partofthe
polarization propagatordirectly;complete knowledge ofthe imaginary partis
suëcient,however,because the function itself follows im mediately from Eq.
(17.16)
rI(
1 l
l
q,q;o,)- q;a,') hœ ,- yj(a; ixt+ ,.,.,(,,- iv :(âa,')
= o Im n(q,
-
(17.18)
l '
x,
1
1
lRR(q.q;œ)= q;tz)') hœ ,- hœ - i.tt+ âf,a,+ xts + j, #(âœ')
.
n. c Im HR(q,
ltispossibletoconstructsum rulesdirectlyfrom Eqs.(17.9)and(17.13).
forweobservethat
x
Jchdf
x'lt
wlko
#2
;
c
w,
)--.
h,
*ImI1(q,q;œ)A
-'
t'l%IJ#(--q)#('
--*lTn)
= (:1'01/1(-:)X--q)1
'1%)- l(T :l)(-Y I
'I%)I2 (17.19)
Itfollowsthatthetotalintegrated inelasticcrosssection directly determinesthe
mean-squaredensityquctuationsintheground state. W ritingouttheoperators
ofEq.(17.19)indetailwehave
(9%l
?1(--q)?(-q)l
'l%)
fe-lq-x('1'ol'
/1(x),&(x)'
g(y)#j(y)I
'l%)eA-'dqxdqy
-
fe-iq-xt'.
l%I4)(x)4z(x)I
'.
l%)(val4,
1(y)4#(y)1 a)e'q-':3x:'y (17.20)
-
Thecanonicalcom m utation relationsim m ediatelygive
#1(x)'
(ktxl,g(y),i,gyl-&=bô(x-y)f4(x),/,gyl+41(x)4;(y)ygyl#zfxl
(17.21)
ahd
Jd'x#1(x)#atxl- Z
(17.22)
wheretheeigenvalueof; isthetotalnumberofcharged scatterersinthetarget.
Thus we can write
-
h *
=
-
o
dœImH(q,q;tM =Z+Je-dQ**g(x,y)e**'#3x#3.
v
(17.23/)
#(x,y)-('l%l#1(x)#;(y)f#(y)#.(x)l
'1%)
('l%I
#1(x)#.(x)i
'l%)('l%i#â(y)#/y)l
'l%) (17.23:)
-
Thefunctiong(x,y)isameasureoftwo-particlecorrelationsinthegroundstate.
192
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
In the specialcase ofa uniform medium,where g(x,y)= g(Ix- yl),we have
P'
â co
-
-
'/F
o
a
dulIm l1(q,œ)=Z + F fe-iq'xglzjd z
uniform system
(17.24)
Thematrixelementsin Eq.(17.23:)canbeevaluated fora noninteracting Fermi
gas(Prob.5.10)andgive
#04p
x-yj
)=-VJg51k
'6F
2j
S
x1
X
--yY
jJ
.
112
X
X
-
- -
-
-
-
b(h.,-(En-Ef
j))- -
--
X
X
(J)II0(t?.to)
Fig. 17.1 Im fl in (J)perfect Fermigas
(d))ringapproximation.
Thustheintegralin Eq.(17.23J) willbecome smallforlarge q because ofthe
oscillations of the exponential. This sam e behavior is to be expected in the
interacting system ,and we can therefore w rite
li
h x'
m -Tr
eco
0
Im I1(q.q;u))#(s = Z
(l7.26)
lnthislimitthescatteringparticleseesjuattheZ individualcharges.' Notethat
thislim itprovidestheonly really meaningfulexpression because ofthe restriction
in Eq.(17.5)(unlessforsomereasonlm l'
llq,q.
'tz?lissmallforco'
lc> t
?1.
W ediscussthreevery briefapplicationsoftheseresultsin theapproxim ation
thatthetargetcanbereplaced byanequivalentuniform m edium withthecorrect
densityand totalnum berofcharged scatterersdeterm ined from therelation
z = Po= j:;)é
F
(17.27)
Thesimplestapproximationto11isjustH0showninFig.17.1J. Theimaginary
part of the diagram retains only the energy-conserving processes in the inter-
mediate state fsee Eq.(17.l2J)). Thusthe inelastic scattering in thissimple
m odelisthe creation ofa particle-hole pair,orequivalently,the ejection ofa
lThroughoutthisdiscussionwehaveassumedthatthetargetparticleshavenointrinsicstructure.
193
LINEAR RESPONSE AND COLLFCTIVE M ODES
single particle from the Ferm isea. In thiscase we can write
1 d2J
-
,
,
)'
3=Z
.
n'
/c)
= - -
tzsyde #f)
Im l7(q,(z))= -
32
x8 -
Im H(q,(s)
4.
14 2
Im rI0(q,(,,)
(17.28)
ey mkr
whichisgiven inEqs.(12.40)to(12.485)and shownin Fig.12.9. Thisfeature
ofthe spectrum isreferred to asthequasielasticpeak. IfqlkF> 2,there isno
Pauliprinciple restriction in the Enalstate, and the m axim um of the curve in
Fig.12.
9 occursat
â
32a2
frmax= lm-*
(17.29)
(seeEq.(12.41)). Equation (17.29)issimply thekinematicalrelation between
theenergy and m om entum transferred to a singletargetparticleinitiallyatrest.
The spread of the quasielastic peak is due to the Ferm i m otion of the target
nucleonsand the half-width isa directm easureofthe Ferm im om entum . Figure
17.2 shows a com parison ofthe theory with electron scattering data in Ca*0.
Asa second exam ple,considerthe polarization propagatorcomputed by
sum m ing the ring diagram s as in Fig. 17.1:. In using the im aginary partof
IIr(q,(,
J),weincludethe propagation ofthe particle-hole pairthrough theinteracting assem bly
1I
=mrltq(
sl--.
n
1Imf
tlootqlq-lgsr
tq
l,
r
z,
l-1q)
1I
(((&(q))-ljj.gt
l
tqlu,(q,
.)-lj)
g-)ImI10(q,
u)j((l-&i-ct
qlReH0(q,
(
s))2
((&(q)Im I10(q,(s))2)-! (17.30)
- - -
.
m
-
-
This im proved approxim ation keeps the quasielastic peak within the sam e
kinem aticalregionswhere Im I70(q,(s)# 0 butredistributesthe strength within
the peak.
In addition to the quasielastic peak,there are also peaksatthe discrete
collective excitationsofthe system . For an isolated resonantpeak atenergy
hto= âf
aares,theintegrated strength givesthe absolutevalue ofthe inelastic form
factor
.
h lover dul jdfdj2
- !z-.o(q)'2
,a
-resenance
O'
M
jye,
.
where
Fa0(qJH paot-'
q)- 1
-E''q-*('1'
-*1'I
)(x)l
'F0.'
)#3x
.
(17.31)
1GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
Atsxed energy lossâoares.theinelastic form factorcan now bem easured fora1l
42(stillwith therestriction q2> oQJ<2),allowing usto map outthe Fourier
transform ofthe transition charge density. By inverting this relation,we can
obtain the spatialdistribution of the transition charge density itself.l For
exam ple,to theextentthatthe uniform electron gasisagood m odel,thecross
1.2
ca40:jqI= 5% Mev/Ac
Fig.17.2 Quasielasti
c peak in Ca40. IP.
%o 1.0
8=O O
Zimmerman, Stanford University Ph.D.
*
0.8
E
Qt: 0.6
z- l
NI
Z
Nz
#'
NNz
/
curves are calculated from a nonintergcting
Ferm i-gas m odel using the experim ental
relativistic elctromagnetic interaction with
-.-
Y
F
g,
/
q 0.4
z/
'
u
/
Qw 0.2 z
y
0
1œ
Thesis,1969(unpublishedl.) Thetheoretical
a
pj.r
2*
h
hI
I
y
yI
the nucleons. (E.M oniz,Phys..
R0.,1M :
1154(1969).) Thedashed cur'
veisobtained
ï
wavenumber was taken as kF= 235 MeV/
j
3*
Electron energylossAta?(M eV)
byassumingan averagesingle-particlebinding
energy -35 MeV per nucleon. The Fermi
hc= 1.19 x 1013 cm-l. The solid curve in
the lower right is a theoreticalestimate of
pionproduction.
section forelectron excitationzofthe plasm a oscillationsin a m etalisgiven by
(seeEqs.(15.8),(15.10)to(15.12),and(17.17))
1 1 #2c = 4 $W)3 (q.j* hl dReI10(q,(xp1 -1 a
.
e '
kos.yfl'de' 3=3(âDpj)4bke) -lmkF Pf
ltt)-Dq)2+yâ
,p
.
(17.32)
where Z isthe num ber ofconduction electrons. The crosssection issharply
m akedatanenergylosshfh,withawidthhys. Sucheflkctshavebeenobserved
in the transm ission of electrons through thln m etallic film s.3 A very sim ilar
treatmentdescri% sinelasticneutron scattering,asshownin Prob.5.13.
PRO BLEM S
6.1. Consider a uniform noninteracting system ofspin-.çferm ions. Reduce
theretarded densitycorrelation function
iDR(x,x')= 0(t- t')(TcII:zf(x),?
'is(x')11
kF'o)h/'
(T-o1T0)
to desniteintegrals. Considerthefollowing lim its'
.
'For a tunmition G tween discrete states,the phase ofFagqlcan l
x determined from timereversalinvariance. (% eT.deForestand J.D.W al
ecka.op.cit..appendix B.
)
2Thisresultneglectstheexchangescattering G tweenthe incidentelectron and theelctronsin
themetal. Italsoassumesasmalldamping oftheplasmaoscillations;however,theintegrated
:trengtha JTdœttt,p- t'
Iv)I+ yD-'Q$=inEq.(17.32)isindependqntofthedamping.
3'n e comparison with exx rimentsisdescri% d in D.Pinesand P.Nozières,el'
rhe Theory of
Quantum Li
quids,''vol.1,sec.4.4,W .A.Benjamin,Inc.,New York.1966.
LINEAR RESPONSE AND COLLECTIVEYMODES
(J) f= t'
(b4 x= x'
(c) x= x'
.
196
al1xandx'
t- t'< âe-l
t- t'>.neF
-1
and interpretthevariousterms.
5.2. RetainthesrstcorrectioninEq.(14.25)andderivetheasymptoticexpansion(compareEq.(14.26))
3()(x))r - ze
2f k/ cos2ksx sin2ksx 2
. (4+.oz-x
((ksx)2-tksx)34+f
x (fln4ksx- 3+ ((y-j))
wherey = 0.577 .' .isEuler'sconstantand therem ainingcontributionsvanish
fasterthanx-*asx -+.œ.1
5.3. DerivetheThomas-Fermiequationsforthepotentialand electron charge
distribution in aneutralatom ofatom icnum berZ in thefollowingway:
(J) Usethehydrodynamicequation ofstaticequilibrium (Eq.(14.17))andthe
boundaryconditionatr.
->.(
x)toshow thatp(r)= (l/e)(â2/2?n)(3>2n(r))1where
@(r)istheelectrostaticpotential.
(â) From Poisson'sequation and the physicsoftheproblem show thatw(r)
satissesV2+ = K+*withtheboundaryconditions:40);>Ze/rand r@trl.
-.
>.0 as
r-+.*,where v is a constant desned by x'= (8V1/3=J!)(e/cn)-1. (Note:
Jt
p= h2(me2istheBohrradius.)
5.4. Verify the dispersion relation for plasma oscillations (Eqs.(15.16)to
(15.18))directlyfrom Eq.(12.36).
5.5. Show thattheplasma oscillation isdam ped above a criticalwavevector
k'
mas determ ined by the equation .p2= (=rs(=)((2+ ylln(1+ 2y-1)- 2).where
y= knz.xlkF. Show thatzerosoundisalsodampedaboveacriticalwavenumber,
givenintheweak-couplinglimitbyy= 2e-iexpL-l=lhljmkrP'
40)).
5.6. Generalize the discussion of Sec.16 to a hard-sphere Fermigas atlow
density,and show thatthedispersionrelation fbrlong-wavelengthzero sound is
given by (compare Eq-(16.8))*(x)= =l4kFa where x= h/vr. W hatis the
resulting velocity ofzero sound?
1Thecontribution proportionalto lnkex wasobtained by J.S.Langerand S.H.Vosko,
J.Phys.Chem.S(#l
W.
A12:196(19*),buttheydidnotretaina11theconstantterms.
!:6
GRouxo-slwzE (ZERO-TEMPERATORE) FonMAuisv
5.7. (J) GeneralizethetreatmentofSec.12 to expressS
'
âl1zj.pctx,.
v')in terms
ofHeisenberg seld operators.
(b) UseProb.3.l5to provethatEq.(l6.5)with P'(4
/)replaced by lz'
ot(
?)correctly
describes zero-sound density oscillations for spin-dependent interactions of the
form (9.21).
(c) Consider a perturbation X ext/).
-(J3vêtx/l.lJexlx;),and prove thatthe
same system can supportspin waves,described by Eq.(16.5)with P'(ç)replaced
by P'1(t?).
.
''
.
5.8. (a) For a uniform spin-.
s Fermi system with a short-range potential
P'(t?);
k;P'(0)(= const),show thata11properpolarizationinsertionswithrepeated
horizontal interaction llnes across the fkrm ion loop can be sum med to give
H*(t?)= f1)())a.I'lt)t,.
c
-...- 110(:)(1.
t-H0(ty)p'(O),
.
'
(2.$-.-1)1-1(seeFigs.12.1b
and 12.3:).
(b) Show thatzerosoundisnow'described bytheequation*(.
v)= nlhltst:lq.g.1
/(0)
wherex = co/'
t'
y (see Eq.(l6.8)1.
(c) Find the corresponding expression for a dilute hard-sphere gas (compare
Prob.5.6).1
5.9. D efinea tim e-ordered Green'sfunction
w'here &z(x)= ,
(1txlttzzlay'
#,txl, and relate Dg(.
v-x') to Haj.as(x,x'). Use
Prob.4.12J to obtain Dclq)= âI1*
g(q) for a spin-! Fermisystem with spinindependentpotentials. W hydoesDediflkrfrom D ?
5.10. Derivetheexpression (l7.25).forthetwo-particlecorrelationfunction of
anoninteractingspin-lFermigas.
5.11.
gas.
5.12. How is the width at l-maximum of the high momentum transfer
(qy.2#s)quasielastic peak (see Fig.12.9)related to the Fermimomentum ?
5.13. Considerinelastic neutron scattering from an interacting assem bly of
atom s or m olecules.
(J)Thehuclearinteractionbetweentheneutronandafreetargetparticlecanbe
describedwiththeaidofapseudopotentialqP'(Ixa- x1)= (4=ah2,
I2mçqq)3(xa- x).
1K.Gottfri
ed and L.Pi4man,Kgl.Danske Videnskab.Selskab M at.-Fys.M edd.,32,no.13
(1960).
liE.Ferm i,Ricerca 5'
cl
'.,7:13 (1936).
'J.M .Blattand V.F.W eisskopf,tiTheoreticalNuclear
Physics,''p.71,John W iley and Sons.New York,1952.
LINEAR REspoNsE AND co uuEcTlvE M o oEs
197
If this potentialis treated in Born approxim ation.it givesthe exactlow-energy
J-wavenuclearscatteringoftheneutronfrom oneofthetargetparticles. ln this
expressionrnreuandal;
k:10-13cm)aretheappropriatereducedmassandscattering
Iength. Show thatthisresultfollowsimmediatelyfrom Eqs.(1l.5),(1l.9),and
(1l.22).
(>)Henceshow thattheinteractionoftheneutronwiththemany-bodyassembly
isW *X= (4=wâ2/2/??rea)H(xn). ThishamiltonianmustbetreatedinBornapproximation.
(c) lftheatomic interactionsamong thetargetparticlesaretreated exactly,how
m ustthediscussionofSec.17bem odised todescribeinelasticneutronscattering?
6
Bose System s
InSec.5wesaw thedrasticefl
kctofstatisticson thelow-tem perature properties
ofan idealgas. Ferm ions obey the exclusion principle,and the ground state
consists ofa 5lled Ferm isea. In contrast,the ground state ofan idealBose
system has allthe particlesin the one single-particle m ode with lowestenergy.
Since the idealgasforms the basisforcalculating thepropertiesofinteracting
m any-particle assem blies,itis naturalthatthe perturbation theory for bosons
has a very difl
krent structure from that previously discussed for fermions.
Indeed,the m acroscopic occupation ofone single m ode poses a fundam ental
dië culty,and itisessentialto reform ulatetheproblem in orderto obtaina welldesned theory.
198
BOSE SYSTEMS
199
IK FORM ULATIO N O F THE PRO BLEM
n eusualform ofm rturbationtheorycannotlx appliedto bosonsforthefollowingreason. Thenoninteractingground stateofN bosonsisgiven by
1*21))= I#,0,0,.
(18.1)
wherea1ltheparticlesarein thelowestenergy mode. Fordefniteness,wehere
consideralargeboxofvolume F with m riodicboundary conditions,wherethis
preferred statehaszero momentum,butsim ilarmacroscopicoccupation occurs
inothersituations(seeChap.14). Ifthecreationanddestructionomrators41
andtu forthezero-momentum modeareappliedtothegroundstate,Eq.(1.28)
im pliesthat
elth(A))- N*t+q(N - 1))
(18.2)
c1l*n(#))-(N+ 1)11
*(/1+ 1))
ThusneitheraonorJJannihilatesthegroundstate,andtheusualseparationof
operatorsinto creation and destruction parts(Sec.8)failscompletely. Consequently,itis notpossible to deâne normal-ordered products with vanishing
ground-stateexpectation value,and theapplicationofW ick'stheorem G comes
m uch m orecom plicated.
ItisinterestingtocompareEq.(18.2)withthecorrespondingrelationsfor
fermions,where the occupation numberscannotexr- d 1,and any single mode
atmostcontributesaterm oforderN -tto thethermodynamicprom rtiesofthe
totalsystem. Ontheotherhand,theoperatorsazand/1foraBosesystem multiply theground stateby N*or(N + 1)*,which isevidently large. Since itis
generally preferable to deal with intensive variables, we shall introduce the
operators
10- F-1az IJ= F-1'c1
(18.3)
with thefollowingproperties
dQ,111= F-1
(18.4)
l:l
*c(x))-(v
#)*I
*n(x-l))
lt
#
hI
*o(x))-(Xp
+'1)*l
*:(x+1))
(1g.5)
Although 1:and l%
oeach multiply 1*n)bya snitefactor,theircommutator
vanishesin thethermodynamiclimit(N ->.*,F ->.*,NlV ->.const). Hence
itisNrmissibletotteattheoperatorslnandIJascnumbers,laslongaswe
:N.N .Bogoliubov,J.Phys.(&SSA).11:23(1947). > alsoP.A.M .Dirac,ten ePlinciplœ
ofQuantum Mechani
cë''24ed..= .63,OxfordUniversityPress,Oxford,1935.
2x
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
consider only stateswhere a fnite fraction ofthe particles occupies the k = 0
m ode. Thisapproxim ateprocedureclearly neglectsiuctuationsin theoccupation num berofthecondensate.
Thepreceding discussion hasimplicitlyassumed a perfectBosegas,where
al1the particles are in the zero-m om entum state. ln an interacting system ,
however,theinterparticlepotentialenergyreduc% theoccupationofthepreferred
m ode,so thattheground-stateexpectation value
t'F0Il
A.
#f
o
A.
a1
'1%)- N.F..j-nz
is less than the totaldensity n= Nj#'. Nevertheless,the Bogoliubov replace-
mentofl(
)anda
?:
tbycnumberscorrectlydescribes,theinteractinggroundstate
in the therm odynam ic lim it whenever the num ber of particles in the zeromomentum state remainsafnite fraction ofN . W earethereforeled to write
theboson field operatoras
(?(x)= #'(
)+ ;'
P'-1edkexck= (%+ (/(x)= nt+ (
/(x)
k
,
(18.7)
wheretheprimemeanstoomittheterm k= 0. Theoperator/(x)hasnozeromomentum components,and(0isaconstantcnumber.
Theseparationof'
;intotwopartsmodifiesthehamiltonianinafundamental
l
w ay. Considerthepotentialenergy
P= .
à.Jd?xdbx'()#(x)(/(x')P'tx- x')#(x')f(x)
(18.8)
lfEq.(18.7)issubstituted into Eq.(18.8),theresultingtermscan beclassifed
according to the number of factors z?4. The interaction hamiltonian then
separatesinto eightdistinctparts
Eù= .
!.
rlàj
-#3x#3x'l
z'
-tx- x')
f-l= èrztlfJ3x#3x'Ftx- x')c
/(x')(
i(x)
(18,9)
(l8.10)
f'2= l.nOJJ3xJ3'
v'(
/A(x)(
/ftx')1.
z
'(
.x--x')
Pa= 2(èrlf
))j#3xJ3x'(
/i(x')P'tx- x')/(x)
f'j= 2(ènj)jJ3xJ3x'Qflx)t
/.1(x')P-tx- x')f/tx)
356= 2(1.
nj)Jdbxdbx't
/A(x)P'tx- x')(
/(x')#(x)
f7= !.f#3x#3x'$#(x)(/'
1(x')Utx- x')Y(x')t
/.(x).
.
(18.14)
(18.15)
(18.16)
Figure 18.1indicatesthe diflkrentprocessescontained in theinteraction hamil-
tonian,whereasolid linedenotesa particlenotinthecondensate(f/)ort
/1'
),a
wavylinedenotestheinteractionpotentialF,and adashed linedenotesaparticle
BosE SYSTEMS
201
belongingtothecondensate(fcorL#- n!). InderivingEqs.(18.9)to(18.16),
wehaveused therelation
f#3x;(x)= P'-+)
'JkJd3x:lk*X= U1X'
Q=0
kj
k Jkdk
(18.17)
so that P containsno term swith only a single particle outofthe condensate.
Theterm Ezin Eq.(18.9)isapnamber
Eo= .
.i
.P'-1Nl1z'(0)=èlz'
nàF(0)
(18.18)
thatm erely shiftsthezero ofenergy buthasno operatorcharacter.
Inanoninteractingassembly,theground stateisgivenbyEq.(18.1)with
No=N. SincetheBogoliubovprescriptioneliminatestheoperatorsaoandJ1
Pj
z
#
z'
#
x'
N
Y'N
&
W
,e
y'N
:5
:6
Fig.18.1 *rocessescontainedin f'forbosons.
entirely,a11rem aining destruction operatorsannihilatethe ground state,which
thereby beeom esthe vacuum
I0)H 1*0)- IA',0,0,
(18.l9)
The vacuum expectation value ofthe totalham iltonian arises solely from Eq.
(18.9)
(01X )0)= En= !.P'-iNlF(0)
(18.20)
whichisthefirst-ordershiftin theground-stateenergyofan interactingBosegas.
The use of Eq.(18.7) removesthe problem assoeiated with the zerom om entum state,butthefollowingdiëcultystillrem ains. Considerthcnum ber
operator
4 -Nz+fd?x4ftxl:(x)=Nz+1,
Jlak
k
(18.21)
V no longercommuteswith thetotal
where Nzisacnum ber. Itisevidentthat.
ham iltonian
(T-+ Eo.
+ Pl+ ...+ P7,XJ# 0
(18.22)
z:2
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
since the various interaction terms alter the number of particles out of the
condensate. Asa result,the totalnum ber ofparticlesisno longera constant
ofthemotion butmustinstead lx determined through the subsidiary condition
N = No+ X'tt/kJk)
(18.23)
k
where the bracketsdenote theground-state expectation value in the interacting
system . In Chap.10,we study an exam ple ofthisprocedure,butitisusually
simplerto reform ulate theentireproblem from thebeginning.
We thereforereturn to theoriginalhamiltonian X = 1%+ P,in which h
and Jc
lare stilloperators. Introducethehermitian operator
k - 4 - Jz#
(18.24)
which hasa com pletesetofeigenved orsand'eigenvalues
2 1.P,)- A,t
'l%)
(18.25)
TheoperatorX commuteswith,
9 sothattheexactproblem separatesintosubspaces ofgiven totalnum ber N. W ithin a subspace,the ground state clearly
correspondsto thelowesteigenvalue ofX
41
:F0(1))= A(/z,F,N)i
'1Cn(N))- Ef(l',A)-p,N)l'l'
-:(A-))
(18.26)
Theserelationshold forany valueofy.. Ifwenow chooseto look forthatsub-
spaceinwhichthethermodynamicrelation(4.3)holds
DE(V,N)
#.=
(18.27)
thenwe willhave found the absoluteminimum ofK
:A'
()t,F.N ) 0E(P'
,N)
DN = ëS' - p,= 0
(18.28)
-
Equation (18.27)may theconsidered a relation to eliminateN in termsofthe
variablesp,and F. In thissubspace,theexpectation value (k1'06: 1:1'0))isthe
minimum value ofthethermodynamicpotentialatzero temptrature (seeEq.
(4.7))andfixedp.and F
r::1-0G)1: $
'F0G ))- fqT- 0,F,p)= (f - pA llr-o
(18.29)
In accordance with general thermodynamic principles, !T%(/z)) therefore
representstheequilibrium stateoftheassem bly atExed F = 0,F,and y.. A llof
thethermodynamic relationsfrom Sec.4 now remain unchanged,forexample,
F,!z)
8 v ( )!
-df1(r,
?pz
r0,p,--y
:,jz# )v (s))-(v,(s)).jp.*)
v,(s))
Fzr
,
-
-
('l%G)lXI'1%G))- N
wherewehaveusedthenormalizationcondition(X1%lN%)= 1.
(18.30)
BosE svsTEu s
20
Aslongasanand2()representoperators,b0thX and* provideacceptable
descriptions ofthe interacting assembly. W hen we use the Bogoliubov prescription,however,thethermodynamic potentialofersa dehniteadvantage,for
itallowsa consistenttreatmentofthe nonconservation ofparticles.l Indeed,
p,m ay be interpreted asa Lagrange multiplierthatincorporatesthe subsidiary
condition(18.23). Wethereforecarryoutthefollowingsteps:
1.ReplacelcandIJbycnumbers
10-*Alà IJ-+n1
a
(l8.31)
Inthisway,# and# become
(18.32)
# -.No+;'
4 au>Nz+X'
k
7
# -..Ev-y.
NZ+ ;'
(e2-rzlallk+J)
(6
k
-l
- Ez- gN.+ #'
whichde:ne#'and#'.
(18.33)
2.Sincea11rem ainingdestructionoperatorsannihilatethenoninteractingground
state1*:2).itmayagainbeoonsideredthevacuum
lP(
))->'10)
(1B.34)
W ick'stheorem isnow applicable,and wem ay usetheprevioustheorem sof
quantum seld theol'y.
3.A11theânalexpressionscontain thetxtraparameterNz%which may bedeterm ined asfollows. Since tlle equilibrium state ofany assembly atconstant
(r,F,p.
)minimizesthe thermodynamic potential,the condition ofthermodynamicequilibrium becomes
êfltr=D0N,F,p,,Nè
o
s,s
= (;
tja.a5)
whichisanimplicitrelationforNéV,y,
j.
I9QGREEN'S FUNCTIO NS
Steps1and2described abovearequitedistinct,anditisconvenienttotreatthem
separately. ln thepresentsection,weintroducea Heisenbergpicturebased on
X'and use the exactsingle-particle Green'sfunction to determine the thermodynamicfunctionsoftheground state. In Sec.20,weintroducean interaction
!N.M .Hugenboltzand D.Pines,Phys.Rer.,116:489 (1959).
2-
GROUNO-STATE (ZERO-TEMPERATURE) FOnMALISM
picture and derive the Feynman rules for evaluating the Gretn's function in
perturbation theory.
TheBogoliubovprescription hasled usto considertheoperator
# = En- Jzyf
l+ X'
(19.1J)
where
7
#'=Jd5x(/t(x)(r-la,
)4(x)+J=1
.
:
(19.1:)
SinceX ishermitian,ithasacompletesetofeigenfunctions,and we shalllet
(O)denotetheground stateoftheoperatorX. ltisessentialto bearinmind
thatlO)isnotaneigenstateofX andthusdiflkrsfrom thestate$
'1'-a)pintroduced
inEq.(18.26). TheHeisenbergpictureisdesnedasfollows
t)x(?)> ei*''%t)se-l*t''
= efRL't1'(9se-tk'f/!
t
(19.2)
wherethec-numberpartofX doesnotasectthetimedependence. ln particular.
thefeldoperator#(x)becomes
.
'
/?
'x(xJ)- ei#'flhfoe-ik't/:+ eik't/h/(
,x)e-ik't.
'
:
(o+ 9Qtxl- nl+ t/xtxl
which showsthatthecondensate partofli'
Kisindependentofspaceand time.
-
Thesingle-particleG reen'sfunction isdesned exactly asin Sec.7
fG(x,y)= OIFIIL + 9Q(
x)J(J1+/f
x(J')J)?O)
(010)
toj/xtxl+ ;;
f
K(1')jO) (O1rE#x(x)/,
x(>'))1O)
f
= ?u + a!
t
y
j
!
o)
.
o
y
o
j
o
jjq.4j
x,
(
wherethe signature factoris+ 1foralltim eorderings.
W efirstprovethatthesecond term on therightofEq.(19.4)vanishes.
Thisresultdoesnotfollow from numberconservation,since!O)isnotaneigenstateofX;instead,theargumentmakesuse ofthetranslationalinvarianceof
thegroundstate. Thequantity(O l/1
x(x)lO)isalinearcombinationofmatrix
elements(OJ41O)fork#0,eachmultipliedbyac-numberfunctionofx. By
desnition,them om entum operator
ê= J
(âkt/kn =J
('âktzlak
k
k
(19.5)
hasnozero-m omentum component. Henceê commuteswith#,which follows
either by directcalculation orby noting thatthe originaloperatorh itself
commuteswithê
(f%,êI= 0
(19.6)
BOSE SYSTEMS
206
so thatthe Bogoliubov replacem entdoes notalterthe translationalinvariance
orthe assembly. Each eigenstate ofX can therefore be labeled by a defnite
value ofthem om entum ,and theground statecorrespondstoP = 0:
f'To)- 0
The relation
(*,alJ=âkt/k
(19.8)
impliesthat4 increasesthemomentum byâk,andtheorthogonality ofthe
m om entum eigenstatesshowsthat
(0 lt/k10)= 0 k # 0
therebyprovingtheassertionthatG(.
x,y)takestheform
iGlx,lq- no+ iG'(x,y)
.
fG'(
tolrEt
/'vtxl4'
x(y))lo)
x,y)> -- (0 1
,0)
-
(19.9)
(19.10)
(19.11)
AsinEqs.(18.32)and(18.33),theprimedpartreferstothenoncondensate. With
theusualdelinition ofFouriertransfbrms,theexpectation valueofX maybe
written as
Ar= (9ïb= No+.P-(2=)-4(d4qiG'(q)piqnn
(19.12)
where the limitT ->.0-isimplicit. Since G'dependson p.and 6,
% through the
operatork 'intheHeisenbergpicture,Eq.(19.12)maybeusedto find .
V(U,p,,Na);
alternatively,thisrelation maybeinverted to flnd /
.
z(Prs.
Y,Va).
The ground-state expectation value of any one-body operator can be
expressed in term s of the single-particle G reen's function. An interesting
example isthe kineticenergy
.
14
..
haa z
r=t.j'y= p'j
-t
'
.
-.T
-4?2mR-.
,c'(
.q)piqap
,.
(
2:
7.)
.
(19.13)
v:hicb
,:shoN&s that the stationary condensate m akes no contribution to F.
also possibie to determ ine the potentlalenergy.butthe detailed proof
-is m ore
ci-m plicatcd than in Sec.n
'. Equation (19.2)m ay be rewritten as
ihJj)x(x) (;s(x).<.,j
t
.
--
(19.14)
Inthethermodynamiclimit.thefields('andf
i%obeythecanonicalcommutation
relations.andthecommutatorisreadilyevaluated with Eqs.(l8.10)to (18.l6)and
(19.1).Aftersomemanipulation.we5nd
'#3x4l
xtxlLi
hj0
/-rssj(!
x(x)=2f'
c.
y.f'
q-1
'
4.
-2Cj-%.
v2f7
(19.15)
ax
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
where PIcommuteswith # and thuscancelsidentically. TheadjointofEq.
(19.15)maylx writtenas
fd,x(-,
:j-r+r
.
z)4l(x)4x
(x)
2Pl+ P3+ P4+ f'5+ 2P6+ 2P, (19
=
16)
.
whiletheiraveragebecomes
!.jdzxtutxltmj
:
j-w+(p
)o(x)+gt-mj
ê
j-w
+p)::(x)j:hx(x))
J'
l'j+ Pz+ P3+ P4)+.
!.
(1% + Pk)+2b
=
.
=
gt-z
2;7- noD-n'
(19.17)
z
wheref-isthetotalinteractionenergy
7
P= Eo+Jw1
)(b'
L
(19.18)
Theground-stateexpectationvalueof(19.17)beeomes
J#3x1im 1im #.ihjê
P,- F(x,)+ p,iG,(x?,x,t,)
-j- F(x)+ /.
t- ihèt
X .x t'-p,+
.
'-
=
2(P)- af
jê
èP
no
(19.19)
therebyexpressing(P)intermsofG'.
Thethermodynamicpotentialatr = 0istheground-stateexpectationvalue
oftheoperatorX
f1(T- 0,P',/z,No)- (0 1
4 ,O)
(19.20)
where iO)isassumed normalized. Thecondition (18.35)ofthermodynamic
equilibrium remainsunaltered,andwefnd
la
Px
f1
n)vp-eN
P,tol
fl
o)-rolap
xo)
o)-4o)d
PNP,-/
z1
o)-o (19.21)
This equation therefore provides a representation for the chemicalpotential
êP
p.= tOlONn lO)
(19.22)
butitmustbenoted thatthestatevectorlO)itselfdependson /
.tand No. A
combinationofEqs.(19.12),(19.19),and(19.22)yields
E=;/+P)=jy.
N+.
if:3xx
l
i
'
mxt
l
j
.
m
,
t+qih(1-a0
t,
j+F(x)jG'
(x,x'
)
(19.23)
BosE SYSTEM S
207
which may berewritten witha Fouriertransform
E-6N+.
iP-J(a
d,
o
*q4ju,.
j.hz
lm
q2jjoy
fg)nj
vav
l
(19.24)
Thetherm odynam icpotential
otr-n)-E-sx--!.
sN+kvj(2
d,
*
4
q.jço+âz
2m
q2)i
o'lqlet
qz,
î(19.
25)
follow s im m ediately. Since Eq. (18.35) (
or Eq.(19.22))determines Nf
j(m),
whereasEq.(19.12)determinesN(y,,Nn),weare now ableto 5nd thethermodynamicpotentialfrom (19.25)andthusobtainthephysicalquantitiesofinterest.
Theremaining problem,of-course,istheevaluation ofG'(x,y),which isconsidered inSec.20.
ZOQPERTURBATIO N THEORY AND FEYNM AN RU LES
W e now usethe techniques ofquantum 5eld theory to study the perturbation
expansion oftheboson G reen'sfunction desned in Eq. (19.11).
INTERACTIO N PICTURE
W etirstintroducethe operator
Xl
l= En- JzAo+ XJ
(20.1)
where
kLa t-p,4'=fd5xt
/1(x)(r-P,J/(x)
(20.2)
and acorresponding interaction picture
ö1.
(f)M t'iXp'790se-'X0t/'
= efXû'f)
'
hOSe-tk0'f,
'
A
(20.3)
JustasinSec.6,theHeisenbergpicturein(19.2)andtheinteractionpictureare
related by an operator
gqt>/0)i
;
:
zzeiko'tjhg-f:'(l-t(
))/Ae-Lkoetolh
(gg*4)
which obeysthefollowingequation ofmotion
Pt'
)'
(/,?()) sx
f/i ?t ==e (,,/,(v,..,g)e-fpo-ta
fsgqt,to)
= etx(
,',/,pj:-y.
:c-,/,p4/,t())
jcjtf)1)(t,t.4
--
(20.5)
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
208
with
7
41- j)f'
J
J=1
# - 40+ 41
(20.6)
N ote that
EXf
),X)- 0
(20.7)
Thusthegroundstate 1
0)oftheoperator#ncanbeconsideredastateofdesnite
numberofparticles. ItisevidentthatI0) isjustthestateintroduced in Eq.
(18.19)where the numberN isdetermined from Eq.(19.12). Furthermore,
Jkl0)=0
(20.8)
for a1lk,which means thatallofthe perturbation analysisofSec.9 remains
correctwith 10)asthenoninteractingvacuum. In particular.weimmediately
concludethatl
I
'
G'(x,p)=
x)
'
m=0
jm 1
=
#/l.'-
c
o
dtm
T m ! -*
-*
x'
t0fF(#l(/I)''-#I(/m)ç%(x)(
J1*
(:')1lolconnected (20.9)
Sincetheoperatort/'
;(x)#)(y)alsocommuteswithX.thediëcultyassociated
withthenonconservationofparticlesisisolatedinthefactorsXj(/f)inEq.(20.9).
The zero-order term becom es
(20.i0)
FEYN M A N R U LFS fN C O O R DIN A TE 6 PA C E
Thereisa factorntforeach dashed lincentering orleavinga vertex. The
totalnumberoflines(solid and dashed)going into a Feynman diagram must
equalthe num bercom ing out.
'The proofthat
Lo
(7j0
..,:
iu.)(0)
.
7O
-V
oxl-k'()lt;((),usx)l().
I
isaneigenstateofk followsjustasintheGell-MannandLow theorem.
BosE svs-rEM s
2Q9
2.ln rnth order.thern operatorskLcan beehosen in /?1!ways. Thisfactorm !
corresponds to the possible ways ofrelabeling the dipkrentinteraction lines.
and itcancelstheexplicit(,17!)-'lin Eq.(20.9). AsinSec.9.thiscancellation
occursonly forthe connected diagram s.
3.The terms f':, P2
.and f'
, are sy'mmetric under
the interchange ofdummy
P
'
*
variables x +-yx . whereas the other tcrm s l'y. 1,r4. lC'j. and I.
A
z6 haNe no such
'
.
J
...
sym m etry. ln consequence.each tim e I
z'1. l'2.or 1,7 appears in a Feynm an
diagram .there is alwaysanothercon
tri
buti
on thatpr
ecisely cancelsthefactor
,
x
.
.
.
!-in frontof
-theseterms. Since l
z,3-k4-kj.and k'f,alreadyhaveafactor1.
-
weconclude thate:erydis?int':FeJnmandiagram need becountedonlyonce.
and the potentialenters with a factorunity.exactly asin Set?.9.
4.Eachn?th-orderdiagram i1:thepertttrbationexpansionofG '(.
v.)')hasa factor
(i;
I
h)m(-i)Csw'
here C is the number ofcondensate factors llo appearing in
the diagram .
5. The absence of backward propagation i1 tinne-or hole propagation- alIou s
us to elim inate large classes of diagram s atthe outset. For eNam ple.thttre
areno contractionsSNithin the fIsincethe)arca1read)'normalordored:thus
there are no contributions in which the hamc particle 1int
t G'0either closes on
itselfor hasits endsjoined b),'the same i1teractlon. l11addition.ue note
that the Feq'nm an diagram is integrated over all i!1ternal Nariableh. which
m eans thatallpossible tim e orderi1)gq ofthe interactions are included. N o
diagrarn can contribute unlessthereit
s.q
-t????ta:1111(-t?,'
Jt.
'r'
/)?qt
z?
'
/?bî'
l?i('hall//q
'!:ar:i(.
lL.
'
//??t:s
-G0rlll:ti?r3x'
t7rJ 111Jl'
??7(n. Forexar'
rlple-Fig.9.8/aild jy'
aniqh ide!
1ticall).
because there arepairsofparticle lines running in oppohite direction>. N ote
that these restrictions elim inate eveo'one of
- the tirst- and second-order
diagram sin Figs.9.7and 9.8thateontribute to theferm ion propagator.
FEYNM AN RU LES IN M O M ENTUM SPACE
TherulesforG'(4/)i1
)nlomenttlrn spacearethesan'
leasbefore.with afactor/71
(,
foreverydaghed 1ine-anoverallfactorof-(1'h)'
n(-?
')t-(2.
77.J4tC''
n1in F/pth order.and
the zero-order Green-s ftlnetion given by Eq.(20.ll). The basic N'
erticesare
Show'n in Fig.20.l and four-m omentunn is conserved ateach vertex. Sint?e a
condensate line carries N'
anishing four-m om enttlm .u hcreas a partic!e line nlust
havek'
h
zu0-u'
eagainconcludethatnointeractionlinecanjoi1
)oneparticleline
and three condensate lines.
A s an exam ple,consider the tirst-order correction G 'f'' The ternls t-'
P2%î,
%!
ktr'
6mZkenocontribution to (;'int
irstorderbecausetheydo notconserv'e
*w
Z /-
Fig.20.1 Basic vertices for bosons.
,
,
21n
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
thenumberofparticles. Theterm containing 97alsovanishesbecausetheonly
possiblediagramswouldinvolveholes. Weareleftwith/4and6,whichlead
l
I
k
A
l
l
l
l
1
l
I
l
Y
1
1
I
l
l
1
(b4
Fig.20.2 Allhrst-ordercontributionsG'tl'
to thediagram sshown in Fig.20.2/ and b,respectively,and the Feynm an rules
im m ediatelygive
G'(l)(t?)= no/j-lG0(ç)(p-(0)+ p'(q))G0(t?)
(20.12)
Thecorresponding analysisin second orderissubstantiallylonger.and we shall
only exhibitthe diagrams(Fig.20.3). Note thatFig.20.3: and e represent
t'
1 Y
Al xl
-
Ak k
f 6
6 6
'
k
f )*v> .
.
/'
f
,&
A
Y
:
A%
(:)
:
A
7,%l
A
(g'
)
Fig-20.3 A 11second-ordercontributionsG'f2'.
difrerentcontractions because the direction ofpropagation,or m om entum flow ,
isdiflkrentin the tw o cases. The diagram s ofFig.20.3J through e are of order
nà,whereas20.3/ and g areofordernfj. Asnoted previously,the absenceof
holesm eansthatno second-order diagram s involve only noncondensate lines.
BosE SYSTEM S
211
Dvso N's EQ UATIONS
The nonconservation of particles arises from the Bose condensation, which
provides a source and sink forparticlesoutofthe condensate. A sa result,the
particle lines need notrun continuously through a diagram ,as opposed to the
situation for ferm ions. Nevertheless,ifa proper self-energy isdefined asa part
ofa Feynm an diagram connected to the restofthediagram bytwo noncondensate
particle lines,then it is stillpossible to analyze the contributions to the G reen's
function in a form sim ilar to Dyson'sequationsfor ferm ions. The structure is
m ore com plicated than indicated in Sec. 9. however,because there are three
distinctproperself-energies.asindicated in Fig.20.4. The hrstone 12'
)'
1(p)
p
p
-#
/
Z
z'
Fig. 20.4 Proper self-energies
bosons.
z
p
)
J,
*!(r)
z
z
z
z
z.
-p'
).:l
*2Lr)
z
-#
.''
p
1:2
*l(r)
has one particle line going in and one com ing out,Fim ilar to thatforferm ions.
Theotheroneshavetwo partic1elineseithercoming out(Z'
!
a)orgoing in (Z%
.'
. j)
and reiectthe new featuresassociated w ith Bose condensatlon.
'the lou'est-order
contributions to these new self-energies m ay be seen expllcitly in Fig. 20..
3:.
(ThechoiceofsubscriptswillbecomeclearinEq.(20.21).q
Correspondingly.A'z m ustintroduce two new exactG reen's function GI
'2
and Gc
'j.representing the appearance and disappearance oftwo particles from
the condensate. They are shown in Flg.20.5.along NKith G'.where the arrows
indicate either the direction ofpropagatien in coordinate space orthe direction
ofm om entum llow in m om entum space. The Dyson'sequationsforthissystem
p
-
Fig.20.5 Noncondensate Green's functions
forbosons.
G'
(J))
-
p
p
G !'
2(pj
G2
'1(p)
212
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
were hrstderived by Beliaev.l They are show n in Fig.20.6 and m ay be written
in m om entum space asfollows:
(20.13t
7)
(20.l3>)
1
qt#
pz
'
Z
J
p
7Z
d
e
z
-
,
p
P
z
Z
Z
-
p
-
p
Z
Z
z'
#
Z
p
-p
-
- /)
p
z
z
z
W -p
/
-p
#
p
Z
.
p
p
Fig.20.6 D l'son'sequationsforbosons.
N ote thatoverallfour-nlomentunl conservation determ inesthe direction ofthe
momentum t
low in Fig. 20,6. An equivalent equation for G'(p) is clearly
'S.T,Beliaev.Sol'.#/l>u.-./f'
'F#,7:289(l958).
BosE SYSTEM S
213
W hen these equations are iterated consistently,we obtain allthe im proper selfenergies and G reen'sfunctionsto arbitrary order in perturbation theory,
TheanomalousG reen'sfunctionsintroduced abovehavea precisedehnition
in term sofHeisenbergfield operators:
iG ,l2(x--).)- C'O
IF(
/y
(-x)
t/'y
T')jO
..
--r
,z
.
k:
.a.(-
(20.14t7)
l'G2'1(-v--;')- --0 rë7).
(.
v)r
/-).(''
)!'o
' ( x 1'
(20.14:)
.
.
.
---.
-
kx5u,:65.1: ',
- .
q
b,öp
where the nonconservation of particle
definitionsim p1y that
(J7j
-.2(.
'
t',1'j.
- (J'j
'2(.è'..v.)
The
'
G2
'i(v.-'k')-.-&,:2l(.;'.-'t-)
so that their Fotlrier
C;'1
-c(/?)--tJ'(
'2(--J?)
G-2,t(p) .
-C;
'c
'k(-.
/?)
The structure of Dyson's equations can be clarificcl by introducing a matrix
notation i1
)U hich
t'
L
'
Rt
.'Y)'
r/s(x)
'
-t
,
y
s(xv)
(20.l7)
() T-f
't1).F.f.'
-.
1
's'(.'
111()
.')f
k'
0 0
-* (v T') z
-* (r ).
E *'(A.,).)= ).1
. N
2''.')
p1''
x-.l
' ),
.
ac,(x.)'
k.
) -j
I(1,.a.)
'
.
.
.
(20.21)
.
and
G0(x,
G0(x.
0
)
.
,
)--g (
;)-) g(
)
(
.
)
,x)
-
.
(20.22)
214
GROUND-STATE (ZERQ-TEM PERATURE) FORMALISM
Weshallgenerallyconsidera uniform medium,whereEq.(20.20)maybe
sim pliied to
G'(p)- G0(#)+ G0(,)Z*(,)G'(#)
G,
(s-jG1'
l(
p;
t)G,
J
(
2
(,
p)j
c(
G0(#) 0
p)-g(
) cot-slj
E*(p)-(Y
X1
'
h'
l(
I#
X)X2'
ht2
(
ç
-*
,)j
(20.23)
()
Itiseasilyverised thatthismatrixequation reproducesEq.(20.13). A matrix
inversion then yields
G,(78- pz+ (,?,- lxlh+ S(p4- Alpt
Dlpj
(20.24)
Gl'c(p)- - 1:)2(J')
D(,)
o'j,(s -
-
X!
1(:)
pl
gj
where
D(#)- Epo- z4(p)22- Eav -vh-,'-.
S'(p)12+ Eh(p)Et(J,)
(20.25)
î'lp)- .
1.(12.
)1(J,)+ E'
r(-;)1
(20.26)
and
A(p4- .
iEY'
h(,)- E'
r(-p))
TheseequationsexpressthevariousG reen'sfunctionsinterm softheexactproper
self-energiesand arethereforeentirely general.
LEHM ANN REPRESENTATION
Before we study specisc approximationsforZ*,itisinteresting to derive the
Lehmann spectralrepresentation forG '. Theproofproceedsexactlyasin Sec.
7,andwefind (compareEq.(7.55))
c,(,)-
0)l
''p)fnp1:1(0)Io)
p'rg(P1
,,4-'(,
-,
(au-x,o.
,.i
,
(0 I*1(0)In,-p)(n,-pI;(0)IO)
pn+ h-bçKn.-p - A-00)- ''
T
wherethecompletesetofstates1np)satissestherelations
êr
z'p)- âptnp) X rnp)h- A'
aplnp)
(20.28)
-
1(20.27)
and each residue isa 2 x 2 m atrix. ltisevidentthatalltheG reen'sfunctions
have the sam e singularities in thecom plexpzplane,occurring atthe resonant
a0sE SYSTEM S
215
frequencies +â-l(Akyp- A%0). The residues of G'(r)are real,while those of
GL
'2(pjandGLj(p)arecomplexconjugatesofeachother.
ZILW EAKLY INTERACTING BOSE GAS
A sourfirstapplicationofthisform alism ,weconsideraweakly interaetingBose
gas whose potential P'(x) has a well-desned Fourier transform U(p). The
p -p
I
I
A
A
j
t
l p
'
1
(J)
-
#
p -#
'h
l. p
,
#
I'-#
#
*'
' -#
p -#
j ! 1 # ' .J. .
LI. I 1 I I 1l 1 I
...-
k m' 1 .'
k 1 1 '
t
(d)
(d)
+l
fc)
k k
(e)
(fb
eh
1
p -p
'
tI'(#)1I $4- 11
G2,1(2ï: +!
+l +1 +p +$
+l .1.
,
A' p -p
(h'
)
-#
(i)
k #
+$ +l
+I
k -# P
(j)
k -p # -#
(k)
(l)
Fig.21.1 A11srst-and second-ordercontributionstoG3
'1andGa'l.
proper selflenergies need only be evaluated to lowest order,and Eq.(20.12)
showsthat
âXtl(X - NnE1z
'(0)+ P'(p))
lowestorder
which isindependentofpv. The Feynm an rulesofSec.20 also apply to Gl
'z
and Gc'I,and we exhibita1lnonvanishing srst-and second-ordercontributions
in Fig.21.1. W esee by inspection thatthe srst-erderproperself-energiesare
â12'
)c(p)= âE!l(J?)= nvF(p) lowestorder
(2l.2)
again independentoffrequency. Thisequality of Xt2(p)and Y1l(p)for a
uniform Bose gasatrestcan be proved to a1lordersby examining the diagram s.
Sincethearrowsdenotethedirection ofm om entum flow,reversingthedirection
ofal1thearrowsisequivalenttotakingp+-+-p. ThusYtc(p)= Y1'
1(-p). The
symmetryofthediagrams,however.showsthatboth)Zt2and12ljmustbeeven
functionsofp,whichalso followsfrom Eqs.(20.16)and (20.24).
216
GROUND-STATE (ZERO-TEMPERATURE) FORMALISM
Before the solution of Dyson's equation can be used, it is essential to
determinethechemicalpotential,which can bedonewithEq.(19.22). lfthis
equationisrewritten in theinteraction representation,weobtain
*
jm j
.- -
h m!
8=
m=0
*
x)
dtj.. .
* -.x)
?-f m 1
r
m
u
ajp
.
#;m (.0tF #jLtj)...xj(/m)..
=. (0)
c'A o
-co
co
cc
.
-
,
.
h- m
--.
y -a)dtj... -rdtg
t(
:
t
0jT(
A'j
(tj.
j...Kj
(u)jj
())
,
m=0 $
(2l.3)
wherethe operator0f-/foA% isassigned thetimef= 0. Thedenominatorserves
tocancelthe disconnected diagrams,exactly asin ourdiscussion oftheproofof
G oldstone'stheorem in Sec.9. Thuswefind
*
y,=
i
.h
-
m
m =0
x
1
dtj..
m ! -x
The lowest-ordercontribution is
:f'
pt- '
ï0ôUNL,10N
p
-
lEoAa
-l+:v(
)l(0pPI+ P2+ I
h + 6 J0)+(2zVc)-1(0pP5+ % 10)
= atjp'(0)
(21.5)
where the m atrix elem ents vanish because they are already norm al ordered.
ComparisonofEqs.(21.1),(21.2),and(2l.5)showsthatthesehrst-orderquantities
satisfytherelation
p,= /i)2tl(0)- âE'
t2(0)
Thisequation isin factcorrectto allordersinperturbation theoryand washrst
derived by H ugenholtzand Pines.;
Thesingle-particleGreen'sfunctionisnow readilyfoundfrom Eq.(20.24).
W enotethatz'
1(p)vanishesidentically,and a straightfbrward calculation yields
G'( po+ h-1gE0
p+ noF(p))
p)=
z
J'J- (f'
p/â)
G''
:-1nop'(p)
12(
:)- GL1(J')= pl- (Fp/â)c
-
where
Ev= ((ep
0+ ngp'(p))2- (n()p'(p)j2j+
= (
:p
02 + 2,6(,)nop'tplj1
:N .M .Hugenholtzand D.Pines,loc.cit.
(2l.8)
BOSE SYSTEM S
217
(2l.9J)
GI
'2(p)- Gz
,1(J')- - --N
lPP
-P
.è'p -F +. --)-t
:
,p .
plj kp4 +l,
rl pe p/h-fj
(2l.9b)
.
-
where
uv=(4.
F-1(ep
0+ nvp'(p))+ Jjl
(2l.10)
l'
p= (JFp
-lg60
p+ n()U(p))- !.)1
and theinsnitesimalszt
zthhavebeendeterminedfrom theLehmannrepresentation.
The m oststriking feature ofthese expressions isthe form ofthe excitation
spectrum Ep. ln the long-wavelength limit,Ep reduces to a Iinear (phononlike)dispersion relation
X
n.(
?.lz'(0)+
;
4:h:
p!g-t
ytj
with thecharacteristicvelocity (com pare Eq.(l6.11)1
nvP'(
C=gm?)j1
-- .
This expression shows that the theory is welldesned only if l'(0)> 0. The
presentcalculation does not allow us im m ediately to identify c-as the speed of
com pressionalwaves,since fp is here derived from the single-particle G reen's
function rather than the density correlation function. N evertheless,a detailed
calculation ofthe ground-state energy (Sec.22)show'sthatc'isindeed the true
speed ofsound. Thisquestion isdiscussed attheend ofSec.22.
The behavior of Ep for large momenta depends on the potential l'(p).
and weassume that P'(x)isrepulsive with a range rv. Itfollowsthat l'(p)is
approxim ately constantfor p .< ro
-l,and we also assum e that
hl
??()P'(0)<:
.
t2n1-q
66
(21.13)
which lim its the allowed range ofdensity'. The dispersion relation f'
p then
changes from linear to quadratic in the vicinity of p ;4:lz?'
nnaP'(0),
'â2J1 and
becom es
Ep;
k;e0
jP'(p)
p + nf
ply>2mnoPr(0)â-2
(21.l4)
for large wave vectors. The lastterm represents an additional potentialenergy
arising from theinteraction with the particlesin the condensate.
2!8
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
Itisinteresting to evaluatethetotalnumberofparticlesfrom Eq.(19.12).
Onlythepoleatpv= -E,/h+ i'
rlcontributes.and weobtain
13= ??o+ (2=)-3 1
-d3p :.2
(21.15)
Thusl.p
2m ay beinterpreted astheground-statem om entum distribution function
for particlesoutofthe condensate. The m ostnotable feature isthe behavior
ofè.p
2atlong wavelengths,where itvariesas Ipl-i. ln addition,noisdesnitely
lessthan?tbecausetheintegrand ofEq.(2l.15)ispositivedehnite. W eseethat
the interaction alters the ground state by rem oving som e particles from the
condensate,exciting them to states of finite m omentum . From this point of
view,the increase oft.2
p as Ip)-.
+0 reiects the macroscopic oceupation ofthe
zero-momentum state. In the limit P'(p)-.
->.0,theenergy spectrum f'
preduces
to eo whileù'
p
2vanishes,properlyreproducing thebehaviorofa perfectgas. A n
p,
equivalentobservation isthatthesecond pole in Eq.(21.9J)arises solely from
the interactions between particles,and G'(p)reduces to G0(p) as F(p)-+.0.
22Z1D ILUTE BO SE GA S W ITH R EPU LSIV E C O RES
W e now considera dilute Bosegas,in which the potentialsare repulsive butm ay
b: arbitrarily strong.l Just as in Sec. ll,the only smallparam eter is the ratio
ofthescatteringlength a to the interparticle spacing rl-1, and wetherefore assum e
naà .
c
s l.
The potential P'(x)no longer has a w'ell-defned Fouriertransform,
t
*
f
tl)Itp)=
#
/
+
!*
t.
ï
!'
t.
'
'
t
:
't. !
ï#. t
t
'
+
/
h
:
t
t
'
!ï.
t
f
ZI2(,)=
#
t'
'
+
k
ï*
+
/
4
!'
y
Z*
21(/7)=
h
+
1
/
't'
+
*
/
t
+
'
!'
!'
ï
#
/
h
/
t
;
ï l
* .4
f $'
I
+
!
!
K.
'
/
+
1
f.
f'
t
1S.T.Beliaev,Sot'.Phys.
-JETP,7:299(1958).
Fig.22.1 Laddersum mation forproper
self-energies.
BosE SYSTEM S
219
and itbecom esessentialto sum aselected classofdiagram sto obtain theproper
self-energy. ln particular,note thatHg.20.3/ and g and Fig.21.1.fand l
representthesecond term sina sum ofladderdiagramsforY* (Fig.22.1). This
sum m ation m ay be evaluated with a Bethe-salpeter equation,exactly as in a
diluteFermigas(Sec.l1),whichyields
x(p,p',#)- (2,,)33(p- p')+ (e+ 2my.h-2-,2+ fn)-l(2,
7.
)-3
s fd'qr(q)xtp- q,p',#) (22.1)
mh-2F'
tp,pza' )= (2.
77.)-3fdàql.(qjy(p- q,pz,p)
(JJ.2)
The/
.tinthelastterm ofEq.(22.1)arisesfrom theform ofG0(p),which depends
explicitly on thechem icalpotential. Theseequationsaresim plerthan thosefor
ferm ions because the theory has no hole propagation;forthis reason,they are
justthosesolvedasà'land 1M(
)in Sec.11,and theirsolution maybewrittenas
(seeEq.(1l.45)J
1->(p,p',#)--,.4=ahl?z?-l
jpjt
z<,
t1,pp'!
.J-,
x1
(22.4)
where a isthe J-wave scattering length. H ence the corresponding proper self-
energiesbecomegcompareEqs.(1l.30)and(1l.40.
à)
/iZtl(P)= N()F'(1-P,1-P.#)v nf
lF(-1'p,àp.#);
kr8'
zr??oahl?A?-l
:Zt2(J'
)= noF(p,0,0);
k;4=.
noahlrn-i
/iY1l(C)= K0F(0,P,0)Q:4nnzahlrn-1
w eagainseethatYtctp)= E!l(p).
lt is interesting to com pare these expressions with those for a weakly
interacting Bose gas. For a short-range potential,the results of the previous
section can be written
â12tl(J')- 2/70F(0)
âYh(J')- âY!I(J')- nzL'(0)
ln Born approxim ation, the Bethe-salpeter scattering am plitude reduces to
Fs(p,P',#);
k0â2X(0,0)/,
zp-l= F(0)=4rraah2m-l
and thesum m ation ofladderdiagram stherefore replacesthe Born approxim ation
csbythetruescatteringlengtha. (CompareEqs.(11.24)and(11.25).1
22c
GROUND-STAJ'E (ZERO-TEMPERATURE) FORMAUISM
The corresponding chemicalpotentialisdetermined from Eq.(2l.4). ln
thepresentapproxim ation,thedom inantterm sarise from theexcitation oftwo
particles out ofthe condensate, where they interact repeatedly and then drop
back into the condensate. Thisprocessisshown in Fig.22.2,where the hrst
term isjustthatstudied in Sec.21. Thernth-ordercontribution mustcontain
one factor 6-2to excite the particles,m - 1factors $7-7to allow them to scatter,
and a factor P1 to return them to the condensate. Since the operator
PP/JAOdoesnotcontain Pc,itmustfurnisheitherthe f'1orthe Pz. Theprecise
'
k'
&
#
/
/
.<
N
*.
T$
'
J#
?
1
M
+
'%
Fig.22.2 Ladderapproximation forchem icalpotential.
num ericalfactorscanbedeterm ined bynoting thatthecontributionsinFig.22.2
areasubsetofthefollowingterm s:
whereEqs.(18.10),(18.1l),and(20.l4)havebeen used. Applying ourFeynman
rulestoeom putethecontributionofthegraphsin Fig.22.2,we obtain the same
laddersum m ation asin theproperself-energy,
'hencetheapproxim atechem ical
potentialbecomes
Jz= nvr(0,0,0);
kJn=noahl,,
n-1
which again satissestheHugenholtz-pinesrelation (21.6).
Itisnow possible to evaluate the single-particle G reen'sfunction in the
region ipl
l<:
t1;thecalculationisidenticalwiththatofSec.21ifweagainmake
thereplacementP'(0)= 4n.
a.hljm ->.4nahljm and yields
G'(p)=
ui
-
r;--
770 Ev/h+ ixî pa+ E.j
% - ixè
-
(22.10)
where
t/j=!.tFp
-'(ep
0+ 4=noahlm-i)+ 1)
(22.11)
th
2
p= J(Fp1(6p
0+ $.
nnoah2m-1)- 1)
and
Ep= (6p
02+ 8/a0ay2ep
om-l)1
(22.12)
BOSE SYSTEM S
221
Thetotaldensity n isreadily found from Eq.(19.12):
(22.l3)
??- n
8 na? 1
.
-y-.
?=yj.
y.
which issmallin thepresentlimit. Notethatwehaveused Eq.(22.l0)foral1p,
includingtherangeIpi
tz%-1:itiseasilyveriéed thatthisapproximationintroduces
'(SnJloab)'
1
'
negligib1e errorin the lim it/7473<:
t 1 because the integralover )'= (qa),
in Eq.(22.l3)converges.
In a similarway,theenergy isdeternlined from Eq.(l9.24):
-
E Z-i= jym +.(32=3)-1fJ3g(6q0- Eq)(Fq
-l(c0
y 4=.nvah1m-1)- lj
Q.
.
=jyn-..hzfbrrllv(7.
)Y(l6=2p?)-l(X
y)24.
y,((.
2.
4,3-;
-.
3.
y7j(.
).
,
24j-.
2)-1
0.
.,2
.
-
=
64'
;r1(>7fz)1'hl
64=1(,7qg)1'hl
jyn- - -.l.
5
.
u?7J-u-- ;
4:i
ua- - -l-ju='
?z7-.
2,,
- .
jj
(22.15)
InbothEqs.(22.13)and(22.15),theintegralrepresentsasmallcorrectionof-order
(?7()tz3)1relativetotheleadingterm.andwehavetherefbresetnfj;4Cn. Thefnal
determ ination ofE and p.is m ost sim ply performed with therm odynam ics.l
Assum e that
p.= tWnahlzn-lgl+ a(nJ3)1)
where a is a numericalconstantthatwillbe determ ined below. Substitution
into Eq.(22.l5)yields
E
27z/2ah2
.
32 na3 '
i'
y,---sy I.(
xqna3)4.-j
j(-#.
-j
,
Thederivativewith respectto ndesnesthechem icalpotential
'E '
/2 =
l 'E
4,
77775:2
8 na? 1
(j(
j'
,1v,=pjj= s? 1+A
4xlna')'
i
'-jn,
!N.M .Hugenholtzand D .Pines,loc.cit.
(22.l8)
a2a
GROUND-STATE (ZERO-TEM PERATURE) FORMALISM
and comparison with Eq.(22.16)showsthat== 32/3=+. In thisway we5nd
E - l=n2ah2
128 na? +
# ()
1+15(-)1
M nnmahljj.
j.s
32jaT
l
r3jl'
j
m
=
(22.19)
(x.
x;
Equation (22.19),which determintstheltading çorrection to theground-state
energy,was frstobtained by Lee and Yang.l N ote thatthe correction isof
order(nc3llandisthusnonanalyticintheinteraction. Thenext-ordercorrection
to theground-stateenergyhasbeen evaluated byW u,2whoEnds
E m 2.
n.
n2ujj2
j28 na3 1
P m gl+j5(.)+stj=-v'
j'
j(na,)ln(na,)+ujj
myjjsro
rjj
butthecoeëcientofthelastterm hasneverbeen determined.
The pressureP and compressibility cl(see Eq.(16.19))areeasily found
from Eq.(22.19)
p=-(e
0f
vls=2=n
m2ahlL
r*
ju
'
.6
j
4
-h
/n=
J3,
j
/+j
DP 1 OP
C 8pm mik 4yr
m
na
lhlL
g*
ju
'
.*
jx
uh
/=
J3j+j
2.
. .
n
(22.22)
(22.23)
Itisclearthatwemusthavearepulsivepotential(J>0)toensurethatthesystem
is stable againstcollapse. To leading order,the speed ofsound agrees with
theslopeofE.inEq.(22.12)asIp1.-.
>.0;inaddition,Eq.(22.23)alsogivesthe
ârst-order correction to c. Beliaev has evaluated Epjhi
p)to next order in
(nJ3)+forsmallIpIand verihed thatitagreeswith thatfound above,buthis
calculationisverylengthy. Indeed,ithasbeen proved to allordersin perturbation theory that the single-partide excitation spectrum vanishes linearly as
lpI-->.0,with aslope equalto themacroscopicspeed ofsound.3 Thislinear
dependencecanbeobtaineddirectlyfrom Eqs.(20.24)to(20.26),thesrstequality
inEq.(21.2)andtheHugenholtz-pinesrelationEq.(21.6)
D(#)-*#I- 2% Xt2(0) p-+0
whereweassumeXt2(p)iswellbehavedasp -->.0.
(22.24)
1T.D .Lee and C.N . Yang,Phys.Rev.,105:1119 (1957);see also K.A.Bruecknerand K.
Sawada,Phys.Ren.,106:1117(1957).
2T.T.W u, Phys.Ret'.s115:1390 (1959). Thisvalue hasbeen verihed by N.M .Hugenholtz
and D.Pines,loc.cïl.ywho introduced the techniqueused in Eq.(22.16),and by K.Sawada,
Phys.Aer..116;1344(1959).
7J.Gavoretand P.Nozières,Ann.Phys.(N.F.),M :349 (1964);P.C.HohenY rg and P.C.
M artin,Ann.Phys.(N.F.
),M :291(1965).
BosE SYSTEM S
223
Theladderapproximation (Fig.22.1)includesa1lârst-and second-order
contributionstotheproperself-energy. Asaresult,thecorrespondingsolution
ofDyson'sequation (20.13)containsa1lthesrst-and second-orderdiarams
forthesingle-particleGreen'sfunction (Figs.20.2,20.3,and 21.1). W henthe
remainingthird-ordercorrectionstoX*arereexpressedin termsofa,theleading
contributionscontainthefactorsn4/3anddonotafectEqs.(22.14),(22.19),or
(22.20).Henceweseethatourmethod correctlytreatsadiluteBosegastoorder
(aJ3)1.
PRO BLEM S
6.1. UseW ick'stheorem toevaluateG'lpj,G'
lc(J),andG114#)tosecondorder
in the interaction potential. Hence verify the numericalfactorsstattd in the
Feynman rules,and obtainthediagramsin Figs.20.2,20.3,and 21.1.
6.2. IteratetheDysonEqs.(20.13)consistentlytosecondorderin F,andthus
reproducethe resultsofProb.6.1.
6.3. (J) Usethe Bogoliubovprescriptionto expresstheIeadingcontribution
tothedensitycorrelationfunction D(k,(s)intermsofG',Gl
'a,andG11. Show
thattheresultingD(k,u))hasthesamespectrum astheexactG'(Kfz)).
(!8 EvaluateD(k,tM explicitlyforadiluteBosegaswithrepulsivecores.
6.4. Provethe Hugenholtz-pinesrelation (Eq.(21.6))to second orderin K
6.5. Considera dense charged spinlessBose gasin a uniform incompressible
background(forchargeneutrality).
(J)Show thattheexcitationspectrum isgivenbyEk= ((âDpI)2+ (d)2)*where
Dplistheplasmafrequency(Eq.(15.4));compareitwiththatderivedinSec.22.
(:)Show thatthedepletionandgroundstateenergyE aregiventoleadingordér
by1 (a-np/n=0.21lr1 and ElN =-.0.803r,
-1'e2/2e, resmctively, where
r2= ql4nnaé,J(
)= â2/-se2,andmpisthemassoftheboson.
(r)Deducethechemicalpotentialand thepressureintheground state. Interpretyour results.
6.6. Suppose Bosecondensation occursin a state with momentum âq,which
describesacondensateinuniform motionwith velocityv= hqlm. Show that
the condensate lines now include a factoret'Q*K. D erive the analogs of Eqs.
(21.9)and(21.10). Find anexpressionforthedepletioninadilutehard-sphere
gas asa function ofv,and compare with Eq.(22.14). Show thatthe total
momentum densityisnmMand notnomy. Explain thisresult.
tL.L.Foldy,Phys.Rev.,1M :649(1961);125:2208(1962).
par: three
Finite-tennperature
Form alism
7
Field T heory al Finite Tem peralure
Z3W EM PERATU RE G REEN'S FU NCTIONS
Ourtheoryofmany-particlesystemsatzero temperature made extensiveuse of
the single-particle Green'sfunction. Knowledge ofG provided 80th thecomplete equilibrium properties of the system and the excitation energies ofthe
system containing one more orone less particle. Furtherm ore,G wasreadily
expressed as a perturbation expansion in the interaction picture. At snite
temperatures,however,the analogoussingle-particle Green'sfunction isessentially more complicated,and itisnecessary to separate thecalculation into two
parts. Thefirststep,whichistreated inChaps.7 and 8,istheintroduction ofa
temperature Green's function @. This function has a simple perturbation
expansion similarto thatfor G at F = 0 and also enables us to evaluate the
equilibrium thermodynamicpromrtiesofthesystem. Thesecondstep(Chap.9)
227
az8
FINITE-TEMPERATURE FORMALISM
then relates @ to a time-dependentG reen's function thatdescribes the linear
response ofthe system to an externalperturbation;thislastfunction provides
the excitation energies ofthe system containing one m ore or one less particle.
DEFINITIO N
In treating system satsnitetem peratures,itwillbe m ostconvenientto use the
grand canonicalensem ble,which allowsforthe possibility ofa variablenum ber
ofparticles. W ith thedesnition
k4 - Jz#
4
thegrandpartitionfunctionandstatisticaloperator(seeSec.4)maybewrittenas
zG= e-biù= Tre-lA
(232)
IG= zo1e-j#= el(f1-A)
(;?.3)
.
whereweagain usetheshort-hand notation #= l//csF. TheoperatorX may
be interpreted as a grand canonicalham iltonian;forany Schrödingeroperator
ös(x),wethenintroducethe(modihed)Heisenbergpicture
dK(xm)- ekr/h0S(x)e-krlh
(234)
.
In particular,thefield operatorsassum etheform
'4.IXTI= ekxkh,
(
ja(x)e-krlh
IJ1 (XT)= ekrlh#1(x)e-*'/h
Notethatf4a(xr)isnottheadjointofQxztxm)aslongasmisreal.l Ifrisinterpreted asa com plex variable,however,itm ay be analytically continued to a
pureimaginaryvalue'
r= it. TheresultingexpressionX .(x,f/)thenbecomes
thetrueadjointof#xa(x,;
'/)andisformallyidenticalwiththeoriginalHeisenberg
picturedesnedinEq.(6.28),apartfrom thesubstitutionofX for.
JZt Forthis
reason,Eq.(23.5)issometimescalledanimaginary-timeoperator.
The single-particletemperatureGreen'sfunction isdesned as
Mxblxr,x'T')H-Tr()cFv(#x.(xT)4x
'jtx''
r')))
(23.6)
whereloisgiveninEq.(23.3). HerethesymbolFzorderstheoperatorsaccordingto theirvalueofm,withthesm allestattheright;T.alsoincludesthesignature
factor(-1)P,where' isthenumberofpermutationsoffermionoperatorsneeded
to restoretheoriginalordering. W eemphasizethatthetrace(Tr)impliesthat
thisGreen'sfunction @ involvesasum overacompletesetofstatesintheHilbert
space,eachcontributionbeingweightedwiththeoperatorlo(seeSec.4).
tToavoidconfusion,theadjointofano-ratort)isexplicitlydenotedby(öJtinthischapter.
1ThiscormKtionwasftrstpointedoutbyF.Blœ h.Z.Physik,74:295(1932).
FIELD THEORY AT FINITE TEM PERATURF
229
RELATIO N TO OBSERVABLES
The tem peratureG reen'sfunction isusefulbecauseitenablesusto calculatethe
thermodynamicbehaviorofthesystem. IfthehamiltonianP istimeindependent,asisusuallythecase,then @ dependsonlyon thecom bination m- '
v'and
noton rand r'separately. Theproofofthisstatem entisidenticalwith thatat
F =0 (Eq.(7.6)Jandwillnotberepeated here.
Considerthequantity
jjF.z(x'r,x'
r+)= trF(xm,x'
r+)
where r'
bdenotes the Iim iting value r+ T as T approaches zero from positive
values.and trrepresentsthetraceofthem atrixindices. Bydesnition,
trF(xm,xm+)=:!LXTr(/cf4z(xm)#x.(xr)J
- me
/'
fz)(TrLe-n*e#T?1.
(1(x)1Jz(x)e-kr/'j
=
qzeb.
f7jgTr(e-I#1
/1(x)fk(x))
=
:F(H(x))
(23.8)
wherethecyclicproperty ofthe tracehasbeen used (Tr(,4#C)= WrLBCA)=
Tr(Cz
4#)J,along with the commutativity ofany two functions ofthe same
operator. Asbefore.ourconventionisthatupper(lower)signsreferto bosons
(fermions). Themeannumberofparticlesinthesystem isgivenby
NIT,F,p.
)= :
2
1:.f#3xtrF(xm,xm+)
(23.9)
and is an explicitfunction ofthe variables specised. Sim ilarly,the ensem ble
averageofanyone-bodyoperatorisexpressibleinterm sof@ . W iththenotation
ofEq.(7.7),we have
#7)- Tr(âc7)
=
):
f#3xI
/jztx)Tr()(;y')(x')y'x(x))
x
p'
xi
'm
-,xv
=
:F)).fd?x lim lim Jj.(x)Fzjtxm,x'T')
xj
-
x'ex'
r'or+
:Ff#3xl
im lim tr(J(x)Ftxm.x'r'))
x'e x 'r'-+T+
(23.l0)
Particularexam plesofinterestare
(*)=mfJ3xtr(cF(xr,xm+))
-
(1-)=+Jd?xXIim
+x
'-
â2/2
rF(x'
r,x''
r+)
lm t
(23.11J)
(23.11:)
zx
FINITE-TEM PERATURE FoaM Aulsu
Two-body operatorsarealso important,buttheensembleaverageofsuch
an om ratorusually requiresa two-particle tem perature G reen's function. In
the slxcialcase thatthehamiltonian consistsofa sum ofkineticand two-body
potentialentrgies (Eq.(7.12)1,however,the mean potentialenergy can be
expressed solelyin termsof@,exactlyasinEq.(7.22). Thecalculation starts
from theHeisenbergequation ofmotion
ê
0
s
-
hà1
X z(XT)= htiEeXT/W (X)e x./A)
T
=
(X,#xa(XT)!
(23.12)
Forsimplicity,weshallassumethatthe potentialisspin independent(compare
Eq.(10.2)),asisusuallythecaseinapplicationsoftheinite-temperaturetheory.
Itisstraightforward to evaluatethecommutatorin Eq.(23.12),which yields
(compareEq.(7.19))
h?
h2:72
y;#xztx'
rl- zm-#x.(xm)+ rz#xatxm)
-
Jd3x'Cxytx'm)#xytx''
r)F(x- x#)yxztx'
r) (23.13)
Thusthesingle-particleG reen'sfunction satisfiestherelation
li
?
,,
# , y:
m hj.
.
jFajtx'
r,x r) =7:Tr)(;fk/xm) y Qxztxm)
TP- T 'F
,
-
.F
'
rr(â(
;f4,(x,-)g(>a
2m
:72+,)4x-tx-l
--
J#3xeYl
xylx'm)'
h ylx',
t
r)F(x- x'
')'
fxatx'
r)
The lastterm isessentially the quantity ofinterest,and wefind
(P)-JJdnxdqxnP'tx-x#)Trgl(;'
(1(x)'
G#(x*)'
/ytx-l1Ja(x)J
p
=
P
h2V2
UFJ Jd3xXl'im,x r'li
m -hV + lm-+ p,trFtxm,x'm')
-,m +
-
(23.14)
where the cyclic properties ofthe trace have been used to change from the
Heisenberg to theSchrödingerpicture. Equations(23.11:)and (23.14)may be
combined to providetheensembleaverageofthehamiltonian,whichisjustthe
internalenergyE (seeSec.4).
E - tz?) = t/+ P)
ê
hlV2
=E
FI.Jd3xAlim.x .'li
)
,.m
.. + -hX - l
m + p.trF(xm,x'r'
'-
(23.15)
Them eaninteractionenergycanalso beused to obtainthetherm odynam ic
potentialf1,by m eansofan integration overa variable coupling constant. If
23I
u. Q.k#.s.. R,s wvg- ws
z
H(
xx)=jv+xl
zd
s, zkxok
s mo +yk,
zx
#6.-'
l@=#.-yv
' az%=Q,a4erw vwuex=îaeamk?eueA d # 123.16)
e.cJ M %,
* 1esmvptveske,- 4>+ @v
:w-v.vwyfw wow-
4.r- à p*v>
.IV >e V '
caaot.go> & w
j-llu,kl .<'< k v '
ww mee o .hfw
.fwn-
.
j> #t>)' V x= e-M K=7ke-/At& , t,1.l7)
% aesvazzœw<
. N e.
a wx lx4..jt- N 3aJx
1>=-&'
rztx?a
-Gx
x '(x?.3t)
T* zhweà'A*jmil'w Lskwv < cacaa œ.> Aygve we..eytg.uw:
e
l/,x-hd #>*A% E1.tttlqlzlh.C (n ')1 L-f)> T.(>k.+xRln
-
.
J: V# qe s G
-1
>a.
3=1 m
:7x
+ .x
) (-j)?ss x o+xA;
xx (x?
- .
.
1. 4e m: +- dW
zx
A
'eyzwl)< iwxvpkvi- 1 V + .4 * *.2 .f
V.nfwa- k.wykl% < .Y àc,ljâwu,m/- :ww u ew:th
JG -îl% a > 44 yrvâ&mws.IQ qcs'cjyywq 4ue<u.uloptk
<wA 4 4œ >Awsa A Le vepzpwex Jz-ovaju .4J y a J,- op.
1.<' *u ,41 G âvm ve 1.c,oe:
::@s
3% .
.
=1 (m?
)'
i(-j)hnTv(/:0+z%,)*-IîI)
V
w
.
-
=-f*
+o((ns)?2-'f-pb-qT.((Aq+yî,)mH$,.
s
)
=-fTx(efytgoj1)= -;
e-Flïszx/j,yx
w
-
wl- 4...1 ûevzelhe-
t.;.I,)
;e .- M4 u)W 4* ejeye &y(:) #
.
x-kxwva 4 Els.(z3.
lî)%/Ejtz1,h)jxwta
ë;s x :-1zynk
31
jyN=x'
d4xv x
-
(xyzp)
Iaejym é.v.,x=p % x=$
t
un-m.=J.x 1jx zs$,yx
K
-
(a3.<)
w.x: dt.u +. 4wwsslh;cjv
o sa y
ue4. w . g.
Mku #e4.>1 ea # eo ai;j.(z3.
x)Ver* #vVoezaevtr.Jwölw W -wwo' g- w S- Jd 4.w w> om 'j< w
JlW *&< :x24% ow la' cm-
.
+ .j- l;xu & yewsx.
a3I
wW E!.(u.)ç)w So.xd ueB' e-,.AJ, Wew',Jwxaz- IXJ.
:
w? .4- < .> oatx)
J7
.
(1v,,)=4,tT,v,l
.)yj.o
xdàJINz
K*.x*
11
.
z.1+tt-'zaz4.
0
aœYy y1.-!>(4a.
,v,z,),q,...)
4w-<hwut!4 em<sao tws(&t.'
.
* 4 .u k < > w kw!4:.
T> u ozi t4. Ev.(z3.œ)yk(>3.az)x're- 19-- I
/ scwt
e t'e,d faujeAevhtes> u,% oe w/ wG 'A ,2** a
z- pvy.(7..,)uz(73,).luu- ,ueup- jnw-t
rG 'J > fzwtkeAwjwo
' < a' -oeewa k- j
w4 .>
% Ef. (zyza).
EXAM/I.
E:ygyzvTgkàcTlyj.&ï&D M
k:w'AV .-+ 4,> aaG w-c' # or ehetv e t
G >'
:
. #*
# (4xz:,=9f'
A e.'
-<
' e - .1.4* G <e
4o/w:d jW
' &,W ew/-/* ?e40 m P ew- ,x 4 <
L -/,
'y.
4+ .
t
% (.7.3J).
.t4?= ?-)x
44qsakx;0
1+(
.
0 k-zl
@ e qs:
) '
kl
k e7.
.
=
.
x
>
.
fzxas)
*- w 4w oaze'a < e w- f- - m
G &.:-
we>
c- G * W
je 4w-zva
e - w!- vm ey4 % !,,.)
*Ae >
' ur e,,<.c- uc.t
m- * twige,w,J
A k&é.. > '- 4- %*Q v* ,W W Yvto i% Ej.(R.%
> % . W< = .;'
r*'
.a.9.// .QeH<%
Ve '
w >
'
>m 4.,- ,Tv
-
taaall
.
z
x,tf,
x.zeA'
4'4r..o*%
. qyse-%*ya,>
14xe
+'kx
é$.
1i
t)= ?-z
%1t'
tzl=k-)zi
k -à
s/
.
x
1< / uer awa..w w/1 c- k *:+ 'f.
71 M L'vu cacwa- ,
y 4+ew w A - -A + .**- 4 %***
/N.
%c
%
l
%
(o
k
/
a k-x -(x3= e
/* = - c:e
k-y)ayxto)
v yg e-<'
.
'
-'
37, D'M '
,
ks wx)
.
233
FIELD THEORY AT FINITE TEM PERATURE
where4 =hlkljlm. Thisdifrerentialequationiseasilyintegrated:
(4 - rz).
v
(23.26/)
JkA(T)= JkzeXP y
-
and sim ilarly
u#kA(,
r)- tzf
kzexp(4 -hp.)r
(23.26:)
NoteagainthatEq.(23.26:)isnottheadjointofEq.(23.26*.
ThenoninteractingtemperatureGreen'sfunction isdehned as
Fyjtxm,x?m')= .
--ebx Tr(e-lXcFr('
#xa(xm)Ql/x''
r')))
(23.27)
and,fordehniteness,wesrstconsiderthecasey.> '
r':
F0
xb(xr,x'r')=-F-lkk
)j'Ajz1l
eltk*x-k'ex'ltyyjlajxjj,jj
'
.+ (e2,-hp).
v' 't:lkzck
f'A,)a
x exp (4 h rt),
.
-
k- p.)(,
r- ,
r/) vxa..kAu.
jA10
jz-jjxbN
gjk.(x-x?):xg (60
-k
v
h
-
m.a .
which follows from the translational and rotational invariance of the noninteracting system . ln addition,the ensem ble average m ay be rewritten as
tJkAJk
fzlg= l:
h:tJk
1l&k>)0= 1:i2Alk
(23.28)
wheretheupper(lower)signrefersto bosons(fermions)andn2isknown from
statisticalmechanicsgEqs.(5.9)and(5.12))tobe
n2-lexpEjtef-p,
)1::
F:1J-'
(23.29)
Theseequationsm aybecom bined to yield
f#'1b(x'
)('
r- m') (1:i:n(
r,x''
r')= -3ajU-1jét?îk*fX-X'1exp (62- p,
k))
-
h
k
(23.304)
A n identicalcalculation for '
r< r'gives
@L
)
' ' = F:
.)(r - m/) ?70
Jajpr-lN
kwtx-x')exp (e0
k- pJ
œj(XT,X T) U
'
t ef
j
k
-
r< r' (23.30:)
Asexpected,f#'
0isdiagonalinthematrixindicesand dependsonlyon the com -
binations (x - x',r- m'). It is interesting to evaluate the mean number of
za4
FINITE-TEMPERATURE FORMALISM
particlesNgandmeanenergyEnwithEqs.(23.9)and(23.15);astraightforward
calculation reproducestheresultsin Sec.5.
Xn(F,U,/z)= Zk Flk= Zk ICXPEl(4 - Jt)1E:
F11-1
(23.31J)
f:(r,F,/z)= Zk fzDî= Zk Ezfexp(j(eî- p,)1:
Y:1)-1
(23.31:)
ZK PERTURBATION THEORY AND W ICK'S THEOREM
FOR FINITE TEM PERATURES
Thetem peratureG reen'sfunction isusefulonlyto theextentthatitiscalculable
from the m icroscopic ham iltonian. Justasin thezero-temperature form alism ,
itisconvenientto introducean interaction picture.which then servesasauseful
basisforperturbation calculations.
INTERACTIO N PICTURE
ForanyoperatorOsintheSchrödingerpicture.weformallydesnetheinteraction
picturedz@)andHeisenbergpictureöx('
r)bytheequations
t)I('
r)Heenrlh0Se-'eT/'
t)x(m)Neer/h()se-kr/h
(24.1)
Thetwo picturesaresimplyrelated gcompareEq.(6.3l)J;
OK(z)= eew,
'ht,-#0m7'Oz(z)ekur''ta-','
/'
- '
#((),c)öJ(.
r).
p
#(m,0)
(24.2)
wheretheoperatoroâ isdesnedby
Notethat# isnotunitary,butitstillsatishesthegroup property
#(Tl.'
r2)#(T2,T3)= &é(T'l,T3)
(24.4)
and theboundarycondition
:
h
#('
rl,'
rl)- 1
(24.5)
In addition,theettime''derivativeofoâ iseasilycalculattd:
0
,
x
hpm T(m,,
r)- e or/1(k()- kje-A'
(z-z.z)/hc-korz/h
= ewcz/,(#0- k)e-xom/,#(z,v,,)
- - 4,
4,
r).
p
#(v.
,z-)
(24.6)
23e
FIELD THEORY AT FINITE TEMPERATURE
where
(24.7)
Xl('r)= eX0T/'#le-hr/h
Itfollowsthattheoperator#('
r,'
r')obeysessentiallythesamedilerentialequation
astheunitaryoperatorintroducedinEq.(6.11),and wemayimmediatelywrite
down thesolution
#
(r,r')=
C
D
n=0
1 a1 m
-j g-j ,
-
--
.
drk--.j.,#,
rnFz(
Xj(,
rI)...Xj(mJ)(24.8)
r
Finally,Eq.(24.3)mayberewrittenas tAhzxm@('
ZY.DQ Y$ :
e kp!h= e-xar/.Qqy,(j)
g4,p
-
lfrissetequaltojâ,Eq.(24.9)providesaperturbationexpansionforthegrand
partition function
ev-jja= a-re-jA
-
Tr(e-i'c#(,â.0))
'''
-n
z
=0(-à
1j*à
1
-ijo
bht
s,...jc
bh#zaTr(e-fforr:41(,
r,
)...#,(a))J
(24.10)
wherea11oftheintegralsextend overasnitedomain. In practice.thisequation
islessusefulfordiagrammaticanalysisthan Eq.(23.22)becauseofdiëculties
associated with counting thedisconnected diagram s.
The exact tem perature G reen's function now m ay be rewritten in the
interaction picture. Ifm > m',wehave
Fajtxz.,x'.
r?)=.
--ebiùTr(e-##Qx.(x,
r)!4/x'm.))
=.
--ebiùTrtc-ifnJ#(#â,0)(#(0,,
r).
#zatx,rlT(.
r,0)!
-
x(J#(p,,..)#l/xzgzl#(,
r',0)j)
Tr(e-/fc#(jâ,m)'
#
;z.tx.
rl#(m.,
r')fl/x'm')'
:('
r/.0))
Tr(e-/xo#(jâ,0))
(24.11)
9)hasbeen used with m=$h.
where Eq.(24.
yields
@(
z/txz,x',
r')=
A similarcalculation for'
r< r'
:
!:TrEc-f'nJ#(jâ,'
r')fljtx'm')J#@',m)f'zztx'
r)#('
r,0))
Try.-jx;wtjpj,oj
(24.12)
236
FINITE-TEM PERATURE FORM ALISM
Equations(24.11)and(24.12)havepreciselythestructureanalyzedinEq.(8.8),
and itisevidentthattheym ay becom bined in theform
As noted in Eq.(24.10).thedenominatorisjustthe perturbation expansion
ofe-lt
33
.in the present form , however,it serves to elim inate a1l disconnected
diagrams,exactly asin the zero-temperature formalism gcompare Eqs.(9.3)to
(9.5)J.
pEnloolcl'
rv oF g
Equation (24.13)shows thatthe integrations over the dum my variables Tia11
extendfrom 0to#â. W eshallseethatitisalsosumcienttorestrictJ'and r'to
thisinterval,sothatthediserence'
v- T'satisûesthecondition-jâ-c:r- r'-ct)h.
ln thislim ited dom ain.thetem perature G reen'sfunction displaysa rem arkable
periodicity which isfundam entalto al1ofthe subsequentwork. Forklel
inite-
ness,supposer'fixed(0< v'< jâ). A simplecalculationshowsthat
@ajtxo,x'm')= zbkebftTrfe-bp;'K
%jtx'r')'
?
Jxz(x0))
= z
bzebOTr(f%a(x0)e-bk'
fx
lp(x'r'))
= 7
zc/'t
'
lTrle-gâ'
t
/xatx)h)#ljtx''
r'))
- ci
ugajtxjâ,x'r')
(24.14/)
wherethecyclicpropertyofthetracehasbeenusedinthesecond line. A sim ilar.
analysisyields
Fajtx'
r,x'0)= +gzjtx'
r,x'#/j)
(24.14:)
so thatthesingle-particletemperatureGreen'sfunctionforbosonst-/prrz?/t
paxlis
periodic (antiperiodic)in each time l
'Jrïtz:/e with period lh in the range 0< m,
r'< jâ. This relation is very important for the following analysissand it
incorporatesthepreciseform ofthestatisticaloperator)ainthegrandcanonical
ensemble.
In the usualsittlation,4 istime independent,and @ dependsonly onthe
combination '
r-.r'. Equation(24.14)maythen berewritten as
Fa#(x,x','
r- r'< 0)= +Fx#(x,x'.T- m'+ nh)
(24.15)
FIELD THEORY AT FINITE TEM PERATURE
237
Thecondition'
r- r'< 0necessarilyimpliesr- 'r'+jâ>0intherestrictedrange
0< r,.
v'< jâ. Itisinterestingto seehow Eq.(24.15)issatissedforthenoninteracting Green's function F0. Comparison ofEqs.(23.30/)and (23.30:)
yieldsthe relation
akej(,k0-B)= 1cszzk
(24.16)
which m ay also beverifed direetly with Eq.(23,29).
PROO F OF W ICK'S THEOREM
It is apparent that the perturbation expansion for the tem perature G reen's
function @ (Eq.(24.l3))isvery similarto thatforthezero-temperature Green's
function G (Eq.(8.9)J. lnthatcase,theexpansion could begreatly simplised
withW ick'stheorem ,w'hich provided a prescription forrelatinga F productof
interaction-picture operatorsto the norm al-ordered productofthesam eoperators. The ground-state expectation value of the norm al products vanished
identically.so thatG contained onlvfullycontracted terms. U nfbrtunately,no
such sim pliscation occursatsnite tem perature.because the ensem ble average
ofthe norm alproductiszero only atzero tem perature. Nevertheless,astirst
proved by Nlatsubara, there exists a generalized W ick'stheorem that aliows a
diagram m atic expansion of @ . This generalized W ick's theorem deals only
with the ensem ble average ofoperators and relies on the detaiied form ofthe
statisticaloperatore'
-îkz. ltthereforedipkrsfrom theoriginalW ickgstheorem .
alid forarbitrary'm atrix elemellts.
which isan operalor /s/t'/?/3
k.1.v'
Before5,
:e considerthe generaltheorem .itishelpfulto exam ine the srst
feu termsofEq.(24.1F), Thenumeratormal'beB rittenas
Herethetirstterm isc-S:
lot<'0
xc.x'c'$and lsexactifX'
-i= 0, lntheusual
xs(
situat:
,t
7n..
X''lcontainsa spatiali1
3tegra1offourtield operatorsin theinteraction
plcture.and the second term ofEq.(24.17)inN'
olNestheensem ble aq'
erageofsix
tield operators. eNaluated with the statlslical operator e-sin. Thfs general
structure occurs in al1orders.and the generalized B'ick'stheorem isdesigned
preciselylo handlesuch problem s.
Although the unperturbed system is frequently hom ogeneous. m any
exam ples ofinterest are inhom ogeneous,and we shalltherefore use a general
single-particlebasis(ç7C
p
?(x))fortheinteraction-pictureoperators, AsinSec.2.
:T.M atsubara,Prog.r/letprer.Phys.(Kyoto).14:35l(1955);therresentrrooffolloB'
sthatof
M .Gaudin,Nucl.Phvs..15:89(1960),
2::
FINITE-TEM PERATURE FORMALiSM
theseld om ratorsarewritten as
?xl- â
g(x)J,
J
(24.18)
#ttx)= )JLW(x)'$
wheretheeigenfunctionssatisfytheeigenvalueequation
KzW(x)= (e0
,-p,)O (x)
(24.19)
and the index j includes both spin and spatialquantum numbers. W ith the
convenientabbreviation
0- M
elX KJ
(24.20)
theinteractionpictureofEq.(24.18)becomes
#z(XT)=ZJ %l(X)JJe-el''h
(24.21)
#1(xT)= Z
t
/Jtxltt/leeJT/9
J
Thtcorresponding singlt-particleG reen'sfunctionisrtadilyevaluated as
F0(x'
r>x',
r')= -: f/(
Jx)+0
#(x')èe-e/(m-'
r')/Ax 1cln
0nn
J
J
+&
(24.22)
where
nJ
o= ebi'
toTr(e-lAna%
/aJ)
= (
e#e?:F1)-l
.
(24.23)
The generalterm in the perturbation expansion (Eq.(24.17))typicall
y
containsthefactor
Trllo:T.L.LAC .../)J- (T.b.ihC .'
(24.24)
where.
'f,A C ...,/ areEeldoperatorsintheinteractionpicture,eachwith
itsown rvariable,and
)G0= vplih-nb
(24.25)
D esneacontraction
J'h'- (r.(,iW!)o- TrlâcnrzlyiWl)
(24.26)
For exam ple,
'
Jzatx'
r)'l/
lljtx'm')'=-F0j(x,
r,x'm')
(24.27)
FIELD THEORY AT FINITE TEM PERATURE
239
Thegeneralized W ick'stheorem then assertsthatEq.(24.24)isequalto the sum
overal1possiblefullycontracted term s
(rvEzi'
/c -../(12h(,- (.
z
f'.
â'C--.../.'-)+ E.
,
i'.
â.-(!'-.../.'')+ .-.
(24.28)
where(zi-'.
:-'C'.../--')isinterpreted as:
ici?f-C-.
J''-../'--). Itisclearly
suëcientto proveEq.(24.28)forthecasethattheoperatorsarealreadyinthe
propertime@)order,becausetheoperatorsmaybereorderedonbothsidesof
.
theequation withoutintroducing anyadditionalchangesofsign. W etherefore
wantto provethealgebraicidentity
l.i.
âc ..-/)()- (,f-â.C--.../.'')+ Evi'
-.
â''ê-.../-.')+ ..
(24.29)
subjectto the restriction '
rx>'rs>rc> '''>'rs,which allowsusto remove
theT.signontheleftsideofEq.(24.28).
.
It is convenientto introduce a generalnotation for an operator in the
interactionpicture,and Eq.(24.21)willbewrittenas
'
;zorW -7)
y,(xm)x,
J
(24.30)
wherex)denotesajorJy
'
tand xJ(x'
r)denotest
po
ltxle-eg,'
/'or+0
J(x)#eelrlh* w ith
thissimpliscation,theleftsideofEq.(24.29)becomes
tZkYC .../):- )g)2)(...I yaz,yc
J b c
.u
.f
xTr()s()n aoa.c'
(24.31)
SinceXacommuteswithX,thetracevanishesunlesstheset(% '''(v)contains
an equalnum berofcreation and annihilation operators;asacorollary, the total
number of om rators must be even. Commute n successively to the right
Tr(p
-ccxax,xc ...ar)= Trtp
Acoltu,t
vlvac...czl
+ Trlp
-ont
xNlaa,œclv '.'az.
l+ '''+ Trlp
mc:xtac '.'l(
u,t
xy.
lvl
zjzTrtp
-ccxbxc'.'ara.) (24.32)
where,asusual,theupper(lower)signsrefertobosons(fermions). Thecommutators(anticommutators)areeither1,0,or-1(compareEqs.(1.27)and(1.48)),
.
.
dependingonthepreciseoperatorsinvolved, and m aythereforebetakenoutside
thetrace. Inaddition,asimplegeneralizationofEq.(23.26)showsthat
xae-bh = aaeàle.
(24 33)
ejA;(
whereà.= 1ifn isacreationoperatorandY = -1ifaaisadestructionoperator.
.
Thisrelation isequivalenttotheequation
a.p
,
..
t;0= p
.
-cox.eh/e.
(24.34)
240
FINITE-TEMPERATURE FORMALCSM
and thelastterm ofEq.(24.32)may berewritten with thecyclicproperty as
+Tr(a,)(;()xbac'''xp = +el.le,Tr()(;()n a.
,xc '.'a.
r)
(24.35)
Inthisway,Eqs.(24.32)and(24.35)leadtotheimportantresult
,)
Trtp
-s,n a,t
xc'-'œy)- (%,aA
jvevrtpc,xc...ç
p)
1+ e , .
+
(
v1v vrtxs,a,ac...)
(t
xa,rzclv vrtpxstjyv . . . a,.)+ ...+ 1(%,
:Fez,;e. p
(24.36)
1:F eA,jea
.
Equation(24.36)assumesamorecompactform withthefollowingdefnition
ofa contraction
l(
x4,sx,(
lv
Xa
'Gj= 1::
Fnenabea
(24.37)
and we5nd
Tr()G0xatx,œc
(24.38)
which deines the traee of an operator expression containing a contraction.
ln practice,m ostoftheterm svanish.and theolïl)nonzero contractionsbetween
thetime-ordered operatorsin Eq.(24..
78)are
e
L
..
p
j--. = ---..--1--.. = ).?a.
'
j.a
c = --at
-a
.
-..t
-.j)
-j aJ:= .:
; e,
nt
'z.
c.
-1 J
.
- v,be.? 1uucîe-
(24.39)
>,,t/,1v.=
l - = 1+ nj
()
t7.ut.= i
a J 1 uce-beJ 1 m e-bej
Both ofthese contractionsare also equalto the ensem ble average ofthe sam e
operators,and weconcludequitegenerally that
K't
xà= lxat
xpè
.0= ttfrlt
xaapll'o
(24.40)
FIELD THEORY AT FINITE TEM PERATURE
because rx > '
rs > Tc > ' ' '> '
TF.
yields
241
A combination ofEqs.(24.31)and (24.38)
tXWC . ../)0
-
i é)X '''.
N
5-Xax,xc'''X,.t-f'rEp
'
-c()xa
'(
x,
't
xc'
a b c
,.
+ Tr(p
''coxa't
x,xc
' '' 'ay1+ .
/'.)v
(24.41)
.
The contraction isa c num berand m ay betaken outofthe trace,leaving a structuresim ilarto thatoriginallyconsidered. Thesam eanalysisagainapplies,and
wethereforeconclude
# ..'u()
+.qX'h ''C ' ... f%-''))()+ . .. (24.42)
. .
.
where 'rx > '
rs > rc > ' . '7'
.ry. By assum ption,the left side is tim e ordered
and m aybewrittenasL.'rzgzi'
éd . ../1'
.a,which provesEq.(24.28).
Thepresentproofshowsthatthefinite-tem peratureform ofW'ick'stheorem
is very general'
,itdescribes a finite system in an externalpotentialaswellas an
inhnite translationally invariantsystem . The only assum ption isthe existence
ofatime-independentsingle-partielehamiltonian Xothatdeterminesthestatistical
operator clscfle-xo'. For deéniteness, we have considered a self-coupled field
in which X refersto asinglespecies. A similarproofcan beconstructed for
eoupled fields,however,aslongasX()isasum ofquadratiutermsreferringto the
separate éelds. In thiscase,thetrace factorsinto products.oneforeach seld,
and the contractions between operators referring to dillkrent lields vanish
identically'. The generalized W ick-s theorem therefore allow's us to study
arbitrary interacting system sigtherm odynam icequilibrium .
25JD IAG RA M M ATIC A NA LYSIS
The preceding analysis showsthat the perturbation seriesfor the temperature
Green'sfunction Fajlxc-sx'z')isidenticalwith thatatzero temperature,theonly
difference being the substitution of$%forG0and thelinitedomain ofthetime
integralsoverr from 0 to jâ. Asa concreteexample,considerthequantity
)'Fg(?
Ja(l)1Jj(2)?
Jj,(2')1
/1#,(1'))='
,:where the number 1 denotes the variables
'
(xI,'
rl).and thesubscriptlhasbeen omitted becauseal1subsequentwork isin
theinteraction picture. The field operatorscan becontracted in two dipkrent
ways.and Eq.(24.27)then gives
'
:
trzl'latl)f,(2);,).(2')1
Jf
a,(1/)))0
= '
fatl)'1
Jj(2)'',
(7
'
j
f,(2')'''
t
/'
t'(l')')()+ x'ifatl)''
t
/'j(2)''('f
j,(2')'('
A,(1')*')(j
x
=
f#jz-tl,l')Ft
/j,(2,2')+f#'
j:
j;-(1,2')F0
ja-(2,l')
(25.1)
z*z
FINITE-TEM PERATURE FORM ALISM
Eachterm inthenumeratorofEq.(24.13)canlyeanalyzedinasimilarway.
Since the algebraic structure of the Enite-temperature W ick's theorem (Eq.
(24.28))isidenticalwiththatofthefullycontractedttrmsinthezero-temperature
form (Eq.(8.32)J,thetempcratureGreen'sfunction F hasthesamesetofall
Feynm an diagram s that G had at zero tem perature. In particular, a given
diagram iseitherconnectedordisconnected,andthedenominatorofEq.(24.13)
precisely cancelsthecontribution ofthedisconnected terms,exactly asin Sec.9.
Asa result,the temperature Green'sfunction hasthesameformalstructure as
Eq.(9.5)
*
@u/1.2)=a.0
1 n l j: ,
#.
-y ?
p jvd
rl-'-Jcdrn
'
.
xTrflsorv(#,(m,
')..-#l(rk'),//1)#,
f(2)!Jco...ct.a (25.2)
where only connected diagram s are retained. For any particular choice of
#ja #l,thedetailedderivationoftheFeynmanrulesisalsounchanged,andwe
shallmerely statetheEnalresults.
FEYNM AN RutEs IN COO RDINATE spAce
ThCmostCommon Sittlation isa Self-coupled seld,in which P1representsa
tWo-particleinteraction
#lM 4,-!.JJd'xjd3xa'
klz
xxllyj
xxc)p'txl-xz)4/xz)4.(xl)
(25.3)
whereF(x)istakenasspinindependent. Thecorrespondinginteraction-picture
operatoriseasilyobtained
XI@l)-!'JJ#l.tl#3.
*2#ltxl'
r1)Mj
ftxz'
r1)P'txl-xc)Qjtxar,)fktx,,
rl)
-!.Jfd3x,dqxz(0
p'tscfctx,z.il4#
jlxavalyz.(,tx,'
rl-xzmc)
o
x#/x2'
rz)#atxlml) (25.4)
wherethesubscript1hasagain been om itted and thegeneralpotential
A etxl'
rl,xc'
rc)= F(xI- xc)3('r1- n)
(25.5)
hasbeenintroduced. TheperturbationexpansionofEq.(25.2)includesprecisely
thesam eFeynm andiagram sasinSec.9,andtheonlym odifcationistheFeynm an
rulesusedtoevaluatetheath-ordercontributiontoFzj(1,2).
1.D raw a1ltopologically distinctdiagramscontaining n interaction lines and
ln+ 1directed particlelines.
2.Associatea factorfq j(l,2)with each directed particle linerunning from 2
to 1.
3.Associateafactor,
F'
c(1,2)with each interaction linejoiningpoints1and2.
4.
Integratea11internalvariables:Jd3xfflt
lâd'
rj.
FIELD THEORY AT FINITE TEM PERATURE
243
5.The indices form a m atrix product along any continuous particle line.
Evaluate a11spin sum s.
6.Multiplyeach ath-orderdiagram by(-1/âP(-1)F,whereF isthenumberof
closed ferm ion loops.
7. Interpretany tem peratureG reen'sfunction atequalvaluesof'
ras
Fotxjgj,xi'
rjl= .rJl
im Fotxj'
rj,xirJ.
.-+.r3+
.
.
As a specifc exam ple,considerthe setofa1lzero-and ûrst-orderterm s
(Fig.25.1). The specific choice of labels for the internallinesis irrelevant
1x
'
X #(1.2)=
lx
3
h p,
à rz4
Fig.25.1 Zero-and first-ordercontribu-
tionstoC#a#tl,2)incoordinatespace.
2#
2#
1x
3 ;
$
A
+
+ '''
4#
#
2#
becausethey representdummy variables. According to the rulesjuststated,
Fig.25.1 im pliesthe following term s:
wheretheuicinthesecondterm arisesfrom theclosedloop. lffq jisdiagonal
in thespin indices(= @Q3aj),then @ isalso diagonal.and thespin sumsare
readily evaluated
F(1,2)=F0(l,2)-h-1fdbxj#3x4J;î
hdvj#,
ugukta
zu
'
y+ 1)F0(1,3)
0
'
G.
x 070(3,2)tf0(4,4)yz-c(3,4)A-@70(1,3)q70(3,4)q$0(4,2)4r0(3,4))A-...
(25.7)
Herethefactor(2J+ 1)representsthedegeneracy associated with particlesof
spinJ. Asnoted inrule7,F0(4,4)isinterpretedasF0(4,4+). ltmayalsobe
identihed asa generalized particledensity
(25.8)
thatdependsontheparameterp ;itthereforedifl
-ersfrom theunperturbed particle
density unlessJ,
tisassigned the value appropriate to thenoninteracting system .
FINITE-TEM PERATURE FORMALISM
2*4
FEYN M A N RULES IN M OM ENTU M SPACE
ThereisnodiëcultyinwritingoutEq.(25.7)indetail,butthediflkrentform of
@0(i,j)forrt;VJsoonleadsto aproliferationofterms. Forthisreason,itis
advantageous to introduce a Fourier representation in the variable '
r,which
autom atically includes the diFerentorderings. This step has been centralto
thedevelopm entofm any-body theory atfnitetem peratureand wasintroduced
independentlyby Abrikosov,G orkov,and D zyaloshinskiislby Fradkin,zand by
M artin and Schwinger.3 It leads to the sam e sim plihcation as in the zerotem peratureform alism .
Thecrucialpointistheperiodicity(antiperiodicity)of@ ineach'
rvarlable
with period jâ (Eqs.(24.14)and (24.15)1. Forsimplicity,we assumethat@
depends only on the diflkrence 'rj- mc,which represents the most com mon
situation :
Ftxl'
r1,xa'
r2)- '
.
#(xl,xz,'
rl- ,
ra)
(25.9)
Forb0thstatistics,@ isperiodicovertherange2jâandmaythereforebeexpanded
in a Fourierseries
F(xj,xz,'
r)= (jâ)-1)
(e-îconrfftxj,xc,(sal
a
(25.10)
T H T I- T2
where
n=
tt)s= --j
(25.11)
Thisrepresentation ensuresthatFtxI,x2,'
r+ 2#â)= F(xIsx2,r)and theassociated Fouriercoel cientisgiven by
lâ d
F(x,,
xz,
(
w)=.
i.j-ls'
ve'
f
t
z
a'
rF(xl,x2,m)
(25.12)
ltisconvenientto separate Eq.(25.12)into two parts
vtxjxz-t
,
?nl-.
!
,
.J0bhdreiœ-.gtxlxc'
r)+!.j(
b
)
'lrei
t
'
-rg(x'
,,xc,.
r)
-+!.j.j,J,
re'
n-rgtx1,xz-.
r+#,)+.
à.j,
b't
sei
u'
a.
rg(xl,
xc,v)
'
,
-0
i
-.
1-(1+,eï.
,a-,,)(0/'âvgeico-.gtxj-xz.o
-
.
(25.13)
wherethesecondand thirdlinesarcobtainedrespectivelywithEqs.(24.15)and
an elementary change of variables. Equation (25.11) shows that e-ioneh is
1A.A.Abrikosov,L.P.Gorkov,and 1.E.Dzyaloshinskii,Sov.Phys.-JETP.9:636 (1959).
2E.S.Fradkfn,Sov.Phys.-JETP,92912(1959).
3P.C.Martin and J.Schwinger,Phys.AeL'.,115:1342(1959).
FIELD THEORY AT FINITE TEM PEHATURE
245
n even boson
n odd
n even ferm ion
n odd
(25.l4)
where
(25.lf)
ferm ion
%0
(X1%. 0(X')'
f(1 - /:S1.
-
y 1
.
v'
w.
u. . .-$t.
.u.u..
y(q-j
.
.
:
h(r)f,.s,jq-..1jt
.
)
(pg;uu,u.4)j .u
c
--Ji x
.
,-.-,.j (!
-
z
'0
-
,
,
--x
%
9' (N.X .t
.
c
pn)u
c
ù ',.-
'
,
.
,
.) ..+
,r.'
-i,
îx jr'.-..
(x q
''
.
ltsu.
- t'j..)(t
rc .- ..)
(2f.l8)
a46
FINITE-TEMPERATURE FORMALISM
and Eq.(25.l7)isseen tobecorrectforb0thstatisticsntheonly distinctionbeing
therestriction toevenoroddintegersinEq.(25.15).
Todevelop theFeynmanrulesinmomentum space,weconsideranarbitrary
vertexinaFeynmandiagram contributingto Eq.(25.2). Thestatictwo-body
potentialhasthetrivialFourierrepresentation
'
F-otxjm,,xz'
rz)= (jâ)-!,1)
)
e-ft
*ntTt-mzl< ;(xl,xz,fw)
'Y'
en
(25.19)
where
< otxl,xc,(snl= Ftxl- xa)
(25.20)
einzbtt',z'
zj
e...jwapy
e- /w&/
r
Fig.25.2 Basicvertexintem x ratureformalism .
and wehaveused the identity
3(T)= (#â)-'?ttZV:n e
'-lf*'**
(25.21)
validintherange-jâ< '
r< jâ. Eachinternalvertexjoinsasingleinteraction
Iineandtwoparticlelines(oneenteringandoneleaving),asin Fig.25.2. The
entire rjdependence iscontained in theexponentialfactorsshown in Fig.25.2,
and theintegraloverh becom es
19#
Jc ,
r,expt
-f((
s-+t
s,.-(
,)
a-).
r/!-jâôa,
.+--,
.--.
(2s.
22)
Equation (25.22)showsthatthediscretefrequency isconservedateachvertex,
exactlylikethecontinuousfrequencyinEq.(9.13). Notethatthisconditionis
independentofstatistics,btcauseboth particleIinescarryevenorodd frequencies,
whereastheinteraction linealwayscarriesan even frequency.
Itis now straightforward to derive the Feynman rulesforthe ath-order
contributiontoFa/xl,xz.(snl,whichwouldapplytoaninhomogeneoussystem
such as an electron gas in a periodic crystal potential. For m ost purposes,
however. it is perm issible to restrict ourselves to system s with translational
invariance,where @ depends only on the diflkrence of the spatial variables
xl- xz. ln thiscase,the Feynm an diagram scan be evaluated in m omentum
space, which greatly simplises the calculations. The transformation of the
Feynman rulesfollowsimmediately from theanalysisin Sec.9,alongwith Eqs.
(23.30). Forsimplicity,only thelimitofan insnite volume(F -+.(
:
c)isconsidered,and thetemperatureGreen'sfunctionscanthen beexpanded asfollows:
@ j(x,x',m)= (jâ)-i(2,
* -3f#3kefketx-x')jge-ft
zuT@ (k(sal
a
(25.23)
247
FIELD THEORY AT FINITE TEM PERATURE
(25.24)
ThephysicalquantitiesN,E,andfl(seeEqs.(23.9),(23.15),and(23.22))become
N = :!
:P-(2*)-3(#â)-lJdàkéleiovn'ltrF(k,f
.
zJa)
(25.25)
E = (4)=upP'(2=)-3(jâ)-ljd?k â& eikon'î.
jtjâf.
t).+ e2+ p trF(k,f
.t)al
(25.26)
fl=flc:
+P'j0
'A-1#A(2=)-3(jâ)-lfd3kX
neiç
unT
xJtï/ir
xpn- 62+ p,
)trF3(k,(sa) (25.27)
and weshallnow statetheFeynm an rulesforevaluatingthenth-ordercontribu-
tiontoF(k,f
x%).
1.D raw al1topologically distinctconnected graphswith n interaction linesand
2n + ldirected particle lines.
2.A ssign a direction to each interaction line. A ssociate a w ave vector and
discretefrequency with each lineand conserve each quantity ateveryvertex.
3.W ith each particlelineassociateafactor
bxp
/j.,(,k-s)
@ljtk,oar
nl-j*m
-
(25.28)
wheref
t
pmcontainseven(odd)integersforbosons(fermions).
4.Associateafactor'
#'-otk,f.
t%lY Pr(k)witheachinteractionline.
5.lntegrate over alln independentinternalwave vectors and sum over alln
independentinternalfrequencies.
6.Theindicesform am atrixproductalonganycontinuousparticleline.
ate a1lmatrix sum s.
M ultiply by (-jâ2(2=)3)-n(-1)F.where F isthe numberofclosed fermion
loops.
Wheneveraparticlelineeitherclosesonitselforisjoinedbythesameinteraction line,inserta convergence factorcfœmn.
Asan exam ple,we onceagain considerthezero-and srst-orderdiagram s
(Fig.25.3). Afterthespinsumshavebeenevaluated.we;nd
FINITE-TEM PERATURE FORM ALISM
248
k&ttln
k:Lùn +
k5u,n
k',a'<,+ k'
&oa
n.
0
k,(.o
k- k'&tsa-a,n,
' k (.
o
Fig.25.3 Zero-and fi
rst-ordercontributionsto F(k,ts)a)
in m om enttlnlspace.
Thisexpressionhastheexpectedform gcompare/q.(9.22))
f#tk,twl= F0(k,(
w )+ F0(k,Ya)E(k,(s.)F0(k,(sa)
(25.30)
where E(k,r.
,)a) is the self-energy. ln particular,the srst-order self-energy is
givenbygcompareEq.(9.23))
X(l)(k,f
.
tG)K X(j)(k)
=
(-â2j)-1(
j
jeiuln''l(2.
/4-3jdqk'F0(k',(z)a.)
n'
x (+42.
:+ 1)P'40)+ P'tk- k'))
l
#3k'
l
J(2,43EV12'
î'
>'1)Z(0)-Z(k-k'
llj-j
=j
t'fu3n''tl
x
a'iœn,- h j(e(k),- /.
t) (25.31)
-
ItisclearthatY(1)isindependentof-u)nandmaythereforebewrittenasY(j)(k).
EVA LUATION O F FREQU ENCY SU M S
The frequency sum in Eq.(25.31)is typicalofthose occurring in many-body
physics,and we therefore study itin detail. For dehniteness,considerthe case
zplane
r
r
C
r
C
C'
7= A' C ,
c#
* 7= l.ttln C #
r
Fig. 26.4 Contour for evaluation of
frequency sums.
FIELD THEORY AT FINITE TEM PERATURE
249
ofbosons,wherethesum isoftheform
X eîu'
nTUtt)a- x)-l
(25.32)
withtz)a- 2nrr/#â. Equation(25.32)isnotabsolutelyconvergent,foritwould
divergelogarithm icallywithouttheconvergencefactor;x)m ustthereforerem ain
positive untilafterthe sum isevaluated.
The m ost direct approach is to use contour integration,which requires a
m erom orphic function with poles at the even integers. One possible choice is
lhlebhz- 1)-1,whosepolesoccuratz= lnnilqh= ialn,each with unitresidue.
IfC isa contourencircling the imaginary axisin thepositfve sense(Fig.25.4),
then thecontourintegral
Bh
dz
p4=
- - .
lni c é''9z- 1 z - .x
?- -
(25.33)
-
exactly reproducesthe sum in Eq.(25.32),becausetheintegrand hasan inhnite
sequenceofsimplepolesatitnnwith residue (jâ)-1eicon'lt/'
f
x>n-x)-1. Deform
thecontourtoC 'and lMshownin Fig.25.4. IfIz'--.
>.vzalongaraywithRez > 0,
thentheintegrand isoforderIz(-1exp(-(jâ- z))Rez),
'if'
;
z -+ x alonga ray
with R ez < 0.then the integrand isoforderh
zl
1
e
x
pt
p
Rez
)
. Sincenh >'
?)>0,
,
Jordan'sIemmashowsthatthe contribùtionsofthelargearcsf'vanish and we
are leftwith the integralsalong C '
eit
zlxn
n
iul?t- x
lh
dz
:nz
2=1 c,e/lz- l z - x
(25.34)
The only singularity included in C'is a sim ple pole atz= .
,
'
t.and Cauchy's
theorem yields
lim
giuaan
)h
=
5-*Qn r!Ve!J1 icon - x ebsx - l
-
(25.35)
where the m inus sign arises from the negative sense of C ', and it is now perm issible to let T -->.0. This derivation exhibits the essentia!role ofthe con-
vergencefactor. Althoughthefunction-J'
?â(p-l$9z- 1)-1alsohassimplepoles
atz= iœnwith unitresidue,the contributionsfrom l>would diverge in thiscase,
thuspreventing the deformation from C to C '.
A similaranalysismay begiven forfermions,where œn= (2n+ ll=/jâ.
Thefunction.
-phlebhz+ 1)-1hassimplepolesattheodd integersz= iulnwith
unitresidue,and theseriescan berewrittenas
f,f(zao'
q
.
sK- x
Fl0dd ïf
-
)h
yz
eTz
l=i c eb'z+ 1z - x
(25.36)
2*
FINITE-TEM PERATURE FORMALISM
where C isthesamecontourasin Fig.25.4. Jordan'slemma again allowsthe
contourdeformationfrom C to C'because#â> T> 0,and thesimplepoleat
z= x yields
lim
elau,l =
pk
:*0atx,d iœn - x e'hx+ 1
(25.37)
Tlw twocasescan btcombintd in thesingleexpression
lim
eltt
h'
ph
:1*: a it
ata - x = :!:e''x:!:1
(25.38)
whichwillbeused repeatedly in thesubsequentchapters.
Thefrst-orderself-energy (Eq.(25.31))can now besimplised with Eq.
(25.38),andwe5nd
âX
dsk'(42.
:+ 1)F40)+ F(k- k'))
(l
)(k)=Jgsj exp(g4.-slj:
j
uj
=
U(0)(2J+ 1)(2.
*-3Jdsk'a2,+(2*-3Jd3k'I
qk-k')ak.
(25.39)
Thisexpressionappliestofermionsata11temperaturesandtobosonsatsuëciently
high temperaturesthatthe unperturbed system hasno Bose-Einstein condensa-
tion. Itisimportantto rememberthatn2dependsexplicitly on thechemical
potentialy.. Forthis reason,(2J+ 1)(2r4-5fdtknL isalso a function of/.
:
and cannot be identised with the particle density. Apartfrom thisone dif-
ference,however,Eq.(25.39)isadirectgeneralizationofthatatzerotemperature
(Eq.(9.24)).
26LDYSO N'S EQ UATIO NS
The structureofDyson'sequationsatzero temperature wasdetermined by the
setofa1lFeynmandiagrams. Asshownintheprevioussection, thetem perature
Green'sfunction leadsto an identicalsetofdiagrams,and itistherefore not
surprisingthatDyson'sequationsremain unaltered. Indeed,thisrepresentsthe
primary reason forintroducing the temperaturefunction,even though @ isless
directly related to physical quantities than the analogous zero-tem perature
function G. In coordinate space,thetem perature G reen'sfunction alwayshas
theform (compareSec.9)
qf(1,2)==qf0(l,2)+J#3#46/041,3)143,4)ùf0(4,2)
(26.1)
wheretheintegralscontainan implicitspin summation and thetimeintegrations
runoverrkfrom 0 to#â. Equation(26.1)desnesthetotalself-energy X(3,4);
itisalso convenientto introducetheproperself-energyX*(3,4),whichconsists
ofallself-energy diagrams thatcannotbe separated into two parts by cutting
FIELD THEORY AT FINITE TEM PERATURE
251
oneparticle linef#0.
- Asin thezero-temperature formalism (Eq.(9.26)),the
self-energy isobtained byiterating theproperself-energy
Y(1,2)= X*41,2)+J#3#4E*(1,3)F0(3,4)Y*j4,2)+ .
(26.2)
andthecorrespondingtemperatureGreen'sfunction obeystheintegral(Dyson's)
equation
F(1,2)= f#'0(1,2)+ .f#3#4F0(1,3)Y*(3,4)F(4,2)
IterationofEq.(26.3)clearlyreproducesEqs.(26.1)and(26.2).
Dyson's equation beconnes m uch sim pler if the hamiltonian is tim e independentand it
-thesystem isuniform . A lthough itiseasyto writedown the
expressionsforspatiallyvaryingsystems(correspondingtotheGreen'sfunction
f#txl,xc,opnll,we shallconcentrate on the m ore usualsituation where the fkll
Fourierrepresentation ispossible (see Eqs.(25.23)and (25.24)1. Thesumsand
integralsin Eq.(26.3)are then readily evaluated,leaving an algebraic equation
Flk,f.
t?al= F0(k,(sa)+ F0(k,(sn)X*(k,t
z)Jf#'tk,f,
enl
(26.4)
where al1 quantities are assum ed diagonal in the m atrix indices. Equation
(26.4)hastheexplicitsolution
Ftk,(5nl- (F0(k,f
.
z)n)-l- Y*(k,(sa))-l
Fajtk,tzynl= bufliu)n- â-1(e2- p,)- Y*(k,tsn))-1
(26.5)
where the lastform has been obtained with Eq.(25.28). Thisexpression for
F(k,(sa)isformallyverysimilarto Eq.(9.33)forG(k). Thereisoneimportant
diserence,however,becauseu)nisadisctetevariable,instead ofatruefrequency
or energy. For this reason,the determ ination of the excitation spectrum ck
fora system containing onem ore oronelessparticleism orecom plicated than
atF = 0s
'weshallreturn to thisim portantproblem in Chap.9.
ThepreviousexpressionsforN,E,and fl(Eqs.(25.25)to(25.27)1can be
simplised with Dyson'sequation:
J3# 1
eîkk
'n'l
N(F,F,
p,
)=:
TEl
z
'
(2.
8i
-l)j(zzp;yFslu.
-a.
-ju
o(yk.s
-j--'
j;w(k,
co
-jj
(26.6)
J3é- 1
j
f(F,U,
J
z)=E
FP'
(2J+1)j(a,
a.
)ajjF.)cwa.
,
y
it
p-1-+ â-1(62+sw
rz)tk
loln-h (e2- /z),.a)
xjh .
d3k 1
i
-urp,
(a,+1)j(a.),sy-e..
,
?
c +.
p x*(k,oa.)
xjâ+j..-y-,(,k-s)-z.(k,
.x)j
. .
262
FINITE-TEM PERATURE FORM ALISM
f1(F,
ldA g d:k j.
y koosyp
P'
,r
z)=fàc(F,F,
Jz)E
FP'
(2J+1)jo
-j-j(zs)u
jm
.
s
!4E*A(k,œJ
s'
!â+ ioln - h-1(ek s). swA(k,(sa) (26.8)
-
Theconvergencefactoragain playsanim portantrolesforitallowsustoelim inate
theconstantterm inthelasttwoexpressions. Fordefniteness,considerbosons
where
(26.9)
The second line ism erely the sum ofa geom etric serieswhile the lastIine follows
for0< '
r)<)h. (ComparethediscussionfbllowingEq.(9.36).) Thefkrmion
sum m ation diflkrsonly by a factorerf47/h and Eqs. (26.
7)and (26.8)therefore
becom e
E=:
!LPr(2J+ 1)(2=)-3(jâ)-1(dbkJ(
eiœnTge2+jâX*(k,(sa))g(k,tt)a)
n
(26.10)
ldv
j
#3k
.
fl=t'
)o:
!
:Pr(2J+1)jou-jjjp()h)j(ej
asyjjj
uwzjkyt
szsAykysa;
(26.11)
where b0th 1:*2 and GA m ust be exaluated forvariable coupling constant à.
Itis also usefulto introduce the polarization A and eflkctive interaction
F ,exactlyasinEq.(9.39). Inthepresentcase,wheretheinterparticlepotential
isspin independent,'
F-tq,(salfora uniform system satissesthealgebraicexpression
'
#?'
-tqsttGl= '
/'
-0(q,t
,an)+ '
/'-0(q,f.
'Q -R(q,œn)'
f/-otq,tz
/al
(26.12)
IftheproperpolarizationJ1*isdeEned asthesum ofallpolarization insertions
thatcannotbeseparated intotwopartsbycuttingasingleinteractionline'
#%,
Eq.(26.12)canthenberewrittenas(compareEq.(9.43*)
< (q,f
.
tGl= A otq,tz)nl+ A otq,tz?2Z*(q,tzG)A tq,f
.
':al
(26.13)
w ith theexplicitsolution
'
#''
tq,uQ = '
#'-0(q,(,)2 E1- '
/'-()
(q,(t?2.
1'
1*(q,tzan)l-1
(26.l4)
Althoughthisequation issimilartothatatF= 0 (Eq.(9.45)),thediscretefrequency œnprecludesa directinterpretation of'
#'-tq,(zulasthe efectivephysical
FIELD THEO RY AT FINITE TEM PERATURE
253
interaction fora given m om entum and energy transfer. Thusthe determ ination
ofthecollectivem odesassociated with densityoscillationsatfinitetem perature
requiresa m orecarefulanalysis,which isgiven in Chap.9.
PR O BLEM S
7.1. D ehne the two-particle temperature G reen's function by
F xj;y:(xlrlsx2r2s
'xi
'r1
',x2
'r2
')
-
Trlp
'-sFr('
t
/
lsatxlrl)fxjtxzr2)7i'lôlxc''
r2
');'ly(x1
'J-1
')1)
Prove thatthe ensem ble average ofthe two-body interaction energy is
'
,x
..
''
NP>- à.fJ3x f#3x')'tx5x')y,à,,sar#zz.;ss,tx'm,xm;x'r--xr--)
.
.
7.2. Consider a m any-body system in the presence of an external potential
U(x)with a spin-independent interaction potential L'(x - x'). Show thatthe
exact one-particle tem perature G reen's funetion obeys the following equation
ofm otion
-
'
g
hlVf
h&t
-l.+ t- &'(xj) $=fj(xjrI,xj
''
rJ)uc.fd?.
vclz-(xj- xa)
-js-?-- /.
x ffay;jy(xIr1,xzr);x1
'rj
',x2T1
-)= hblxI- xL
')b(rl- r/)bx;
where the two-particle G reen'sfunction isdesned in Prob.7.1.
7.3. Assuming a uniform system of spin-! fermions at temperature F,and
using the Feynman rulesin m om entum space.
(J)writeoutthesecond-ordercontributionstotheproperself-energyinthecase
ofa spin-independentinteraetion ;
(:)evaluatethefrequencysums.
7.4. Considera system ofnoninteracting particlesin an externalstaticpotential
withahamiltonian#ex= j#3x.
?
J)(x)Pr
a#txl'
t
;jtxl.
(J) UseW ick'stheorem toevaluatethetemperatureGreen'sfunctiontosecond
orderin Pex. HencededucetheFeynmanrulesfbrFe
axj(x-rsx?r')toaIIorders.
(b) DefnetheFouriertransform
Fex/xmx',
r')=(jâ)-1j.f(2x)-6#3/
cJ3k/el(k*x-k'*x')
x
j
(c-tuln;r-r')Fe
ax
jl
/u
b.'a
a
lsz,
.wn
,
=..?
à
n
,
Find@'
xx
j(k,k''(,)nltosecondorder,andhenceobtainthecorrespondingFeynman
rulesin m om entum space.
(c) Show thatDyson'sequationbecomes
Fe
zxptk,k';tozj= Fkjtk,fzlnl(2033(k- k')
+ (2,
0 -3â-1fJ3pF;A(k,(
sa)PrAA,(k- p)Fqljtp.k''
.(.
,
anl
asl
FINITE-TEM PERATURE FORM ALISM
(#) Expresstheinternalenergyandthermodynamicpotentialinaform analogous
toEqs.(23.15)and(23.22).
7.6. ApplythetheoryofProb.7.4toasystem ofspin-!fermionsinauniform
magneticheld,wherePk/x)=-Jz;.e''laj.
(u) Expressthemagnetization M (magneticmomentperunitvolume)interms
ofG**(forF= 0)andFex(forF> 0).
(bjSolveDyson'sequationineachcaseandfindM ;henceobtainthefollowing
limits yp= 3p.
!n/2es as F-+.0 (Pauliparamagnetism) and yc= Jzln/ksr as
T -.
>.cc(Curie'slaw),wherenistheparticledensity.
(c) Hz'
/l
y doesthezero-temperatureformalism gfppthewzwz?ganswer?
7.6. Prove
1
y(z)tanhzJz= *
2.
p'
fc
1
... .
2n+ 1
/-(a inj
.
1 /(z)cothzJz= *
ljn-i=
2=i
)((of c
a.whereC isthecontourshownin Fig.25.4. Stateclearly anyassum ptionsabout
theanalyticstructureof/tz).
7.7. Use Eqs.(26.10)and (26.11)to compute thehrst-ordercorrection to E
and (1forboth bosonsand fermions.
8
PhysicalSystem s at Finite
T em perature
Z7LHA RTREE-FOCK APPROXIM ATIO N
A sdiscussed in Sec.10,there are m any physicalsystem swhere itism eaningful
to talk aboutthe motion ofsingle particles in the average self-consistentheld
generated bya11theotherparticles. Thesim plestoftheseself-consistentapproxi-
mationsisshown in Fig.27.1(compare Fig.10.3),where theheavy linesdenote
@ itselfandnotjustF0. Thisapproximateself-energyyieldsasnite-temperature
1
Fig.27.1 Self-consistentHartree-Fockapproximation to theproper
self-energy atfinitetemperature.
t
1
2B6
zs6
FINITE-TEMPERATURE FORMALISM
generalization ofthe Hartree-Fock equationsthatisvalid forboth bosonsand
ferm ions.
W e again considerasystem in astaticspin-independentexternalpotential
&(x). Thegrand canonicalhamillonianisthengivenbyEcompareEqs.(10.1z8
and(10.1!8)
#
hlV2
4
)-J#3x#)(x)- am + &(x)-/.
t'
#atxl
#1= !.JJ3x#3x'.
C (x)ij
ftx')Ftx- x')?
/j(x')'
(ktxl
(27.2)
.
where the interparticle potentialisagain assum ed to be spin independent.. ln
thepresentapproxim ation,D yson'sequationtakestheform show nin Fig.27.2,
t-)+. *$)
0
Fig. 27.2 Dyson's equation for (.
q.
# in Hartree-Fock
approxim ation.
which isformallyidenticalwiththatatzerotemperature(Fig.10.5). SinceX
istime independent,itispermissible to introducea Fourierseries with respegt
tothe'
rvariables:
(27.4)
xa'j)eicon'n@(xj,x'
j,o,a,) (27.5)
g(x,y,(w)= C#0(x,y,(s,)+ .
f#3xI#3x'
lF0(x.xl.(,),,
)Y*(xl,x'
l)F(x'
l,y,(.
n)
(27.6)
wheretheself-energy X*(xl,x'
l)isindependentofthefrequency (sa.
The unperturbed tem perature G reen's function f#0 can be expressed in
termsoftheorthonormaleigenfunctionsofHf
j(Eq.(10.8)1,and wefind (com pare
Eqs.(10.l0)and (25.l7)2
F0(x,x',
-q +l(
X)çI
?î
)*
-j((
rX'
y-.
-s-)
(sa)-/.
j
t
s
J
a.
(27.7)
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
257
In a similarway,theinteracting temperature Green'sfunction @ isassumed to
have the expansion
F(x,x',
@k(x)çn(x')*
fw )- J iœn- h-:(0 - p,)
(27.8)
where(yb)denotesacompleteorthonormalsetofsingle-particlewavefunctions
with energy EJ. The mean numberdensity can beevaluated with Eqs.(23.8)
and(25.38):
(H(x))= :!
u(25,+ l)(jâ)-1j)etu,n'lr#tx,x,f
.
sal
-
(2,+ 1)zJ !(s(x)12,,,
where
nJ= leblEl-l
at:J:l)-l
(27.10)
istheequilibrium distribution function fortheyth state. Correspondingly,the
mean num berofparticlesisgiven by
N(F,#',/
.
t)= (2.
î+ 1)X ni
(27.11)
J
which can (in principle)beinverted to *nd p,(F,V,NjifN isconsidered sxed.
Thefrequencysumsin Eq.(27.5)can now beevaluatedimmediately,with the
result
âE*(xj,xl
')= (2J+ 1)3(xj- xl
')J#3xaUtxj-xz)X
I
gb(x2)12nl
#
:i:P'txl- xl
')2J)+,.
(xl)m.
Jx;)*n.
i
-
5(xl- x;).fy3xzprtxj- xc)(H(xz))
:t:P'(XI- Xl
')Z
9b(X1)îb(Xi
')*n.
i (27.12)
J
A combination ofEqs.(27.6)and (27.12)yieldsa nonlinearequation forJ;Jin
termsof(/.
J
Itisconvenientto introducea diflkrentialoperator
h2V2
F j= ihu)n+ 2 '+ p,- U(xl)= ihu)n- Kn
t
,,
m
which isthe inverse ofh-3@%. The subsequentanalysis is identicalw'ith that
ofSec.10,and weshallonly statethesnalequationfor%y:
-
hl7'(
,
2m 1-U(xl)+y(xI)+f#3x'
,âX*(xI,x'
l)+J(xl
)=œjço(xl)
-
(27.13)
asg
FINITE-TEM PERATURE FORM ALISM
wherehX*now dependsexplicitly on F and y,. Thissetofself-consistentequationsisa generalization ofthe Hartree-Fock theory to hnite temperatures;the
tem perature aflkcts the distribution function nl directly and also m odihes el
and J'
J through the self-consistent potential. Although the theory loses its
physicalcontentfor bosons below the condensation temperature,itremains
valid forfermionsatalltemperatures. In particular,theFermi-Diracfunction
nlreducestoastepfunction%y.- eJjatr= 0,sothata11stateswithenergyless
than p,areElled. A sexpected,thisH artree-Fock theory forferm ionsfixesthe
totalnumberofparticlesatF= 0bythenumberofoccupiedstatesEEq.(27.11)1.
The internalenergy in the Hartree-Fock approxim ation can be evaluated
withageneralizationofEq.(23.15)
E(F,P',p,
)= :F(2J+ 1)J#3%l
im (jâ)-1jnlelOn'l
x'-yx
hlV2
xJtf/jf
xu- --jy,u + &(x)+P.JF(x,x',(
w)
=
(2J+ 1).
'
j
)eJnp- !.(2J+ 1)Jdbx#3x'
#
>:Z
+.
/(x)*âZ*(x,x')+,(x')r,, (27.14)
J
wherethefnalform hasbeenobtained with Eqs.(27.13),(25.38),and (26.9).
Thisexpression can beinterpreted astheensem bleaverageoftheself-consistent
single-particle energies eJdetermined from Eq.(27.13),while the second term
explicitlyremovestheeffectofdoublecounting(seethediscussionfollowingEq.
(10.18)j. A combinationofEqs.(27.l2)and(27.14)yields
X ((2.
î+ 1)l
t
/1
J(X)
.12i
ç)
k(X')2+:y)J(x)*+k(X)Tk(X')*ç)y(.X/)j
which (for fermions) reduces to the usual Hartree-Fock expression at zero
tem perature,apartfrom thedependenceon/,
Linstead ofN.
Itisinterestingtoconsidertheform oftheseequationsforauniform system ,
where &(x)= 0 and E*(x.x')= Y*(x - x'). The self-consistency conditions
then become m uch simpler,since@,(x)may betaken asa plane wave U-it?îkel
and only ek rem ains to be determ ined. D irect substitution shows that pfk*l
t
indeed represents a solution of Eq.(27.13);furthermore,the self-consistent
single-particle energy becomes
eg= 4 + âY*(k)
(27.16)
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
Theself-consistencycondition reducesto
1
nk = (:
:-#)/ka'r :
!:j
2B9
(27.l8)
in which ekboth determines,and isdetermined bynk. N otethatnkand nkboth
depend on p.,which maybeExed bytherequirement
NIT,P',p,)= (2J+ 1)F(2=)-3Jd'knk
(27.19)
Finally,the internalenergy becomes
E = (2J+ 1)F(2zr)-3Jd?k(q - 1âZ*(k))nk
= (
2J+ 1)F(2zr)-3fd3k(4 + !4E*(k)Jnk
(27.20)
W eemphasizethatek,nk,andZ*(k)intheseexpressionsalldepend onF and p,
throughEq.(27.18).
Z8LIM PERFECT BOSE GAS NEAR Fr
A san exam pleoftheself-consistentH artree-Fock approxim ation,we consider
a spin-zero im perfect Bose gas near its condensation tem perature Fc. lt is
helpfulfrstto recallthe situation in a perfectgas,where there are only two
characteristic energies: the thermal energy ksF and the zero-point energy
hlnkjm arisingfrom thelocatization within a volumen-1. Thecondensation
temperatureFoinanidealBosegai1isdetermined bytheconditionksFcœ hlnklm
(seeEq.(5.30)),whichisevidentftom dimensionalconsiderations. Incontrast,
the introduction ofinteractions èomplicatesthe problem considerably,since
thepotentialU(x)hasboth a strength and a rangea. Asshown below,the
presentcalculation isvalid when
hlnk h2
ksrc= ksTo ;
k:
<< z
m
(28.1)
MJ
h2
nF(0)x:
tmaz
where P'(0)H P'tk= 0). Thesrstcondition showsthatthisisa low-density
approximation (na3< 1),whilethesecond condition limitsthestrength ofthe
potential. N ote,however,thatwe do notrequire the usualcondition forthe
Bornapproximation(1z'(0)c4hzalmj,whichismorestringentbyafactorna3<t1.
The mean particle density and self-consistent excitation spectrum are
given by
#3k
1
(28.3)
nçl'Z'X = (2= )3ele:-pp/tar- 1
:2k2
ek= -lm +
k'
F(0)+ F(k- k')
(#3
2=)
) e(r:,-pjyksr j
. .
(28.4)
a6c
FINITE-TEM PERATURE FORM ALISM
Equation (28.3)specitiesthedensity asa function ofF and /.
z,butitismore
convenienttol
ixaand theninverttofindp,(F,F,1). Inthiscase,thechemical
potentialislargeandnegativeathightemperatures(seeEq.(5.26)J,butitincreases
toward positivevaluesasF islowered. Exactly asforaperfectBosegas,the
tem perature Tc for the onset ofcondensation isdeterm ined by the condition
ek- p,= 0 atk = 0,when a hnite fraction ofthe particlesstartsto occupy the
lowestenergy state eo(see thediscussion following Eq.(5.30)1. The present
calculation ism orecom plicated,however,because both ekand Jzdepend on F.
WeassumethatP'(x)hasaFouriertransform
F(k)= J#3xF(x)p-ik'x
(28.5)
whosesniterangeallowsanexpansion oftheform
U(k)= 1
-d?x P'(x)(1--ïkmx- J(k.x)2+ ...)
(28.6)
For a spherically sym m etric potential,the linear term vanishes. lf the mean
square radius alisdefned by the relation
x P-(-x)
.
x2
a2 = -f#3
- .--
J#3x Pr(x)
(2g.J)
Eq.(28.6)canthen bewrittenas
P-(k)= F(0)(1- à(kw)2+ ...)
(28.8)
where F(0) and alare both positive if U(x) iseverywhere repulsive. The
energy spectrum can also beexpanded in powersofkl:
(28.9)
where,from Eqs.(28.3)and(28.4),
t/3k/
k'l
et
l=lnF(0)-.
à.
U(0)aljjj,
szgf
r:,-j
t
ws
-z--j
=2nF(0)gl+O(ma2
yk
zsFjj
(28.10)
and 1 1gj.nF(0)ma2
(28.11)
Thesecondterm inbothoftheseexpressionsrepresentsasm allcorrectionbecause
ofconditions(28.1)and (28.2),respectively. Since the eflkctive mass arises
entirely from the exchange interaction in Eq. (28.4),it clearly represents a
quantum -m echanicaleflkct.
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
261
The transition tem perature and corresponding chem ical potential are
determ ined by thepairofequations
n-
dqk
1
J(2,
43exp(â2k2/2-*ksrc)-l
(28.12)
pïrc)- etl
(28.13)
Equation (28.12)isidenticalwiththatfora perfectgaswith massm*,and we
5nd (secEq.(5.30))
lnhl n 1
ksFc= v
rn ((1)
(28.14)
lfTv denotes the transition temperature for a noninteracting gasofthe sam e
density and m assm,theinterparticle potentialshiftsthe transition tem perature
byan am ountl
LTc= rc- To= m 1malnF(0)
Tz
Fo m. 1= -j ha
(28.15)
N ote that a purely repulsive potentiallowers the transition temperature. The
constantsaland F(0)arereadilyevaluated foranyspecificchoiceofF(x);in
particular,LTC= 0forapointpotentialF(x)= L 3(x).
Z9LSPECIFIC HEAT OF AN IM PERFECT FERM I GAS
AT LO W TEM PERATU RE
The H artree-Fock approxim ation also representsa usefulm odelforfermions;
asa specifcand nontrivialexam ple,weshallevaluate the entropy and specisc
heatin the low -tem perature lim it.2 O ne possible approach isto com pute the
thermodynamicpotentialf2(F,P-,p,
)from Eq.(26.11)butitiseasiertoworkwith
Eqs.(25.25)and(25.26):
K =E-P.
N = JF(2zr)-3(jâ)-1j#3k)(elulnïlgjâtx)a+ek-jz)trF(k,f
w)
(29.1)
ThefundamentalrelationistheidentityK = fl+ TS (Eq.(4.7)),sothat
(:
Jw)---(z
%f')FM+s+r()
%S)Fp
lThisresultwasobtained by M .Luban,Phys.Aen.,1M :965(1962)andbyV.K.W ong,Ph.D.
Thesis,University ofCalifom ia,Berkeley,1966 (unpublished).
'Someofthetechniquesused here wereintroduced by A . A.Abrikosov,L.P.Gorkov,and
1.E.Dzyaloshinskii,çeM ethodsofQuantum Field Theory in StatisticalPhysics,''sec.19,
Prentice-Hall,Inc.,Englewood Clifli,N.J.,1963,butour calculation difl
krs from theirs in
severalim portantways.
z62
FINITE-TEM PERATURE FORM ALISM
Thelirsttwotermsontherightcancel(Eq.(4.9)),leaving
DK
DT
=
zs
95-
F CY p.s
(29.3)
Thisexpression diflkrsfrom the usualspecisc heatbecausep,isheld sxed.but
itallowsusto com pute theentropybyintegratingatconstant#'and y,.
LOW-TEM PERATURE EXPANSION OF F
Equation (29.1)iscompletelygeneral,butthepresentcalculationcan besimpli5ed considerably by studxping only the leading snite-tem perature correction.
TheexactG reen'sfunction @ dependson F b0ththrough thediscretefrequency
zon= (2n+ llrr/jâ and through the self-energy E*(k,(sa,F) (see,for example,
Eq.(27.17)). This functionaldependence may be made explicitby writing
D yson'sequation as
F(k.a)n,F)= F0(k.(
sn)+ F0(k,tt?a)Y*(k,(oa,r)F(k,(,pa,F)
(29.4)
where @Qdependson F only through u)nand thematrix indicesaresuppressed.
TheinversefunctionsF(k,tw,F)-1andF(k,tw,0)-isatisfytheequations
F(k,a)a,F)-1= F0(k,(z)a)-!- E*(k,(s,,r)
(29.5)
F(k,œ.,0)-l= F0(k,(sa)-l- X*(k,(sa,0)
whosediserenceyields
F(k,(sa,r)-l- F(k,œu,0)-1= -X*(k,*n,F)+ X*(k,(,)n,0)
(29.6)
M ultiplybyF(k,a>n,0)ontheleftandF(k,fw,F)ontheright;
F(k,t,)a,F)= F(k.u)n,0)+ F(k,tt)n,0)(X*(k,(x)n,F)- X*(k,u)a,0))F(k,f
w ,F)
(29.7)
H ere the last term explicitly vanishes as r -->.0, and this exact equation can
thereforebeapproxim ated at1ow tem perature by
F(k,(z)n.F)œ F(k,(z)a,0)+ F(k.f
.
,aa,0)EX*(k,(.
t
)n,F)- Y*(k.(sa,0))F(k,(,)n,0)
(29.8)
HARTREE-FOCK A PPROXIM ATION
TheonlyassumptionusedinderivingEq.(29.8)isthatof1ow temperature. The
subsequent analysis is less general.however,because we shall now restrict
ourselvesto theHartree-Fock model,in which E*isindependentoffrequency
and isdetermined from thediagramsin Fig.27.1. Assumingspin-!fermions
and spin-independentinteractions,wehavefrom theFeynm an rules
âE*(k,r)= (jâ)-l(2*)-3J#3g(
j
(eia:a'n(2F(0)- P'tk- q))F(qs(z?a,,F)
n'
(29.9)
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
263
where we now write @ug= 3.j@. Equation (29.8) may be used to rewrite
thisexpression as
âY*(k,F)=(jâ)-1(2,4-3Jd3q)
getuG'9(2F(0)- F(k-q))
n'
x lf#(q,oan,,0)+ (f#(q,oaa'.0))2(E*(q.r)- Y*(q,0))J (29.10)
H erethe second term in bracesapparently becomesnegligibleasr -.
>.0,and it
istempting to replacethediscretesummationoverœn'by an integralEseeEq.
(25.15)):
(#â)-1j
-->.(2.
v)-1J(*
œn,
a#)
*X d
(29.11)
Such a procedure isperm issible only ifthe sum and theintegralboth converge
to the sam e lim it. In the presentcase,however,the resulting integralis too
singularto perm itthesubstitution.l
To dem onstrate thisrathersubtle distinction,we shallevaluate the sum
explicitly and then compare it with the approximate integral. Consider the
quantityoccurringin Eq.(29.10):
(jâ)-'Z
df*a'9(F(q,(z)n's0))2= (j#)-1j
)e'L
ûn'YlLil
-on,- â-1(e:- jz))-2
n'
a'
ê
u
-
- ho ((jâ) I)2ej.a-.
,
?tj(sn,.,-j(rg- slj-lj
=
a'
h?a0/F)
%
where wehaveintroduced the excitation spectrum atzero tem perature
tq = e0 + J
lE*(q,0)
q
(29.12)
(29.13)
and
nq(F)= (ebç'q-Hb+ 1)-1
(29.14)
dependsonFonlythroughtheexplicitappearanceofj. Equation(29.12)does
notvanish atF= 0;instead,itreducesto -â3G - eJ. W e now turn to the
corresponding zero-tem perature integral
* dœ
edf
i'
x
vx 2=lf(.
&)- â-1(e:- p,
))2
(29.15)
The double pole at (
.
o- ,4-1(/,- e) with residue -f,?en'-'t'.-'z' apparently
ensures that the integralvanishes as T .-.
>.0. Closer exam ination shows that
the integraldiverges at îq= y.. A lim iting procedure istherefore required to
defneitsvalueatthatpoint,andthediscretesummation oftheânite-temperature
theoryexaminedinEq.(29.12)servesjustthispurpose.z
'ThispointwmsfirstemphasizedbyJ.M .LuttingerandJ.C.W ard,Phys.Aer.,118:1417(19K ).
2Note thatthe adiabatic damping terms iiz iz.the denominators ofthe corresponding reealfrequency integralsin thezero-temm raturetheory sel
'veexactly thissamefunction.
264
FINITE-TEM PERATURE FORM ALISM
- '
--
:eexplicitform oftheself-energy Y*(k,r)atlow temperaturecannow
befound
âE*(k,r)= (2=)-3fd3q(2F40)- U(k- q))nall'j+ (2=)-3fd?q
x(2P'(0)- Prtk-q))UY*(q,F)- âE*(q,0))&7
j#j0
-)
œq
whereitispermissibletoreplacenq(T)byz?q(0)inthesecond(correction)term.
Atzero temperature,Eq.(29.16)reducesto the familiarform (compare Eq.
(27.17))
becausen/0)- 0(y.- eç). In thepresentapproximation ofretainingonlythe
leading low-temperature corrections, Eqs.(29.16) and (29.17) together yield
(29.l8)
whichmaybeconsideredanintegralequationforX*(q.F)- Y*(q,0).
Thefundamentaltherm odynamicfunction KIT,1
,',J.
t)can now berewritten
bycombiningEqs.(29.1)and(29.8)
x (Y*(k,F)- X*(k,0)) (29.19)
Herethesummationinthesrstterm iseasilyevaluatedwith Eqs.(25.38),(26.9),
and(29.l3):
(jâ)-1(
j)elconTlihu)n+ ek- p,
)glua- â-l(E'2- p,
)- Y*(k,0))-1
=
(2(e:- p.
)- âE*(k,0))nktr) (29.20)
Thesecondterm in Eq.(29.19)formallyvanishesasF-.
>.0,butthesummation
isagain too singularto replaceby an integral;a directevaluation yields
(jâ)-lj)elœnTlihu;n+ eî- y,
)(F(k,tsa,0))2
=
T) (29.21)
hnkll'
)+ (2(e:- Jz)- âE*(k,0))h8nt
-k
î(
ek
26s
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
and wem aynow takethelim itr -+ 0. A combinationofEqs.(29.19)to(29.21)
gives
#(F,l'',rz)-*-(0,F,rz)- F(2=)-3Jdbk2(E'
k-Jz)LnklTj- a:(0))
lçb .
+ 7(2=)-3f#3k2(6:- p)BngkC
(âE (k,r)- jjxwtk,(j
lj
k
+ P'(2=)-3f#3:nk(0)(âY*(k,F)- âY*(k,0))
F(2=)-3f#3:âE*(k,0)(rlk(r)- rl:(0)
Dnq(0) wk r)
+ 0Ek px ( , ,x.(k,o)))
(29.22)
-
--
-
The second term vanishes owing to the factor 6k- Jz,thatmultiplies the delta
function onk(Qlloçk= -3(6k- p.). Thiscancellationoccursbecauseallquantities
have been expressed in termsofthe exactspectrum ek,showing the necessity of
retaining thefullself-consistency in the H artree-Fock theory. In addition,the
lasttwotermsofEq.(29.22)alsocancel,whichcan beseenbysubstitutingEq.
(29.18)intothethirdterm ofEq.(29.22)andthenusing Eq.(29.17). Equation
(29.22)thusreducestotheextremely simpleresult
A(F,F,/z)- #40,V,y.)= P-(274-3(#3#2(s:- y)Lng(T)-z4(0)J (29.23)
which is ourhnalform . This result indicates thatthe only low-tem perature
corrections to L'
è- rzxlaarisefrom astatisticalredistribution oftheparticles
.I
am ong the zero-tem perature energy levels ek determ ined from the interactions
in theground state.
EVALUATION oF TH E ENTqo py
Theentropycannow becomputedfrom thethermodynamicidentitygEq.(29.3))
w(J
8.
S
?
u
'
)Fp-vlvj(/
2
8,
31'
0
32(,.-slnktr)
-rj:
.
yj(a
#.
3/
)
c)(..
k-s)(j-tanj
az
ek.v
#j
,
p'
djk
z
p.
2ksvn J(
2,
r)3tek-?
*'SeCjjze1kk-.r
,
-
(29.24)
which isan explicitfunction of(F,F,p,
). The angular integrations are easily
perform ed :
W()
asjp
/
,
s..
irlk
p.
'r2(
*
)kzvt
klc,-s)2sechze
2k.T
M
(29.
25)
which leaves a single integralover k. A tlow tem peratures,the integrand is
peaked atthe pointek= p,
,with a width thatvanishesasF -.
>.0. Ifwechange
aee
FINITE-TEM PERATURE FoqM AujsM
variables to tH (l/2ksF)(e:- /z),thelowerlimitmay beextended to J= -cc
with negligibleerror:
r(P
j))FJ
Z-p'
kàwL
gkz*
w'
d
k
%
'
<J
1ek=J
z.
'
2
#J(-c
ot
qfzsechcJ
=àp.
/
ciwgkzk
tjfà=/
z
(29.26)
where the slowly varying factor kldkjdqk has been taken outside the integral.
Theintegration atconstantP-and p,istrivial,and we 5nd
dz -1
SIT,P-,/z)= .
jl
z'
/cjF k2 àk
-!
(29.27)
rk-p
The entropy isa therm odynam icfunction ofthe state ofthe system ,and
itisnow perm issible to change variablesfrom fxed p,to fixed N . To obtain
the leading order in the low-tem perature corrections,we m ay use the zerotem perature equation
N = 21/-(27.
4-3Jd3kt?(p.- Ek)
=
2F'(2r)-3jdàkjtks- k)= ;Q)(3,
rr2)-1
(29.28)
wheretheFermienergyisnow definedbytherelation(seeEqs.(29.13)and(29.17))
2
/c/+ âX*(Vs,0)
p,= eks= h
-j
-2uk;+ t(2=)-3.f#3t
-h
-j
?L2P'(0)- P.(k - q))ptks-t
?))):.ks
(29.29)
The derivative(J6k/#/f)tksatthe Fermisurfacedesnestheeflkctivemass
#E#'
i'
iiik =
,
blk
F
rn*F
(29.30)
and the low-tem perature entropy and heatcapacity becom e
iDs j
. 2m * =2
S(TsVsNj
,=Ck=Tjjy/vy=NkLFjj-yj'
-j
(29.31)
These expressions are form ally identicalwith those for a perfect Ferm i gas,
apartfrom theappearanceoftheefectivemassm*(seeEqs.(5.58)and (5.59)J.
A sim ple calculation yields
1
l
l PE*(k,0)!
mw
-z=m
- +n
za
s.
j
y. a
v'as rks
(29.32)
Itisnotable thatthelow-temperaturethermodynamic functionsare determined
solely by thezero-tem perature excitation spectrum . Thissim ple result,which
pHyslcAu svs-rEM s AT FINITE YEM PERATURE
267
isin faetquitegeneralylarises here from the specialform ofthe Hartree-Fock
self-energy. Since X*(k,F) is independent of frequency, the spectrum ek
merely shiftsthe energy ofeach single-particlelevel,and the interacting'ground
state stillconsists ofa slled Fermisea. The low-temperature heatcapacity is
determined by thoseparticleswithin an energy shellofthickness;
aJksr around
theFermienergy erEseks. Ata temperature r.theincreasein thetotaleneror
LE isproportionalto theenergy change perparticle ksT times the number of
excited particles
hEc(ksr)p'(2,
rr
)-3js:3k
(29.33)
where the subscripts denotes the integration region Iek- eFlf ksF. Since
ksF <:
<:s,weobtain
''
jz#
ktjkr-(ur)2(a
3)j')*)F)
hE cc(ksr)2arrj
(29.34)
wherem*isidentisedwiththehelpofEq.(29.30)andEq.(29.28)hasbeenused.
Thusweseethat
C #(àF) NkkTm*
v= dT Cc hzkà
(29.35)
and the constant ofproportionality m ustclearly be the same asfor a perfect
Ferm igaswith m assm *.
3OLELECTRON GAS
In the previoussections,we studied the H artree-Fock approxim ation atfm ite
tem perature,which applies to system s with sim ple two-body potentials. For
example,theFouriertransform F(q)mustbewelldeflned andboundedforall
q;theserestrictionsprecludebothahardcore(U(x)->.coforx< J)andalongrangecoulomb tailgU(q)--.
>.azasq-->.0). M ostphysicalsystemshave more
com plicated interactions,however,which m ustbetreated by sum m ing selected
classesofdiagrams,exactly asin thezero-temperature formalism (Secs.11and
12). Fordefiniteness,we studythethermodynamic propertiesofan electron
gas in a uniform positive background;this system is particularly interesting,
because the hnalexpressionsdescribe both thehigh-temperatureclassicallimit
and thezero-tem peraturequantum 1im it.2
'J.M .Luttinger,Phys.Rev.,119:1l53(19* )hasconstructedageneralproofvalidto a11orders
in m rturbation theory.
2Thispointwasfirstnoted by E.W .M ontrolland J.C.W ard.Phys.F/UI
W.
1.1:55 (1958),who
derived theresultspresented in thissection. Ourtreatmentdifl-ersin detail.butnotin spirit.
268
FINITE-TEMPERATURE FORMALISM
A PPROXIM ATE PROPER SELF-ENERGY
TheAamiltonian isthatstudiedpreviouslyinSecs.3and 12,inwhich theuniform
positivebackgroundcancelstheq= 0componentofF(q). Thusthediagram-
matic expansion of F has no terms containing '
#%tq=0,(sa)= U(0). The
equilibrium behaviorism osteasilycalculated from thethermodynam icpotential
(seeEq.(26.11))
!d,
j
dqk 1
i
(1(F,P'
,
p,
)=Dc(F,P'
,J
z)+P'joj-j(,,
a.
);y(e.vvyj
uwztkj
oa)r
yz(k,
(s.)
(30.1)
where E*2 and F2are the appropriate functions for an interaction potential
AF(x)andthespinsum givesrisetoanaddedfactorof2. W ewould normally
evaluate f2as a powerseriesin the coupling constantc2,butthe second-order
term diverges,justas atF= 0 (see Prob.8.4). ltistherefore necessary to
tk'a'n
'
j;(*
l)(k&
o,n)= q,tsa,
k-q,(
s.-oln,
'
tk.o,a
Fig.30.1 First-ordercontribution totheproperseifenergy foranelectron gas.
include a selected class of higher-order diagram s, whose sum yields a linite
contribution. Thechoice ofdiagram scan bem adebyexam ining the perturbation expansion,and we now turn to the Feynm an series for Z* and @ . Itis
convenientto isolatethe eflkctofthe interaction in the properself-energy;we
shallwrite@ = F0+ @0X%.@0+....,andtheintegrandofEq.(30.1)becomes
X*F0+ X*F0E*r#0+ .. ..
The condition F(0)= 0 means that all tadpole diagrams vanish. ln
particular,there isonly one srst-orderproper self-energy (Fig.30.1). This
contribution iseasily evaluated with theFeynm an rulesofChap.7:
XC)(k,t,&)
â-1(2rr)-3jd3q(#â)-1j
letûon'T'F-t
ltk-q,(z)a-tt)
a,)F0(q,tt)s,)
n'
- â1(2rr)-3Jd% P'tk-q)nî
(30.2)
= -
=
where Eq.(25.38)hasbeen used to evaluatethesum and10= (ençnq'-l
at+ 1)-1
isa function ofthe chemicalpotentialy.. Equation (30.2)diflkrsfrom Eq.
(25.39)becausetheuniform positivebackgroundcancelsthedirectcontribution.
The corresponding frst-order term in the therm odynam ic potential is
given by
t'ljtr,F,p)= 1,'(2,
*-3jd3k(#â)-1jngeiconTâlzlkjtk,(z)alF0(k,f
x)a)
(30.3u)
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
269
f1j(F,F,/z)= -F(27r)-6Jd?kd?qF(k- q)n2n0
q
(30.3:)
where the A integration has already been perform ed. A partfrom the explicit
dependence on rzinstead ofN = Vk((à=2,which isdiscussed in detailin this
section, this expression is a direct generalization ofthe hrst-order exchange
contribution to theground-stateenergy !Eq.(3.34)4. lf(1isapproximatedby
jltl+ Dj,a directcalculation (seeProb.8.1)predictsthatthelow-temperature
specischeatbehaveslike.-rllnFJ-l,whichdehnitelydisagreeswithexperiments
on the electronic specifc heat in m etals.l The sam e divergence has already
tk.ttl
n
R>P
k- q- p
p+ q,tsj+ v
tA)n* V - (Al1
k - q. p+ q
ut&- v o))+v
tks
/w
tk,tl
p,ttll
Qy-j
q-p
k- q
p,œ2
p,
tzj
œ.- œ l
t/l-fzN
q&v
q,v
k - q,fxla-v
tk./
.
o
(J)
'
tk.ts
4.*1
tk..
xv
'
(:)
(c)
Fig.30.2 Second-ordercontributionsto the proper self-energy for an
electrongas.
appeared inthe-l-lartree-Focktheoryof-ashieldedpotentialP'(x)= Fcx-le-x/a
where the eflkctive m assm* and low-tem perature specif!c heatboth vanish like
(1n(ksJ)1-1asa--+.cc(seeProbs.4.1and8.2).
The unphysicalbehaviorpredicted by the first-ordercontribution necessitatesan exam ination ofthe higher-orderterm s. AtF = 0,the second-order
proper self-energies have already been enum erated in Fig.9.16,and the sam e
diagram soccurin thehnite-tem peratureform alism . In the presentcalculation
however,threetermsvanish identically EP'(0)= 0J,and theonly second-order
contributionstoXAjareshowninFig.30.2. Hereandthroughoutthissection,
weuse v and f.
o to denote even and odd frequencies,respectively. Thecorre-
sponding analytic expressionsare(thesubscriptrdenotesthe ring contribution
ofFig.30.24)
Y!2)r(k,t
.
tV)= (-â)-2(-2)(jâ)-2(j
)(2,
0-6Jd'pd'qjF(q)j2
1)jF'
x'
.
4*(p.(sI)F0(q+ p,(
.
,,1+ v)gotk- q,œn- v) (30.4c)
Y?i),(k,(z?n)-(-â).
-2(jâ)-2tté
((2=)-6Jd?pd?qP'(q)F(k-q-p)
l1:'
x F0(k- q,u)n- p)F0(p.tz)j)gtptp+ q.(sl+ p) (30.4:)
:J.Bardeen,Phys.Rev.,50:1098(1936);E.P.W ohlfarth.Phil.A.
ft7
#..41:534(1950).
270
FINITE-TEM PERATURE FORM ALISM
I:?iptk,o,al- (-â)-2(#â)-2ttlZ
(2*-6Jdqpd3qF(k- q)F(q- p)
lLt)z
x F0(q,t,?,)F0(p,(sz)efa'ang0(q,(sj) (3p.4c)
wherethefactor(-2)inZ?i)rarisesfrom thespinsum andtheclosedloop,while
thefactorelu'
nTinZAlcarisesbecauseaninstantaneousinteractionlineF(q- p)
connectsboth endsofthesameparticlelineF0(p,(,)a). Although a11threeterms
.
areform allyofordere4,thehrstdiFersfrom theothertwoin thefollowingway.
In each term,the frequency sums yield various com binations of Fermi-Dirac
distribution functionsnobutdo notqualitatively alterthem om entum integrals
forsmallpandq. ItisthereforeclearthatZ?i),divergesas<-->.0gcc c4fd3qq-*
),whereas12â,,and Yxlcconverge. Forthisreason,anycalculation that
' ' '
includesonly irst-and second-order terms in X* cannot be considered satisfactory,and itisessentialtoexam inethehigher-orderdiagrams.
ThesourceofthedivergenceinE(
t)ristheoccurrenceofthesamemomen-
tum transferhq on each interaction line;in contrast,theotherdiagram stransfer
diserentmomentum on the two interaction lines. A similar structure persists
to allorders. For example,the third-order proper self-energy has one (and
only one)diagram E!)rwith thesamemomentum hq transferred by allthree
interactionlines(Fig.30.3J). Thisterm containsthemostdivergentthird-order
'
tk
q
tk
pz+ q
q
:
:q
pz
:
:
:
q
k-q
tl
k -q
q
p+4
p
q
tk Q
tq
k
(X
(:)
Fig.30.3 Ring contribution to the properself-energy in (J)third order
(b)higherorder.
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
271
term cce6jd% q-6.al1otherthird-ordertermsarelessdivergent(ebjd3qq-4
at most). Correspondingly in ath order,there is always a single diagram
l2â)ra:Jd3q(P'(t?))nthatismoredivergent(byafactort?-2)thananyotherterm
(Fig.30.3:). Thefundamentalapproximation inthetheoryoftheelectrongas
isto retain thisselected class ofm ostdivergenthigher-orderdiagram salong with
the com plete hrst-and second-ordercontributions
*
E*;zE*
(l)+ E*
(2)b+ Y*
(2)c+ n)=(
)2(
*n)r
2
=
I:(
*l)+ Y*
(2)b+ )2*
(2)c+ X*
r
(30.5)
whereY)'isthesum ofa1lself-energydiagramswiththestructureofFig.30.3:
(ringdiagrams).
SUM M ATIO N O F RING DIAG RAM S
Theevaluation ofY)ismostsimplyperformedbyintroducinganapproximate
eFective two-body interaction '
/''
rthatincludesthepolarization ofthe m edium
(compareSecs.9.12,and 26)associated with theclosed loopin 12;*. Figure30.
4
q,vn
q,G
q.va
q+ p
* = vw - vqzvq + vn+ .1
.
p,
a,j
q&G
Fig.3û.4 Ringappro+ ation totheefectivetwo-bodyinteraction.
shows the relevant diagram s and the corresponding analytlc expressions are
'
#'-rtq,wl--Y'-(,
(q,y,
n)+ '
/'
-o(q,'za)J10(q,vn)'
#'-o(q,y,n)+ .''
- '
#'-o(q,p'
2+ '
#'
-o(q,v2JIQ(q,r2 '
fz
--rtq,ra)
(30.6)
The function J10(q,va)representsthelowest-orderproperpolarization insertion
and willbeevaluated in detailbelow. Forthem om ent,however,itissu/ cient
to solve D yson'sequation
'
/--rtq,p'
n)- '
#'-(,(q,'
za)(1- '
#-o(q,'za)-110(q,vn))-1
-
p'(q)Il- p-(q).
rIqq,p,,
))-1
(30.7)
Thissolution isformallyidenticalwith thatatzero temperature (Eq.(12.22))
exceptthat'
/Gtq,valdependsonthediscrete(even)frequencyy'
n.
272
FINITE-TEMPERATURE FORMALISM
Theanalyticform ofJ10(q,va)iseasily determinedbywriting outthesrst
two term sofFig.30.4
'
#'-r(q,va)-< o(q,v,)+ (< n(q,'z2l2(-â)-.(#â)-'(-2)
xI
(2,4-3JdbpF9(p,œl)F0(p+q,fsl+ va)+...
œl
wherethefactors(-â)-1and(-2)arisefrom theextrapowerofe2and thespin
sum aroundtheclosedfermionloop. BycomparingwithEq.(30.6),weidentify
J10(q,v,)=2(jâ2)-1)
((2=)-3Jd% F0(p,(sj)F0(p+q,(sj+ w)
œl
2
d3p l
= à (2,)3j
v
1
œ1
1
flsl- â-t(e0
? - p,
)iolk+ ivn- â-1(:P+Q
0 - /.
t)
(30.8)
whichisverysimilarto Eq.(12.28). A typicalterm ofthefrequencysum isof
ordert(ult-2as1(t)1i->.*,and thesum thereforeconvergesabsolutely. Itcan
beevaluated directlywith a contourintegral,buta simplerapproach isto insert
aredundantconvergencefactorctu'ln,which pcrmitsadecomposition into partial
fractibns. Eachterm maythenbesummedseparatelywithEq.(25.38)andgives
2
d3p
1
1
Jl0(q,vn)- j (a'
z)3jva- ,-1(:p
0+q- el
pj
--j (sj s
-yajl,?
X
#3p
1
1
iœj- h j(%()- p,)- jgsj+ ivn- h-j(6p
()+q-'p.
)
-
n0 - nn
= -2 (2 j p+q() p (jj
=) ihvn-(Ep+q- ep
(3:.9)
wheretheidentityeiîhvn= 1hasbeenused. W eemphasizeagain thatnp
odepends
on the parameterp.,which can be related to theparticledensity N/P'onlyatthe
end ofthecalculation.
A
+
+ ''' =
Fig.30.5 Ringcontributi
on totheproperself-energy.
-
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
273
ThecontributionY)(k,(
z)n)to theproperself-energycan now beevaluated
directlyintermsof'
iz'
h (Fig.30.5)
X)(k,f
.
t)n)=(-â)-1(278-3jd?q()h)-$j
lj'
Fvq,yz
a)
Ua
'
f--otq.rall'
#0(k- q,o)n- '
.
%) (30.10)
Equation(30.9)showsthatJ10(q,pa)vanishesatleastasfastas)w1-tforlpnl-.
>.x.
-
In consequence,the diflkrence $-r- '#-()also has this behavior,whiuh ensures
theabsoluteconvergenceofthefrequencysummation forY). Thisconvergence
maybemadeexplicitbyrewritingthesquarebracketinEq.(30.10)asfollows:
'
#'
-,(q,v,)- y'
-o(q,u)-s -stV'tq)- ---- p'(q)
qluyq,s)
(P'(q)12Jl0(q,'y)
.
l- p-tqjhotq,r)j
(30.11)
APPROXIM ATE THERM O DYNAM IC POTENTIA L
It is now possible to evaluate the corrections to the therm odynam ic potential
arisingfrom thetermsinEq.(30.5). Theintegrand ofEq.(30.1)correspondsto
IV W
t)2b
Fig.30.6
gaS.
274
FINITE-TEM PERATURE FORMALISM
adding a factor 10joining the two ends ofX*+ Y*F0E*+ ...,thereby
makingaclosed loop. ThusX1),X?i),,XC)c,and X) taken oncelead tothe
termsfl),Dzp,fàzc,and D,shown in Fig.30.6,wherewe havealreadyevaluated
fàlwiththecorrectconvergencefactorsinEq.(30.3). Thereisalsoanadditional
second-ordercontribution (22,arisingfrom theiteration ofY?kt. Theterm D,
contains the m ost interesting physicalesects and is studied ln detail in the
followingdiscussion. Theother(explicitly)second-ordertermscanbewritten
outbycombining Eqs.(30.1)and (30.4). Itisevidentfrom Fig.30.6thatDac
and j'
ln are topologically equivalent,and a detailed evaluation showsthatthey
areequal. A straightforward calculation yields
(1a,
d?pd?qP'tql;'(k+p+q)n2nk(1-nk-jq)(l-nk+q)
(F,P'
,/
z)=P'j#3*(a
.
o:
skyls.sko.
q.sk-r; (30.13J)
('
Lactr,p-,rz)- Dz2F,P'
./
.
z)
= -
!.p'
j(2x)-9fd3k#3,#3:)'(k-q)F(p-q)nkrjnk(1- rj)
(30.13:)
and the totalcontribution to the therm odynam icpotentialbecom es
fà= flo+ f1l+ flr+ D zy+ 2f1zc
(30.14)
Notethatthecoupling-constantintegrationinEq.(30.1)leadstoanadditional
factorn-iforeach l/th-ordercontribution to f1,which isautom aticallyincluded
in ourcalculationalprocedure based on the properself-energy and the singleparticleG reen'sfunction. Asanalternativeapproach.D issom etimesevaluated
directlyfrom Fig.30.6withasetofmod/edFeynmanrules,butthecountingof
topologically equivalentdiagrams and the factor n-imakes such a calculation
quite intricate. Ourprocedure,however,requiresonly theFeynm an rulesand
diagramsdeveloped previouslyfor$ .
The preceding expressionsapply to an arbitrary two-body potential,and
thespecialfeaturesoftheelectron gasbecom eapparentonlyin theevaluation of
-1
!'
1r(r?Iz
'1rz)= r//?-l(2rr)-3j(jA-lJAJdïkZ cia'nnZ4A(k%(
w)F0(k,(w)
-v
-- -
œa
1#A j
- d% 1
A2(F(q)j2J10(q,pa)
joA
.,(2,
.
43#jvl-A)'(x-j'J10(q.va)
J3: 1 x-x j
'IJ(2,r/phz
,
- s.,
;gotk-j,(o,-valgotk,
t
sal (3:.)j)
1-
.
e
.
,
0 1: 1. Jhe sura'lation overo)nconverges
where Y'
r
k' la leer ta ken fr lm Eq. '
.3
evenif'?= 0.fnd :
o11pa.ispnwit? Eq.(30.8.slc'wstnatthequantityinsquare
bracket.is.
!iJ 0(q.vn). gB'eE)oM in Eq.(3b.18)thatJl0isan even function of
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
275
itsarguments.) Since F0isindependentofA,the integration over A is easily
carried out,and we find
fL, =
=
-
Z
J*d3q gldhà2jF(q)J'
I0(qyprljz
.
z) sa .)jjxjyjj-j-j---jpq-l.j.
jjq,
-pal
P'(2j)-1(2=)-3jdbq(
j((1ngl- PrtqlJ10(q,s))+ P'tq)J10(q,s)j
Pa
(30.l6)
If Eq.(30.16)is expanded in a power series in c2,the leading contribution is
form ally of order :4 owing to the explicit rem oval of the ûrst-order term . As
shown below ,however,the sum m ation ofthe infnite series m odises this sim ple
pow er-law dependenceon el,and,indeed,e-4j1rdivergesasel--.
>.0.
Furtherprogresswith Eq.(30.16)dependson theexplicitform ofJ10(q,ru),
and we hrstprove that
J10(q,pu)=J10(q,-s)
Add and subtractrlo
p-qr?0p inthenumeratorofEq.(30.9)
JI0(q,vu)= -2 (d3
;)
n0.(1---p
n0)--n
0(
np
o,'
,)
2=)
.
u
-1
-/-p3-?.-9.I.
fvn- (ep
oyqP.
)--Thefil'
stterm mayberewrittenwiththesubstitution(p+q ->.-p)alongwiththe
assumedisotropyofthedistributionfunction(notethatE'
p= 60
o
-p= 60
p)
0
'
. d3,
j
j
(q,pa)= -2 jy-.
p(1- np
0+q) ijjv, (rp
(.
)jn0
(
).sj.
y.
g;-yy,
u..(sp
().j
y. rj;
-
.
- -
'
* d3n
s0- <.
0
4j(2='
)jn0
p(l-?
:p
o+q)t
-y
jssja
-p
-ts
-p
Jtqjj
p
.q
-j.
j
(30.18)
which proves Eq.(30.17). ltisalso clearfrom the above calculationsthatJl0
isan evenfunction ofq and oforderr,
-2as'pn1->.cc:
J10(q,vn) '-- 4 c d5p3np
0(1- n0
p.q)(eD
p- so.
hql
(30.19)
ivas-'c
o(â%) (2=)
cLAsslcAL LIM IT
Thering-diagram contribution to the therm odynam icpotentialD rcan now be
used to study two lim iting cases,and we srst consider the behavior athigh
tem perature and 1ow density, when the quantum -m echanical Ferm i-D irac
distribution may beapproximated bytheclassicalBoltzmann distribution (see
Eq.(5.23))
n0p= expL)Ly - vp
0)j= eH/kBT/-92p2/2mksT
(?g.J())
276
FINITE-TEMPERATURE FORMALISM
ThisapproximationisjustisedwhenevereH/kB'
r<t1ortquivalentlyforJz/ksF -+.
cc). Inthislimit,Eq.(30.19)showsthatJ10(ç,p,j)vanishesasF-2for /# 0.
Thefrequencysum inEq.(30.16)separatesintotwoparts
-
f1,=.
è.VkpF(2=)-3jd3qtlngl- Pr(t
?)Jl04ç,0))+ Pr(t
?)Jl0(g,0))
*
+ Fksr(2=)-3jd?q1j-j1(ln(1- P'(t
?)J10(t
?,2=/ksFâ-1))
+ P'(:)Ahq,l=lkzFâ-l)) (30.21)
corresponding to /= 0 and I# 0,respectively,and the divergence at sm allq
hasnow beenisolatedinthefrstterm (/= 0). Toverifythisassertion,welook
atthe contribution to the second term from a sm allregion around q= 0. In
thehigh-temperaturelimitEqs.(30.19)and(30.20)give
Aoqqnlnlk.Fâ-1).
-+.4(2=/#sF)-2(2,
0-3j#3/(6k- ek+q)nk
= 2e1no(2'
zr//csF)-2 F-.
+ctl
wherewehavedesned
no= 2(2g4-3Jdqpnn
Furtherm ore,theproduct
ZW)Aolqgl=lk.F/i-1)-.
>.-4=e%l=IksF)-2hlnam-l
remainsbounded asq-.0. Thelogarithm in thesecond term in Eq.(30.21)
can now beexpanded asapowerseriesine1,and theintegrand becomes
z-'(lngl-p-tvlao(.
l
?,2'
,
r',s?.
7)j+,
ztt
?lno(ç,2=',
'b''
j)
,
1 * 1 e 4 hlnv l
c
.
i
T r p (,-,)(--)
=
1
,. ,
Thesum overlconverges,andthesingularbehavioratt
?;k:0hasthusdisappeared.
Hencethesecondterm ontherightsideofEq.(30.21)contributestothethermodynamicpotentialinEq.(30.14)inorderp4,justlikef-la,+ 2j1zr.
TheleadingcontributiontoflrthereforerequiresonlyJ10(ç,0).which can
beevaluated with theoriginaldefnition
#3p nz
p+q- nz
p.
z z.
p
Jl%ç,0)= 2 (2=)3ec
p..
q- e;
p
:3 nn - nn
..
2 p)aap+q p
....
(= p+q ap
-
(30.22)
Sincethenumeratorvanishesatthesameplaceasthedenominator,wecan keep
track ofthesingularitybytreatingtheintegralasaCauchyprincipalvalue. The
277
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
rem aining integrationsare elem entary and give
(30.23)
where
A. z'
nphl'1
l=hl 1
(. j.(p,yv)
(30.24)
isthe therm alwavelength.
(30.255)
with the lim iting behavior
(30.25:)
(30.25c)
-
x-2wgztptzzrjt26MA-l)1.
x1) (30.26)
.
Wherethesecondlineisobtainedwiththechangeofvariablesçz= 8,
77.,
62-3ebl
xelxz.
This equation contains the coupling constant :2 both in an overall coeëcient
and in the argum ent of t
p ;we m ay obtain the leading contribution by setting
e= 0 in f/ltaintegrand,since the term s neglected are ofhigherorder in E'2. A
'
combination ofEqs.(30.255)and (30.26)yields
(30.27)
wheretheconvergentdesnite integralhasbeen evaluated by partialintegration.
ltisnotable thatfl,risoforderta3,although the lowest-orderterm in Y*
r
is form ally oforder e4. This behavior can be understood by exam ining the
z7g
FINITE-TEM PERATbRE FORM ALISM
perturbation seriesforDr,whichisobtainedbyexpandingEq.(30.26)
DrQ$Z(2j)-1(29-3Jff3t
?(.-.)(8'
Jr#el81-3)2(eg-l)4(y4#1))2
+ jlu=qebHA-3)3(eq-1)6(p(éA))3+ ...j (30.28)
Theleading term (e4)divergeslinearly,the next(e6)cubically,etc.,and each
integralmustbecutofl-atalowerlimitçmin. Itisclearfrom Eq.(30.26)(see
alsoSec.33)thatthenaturalcuto/çmjnisproportionalto e,whichmeansthat
each divergentterm is really oforder,3and mustbe retained in a consistent
calculation. Our procedure for evaluating Dr provides a convenient way to
include al1oftheseterms.
The therm odynam ic potentialfor a high-tem perature electron gas can
now bewritten as
Dtr,F,p,
)= fln+ f1l+ Dr+ O(e4)
(30.29)
because the remaining (hnite)second-order termsare explicitly ofordere4.
W eshow,in thefollowingdiscussion,thatthe hrst-orderexchangecontribution
f11isalsonegligibleintheclassicallimit,andEq.(30.29)reducesto
D(F,F,/.
t)= Do(T'
,F./.
t)+ DrIF.F,X
(30.30)
Athigh temperatures.the thermodynamic potential(1ntr,Pe,p,
)fora perfect
(classical)gasisgivenby(Eqs.(5.24)and(5.25))
flotr,P',/z)= -2Fj-lel8A-3
andacombinationofEqs.(30.27)and(30.31)yields
2P': w
D(F,F,p)=-p-ses/ksz'gl+j(2=)+(k
C2/
p
Aj1es/2uz.
j
(30.31)
(30.32)
ThethermalwavelengthAisgivenintermsofFbyEq.(30.24);thusf2isproperly
expressedintermsof(F,P'
,p,
).
onlyatthispointisitpossibleto5ndthem eanparticledensityasafunction
ofJz
N(T,p'
,
p,
)--(a
0p
t
',
1)z.
'
z-2
AZ
aes/
ksrgl+(2,
01(y
d
(
.
2/T
A)êep/zks'
r) (3:.33)
Asusual,however,wepreferto considerasystem atfixeddensity'
,Eq.(30.33)
iseasilyinvertedtoirstorder.whichprovidesanequationforg N)
Y
ks/;
kllnt123(1-n.
1(C
:2
sNy
1j'
jj
(30.34)
wherethefirstterm inbracketsistheresultforaclassicalidealgasEq.(5.26).
ThecorrespondingpressureisgivenbyEqs.(30.32)and (30.34)
Dtr,F..
N) k y.j
#(F,V,
N4=- p, ;
kJns '
r*(k
y
e2
sa
y
l.
)!
'
-
(30.
35)
pHvslcAt- SYSTEM S AT FINITE TEM PERATURE
279
which isthe Debye-l-liickelequation ofstatefora classicalionized gas.l The
Ieadingterm describesaperfectgas,whilethecorrectionterm reducesthepressure
slightly. Our approximations require that (e2n1/ksF)1<:
.1, which ensures
that the average potential energy per particle e2n% is m uch sm aller than the
therm alenergy perparticleksl'
. Thiscondition restrictsthepresenttheory to
high tem perature and low density. N ote that the leading correction to the
perfect-gas law is oî order e? and cannot be obtained with any snite-order
perturbationseriesintheparametere2. Furthermore,Eq.(30.35)isindependent
ofh,asbeftsaclassicalexpression.
TheD ebye-Hûekelresultisobtained classically bysrstexam iningthecharge
density and potentialin the vicinity ofa single electron (compare Eqs.(.14.16)
to(14.23)). lfthemeanelectrondensityisnv(exactlyequalto thatoftheuniform
positivebackground),then the Boltzmann distribution gives
n
'k w
= ee+a
,s
nu
(3:.36)
-
where (
p isthe electrostatic potentialin the vicinity ot-the electron (note that
+(x)-->.0asx -->.csbecauseoftheneutralityofthemediuml. Furthermore,;?
isrelated to thechargedensity through Poisson'sequation
V2p = 4xeta- no)+ 4zre3(x)= zWenokee*'kBT- l)+ 4,
zrp3(x)
nz+ + 4gre3(x)
;k;4=eck
(30.37)
aT
when the last equality holds under the conditions discussed above,thatis,
ew .c
4ksr. Thisequation can be rewritten
(V2- qj)(
/?= 4=e3(x)
(30.38)
whereqo isthereciproealoftheD ebye shieldinglength
4=nzel
t
?lH k F-
(30.39)
s
SinceEq.(30.38)hasthesolution
%(.
X)= -eX-'f'
-OX
(30.40)
thechargecloud around theelectron isgiven by
1
eln
e3n
P
= -
clouu
V2+
yg
=-
x>0
(
)%'=
0 ewqox
kBj. kBpy
(30.41J)
oralternatively
e-qDx'
Pclouu= eqn
2 4=x
'P.Debyeand E.Huckel,Physik.Z..24:185(1923).
(30.41:)
ax
FINITE-TEM PERATURE FoRuAl-lsu
The w ork necessary to bring an inhnitesim alcharge elem ent -de from
inEnitytothecenterofthischargecloudisgiven by theelectrostaticpotentialat
theorigin. ThusifdW isthework doneby thesystem when thechargeon each
oftheN electronsisincreased by-de,we5nd
dW = y#eecloudto)= Ndefpclouutx)x-1#3x
=Nqpede=Nju4=w
Nyz
j+ezde
(30.
42)
Thework donebythesystem in building uptheentirecharge,
-eoneachelectron
istherefore
N e5 4zr1 *
e
l
'
p'
e1-jodW-a(pwsr)
(30.43)
From Eq.(4.4)thechangeinHelmholtzfreeenergycanbewritten
JF= dE - TdS- SdT= -dW - SdT
#FIw= -dW tr
(30.44/)
(30.44:)
wherethelastform ofEq.(30.444)followsfrom thesrstlaw ofthermodynamics.
Thusthechangein the H elm holtz free energy oftheassem bly due to electrical
work is
Ne3 4=N +
(30.45)
gw
F.k=-p'
ey=- a(vg
Thecorrespondingchangeinpressureisobtainedfrom Eq.(4.5)
l
F
P .t= - ppFe
..S
P' zw = .2
PP el
rrl e2al
0 1
no/csT
(30.46)
(30.47)
-j- kaT
whichistheresultgiveninEq.(30.35).
W ecannow verifythatf11isindeednegligiblein theclassicallim it. W hen
Boltzmannstatisticsapply,Eq.(30.3:)mayberewrittenas
D
#â2(p2+q2j
-
(F,U,Jz)=-4=elFe2l#t(2=)-6jdlpd3qIp-q!-2exp
l
--
F e2
)y e2sxsrjjdqxdqy e-x2-#2a
(,rA)
lx- yl
lm
(3().4g)
where the dimensionlessdefnite integralconverges. A straightforward calcu-
lation showsthatDl/D,isoforder(hlnkjmjlelnkksTj-k,whichvanishesinthe
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
281
classicallimit(h .
-.
+0)or asF -.
+co. Thequantityhlnklm isan averagekinetic
energy perparticle,and f1lthusbecom esnegligibleifthisenergyism uch sm aller
than the geom etric m ean oftheaverage therm aland potentialenergy perparticle.
Finally,we rem ark that the lirst quantum -m echanical correction to the
classicalequationofstateforaperfectgas(Sec.5)isoforder
l l=hlnk ê
ebHQs.
in/3;kr1 mkst
'
which is sm all at high tem peratures and low densities. For com parison,the
Debye-ldiickelterm included in Eq.(30.32) is of order (c2/V sF)êt>1l8;
k;
llelnkjlkslhà',which is again smallathigh temperaturesand low densities.
Itisevidentthatthe quantum correction isnegligibleaslong ashln%jm .
,
:
telnf,
which guarantees thatthe m ean kinetic energy is m uch sm aller than the m ean
potentialenergy. In summary,the three relevant energies (kinetic,potential,
and thermal)m ustsatisfy the setofinequalitieshlnkl
!m <x e2nk<:
4ksF;thefrst
allow stheuseofBoltzm ann statisticsand rendersD lnegligible,whilethesecond
ensuresthatthe D ebye-ldiickelterm representsa sm allcorrection to the perfectgas law.
ZERO-TEM PERATU RE LIM IT
The preceding section considered only the classical lim it, but the sam e ring
diagram s m ust be retained at al1 tem peratures to yield a convergent answer.l
Asan interestingexample,weshallnow turnto the opposite(zero-temperature)
limit,when the distribution function becomes a step function n0
17= t?(/
.
z- 6p
f
)).
O nce again,it is im portant to rem em ber that p.is an independent param eter,
Thus the mean particle density and the Fermiwavenumber kyHEEL?nlN;
'P')1
cannotbelixed untiltheend ofthecalculation,w hich diflkrsconsiderably from
thepreviousground-stateformalism (Chaps.3to5).
Thetermsf1l(Eq.(30.3)1,flzp(Eq.(30.13J)1,and Dcc(Eq.(30.13:))in
the therm odynam ic potentialhave already been evaluated in a form thatis
convenientat1ow tem perature. The rem aining dië culty isthe evaluation of
Dr,which givesthedominantcorrection to f1lbecauseitcorrectly incorporates
thelong-wavelengthbehavior. SincetheintegrandinAolqnvn)hasonlyasimple
pole asa function ofvn(Eq.(30.9)1,itispermissible to replacethe discrete
frequency sum (jâ)-1)
(in Eq.(.
30.16)by a continuousintegral(2,8-1fdv,
Pa
becausethediflkrencevanishesatF .
->.0(seethediscussioninSec.29):
DrIF=0,F,p,
)=!.P/42rr)-4j*dvJdht1n(1-U+(t
?P)'J
l0(ç,
p))z)l (30.49)
(t?)J10(ç.'
'M .Gell-M ann andK .A.Brueckner.Phys.Rev-,1* :364 (1957).
282
FINITE-TEM PERATURE FORMALISM
Onceagain,itismostimportanttoevaluateJl0(ç,$ accuratelyforsmallq,and
Eq.(30.9)immediatelygives
o
2
Jl(:,$ = -
#3p
n0
p+q- a0
âv- qyzjnt)(pwp
(2*)3l.
qs jqa)
x, a #3p) q.vp
znl
q.+0
(2*)ihv- (h Im4p.q
((
p .5(,)
-
since thecorrectionsoforderq2in thedenominatorcan beneglected aslong as
vishnite. A tzero temperature,n0
preducesto a step function,and itsgradient
becomesVpnl=-p&(p-k0),wherekoisdehnedbytherelation
hk.= (2-p,)+
TheintegrationsinEq.(30.50)arereadilyperformed,andwe5nd
J10
kom
1
(30.51)
dzz
(4,r)= -N/z-j
h -jz- imvlhqko
= -
kont
.
n.aha A(x)
(30.52)
where
mv
X = âçk
(30.53)
o
and
&x)=
.
1 Jz22
-+ xz-=
()zz
1
l- .
xarctanx
(30.54)
Wenow returnto Eq.(30.49)andintroducethedimensionlessvariablesx
(Eq.(30.53))and (=q/ko
o =Vh2/CJ4% gxdyrX(qa jjugj.4=e2
r 2-(2,44j-. :(
)- $L kà(2a(
)y
(go(,â/
cm
lxtjj
4.
:,e2
+kl(zat
jy(
)t,âkm
dxljj(a(j.
55)
.
Althoughwereallywantonlythedomirvntterm inEq.(30.55)forsmalle2,the
divergentbehavioroftheintegrand precludesadirectexpansion in powersofe2.
Instead,the(integrationwillbesplitintotwo parts:from 0to (0<tland from
(0 to ç
n. For (< (a.itispermissible to approximate J'Io/ko(,hkixtjm)by
J1040,hkkxl/m)= -(kom/h2=2)R(x),whilethefull(dependencemustberetained
for(> (o. Aslongas(0isEnite,however,theintegrand intheregion (> (:
can be expanded in powersofe1,retaining the leading term ofordere*. This
procedureyieldsEcomparethetreatmentofEq.(12.56))
Dr= Drl+ Dra
(30.56)
283
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
w here
12
o-l- 1-8. 3
k
MlJ
(--xdxJ(0
te(3d(tIngl+ko
zj,(t)2atxj
-k,
4=
m#i(t
cj2stxjj(y()
.j,
yu)
or2'
,
.
wz.g
x
.
1g
V,
h3
m
lp
/
cy
)j(
**
-:
rtscg(
:
r(
)t.
aC;js
4j
rt
e.
zao(
j
q
j
;,(,hkm
àxtjjz (y(;,ygyj
The(integrationlnEq.(30.575)cannow beevaluatedexplicitly:
*
(,
'
(-.
:3d(tlngl+d
te
szq-d
gzj-.
taze.tvit,gIn(l+-tt
jz,
l-'
dte
jz
-j
1,(,+-td-))
-
*.
$.12:4(1n(ad,2)- J.- 2ln(o)+ O(e6)
(30.58)
where
4mR(x).
,
dtxl=ko'
rr
hz F
co
4'
m
râc(j..xarotanx
1.
)
(30.
59)
Thisexpression exhibitsthenonanalytic behaviorofflr. A lthough the desnite
integralis fnite for any (0> 0.each term of the formalperturbation series
diverges:
.
(0
1B(3#()lngl+'
/j-'
X(C
c2)
=
-
-((,dy
1
.
*(0dï
J,42e4j0y+
>+jz
43e6.1
j0y*
j+. (30.
60)
ThisbehaviorissimilartothatofEq.(30.28)describingaclassicalelectrongas.
The high- and low-tem perature lim its diflkr in one im portant way, however,
because the leading term here diverges logarithm ically rather than linearly.
Inconsequence,whenEq.(30.60)iscutofl-atalowerlimit(mincce,weseethat
thefirstterm isofordere4lnewhiletherem aining onesareofordere4,in contrast
totheP dependenceofeachterm inEq.(30.28). Itisthisisolationofthe:41nEz
behaviorthatallowed usto determ ine theleading term in thecorrelationenergy
directly from the second-order term in the ground-state energy (Prob. 1.5).
The contribution Drcalso exhibitsa logarithmic singularity as (c--.
>.0,
becauseJ10(/c0(,hkixljm)approachesaconstantvalue as(-.
+.0. Itiseasily
verifedfrom Eq.(30.52)thatthedivergenceisidenticalwith thatinEq.(30.58),
and the quantity
6 li
1(x4M-4.
5(,.(
jtA2(x)ln(c+jâ
s2
y
:
7(
2
)
.
)2.((
X,6;gno(kn(,'',
t''
t)j2)(30.61)
m
284
FINITE-TEM PERATURE FORM ALISM
is:nite. A combinati
onofEqs.(30.56).(30.58),and(30.61)yields
kp *
DrQ;P
:32
= 3m
j-.dx)'
l'
E
faa(x))ajngezxtxlj- jyaxtxljz
+-4j
=
-(â
mzk
el
z42uxjj(g
p.
6p
correct through order e4. The calculation has now been reduced to a one-
dimensionalintegralcontainingthefunctions:(.
x)and1(x)sgiveninEqs.(30.59)
and(30.61).
Beforewecompletetheevaluation off1,,itisusefultocollecta1ltheterms
of(1through ordere4lndand e4:
D(F,F,p)= flc+ Dj+ t'
1,+ flz,+ 2f1zc
(30.63)
To the same orderofapproximation,the mean numberofparticlesisgiven by
#(r,
fI
flc pfll p(f1r+ (1a,+ 2f1cr)
F,
p)=-tMuj
ss=-êa
s-og- as
(30.
64)
which expressesN asafunction ofy.. These/w't?equations(30.63)and(30.64)
providea valid and complete descri
ption of a degenerate electron gas.forthey
constituteaparametricrelation betweenN and f1. Nevertheless, itisfrequently
convenientto eliminate p,explicitly'
,thisisreadily performed by expanding p,
asaperturbation seriesl
M= /z0+ JLl+ /12+ '''
(30.65)
wherethesubscriptdenotesthecorrespondingorderine2,andthen byexpanding
each term on therightsideofEq.(30.64)asa Taylorseriesaboutthe value
Jz= n . Equation(30.64)cannow beinvertedorderbyorderine2,andthesrst
two term syield
N=
-
dfl
c
.
drz s-so
(pf1l/ap,
),,-,,,
#'l= -(azfto/arz2j.-..
(30.66)
(30.67)
Herethesrstequationdeterm inespmasafunction ofN,whilethesecond detcr-
minesp,1in termsofpm(and thereforeN4. Notethatp,()isjustthechemical
potentialforanidealFermigasattemperaturerwithdensityNIF.
The change ofvariable from p,to N indicatesthatthe relevantthermo-
dynamicfunctionistheHelmholtzfreeenergy (seeEq.(4.5))
F(r,V,Nj= E - TS= D + IJ.N
lThepresenttreatmentfollowsthatofW .KohnandJ.M .Luttinger,Phys.Rev.,t18:41(19648.
PHYSICAL SYSTEM S AT FINITE TEM PERATURE
285
which m ay beform ally expanded through second orderin powersof,2:
pfl
?2j1
F=Dotp.ol+(JzI+rzzl-sf
p. .-.0 +1./,1-i-r- s-ya +DlGol
z)
?
- -
+P'
1,
?
+/
tj
,
p-lxll
Thesecond and lasttermscancelbecauseofEq.(30.66)so thattheexplicitform
ofy.zisneverneeded. Equations(30.67)and (30.68)can becombined to give
F(F,V,N)= Fo(F,P'
..
N)+ D I(/
.
ta)+ f'
lrtrzol+ f22,(/.
t:)
l (8(a1/
'8rz)2.,
+ 2f22c(/z0)- j(
-jk-o--o aP-'
?'
?- (30.69)
()/ /
.
zl,
x-,.,
wherep,
aisa function ofN,and Fv(T,U,A')- fàotp,el+ p,oN isthe Helmholtz
free energy ofan idealFerm igas.
Thepresentdescriptionbecomesespeciallysimpleatzerotemperature(see
Eq.(5.53)1:
(8f?
.
p.
(
e
0g
,I
qjj)y
js-s,
/92()a(0,#',rz)
!, as2 )
,-s,
(30.70)
In thislim it,the zero-order term p,oisgiven by
hl 73,
77.2N '
i h2/(.k-
#'
o(X )= gsy
( p-' K-j///-=61
(30.71)
where kr is the usualFerm iwavenum ber. The subsequentdiscussion shows
thatthe lastterm ofEq.(30.69)(in square brackets)vanishes atF = 0. Consequently,the ground-state energy ofthe Atparticle interacting system hasthe
followingexpansion
E - Ez+ .
t'
1,(E'k.)+ Drte'yl+ t'
12,(e'
î.)
(30.72)
because F - E - TS ..
->.E asF -.>.0. Here the firstterm foisthe ground-state
energy ofthecorrespondingperfectFermigasgf'
o= j.k
ve/l.whilef1l(6k.)isthe
srst-orderexchangeenergy(compareEqs.(3.34)and(30.3:))
f1l(6))= -4=e2Pr(2=)-6jd?p(/3(y4p- qj-2jjks-pjpjks- qj
(30.73)
TheremainingtermsofEq.(30.72)clearlyrepresenttheleadingcontributionto
the correlation energy
fcorr= Dr(6/)+ (11,(eî.)
(30.74)
28e
FINITE-TEM PERATURE FORM ALISM
Thedom inantterm in the correlation energy com esfrom the long-wave-
length partof(1r. Introducing the usualdimensionlessunits(see Sec.3)and
identifyingkowith ks,wecan write
Nel
.
fcor,= 2 ecorr
(30.75)
wherefrom Eqs.(30.59)and(30.62)
3
co
z
ecor=4jlnGJ-.I
A@))dx G*0
(30.76)
TheintegralismosteasilyperformedwiththeintegralrepresentationEq.(30.54)
l
l
co
j*
x(R(x))2=jvdyjt
j(f
zJ #x(yz.y.x:
,
2
2z.j.azj
-Xd
z
;z
(z
.
..
=rjo
'dyj;
'#zyA
.
'
j
Z
.y
=
3=(1- ln2)
(30.77)
Thus
2
ecorr= 4-1(1- ln2)lnr,+ const
r,->.0
(30.78)
whichagreeswithEq.(12.61).
Theconstantterm canalsobeobtainedfrom Eq.(30.74),buttheevaluation
is considerably m ore dië cult. Introducing the same dim ensionless unitsinto
Eqs.(30.13c)and(30.62)gives
D
Ne2
(4)= za0 t'l
(30.79/)
2,
N el
.
f1r(6k)= za
-o (
3*-3j-wdxïRlxbjl(
ln(4aGzr-1)+lnA@)-è)+3j(30.
79:)
wherex= (4/9,
0 1,elisadesniteintegralgiveninProb.1.4,and
-
3-J--dxllp-(
I
,
im
.(-=
4a(l-I
n2)ln(,-a.
3,j(
*aq
.
#
-.
î
z
d3:d3,061- k)$ 1-p)#(lp+ q
xjj- gz.
y.q,(p.
y.k;
j-1)%I
k+qI-1)j
.
s
(30.80)
isindependentofr,. Thelastexpression for3isjustETwith thelogarithmic
singularity removed (see Probs. l.4 and 1.5),
.itsderivation isoutlined in Prob.
-W
7
.
'
'
$'
!1
h
PHYSICAL SYSTEMS AT FINITE TEM PERATURE
8.7. substitution ofEqs.(30.77)and(30.79)intoEq.(30.74)yields
2 (1
fcorr==j;i l
n2)ln(4œ
=G)+(jnR)..-j
l+3+4
-
287
(3O.
gl)
Herethenumericalconstants(lnAlavand 3mustbefound numericallyl
J-.
*#xRllnR
(1nAlav> J. dx Rz o.
jsj
(30.825)
& = --0.0508
(30.8%)
= .-
--
whilethenine-dimensionalintegralelhasbeenevaluatedanalyticallybyOnsagerz
3
el= .
à.ln2- 2= z(43)= 0.048
(30.83)
Thesnalexpressionforthecorrelation energy becom es
ecorr= 0.0622lnr,- 0.094+ Otrylnra)
(30.84)
in complete agreementwith Eq.(12.62) derived from the zero-temperature
form alism . Itisinterestingthatthepresentzero-tem peratureapproxim ationis
valid athlkh densities (n1e2<<
:hznhjmj,in contrastto the previousclassical
calculation. Inbothcases.however,thepotentialenergyn1e2issm allcompared
to theotherrelevantenergy(hlnhjm atF= 0,ksratF-.
>.*).
To com plete thiscalculation,3itisnecessary to show thatthelastterm in
Eq.(30.69)indeed vanishes. Thesecond-ordercorrection Dzr(Eq.(30.135))
can berewritten foralItem peraturesas
Dac
d5k:3p:3:
a?4
(
F
,
F
,
p
,
)
=
'
I
'
F
J
(
2
,
0
9
Z
(
k
q
)
F
(
P
q
l
d
$
:4
because
-
(30.85)
P?
4î
ja1(1-n2)=ê
e
Inthelimitr-.>.0,thefactorênj/êelreducesto-J4p,- 4),andwecanwrite
Q2e(0,F,p,
()= -.
i'F(2=)-3fd%3(pm-4)(./:2
(30.86)
H ere
fq-(2,4-3J:3kP'tk-qlakjswjji,r.o
=
(2x)-3jd5kprtk-q)ely.o- 4)
'M .Gell-Mann and K .A.BruK kner,loc.cit.
2L,Onsager,L.M ittag.and M . J.Stepiwn,Ann.Physlk.1*:71(1966).
3Thispointwassrstemphasized byW .Kohn and J.M .Luttinger,Ioc.cit.
(30.87)
28:
FINITE-TEM PEnATURE FOaM ALISM
and Jzhasbeensetequaltop,
().asrequired by Eq.(30.69). A similarcalculation
leadsto(compareEq.(30.3:)forF= 0J
'?D
#3k#
(-j.
s
-!)s..,--p'J-yj.),
3yp'
(k-q)l
?2.
o
jn
..
s
6
0+A
?zD
jn
sk
-s.sa
....
, v.,
= -
2P'(2,7.)-3f#3t?3(jza-62)A
(30.88)
Finally.a directevaltlation yields
(.
0j
2/
.
(2
7
)1j
'#(-vo=-2p-*(y
dx
.
3
.
6
y3(s(
/
)-e(
:
)
.
(30.89)
which isequivalentto thelastlineofEq.(30.70). W ith thefollowingdehnition
ofan average overthe Ferm isurface
(2=)-3 $
'(134
3ts()- eq
0)
(*
'
i
-'
.)-3 jd3:blgv- :b
)
q
.
,
''
. .
.
(30.90)
.
thelastterm ofEq.(30.69)assumesthetransparentform
2(1zc
l (t
'
?f)(0)
j.)v
2.
yo
(J,,o)-2(-#ijj()yjszj..s,
- -
U(((/q- x.
/k)s)2-s)(2=)-3fd?q3(Jz()- e0
q)
(30.91)
Thiscontribution to the ground-state energy isproportionalto the mean square
deviationof
-fqoverthellnperturbedFermisurfaceandcanneverraisetheenergy.
Furtherm ore.the correction evidently vanishes fora sphericalFerm isurface,
which is the case foran electron gas in a uniform positive background.
The foregoing cancellation is a specihc exam ple of a general theorem l
forspin-l l
-ermionsthatthe F -+0 limitofthe tinitetemperature formalism
always gives the sam e ground-state energy as that calculated with the r = 0
formalism (eitherin Feynman orin Brueckner-Goldstone form),aslong asthe
unperturbed Ferm isurface is spherically sym m etric and the interaetions are
invariantunderspatialrotations. Thisresultisnotata11obvious,becausethe
two approaches describe the interacting system in very diflkrent ways. The
F + 0 form alism com putesthetherm odynam ic potentialD asafunction ofthe
param eter Jz, and its F .
-->.0 lim it involves integrals over Ferm i distribution
functions that are singular where the energy is equalto y,. A t the end of the
calculation,/.
tmay beeliminated infavoroftheparticledensityNjP',and p,then
dehnesthe Ferm ienergy es ofthe interacting ground state. O n the other hand,
the usualF = 0 form alism considersa fixed num berofparticles N from the start
and evaluatesthe ground-state energy as a seriesin the coupling constantofthe
tW .Kohn and J.M .Luttinger,Ioc.cit.;J.M .Luttingerand J.C.W ard,Ioc.cit.
pHvslcAt- SYSTEM S AT FINITE TEM PERATURE
289
two-body potential. Each term in this series involves integrals over the unperturbed Ferm idistribution function,which has its discontinuity at the un-
perturbed Fermienergye/= (hl(2m)(3=2A'
/F)9.
ln com paring the two form alism s,we im m ediately note that the F # 0
expansioncontainsterm sthatare neverconsidered atr = 0. Forexam ple,the
zero-temperature version ofXtilccontainsan integraloveroçkr- qj#(ç- ks)
andthereforevanishes(Prob.3.12). Athnitetemperature,however,thethermal
width yieldsanonzero valuethatrem ainssniteeven in thelim itF = 0. A 1lof
theseadditionaldiagram scontain singularfactorsatzero tem perature,such as
3(e2- J.
t)oritsderivatives,whereas the diagrams thatare common to both
form alism s contain only step functions at F = 0. Since the F = 0 form alism
antedatesthe F + 0 one,these additionalterm sareconventionally described as
anomalous. Thetwo form alism salso diflkrbecauseoneusestheexactchem ical
potentialJzH es,while theother usesthe unperturbed Fermienergy e/M y,n.
The Taylor series for f1(F= 0,es)aboutthe value f1(F= 0,eî.
) involves an
expansionofstepfunctionsatesintermsofthoseatek;thisexpansionleadsto
additionaldelta functionsand derivatives ofdelta functions. The contentof
the K ohn-l-uttinger-W ard theorem isthattheextra contribution incurred inthe
shiftin Fermienergyfrom esto e/preciselycancelstheanomalousdiagrams,
leavingtheBrueckner-G oldstoneseriesfortheground-stateenergy.
lfperturbation theoryprovidesa valid description ofan interactingsystem ,
then the F = 0 lim itofthe tem perature form alism necessarily yields the true
ground state for any value ofthe coupling constant. In contrast,the F = 0
form alism m erely generates that eigenstate of the ham iltonian thatdevelops
adiabatically from the noninteracting ground state. Foran arbitrary system ,
these two approaches m ay yield diflkrenteigenstates,as shown by the sim ple
exampleofa perfectFermigasin auniform magneticseld (Prob.7.5). The
Kohn-l-uttinger-W ard theorem can therefore be interpreted as specifying
suë cientconditionsto ensure thatthe F = 0 form alism indeed yields the true
ground state. U nfortunately,the very interesting question ofnecessary conditionsrem ainsunanswered.
PRO B LEM S
8.1. Verify thatCk for an electron gas in the Hartree-Fock approximation
behaves like - F(1nF)-1 as F .
-+.0. (Comparethe discussion following Eq.
(30.3).)
8.2. (J) UsingtheresultsofProb.4.9forthelowest-orderproperself-energy
ofanelectrongashtïi)(q)andeflkctivemassm*,show thatthecorresponding
heatcapacity atzero temperaturesatissesCvlT= 0.
(b4The long-wavelength coulomb interaction ismodised by thepresence of
themedium accordingto Eq.(12.65). Show thatthecorrectlow-temperature
290
FlNITE-TEM PERATURE FORM ALISM
heat capacity is given by Cv(C;- (l- (ars/2=)llntt
xrs/'
n'l+ 2)+ ...)-1 as
rs.
-+.0,whereC(istheheatcapacityofanoninteractingFermigas.l
8.3. RepeatProb.4.4withthesnite-temperature Green'sfunctions.
8.4. (J) Use the Feynman rules to evaluate the second-order self-energy
contributions to the tem perature G reen's function show n in Fig. 30.2/ and b
forauniform system ofspin-!fermions. Findthecorrespondingcontributions
to thethermodynamic potentialand evaluate the necessary frequencysums(see
Prob.7.3)to obtain
Dzg=-2Fjj-j(20-9#3gJ3pd3kjFtql2aka0
p(1-nk-uq)
x (1- np
0+q)(62.q+ c0
p.a- 62- ep
nl-'
and Eq.(30.I3J).
(:)Consideranelectrongasathightemperatureswhererlk= eb?expt-jek)<<
:1.
ByusingcylindricalpolarcoordinatesEk=tlkà
,+ kulshow that
-
m Zé'4
l'
n
2 d3q
*x
cc
flza---vt
p,2/8(znpz q..
'
#j.
-wti
pb
'J-.A.
.
jâ2(/c2
,+p2
,)
1
'
im
t?(pI
I+ /c,
!)+q
H ence conclude that fl:a is Iinearly divergent at sm all m om entum transfer
(q -+0).
8.s. (J)In theclassicallimitwhererlk-explj/.
t-jek),show thatpR0(q,vI)
hastheasymptoticform J10(q,rl),.
w-L6=$eblx(ql(;j)((4
/A)4+ (8=2/)2j-lforlarge
q,
j-qlznhllmkzTlkand 1/1.
(b) If3f1,denotesthesummation oftermsforl>0 inEq.(30.21),verifythat
3tlr/f1r= &((e2ai)1(n1â2/rn)1(1/#sF))wheret'
1,istakenfrom Eq.(30.27).
8.6. (J) Use Eq.(30.32)to computethe specifcheatofan electron gasin the
classicallimitCpz=lN/csll+J'
rr1(e2n1/#sF)ê).
(b)Derivethisresultfrom Eq.(30.45)intheDebyetheory.
8.7. Evaluate (1/jâ)(
)
((J10(q,y,)12with theintegralrepresentation (30.9). In
P
the zero-tem perature lim it,this sum m ay be approxim ated by an integralover
acontinuousvariable. HenceevaluateJX
-xdxllx),where1(x4isgiven in Eq.
(30.61),andverifyEq.(30.80).
1M .Gell-M ann,Phys.Re!
,
'.s106:369(1957).
9
Real-tim e G reen's Functions and
Linear R esponse
In the zero-temperature formalism ,the poles of the single-particle Green's
function G(k,œ)yield theenergyand lifetimeoftheexcited statesofa system
containingonemoreorlessparticle. Similarly,thefunctionDR(k,(s)determines
thescreeningofanim purityinanelectrongasaswellasthespectrum ofcollective
densitymodessuch asplasmaoscillationsorzero sound. ThroughoutChap.8,
however,thetemperature Green'sfunction @ wasused onlytocalculateequilibrium therm odynam ic properties. W e shallnow com plete the description at
snite temperature by introducing a real-time Green'sfunction C thatcontains
thefrequenciesand lifetimesofexcited statesatsnitetemperature.
291
292
FINITE-TEM PERATURE FORM ALISM
3IQGENERALIZED LEHM ANN REPRESENTATION
W e startourdiseussion by generalizing the notion ofa G reen'sfunction from
the ground-state expectation valueofa time-ordered productofheld operators
to theensem bleaverage ofthissam equantity.
DEFINI
TION oF ö'
The real-timeGreen'sfunction isdefined in directanalogy with Eq.(7.1)at
F = 0:
icxblxt,x't')ëeTrt/sF(#x.(x/)#f
xjtx't')))
where)(;isthestatisticaloperatorforthegrandcanonicalensemble(Eq.(4.15)1
andgxulxt)isatrueHeisenbergoperator
9Xa(X/)H t'iâl/'9x(X)e-iktlb
.
with respect to the hamiltonian X. As in Chap.3,the ordering operator F
includesafactor(-1)Pforfermions. Equation (31.1)hasoneimportantnew
featurebecause C dependsexplicitly on F and p.in addition to the usualspacetim evariables.
In m ost cases,the ham iltonian is tim e independent,and the resulting
Green'sfunction contafnsonly the com bination t- ?'. Furtherm ore,we shall
consider only hom ogeneous system s, and the G reen's function assum es the
sim pleform
Cajtx/,x't')= Czjlx- x',t- t')
(31.3)
Finally,we exclude externalmagneticfeldsand ferromagnetism so thatCaj is
diagonalin the m atrix indices
0aj(x,l)= îajC(X,t)
Each oftheseassum ptionscan berelaxed,butthesubsequentanalysisbecomes
considerably m ore cum bersom e.
Assumethat/ispositive. Equation (3l.1)then becomes
iC>(x,t)-(2J+ 1)-1Trtâs('x=lxt)'
X a(0))
In the presenthom ogeneoussystem ,the Heisenberg operators m ay be rew ritten
a.sEcompareEq.(7.52))
jXa&
f'
vt7
a*
j..<
u-/Pwxlhu
w/Xl/â.
Jf
F
œN
f'
/
v
M/v
u-2kt,
'h<
uiê'.x,
?5
,
/'
N
)A
'
1.6)
becausef:eommuteswith #. A combination ofEqs.(31.5)and (31.6)yields
ïd>(x,/)- (2.
s+ 1)-1Trleldfz-f'c-i'-x/.eiktl'#a(0)e-iet,
'eip-x/,#:(0))
(31.7)
nEAL-TIM E GREEN'S FUNCTIO NS AND LINEAR RESPONSE
293
wherethetracecan beevaluated in anybasis. A particularlyconvenientchoice
istheexacteigenstatesof/?,ê,andX :
X (rn)= Kmfral= (Em- y,
Nm)fm)
(3l.8)
ê(v?)= Pm(&l)
and wehnd
,'
G>(x.f)- (2.
s+ 1)-1eôil)(e-lA-e-îï'm-xiheixpntlhikm3v.
-(0)1a)
lnp
x e-ixntlheirawx/.tnjy-ttollrn)
= (2:.+ l)-1ebt
'
l)(e-bKmei(P,-P,
n'*x/he-itA:n-Amlr/'1(m jy'
AaIn)!2
(31.9)
In asim ilarway,thecorresponding function forf< 0 becom es
ic<(x,?)- +(2J+ 1)-lebt'
tTrfe-bkf4.(0)#aa(xr))
= +:
(2.
j
.+ 1)-1ebt
'
ljge-ôKnef(P.-Pm'*x/'e-f(Xa-X'
.)'/âj(pl!<zIa)12
(31.10)
ThetotalGreen'sfunction isthesum ofthesetwo terms
tRx.?)= #(?);>(x,?)+ p(-J);<(x,?)
(31.11)
anditsFouriertransform maybecalculatedexactlyasinEq.(7.54)
(;(k,(
s)=(2J+1)-1ebi
ùmat(2rr)33(k-â-l(P,-Pm)1l
.
(?
n!1
'
/)
zt
n)1
2
e-qKm
e-lA.
,-- ,-.(a--- xo + iT* - â-'(x-- x-)- i, lt
3''l2)
x
-
Equation(3l.12)showsthat(Rk,f
.
s)isameromorphicfunctionofhulwithsimple
polesatthesetofvaluesKn- Km> En- Em- y.(Nn--Nm);thecorresponding
residueisproportionalto ltrrl1'
(,!n)12and vanishesunlessNn= Nm+ l. The
ensem ble average atsnite tem perature clearly generalizesthezero-tem perature
expression becauseboth 1zn)and 1a)canrefertoexcitedstates. Forfermions
atF= 0,however,itiseasilyproved(Prob.9.1)that
Ctœ - Vâ)l'
r-o= G(tz9
whereG((s)istheground-stateGreen'sfunctionfrom Chap.3.
(31.13)
FINITE-TEM PE9ATUBE FORM ALISM
2%
RETARDED AND ADVANCED FUNCTIONS
If(z)isreal,therealand imaginary partsofd(k,f,))arereadilyfound with Eq.
(7.69):
:(k,a))- (2x+ 1)-1eln)2e-/n(2,)33w - â-i(Pa- 1u))ltrz'l#zlnl12
x(.
'
:
.
#(f.
&)- h-$(Kn- A-,
21-1(1:
FF-#tR-X'e')
f.
p.
3lo,- h-3(Kn- Am))(1+ e-#tA.-A'
.')J (31.14)
where.
' denotesa principalvalue. Theimaginary partofEq.(31.14)maybe
-
rew ritten as
Im <tk'ts)- -(2J+ 1)-1'
nebtz(2e-/A'n(2=)3ôgk- â-l(Pa- Pm))
rrln
x J(?'
n!#.In)l23(oa- â-1(A'
a- .
&ml!(1+ e-bhu'l (31.15)
and itiseasilyverised thattherealpartthen becomes
RetRk,(z))= -.
t
.
#
* dœ'Im Jtk oa')1:T:e-nhu''
-X
(z;- o'
A' l :
i:e-eh,,
=
a/lm tRk,f
x
# *X #o
,)/)
.(tanh(Jjâf,)'))tt
=
O - œ
= - .
(31.16)
-
which wassrstderived by Landau.l
Formanypurposes,itismoreconvenientto dealwithretardedoradvanced
real-timeGreen'sfunctions(compareEq.(7.62)):
fGljtxr.x't')Y e(t- t')Trtlsl#xatxfl,&11jtx?//)1v)
(31.17)
iclblxt,x't?)- -p(f'- t4Tr()G(#x.(xf),f4,(x'r'))v)
W e shallagain consideronly homogeneoustime-independentsystems with no
magneticNelds. InthiscaseCRand(P havethesamestructureasinEq.(3l.4),
and theirFouriertransformsareeasily found tobe
CR(k9(t))= (2J+ 1)-1elfljé(e-#Xm(2rr)33(k- â-l(PFl- Pm))jtrnjjk nj12
mn
x (1:F:e-'çKn-K-t)lo?- â-1(#n- Km)+ 1*
.
r/)-1)
(31.18)
(P(k,tM = (N + 1)-iebfb72(e-#X''
(2'
zr)33(k- â-1(Px- Pm))j(?n1.
fa1a)l2
mn
x(1:
Fe-#tKn-Xp,))gt
.
s--h-1(n/
K - um
K )- iyj-1)
ltisevidentthatboth CRand GA are meromorphicfunctionsof(,);in addition,
CRICA)isanalyticintheupper(lower)halftz
aplane.
lL.D .Landau,Sov.Phys.-JETP,7:l82 (1958).
REAL-TIM E GREEN'S FUNCTIONS AND LINEAR RESPONSE
29:
The retarded and advanced G reen'sfunctionsare closely related. W ith
thedesnition
p(k,to)- (2.
î+ 1)-ielfz)g(e-lA-(2,r)38w - â-1(Pa- P.,)1
x 2=3(ts- h-t(Kn- AQ )(1:Fe-#'œ)I(vI
#.ia)12J (31.19)
which dependson bothFandJz,theimaginarypartsofCRand CAmaylx written
as
Im tP(k,œ)= --1p(k,tM
(31.20)
Im (P(k,œ)= .
i.ptk,œl
Furthermore,a combination of Eqs.(31.18)and (31.19)yields the integral
representations
JR(k
*)=
,
* dœ' p(k,o,')
jrr(,,- (.o,+ i.b
(31.21)
6X(k't.
p(k tt?')
tll= X d2œ'
= (.o - œ'
' i'
q
-
whose realparts are formally identicalwith the dispersion relations at zero
temperature (Eq.(7.70)J. Ifweintroduce afunction ofa complexvariablez
F(kZ)= co d2œ'p(k,ts')
,
(31.22)
,
.n z - f.o
thent7R(k,oa)and (P(k,œ)representtheboundaryvaluesofr aszapproaches
therealaxisfrom aboveand below ,respectively:
GR(k,o)= P(k,(
.
o+ iTl
(31.23)
(P(k,œ)= l>(k,u)- iTj
Inview ofthegeneralrelationbetweenG,GR,andGAatF = 0(Eqs.(7.67)
and(7.68))itisnotsurprisingthatptk,(t)lalsodeterminestheFouriertransform
ofthe tim e-ordered G reen'sfunction. A straightforward calculation with Eqs.
(31.14)and (3l.19)showsthatC(k,oy)hasthefollowingalternativerepresentations:
- f'œ'
(Rkq
f
z)
l=J..axptk,
u,'
lt.
'
#t
,
s.10,-f'
mEtanht.
ijâaplpl:(f
.
,-a,))
=
(1:Fe-x'
*'l-iCA(k.(s)+ (1:FelAœ)-ltP(k,œ)
(31.24)
Forrealf,),allthreeGreen'sfunctionshaveequalrealparts
ReG(k,a,)= RedR(k,(s)= RetP(k.(s)
(31.25)
ae
FINITE-TEM PERATURE FORM ALISM
whiletheimaginarypartsaregivenbyEq.(3l.20)andby
Im (:(k,f
a))= -èltanh(à.
#âf
.
,)))71ptk,f
.
t)l
(31.26)
In the specialcase offermionsat zero temperature, Eq.(31.
24)assumesthe
fam iliarform
tRk,
(/
-(s)
-)ir-o-J*..Jau.
r,'Ptk,
a',)1
'
r-0(
,-p(
.
,.s,)+.jj+.-9((
,,,-iq (31.
27)
whichshouldbecomparedwith Eqs.(7.67)and(3l.13).
Theweightfunction ptk.(slcontainstheimportantphysicalpropertiesof
thesystem . A lthough thepreciseform ofpcanbeevaluated onlywithadetailed
calculation,there are certain generalproperties that follow directly from its
desning equation (31.19). Each term in the sum is positive if(
.
v is positive;
m ore generally,p hasthe following positive-definite properties:
(Sgnaklptk,ttJl> 0
bosons
(31.28)
plk,tzal> 0
fermions
ln addition,p satissesanimportantsum rule,which we now derive. Consider
thefollowing integral
(31.29)
wherethe(
.ointegralisevaluated by closing thecontourin thelowerhalfplane.
Equation(3l.29)canalsobecomputeddirectlyfrom thedesnition gEq.(31.17))
* dœ
-*
2,/.l
'(;R(k'(
z)le-2*T= JdTxe-îkexï(;R(x,'
r/)
=
(2&+ 1)-1fd3xe-iksxTr(âcg<xa(x0),'
#)a(0));:)
=
j#3xe-fk*x3(x)Tr)s
=
1
(31.30)
wherethecanonicalcommutationrelations(2.3)havebeen used in arrivingat
thethirdline. Comparison ofEqs.(3l.29)and(31.30)immediatelyyields
* d
l'(
2u
P k,(s')= l
=
(3l.31)
REAL-TIM E GREEN'S FUNCTIONS AND LINEAR RESPONSE
297
whichiscorrectforbothbosonsandfermions. Thissum ruleExestheasymptotic
behavioroftheGreen'sfunctionsforlarge1*l:
JR(kœ)= CA(k,œ)'w1 * #tt)'p(k,œ')'w-1
,
(.
o -. 2=
(,)
(31.32)
which isusefulin establishingconvergenceproperties.
TEM RERATURE GREEN'S FUNCTIONS AND ANALU IC CONTINUATION
Intheprevioussection,theweightfunctionservedonlytodeterm ineandçorrelate
thevariousreal-timeGreen'sfunctions. Although such relationsarevaluable,
they would notby themselvesjustifyourextensivediscussion oftheLehmann
representation atsnitetem perature,andweshallnow provetheim portantresult
thatthesameweightfunction also determinesthetem perature Green'sfunction
@. By thismeans,the Lehmann representation providesa directconnection
between@ and C andthusplaysacentralroleinthefnite-temperatureformalism.
ltissuëcientto consideronly positiversand F then becomes
@(xr)=-(2J+ 1)-lTrg)(;'
lhxztx'
r)f4..(0))
= -(
2.
:+ I)-lebflTr(e-#Ae-in-x/he#'
r/1,
4
/a(0)e-krl'eïp*x/âyJ((j)j
= -
(2.
9+ 1)-1ejflJ;Le-pxm:ïfI%-Pm)*x/1(?-tA'.-Au)m/Als
tra(.
.
f
/zj
n)t2)
mn
(31.33)
ThecorrespondingFouriercoemcientisgivenbyEseeEq.(25.14))
F(k,
j,
(
.
&
),
)=jv#'
rei
u''jd3xp-f
k*xF(x.
r)
-
(2.
s+ 1)-1e't'
t)((e-#A'.
(2zr)331 - â-1(Pn- Pm)j1(rrIl#.Ia)12
mn
x(1:!ne-/(Kn-xm')(f(,a,- h-,(Kn- .
&'
m))-1) (31.34)
wherç u)t= 2/'
v/#â for bosons and (2/+ 1)=/#â for fermions. Comparison
withEq.(31.19)immediatelyyieldstheimportantrelation
F(k,
= lts,p(k,(s')
œ2=J..2xi
co.-(
z)
'
(31.35)
which showsthatthefunction rtk,z)
,IEq.(31.22))determinesthetemperature
Green's function as wellasCR and CA. ln any practicalcalculation,we Erst
evaluate Ftksf.
saland thereforeknow r(k.z)only atthediscretesetofpoints
Liœnj. Itisthen necessaryto perform an analyticcontinuation to thew'
hole
com plex z plane. W ithout t
-urther information.such a procedure cannotbe
unique. Suppose thatP(k,z)isone possiblecontinuation:forany integerp.
the function elnpziœ'.P(k,z) is another possible continuation because it also
reducesto P(k,r a)atthepointsiuu. Nevertheless,thesevariouscontinuations
29:
FINITE-TEM PERATURE FORM ALISM
diFerfrom each othereverywhereelsein thecomplexzplaneincludingthepoint
atinsnity. Sincethesum rule(Eq.(3l.31))requiresthatP(k,z)'wz-3asIzI->.cc,
weare thusable to selectthe properanalyticcontinuation,which isguaranteed
to be unique.l
In practice,itisusually sim plestto com putep.(k,(
s)directlyfrom Ftk,(sal
by form ally considering iulnasa continuous variable. The weightfunction is
thenobtained asthelimiting value
p(k,x)= ï'
-1(Flk.œalliuu-x-in- f#lk,t/
anltif
au--x.inl
(3l.36)
Hence any approximation for C#tk,(snlimmediately provides a corresponding
ptk,f.
slandtherebyC,CR,andCA. Asaparticularlysimpleexample,consider
thenoninteractingtemperatureGreen'sfunctionF0(k,t
z)a)= (jtz?a- h-1462- p.))-1.
Thenoninteracting weightfunction poisgivenby
1
p0(k,x)= i
=
1
l
w-w-------w - --- -, -- v.
x- h '(< - p,
)- t'
rl x- h '(e2- Jz)+ In
2=3(.
x- /j-1(..
2-)t)J
and somesimplealgebrawith Eq.(31.24)givesthetime-ordered function
(;otkjtuj= l+ - -1 ---- ----. r
1 - --)
-t
expl-i
j'
(d -p'ilo,-j'
Z'
et-Jz)
,-l.
q
1
1
+-j
-.
'!
-nex
--p
--ytet--.
-p.ô1(
a,----y.
uj(e
-(-.
---Jz
-j.
--.-iq (3j.J8)
Equation (31.38)can also be obtaineddirectlyfrom thedesnition (Eq.(3l.l)1
withtherelationTrtlct
zluk)= /12= (expEj(E2- p.
))D.
r1)-l.
32LLINEAR RESPONSE AT FINITE TEM PERATURE
ln Sec.31 we saw how the temperature Green'sfunction Ftk,f
z)alcan be used
to determ ine the behavior of excited states obfained by adding or subtracting
oneparticlefrom asystem intherm odynam icequilibrium . AsnotedinChap. 5,
however,therearem any otherkindsofexcited states,them ostim portantbeing
those thatconserve the num ber of particles. The theory of linear response
provides a convenientbasisfordescribing such excitations,and we shallfirst
extendtheprevioustheory to snitetem peratures.
GENERA L THEORY
Ifa system isperturbed from equilibrium att= to by an externalham iltonian
XeA(/),thesrst-orderchangeinanarbitrarymatrixelementofanoperator0 is
1G.Baym and N.D.M ermin,J.M ath.Phys.,2;232(1961).
299
REAL-TIM E GREEN'S FUNCTIONS AND LINEAR RESPONSE
given by Eq.(i3.9). ln particular,the change in a diagonalm atrix elem ent
reducesto
i î
bijN I:(l)l/N)- j
.
dt'lj.
N 1EW7(!')-ö,?(?)1l/.N')
.
(32.1)
f0
where the subscriptH denotesthe H eisenberg picture with the fullunperturbed
hamiltonianX,and 1jN)isanexacteigenstateofX and 9 with eigenvalues
.
EjLN)and N. Fort< to,thesystem isin thermodynamicequiiibrium.and the
occupationofthediflkrentstateskjN$isdeterminedbythestatisticaloperator
?o. Since Eq.(32.1)isalreadyproportionalto Xex,thesrst-orderchange in
theensembleaverage of0 may beevaluated by adding thecontribution ofeach
state I.
jN),weightedaccordingtotheunperturbedensemble
3(d(/))rx= pâ
EbLLI-EJCN3'
PI
iN?34/.
N 10(/)bjN)
'A'
.
l' t
-j
JJ,'
rrtâcl/kt
xf,),ys(?))j
t:
ThisequationisadirectgeneralizationofEq.(13.10)atF = 0.
Tobespecifc,assumethat4e'(/)takestheform
4 ex(?)= fdqx:(x?)Sextx?)
(32.3)
whereE*x(x?)isageneralizedc-numberforcethatcouplestotheoperatordensity
d(x?). Thelinearresponseofötxf)isgivenby
j t
3(:(x?))ex- -j
dt'fdqx'Trtlsltqstxrl,tgstx'r')))Eex(x'/')
.
r:
=
j $
j
dt'J#3x.DRlxt,x't'
)Eextx't')
f;
where D Risa retarded correlation function
I
'DRLXt,x't')> Tr(âG(ds(x;),ds(x'/:)1)0(t- ?')
(32.5)
evaluated in the equilibrium grand canonicalensem ble. Theanalysisoflinear
response is thusreduced to the calculation ofa retarded correlation function.
Since the unperturbed ham iltonian is time independent, D R takes the form
DR(x,x',t- ?'). Furthermore,O usually commuteswith X,which allowsus
to reinterpretthe H eisenberg operatorsin term softhe grand canonicalham il-
tonian# = W'- p.
X:
Onlxt)= eiktlhd(x)e-6ktlh. Jx(x/)
(32.6)
The Fouriertransform DR(x,x',*)hasasimpleLehmann representation,which
showsthatDR(x,x',u))isanalyticforImop>0.
3*
FINITE-TEM PERATURE FORM ALSSM
Itisinconvenientto calculate D R directly.
.instead,weintroducea corresponding temperature function V thatdependson the l
'
maginary-timevariables
T:
Vtxmsx''
r')- -Trfâ(;F.(tk(x'
r)Oxtx'm')J)
HeretheHeisenbergoperatorisgivenby(compareEq.(24.1)J
öxtxr)= e*%/'d(x)e-kr/h
Since9 isoftheform .
@ (x,x',r- '
r').itsFouriercoemcient.
f?(x,x',pn)also has
asim ple Lehm ann representation. Justasin Sec.31,thisrepresentationisvery'
importantbecause the same weightfunction determines both DR(x,x',o))and
# (x,x',va). Furthermore,.
@ (x,x',L)can be evaluated with the Feynman rules
and diagram m aticanalysisofSec.25. A n analytic continuatïon to the upper
sideofthereal(,aaxisthenallowsustocalculatetheretarded correlationfunction.
DENSITY CO RRELATIO N FUN CTION
The precise form of-the Lehm ann representation depends on the particular
operators involved. As an exam ple of great interest, we now consider the
particle density and carry through the preceding analytic continuation in detail.
Forsim plicity,thesystem istakenashom ogeneous,butthesam egeneralmethod
applies to m ore com plicated situations. The operatorin question isthe density
deviation operator
H(x)= H(x)--.
xH(x))
'
whereq
xHtxllistheensembleaïwrageofthedensityoperatoranddependsexplicitly
on F and y.. Theretarded andtem peraturefunctionsaregiven by
iDRlxt,x't')= Trl)c(?
'
ix(x/),#x(x'/')J)#(?- t')
glxrsx'r')= -Tr1/GFv(:x(xm)hK(x'r')))
(32.l0)
(32.l1)
and havethe usualFourierrepresentations
DRLxtx';')= (2zr)-4Jdhdcoefq*tx-x'):-ïtz
a(!-l')oR(q(z)l
glxr,x',
r')= (274-3Jd3qlphl-')
yefq*tx-x')e-ï'
zatm-r')5>(q,pa)
11
(32.12)
(32.l3)
whereva= lnn'
llhdenotesaneveninteger. ltisstraightforwardtoevaluatethe
Lehmann representation ofeach ofthese functions,and wefind
DR(
., dco' h(q,(s')
q,tz,)- h '
1V (.o - o),+ ix-l
(32.l4)
9(
co dl.
o'a(q,(s')
q,?
%)= h X 2,v ivn- to,
(32.15)
-
301
REAL-TIM E GREEN'S FUNCTIONS AND LINEAR RESPONSE
Where
âzâtqqtsl= -â.
â(-q,-(,?)
= eçC1j((e-#Xl(2.
rr)3jlq- â-l(P - Pj))z.
rrlg(s- h-1(#m- A'jlj
lln
x(1- e-bhcvlill(
J1
-)i2) (32.16)
Equations (32.l4)and (32.l6)togethershow thatD&(q,f
.
,))isa meromorphic
functionoff.
,
ywithitspolesjustbelow therealaxis. Eachpoiecorrtspondsto
a possibletransition between statesthatareconnected by the density operator.
If.
@ weregivenexplicitlyinspectralrepresentation (Eq.(32.15)),thenthe
analyticcontinuationwouldbeelementary. lnpractice,however,theexpressions
takea difrerentform ,and itisnecessary to exam ine the perturbation expansion
forfi'inmoredetail. Equation(32.l1)may berewrittenas
N'
txm,x'r')- vHx(xm)))tlixtx'z'))
- -
Trt)cFmEfhx
l.(xr)v'Kxlxr)f4/x''
r')yV#(x'r'))) (32.I7)
which can be transform ed to the interaction picture. The stepsare identical
with those in Chap.7,and we merely state the hnalresult
*
-
V''(xm,x'm')= '
)éxtx'
rll(z
ixtx''r'))-
j l j bh
u- s
1= () '4
:' 0
jh
#ml'''
0
x Trjt/fCl(à-ka)rT(#j(.
rj)...#jtzj)t
/t
yztxz)
x'
fzztx'
r)'
Jf
zjtx',--)v-kblx'g''lllconnected
drl
(32.18)
wherethesubscriptm eansthatonly connected diagram sareto beretained.
Itis im portantto rem ember thata connected diagram is one in which
everypartisjoinedeithertothepointx'
rorthepointx'r'. Thusbothdiagrams
in Fig.32.lareconsidered connected. Nevertheless,theyhavea quitediflkrent
structure.because Fig.32.1: itselfseparates into two distinct parts. The sum
of a1l such separable contributions isjust the perturbation expansion of
(?
ix(xm))(Hx(x?'
r')),and precisely cancelsthe f
lrst term on the right side of
Eq.(32.18). Consequently V'txc-,x''
r')consistsofa1lconnected diagramsin
which the pointsx'
rand x'r'arejoined by internallines. Forexample,the
-
Fig.32.1 Lowest-ordercontributionsto (J)1(x'
r,x'r')
(:)'
rzixtxrl)'
tâxtx''
r.
')).
XT
17
X'm'
x''r'
(J)
(h)
x:
FINITE-IEM PEHATURE FORM ALISM
only zero-order contribution is that given in Fig.32.1J,and W ick's theorem
applied toEq.(32.18)yields
#0(xm,x',
r')=:
!:F.
Q#(xm,x'm')F(
/atx'r',xml
=
:4:(2.
9+ 1)F0(xm,x'm.)@0(x'r',xml
(32.19)
Itisclearthat9 hasthtstructureofa polarization part,and,indeed, ,V'(l,2)is
proportionaltothetotalpolarizationJl(1,2). Anargumentexactly analogous
tothatusedinobtainingEq.(12.l4)gives
T(1,2)= âJ1(1,2)
(32.20)
Inanyspeciscproblem ,itisalwayseasiertoevaluatetheproperpolarization
J1* and then to determine Jl and .
@ from Dyson'sequation. The analysis is
particularlysim pleforauniform system ,when thesolution ofD yson'sequation
reducesto
J1(q,w)= J1*(q,p'2(1- A t)tqyyt2J1*(q,v21-1
= J1
*(q,v2 (1- F(q)Jl*(q,p'
a))-1
(32.21)
SinceJl*(q,va)isaparticularpolarizationinsertion,itsLehmannrepresentation
musthavethesameform asthatforJ1(q,%)and.
@(q,p.):
JI*(4,
= Jftt
p'à*(q'f
s.)
L%)= -x ,
xr IzPa (,)/
(32.22)
zl
-
whereA*(q,o?'
)= -1*(--q,-(zg')isreal.
W e can now perform the analytic continuation from @ (q,pn)to .
DR(q,(,p)
(Eqs.(32.14)and(32.15)j. Itisconvenienttointroduceafunctionofacomplex
variablez
F(<z)= co d
l*(q''u,
-2ol';')
,
-* .a. z - ttl
(32.23)
thatredtlcesto .J1*atadiscrete setofvaluesz= ivn:
J1*(4,1
G)= F(q,i>a)
(32.24)
i
Analytic
'% continuation
i.2
i iv
v 1
*Z= oz+ j5
iv- )
iv-z
iy'-3
Fig. 32.2 Analytic continuation from
Z(q,&%)to 11R(q,a)+ iTs.
REAL-TIM E GREEN'S FUNCTIONS AND LIN EAR RESPONSE
303
In thisway,thepolarization can bewrittenas
J1(q,'%)- â-1@(q,w)- Flzbivn)(1- P'(q)F(q,ï%)J-1
(32.25)
Sincethezdependenceandanalyticpropertiesof-F(q,z)(1- PrtqlF(q,z)j-lare
explicitin both the upper and lower z plane,this function can be imm ediately
continued onto therealaxisz -.
>.(.
o+ l
h (see Fig.32.2),where itgivesthe corresponding retarded functions
l-lR(q.(s)= â-1DR(q,f.
t))= F(q,(
.
o+ j.
?yl(1- )'(q)F(q,u)+ ,'
?y)j-1
.
(32.26)
3D SCREENING IN AN ELECTRON GAS
A san exam ple ofthistheory,we shallstudy the responseofan electron gasto an
applied scalar potential(pextxr). The externalperturbation is the same as in
Eq.(l3.ll)(thechargeon theelectron is-e')
Ièqxlt)- --(dbxHs(x/)tatpextx/)
(33.j)
where the subscriptH now denotes a Heisenberg picture with respectto X =
W - y,,
9,asin Eq.(32.6). For a uniform system,the induced density atwave
vectorqandfrequency(
.
oisgiven by(compareEq.(13.l8)j
3(?
i(q,(,a))= -â-1DR(q.(s)ty extq,t
=)= -I-IR(q,t.
s)gvextq,tz?)
Since F(q,(
.
o+ iT4isthe continuation ofp'
1*(q,p'
o),Eqs.(
.32.26)and (33.2)relate
thelinearresponseofa system intherm odynam icequilibrium tothetotalproper
polarization evaluated in the tem perature form alism .
lnpractice:J1*mustbeapproximatedbysomeselectedsetofdiagram s,and
we now consider the sim plestchoice J10,studied in Sec.30. Com parîson of
Eqs.(30.9),(32.22),and(32.23)showsthatF0(q,z)isgivenby
F0(q,z)- -2 -:3
O j--no
.*-1- no
p g(à=)hz- (ep
o+q- e,)
Thusthecorresponding retarded function becom es
o
613P
n0 - nn
F (q,(z)+ iTl= -2 (2 aj- .p+q op c
=) (
.
o+ IT- (epy.q- ep)
= -
2
d3p
nn
l
(2z43 >+1Q hut+ ixl- h2p.q/ra
-
1
hut+ IT + h2p-q/rn (33.4)
,
wherethelastform isobtained with asim pleehangeofvariables. Thisequation
applies for allF and p,and clearly reproduces the zero-tem perature retarded
a-
FINIAU-TEM PERATURE FORM ALISM
function l70R(q,rs)asF -->0 (Eq.(15.9)J. Although a completeintegration of
Eq.(33.4)would bequiteintricate,theexpressionsalso becomesimplein the
classicallimit;where the distribution function reduces to a gaussian function
np
0= ebHe-bh2/-/2m. Equation (33.4) can then be evaluated using cylindrical
polarcoordinatesp= ps + k 1with4asthepolaraxis:
F0(q,(
.
o+ ï'
,?)- -le$. #2p
.
J(2,42
srxp(-/J
2
'
a
2
?
.7
A
â2)
= #,1
.
'j-.2,.expg-#(:12
+
m!'
ç)2
â2j
1
1
c
n
.i
,
.
,
.,c,,
,qjm)
'
<lnf
.
s+i,-Jk,,q;m
--c-oz-zf- dp,evnr./(p.
.+14)212
- J-. 2,
7.
,--rL
zm
1
1
Xl/
j(
s+i
y-:2,,qjm J
joa+i
.
r
l+l/j2p,qjmj (33.5)
.
-
whereA= lznphzlmlkisthethermalwavelength(seeEq.(30.24)2.
Itisnow convenientto separateEq.(33.5)into itsrealand imaginaryparts:
F0(q,(.
o+ iT)= Ff(q,f
z))+ ïFî(q,tz))
(33.6)
z-?(q,
(
s)--2,p,z-c.
,j-.a
A.expg-/
xr+
zmbqtzhlj
(à lz
'
b-oà-hol+llpqlmj(33.
7*
z-@(q.
r
.
,)-lejsz-z.
(o.t
yexpg-#(#z
f
m14)2â2j
jj
laj(
s-fmpq/
j-z
)jy
(y(
.
v+hl
mpq)j(aa.,
7,)
'
x
Furtherm ore,w'
e shall use the thermodynamic relations obtained previously
gEq.(30.34))to elinninatethechemicalpotentialrtin favorofthedensity n.
SinceU(q)F0(q,(s)isalreadyofordere2,itispermissibleto retain onlytheleading
term ebî
i;kù.
!.nA3. TheimaginarypartFîiseasilyevaluated,andwe5nd
/-y(t
?,
u?)--rl
qa
'(jnpm)Aexp(-la
.
Tqt
'
.
/
'-'h
àm
lq'jSinh
jj
qsp
.ho.
,
l
(33.8)
ItisclearthatFîisan odd functionof(.
oand vanishesat(
.
o= 0,in agreement
w ith theantisym m etry oftheweightfunctionà*.
REAL-TIM E GREEN'S FUNCTIONS AND LINEAR RESPONSE
305
In contrast,the realpartFfcannotbeexpressed in termsofelementary
functions,buta changeofvariablesyields
Fîçq,
o,b--Iq(!qml'(*g(
.
i,
z4)+()+.
a
/
i
fylj-*gt.
s
lnkpl.('
4
2'-),
T)1)
(33.9)
where
isthe realpartofthe plasma dispersion function.l lftheintegrand ism ultiplied
by (x + y),
I(x 4-y),theresultmay berewrittenas
(33.11)
which shows that*(x)is an odd function. The asym ptotic form is obtained
by expanding the integrand for large x
*(.
X)- X-1(l+ 1.
1-2-i
-''
(33.l2)
butthe behavior for small.
,
'
t requires a little more eflbrt. Equation (33.10)
showsthat*(0)= 0,beuausetheintegrand isthen an odd function ofy. DifferentiateEq.(33.10)with respectto x and integrate by parts. Theresulting
.
expression m ay be rearranged to yield
*'(x)= 2 - 2x*(.
x)
(33.l3)
which hasthesolution
*(x)=2e-x2jx#>'p'2
(33.14)
Weseethat*(x)isanentirefunctionofx,andadirectexpansiongives
tD(Y)1
k72.
X(1- i'
.
X2+ .'')
To be speeific, assum e that the external perturbation is a positive point
chargewith potential+*'(q.tz))= 4=Zeq-22=(
5((z?). W ethereforeneed onlythe
zero-frequency compohent of DR(q,o,);a combination ofEqs.(32.26),(33.2),
1Thisfunction istabulatedinB.D.FriedandS.D.Conte,'e-l-hePlasmaDispersion Function.*'
AcademicPress,New York,1961,
3g6
FINITE-TEM PERATURE FORM ALISM
(33.8),and(33.9)yieldstheinducedchargedensity
ô()(x))= -e3(?i(x))
#% fq-x P-(t?)F%ç.0)
= ze
= -
j(a,
o)e j-stt
y;sytoo
#3: f
).
g'l(çA)
ze
j(z.jye gz(?./)
.
g
jgjfqy
q-x
(33.16)
where qy= (4owc2j)1 is the reciprocal of the Debye shieldin: length and
A= (2=h2#/,A;)1isthethermalwavelength (seeEqs.(30.39)and (30.24)). Here
thefunctiongjty)isgivenby
#1(.
$=2'
zr+.
F-1(
II(zJ11
(33.17)
and hasthefollowing lim iting behavior
:1(.
$;
kê1+ 0(.
F2) y.
,
c
:1
(33.18)
gl(y)- 8.
71-2
y> 1
(33.19)
Equation (33.16)isclearly very similarto Eq.(14.14),and mostofthesame
rem arksapply.
l.Thetotalinduced chargeis
3: =J#'x3()(x))=-ze
(33.20)
so thattheim purity iscom pletely screened atlargedistances.
2.The integrand vanishes Iike q-4 asq.-+.cc),which ensures that3()(x)) is
bounded everywhere,including x = 0.
3.Thesingularg-2dependenceatsm allg2iscutofl-attheinverseD ebyescreening
length qn= l4=nezpl'
t,which justises ouruse ofa cutofïçmjncce in Eq.
(30.28). Sincegl((
?A)isanentirefunctionofqh,itisinsnitelydiflkrentiable
throughout the complex q plane. The asymptoiic behavior of 3t)(x))
thereforecanbeobtainedwiththeapproximationgjtçà)a;gl(0)= 1,because
thetermsneglected areoforder(çsA)2(
x:(ae2j)(/i2j/n1)= (n1elljlhznkpjl.
nj
<:
t1gseethediscussionfollowingEq.(30.48)):
3f:?(x))- -Ze #3îjei
qex /)
, -Xj
(2*) q + qo
= -
ZtNà(4'
rrx)-'e-qBx
(33.21)
Thisexpression exhibitsthe role ofgoiasa classicalscreening length and is
identicalwithEq.(30.41:)which describesanegativelycharged impurity. At
zero tem perature,the sharp Ferm isurface m odiied the asym ptotic charge
density byintroducing additionaldominantoscillatory terms(Eq.(14.26)J;
REAL-TIM E GREEN'S FUNCTIONS AND LINEAR RESPONSE
307
no Such behavior occursin thepresentclassicallim it,becausethe distribution
functionsare5m 00th.
3K PLASM A OSCILLATIO NS IN AN ELECTRON GAS
The presentform alism also can be used to study the collective oscillationsofa
system in therm odynam ic equilibrium ,and w'e now exam ine the plasm a oscilla-
tionsinanelectrongas. lfthesystem issubjectedtoanimpulsiveperturbation
@extx/l= @()t?iq-x3(/),the associated induced density becomes (compare Eq.
(15.7))
3(H(xr))
F.(v,t
.
,a)+ iFblq,œt
tq-x dko ioaf
=.
-epoe
y-(s)..jst(
y)sztt
y= e 1-ur(q)sjg,
jo l
whereF(q,(
.o+ iT)hasbeen separated into itsrealand imaginary parts,asin
Eq.(33.6). Thenaturaloscillation frequenciesaredetermined bythepolesof
.-
.
the retarded density correlation function,w'hich occuratthesolutionsfzq- iyq
oftheequation
1- 1,'(t?)Fl(ç,Dq- iyq)- 1
*F'(4
2)#c((?,t'lq- iyq)= 0
This description is entirely general. lf the exact proper polarization
A*lqtvnlisapproximatedbythezero-orderpolarizationJ-10((
?,s),weobtainthe
fnite-tem peraturegeneralization ofEq.(15.10). Thetheorybecomesespecially
simplein theclassicallimit,when the previousexpressionsforFfand F?are
applicable. Furtherm ore,we assum e thatthe dam ping is sm all,so thatthereal
andimaginarypartsofEq.(34.2)becomc
1- p'(:)Fîlq.fjq)
.'
3 -'
yq- F@(ç,f1q) 0Fî
)(
-C:f
CO
(34.4)
o
A sshown below,thisisa good approxim ation for(
1c.
tqo,when itispossible to
evaluateFfwith theasymptoticform given inEq.(33.12). A straightforward
calculation yields
nq2
3V2
#î(t
?,f
.
'J)QJmœ'
j 1+ jznf
xpz-1-'''
q-*()
Equation(34.3)thenreducesto
j=
4=nel
3g2
y j.
j.
.
y
mûq
jrntlq
w ith the approxim atesoltltion
flq-'
in
t
aplgl+j
3jq
.
#n
-)21
(34.
7)
ats
FINITE-TEM PERATURE FORMAUISM
whereflpland qoaregivenin Eqs.(15.18)and (30.39). Thecollectivemode
again representsa plasm a oscillation,and the dispersion relation hasthe sam e
form asatF = 0(compareEq.(15.17)).
Thesnitetem peratureintroducesone new feature,however,becausethese
collective m odesare now dam ped,even in the lowest-order approxim ation of
retaining onlyJlO. SinceJâflpl= Olqohj< 1athigh temperatures,Eq.(33.8)
maybeapproximatedby(q<:
.
:qo)
Fî(ç,(
s)=-Klt
qzl(Jrrjr
nllexp(-Vl
Kq
1t
*
22j
(34.8)
ThederivativeofEq.(34.5)can becombinedwith Eqs.(34.4)and(34.8)to give
v
SQ i -'
g-J'j(g,ci
nopj)gOXIaI.
;
..;opy
j
=q3fàcl
t'''
*lti'
rl'exp(-Dl
Jq
'm
2îj
0-,(
.
4.
,,
)+(%;)'expg-j
'(%
q1
?)2j
-
-
As expected from generalconsiderations,w ispositive,and both poles of DR
1ie in the Iower half plane at u);
k;uinfàp!- iyq. The approxim ation of sm all
dampingisfullyjusti:edatlongwavelengths.because l
yq/fhvlvanishesexponentially. Thisweak dampingisknownasLandaudam ping,lbecauseLandau was
thefirsttonotethatthesolutionsofEq.(34.2)arecomplexinsteadofreal. Itis
interesting thatthe tem perature aFects both the dam ping and the ql correction
to the dispersion relation,butdoes notalter the fundamentalplasm a frequency.
PRO BLEM S
9.1. IfEo(Njisthe ground-state energy ofa Fermisystem with N particles.
show thatthegrand partitionfunction atlow tem peraturem ay bewrittenapproxi-
mately as e-bt
'
lœ e-bïEzçNQb-l
x'
%'z?g2rr/jQ'
'
(A%))+,where .
N'
e(p,
)isdesned by the
relation Eo'(Nft)= rz and the primes denote diflkrentiation with respect to N.
W hycanNobeidentisedasthemeannumberofparticles? EvaluateEq.(3l.12)
inthesameapproximation and provethat6(k,(
.
o- ghlv-o= G(k,tz?),whereG
isthezero-tem perature function ofChap.3.
9.2. Evaluate the weight function p(x,x'.(s) for the Hartree-Fock Green*s
function (27.8). Find thecorrespondingreal-timeGreen'sfunction C(x,x',Y)
and theretarded and advanced functions.
1L.D.Landau.J,Phys.(USSR4.10:25(1946).
REAL-TIM E GREEN'S FUNCTIONS AND LINEAR RESPONSE
309
9,3. (X lftheproperself-energyX*(k,con)inEq.(26.5)takestheform
Y*(k,œa)= Xo(k)+ (* d2ul' tztk't
.
,
a.)
= iuln- Q
with lztlandcreal,usethedeEnitionEt(k,(.
t))+ fE1(k,(,
p)O Y*(k,tz?n)1ic
oa-u,-i,
tto
5ndthecorrespondingweightfunctionptk,ct?lin Eq.(31.35).
(b)Expand ptk,tz?laboutthe pointf
.
,
a= cokdetermined by the self-consistent
equation hulk= e2- p,+ âY)(k,t
sk),and derive an approximate quasiparticle
(Iorentzian)weightfunction. Compute theapproximate(;qptk,f
xll,and prove
thatthe excitation energy and dam ping ofsingle-particle excitationsare given by
ek= hœk+ p.and yk= (1- ?Yt(k,f
.
t))/'0(.
t)Iua.)-1Xllkstt)
kl(Compare Prob.3.14).
9.4. R epeatProb.5.1 for the retarded density correlation function at hnite
temperature iDR(x,x')= 0(t-./')Trt)(;(?
'is(x),9s(x'))). Considerthefbllowing
equal-tim e lim its:
(J)low temperatures(/t'sF<t6s)and lx- x'Iy>kFi,
(b) classicallim itand '
Ix- x'r>>A = (lrrhllmkaF)1.
Com pare the discussion atthe end ofSec.l4.
9.5. Use the spin-density operatorJz(x)- '
?
/ltxlttzzlxj?
k
S
?j(x)to constructthe
retarded correlation function
iDl
llxt,x't')=Trtlclêsztx/lsêpztx't')J)0(t- /')
and the corresponding tem perature G reen's function
9.gtx'
r,x'r')= -Trt/(;Fggêxztx'
r)êxzlx''
r')))
.
D erive the Lehm ann representation forthese two functions,and show thatthey
arerelated through thespectralweightfunction asin Eqs.(32.14)and (32.15).
9.6. Study the linearresponse ofa uniform spin-l Fermisystem to a weak,
externalmagnetic held .#'(x?)where the perturbing hamiltonian is given by
Xex= .
--y,vj#3xêz(x).
Y#(x/).
'
(t7)Show thattheinduced magnetization isgiven by
(4 Z(x/))= -J.
t2
()â-1jdbx'dt'Djtx- x',t- t')./#'
(x'r')
tXz(k,œ))- -Jz2
()â-1.
&j(k,(z))X (k,t,))
whereD R
eisdefned in Prob.9.5.
(b) Use W ick'stheorem to evaluate 9.
'
. fora noninteracting system,and determ ine DR
v with the results ofProb.9.5.
(c)Find the generalized susceptibility ofa noninteracting system in a static
magneticseldy(k,0)= (X z(k,0))/,#'(k,0),andverifythat
30,2
,,n
lim z(k,0)-
k-+o
-'
i
%6.
XèF
p.
à-/n
k
B
F= 0(Paulispinparamagnetism)
ww cctcurie'slaw)
a1:
FINITE-TEM PERATURE FORMALISM
9.7. Repeatthe calculation of Prob.9.6 in the zero-temperature formalism.
W hy does tbis zero-tem perature calculation w ork,whereas thatin Prob.7.5
fails?
9.8. (J)Use Probs.5.9 and 9.6 to show thatthe staticsusceptibility for a
spin-àFermisystem withspin-independentpotentialsisgivenexactlybyy(k,0)=
-
MlZ)(k,0)= -p,
!I1e
*(k,0),
(:)W ith the approximation used in Prob.5.8,derive the zero-temperature
magnetic susceptibility of a dilute spin-è hard-sphere Fermi gas zlzp=
(1- 2ksc/'
r)-twhere Xpisthe Paulisusceptibility. ComparewithProb.4.10.
9.9. Discuss the asymptotic form ofthe screening cloud around an im purity
inadenseelectrongasat1ow butsnitetem peratures. Henceverifythediscussion
attheend ofSec.14.
9.10. Show thattheringapproximationto theplasmadispersion relationata11
temperaturescan be written to orderq2asf1g
2= f1p
2j+ q2(p2)ywhere (p2) isthe
m ean squarevelocityofparticlesin anoninteracting Ferm igasattem peratureF.
Verify thatthisequation reproduces Eqs.(15.17)and (34.7). Find the srst
low-temperaturecorrectiontoEq.(15.17).Repeatfornoncondensed bosons.
9.11. Evaluate F%ç,f,y)(Eqs.(33.4)and (33.6))forallr and Jtand rederive
Eqs.(12.41),(12.43),(12.45),and(33.8). Findthedampingofplasmaoscillationsand zero sound atlow tem perature. W hathappensto zero sound in the
classicallim it?
9.12. Use the ring approximation X*(k,(sa)= Zh)(k)+ E)(k,au), with Y)
takenfrom Eq.(30.12),tostudythesingle-particleexcitationsinadenseelectron
gas atzero tem perature.
(J)Show thattheexcitationspectrum (seeProb.9.3:)isgivento orderr,by
rs , p(l- IK+ qI)
ek- e2- eyx
zra d qqc+ gyv,ytj-u .j.jt
yl
wherea= (4/9rr)1,/isdesnedinEq.(12.58),and K = k//cs.l Hencedetermine
theehkctivemassm%(seeProb.8.2*.
(b)Show thatcloseto theFermisurfacethedamping constant(Prob.9.3:)is
givenbyâykQJeXœG)*(TX/16)(k/kF- 1)2.
tTheresultsforProbs.9.12/and9.1% werederivedbyJ.J.QuinnandR.A.Ferrell,Phys.
Rev.,11::812(1958).
part four
C anonical
T ransform ations
10
C ano nicalT ransform ations
In thepreviouschapters,ourdiscussion ofinteracting m any-particleassem blies
em phasized theuse ofquantum held theory and Green'sfunctions. Form any
system s,however,thephysicsbecomesclearerin am oredirectapproach,where
we sim plify the originalsecond-quantized ham iltonian and obtain an approxim ate problem thatis exactly solvable. This chapter studies a class of such
problem sthatcan besolved with acanonicaltransform ation ofthecreation and
destruction operators in the abstract occupation-num ber H ilbertlpace. A s
notedin Chap.1,thecom m utationrelationscom pletely characterizethecreation
and destruction operators. Since,by desnition,a canonical transformation
does notalter these com m utation relations,the transform ed operators again
satisfy Eqs.(1.28)inthecaseofbosonsorEqs.(1.50)and (1.51)forfermions.
Asa frstexample,weconsidertheinteracting Bosegas(Sec.35)following a
treatmentdueto Bogoliubov,!andthen studyan interacting Fermigas(Secs.
36and37).
'N.N.Bogoliubov,J.Phys.(&u
$WA),11:23(1947).
313
314
CANONICAL TIqANSFOICMATIONS
3SLINTERACTING BOSE GAS
An interacting Bose gas at zero tem perature has previously been considered
withintheframework ofquantum Eeldtheory(Chap.6). W eheretreatessentially the same problem as an exam ple of a m odel ham iltonian thatcan be
diagonalized exactly,thereby yielding al1thephysicalpropertiesofSec.22in a
directand intuitivefashion. lftheassemblyisdilute,then mostoftheparticles
occupy the zero-momentum state, and only two-body collisions with small
momentum transfers play an importantrole. tn Secs.1l and 22 we have
already seen thatsuch collisionscan be characterized by a single param eterc,
the J-wave scattering length. For this reason,we shall introduce a model
hamiltonian consistingofa kinetic-energy term andanartiscialpotentialenergy
X =U)âf
-tlkllck+C 1) t4
4dkl-ykzykz4k4
2F k,kzkak4 l4zJk,Jk
k
(35.1)
inwhichtheactualpotentialU(k)isreplacedbyapseudopotentialg.
The constantmatrix elementg can be determined by requiring thatX
correctly reproduce the two-body scattering properties in vacuum . This
problem hasalreadybeenstuditdinSec.11,whereitwasshownthatthescattering
amplitudewasrelatedto thetwo-bodypotentialbyEq.(11.14). Inthepresent
case,the Fouriertransform ?
Xk)M mvlkjlhzisreplaced bymg/hlsandwe5nd
/(k',
(mglhljl
k)=-4zr
/(k'
,
k)=mg
yz+J(d3
2,
aq,
)3k2
-q2+iy+'''
Theleftsidereducesto 4=a atIong wavelengths,which yieldstherelation
lW ahl
-../.
2 #3g 1
i+ '''
m - #- â2 (2=)3q'
..
Thehrst-orderresult
4nhla
ç- m
.
(35.
2)
(35.3)
(35.4)
iswelldesned. lncontrast,thesecond-orderintegraldivergesatlargem om enta.
Thisartiscialdivergence arisesfrom the substitution ofg for F(k),and we
thereforecutofl-theintegralatsomelarge wavevectorQ. W eshow,inthe
followingdiscussion.thatasim ilardivergenceoccursin theground-stateenergy,
andthetwoexpressionscanbecombinedtoyieldasniteanswer,evenforQ ->.q
n.
W e return to thisquestion attheend ofthissection.
The scattering propertiesm ustbeevaluated with carebecause the J-wave
scatteringlength ahasbeendefned asiftheparticlesweredistinguishable. For
identicalbosons,the overallwave function is symm etric,and the diflkrential
crosssection isobtained from asym m etrized scattering am plitude
Jtr 2
1./-(*)+f(= - #)l
(35.5)
315
CANONICAL TRANSFORM ATIONS
Each term reducesto -Jin thelow-energy Iim it,giving
#c .-
a
(35.6)
#Ii sI2cl
.
whichisfourtim esthatfordistinguishableparticles.
The param eterg hasnow been related to observable quantities,and we
return to the originalmany-particle hamiltonian,Eq.(35.1). In view ofthe
specialroleofthezero-m om entum statesitisagainnaturaltoreplacetheoperators
aoand t/()by cnumbers
tp(),f/o-.
>.Nt
(35.7)
exactlyasin Sec.18. Theterm softheinteraction ham iltonian canbeclassihed
accordingtothenum beroftim estzaandJ'
oappear,and we shallretain only term s
oforderNlandNo
Xint;
kJg(2F)-i(t
/0t7(
1)t7;av.
4-j)'g2((7l(zu(4av+ Z kJ-kA J0)
k
+ JlY-kJ0tk V JY
OY()JkJ-k1) (35.8)
.
where the prim e m eansto om itthe term sk = 0. This truncated hamiltonian
clearly neglects the interaction of particles out of the condensate; it should
provide a good approxim ation as long as N - N.<:
4N . The validity of this
assum ption isexam ined below. ltis plausible,however,thatthe term som itted
can only contribute to theenergy in third orhigherorderofperturbation theory,
fortheyinvolveonecollision to gettheparticlesoutofthecondensate,asecond
collision above the condensate,and a third collision to return the particlesto
thecondensate (see thediscussion following Eq.(11.22)2. A combination of
Eqs.(35.7)and(35.8)gives
Xint= g(lF)-1EAT!+ lNvX'(t4Jx+t/-kJ-k)+Xtl1l/(t4Y-kVJk&-k)1
.
k
k
(35.9)
while the num beroperatorbecom es
# -Nz+.
è.ï'
(clck+Jtkl-k)
k
(35.10)
Theproblem ofparticlenonconservation thatwasseenin Sec.18evidently
occurshere aswell. Although itis possible to introducea chem icalpotential
(Prob.10.3),weprefertoconsiderN = vj
9'
sgivenandtoelim inateh'0explicitly.
,;a
IfonlytermsoforderA'2andN arekept,substitutionofEq.(35.10)into(35.9)
yieldsourfinalm odelham iltonian
/?=.
i.Vgnl+J;'
((e2+ng)(t4tu +JtkJ-k)Vnglsatk+JkD-k)1 (35.11)
k
.
where n= N/U is the particle density. ln obtaining this result, terms like
()
('ï/ktzklz have again been neglected on the assumption that N - No<:
<N.
k
a16
CANONICAL TRANSFO RM ATIONS
Equation(35.11)hastheimportantfeaturethatitcanbesolvedexactlybecause
itisa quadraticform in theoperators,and thereforecan bediagonalized with a
canonicaltransform ation.
Thediagonalization ofP ismostsimplycarriedoutbydesninganew set
ofcreationand destruction operatorsl
au= uklk- Ukœt
-k
t
/k= ukt
zk
#- vkl-k
(35.12)
wherethecoeë cientsukandt'kareassum ed to berealandsphericallysym m etric.
The transform ation iscanonicalifthe new operatorsalso obey the canonical
com m utationrelations
lt
xk,t
x#
k'l= 3kk'
.
ltxksœk'l= Et
xlyd'l=0
(35.13)
and itiseasilyseenthatthisconditionm aybesatisEed byim posingtherestriction
/
.
/1- !)1= 1
(35.14)
foreach k. Equation (35,12)may besubstituted into X directly,and we5nd
# = !.Vgnl+ X'
((4 + ng)tj- nguk?
Jkl
k
+11'
tE(e2+nr)(?
zl+t'ï)-2?z.tyngl((
4th +atka-k))
k
+1'Z'
f/#(&1+/
71)-lukrktez+Fl#lltt
xl(Zk+œkJ-k)l (35.15)
k
Although theparametersukand vksatisfy therestriction ofEq.(35.14),
theirratioisstillarbitraryandcanbeusedtosimplifyEq.(35.15). Inparticular,
wechooseto eliminate the lastline of X. The resulting hamiltonian is then
explicitly diagonalin thequasi
particle numberoperatorsthtt
xk,which allowsus
todeterm ineallitseigenvectorsandeigenvalues. Theconditionontheparam eters
ukand vkbecom es
ngluk+ rl)= 2ukh(4 + ngj
(35.16)
Theconstraint(35.14)can beincorporated with theparametricrepresentation
uy= coshJu
vk= sinhn
whichreducesEq.(35.16)to
t
ng
anh2+k=4 + ng
Since the leftside lies betw een -1 and 1,thisequation can be solved forallk
only ifthe potentialis repulsive (g>0). The use of standard hyperbolic
identitiesgives
t,z- t4l- 1= !.
E.
E'
:
-l(4 + ng)- 1)
(35.17)
1Although thisstepisusuallyknown asaBogoliubovtransformation,itwasusedearlierby
T.Holstein and H.Primakoë Phys.Aep..58:1098 (1940)in a study ofmagnetic systems.
317
CANONICAL TRANSFORM ATIONS
Where
EkO ((4 + ngjl- (ng)2)+
(35.18)
A combinationofEqs.(35.14)to(35.18)yields
X'= !.Vgn2- JX'
(62+ ng- Ek)+ !.X'
Jk(œ:txk+ Z-kt
x-k)
k
k
(35.19)
The operatorQ akhastheeigenvalues0,1s2,....Consequently,the
groundstate l
O )ofX isdeterminedbythecondition
aklO)= 0 allk;
#.0
(35.20)
andmaybeinterpretedasaquasiparticlevacuum. Notethatlolisacomplicated
combinationofunperturbedeigenstates,sinceneithertzknOr4 annihilatesit.
Theground-stateenergyisthen given by
E = (0(4 (O)= !.Pk2g+ !.X'(Ey- d - ng)
k
(35.21)
Furtherm ore,al1excited statescorrespond to variousnum bersofnoninteracting
bosons,each with an excitation energy fk. This sped rum has the sam e form
asthatobtained in Sec.22foradilutehard-coreBosegas:
X
-.
E (zn
Y)
11
'
s'
Y
!
Y
-=$
I
'
.
4=m
t
m2hl1
$%%k
4=anh2
(35.22/8
k Q$
62+
(35.22:)
Atlong wavelengths,theinteracting spectrum ischaracteristicofa sound wave
with avelocitygiven by(4nanhzjmljk. Itisagain clearthattheseresultsare
meaningfulonlyforarepulsive interaction (g > 0,J > 0).
ThedistributionfunctioninthegroundstateIO)isgivenby
nk= (0 lt
/kJk10)= rl(O lœkQ 1O)= b'
i
(35.23)
whichvariesask-ifork ->.0. Atlargewavenumbers,t'
ïa:k-4,thusensuring
thatthetotalnumberofparticlesoutofthecondensateremainssnite. W esee
that the interaction rem oves particles from the zero-mom entum condensate'
,
indeed,there is a inite probability of fnding a particle with arbitrarily high
m om entum . ltisinteresting to 5nd thedepletion,defned by
N - No 1 '
l #3#
N = y k t,j= n
- (2=)yr,k
4j2n
=J.
)j1j(
r
jyzyyjjj.
y+
z+2.
y
lzj,.jj
=8j/ynnj+
.
a3
.
.
(35.24)
3,8
CANONICAL TRANSFORMATIONS
in complete agreementwith Eq.(22.14). Note thatthisexpression isnonanalyticina (org)and thuscannotbeobtainedin sniteorderofperturbation
theory.
Theground-stateenergyhasbeenfoundinEq.(35.21),butthesum diverges
like72
'k-2ask-+.œ. Thisdivergencerecectsthefailureofperturbationtheory
k
fora Bosegas. In fact,ifweevaluatethesecond-orderterm intheground-state
energy using our pseudopotential,the answer di
.v
ergesin justthe same way
(ste Prob.1.3). Thusthedivergenctisnotvtry basic,foritaristsfrom the
assumption thatthe potentialhasconstantm atrix elementsasa function ofthe
relative m om entum . The Fouriertransform ofa m ore realistic potentialfalls
ofl-at high momentum, which renders the resulting expression convergent.
Thisprocedureis unnecessary,however,since the expansion for the scattering
lengtha to orderg2in Eq.(35.3)containspreciselythesamedivergence. We
may thereforeelim inateg entirely,which givesa convergentexpression forthe
ground-stateenergyintermsofthedirectlymeasuredquantitya(seeEq.(35.6)1.1
Toverifytheseassertions,Eq.(35.21)mayberewrittenbyaddingandsubtracting
the second-orderenergy shift
,
E= '
l
,
n
mn2g2
l'Zrl2g-Vng)2Fk jzkzj
tn+!.k Eg-ek-ng+ yzy;a
'
Itisreadilyseenthatthelastsum converges;furtherm ore,thesrsttwo term sare
justthoseintheexpansionofEq.(45.3)forthescatteringlengtha. Inthisway
weobtain
E-
A !.
n(g-jj2j(d
2*
3k3Vl)j
-.
jgj(d
z.
3j
)jt
-g
j-X
n
e
g-j+g
A
ndj
2zrm
t
02nj
lj.
j
.gt
yza
=t
p/
j+Jg0
*yzysjty.+zyzll
'
.-y,.j+slzjj
zzrm
ahznjj+1
l
2
58jn=
c3jlj
(?5.
a5)
.
which is precisely Eq.(22.19). Note thatthese two calculations are quite
diflkrent.becauseweevaluated E - jy,
N inSec.22,whereashereweevaluateE
directly.
Itisinteresting to study the magnitude oftypicaltermsomitted from Eq.
(35.25). Considerfrsttheinteraction ofthe particlesoutofthecondensate.
A san estim ate ofthiscontribution,we m ultiply the strength ofthe interaction
by thenum berofpairs:
1(A
2Y-p'i
X0)2=$a#Xg1(X=
&î)112-Nl=h
mldni
1g3
!(N=
&'j1j2
-
-
which shows that these terms represent a higher-order correction to EIN.
1ThisobservationwasmadebyK.A.BruecknerandK .Sawada,Phys.Rev.,106:1117(1957).
CANONICAL TRANSFORM ATIO NS
319
AnotherapproximationoccurredinEq.(35.11)withthesubstitution
gNl #A2 ININ - Nz4
2F
2P'
P'
Thisexpression om itsterm soforder
WU(X-Xo)2=NlR%
2
ml&X(à
8(X
-=
&3jij2
which are again negligiblein thisapproxim ation. Finally,we com m enton the
useofEq.(35.3)to eliminateginfavorofa. Theleadingterm intheenergyis
oftheform
E y#
A = 27+ .
- .
(35.26)
whichmayberewrittenwith Eq.(35.4)as
E
2.
14 2na
X
m
(35.
27)
Thisterm can becom pared with thesrst-ordercorrection foradiluteFerm igas
(Eq.(11.26))
E j.eî.= gl kra:2
k;
2an(2- 1)
A
za
y = =hnl
-
.
(35.28)
where the numericalfactor (2- 1)arisesfrom the directand exchange terms,
respectively. Apartfrom the diFerentdegeneracy factors,Eqs.(35.27)and
(35.28)areidentical,and they both can be interpreted in termsofan optical
potentialEsee the discussion ofEq.(11.26)). The correctionsto Eq.(35.26)
requirethesecond-ordertermsin Eq.(35.3),aswellastheadditionaltermsin
Eq.(35.21). Once we have eliminated the divergences,however,itisthen
permissible to set g= nnv
ahllm in the remaining correction terms. Since the
answeriswelldehned,the error introduced by this lastapproxim ation is of
higherorderand thusnegligiblein the presenttreatm ent.
The above results for the ground-state energy and depletion ofthe condensate reproducethoseobtained in Sec.22 with them ethodsofquantum seld
theory. A lthough the canonical transform ation provides a m ore physical
pictureoftheground stateand excited states.itislesswellsuited forcalculations
to higherorder. ln principle,ofcourse,itispossibleto retain allhigher-order
termsin the interaction hamiltonian (35.8);the two approachesmustthen lead
to identicalresultssince they are based on the sam e physicalapproxim ations.
N evertheless,practicalcalculationshavegenerallyrelied on them ore system atic
methodsusingG reen'sfunctionsintroduced in Chap.6.1
lSee,forexample,S.T.Beliaev,Sov.Phys.-JETP,7:299(1958).
CANONICAL TRANSFORM ATIONS
320
36LCOOPER PAIRS
A lthoughthemostrem arkablepropertiesofsuperconductorsarethoseassociated
withelectromagneticfelds(seeChap.13),superconductorsalsoexhibitstriking
therm odynam ic eflkcts,which played a centralrole in the developm ent ofthe
m icroscopictheory. TheelectronicspecischeatCe3variesexponentiallyatlow
tem peratures
Cetccexp -p-.
p
'*B =
F ->.0
(36.1)
w hich istypicalofan assem bly withanenergygap,
à separatingtheground state
from the excited states. A second im portantexperimentalobservation is the
isotope eflkct,where the transition tem perature Fcofdiflkrentisotopes ofthe
sam e elementvarieswith the ionic m assM as
FcccLV -k
(36.2)
This result indicates that the dynam ics of the ionic cores aflkcts thesuperconductingstate,eventhoughtheionsarenotespeciallyim portantinthenorm alstate
(Sec.3).
ln the presentsection,we study a sim ple m odeldue to Cooper,lshowing
thatan attractive interaction between two fcrm ionsin the Ferm isea leadsto the
appearanceofaboundpaix
'. Thenoninteractinggroundstate(filled Fermisea)
thus becom es unstable with respect to pair form ation,and the linite binding
energy ofthepairprovidesa qualitativeexplanation forthegapin theexcitation
spectrum . Before Cooper's m odel can be considered relevant to superconductivity.however,itis necessary to show thatthe esective interaction between
electronsisattractive,and itisherethatthe isotopeeflkctgivesan im portantclue.
Although the shielded coulom b potentialofSecs.12 and 14 is repulsive,there is
also a virtualelectron-electron interaction arising from theexchangeofphonons
associated withthecrystallattice. Astirstnoted byFröhlichjzthisinteractionis
attractive forelectrons near the Ferm isurface,and ittherefore gives a physical
basis forthe attractive interparticle potentialin Cooper'sm odel. The electronphononinteractionisstudied indetailin Chap.l2,and weshallnotattem ptany
furtherjustiscationofCooper'smodelatthispoint.
Consider the Schrödinger equation for two fermions in the Ferm i sea
interacting through a potentialAFtxj,xc). The many-particle medium afects
these two particlesthrough the exclusion principle,which restrictstheallowed
intermediatestates,exactly asin Sec.11. TheSchrödingerequation
EFI+ Fc+ AF(1,2)J/(l,2)= f/(l,2)
1L.N.Cooper,Phys.RFt,..104:1189(1956),
2 H . Fröhlich,Phys.Rev.,79:845(1950).
(36.3)
CANONICAL TRANSFORMATIONS
321
can berewritten in a slightlydiferentform asfollows:
441,2)- @a(1,2)+ é)@a(l,2) l (ealAUI/(1,2))
a#o
E - E.
(36.4)
E - En= (@olAP'I/(1,2))
(36.5)
wheretheeigenstatesn areeigenfunctionsofH o
S0n = (L + Fz)Tn= EnTa
(36.6)
The equivalence ofthese two form scan be verised by applying the operator
Ih - E = L + Fc- E toEq.(36.4)andusingthecompletenessoftheeigenstates
ofHn. Theremainingequation(36.5)issimplyanormalizationconditionfor#:
(9701/)= 1
(36.7)
and we m ust naturally com pute a1lotherexpectation valuesaccording to the
relation(O)= (/101/)/4/1/).
Ifthe system isconined to a large box with volume F,the unperturbed
wavefunctionsareplanewaveswith periodicboundaryconditions
PkTkz(j,;)= pr-+pfklpxlpr-+elkz.xz
(J6.8)
To sim plify thediscussion we shallneglecttheeFectofspinsand treatthetwo
initialparticlesasdistinguishable. Thisisperm issibleifthetwo particleshave
opposite spins,while P'mustbe spin independent. The many-body aspectsof
the problem are now incorporated by restricting the sum over interm ediate
statesinEq.(36.4)inthefollowingway
Zn -*klkzZ>kF
(36.9)
becauseal1otherstatesin theFermiseaarealreadyslled.
In a hom ogeneousm edium ,thetotalm om entum ofthe pairwillbeconserved,and wetherefore introducethefollowingdeEnitions
P = kl+ kz k = ètk1- ka)
R = ètxl+ xg x= xj- xa
v= m Fâ-2
E = hl,2:n-1+1/2#2,n-1
(36.10)
(36.11)
(36.12)
(36.13)
Thesolution to theSchrödingerequation takestheform
/41,2)= F-+cdP*RF-+4.k(x)
(36.14)
where the srst factorcontains the center-of-mass motion,while the second is
theinternalwavefunctionoftheinteractingpair. IncontrasttotheSchrödinger
equationin freespace(Eq.(11.8)),thetotalmomentum hp asectstheinternal
a22
CANONICAL TRANSFORMATIONS
wavefunctionbecausetheElled Fermiseaprovidesapreferredframeofrtference.
W enotethat# cannotexceed lkFiftheparticlesareinitiallyinsidetheFermisea.
SubstitutionofEq.(36.14)intoEqs.(36.4)and(36.5)yields
d3t
1
(;v'k(x)= e'tk*f+ A r(2=)3ctt*xKl- t2(t1t)1/p'k)
1.
iP + kl< ks F'O I
èP + tl>kr
.2 k2= zz-l
tklpl/p,k)
-
(36.15)
(36.16)
which isknown astheBethe-Goldstoneequation.l ItissimplytheSchrödinger
equation fortwo ferm ionsin a Fermigas,wherethePauliprincipleforbidsthe
appearance ofintermediate statesthatare already occupied by otherfermions.
Since the interacting pairinitially liesinside theFerm isea,itcannotm akereal
transitions. N evertheless,itcan m ake virtualtransitions to al1states outside
theFermisea,asseeninthelastterm ofEq.(36.15),wheretheenergydenominator
nevervanishes. In consequence,the solution ofthisequation hasm om entum
com ponentscorresponding to a11theunhlled statesaswellasthe originalcom -
ponentsI.
P +k andI.
P - k.
Ingeneral,theBethe-Goldstoneequationcanbesolvedonlywith numerical
techniques. A lthough straightforward in principle,thisapproach isnotalways
suëciently accurate to uncoverthe rather subtle features associated with the
Ferm isea,and weshallthereforeintroducea m odeltwo-particlepotentialthat
allowsusto obtainan exactsolution. Theconceptofapotentialisirstgeneralized to includenonlocalpotentials
r(x)-.
>.p(x,x')
(36.17J)
J#3xt?-fk'xp(x)/(x).
->.f#3xJ3x?c-ik*xtjxyx'l/(x/)
(36.17:)
A localpotentialisthen obtained asthelim it
l.(x,x').
-+.n(lx1)3(x- x')
W enow chooseto consideranonlocalseparablepotential.which takestheform
!7(x,x')= l
z(Ixl
)&(1x'I)*
(36.18)
with the Fouriertransform
J#3.
xt'-tk*xu(x)= ulk)
ltisevidentthatthe only Iocalseparablepotentialisa deltafunction
r(x)= v3(x)
butwemayexpectournonlocalapproximationEq.(36.l8)toprovideareasonable
description ofa short-rangepotential.
!H.A.Betheand J.Goldstone,Proc.Roy.Soc.tf-t/alt?al,A238:551(1957).
323
CANONICAL TRANSFORMATIONS
W iththeseparablepotential,Eq.(36.16)becomes
x.2 -
k2= àpr-lu(k)jd3xu(x)*4p,k(x)
(36.20)
SubstitutionofEq.(36.15)intherightsidethenyields
<2 -
u(k)12
k2= l1
t
1
jz +àjpgd3,
a.
),ultj*sa.
jzult)(xg-s;
(36.21)
which m ay berearranged asfollows
1 1 Il4k)I2
d5t 1l4f)l2
;
î=f',2-k2Xlr(2
,
43,2-t2Kfç'
cg)
(36.22)
Thisequation determ inesthe eigenvalue s2,and hence the energy sltiftN rpair
through the relation
LE = hlv-1(s2- k2)
(36.23)
r
k
JP
kr
Fig.36.1 Integrationregioninmomentum spacefor
Bethe-Goldstoneequation.
V
IJP :
i:k1<kr
Equation(36.22)ismosteasilystudiedgraphically,andwedenotetheright
sidef(<2),although italso dependsparametrically onP and k. The integral
in f(<1) decreases monotonically as K2 increases,becoming logarithmically
singular when the denominatorcan srstvanish. The integration region l''is
illustratedinFig.36.1,whichshowsthatthisdivergenceoccursatx2= k/- $#2.
Inaddition,thefrstterm of/(x2)issingularatKl= k2,and itisnow easyto
sketch ftsz)asshown inFig.36.2. Thesmooth background curverepresents
-
the integralterm,which isindependentofk2. Furthermore,the Erstterm of
/(x2)contributesonlyin theimmediatevicinity ofk2,Gcauseitscoeëcientis
proportionalto F-!and thusbecom essm allfora m acroscopic system . A sa
result,/(s2)consistsofa singularity with narrow width atthevariablepoint
s2= k2,superposed on thebackground curve thatislogarithmically singularat
K2= k/- jPZ.
Theeigenvalueisdeterminedbytheintersectionof/(M2)withthehorizontal
line A-i. Itis evidentthat there is only one solution forA > 0,occurring at
K2 -
k2;
k;àF-1jutkljz+ &(à2)
(36.24)
324
cAsoNlcAu TRANSFORMATIONS
Here the integral term has been neglected,because itis fnite at<2= k2and
contributes only to higher orderin A. W e conclude thatthe only esect ofa
repulsivepotentialistoshifttheenergyofthepairbyasmallamountproportional
to Ilz(k)l2F-1,asexpected foran interacting medium. In contrast,Fig.36.2
showsthatan attractive potentialA < 0 alwaysleads to /wt?solutionsforeach
k2,aslongasIAIdoesnotbecometoolarge. Thusanattractiveinteractionalters
the energy spectrum in a qualitative m anner. Although the ordinary solution
ofEq.(36.24)stilloccurs,wealso5ndanew (anomalous)solutionarisingdirectly
from thelogarithmicsingularity of/(s2). W hich ofthe two eigenvalueslies
lowerdependsontherelativevalueofkland IAl. IfIA1issxed,thenthereisa
Fig. 36.2 Eigenvalue condition Eq.
(36.22)forBethe-Goldstoneequation.
corresponding criticalvalue kc
l such thatthe anom aloussolution is the lower
eigenvalue for a1lk > kc,while the ordinary solution islower fork < kc. A n
approxim atevalueforkcisobtained from thesolution ofthe equation
(2.
*-3Jrd3t1
N(1)1
2(/
cJ-/2)-i=A-1
(36.25)
which istheintersectionofthelineA-1withthepartoff(Kl)arising from the
integralinEq.(36.22). Itisclearfrom Fig.36.2thattheanomalouseigenvalue
isessentiallyindependentofk2,and thustheground-stateenergy of thepairis
independentofitsinitialrelativewltl
cvectork asIong ask >kc. Thisbehavior
isverydillkrentfrom thatoftheordinarysolution(36.24),where<2;kJk2apart
from correctionsoforderF -1.
W e shallnow study the anom alous eigenvalue in detail. The srst term
ofEq.(36.22)isnegligibleunlesskisexactlyequaltokr,sothattheeigenvalue<
isessentiallyequaltokcandobeysjustthesameequation(36.25):
I
A1-1-(2*)-3Jy,d5tlu(f)l2(/2-a:
2)-1
(36.
26)
CANONICAL TRANSFORM ATIONS
:26
Although thisequationcan bestudied foral1# < 2ks,itissim plestto setP = 0,
whentheeigenvaluecondition becom es
1 * t2dt11
z(/)12
$jI= kr 2=2t2- <2
kF *xcdx :I
?z(ksx)I2
- 2 z
= I
x - (s/ks)z
(36.27)
Thelogarithm icsingularity ofthisintegralcan beextracted through an integration by parts
F I?
Q11l;k$4k.2
z(/fF)Iclnk/kl<c
-
(36.28)
InarrivingatEq.(36.28)thepotentialhasbeenassumedtobeasmoothfunction
ofthe m omentum and the remaining snite integralhas been neglected. Since
Klislessthank/,wewrite
Kl =
pt'/- möh-l
(36.29)
and asim plerearrangem entyields
l h2ky4=2
= m exp - /
cslA1IN(k21z
(36.30)
Asnoted above,Eq.(36.29)determinesthe ground-stateenergy ofthe pair
wheneverk > kc. Thecorrespondingexpression for.
(
ïhasseveralveryremarkable features:
1.TheenergyshiftofthepairLE = â2?.
n-l(x2- k2)= 2(eks- 6k)- :
'
Xisnegative
nearthe Ferm isurfaceand isindependentofthevolum e.
2..
t
ïhasan essentialsingularityinthecouplingconstantandcannotbe obtained
with perturbationtheory.
é(#)isgreatestforthosepairswithP = 0.becausethephasespacewherethe
denominatorof(36.26)vanishesisthen maximized. If# = 0,weseethatt
attains its m inim um value everywhere on the surface ofthe Ferm isphere
(Fig.36.1),
.forsnite1Pt,however,thisvalueoccursonlyonacircleofradius
(k/- $#2)1.
4. Theoccurrenceofa bound pairforan arbitrarilyweakhnite-rangeattractive
potentialdependscrucially on the presence ofthe m edium ;two particles in
free space willnotform a bound state unlessthe strength ofthe potential
exceedssome criticalvalue. This resultalso can be seen in Eq.(36.30),
because'
â vanishesexponentially askF -.
>.0.
Theforegoingcalculation im pliesthattwoparticleswith oppositemomenta
and spins nearthe Ferm isurface willform a bound pair,as long asthe inter-
z26
CANONICA: TRANSFORM ATIONS
particle potentialis attractive.l ln thisway,the system lowers itsenergy by
an amounté,and the originalunperturbed ground state clearly becomesunstable. Unfortunately, Cooper's model is restricted to two particles'
, it is
thereforeincapableofdescribingthe new ground state,which evidentlyinvolves
m any bound pairs.
Nevertheless,thecalculation hasprovided a qualitativedescription ofthe
instability,and italso indicatesthat the new ground state cannotbe obtained
with aperturbation expansion. InSec.37weshow how theBogoliubovcanonica1transform ation allows us to study the m any-body ground state of such a
system .
37UINTERACTING FERM I GAS
W e now discusshow theformation ofCooperpairscan beincorporated into a
consistentmany-body theory. The basic idea isthatpairing between particles
inthestates(kt)and(-k))canmaketheFermiseaunstableiftheinterparticle
potentialis attractive. ln consequence,these states play a specialrole, and we
thereforem ake thefollowing canonicaltransformationz
=k= 1z..Jk!- rkt
/-kt
8-k= &kJ-ktV'V Yk!
,
(37.1)
Thec-numbercoeëcientsukandrkarerealanddbpendonlyon I
k(. Thislinear
transformation iscanonicalifand only ifthe new operatorsobey the relations
lt
zk>o1'l=(Vk,X'l=Qk'
Allotheranticom m utators= 0
(37.2)
G iven theoriginalanticom mutation relations
1/kâ,&1'â'1=t
skk't
sâ/,
(37.3)
itisreadilyseenthatEq.(37.2)implies
uk
2+ (,
k
2= j
(37.4)
Theseequationscan beinverted to give
Jk?= ukœk+ t'k#1k
J-kt=uk#-k-rkœl
(37.54)
(37.5:)
1In principle,a bound paircan also lx formed by two particleswith parallelspinsin an antisymmetric spatialstate. To the extentthatthe attractive interaction is of shortrange,we
exr< ttheefrectsto l
x largestin (symmetric)relative.
îstates.
2N .N .Bogoliubov,Sov.Phys.-JETP,7:41(1958);J.G.Valatin,Nuovo C'
l'
znealtp.7:843(1958);
S.T.Beliaev,IntroductiontotheBogoliubovCanonicalTransformation M ethod, in C.DeW itt
(ed.),Ab'
f'
heM anyBodyProblem.''p.343,JohnW ileyandSons,Inc.,New York,1959;S.T.
Beliaev.Kgl.Danske Videnskab.Selskab.M at.-Fys.Mee ,31,no.11(1959).
CANONICAL TRANSFORM ATIONS
327
A s in Chap.6,we shallconsider the therm odynam ic potential at zero
temperature D(F= 0,P,p,).which is the expectation value of (compare Eq.
(18.29))
k - # - /z#
=
)(c1zJkA(E2- /z)-'
l' Z tklAIkzlalFIk3A)k4l4)
k2
kll,
kzz
,l,
kl+
.k4
XJ1là,JlzàzJuz.lkzA, (37.6)
The use ofthe therm odynam ic potentialallows usto treatassem blieswith an
indesnite num ber ofparticles. In the end,ofcourse,the chem icalpotential
willbechosentoensurethat(X)= N. Wealsoassumeanattractiveinteraction
potentialF > 0.
Thetherm odynam icpotentialwillnow berewritteninterm softheom rators
akand ju,arranged in normalorderwith a1lthe destruction omratorsto the
rightofa11thecreation operators. Although thisprocedurecan becarried out
directly,it is m uch simpler to use W ick's theorem. Consider the operator
4:ck,!,whichcanbeexpressedasfollows
t/k!Jk'l=X(4!Jk'!)-F't4ïJk't
(37.7)
HereN standsfornormalorderwith respectto the operatorsx and j.1 If
1O)isthenew vacuum characterizedbytheconditions
JklO)=#k)O)= 0
thevacuum expectationvalueofEq.(37.7)yields
(37.8)
t/k'
yG',=(0141zkœl+pkj-k)(uk'Jk'+Lk'#!k11O)
=
ôk.k,p2
k
(37.9)
and sim ilarly
Jtk)Jl.
k't= Q ,k'1ll
ThustheErstterm ofEq.(37.6)becomes
1k2(4 - 8)l2ktJk!+ tzk)&-ktl
(37.10)
=
Zk (4 -rt)(2/71+A(J1tJk!)+N(Jtk)J-k))1
=
X
(4 -M)7171+(u1- &I)(Q ak+#!k/-k)+lukr:tj-kœk+œ1/!.
01
k
(37.11)
1To maketheformalconnection with W ick'stheorem complete,wemayconsiderthex o> l'atorsto betimedependentwiththetimeoftheoperatoron theleftinhnitesimallylaterthan that
on the right;howtver,a littlereqection on thereader'spartwillconvince him thatthisartifice
isunnecessary.
a2:
cAsoNlcAu TRANSFO RM ATIONS
wherethenormalproductshavebeenevaluatedexplicitlywithEq.(37.5).
The potentialenergy in Eq.(37.6)ismore dimcult. Forsimplicity,we
assum ethatthepotentialisspin independent'
.
(k1AIk2A2IFI
k3A3k4A4)= 3A,A,3AaA.tklk2IP'lk3k4)
(37.12)
ln addition,theexplicitexpression
tk1k2Ilz-lkqk4)= F-2JJ#3x#3y('-i(kl*x+ka*y)p'tx>y)eilkz*x-Fk4*y)
showsthatthe m atrix elem enthasthefollowingsym metryproperties:
(kIk2IP'Ik3k4)- (k2kl1Uj
k4k3)= (-k3-k41UI-kl-k2)
(37.13)
(-kj-kzt#'p-k3-k4)
wheretherelation F(x,y)= P-(-x,-y)hasbeen assumed inarrivingatthelast
equality. The potential-energyoperatorcan now berewritten with Eq.(37.13)
as
P= f'
.+ Vi
(37.14)
where
(a= -j.)é tkk'(FIk+ q,k'- q)gt/k,t/k'!Jk'-q!Jk4.
q!
kk'q
(37.l5J)
(77.15bq
Thetwo termsofIh diflkronlyinthesubscriptson theoperators,and
comparison of Eqs.(37.54)and (37.5:) shows thatthe second term can be
obtainedfrom thesrstwiththesubstitutionlau+..
+j-kandt'4-+-r). Wetherefore concentrate on the frst term ; the corresponding operator product can be
rewritten with W ick's theorem
dk
%tf/kz!Jkz-q!Nk-hq!= Xlt/ktt/k'!Ck'-q!Ck/.
q!1V îq,0Dk
2X(t/k'rJk,t)
0
::
2
k+q
.
+ 3q, Xtlf
k!&k!)- 3k', 11:2.XtJ)yJkrl
dk'k+n112
:X(JY
k',au't)-h30oPJ?
7.
2,--8u,k+qr2
:!Jk
2,
(37.16)
--
wheretheonlynonzero contractionshave been evaluated with Eqs.(37.9)and
(37.10). Inthisway,Eq.(37.15/)reducestothefollowingexpression
Pu- #(P2 - '
ikk'
Z (fkk'l#'lkk')- fkk'lFlk'k))
x((ryl?) + 2?
é,.
N'(< TJk!)J+ lak*-+j-k,1
7+->-lC11
CANONICAL TRANSFORM ATIONS
329
which has been simplihed slightly with Eq.(37.13). The normal-ordered
productsinthesecond term havealreadybeenevaluatedinEq.(37.1l),andwe
therefore;nd
Pa= N(P,)- kk'
)((tkk'IFlkk')- (kk'!P'1k'k))(?irj,+ t7j.
(l4 - I72
k)
X(Zkœk+jik/-k)+ Pl(2l4rk)(aljl
-'k+j-k(V)l (37.17)
Theremainingcontribution P:can betreatedinasimilarway. W eneed
thefollowing contractions
kt -k t
t/kiJlk-t= Z-kttïb.T= 0
and theoperatorin f'
,becomes
(37.18)
.
Yk!Y-k'tU-k'-qtflkhq!= XlYk!Y-k'tV-k'-q)Qk+q!)'
Xfq,
0lliXlt
/-kztJ-k't)
+ âq.0&ieA(J1!Jkll+ 3k.k'l
lkWX(D-k-q)&k+q!)
+ îk.k'Nk+qlk4.
qX(Yk!Y-ktl-f-3q,01lP1,
+ îkk,ukt4l
zkl.qL'kq.
q
ThislastequationmaybecombinedwithEq.(37.15:)togive
bh = N(6)- kk'
U((k-k'Jl'J
k- k')r2,(?é+:V(4,Jky)+.
N(t/.
-k)J-kJ))
-
)j(k- k1P-1k'- k')uk,!7k,(h vk+ At(
/krf
/-kt)+ A'tl-ktJk!)1
kk'
wherewe havemade somesimplechangesofvariablesand used Eq.(37.13).
Thevariousnorm al-ordered productsarereadilyevaluated,and we*nd
% = NL%)-kk'
jl(ukpkuk,pk,lk-kjP'Tk'-k?)+ plYtk-k'JP'Jk- k'))
Z ft=lak+#tk/-k)((&1-&l)&i'(k-k'llz
'lk-k/)
kk'
-
-
2uktl
kuk,th,tk-klPrJk'-k'))J- )
g((4#!k+#-kœk)
kk'
x R14 - &l)uk.rk,tk- klP'Ik'- k')+ 2ugvk!?
ltk- k'IP-lk- k')))
(37.19)
Itisnow possibletocombineEqs.(37.11),(37.17),and (37.19)to obtain
the therm odynamicpotential
#- &+41+ zh +x(l
2)
(37.20)
where
& = 2)k((4 - p)!?l- kk
72
(kk'1Plkk')rlr)
'
-
ï (k- ktF!k'- k')ukvkh,t'
k' (37.214)
kk'
330
CANONICAL TRANSFORMATIONE
Xl= Z
(Q ak+#!kl-klllfî-/*-Z
(lkk'lPlkk')?
é,
))(d - 17k)
k
k,
+ luktyI
((k- klFIk'- k')uk,t4-)) (37.21:)
k#
X2=Zk (Q/!k+/-kœk)f2(4 -/*- X
((kk'IPI
kk')tlllllzkpk
k'
-
(l4 - tj))
g((k- k!FIk'- k')uk,tv)) (37.21c'
p
k;
and wehaveintroduced theabbreviation
tkk'lPlkk')> tkk'!P'gkk')- (kk'lP'!
k'k)+ (k- k'JFI
k- k') (37.22'
Itisconvenientto desne anew single-particleenergy
ekY eî- )
j(kk'1F I
kk')t'
)
k'
(37.232
which willturn outto be theH artree-Fock expression,and to m easure nkfrorr
thechem icalpotential
Q H ek- rz
(37.24'
Finally,weintroducetheenergy gap bytherelation
Ake 12
Flk'- k')uk,p:,
k: (k- kl
(37.25'
'
and thevarioustermsofk become
U = 2X
(krl+kk?
72!?Ipltkk')F (kk')- 1)
ukvkzk
k
k
(37.26/)
Pl= Z
lœlck+#!.kl-k)RNl-L7l)1k+2uktkZkl
k
(37.26:)
X2= Z
(œ1#!k+v-kœklE2NktlkL -(Nl-r1)Zkl
k
(37.26t$
ltmustbeemphasizedthatEqs.(37.20)and(37.26)togetherconstituteanexact
rearrangem entoftheoriginaloperator.
Untilthispoint,the only restriction on ukand !7kisthatin Eq.(37.4),and
weshallnow im posetheadditionalconstraint
I(kukllk= 1ktd - ri)
(37.27)
to makePcvanish. Thecondition uk+ t?i= 1ismosteasilyincorporated by
writing
uk= cosy:
!l:= sinzk
and Eq.(37.27)thenbecomes
(ksin2zk= zk cos2zk
(37.28)
(37.29)
331
cAsoNlcAu TRANSFORMATIONS
or
tan2zk= à:fk-l
(37.30)
Simple trigonometry gives
sin2y:= +N Ek
-t= 2ukty
(37.31)
cos2yk= +& Ek
-l= W - rj
whereeithertheupperorlowersignsmustbetaken throughout,and
(37.32)
E.- (ài+ fi)+
Theterm 41maynow berewrittenas
Xl=+zk Zktœlœk'Y'K lk)
(37.33)
which shows that the upper sign must be chosen to ensure thatthe energy is
boundedfrom below. W iththischoice,Eqs.(37.31)and(37.25)become
UkW = 2E
k
-I-j
l(1+jk)
-I-)(l-jk)
é
(37.34)
à:.
k= .
!ï
k - k')Ev
k'(k- kjFI
(37.35)
ThislastrelationistheBCS gap equation,which isa nonlinearintegralequation
forthegap functionAk.l
Thezero-temqeraturethermodynamicpotentialk now consistsofthree
terms& + 41+ N(F),where& isacnumber,41isdiagonalinthequasiparticle
numberoperatorsafxandp%p,and #(P)isanormal-ordered productoffour
quasiparticle creation and destruction operators. This last term m akes no
contributioninthegroundstateofU + 41
(o I#(P)1O)- 0
(37.36)
and it clearly describes the interaction between quasiparticles. For many
assemblies,itisagood approximationto neglect.
N'
(P)entirely,in which case
weobtain
#:> U+Xlr'JU+ )
gZktllak+X xj
k
(37.37)
!n isgapequation wassrstobtained byJ.Bardœn.L.N .Coom r,and J.R.A hriefer,Phys.
Aep..108:1175(1957). Thepresenttreatmentiscloserto thatofBogoliubov,Val
atin,and
Beliaev,Ioc.cit.
=2
CANONICAL TRANSFORM ATIONS
Evenwhen#(P)isnotnegligible,theoperatorkoprovidesabasisforaperturbation expansion,and we shallnow study the properties of #:in detail. Itis
evident that U is the therm odynamic potential of the ground state, while
Ez= (àI+ fi)1 represents the additionalcontribution ofeach excited quasiparticle. Note thatEk> Ak,which accountsforthe namegap function because
theexcited statesareseparated from theground statebyahnitegap. Themean
numberofparticlesinthegroundstateisgivenby(compareEq.(35.23))
N=Z
(0IJZJkAIO)
kz
=
2)
jpi= )
((1- (kE:
-1)
k
k
(37.38)
where Eqs.(37.9)and (37.10)havebeen used to evaluatethematrix elements.
In a sim ilarway,thetotal-m om entum operatorbecom es
f:= U
)âkzkzJkz= 7
)âktf/k,Jk!- Z klJ-kt)
kz
k
=
7
2âk(#(4tckl)-AtltktJ-kJ)1
k
-
7
2âktal
kœk+K /k)
k
(37.39)
whereEqs.(37.9)and(37.10)havebeenusedtoobtainthesecondline. W esee
that
(Xn,ê1= 0
(37.40)
so thattheexcited statesobtained by applying quasiparticle creation operators
a
#andj'to 1O)areeigenstatesofbothknandê.
Further progress depends on a detailed solution of the gap equation
(37.35). Sinceitisahomogeneousequation,thereisalwaysthetrivialsolution
à:= 0 fora11k normalsolution
(37.41)
whichdescribesthenormalgroundstate(flled Fermisea). Thisidentifcation
followsimmediatelybecauseEqs.(37.32)and(37.34)thenbecome
Ek= 1f:1
ukp:= 0
uk-j
1(j.
j.t#
.j
).:(e.-s) normalsolution
,
?1-j
1jj-l
.
(kj
j
)-ocs-0)
(37.42)
whileEq.(37.1)reproducesthecanonicaltransformationtoparticlesand holes
(comparewithEq.(7.34)J. Furthermore,thelastterm ofEq.(37.26/)vanishes
identically,and thetherm odynamicpotentialin theground state reducesto the
Hartree-Fock value studied in Sec.10.
CANONICAL TRANSFORM ATIONS
333
In addition to the foregoing norm al-state solution,thegap equation also
hasnontrivialsolutionswith tx# 0,which weshallcallsuperconductingsolutions.
Asa specihcmodel,assumethatthematrixelementsofthepotentialareconstant
in the region neartheFerm isurface and vanish elsewherez
(k- kfU41- 1)= gV-1#tâoao- lf:l)elhu;n- Ifll)
(37.43)
whereholnisacuto/ introdueedtorendertheintegralsconvergent. Thism odel
isapplicableto m etalswhere the interaction with thecrystallatticecan lead to
an attractiveinteraction between electronsneartheFermisurfacel(seeChap.
12). Inthisway,thepotentialbecomesseparable,andthegapequationmaybe
solved exactly. Itisreadily veritied thatthe gap function reducesto the form
Ak= Zhhœn- llk1
)
(37.44)
w here21 isaconstant,given asthesolution oftheequation
1= g(2F)-1X
olhu)n- îfk()(Z2+ fi)-1
k
=
l.gJd?k(278-3olho)n- l
J:1)(A2+ (:2)-1
(37.
45)
In allpracticalcaseshutnism uch sm allerthan !z,and wem ayw rite
(2,
/8-3#3k= 4r429-3:2dk= S(0)X
(37.46)
where
k
v(0)-2=
1cjap
dkjkkv,
v
(37.47)
isthedensity ofstatesforonespinprojection attheFermisurface. Equation
(37.45)can now beevaluated as
1- gNI?.
I â-'p
2
q
J-,-s(l2d+lJ2).-#Xt0)J.o-'p(u
s2'+
;2)+
lhœn
;
kJgNfQ)ln
(37.48)
wherewehaveretained onlytheltadingterm forhulol.
f
s>.l. A simpletransform ation yields
a-zâoasexp(-:(1
(j)g)
(37.
49)
which exhibitsthesamenonanalyticstructureseenin Eq.(36.30). Fortypical
metals,huto can be taken as a mean phonon energy hulo= k.0 (the Debye
energy)andNlQjgQ;0.2-0.3(seeTable51.14.
Thecorrespondingquantitiesuiand&,
Ibecome
W =.
i(1+ (k(L2+ JI)-+)
superconductingsolution
t'l= .
!.
(1- J:(é2+ J2)-1)
1H.Fröhlich.loc.cit.
(37.50)
CANONICAL TRANSFORM ATIONS
334
uk
1
l
t?2
1
k
> a
8G
'k
Fig.37.1 Distributionfunctiont'
l= 1- 1/1for
superconductingsolution.
andareshownschematicallyinFig.37.1. Sincerïisthedistributionfunction
for quasiparticles.we see that the sharp Ferm isurface ofthe normalstate is
smeared throughoutathicknesstâin energy. NotethatEq.(37.50)isinsnitely
diFerentiable and thuscan neverbe obtained in perturbation theory from the
discontinuousstepfunctionsofEq.(37.42).
Atsxed chemicalpotentialrt,the resulting excitation spectrum of#(
)in
thesuperconducting stateisshown in Fig.37.2,which clearly indicatesthe role
ofthe gap A. In the limit 2î .
-+.0 we recover the excitation spectrum in the
norm al state, shown by the dotted line. The apparent paradox that Ek is
positive,even forek< rzin the norm alstate,iseasily explained by rem em bering
thatallenergies are here m easured relative to thechemicalpotentialorFerm i
energy(recallk M X - p,
X ). ThusthegroundstateofN - lparticlesand one
hole isa Elled Ferm isea containing N - 1 particles,and the creation ofa hole
withe:< J.
tthereforerequiresaminimum energyJ.
t- ek= ffkI>0(comparethe
discussionfollowingEq.(7.61)).1
Itisinterestingto com parethephysicalpropertiesofthenorm alandsuperconducting ground states. ForNxed p.
.the numberofparticles isdetermined
byEq.(37.38),and we;nd
Ns- Nn= 2)
((?
Jj1,- riIa)
k
P'42=)-3J#3/cJk(IJkI-1-((L+Z2)-+j
= PW(0)J#ff(1ft-1- (f2+ é2)-1)
=
0
(37.51)
becausetheintegrand isodd in(. Herewenotethattheonlycontributionto
=
Ek
Sum rconducting
I
l
I
I
NN.1Z-- NOI'ITIaI
#
y
ek
Fig.37.2 Comparisonofexcitationspectrum for
normaland superconductingsolutions.
'lfp.
(N)isdeterminedfrom Eq.(37.38)then Ekistheexcitationenergyat:xedN,asdiscussed
indetailin %c.58.
cANoNlcAu TRANSFORMATIONS
335
the integralarisesfrom the im mediate vicinity ofthe Ferm isurface,and itis
therefore permissible to use Eq.(37.46). As a result,the transition from the
norm alto superconducting statedoesnotalterthem ean num berofparticlesin
thisapproxim ation. A lternatively,ifthe numberofparticlesN isconsidered
éxed,then thechemicalpotentialsdiFerbyonlyaverysmallamounttrzs- ynlf
'y.
n
=
3.
Thequantityofdirectphysiealinterestisthechangein ground-stateenergy
atfixed N
fs- En= UXpa)- Un(p,
n)+ lpa- l
hqN
&s(m)-&n(p.
a)+bl
h(lV
J
.
t)gn+N+0(32)
.
,.
k:&,(p.
n)- UntJznl
(37.52)
wherethelinearterm vanishesbecauseofEq.(4.9). Since & issmall,we need
only computethechangeinthethermodynamicpotentialatlixed /.
zgcompare
thederivation ofEq.(30.72)). Thisexpression iseasily evaluated with Eqs.
(37.26*,(37.34),and(37.42). Assumingthatthematrixelementsf
'
xkk'1F 1kk')EE
gIVareconstantsinthevicinityoftheFermisurface.wehave
Es- En- 2é
)fklrlls- t'il
al- 7k).
'
.
ï(ukrklfs+gV-2)
(((t'inllls- (riLf,)pn1
k
kk'
Si
-Xkl1
s1(fk+(ià-ipg-i
1yktt'
l+
a2
a2).
+ 4'
v7
kk'
)(1
)1-(u+&a2).1()
1-((i,tkspl
(1-j
lkl(,-,
jk.
,!
))
(1
(2
1 12
gU(X(0)j2
;
epwtolfJt'gt
yi-((,.
,.aip-jtlco-spj+ 4
xJfdt#f'tg1-(y2ofaclljgl-(y,c.
f'
ac).1
(1-t
yt
f
)(1-l
#J
'I
))
-
-
-
= -
jVN(% 12= fk - Da
(37.53)
where thedouble integralvanishesby sym m etry.l Recalling ourdiscussion of
:The thermodynamic identity ofEq.(4.3)now allowsan explicitcalculation ofJza-ym.
Assumingasingle-particlespectrum e:Q:6î,weNnd &=r-i.
(A/eî.)2(1+ 6(PlnA/êlnN)),which
justisesthe omission ofthe82correction in Eq.(37.52). Note.however,thatz
V(Jz,- rtJ/
(Es- En)=!.I1+.6(PInA/
'ê1nN)1iscomparablewith oneso thattheseparatecontributions
of order .
3 in Eq.(37-52) are not negligible. If o)o and g are independent of Ns then
6(?lnA/PlnN)reducesto2ln(2âtt)p/A).
3a6
CANONICAL TRANSFORM ATIO NS
Cooperpairs,wecan interpretthisexpression asthe binding energy ,
â perpair
multipliedbythenumberofpairs.
è.U.
NIOI.
XIyingwithintheshellofthickness2î
'
around the Ferm isurface. Itisthepairsin thisshellthatcan lowertheirenergy
by forming theCooperbound state.
Equation (37.53)showsthatthesuperconductingstateindeed hasalower
energy and therm odynam ic potential than the normal ground state, and is
therefore a better approximation to the true ground state or the interacting
system . This conclusion follows from the same variational prineiple that
determinestheground-stateenergy,forEq.(37.36)show'
sthatfq andtlsarethe
expectationvaluesoftheexactoperatorX (Eq.(37.20))inthenormalizedground
states(normalandsuperconducting)ofXc(Eq.(37.37)).
PROBLEM S
10.1. Considera Bosesystem withm acroscopieoccupation ofthesinglem ode
with m om entum h%. Find the depletion ofthe condensate as a function of
&'= hzjm assuming the pseudopotentialmodelof Eq.(35.1). Compute the
totalmomentum P,and compareitwith thevalueNnhz (compareProb.6.6).
10.2. C onsidera densecharged spinlessBosegasin a uniform incom pressible
background. U sing a canonicaltransform ation,show thatthe depletion and
ground-stateenergyaregiven by (n- n()/n= 0.21lrfand EOIN = -0.803rs*el(
2J(). Intheseexpressionsr,
3= ?jn=natandtzl= hljmpel,wheremeistheboson
mass(comparewithProb.6.5).
10.3. Treat the particle noneonservation arising from the substitution
av-->.NtbymakingaLependretransformationtothethermodynamicpotential
atzero tem peratureX = H - p,X. Assuming a nonsingularpotential,carryout
a canonicaltransformation;rederive Eqs.(21.5),(21.8),(21.15),and 5nd the
ground-stateenergy.
10.4. ((8 Solve Eq.(36.26)forP.c
tlky,and show thatthe binding energy
LIPIforapairwithcenterofmassmomentum hpisgivenbyé(#);4sA(0)- ht'yp,
wheret's= hkr/m and /
.
X(0)isgiveninEq.(36.30).
(b) lf:â
'(0)/ks;4JIOQK,estimatethecriticalvalueofP whereà(#)vanishes.
10.5. Show thatthe anom alouseigenvalue corresponds to a solution ofthe
hom ogeneous Bethe-G oldstone equation.
(J) UseEqs.(36.15)and(36.18)tofndtheasymptoticform ofthewavefunction
4o,k(x)for a bound pair nearthe Fermisurface with P = 0;explain why it
difersfrom theusualexponentialform .
(b)Show thatthe form factor(Fouriertransform ofthedensity)forthisstate
isF(q);
k:1- hvrqllh asq-+.0.Interpretthisresult.
(c)lfA/k,;
k;l0OK,estimatethepairsizeand comparewith thecriticalwavenum berderived in Prob.10.4.
CANONICAL TRANSFORMATIONS
337
10.6. Com pute the expectation value ofthe operatorX 2in the ground state
)0)ofEq.(37.8),andshow thattheQuctuationsaregivenby
'N-2x
Y- (ukl-k)2
y;N-)2 = -..j
w
.
(X /2
Z t.'k2 2
k
Discuss the difference between the norm aland superconducting ground states.
10.7. Com pute the pairing am plitudes Fk
*- '
.O '
t
/k-t/- k: O ' and Fk>
t'O f
J-klt
zk!10 '
,hin theground State ofEq.(37.8). Sketch their behavior asa
.
function ofk.and show thatthey vanish in the norm alground state.
10.8. Reducethecorrelation function in thesuperconductingground state
C)A.(x.x')==..0 t
liatx)gé,(x')io '
to detinkte integrals. Evaluate the expressionsforantiparallelspins,and com pare w'ith the corresponding situation in the normalground state.
10.9. The superconducting ground state u'
as originally derisr
ed U ith a s'ariationalprinciplel
tby considering the state
!
%',= l-k1(?
.
/k+ t'
kt
/k:Y-kt)10)'
'
where the productisover a11k,and '
!0 isthe no-particle state.
)isnorm alized if ug
$ z-t'k2= 1.
(t8 Show thath%h,
.
(b) Show that the expectation value of X gEq.(37.6)1in this state is t.'gEq.
(37.21J)J.
(c)Varying s?kand l;ksubjectto theconstraintu2.-@'
2= l.show thatthegap
equation (37.35)isthecondition fbrminimum thermodynamicpotential.
(#) Apart from normalization,verify thatt
/k!,
'ç7).and J-k. (/' both represent
the same state which isorthogonalto (f
p),. Evaluate theexpectation value of
X in thisstateand show thattheincrease in the therm odynamicpotentialisfk.
part five
A pplications to
P hysical S ystem s
11
N uclear M atter
The study ofatom ic nucleirepresentsan im portantapplication ofthe techniques
developed in the preceding chapters. The detailed properties of hnite nuclei
are discussed in Chap.lf. This chapter,however,concentrates on the sim pler
problem of understanding the bulk properlies of nuclei(nuclear matter) in
term softhe interaction between two free nucleons. h'
'e introduce thediscussion
by giving a very brief review of the nucleon-nucleon force and by precisely
desning nuclearm atter.
381N U CLEAB FORCES : A R EVIEW
In thissection wesum m arizethem ainem piricalfeaturesofthenucleon-nucleon
interaction.
l. Attractive:The existence ofthe deuteron w ith J = 1 and even parity
indicatesthattheforce between the proton and neutron is basically attractive,
341
a42
APPLICATIONS TO PHYSICAL SYSTEMS
atleastinthespin-tripletstate(thatis,the3&1state). Furthermore,theinterference between coulom b and nuclear scattering in the proton-proton system
showsthatthenuclearforcebetweentwoprotonsinthe1u
L stateisalsoattractive.
Finally,itisclearfrom theexistenceofstableself-bound atomicnucleithatthe
interaction between any two nucleonsisessentiallyattractive.
2.ShortrangekForincidentnucleonenergiesupto œ 10M eV inthecenterof-m om entum fram e,thediFerentialcrosssection forneutron-proton scattering
is isotropic. W e therefore conclude thatscattering occurs in relative J-wave
states. This resultallows a rough estim ate ofthe range ofthe nucleon-nucleon
force from the classicallim iton the m axim um angularm om entum â/max= rp
that can contribute to the scattering am plitude. Substituting the relation
betweenenergyand m om entum gives
zrnreuE +
E
+
1..,-r y, =rtFermi)40uevl
(38.1)
,
wherernreaisthereduced mass,and thefollowing relationshavebeen used
1Fermi= 1F H 10-13cm
:2
=
20.8MeV F2
(38.2)
(38.3)
Equation (38.3)isa very usefulresult,foritsetstheenergy scalein nuclear
physics. Sincefmax< 1forenergiesup to 10 M eV,itfollowsfrom Eq.(38.1)
thattherangeofthe nuclearforceis
rQ;few Fermis
(38.4)
3.Spin-dependent:The neutron-protoncrosssection c,
,pism uch too large
atvery low energiesi
csp(0)= 20.4barns= 20.4x 10-24cm2
to arisefrom apotentialchosen to fitthepropertiesofthedeuteron. Since the
m easured neutron-proton crosssection is the statisticalaverage ofthe triplet
and singletcrosssections
aip= i(3t7'
)+ Xic)
(38.6)
itfollowsthatthe singletpotentialmustbediferentfrom thetripletpotentialof
thedeuteron. A low-energyscatteringexperim entm easuresonlytwoparam eters
of the potential. These can be taken asthe scattering length a and efective
rangeradesned by
k
1
2
cot3a= -+ èrnk
a
(38.7)
1M .A.Preston,S.physics ofthe Nucleus,''p.25,Addison-W esl
ey Publishing Co..Reading,
M ass-,1962.
NUCLEAR M AU ER
343
where3:istheâ'
-w avephaseshift.l A nextensiveanalysisoflow-energyneutronProton scattering yieldsthe follow ing param eters2:
'J =
23.71c!c0.07 F
àa = 5.38 ...+ 0.03 F
'ro= 2.4 + 0.3 F
3ro= l.71 u
i
n0.03 F
-
(38.8)
Thesingletstatehasa verylargenegative scatteringlength and thereforejust
fails to have a bound state. (A bound state atzero energy impliesa = -cc.)
In contrast,thetripletsystem hasonebound state,thedeuteron,with a binding
energy of2.2 M es'. A lthough there isa large diflkrence in scattering lengths
and zero-energy crosssections,the singletand tripletpotentials are in factrather
sim ilar,both essentially having a bound state atzero enerp '.
4. Noncentral:Since the deuteron has a quadrupole m omentathe orbital
angular m om entum cannot be a constant of the m otion. ln fact the ground
state ofthe deuteron m ustcontain both l= 2 and l= 0 to yield a nonvanishing
quadrupolemoment(theeven parityforbidsl= 1). Hencethenucleon-nucleon
potential cannot be invariant under rotation of the spatialcoordinates alone.
The most generalvelocity-independent potential for spin-l particles that is
invariantunder totalrotations generated by J = L - S and under spatialreoectionsisgiven by
x EB x j- x2
where the tensor operatorisdesned as
s'j2H 3(cj..
f)(nz.f)- cj.nz
(38.10)
Anyhigherpowersofthespin operatorseanbereduced totheform ofEq.(38.9)
u
through the properties of the Pauli m atrices. The total spin of the nucleon-
nucleonsystem isgivenbyS = !.
(cl+ cclaandthesquareofthisrelationyields
3 singletstate,,
S'= 0
(38.11J)
G I*l 2 =
+1 tripletstate,S = 1
(38.l1:)
The totalhamiltonian constructed with Eq.(38.9)issymmetric underthe inter-
change ofthe particles'spins,w'hich m eansthatthewave function m ustbe either
symmetric (5'= 1)orantisymmetric(5'= 0)underthisoperation. As a results
the totalspin S is a good quantum num ber forthe two-nucleon system . Since
thesingletwavefunctionlyisannihilatedbythespinoperator,1.(c1+ ca)lx= 0,
1Fora review'ofefrective-rangetheoryseeL.1.Schifr.''Quantum Mechanics.''3d ed.,p.460.
M cGraw-HillBook Company,New York.l968.
2M .A .Preston.op.cit..pp.26-27.
3*4
APPLICATIONS TO PHYSICAL SYSTEM S
itfollowsfrom Eq.(38.10)that
S12ly= 0
(38.12)
Thusthe tensoroperatorannihilatesthesingletstateand actsonly in thetriplet
state.
5. Charge independent:The nucleon-nucleon force ischarge independent,
which m eans thatany two nucleonsin a given two-body state alwaysexperience
the sam e force. The Pauliprinciple,however.lim its the neutron-neutron and
proton-protonsystem sto overallantisymm etricstatesbecausetheyarecom posed
oftwo identicalfermions. A com pletesetofstatevectorsfortwononinteracting
nucleonsisobtained by specifyingthem om entum ofeach nucleon and the spin
projection Ipj5'lpclz). In the interacting system there are stilleight good
quantum num bers,which can be taken to betheenergy,totalangularmom entum ,
z-projection ofthetotalangularmomentum,thespin,theparity,and thethree
components of the center-of-mass momentum, 6EJMJSxpcml. The parity
ofthe variousstatesarisesfrom the behaviorunderspatialinterchange,w hich
need notbethesam easthebehaviorundercom bined spatialand spin interchange
(particleinterchange). These relationsare shown in Table 38.lalong with the
typesofpairsthatcan exist in any ofthe states. C harge independence im plies
thatthe forcesare equalin those statesthatcan be occupied by a1lthree kindsof
pairs:nn,pp,and np. ltisim portantto realize,however,thatcharge independence does not im ply the equality ofscattering am plitudes and scattering cross
Table 38.1 Low /states ofthe nucleon-nucleon system
States
Particle interchange .
Particles
N#
sections forthe various pairs.since the statesavailable arerestricted by the Pauli
principle. Forexam ple,atIow energy we have
(38.13X
-
t/(:5'
c)?
.2
(38.13:)
346
NUCLEAR M ATTER
and charge independence m erely requiresthat
L)r('ékollnp--E?r(1éko)1,p--(?C(1éko))an
(38.14)
6. Exchange characterï As the energy increases, m ore partial waves
contribute to the scattering am plitude and the analysis becom es very dië cult.
A tsuë ciently high energies,however,the Born approxim ation suppliesa useful
guide to the diserentialcrossseetion
da
Im ,
tfc.
z
rm
.
(+) j
g
yy 12
(38.l5J)
dûcm=jxjzjc-f
kr*xôz(x)/kj(
.
x) a'
j
i2
df'
j ='I4v
'ât'. cfq'x k'(x
')J3x1
Cm
(38.15:)
where
q2u (ky- ky.
)2= 4k1sin2(à.é?j
(38.16)
Forlargemomentum transferhq.theintegrandinEq.(38.15:)oscillatesrapidly,
and the Fouriertransform willtend to zero. Thusthe scattering from a potential
P'(x)should yield a diflkrentialcrosssection thatfallsof'
fwith increasing (
3.
In contrast, the diflkrential cross section for neutron-proton scattering at
laboratory energies up to 600 M eV is shown in Fig.38.1, N ote that there isa
greatdealofbackward scattering;indeed,the m ost im pressive feature of these
results is the apparent sym m etry about 90*. If this sym m etry is exact
ffl.
rr- é?)=.
/'(#)J,thenonlytheeven /'scontributetothescatteringamplitude,
forodd l'sw illdistortthecrosssection. To explain thisbehavior,theconcept
ofan exchangeforcehasbeenintroduced. Thisexchangeforcedependson the
l(X)
'
j
xev
l4.1-'1
1
7.
8
6
9$
27
42
R
#
137
b l0
N
%
Q
300
Fiq.38.1 Experimentaln-p diFerentialcross
section in the center-of-momentum system at
M .A .Preston,e'PhysicsoftheN ucleusy''p.92,
Addison-*lesley Publishing Co., Reading.
M ass..1962. Reprinted bypermission.)
172-21j '
130
15
7
400
various laboratory energies (in MeV). (From
3:0-380 92
3
1
0
400
580
215
K
80
110
0cmdegrees
160
346
AppulcATloNs To PHYSICAL SYSTEMS
sym metry ofthewave function and iswritten as
I
''sr= P'(x)PM
(38.l7)
wherePM istheMajoranaspace-exchangeoperatordehnedby
Pnib@(xf,x.
)= +(x,,xï)
(38.18)
W hen operating ona stateoforbitalangularm om entum /,wehave
PM 1$m(.
Q)H F.,,(-.
Q)= (-1)1Fî,
n(.
f)
(38.19)
and the odd lin the scattering amplitude can therefore be elim inated with a
Serberforccdefined by
lz'> I
,'(-v)J(1+ PM)
(38.20)
10O
M ek'
97
.
V 1
.
-- -
-
b
29.41:
1
!5
j4
9F
25.
?
31.8 30.1
*
.4
51
2
N 10
u5
.
Q
.
188
19.8
7o
Fig.38.2 Experimentahp-p diflkrentialcross
section in the center-of-npomentum system at
j7o-4jp jy4 yI various laboratory energi
es (in M eV). The
f
or
war
d
peak
i
s
due
t
o
coul
omb scattering,
r
t
1' I 1 I i . 1 . â .t .: j I (FronnM .A.Preston,**physicsoftheNucleus,*'
0
30
60
9û
p.93,Addlson-W esleyPublishingCo..Reading,
#cm degrees
M ass.,1962. Reprinted by permission.)
w
330?4: ,7:.4:9
*
)h.
5.:5
*147
Thedifferentialcrosssection forhigh-energy scattering from such a potential
can be calculated in Born approxim ation
(38.21)
and is evidently sym m etric about900. Phase shiftanalysesconfrm thatthe
nuclear force has roughly a Serber exchange nature and is weakly repulsive in
theodd-/states.k
7. Hard core:Thepp diFerentialcrosssection forlaboratory energiesup
to 500 M eV isshown in Fig.38.2,whereitisplotted onlyfor8cm< '
zr/2 because
tSee.forexample.M .H.Hull.Jr..K .E.Lassila.H.M .Rugpel,F.A.M cDonald,andG . Breit,
Phys.Ret'
.,l22:1606(196l).
NUCLEAR M ATTER
347
the identity ofthe particlesrequires
dc('
c - #)
kij
Jc(é?)
cm
àti cm
(38.22)
A lthough thedifl
krentialcrosssectionsforthenp andpp system slook com pletely
diflkrent,itispossibletom akeacharge-independentanalysisof-theseprocesses.l
The isotropy of the nuclear partof the pp crosssection m ightsuggestthatonly
s waves contribute even up to these high energies. This conclusion can be
ruled out,however,by the unitarity lim it/
:'-2 on the u
v-wave diflkrentialcross
sectfon,whfch fssmallerthan the observed 4mb/sr. The hfgherpartfazw'
arzes
m ustthereforeinterfere to givea llatangulardistribution. ln particular,Jastrow
observed thata hard core in the singletpotentialwould changethesign ofthe
y-wave phase shift at higher energies. The :D -iS interference term in pp
scattering could then givea nearly unifbrm distribution.z (W ith a Serberforce
inpp scattering,theonlycontributing statesoflow /are iSv,1.
Dc,and so forth.)
Jastrow 's suggestion was subsequently confirm ed by detailed m easurements.
which show thatthe y-wave phase shift becomes negative atabout 200 M eV
and indicate thatthe singlet nucleon-nucleon potentialhas a hard core with a
range
(38.23)
Further phase-shift analyses im ply the existenee of a sim ilar hard core in the
tripletstate.3
8.Spin-orbitlhrce:Largepolarizationsofscattered nucleonsareobserved
perpendicular to the plane ofscattering. These eFectsare dim cultto explain
withjustcentralandtensorforces,andanadditionalspin-orbitforceofthetype
p-==-p'
x.ta.s
(38.24)
is generally introduced to understand these polarizations. The spin-orbit
operatorcan be written
(38.25)
Itisobviousthatthespin-orbitforcevanishesinsingletstates(5'= 0,I= J)and
also in Jstates(/= 0,S = J). TheusualphenomenologicalU,.hasa veryshort
range. Thusthespin-orbitforce iseflkctiveonly athigh energy,foritvanishes
in â'states,and the centrifugalbarriertendsto keep the high partialwavesaway
from the potential.
In sum m ary,ourpresentem piricalunderstanding ofthe nucleon-nucleon
force isthe following:
tIbid.
2R.Jastrow,Phys.#t?t'..81:165(1951);seealso M .A.Preston,op.cfJ..p.97.
3See.forexample.R.V.Reid,Jr.,Ann.Phys.(N.F.),50:411(1968).
348
Appulcv loss To pHYslcAL SYSTEM S
1.Theexperimentaldata can be fitup to ;
k;300 M eV with a setofpotentials
depending only on thespinand parity 1P'),3F+
C,1/'C
-'3Vc:3jz't'3p's,and
so forth.
2.Thepotentialscontain ahard corerc;
kJ0.4to0.5 F.
3.The forcesin the odd-/statesare relatively weak atlow energiesand on the
averageslightly repulsive.
4.The tensorforce is necessary forthe quadrupole mom entofthe deuteron.
5.A strong short-rangespin-orbitforceisnecessàrytoexplain thepolarizations
athigh energies.
The best phenom enological nucleon-nucleon potentials are those of
H am adaand JohnstonsltheY alegroup,zand Reid.3
39LN U C LEA R M ATTER
Nuclearscatteringofshort-wavelength electronshasbeen studied extensivelyby
H ofstadter and his collaboratorsi; these experim ents form the basis for the
currentpictureofthesizeand chargedistribution ofnuclei.
NUCLEAR RADII AN D CHABGE DISTRIBUTIONS
A phase-shiftanalysisofelastic electron scattering indicatesan averagecharge
distributionofthetypeillustratedinFig.39.1andgivenbyp= pa(1+ ctr-R'/J)-1.
L
F()
1.
0
0.9
0.
5
t
R
0.1
0.
0
' Fig. 39.1 Tiw nuclearcharge-densitydistribution.
r
Herea determinestheskin thicknessand R isthe pointwherep= Jpc. The
param eters show the following system aticbehavior:
1.Thecentralnucleardensity desned by APZIZ isconstantfrom nucleusto
nucleus.
2.TheradiusR isgiven by
R = rft.
,
11
withro;
kJ1.07F
(39.1)
'T.Hamadaand1.D.Johnston,Nucl.#/1#.
ç..M :382(1962).
2K.E.Lassila,M .H.Hull.Jr.,H .M .Ruppel,F.A.M cDonald,and G.Breit,Phys.Rev-,
126:881(1962).
3 R . V.Reid,Jr..Ioc.cit.
4R.Hofstadter,Rev.M od.#/I.y'J.,2,
$:214(1956);seealsoR.Hofstadter,'*ElectronScattering
andNuclearandNucleonStructurey''W .A.Benjamin,Inc-,New York,1963.
NUCLEAR M ATTER
349
W eshallestimatethevolume P'ofthtnucleusas4*R3/3. Asan immediate
consequence,the particle density in nuclearm atter
d
.'
3
#'= 4 = 1.95x 1038particles/cm3
,
rrrg
isaconstantindependentofthesizeofthenucleus. (Thisisnottrueinatoms,
forexample.)
3.The root-m ean-square radiusofthe protonlis rp;
kr0.77 F,while the m ean
interparticledistancein nucleim aybecharacterized b)'
p = l-3
w'ith /,
t J.73 F
Since/> lrp(butnotbyverym uch),wemayhopetounderstandtheproperties
ofnucleiby exam ining the behavior ofa collection ofnucleonsinteracting
throughtwo-bodypotentials. W'eshallcertainlyproceed underthisassum ption,butitm ustbe rem em bered thata11ofnucleartheory dependson this
verybasicapproxim ation.
4.The surface thickness t= 2cln9,desned to be the distance oser Bhich the
chargedensityfallsfrom 90 to 10pereentofitsvaluepoattheorigin,isfound
to be
for nucleiranging from :254g24 to g2Pb208.
W e m ustem phasize thatelectron scattering m easures th: proton distribution or charge distribution,and the m atter distributien need not be identical.
The nuclear force extends outside of the charge distributfon :therefore.purel)
nuclearm easurem entsgenerally yield slightly largerm ean-squareradii.
TH E SEM IEM PIRICA L M ASS FO RM ULA
W e next study the energy ofa nucleuscontaining al nucleons.A'neutrons.and
Z protons. A firstapproximation.suggested by B'eizsxcker,zisto considerthe
nucleusa liquid drop. lfa drop contains twice as m uch liquid.then there will
be twice asm uch energy ofcondensation.orbinding energy. Thisresultm eans
thatthenuclearenergy m usthavea term oftheform
Fj= -tz1,4
(39.5)
which isknown asthebulk propertl'ofnl
tc/t'lrmatter. Thereare,ofcourse,
m any other contributions to the total energy. The nucleons at the nuclear
surfacewillbeattracted onlybytheparticlesinside,leadingto asurface tension
1E.E.Chambersand R.Hofstadter.Phys.Rtu'.,103:1454(1956).
2C .F.vonW eizsàcker, Z.#/1yç/
*
:'.96:43l(l9351.
3s0
APPLICATIONS TO PHYSICAL SYSTEMS
and asurfaceenergythatdecreasesthebinding. lfc isthesurfacetension,this
surfaceenergycan bewritten as
E2= 4=R2tz= 4=rkaAk% aga#l
(39.6)
which varieslinearly with the surface area or .
x9'. There is also the coulomb
interaction ofZ protons,which can be calculated approxim ately by assum ing
thechargesto beunifbrm lydistributed overa sphereofradiusRcM ?bcWl. A n
elementary integration from electrostatics then yields the interaction energy of
theèztz - 1)pairs
f 3z(Z - 1)e2 3 e2 Z(Z - 1) Z2
3= '
j Rc = '
j rac a.
j ;kê/3al
(39.7)
Som e nucleareflkctsm ustnow beincluded in theenergy. First,wenote
em pirically thatnucleitend to have equalnum bers ofneutrons and protons
N = Z,and a corresponding sym m etry energy .54 willbe added to the m ass
form ula. The bulk properties of nuclear m atter imply that twice as m any
particleswith thcsameratio ofNIZ willhavetwicethesymmetry energy. This
observation suggeststhatE4isproportionalto .4. A sa Erstapproxim ation to
thenum ericalcoeë cient,weshallretain onlythequadraticterm inanexpansion
aboutequilibrium ,and wehnd
1
z 2
c
f4= c i- N + z ,1= 4.4zt.4- 27)2,4
H c4X-1(A - 2Z)2
(39.8)
Nextwenoteempiricallythatnucleitend to havcevennumbersofthesamekinds
of particles. For example, thert are only four stable odd-odd nuclei lH2.
3Li6, jB10,and 7N14. For odd-,l nuclei,there is at most one stable isobar
(nucleuswithagiven,d),whileforeven.
,
1theremaybetwoormorestableisobars
with even N and even Z . The experimental energy surfacesdescribing the
situation fora seriesofisobarsare shown in Fig.39.2. W ithin such a series,
f
odd ,
4
Even ,
4
A
A
cdd-e d
e.C.
''x.
#- #-e
,c.#+
#+.
,Z
N
e C.
#-
R
#- ##
C C.
##
.
CVCn-CVCn
.
l ..l
I. I
--IL .t:
I
I
1
1
1
)..
Z
Z
Fig.39.2 Odd-W and even-A energysurfaces,showingtheallowed
jdecays(e.c.standsforelectroncapture).
NUCLEAR M ATTER
x1
theonlypossibletransitionsarebyelectron orpositronem ission,and byelectron
capture. (ltis a generalrule,following directly from energy conservation,
thatoftwonucleiwithZ diseringby1andthesameW,atleastoneisj unstable.)
Two adjacenteven-even nucleion thesameenergy surfacecan both bestable
becausetheonlyallowed transition between them would l)ea double# decay,
which isbelieved to beabsentoratleastextremely rare. Thesplitting ofthese
energy surfacescan now be included in the massformula by adding apairing
energy oftheform
E5= XB,4-ê
(39.9)
TheparameterA isdefned by
Ae
+1
0
odd-odd
odd-even
1
even-even
-
(39.10)
while the .
,
4-1 dependence is empirical. Com bining these resultsleadsto the
W eizsâckersem iem piricalm assform ula
E = -Jla
d+ azW1+ aszlW-1+ J4ta
d- 2Z)2W-1+ M jA-k
(39.11)
where thefollowingbest-stparametershavebeen given by Greenl'2
Jj= 15.75 M eV
a.= 23.7M eV
az= 17.8M eV
as= 34MeV
(39.12)
as= 0.710 M eV
Thisexpression hasonly two termsdepending on Z:the coulomb energy
andthesym m etryenergy. ThestablechargeZ *can befound bydiferentiation,
OE/DZIA= 0,whichimplies
*' 21 -î
Z *= W 2+ *
*301 *
'
2J4
(39.13)
Forsmallnucleitheequilibrium valueisZ*= a4/2,whichisindeedobserved up
to ,4 ;
k;40. Thisexpressionclearlydoesnotaccountforlocalvariationsdue to
shellstructure around this equilibrium value. N evertheless,the overallfit is
excellent.
Equation (39.11)isvery usefulin studiesofsssion,foritallowsoneto
determ ine when theenergy oftwo separatepieceswillbelessthan theenergy of
theoriginalexcited nucleusand even to calculatetheenergy releasein thesssion
process. Itisalso usefulinpredicting massesofnew nuclei.
'A.E.S.Grœn,Phys.Rev.,95:1(G (19541;A.E.S.Grx nandD .F.Edwards,Phys.Rev.,
91:46(1953).
2Thecoemcientasimpliesthatr4
k= 1.22F Isœ Eq.(39.7))inagrxmentwith themuivalent
value measured inelK tron scattering.
asa
AppulcA-rloNs To pHyslcAL SYSTEM S
W ecannow deineasubstaneeknown asnuclearm atter. Ifwelet.
,
4 -+.co
in Eq.(39.11)and atthe same time setN = Z and turn offthe electric charge,
thenthisnuclearmatterhasa constantenergy/particlegiven by
E
li;4J-15.7MeV
(39.14)
Thfs value therefore represents the energy/particle ofan insnite nueleus with
equalnum bersofneutronsand protonsbutwithnocoulom b efrects. Equations
(39.14)and (39.2)exhibitthesaturation t?/'nuclearw/brctaé',becausethe binding
energy perparticleand the nucleardensity are both constantsindependentofA .
N uclearm atterisauniform degenerate Ferm isystem and m ay becharacterized
by itsFerm iw'
avenum ber. Sinceeachm om entum statehasadegeneracyfactor
of4 (neutrons,proton,spin-up.and spin-down),the particle density becomes
(see Eq.(5.47)1
4
2
,
... 1.3
)r= 3
=i'
eF
--
.
(39.l5)
(39.l6)
The experimentalvalue ofrvfrom Eq.(39.1)yieldsthe Ferm iwavenumber of
nuclearm atter
kF :
k:1.42 F -1
4OLINDEPEN DENT-PARTICLE (FERM I GAS) M ODEL
W e wish to examine the bulk properties ofnuclear m atterasdeined above in
thelim it.
,
'
f-->cr. Forauniform system itisappropriateto usea box ofvolum e
P- and apply periodic boundary conditions. Trarislational invariance then
im pliesthatthesingle-particleeigenfunctionsareplanewaves
(
pktxl= U-1'dfk*x
(40.1)
which are already solutions to the H artree-Fock equations'
,they are the eçbest''
single-particle wave functions that can be found (see Sec. 10). This result
accountsforthe appealand sim plicity ofnuclear m atter. The starting single-
particlewavefunctionsarekaownandsimple. Suchisnotthecase,forexample,
with hnite nucleior atom s.where it is a very dië cultcalculation m erely to
generate the starting Hartree-Fock wave functions.l
The present Ferm i m edium consists of both protons and neutrons with
spin-up and spin-down. W e know that protons and neutrons have the sam e
!W econsiderthe theory ofsnitenucleiin Chap.15.
NUCLEAR M ATTER
3:3
nuclear interactions, which is the statem ent of charge independence. H ence
itisconvenientto treatthem astwo diflkrentchargestatesofa singleparticle,
the nucleon,and to introduce the conceptofisotopic spin. In this way,the
nucleon acquiresanadditionaldegreeoffreedom thatcantaketwovalues'
,these
values indicate w hether the nucleon isa proton ora neutron. Introduce the
two-com ponentisotopic-spin wave functions
(p = 0
l
(a= 0
1
(40.2)
justasforangularmomentum !.. The operatorsin the space ofthese twocom ponentcolum n vectorswillbe 1andm wheretheTdenotethePaulim atrices.
Thecomplete single-particlewavefunctionscan then bewritten as
+k(X)Tà(p
(40.3)
where '
:A is the ordinary-spin wave function. The charge operator that distinguishesa neutron from a proton isgiven by
q= V 1+ T3)
(40.4)
Second quantizationcanbeintroducedexactlyasbefore,andthecanonical
anticom m utation relationsbecom e
tlkAont/k'A'p-l= 3kk'3Az'îpp'
(40.5)
'
Herethe anticom m utation relationshave been written in term sof-a generalized
Pauliprinciple:The statevector,orw ave function,ofa collection ofnucleons
m ustbeantisym m etricundertheinterchangeofallcoordinatesincludingisotopic
spin. lftheinteraction doesnotcause transitionsbetween neutronsand protons,
Eq.(40.5)representsno lossofgenerality since itisthen irrelevantwhetherthe
corresponding operatorscom m ute oranticom m ute. Ifthe interaction can cause
such transitions,then this choice is im portant,and in al1 the theories so far
developedtheanticommutationrelationsofEq.(40.5)havebeenimposed.
Theexpectationvalueoftheham iltonianinthenoninteractingFerm isystem
givesthefrstapproxim ation to theground-stateenergy ofnuclearmatter
Eo+ EL- (:F14 IF)
(40.6)
Thesrst-ordercalculationisalso avariationalcalculationbecausethevariational
principle show sthat
E < tF!4 1F)
(40.7)
W e assum e initially thatthe potentialis nonsingular,and the corresponding
expectation value can be com puted as
kr h2.k2
Ev+ E,- 4
2m + .
i :
k
klAIpj
éz
k4:4p4
x'rklAlplk2Aap2I#')k3A3p3k4A4p4)
X t.Fl
l1,A,p,t/kzAzpzJk4A*p4&k3A3ps1F1
(40-8)
3:4
APPLICATIONS T0 PHYSICAL SYSTEM S
(Wereturntoadiscussionofthesingularcaseshortly.) Justasinthecalculation
withtheelectrongas,alltheoperatorsinthematrixelementin Eq.(40.8)must
referto particlesin theFerm isea orthem atrixelem entwillvanish. Them atrix
elem entofthecreation and destruction operatorsbecom es
(#lll
kyAlplt/kaâzpz&k4A4p4&k323p31F)= (t
skjkzîA:Aaîp:pzlEbkak43Az24îpapxl
(3klk43AI/43p1p41(t
skzk;3Azza3pap3l
-
which gives
kr h2k2
Eo+ Et- 4
k
kr kz'
lm +àkAp
X k'I
ttkApk,A,p,(P'l
kApk,A,p,)
.
A'p'
-
(kApk'A'p'1P'!k'A'p'kAp)) (40.9)
This expression for the ground-state energy is represented by the frst two
G oldstonediagram sin Fig.9.22.
A san exam ple.considera potentialthatisan arbitrary com bination ofan
ordinarycentralforce(W ignerforce)anda Majoranaspace-exchangeforce
I,
'= U(x)kw,-z.au#,
u)
(40.10)
where Jw.and askare positive constants. W'e assume that U(x)is attractive,
nonsingular,and for sim plicity- spin independent. In fact,the spin depen-
denceofthenucleon-nucleon forceisquiteweak;the1.S'
0stateisjustunbound
andthe3u
$'jstateisjustbound,asdiscussedinSec.38. Thedirectmatrixelement
(frstterm inE1)thenbecomes
Zo= Z 2$
*djA'j
-d(
)ye-ik.x(7-Lk'.yTA
j
'(jjTt
A'(g)tp
1(jj(p
#',jg)jz(j.g)
xgjk.xyfk'.yyyztjl.
q,
j,(g)(ptjl(pw(g)
-
=
F-1kp,j F(z)#3z+ asfje-ftk-k'l*zF(z)#3z)
(40.11J)
and,similarly,theexchangematrixelement(second term in F:)isgiven by
P'E= Z-2J#3xj#3ye-ik*xg-ik'eyTA
'
t(j)TA
Y'(2)t'p
Y(1)(p
Y,(;)Fjj,ij
x eik..xcik.y
TA,(j).
r)A(J)(p,(j)(p(2)
=
U-13zA,ôpp,(Jw.Je-itk-k'l@zP'(z)#3z+ aM .
fU(z)#3z)
(40.11:)
W hentheseexpressionsareinsertedinEq.(40.9)andthesumsareconvertedto
integrals,theexpression fortheenergy becom es
3klj./
pr 1
ks z ks j,jyy
Ez+Ek-jzm,
4+y(a.)6J dkj dkt sjo
s,
+ au jP-itk-k')*ZF(z)#3z)- 4(JséP'(0)+ Ja.j(?-itk-k')*7F(z)#3Z)l
(40.12)
NUCUEAR M AR ER
z:B
whereF(0)* Ftk= 0). Themomentum integralscanbeevaluatedasfollows:l
lkFdjkeîk.x=4n.jt
k
'
Fy
rzyajk.
xlgk.403
W)?jk
L(k
Fx)
rx
4:jg)
t '
Finally,therelationbetweenthevolumeandthenumberofpartieles(Eq.(39.15))
allowsusto rewritetheenergy
Eo+ Fl 3hlk; #/
x -Jzm+12,2((4c'
z-autfP'
tzl#)z
+ (4uv - cp,)
?jtlkFz)2
a
k z F(z)d z (40.14)
F
A s kr--+.cc, the integrand in the second integral goes to zero alm ost
everywhere,and thefirstterm thuscan be expected to dominate athigh densities. Foran attractivepotential,itfollowsthattheassem blycollapses,thatis,
theenergybecomesm oreandm orenegativeasksincreasesunlessthecoeë cients
oftheforcesatisfy theinequality
au >4Js, to preventcollapse
(40.15)
W ith the present nonsingular potential, Eq.(40.15) represents a necessary
conditionfortheexchangeforcetoprovidesumcientrepulsionintheoddangular-
momentum states (recallEqs.(40.10)and (38.19)). Ifthisinequality is not
satissed,the potentialenergy willalwaysdom inate the kinetic energy athigh
densities,because the potentialenergy varies asi') while the kinetic energy
variesas/c/.. Thetrueground-stateenergythusbecomesmoreandmorenegative
asthe densityincreases. In particular,the experim entalnucleon-nucleon force
isroughlyofaSerbercharacter(seeEq.(38.20))with
tzp,;k;asf experiment
(40.16)
and itisclearthatsaturation cannotoccurfornonsingularSerberforces.
A single-particlepotentialcan bedesned in thefollowing way:
UAp(k)O âXô)(k)
kF
=
J( ttklpk'A'p'IFlklpk'A'p')- (klpk'/.
/p'jFIk'A'p'kAp))
(40.17)
This quantity represents the srst-order interaction energy ofa particle in the
state lklp) with allthe otherparticlesin theslled Fermisea;itistheusual
H artree-Fock potential. ltm ay also be interpreted asthediagonalelem entin
bothspinandisospinofthelowest-orderproperself-energyforthesingle-particle
Green'sfunction given by the frsttwo diagramsshown in Fig.11.3 (see the
!seeL.1.Schift op.cit.,p.86.
x6
AppulcATloss To PHYSICAL SYSTEM S
discussioninSec.11). Therelevantmatrixelementshavealreadybeenevaluated
inEq.(40.11),andwe5nd
1
U(k)= (2=)a
kr
#3/c'(4(Jsr-fU(z)#3z+ au jp-ftk-k'l@zFtzlJ3zj
-
gt
zsfJU(z)#3z+ Js,.
fe-itk-k'l*zF(z)J3zjj (40.18)
w hich isindependentofspin and isospin. The angularintegralscan becarried
outasbefore:
c(k)-6
k
,,
l
aj(4a.-as.4jp't
zll3z+t4us,-uw.
)
>:)7
-
Hjlkpz)
3
ks z P'(z)d z (40.19)
0(
/ûz)
Forsm allm omenta,thesphericalBesselfunctionappearingintheexchange
integralcanbeexpanded-/otkz)= l- :2z2/6+ .. .,which leadsto a parabolic
single-particlepotentialatsm allk
:2/c2
&(k);
kJUv+ lm U1+ ''.
(40.20)
Thisquadratic m om entum dependence m ay be used to define an eflkctive mass
through therelation
hlkl
hlkl
6u= e2+ U(k);kûlm (1+ Uj)+ UfjN Uo+ -j;ju
(40.21)
where
m
O
=
l+ 171
(40.22)
ln the present H artree-Fock approxim ation, the m omentum dependence of
U(k)and eflkctive-masscorrection ariseentirely from the exchange term for
thedirectforce Pk and from thedirectterm fortheexchange force Pk.
ln the opposite lim it of large k.the sphericalBesselfunction oscillates
rapidly,and theexchangeintegralvanishes. A saresult,theasym ptoticexpression forthe single-particle potentialbecom es
&(k) >'gklj(4Jp,- aMjj-U(z)daz
--
=
(4().a?)
The resulting single-particle potential is sketched in Fig. 40.1. N ote that
&(k)automaticallyincludesthePauliprinciplesincetheparticlein state (klp)
hasbeen antisym m etrized with a11theotherparticlesin them edium . lfk < #s,
then U(k)represents the interaction between nucleonsactually presentin the
Fermisea. In contrast.ifk >Ft's,then &(k)representsthe interaction ofan
NUCLEAR M ATTER
357
Fig.40.1 Sketch ofthe singl
e-particle potential&(k)in
Eq.(40.19). Seealso Eq.(40.23).
additiollalnucleon innuclearmatters
'thisquantity isjustthe opticalpotential
seen by theadded particle.properlyincluding thepossibilityof-exchangeefects.
So far we have assum ed a nonsingular nucleon-nucleon force of a Serber
exchange nature and have calculated the ground-state energy shift to lowest
orderin the interaction. Thisresultisvery powerfulsinceitgivesa variational
bound on thetrueground-state energy and showsthattheassem bly isunstable
againstcollapsewithsuchaSerberforce. W earenow facedwith twoproblem s.
First,how do we explain nuclearsaturation'
? The answeristhatthe potential
hasbeen assum ed to be nonsingular,NNhereasnuclearforcesare actually singular.
As seen in Sec.38.there is evidence for a strong repulsion at short distances.
whiehm ustbeincludedinthecalculation. Thesecondproblem isto understand
the successofthe independent-particle m odelofthe nucleus. ltisclearthatthe
singular nuclear forces introduce im portant correlations. Nevertheless, the
num erous trium phs of the single-particle shell m odel of tlte nucleus and the
accuratedescription ofscattering through a single-particleopticalpotentialshow
that the independent-particle m ode!frequently represents an excellent starting
approxim ation in nuclear physics. ln Sec. 41 we attem pt to answer these
questions with the independent-pair approximationb in which two-body
correlations are treated in detail.
4ILIN D EPEN D ENT-PA IR A PPRO XIM ATIO N
(BRU ECKNER'S THEO RYII'Z
*
The force between two nucleonsis singular. This pointis crucial,for itm eans
thatthenuclearpotentialhasafundam entaleflkctonthetwo-body wavefunction.
lK.A.Brueckner,C.A.Levinson,and H.M .Mahnloud.Phys.Aez
ë'
..95:217 (1954);H .A.
Bethe,Phys.Rct.,103:1353 (1956);K.A.Bruecknerand J.L.Gammel,Phys.Ret'.,109:1023
(1958);K.A.Brueckner,Theory ofNuclearStructure,in C.DeW itt(ed.)aSsrhe M any Body
Problem q''p.47.John W ileyand Sons,lnc.,New Y ork.1959.
2Thepresentdiscussion ofBrueckner'stheory intermsoftheindependent-pairapproximation
isbased on L.C.Gomes,J.D.W alecka.and V.F.W eisskopf,Ann.Phys.(X'.F.),3:241(1958)
and J.D.W aleckaand l-.C.Gomes,Ann.da Acad.BrasileiradeC'
ï/pcf
'ç.39:361(1967).
au
AppulcA-rloNs To pHvslcAu SYSTEM S
asdiscussedindetailinSec.11. lnterm sofdiagram s,wem ustretaintheladder
contributions to the proper self-energy indicated in Fig. 11.3,and the correspondingcontributionsto theground-stateenergy. Forsimplicity,the present
section describestheinteracting pairwith the Bethe-Goldstoneequation,which
containsal1theessentialphysicalfeaturesoftheproblem. InSec.42wediscuss
therelation to theGreen'sfunctionsand theGalitskiiequation ofSec.11.
sEl.F-coN slsTENT BETH E-GOLDSTON E EQ UATION I
The Bethe-Goldstone equation fortwo interacting nucleons in the Fermisea
wasstudied in Sec.36. Thefundamentalapproximation wasto concentrate on
the two particlesin question,omittingentirely the interaction oftheseparticles
The region r
I1P :!
:kl> #s
/
zA/'
1.P
/
Fig.41.1 M omentum regionsin the Bethe-Goldstoneequations.
The region F
fiP t kI< kr
with therestoftheFerm isea. Such apictureisclearlyincom plete,and wenow
m odify theenergyoftheinteractingpairby includingan eflkctivesingle-particle
potential&(k)comingfrom theinteraction witha1ltheotherparticles. Inthis
way,thekineticenergyE2isreplacedbyek= e2+ U(k),andtheBethe-Goldstone
equations(36.4),(36.5),(36.15),and(36.16)become
#3t
1
4Pk(x)= efk*x+ r(gx)3eit*xEvk- sus.jj- sup..jjd?ye-dt.yp-tyl4,,k(y)
.
.
F
- l!P :lukl< ks
r- I!.P + tl>ke
(41.1)
ZepkX EP.% - C1P+k - f1P-k
=
F-1J#3xe-îk*xF(x)4p,k(x)
(41.2)
where the excluded region in momentum space (that occupied by the other
nucleons) is shown in Fig.41.1. These equations now contain an unknown
function&(k),whichwillbedetermined self-consistentlyfrom theinterparticle
potentialF(x)and thetwo-body wave function l
/+,x(x). ThepotentialF(x)
willbe takenasspin and isospin independent,butthisisnotan essentialrestriction.
!H.A.Betheand J.Goldstone,Proc.Roy.Soc.(London),A238:551(1957).
NUCLEAR M AU ER
369
Consider a pair characterized by center-of-m ass momentum hp and
relative momentum âk as in Eq.(41.l). The interaction between the two
nucleons m ixes in tw o-particle states above the Ferm isea,which produces a
corresponding shift in the two-particle energy given in Eq.(41.2). In the
independent-pairapproxim ation,the totalenergy shiftofthe whole system is
obtained bysum m ingovertheenergyshiftsofa1lpairsofparticlesin theFerm i
sea. (Fortwo identiealparticles,forexamplepïandpt,thewavefunction in
Eq.(41.1)must,ofcourse,beantisymmetrized.) Theindem ndent-pairapproximation hastheimportantfeature thatitautomatically givestheenergy shiftof
an interacting Ferm isystem exactly to second order in the potential. This
resultisevidentfrom Fig.9.22,sincethereareno othersecond-orderterm sin the
ground-state energy (see the diseussion in Secs.9 and 42). In addition,the
Bethe-G oldstoneequation m akesitpossibleto includetheeflkctofthepotential
onthewavefunctiontoal1orders. Equation(41.1)isstillanintegralequation
forthewavefunction,and thepotentialappearsonly in thecombination F/,
which iswelldesned evenforsingularpotentials.
Tosimplifythediscussion andto gain someinsightintothephysicalaspects
ofthe problem,we shallassume that U(k);
k;Un+ @2kl(2m)Ujand use the
eflkctive-m assapproxim ation. A sseenin Fig.40.1,thisisagood approxim ation
overlim ited regionsofthespectrum ,butitcannotbecorrectfora11valuesofk.
Thesingle-particleenergiestherefore becom e
hlkl
/12kl
hlkl
hlkl
ek= lm + U(k)Q; lm + Un+ -lm U1Q;lm++ Uv
(41.3)
A lthough thisapproxim ation leadsto a greatsim plihcation,itstillcontainsan
elem ent of self-consistency, because m * afl
kcts the two-body wave function
through Eq.(41.1). Itthusaltersthesingle-particlespectrum U(k),which is
calculatedasthetotalinteraction energyofaparticleinthestate Iklp)withal1
theotherparticles. TheconstantpotentialUvcancelsidenticallyin Eqs.(41.1)
and (41.2) because these relations only involve energy diFerences. ln the
eflkctive-m assapproxim ation,itfollowsthattheself-consistentBethe-G oldstone
equationsreducetothosestudiedinSec.36(Eqs.(36.15)and(36.16)),butwith
theinteraction potentialnow given by
!l(x)= zrrlr
*euâ-2F(Ixi- xz1)
(41.4)
where mr
*.n= m*Ilisthereduced ef
lkctivemass.
The energy shiftgiven by the Bethe-Goldstone equation in Eq.(41.2)
variesas P'-1. Since K.2- t2in thedenominatorofEq.(36.15)cannotvani
sh
exceptcloseto the Ferm isurface,wem ay m akethereplacem entKl-+.klin the
equation for the wave function. This approxim ation is essentially exact for
particlesdeep in thc Ferm isea;asw ehaveseen in thediscussion ofthe Coom r
pairs,however,itm ay be incorrectcloseto the Ferm isurface. Indeed,itwas
36:
APPLICATIONS TO PHYSICAL SYSTEM S
justtheappearanceofK2inthedenominatorin Eq.(36.15)thatledtotheeigenvalueequation withtheexceptional(bound Cooper-pair)solution. W eshow
in Sec.43 thatthe gap A in nuclearm atterisvery sm all.l Forthisreason,the
possibilityofbound pairsm aybesafelyneglectedindiscussingthebindingenergy
and density ofnuclearm atter.
w e shall now exam ine the solutions to the Bethe-G oldstone equation.
The e/ective-m ass approxim ation allows us to convert the integralequation
(41.1)intoadiFerentialequationbyapplyingtheoperatorV2+ kl:
d3t
(V2+ k2)4(x)x r(..
yr?
(y)/(y)
2'
zrPweitexjdjyp-ite
d3t
= I?(x)/(x)- r.
ait*xjJ3ye-it.y?J(y)/(y)
(j'
r8-jt
(41.j)
where P isthecom plementoftheregion f'inFig.41.1(P isdesned astheunion
oftheresionstèP +.tl< /cs). Westartbyconsideringthesimplestcase(P = 0)
and look forJ-wavesolutionsto thisequation in theform
4(x)= x-iN(x)
(41.6)
TheJ-wavesolutionsaretheonly onesthatpenetrate to sm allrelativedistances
wheretheeflkctsofthesingularpotentialarestrongest. Inserting Eq.(41.6)
into Eq.(4l.5)yields
d2
f.x
j+ k2 lz(a'
)=t'
(x)lf(x)-J()Zx,.
p)r(y)1,
/(.
p)Jy
..
.
(41.7)
where thekernelappearing in thisJ-wave Bethe-G oldstoneequation isgiven by
so LuTlo N FoR A NONSING ULAR SG UARE-W ELL POTENTIAL
Asa hrstapproxim ation,we shalleonsidera nonsingularsquare-wellpotential
that5tsthe low-energy 1.
% scattering. Ourapproximate potentialissketched
in Fig.41.2. In accordancewith thepreviousdiscussion,itsparam etersareto
'Thisissimilarto the situation in a metal,where L,
l
k.= l0OK,es.,''/cs= 104OK . Nevertheless,
there isa fundamentaldistinction between the two systems,forwe are here interested in an
absolute determination ofthe bulk properties ofnuclear matter,ratherthan the very small
energy diflkrence Es- En evaluated in Eq.(37.53). This difrerence in attitude rtiects the
physicalfactthatthetransitionbetweenanormalandsuperconductingmetalisreadilyobserved
andstudied,whilethereisno obviousw'ay even to decidewhetheranucleusisnormalorsuperconducting.
NUCLEAR M AU ER
361
bedetermined from the behavioroftwo nucleonsinteracting in freespace. At
zero energy,therelativewavefunction insidethepotentialisgiven by
uinlx)t
fsinEtzznredl
zr
câ-2)1x)
Sincethe1.
% potentialhasaboundstateatessentiallyzeroenergy,theremustbe
exactlyaquarterwavelengthinsidethepotential(seeFig.41.2),whichprovides
the condition
(zvreuPkâ-2)1#= J,
zr
The free-nucleon reduced massmzzn= m/l can be used to rewritethisresultas
Pk=
h2.
n.2
c
4m d
(41.9)
Fig.41.2 Nonsingularsquare-wellpotentialfitto
low-energy !x
sk scattering.
In addition,ifa squarewellhasa bound stateatzero energy,then theeflkctive
rangeisequaltotherangeofthepotential(seeProb.11.l)
eflkctiverange
(41.10)
W e shall take the singlet efikctive range obtained from p-p scattering datal
lr(
)= 2.7F
(41.11)
and thedepth ofthenonsingularsquare-wellpotentialbècom es
P'o= 14 M eV
e(41.12)
N otethatourapproxim atenuclearpotentialisactually very weck:
L <:
te;= h2k;= 42MeV
lm
(41.13)
where6kiscomputedwiththeFermimomentum appropriatetonuelearmatter
given in Eq.(39.l7).
In the specialcase ofthe nonsingularsquare-wellpotential,itiseasierto
calculatethechangei.
S$inthewavel
-unctiondirectlyfrom Eq.(4l.1)ratherthan
1M .A.Preston,op.cit..p.33.
362
APPLICATIONS TO PHYSICAL SYSTEM S
from Eq.(41.7). Considerthelimitk -.
>.0andassumeinitiallythaté/ issmall.
Itisthen m rm issible to set
/(x)= x-lu(x)khlkx).
-+.1
(41.14)
ontherightsideofEq.(41.l);thecorrespondingmodiEcationofthewavtfunction
isgiven by
'ztxl
x/-ji
.
zgx yotkxlj-l
zt
xl-1
=c
2%jk
erdtjétpj0
ësllylyzgy
-
0.
03
0.01
0.01
0.œ
g.
(jj
0.02
- 0.
03
1
2.
(41.15)
Fig.41.3 M odiscation A/awIEq.(41.17))of
4
5
the J-wavetwo-body wave function caused by
kyx the potentialin Fig.41.2. Thiscalculation is
for a pair with k = P = 0 in nuclear m atter.
krd
(The authors wish to thank E. Moniz for
preparingthisfigure.
)
They integralcan beevaluated explicitly
-d
a
f
tj.
jj(
?
hltV)y JA,- c
d#
-)
0
t
(41.16)
.
.
Thusthe m odiscation ofthe wave function dueto thenonsingularsquare-well
potentialis
2> (1 #lc -' #4
Lssw=n
'
.
kl às;pj,(p).
&jpa
''
j
--
s
.
(41.17)
where a dim ensionless integration variable has been introduced and kFd = 3.8.
Thedimensionless potentialis given by
!'0
k
'
'
=
P'
0
.. -- -- ..--
m*
.
=2
m*
= -.
-- . -u = 0.j7--.a:().j()
/ hzkF
il
'lm.
red m ktt's(?)
m
(4j.j8)
wherewehaveused the valuem*/m :
k:0.6,which isderived below. Thecentral
resultofthiscalculation isthat
(4l.19)
HenceêheattracîivepotentialhasalmostnoE'
/-/cc/onîhetwo-particlewavefunction
ulxl/x= 1+ Z$ gsee Eq.(41.l5)). There are two reasons forthisbehavior.
First.the large Ferm im om entum m akesitdië cultforthe nonsingularpotential
to excitetheparticlesoutoftheFermisea.and second m*/m < l. Thefunction
NUCLEAR M AU ER
363
A/,wofEq.(4l.17)isplottedinHg.41.3,alongwiththerangeofthesquare-well
potentialitself. W e seethatthecorrelationsinduced by the potentialoscillate
with wavenum ber ;
:
r;ksand fallofl-forlarge distanceslikex-2. In conclusion,
the long-range attractive partofthe nucleon-nucleon force scarcely afectsthe
two-particlewavefunction;instead,them odihcation ofthewavefunction arises
from the hard core com bined w itb any strong short-range attractive potential
lyingjustoutsideofthehardeoreal
soLuTlO N FOR A PURE HARD-CO RE POTENTIAL
The Bethe-Goldstone equation (41.7)can also be solved fora pure hard-core
potential,asoriginally doneby Betheand G oldstone.z ltisconvenientErstto
rewriteEqs.(41.7)and (4l.8)intermsofdimensionlessvariables
r= krx
r'= ksy
k
K =F
s
r(x)
P'(x)
vtr)= /c/ = haks
s/zrnv
ret
j
(4l.20)
u(r)= ksu(x)
(41.21)
(
l sin(r- r') sin(r+ r')
;
t'r.r')= =
r.
-. r' - r+ r,
(41.22)
A sindicated in Fig.41.4,thehard coreintroducesa discontinuity into theslope
of the wave function at the (dimensionless)distance c. Thisresultiseasily
verised by exam ining a barrier ofNnite heightand then letting the barrierheight
becomeinhnite(seeProb.11.5). Sincethewavefunctionmustvanish insidethe
inhnite potential.the productru can thusbe written
H ere the srstterm givesa énitediscontinuity in the slope of-the wâve fknction
atthecoreboundary,ascan be seen by integrating the Bethe-Goldstoneequation
(41.5)acrossthe core boundary. The remaining term u.
tr)cannotcontribute
Outside the core.where the potentialvanishes.butit m ay be finite inside the
core region because of the lim iting procehs v'-.
vz
csu -v0. This extra ''leak''
term m ust be chosen so that
u''-4-A'2u= 0
(41.24)
'Sofaronlythe1s'oattractivewellhasbeen considered. ln fact,thisu'illsumceforthepresent
analysisofnuclearmatter(seethediscussionfollou'
ingEq.(41.39)).
2H .A .Betheand J.G oldstone,Ioc.c#.
364
APPLICATIONS TO PHYSICAL SYSTEM S
W hen Eq.(4l.23)is combined with Eq.(4l.2l),the condition of Eq.(4l.24)
becom es
.*i-
p;-iw(r)= y(r,c)+ .
W -tJ0
j ytcr')w'tr')dr'
.
(4l.25)
The rem aining analysiscan be sim plised by noting thatthedim ensionless
constantc issm all. Forexam ple,atthe observed density in nuclearm atter
c= l.42F-lx0-4F= 0.57
(41.26)
wherea hard coreofrange0.4 F hasbeen assum ed. In theIim itwhereboth of
its argum ents are sm all, the kernel in the J-wave Bethe-G oldstone equation
becomes
2.
rr'
ytr-r')->.-jx-*
r< c,r?< c
(41.27)
Equation(41.25)showsthatB'
(r)isoforderc2,whileanyintegraloverw(r)w'
ill
beoforderc3. Consequently,theleak in Eq.(41.23)can beneglected forsmall
c,and acombinationofEqs.(41.21)to (41.23)yields
jt
d2
s
j+A2)
hl
/
(r);
k;,
.
Vt3tr-c)-y(r.c
.
))
2
-.
ç
#'.=cj'
x
k
'yo(tr)joltc)t2dt
r
- -
- F(r)
(4l.28)
whichdefinesF(r). Itisclearthattheterm xtrsc)servestocancelthoseFourier
componentsof3(r- c)thatlieinsidetheFermisphere (see Eq.(4l.8)1. The
rightsideofthisequationisaknownfunctionofr,andtheonlyunknownquantity
is the normalization constant..
W . Since 140)= 0,the generalsolution ofthis
equation isgiven by
1 r
?
,
?(r)= k
0
sin@A-(r- .
&))F(s)ds
(4l.29)
foritiseasily verilied directlyfrom Eq.(41.29)that
&'
'(r)+ Kl?,
/(r)= F(r)
(41.30)
IfthesineontherightsideofEq.(41.29)isexpandedwithtrigonometricidentities.thesolutionto theJ-wave Bethe-G oldstoneequation becom es
sin Kr r
cosKr '
u(r)= K - fjcosKsF(&)#J- --=A.
whereF istakenfrom Eq.(41.284.
sinKsF(J)ds
(h
(4l.31)
365
stlcuEAn M ATTER
W'e shallnow prove an im portantresult
(
*sinKsF(sJds=0
'0
(4l.32)
.
ThelastintegralvanishesidenticallybecausetliesoutsidetheFerm ispherewhile
K lies inside,as illustrated in Fig.41.1. The foregoing derivation shows the
im portance ofthe fslled Ferm isea.which rem ovesthe com ponents t< lfrom
the interm ediate states. Itisclear that this resultfollows quite generally from
thefbrm oftherightsideofEq.(4l.f). Equation (4I.32)ensuresthattheexact
solutionr-1l
?(r)gEq.(4l.3l))isproportionaltotheunperturbedso1utionj0(Kr)
.
as r -> '
,t.so thatthe J-wave phase shiftis zero. Ifthe wave function isto be
norm alized in a box of volum e 1,*.it m ust approach a plane wave with unit
am plitudeastherelativecoordinategetslarge:
bblxj->.cik*x
x .-sx
(41.34)
This condition im plies thatIz(-v).x = ulrlf
'r--'hlh'r)as1---'.:
c,and the i-wave
normalization condition in Eq.(41.31)becomes
#
awcosKsF(s)ds.
-.l
(4l..
3.
5)
J0
..cc
.)-j
V=g
cosKc-9acosKs:(.
s.6
.
)Jâ
'j
..
(41.
36)
Equations (4l.28),(4l.311.and (41.36)eompletely determine the â'-wave
relativewavefunction fora dilutecollection ofhard sphepes. Onetypicalcase
is plotted in Fig.41.4. This resultexhibits severalinteresting features. A s
Fig.41.4 TheBethe-Goldstoneu'
aN'efunction (Eqs.
(41,31'
).(41.28).and (41.36)Jforan ç-was'
e pairwith
k = P = 0 interactlngthrougha hard-core rotentialln
nuclear nlatter. The lourer lim it on the integrals in
Eq.(41.31)u'astakenascso thatl/tcànEE0. TheNN.av'e
function for a noninteracting pair ls Sheu'n by the
dashedlineandthecross-os'erpolntdetinesthehealing
distance. Also indicated aretheavrerage interparlicle
distance krl (Eq.(41.38)4and the range krd of the
rotentialin Fig.41.5.
ae6
APPLICATIONS TO PHYSICAL SYSTSMS
exptcted,the relativew ave function vanishesatthe hard-coresurface. ltthen
very rapidly approachesthe unperturbed valueofl,crossingoverthatvalue at
a 'shealing distance''ofabout
kyx ;r:1.9
healing distance
(4l.37)
Byexaminingthemoregeneralexpressionarisingfrom Eq,(41.5)gsceEqs.(41.7)
and(41.62)),wecanshow thatthishealingdistanceisessentiallyindependentof
kand#.1Furthermorethecorrelationfunction,desnedby,
z
'
&'
Jc= N(r),
/r-h(Kr),
isalso nearly independentofk and #. Beyond the healing distance,theBetheG oldstonewavefunction oscillatesaround theunperturbed solution,approaching itwith dam ped oscillations as ksx -.
+:
x:. The average interparticle distance
/in nuclearmatter,desned by theexpression l//3n Njl
''= lk((?=l,hasalso
been indicated in Fig. 41.4. The corresponding dimensionless parameter
characterizingtheinterparticledistanceinnuclearmatterbecomes
3.2 1
kpl= 2
= 2.46
interparticledistanee
(41.3B)
and we notetheinteresting resultthatthe healing distanceofEq.(41.37)isless
than thcinterparticlespacingofEq.(41.38),asillustrated inFig.41.4. Bythe
tim e thatone ofthe two colliding particleshasarrived atanotherneighboring
particle,the ttwound''in the wavefunction ofthe originalinteracting pairhas
healed back to itsplane-wave value.Forsubsequentcollisions,itfollows that
thecolliding particlescan again be assumed to approach each other in relative
plane-wavestates. Thisresultjustisestheindependent-pairapproximation.
Theseobservationsalsoprovideasimplequalitativebasisfortheindependentparticlemodclo-fnuclccrmcllcr. ThePauliprinciplesupprtssesthtcorrelations
introduced by the hard core and restricts its eflkcts to short distances. In
particular,the hard core cannotgive rise to long-range scattering because all
available energy-conserving statesarealready occupied. Exceptforthe shortrangecorrelations,anucleon m aythereforebeassum edtom ovethrough nuclear
m atterin a plane-wave state. Thedom inantrole ofthe exclusion principle in
explaining how an independent-particle modelcan describe a dilute strongly
interacting Ferm isystem at or near its ground state w as hrstem phasized by
W eisskopf.z
PROPERTIES OF NUCLEAR M AU ER W ITH A ''REA LISTIC'' POTENTIAL
W enow com binethe previousresultsto study am orerealisticm odelofnuclear
m atter,wherethetw o-nucleonpotentialconsistsofa hard coreplusanattractive
square-wellpotentialshown in Fig.41.5. Thispotentialgrossly oversim pliâes
'J.D .W aleckaand L.C.Gomes,loc.cit.
CV.F.W eisskopf,Helv.Phys.Hclt
z,23:187(1950);Science,113:101(1951).
NOCLEAR M ATTER
367
the actualnucleon-nucleon force discussed in Sec.38;thusitcan only provide a
qualitative,or at best sem iquantitative,description ofthe properties of nuclear
m atter. The m odelpotentialisgiven by
+.'
x
r<-b
P'= -Iz%!.
(l-hPM)
0
b<cr.
,:bc-bw
b +.bw < r
with a hard core in allstates(we shallcalculate the nuclearenergy both with
and withouta hard core in the odd-/states)and an attractive Serber force as
indicated. W e have here neglected the dipkrence between the 1So and 35-)
Fig.41.5 Hard-coresquare-wellpotential.
potentials,whichcanbejustisedifthisdiflkrencearisesfrom atensorinteraction
(asistrue,forexam ple,in theone-pion exchangepotential). Theground-state
expectation value ofa tensor force w ith Serber exchange nature vanishes in a
noninteracting Ferm i gas because the spin average of the tensor operator
jtrjtrz5'lziszero and there are ne exchange matrix elements. Thusthe epkct
ofthe tensorforce ism uch reduced in nuclear matter.
Thecondition thatthepotentialinEq.(41.39)haveabound stateatzero
energy isreadily derived as
hl,2
I
z'll= 4mb-j
w-
(41.40)
whichagainensuresthecorrect1u
S scatteringlength. Furthermore,theefective
range rt
lfor such a potentialwith a zero-energy bound state is given by (see
Prob.ll.1)
ro = 2b + bv,
W ith a hard core of0.4 F,theparam etersofthe com bined hard-core square-well
potentialbecom e
bw= 1.9 F
b= 0.4 F
(4l.42)
w hile the totalrange # ofthe potentialis characterized by
krlb + >w,
)e /1
's# = 3.27
(4l.43)
a68
AppulcA-rloNs To pnyslcAt- SYSTEM S
and is indicated in Fig.41.4. From the previous discussion,it isclear thatthe
attractive part of this potentialagain has little eFect on the w ave function,
tending to enhance its value slightly w'ithin the potential. A s a result, the
correlation function -X/ isessentiallythatofthehardcorealoneand isofthe
type shou n in Fig.41.4.
To com pute the interaction energy ofa pair ofnucleons in nuclearm atter,
we shall evaluate the energy shift Ae'e,k with the Bethe-G oldstone equation
(4l.2)
l6.
y,,kz,J
z'-1(c,-ik.xjp'
t'/.
;
-JzC)4p,k
tx)J3.
'
r
(41.44)
Here Iz'
aistheattractive partofthe potentialin Eq.(41.39)and P'
cisthehard
.
.
core. ln the independent-pairapproxim ation.the single-particle potentialand
total energy'of the assem bly are then determ ined trom Aee k by the relations
(4l.45)
(4l.46)
wherethesum srun overtheinteriorofthe Ferm isphere. The self-consistency
of the theory is now particularly evident, because the single-particle energies
jpakappearing in Eqs.(41.1)and (4l.2)areevaluated with Eqs.(41.44)and
(41.45).
er.
Since the hard core is m ost im portant in determ ining the solution.it will
be a good approximation to replace the exactwavefunction in Eq.(4l.44)by
thatderived in the pure hard-core problem :
(41.47)
(41.48)
For the present discussion w'e shall.in addition.approximate the hard-core
solutionintheregionoftheattractivew'ellbyaplanew'
ave(Bornapproximation)
(41.49)
Although the relative wave function vanishes inside the hard core (see Fig.
41.4),itgrows rapidly for r7-b and exceeds the plane-wave value within the
range ofthe potential. Since the weighting factorin the integralisr2#r,the
Born approximation should provide a reasonable estim ate of the attractive
energy. N ote that this is true only for the very sim plifled potentialused here.
A m ore realistic m odel.including a strong attraction concentrated outside the
hard core, w'
ould require a m ore sophisticated treatm ent. For exam plea Eq.
(41.48)could be evaluated with no furtherapproximation or,in principle.the
NUCLEAR MATTER
369
coupled problem ofthe hard core plusnearby attraction could be solved exactly.
(Onetractable approach to thiscombined problem can befound in thew'
ork of
M oszkowskiand Scott.l)
W'ith Eqs.(41.48)and (41.49).theenergyshiftofan interactingpairreduces
to the sum ofthe energy shiftfora pure hard-spheregasand an attractivecontribution calculated with the independent-particle m odel in Born approxim ation
xcp,k= Zt'
;,k - Z6$,k (independent-particlemodel)
u
(41.50)
The latter contribution has been discussed in Sec.40,where we calculated both
the energy shift of the whole assembly (Eq.(40.14)j and the single-particle
potential(Eq.(40.19)1fbra purely attractive interparticle potential. Thislast
quantity is readily evaluated with the param eters appropriate for Fig.41.5 and
Eq.(41.39)
9
l't
l/f). 3 j3)u
(yalk)=-3
,
7..(J- tsj
-dl
,
v(pgjyjjysgj
,z(
yg
.
..
.
t,'
becauseaSerberforcerequirest
za.= au = !.,and theintegrationrunsfrom the
hard-core range b to the outeredge ofthe attractive potentialb-L.:w.= J (see
Fig.41,5). ltisevidentthatUJisisotropicand dependsonly on k2.
W e arenow ready to com pute the contribution ofthe attractiveinteraction
to the eflkctive mass. A s indicated previously.a single eflkctiN'e m ass cannot
approxim ate the single-particle spectrum over the whole range of 1'2. and it is
therefore necessary to choose the relevant region in m omentun: space. One
possibility isto assertthatthe m ostim portantvirtualtransitions occtlr neartl
-ie
Ferm i surface. choosing ?37* to fit the s'alue and $1ope uo.
-:ht' s:nglt'-par:icle
spectrum atky(com pareSec.29b
,. lfthesingle-particlepotc:1t1u
lllsappro'qinqated
near a generalvalue kv as
hl
U(k2):
kSU(k0
2)+ L.(kl- k(
2
))&.
Jj
Jn
then the corresponding efl
kctive m assbecom es
n)*
1
r,? l+-P1
(4l.53)
To reproducethespectrum neartheFermisurface(/fa= /s
-s),U:mustbechosen
as
U
dU
j= p
.-nj
-kF -'
dk''k-ks
(41.54)
a7o
AppulcATloNs TO PHYSICAL SYSTEM S
Thisexpressioncanbeevaluateddirectlyfrom Eq.(41.5l)andweEndi
3
)
''
L-1- j k2si0,4-
p;
-
.- .
'x
= n &s/=>7 Z
krJ
(J'f(p)-7'
e(p)J'z(p))
ks&
(4l.55)
where the quantity ln brackets isto be evaluated between the indicated lim its.
Numericalvaluesforthesinlplifed potentialofEq.(4l.39)are givenin Table
41.1.
Table 41.1 Effeclive mass atk = kpand k = 0 fortwo nuclear
densities
(m*jmlk-kp
1.25F-1
1.48F-'
(7,(0)j
0.65
*.65
A nother way ofchoosing the eflbctive m ass isto expand the single-particle
potentialaboutk2= 0,
'substitutingjolkz);
k'l- k2zl6in Eq.(41.51)immediately yields
'
t71- jl-k'
y-cPo2/
.
a Ep3/
.atpllts
f'
j
r n-'lk#/ .
-
(4l.56)
N um eriealvaluescom puted from thisexpression are also shown in Table 41.1.
W e note from Table 41.1 that the eflkctive m ass is rather insensitive to changes
in the density about the equillbrium value, and that the effective m ass at the
bottom ofthe Ferm isea isslightly sm allerthan theeflkctive massatthe Ferm i
surface.
So far,the hard core has been neglected entirely in evaluating m *. To
justifythisomission,werecallthediscussionoftheGalitskiiequation wherethe
hard coredoes notaf
fcctthe efl
kctive m assto tirstorderin c= kt
-b.
,to a good
approxim ation.itfoliow'
sthatthe efl
kctive m assarisessolely from the attractive
welland exehangenatureoftheinteraction. ln fact.Eq.(1l.68)show'
sthatthe
leading contribution ofa hard core c= 0.57to the Ferm i-surfaceeflkctivem ass
ofanassemblyofidenticalspin-lparticlesisobtainedfrom .bU(;k:-(.8c2/15=2)x
(7ln2--1)= -0.067. whereas the same hard core, if present in the p state.
contributes the following amount to f71 (Prob.4.6) âUL;k:+c?j
l= = +0.059.
1Thenecessary integralsofthe sphericalBesselfunctionscan befound in L, 1.Schift op.ci/.,
17.86.
NUCLEAR M ATTER
371
These contributions tend to cancel,and the correction isindeed sm all.l W e
shall henceforth neglect the hard core in evaluating m *. This sim plises our
Presentdiscussion of nuclear m atter considerably,as the self-eonsistency condition is no longer a problem . The eflkctive m ass is now dcterm ined from
Eqs.(41.51)to (41.54),whichareindependentofthe valueot
-m*.
ltisnow'possibleto estimatethe nuclearbindingenergyfrom Eq.(4l.46).
The contribution Eiaïofthe attractive wellis given in Eq.(40.l4);it may be
evaluatedjustasinEq.(41.51)andgives
*p-(a)
- --- =
.X
pz k3
-
a'
y d
J6.
?*Z (J3- b3).c.17F
2
w. b
j2
(kg:r)Jz
i
E (i) ?h2k2
F'
,4
f lm
N ote thatthis expression containsm and not??lï'.because ???'.is mereis'an oversimplified way of treating the single-particle potential t.'(kj that is needed i1)
evaluating the interaction energy.
The tinalcontribution to the nuclear binding energy arisezfrom the hard
core in Eq.(4l.50). Thls quantity can be com puted from the solutik)n tt3the
Bethe-G oldstoneequation. The energy shit'
tofa pairin the hard-core problem
fol1ow sfrom Eq.f
2)
.41.
X6I
a,k = l'-:.k r'
c $F',k
.-
= pz-;$
.:,-ik.xj
a.
gpk
x)t/rl.
v
C(
, (
.'
A sdiscussed in detailin Sec. 1l.only .
5-waves contribute to the totalhard-core
interaction energy E tf'' through order (-2. For î waves. ît follow s from Eq.
(41.2.
3àthat
2/'
)7*
re2
d
s .c
..a
h.2
.k
-j.t.jkP'kjy.j.
.;j
(t.. cj..yj,
jt.j
.
F
;
k:vr
.
n(
/
$(r- c)
(4l,60)
lThefourspindegreesoffreedom fornuclearm atterlead toanadditionalfactorof3and 5 3,
respecti
vely.in theserelations. An examlnatlon oftheslngle-parti
cl
epotential(Eq.(41.45))
derived fl'onnEq.(41,65)contirnlsthatthiscorrectlon issnnallfornuclearnnatter. W hen this
hard-core energ) is included in L'(k).the resultlng ettkctiv'
e mass at the Fernlisurt-ace ls
m*Frrl= 0.68 forkr= l.25 F-l(J.D.B'alecka and L.C.Gomes./tpc.c?'J.).which should be
com pared with thes'alueltl. ???=z0.65ln Table41.Icom ingfronntheattractivewellalone.
a72
Appulcv loNs To pl-lyslcAt- SYSTEM S
where the second iine isobtained by noting thattheJ-wave leak contributesto
the totalenergy only in orderc4. The overallnormalization constantisgiven
byEq.(4l.36)
c/(P,K)-g
cosKc-q'
RcosKsyz
0
,
(5'
,c)Jîj-l
.
(41.61)
where we now considerthe generalcase ofan interacting pairwith center-of-
massmomentum P (here and henceforth in thissection itis assumed thatP is
dimensionlessand measured in unitsof/cs). Itisevidentfrom Eq.(4l.5)that
thegeneralization ofEqs.(4l.8)and (4l.22)toP + 0 is
2rr'
dLj
ys(r',r')= =.
-. pjvltr)joltr') 4='tldt
(4l.62)
where1%isthecom plementof1Mshown in Fig 41.l. The angularintegration in
Eq.(41.62)gives
.
for?<gl-(
'
y
#,
j2j1
4j
( yyj
forgl-(y
#j2j1ojcjm#
j
o
whiletheintegralin Eq.(41.61)can beperformed with therelation
a
(c
oeOSKsSinJJds=..
#--K-j
0
ta-t
.
vshere@ istheCauchy principalvalue. Thuswefind
c.
V(P,K)= cosKc- 2
;.p
.
jvltc) #f1, a -1
?(fi(a.gz4.
g.tdt
(41.64)
.
A combination ofEqs.(41,59),(4l.60).and (4l.
64)yields
(4l.65)
Thefactor 1- 3,$!,$:3p:p:arisesfrom theantisym m etrization ofthewavefunction
for particles of identical spins and isospins and prevents such particles from
being in relative s states. The hard-core interaction energy is therefore given
by Eq.(41.46)as
(41.66)
NUCLEAR MATTER
373
Convertingthesumsto integralsand using Eqs.(41.65)and (39.l5),we t
ind
g.
(c) li2/k
-;
(2c * * J3K d3K
?,
=0 ;:r .-- - 1..
.
.
1
2/777 '
n'
=
. . ,1 ... 2 j
)(Kc),yvxr(p,jkj
(4=:
/3)(4=73) (
.
(F)
ii2k2
y.
*, (y3K Jàp
F 2(
- .. ..
l
'oqsKcj-c/(P.K)
252dt ,
rr . (4=../3)(4=/
/3)'
. . -- .
.-.
-- - .
(.!:J
where F is deéned by the intersection ofthe tuo Fermispheresx(compare Fig.
41.1)and.V(P.K)byEqs.(41.64)and(41.63). Theresultingvalueofthisdetinite
integralobtained by num ericalm ethodsisshown in Fig.4l.6 as a function of
c= kyb. lf Eq.(41.67) is expanded as a power series in c.the first few coeflicientsmaybeevaluated analytically. Equations (41.64)and (41.63)yield
(.
41.68)
and Eq.(41.67)becomes
& (c)
'&'
.-
lui2!.2
'
(41.69)
1 - . t'
V'
x
'gac..-j(.2. 0((y3)j
?*?
-
g . . yqy
y z=
g;p
p
(41.70J)
jjj
.
jy
-)yjjjg.
j
yjj
.= y
.m V
(F.%)
,
(l-i-/7/2)2--K2
xlnj--
l2
--3q=j.(1)- 21n2)
-jp
-t
j
-j
g..gjj
(41.70:)
Hencethe power-seriesexpansionof-Eq.(41.67)in cgives
EIrCJ'-- hl
/
t'l2
2-(
7 l
2-c.2
-.
-6r3-j-...
4 -bnl
.z -.
=.+ .
25.
:7.
.1(jl- 2ln2)+ 0.2,
.-. .
,
Thefirsttwo term sareexactlythoseobtained inthediscussion oftheinteracting
hard-sphere Fermi gas and the Galitskii equation (compare Eq. (11.72)),
a74
APPLICATIONS TO PHYSICAL SYSTEM S
generalized to include the possibility of an esective m ass com ing from the
attractivepartoftheinteraction. Thethird term ,which hasnotbeen com puted
here,wasfirstevaluated by DeDominicisand M artinland isjustthatpartof
their result due to J-wave interactions in the independent-pair m odel. The
powerseriesexpansionofEq.(41.71)isalsoplottedinFig.41.6.
Fig.41.6 Hard-coreenergyEt
lt
ojA computed from Eq.
(41.67)andthepower-seriesapproximations(Eq.(41.71)).
Also shown isthep-wavecontribution FJ*
.
'?!.
1
'
A from Eq.
(41.72). (TheauthorswishtothankE.M onizforpreparing
thissgure.)
Thecontributiontotheenergyofa hard corein thep statecan beobtained
from an exactly analogous treatment of the r-state Bethe-G oldstone equation
(seeProb.11.7)
sl
(c
3
-.
*F2:.
r-)
!= Ac
'= .'1
-
li
)/nv-zr
(4I.'
:2,
)
A snoted in Sec.38,the nuclearforce isrelatively weak in p states butthere is
some evidence foran overallp-state repulsion.
TheenergyofnuclearmatterasgivenbyEqs.(41.57),(4l.58),and(4l.67)
is shown in Fig.41.7. The results are plotted b0th with and withoutthe contri-
bution ofEq.(41.72)arising from ap-statehard core,and theefl
kctive massat
the Ferm isurface hasbeen taken from Table 41.1
(41.73)
lC.DeDolninicisand P.C.M artinaPhys.RcL'
.&105:14l7 (1957)and privatecommunicati
on.
NUCLEAR MAU ER
376
W e note severalinteresting features ofthese results. The m edium obviously
saturates,which isevidentfrom the variouscontributionsto the energy because
the hard-core repulsion will always dom inate at high density. Indeed, for
elose-packed hard spheres,the uncertainty principle requiresthatthe hard-core
energy become int
inite.l ltisclearfrom Eq.(41.71),however,thatthe hardcore contribution to the energy becom esim portantatdensitiesm uch lowerthan
E1A (MeV )
()
Hard core of0.4F
inpstate
2
-
4
-
kF = 1.25F-l
El.4 = - 6.2 MeV
)
- '
y
L
I
xoj
aar
dcor
e
+- inp state
6
-
l
-
-
8
j0
l
1.0
:
l.25
kl'= 1.45 F-1
EIA = - 9.1M eV
i
t
r
l.5 l.75
kF (F-')
Fig.41.7 The energy per particle in nuclearm atterasa
function ofkrcomputed from Eqs.(41.57),(41.58).(41.67),
and (41.72)forthe two-body potentialofEqs.(41.39)to
(41.42). The resultsare shown b0th with and withouta
hard core in the p state. (The authors wish to thank
E.M onizforpreparing thisfigure.
)
thatforclosepacking. ThehardcoreforcestheA-bodywavefunctiontovanish
whenever pxi- x,/< >,therebyprovidingextracurvaturein thewavefunction
and increasing the kineticenergy. Itisnotable thatthisvery sim ple picture of
nuclear m atter gives saturation at about the right density. The calculated
binding energy is too sm all;this result is to be expected because we assum ed
thatthe diflkrence between the tripletand singletforce com es solely from the
tensor force,whose eflkctsvanish in lowestorder,and used an attractive well
fit only to the 1.
% scattering. The tensor force stillm akes a second-order
contribution to the energy,however,and,like a1l second-order eflkcts,these
alwaysinerease thebinding.z
The binding energy isthediSerence between a large attractiveenergy and
a largerepulsiveenergyand,assuch,isverysensitivetoany approxim ationsthat
!Foradiscussion ofthispointseeR.K.Cole,Jr.,Phys.Rev.,155:114(1967).
2Notethata valueofm*/
'm closerto 1also increasesthe binding.
376
APPLICATIONS TO PHYSICAL SYSTEM S
have been m ade. To dem onstrate this point,we list below the various contributionsto the energy atkF = 1.50 F-1
E fa)a-1= -:6.9 M ev
s(y)-4-1= -r.2.8.()M ev
Elr'oA-1= +39.9M eV
Fl=
c)1,4-1= -4-4.9Mev
VT
(J)
Fig.41.8 Sonne tqpicalhigher-orderG oldstone
diagranlsfornuclearmatter:(t?)three-bodycluster.
(/p)hole-holescattering.
T'
w o tkaturesofthe nuclearforce tend to keep the density ofnuclearm atterlow :
the hard core and the Serber nature of the interaction. w hich decreases the
attraction and hence also m oNb
es the m inim um in Fig.41.7 to lower densities.
The resulting interparticle Spacing allows enough space for the wound in the
w ave function to heal. This resultaccountsforthe success ofthe independent-
particle modelofthenucleusandjustiéestheindependent-pairapproach to the
propertiesofnuclearm atter,
The discussion in this section m ay beconsidered an oNrer-sim plitied version
ofthe theory ofnuclear m atterdeveloped by Brtleckner,1Bethe,z and others.
Thuswe ha&e retained only the m ostessentialphysicalfeaturesand m ade m any
approxim ationsthatm ustbe im proved in any m orerealistic treatm cntofnuclear
m atter. Forexam ple.the present form ofthe independent-pair m odelom itsa
greatN'ariety ofhigher-ordercontributions.such asthree-body clusters(dehned
to be Goldstone diagramscontaining three hole lines),or contributions from
hole-hote scattering,which really involve three or m ore particles,as seen in the
discussion of the G alitskii equation.3 Some typical higher-order G oldstone
diagranns are indicated in Fig.41.8. Allofthese proeesses involve the sim ultaneouscollision ofm orethan tNso particles,and the shorthealing length ensures
that such processes are very im probable. These m any-particle cluster contributionshavebeenanalyzed in greatdetailby Betheand hiscoworkers.w ho use
lK A .Brueckner.loc.('1't.
? H . A .Bethe,/t)c.('i:.
3The preclse relation ofthe presentcaleutatlon to the diagrammatic analysisof)J,* and the
ground-state energy shift isdiscussed in Sec.42.
NUCUEAR M ATTER
377
theFadeevequationstoevaluatethethree-bodyclustercontributionscorrectly.l
The net efrect of these higher-cluster contributions is to decrease the binding
energy of nuclearm atterby about 1 M eV.
In addition to including higher clusters, it is necessary to im prove the
evaluation ofthe tw o-particlecontributions. Forpurposes ofillustration and
sim plicity, an eflkctive-m ass approxim ation has been used ; unfortunately, the
exactanswer isvery sensitive to the precise value chosen form *. ltisalso true
that the eflkctive-m ass approxim ation is not very good. A better approach is
thereference-spectrulnmethodofBethe,Brandow,andPetschek,zwhichchooses
the single-particle potentialto m inim ize the higher-order contributions. A s a
result,particles and holes are treated diflkrently,holes being assigned the selfconsistent single-particle potentialdiscussed here,and particles being assigned
the free spectrum . In this approach, a single-particle potential is merely a
calculationaltool. ltcorresponds to the freedom ofrewriting the ham iltonian
as
H = F-i-)i(&(i)+ i2
,
i
l
z'(l
j)- 2it&(i)
'r'
kj
and then choosingthesingle-particlepotentialL'
'(,
')to maximizetheconvergence
ofthe expansion for the energy.
A n additionalproblem in the study ofnuclearm atterislaek ofknowledge
ofthe nucleon-nucleon force in the odd angular-m om entum states. Thisisa
largeeflkct,ascanbeseenfrom Fig.41.7,anddiflkrentpotentialsw illgivediflkrent
valuesforthe binding energy. A nother question istheeflkctoftrue m any-body
forces,w'hichm ightoccurintheoriginalham iltonianbecausethemeson-exchange
processesbetween nucleonsare m odised by the presence ofadditionalnucleons.
A ny analysis of this problem is very dim cult-and the current philosophy is to
calculate the bestpossible binding energy and density using two-body potentials
fitto nucleon-nucleon scattering. Only ifa discrepancy rem ainswould we be
forced to introducem any-body forces. The presentsituation isthatthetheoretica1valuesobtained % ith two-body forcesalone are close to the observed binding
energy of-16 M eV and density kt-u-= l.4 F -1.3
MZU RELATIO N TO G REEN'S FU N CTIO NS
A N D B ETH E-SA LPET ER EQ UATIO N
ln the preceding sections, nuclear m atter has been discussed in term s of the
Bethe-G oldstone equations,and we now'relate this treatm ent to the G alitskii
equationsand the G reen'sfunctions. For sim plicity,we shallfirstneglectal1
1SeeR.RajaramanandH.A.Bethe,Ret'
.M od.Phvs..39:745(1967).
2H .A.Bethe,B.H.Brandow,and A.G.Petschek,Phys.Rc?
J.,129:225 (1963).
3For a review ofnuclear-tnatter calculations,see B.DaysRev.u od.Phys.,39:719 (1967)
andR.RajaramanandH.A.Bethe,Ioc.cl
'
r.
a7:
AppulchTloss To pHvslcAu svsu M s
self-consistency in the Bethe-G oldstone equationsand assum e thattheenergy
denom inators contain only free-particle kinetic energies. In this way, the
Bethe-G oldstoneequationsreduceto thesim plerform studied in Sec.36,which
is moreclosely analogousto Galitskii'sequationsofSec.11. Recallthatthe
GalitskiiapproachcalculatestheGreen'sfunction bysummingtheladdergraphs
asFeynm an diagram s. W e frstobservethatifweareinterested in calculating
only the ground-state energy shift:the G alitskiiequations can be sim pliied
considerably. The G reen'sfunction isrelated to'the ground-state energy shift
byEq.(9.38)
.-
ivh
ld,
j
4
s
((
)-;
.
('JdJ'trEwzts cats;eipo,t
E - f(
)- a.
(2.)4.
(42.1)
where the coupling-constantintegration isstillto be perform ed. To be con-
sistentwiththeappearanceofG0inEq.(11.28)forX*intheladderapproximation,
wereplace G(p)by G0(p)in Eq.(42.1). Alltheladdercomplexityand Adependence is thus put into Y*(p).l Our starting expression for the ground-state
energy shiftintheG alitskiiapproach istherefore
- y . .
-ivh
1(/A
,
j
kJ(
)-j-J#PtrLaygjswzjs jefp,,y
Z - 0 jtzrr
(42.2)
This expression now allow sus to draw a setof Feynman diagram sfor the
ground-state energy shift. A generalterm in the ladderapproxim ation isdrawn
in Fig.42.1. N ote that these are Feynm an diagram s;consequently,only the
topology of the diagram s is im portant, and we m ay draw any pair of lines as
returning lines,and any pairoflinesascrossed linesin the exchange diagram .
SincetheFeynm an G reen'sfunctioncontainsboth partieleand holepropagation,
thediflkrentrelativetim eorderingsinthesediagram scandescribeveryeom plicatedprocesseswithm anyparticlesand holespresentatanyinstant. ltwasshown
in Eq.(11.35),however,thatanypairofpartielesin aladderpropagatesbetween
interactionseither with both particlesabove the Ferm ikea orwith both particles
below. In com puting the energy shift,every pairofhole lineswilllead to an
extrapairoffactorsj
-krdhk'jkF#3k' At1ow densityitisthereforemeaningful
to classify thecontributionsby the num berofholelinesretainedz(comparethe
discussion attheend ofSec.l1). Theminimum numberofholesisclearlytwo;
thus in the Iow-density lim itweneed keep only oneintermediatepairpropagatl
ng
asholesandmayusetheparticlepartoftheGreen'sfunctionforalItheotherpairs.
Innth ordertherearenpossiblewaysofchoosing thatpairwhich contributesas
holes,and thesym m etry ofthediagram sshowsthatal1thesenterm sareidentical.
1Thehrstcorrection to thisapproximation involvestheintegralfd4p(Y*(p)G0(#))2,which
vanishes whenever Y* isindependentof frequency. This is true ofthe leading termsboth
foranonsingularpotentialEEq.(10.21)1and forahard-spheregasEEq.(11.57)1.
2Thisobservation waSfirstmade byN .M .Hugenholtz,Physica,23:533(1957).
a79
NUCLEAR M ATTER
W e can now carry out the coupling-constant integration for the nth-order
contribution
1dj
J--,--jy
j..j
y
VV
(1)
Hencewecan keepjustasinglegraphwithn.
- 1pairspropagatingasparticles
and onepairpropagatingasholes. ltism ostconvenientto assum ethatallthe
intermediatepairsin F gseeEq.(11.30))propagateasparticles,VNiththeholepair
coming from thetwo extrafactorsofG0in Z*and E - Ev.
ln thisway theG alitskiiequationsfortheground-stateenergy shiftinthe
low-density lim it can be rewritten as follows. The ground-state energy is
obtained from
where the coupling-constantintegration has now been perform ed. The corre-
sponding properselflenergy isgiven by Eq.(11.49)
hLj*lpq= -ï(2zr)-4fd*kG0(V)g4rtpk;p#)- Ftkp.
,p/())é'iknn
(see Fig.42.2),where we have used the fourfold degeneracy ofnuclear matter
associated with spin and isospin. The scattering amplitude in the m edium isa
p
k
k
k
#
p
âs*(p) O
N
Nx
Z
=
Fig. 42.2 Structure of l1* in Galitskii
approach.
'
,
z'
N
e% o +
z,
N
380
APPLICATIONS T0 PHYSICAL SYSTEM S
convolution ofamod@edGalitskiiwavefunctionzmwiththepotential(seeEq.
(1l.40)1
l-tqqq';#)= h2m-142.
/$-3f#3trjtlymtq- t,q';#)
where the variables in the scattering problem are indicated in Fig.42.3,while
ptx
Z
93
z p2
N
q- 1.(p1-p2)-)lp-k)
q'- 1-(p3-p4)- J.(p-k)
P= p3+ p4= p!+ pz= p+ k Fig.42.3 Momentum variablesforF
#4
in Fig.42.2.
ymisinturnasolutiontothefollowingintegralequation(compareEq.(1l.39)J
,
3j(q- q,
)o #(I'
P + q!- kr)#(II'P - qr-/fs)
ymtq.q :#)- (2:.
,)
+ j.
m gp
/ qc-,y
.
.
d't
xj(2.)3Dtt)i
tr
atq-t,q,;p)(4a.
6)
In accordancewith ourpreviousdiscussion,themodised G alitskiiwavefunction
satishesan integralequation w'ith only the particle-particlepartofthe G alitskii
kernel. The energy appearing in the denominatorsis the totalenergy in the
center-of-momentum frame,defined by Eq.(1l.36)
h2P2
hlk2 h2p2 h2tl/2
E = â#;- -jm = hkv+ hpo- 2m - -lm + m -
(42,.p
W e shallnow write the corresponding equations from ourdiscussion of
Sec.41. lftbe Bethe-Goldstone equations(41.l)and (41.2)for m*= m are
rewritten with a Fouriertransform,theenergyshiftofa pairbecom es
letq/,q';P)= hllnlP')-lfe-i*'*A4.
(x)t
/zpq,(x)#3x
= hz
lm).
')-:(2.
7$-E
$fJ3/?
-'
(t)/(q'- t,q';P)
(42,8)
wherewenow usea notation similarto Eq.(42.5),whilethewavefunction in
m om entum space satissestheequation
4(q.q,:P)- (2=)z,
!tq- q,
).y.#(1!'
P +ql-,aà'slfa(I1'Pj- ql- ks)
q -q + n
d5t
'
<j(2.)'
jL
'
lt)/(q-t,q,;p)(z
u.
q)
Thisis the free scattering wave equation with the Ferm isphere excluded from
thevirtualstates (compareEq.(1l.11)q. Thesingle-particleenergy spectrum is
NUCLEAR M ATTER
381
(42.10)
wherethevariablesaredesned asin Fig.42.3. Herethesecondterm in brackets
m akes explicit the exchange contribution that arises from the use of antisym m etric wave functions forpairs in identicalspin and isotopic-spin states.
Finally the energy ofinteraction o'fthe asscm bly is obtained asone halt
-the sum
ofthe interaction energy &'(p)forallparticiesin the Fermisea (compare Eqs.
(41,4f)and(41.46))
Forsim plicity.u'
eassume5pïn-and isospin-independentforces.
ltis now possible to prove thatthe Bethe-Goldstone equations(42.8)to
(42.l1)and the Galitskiiequations(42.3)to (42.6)are identical. The proofis
asfollows. Themodised Galitskiiwavefunction ym in Eq.(42.6)isan analytic
function in the upper halfko plane. (N ote that this is nottrue ofy itself.)
Since1-in Eq.(42.5)isjustaconvolution ofy,,withthepotential.F isalsoan
anall'
tic function in the upper-half-kvplane. Asa result,when thekf
jintegral
intheproperself-energl'gEq.(42.4))isclosedintheupper-halfkoplane,theonly
contributioncomesfrom thepoleofG0(k)atkv= coo
k= /ik2.2n?. 5Venow observe
thatY*(p)isalso an analyticfunction ofpvin theupper-halfpk
jplane;w'henthe
conteurin Eq.(42..
3)isalsoclosed intheupper-half'plane,theonlycontribution
again arisesfrom thepoleofG0(pjat;)v= ct?o
p==hpl:
'
2m. Theexpressionforthe
energy shifttherefore becom es
*kr
E-Ev=44L'(2rr)-3j d?pâY*(
.p,(z?p
0)
(42.l2j
precisely reproducing the Bethe-G oldstone results, From the preceding
discussion.itfsevidentthatthe Bethe-G oldstoneexpression fortheground-state
energy isobtained bysum m ingtheladdercontributionsinterpreted asGoldstone
J2
'J.cra???5 (see also Preb, 11.11b.NNhere the interm ediate pair always propagates
asparticles.
In sum m ary.NNehaNe demonstraled thattheground-state energiesobtained
from theG alitskiiequationsand from the Bethe-G oldstone equationscoincide
at los: densitq. Itisim portantto note.however.thatthiscorrespondence need
nothold for other physicalquantities. ln particular,the proper self-energy
difers from the single-particle potential L'(k) by the inclusion of hole-hole
scattering.l Sinceitisâl1*(k,l-())thatgivesthe correctsingle-particle energies,
we see again thatr7(k)merely representsa convenienttoolforcomputing the
ground-state energy.
!This pointwasfirstemphasizt-d
' by N .M .Hugenholtz and L.Van Hove.Physica,M :363
(1958)andbyD.J.Thouless.Phvs.Rev..112:906 (19584.
a:2
APPLICATIONS TO PHYSICAL SYSTEM S
TheBethe-G oldstonetheory described above stilldiflkrsin principlefrom
the Bruecknertheory because the Bruecknertheory relies on a self-consistent
single-particle potential. In term s of G reen's functions. this result can be
achirvedbyreplacingG0(p)withaG(p)thatincludesself-energyeflkctsassociated
with r. Furthermore,l-'mustitselfbe determined with G and notG0. The
equationsfor this self-consistenttheory are show n schem atically in Fig.42.4.
Self-consistentG :
=
+
+
+ *. *
+Exchange
Self-consistentla:
=-
+
Fig.42.4 Self-consistenttheoryforG and l''.
A s they stand, thcse equations are quite intractable because the frequency
dependence of-X*(p,#(
)) complicates the integral equation for l' immensely.
(Thisdiëculty issometimesknown kspropagation 07.
/-theenergy shell.j The
sim plerBrueckner-G oldstonetheory can beobtained from these equationsin a
series of approximations. First,fhe self-consistency is treated only on the
average,and we use a frequency.independent self-energy Ez(p)- Z*(p,v/â),
obtained by settingpn= ep/â,whereepsatissestheself-consistentequation
e'p = 60
(p,ep/â)- e0
p+ âtlytpj
(42.13)
p + âE*
Inthisway,theG reen'sfunction isgiven approxim atelyas
G
#(IpI- k,) hkr- IpI)
lp,po)-J,
tl- %/yjj.ixl+;o- spj)t- ixl
sc
..
(42.14)
Second,this G reen'sfunction is used to evaluate both the properself-energy
(Eq.(42.4))and the scattering amplitude (Eqs.(42.5)and (42.6)). W eagain
obtain xm by om itting the hole-hole scattering,which ispresum ed sm allin the
low-density lim it. The only eflkct on the self-consistent wave function is to
changethedenominatorinEq.(42.6)from mpnlh- !.
(!.
P + q)2- JIJP - q)2+ iyl
NLICLEAR M ATTER
383
to (m/hljlhpo- E'
lp+q- qe-ql+ iT. This set of equations determines the
approxim ate self-consistent spectrum 6p. Third, the ground-state energy is
evaluatedfrom Eq.(9.36)usingGsclp,po)andZ).
(p)
E=-4ïlzr(2=)-4j#4pefpong6k+j.
âE).(p))Gsclp)
=
4U(2=)-3fd3pgek+.!4Ek
t.
tpljptks- jpr
,)
=
Eo+ 21zz(2=)-3Jd?pJiE).(p)hky- Jpl)
(42.15)
Thisresultisidenticalwith thatobtained by substituting G,c(p,;()and Z).(p)
into Eq.(42.3). W enote,however,thatitdoesnotfbllow im mediatelyfrom the
originalform ofEq.(42.1)becauseofthecomplicated Adependence ofXJ(p)
and t'
p.
43rTH E EN ERG Y G A P IN N U C LEA R M AU ER
A sa fnaltopicin thischapterwestudy theenergy gap in nuclearm atter. The
semiempiricalmass formula indicatesthatthe lastpair oflike particles(pp or
nn)contributesanextraamount
Epair= 34 M eV .
,
4-'
i'
to the binding energy of nuclei. ln addition,Bohr, M ottelson, and Pinesl
observed thattheenergy spectrum ofeven-even nucleishowsan energy gap of
about1M eV (weshallreturn to thisquestion in Chap.15). Thusthereissome
em piricalevidencethatlikenucleonstendto pairup,and itisinterestingtolook
foran exceptional(orsuperconducting)solution to the gap equation in nuclear
m atter.
ThegapisdeterminedbyEq.(37.35)
l
uk,
k= !.)((k- ktPrjk'- k')
.-1
k,
(1)+ #))
where we now use theconvention ofChap.10 that l' > 0 foran attractive poten-
tial. W erestrictthe pairing to like particles,thatis,(ptp$)or(rl1r?$),and
assum ethat1z'isindependentofspinandisospin.z Toobtainanexplicitsolution
ofthisnonlinearintegralequation,itisconvenienttom akethefollowingapproxim ations:
lA.Bohr,B.R.M ottelson,and D.Pines,Phys.ReL'.,110:936(1958).
2Asdiscussed in detailin Chap,15.the pairing comesfrom the lastvalence particles. In al1
butthelightestnuclei,thelasthlled statesarequitediflkrentforneutronsand protons. Thus
theoverlap in thematrix elementsofthe interaction between valence neutronsand protonsis
generallysmallerthan thatbetween likeparticlesinthesamestates. Thisistheargumentfor
consningourattention to pairing between likenucleons.
ag4
AppulcA-rloNs To pHyslcAL sys-rEM s
1.The single-particle excitation energies fk measured relative to the
chemicalpotentialp.arewritten in theeflkctive-massapproximation
L = e:î- 12tkk'iPlkk')pl,- p.
k'
;>e2+ &tkl- (4,+ &(ks))
a;â2(2rn*)-l(k2- k/)
(43.3)
2.Since the resulting gap t
â ismuch smallerthan es,theintegrand in Eq.
(43.2)issharplypeakednearJ= 0,anditisthenpermissibletoset
z
k aràks> ,
'
X
(43.4)
In thiscase,thegap equation becomes
1=
1 d3k
:-îkF*XZ(x):ik*Xd3x
i (2=)) (.
âa+ (âc(kz ypyzmvjzjl
-
where ksisan arbitrary wave vectorlying on the Ferm isurface. The angular
integrationsoverdûkand lDxcan now be evaluated to give
1 zm. c
o
c
o
sin(x(ksx))
= 'rrh2kj okFdxsin(kFx)Prtxl f)KdK (zX
y
.2+ (Kz- 1)zJ.
(43.6)
wherethefollowing dim ensionlessvariableshave been introduced
ïâ
u:
2!H hzk/ +- a
k
K =g
/2rn
s
s
W eareinterestedin thelimitofthisexpressionasâ ->.0whenthesecondintegral
inEq.(43.6)canbeevaluatedas
=
gdg sin(A-(k,x)) .
lc (
â2+(Kz-1)2)+;-.
(
,jjn(,
8j.jo
lkrxQ
JA(j.cosAljsjnksx
* dl
+(jzgyy-j-sinjcosksx(43.
8)
ThefniterangeofthepotentialF(x)ensuresthatthequantitykex isbounded;
hencethedominantbehaviorofEq.(43.8)arisesfrom theflrstterm,andweshall
Write
co
lj
1
jg j 8
Forany smallhnite .
i,the validity ofthis approximation can be verifed with
Eq.(43.8)(seeProb.11.12).
w e can now complete the solution ofEq.(43.5). W ith the dehnition
kpj0
*sinksxF(x)sinkpxd
.x-(ks!I
,
'I
(
?
us),
(43.10)
NUCLEAFI MAU ER
38B
Eqs.(43.6)and (43.9)become
-
1
-
x
2m *
ln
8
(43.11)
(/fFIP'+k,)1-+0=hlV i
The exponentialofthisresultyields
i;
4;8exp - =h2k//2v*
.
(43.12)
Lkt,IFrn,)
and theenergygap in nuclearm atterisgiven in usualunitsas
/
=h2/c//2rn*
â = 8hl
ztyk
tv exp - )g jyzpyy
-j
-
.
y
j
(43.13)
,
A crudeestim ateofthisquantitycanbeobtainedwiththenonsingularsquare-well
potentiallitto 1S(
)scattering (Fig.41.2):
(43.14)
W e take thevaluem *t'
m = 0.65 from the discussion ofnuclearm atterand End
theenergygapshowninTable43.1. Sincees
*= (m(m*)esQ;65MeV,thequantityu
;
îdehnedinEq.(43.7)isindeedverysmall.
The resulting energy gap at the equilibrium density of nuclear matter
(/cs= 1.42F-1)isverysmall,thusjustifyingourprevioustreatmentofthebulk
properties. ln particular,the calculated .
t
ï ism uch sm allerthan both the gap
observed in thespectra ofeven-even nucleiand the pairingenergy in the sem i-
empiricalmassformula (forthe heaviestknown nuclei). Itmustbenoted.
however,thatthegap hasbeen evaluated in nuclearm atter,whereasthe actual
pairing energy in snite nucleiarisesfrom thenucleonsin the surface region of
m uch lowerdensity. Table 43.1 shows thatthe gap depends strongly on the
density.and becom esaslarge as2.5 M eV atks= 1.0 F-'. Thisestim ate is,of
course,only very crude. Emery and Sesslerlhave used the Bethe-G oldstone
equation to obtain much more realistic valuesof(/cslFlnsl. Nevertheless,
theirvaluesforu
'
X arevery similarto thosein Table43.1.
Table 43.1 The enerqy gap in nuclearm au er
#ortw o nucleardensi:ies
i,M eV
1.42
1.(X)
9.3x 10-2
2.5
iV.J.Erneryand A .M . Sessler.Phys.Retl.,119:248(19* ).
APPLICATIONS TO PHYSICAL SYSTEM S
386
PRO BLEM S
11.1. (J)Ifasquare-wellpotentialofrangedhasaboundstateatzeroenergy,
usetheefective-range expansion kcot3c= -1/c+ !.r(
)k2to provethatrf
j= #.
(b4Ifthehard-coresquare-wellpotentialshownin Fig.41.5hasabound state
atzero energy,provethatro= 2b+ bw.
11.2. (J)Assumethenuclearinteractionsareequivalentto a slowlyvarying
potential-U(r). Within anysmallvolumeelement,assumethattheparticles
form a noninteracting Ferm igas with levels Elled up to an energy -B. In
equilibrium ,B m ustbethesam ethroughoutthenucleus. From thisdescription,
derive the Thomas-Fermiexpression for the nucleardensity n(r)= (2/3*2)x
(2v/â2)1 (U(r)- .
8)1.
(bjDerivetheresultsofpartabybalancingthehydrostaticforce-V' and the
forcefrom thepotentialnT U.
11.3. Thesymmetryenergyf4/a4 (Eqs.(39.8)and (39.12))maybeestimated
asfollows. AssumethenonsingularpotentialofEq.(40.10)and computethe
expectation valueofX in the Fermigasmodelfor4= Z + N nucleonswith
3= (N - Z4lA # 0.
(J)UseEq.(40.9)toprovethat
E
....
j.32(h
zz
mk.
/-k
=),
sF(*RM+f
'w'
.
/dtkz'zllza(
sj
wheretheeflkctivemassatkFisgiven byh2kF/m*= (#(e2+ Ulkjjjdktk.kyand
t7(k)isdesned by Eqs.(40.17)to (40.19). Discussthephysicsofthisresult
and comparewith Probs.1.6and l.7.
(:)W ith m.Q$0.65 (Table 41.1) and the potentialof Fig.41.2,show that
th = 37M eV.
11.4. Prove thata two-body tensor force with Serberexchange F = P-ws'
jzx
M1+PMjmakesno contributiontotheenergyofaspin-è isospin-èFermigas
(i.e.,nuclearmatter)inlowestorder.
11.5. Given arepulsivesquare-wellpotentialofheightFcand rangeb,ifuIr
istheI= 0wavefunction,prove thatFN-->.W3(r- bjforFc->.c,whereW is
aconstantthatcan depend onenergy.
11.6. Carry outan exactpartial-wavedecomposition ofthe Bethe-Goldstone
equations(41.5)and (41.2)forarbitrary P. Show thattheeven-/wavesare
coupled,asaretheodd-lwaves. Estimatethelowestorder(inc= krajinwhich
thiscoupling aflectstheground-stateenergy ofa hard-sphereFermigas.
11.7. U setheBethe-G oldstoneequation forapurehard-corep-wavepotential
toprovethatELC
-LIA = (â2k//2v)(c3/=). ComparewithProb.4.7b.
NucuEAR M AU ER
:87
11.8. Thecom pressibilityofnuclearm attercan bedehned as
Kv
-lO k/Wztf/Wl/tf/c/lequilibrium
(u)EvaluateKvapproximatelyfrom Fig.41.7forthetwocasesshown. Relate
K vto the usualtherm odynam iccompressibility.
(b)Comparewith thecorresponding valuefora noninteracting Fermigasat
the sam e density.
11.9. TheA particleisabaryon w ith strangeness-1,and henceisdigtinguishable from the nucleon. Show thattheenergy shiftdueto theintroduction ofa
hard-sphereA particleinto a nucleargasofhard spheresisgiven by
Ec- h2
z.
/ bksa 8(/k
.sa)2 l l (?,)2 jl- (,
r
j241)2 1+,
r)
2/, 3,
x)2 lnl ,r)
7.H- =z j-kl-,
?)1 6,7 - '
.
,
-
+OtksJ)3)
wherea is the rangeoftheA-nucleon hard core (compare Fig.11.1),l/Jz=
l/vx+ l/va,and'
?
7= (An,
x- mslllm;k+ ms).
11.10. Considerthebindingenergy ofaA particlein the nucleus.
(J)Ifthenucleusisconsidered a square-wellpotentialofdepth Uzand range
R = rozll,show that the binding energy Bh ofthe A particle is given by the
solution to theequations
s(1-x)-+cot-'-(j.
x.x)*
h2=2
2 3
B a - Uz- a
sw ac(
l-.
y+p+''
-
where s= ((2p,
aA2/â2)Uc)1,x- Bhjlh, and l//u = 1/?'
?;A+ L(Ams. Explain
how tousethese resultsto identify thebindingenergy ofa A particlein nuclear
matter.
(b4 SupposetheA-nucleonpotentialisoftheform showninFig.41.5. Discuss
the calculation ofthe binding energy ofa A particle in nuclearm atterwithin
theframeworkoftheindependent-pairapproximation.t
11.11. Starting w ith G oldstone's theorem , show that the Bethe-G oldstone
equations(42.8)to (42.11)sum thatpartofthe laddercontributions to the
ground-state energy shift where the interm ediate pair always propagates as
particlesabovethe Ferm isea.
::Forareview ofthissubjectseeA.R.Bodmerand D.M.Rote,ttproceedingsoftheInternationalConferenceon HypernuclearPhysicsy''vol.II,p.521,Argonne NationalLeaboratory,
Argonne,111.,1969.
agg
APPLICATIONS TO PHYSICAL SYSTEM S
11.12. (X VerifyEq.(43.8).
(b) Retaining allthetermsin Eq.(43.8)andusing thenonsingularsquare-well
l5'
t
)potentialofFig.41.2,solvethegapequation(43.6). Show thatthenumbers
in Table43.1arereduced byapproxim ately afactorof4.
12
Pho nons and Eleclrons
O ur previous discussion of the interacting electron gas treated the uniform
positive background asan inertsystem whosesolepurpose isto guarantee overall
electrical neutrality. In this chapter we now investigate the dynam ics of this
background and the interaction between itsexcitation m odesand theelectron gas.
The resalting Debye m odel of the background form s the starting point for a
generaldiscussion ofcrystals.while the m odelelectron-phonon system provides
an excellent basis for the study of metals. In realsolids.of course.the ionic
background form s a lattice.
'fortunately this discrete structure is unim portant
for acoustic norm al-m ode excitations w ith a uavelength Iong com pared to the
interionicspacing(whichwillbetrueofalmosta1loftheexcitationsofimportance
here).
389
390
APPLICATIONS T0 PHYSICAL SYSTEM S
4K THE NONINTERACTING PHONON SYSTEM
W eapproximate the background bya hom ogeneous,isotropic,elasticm edium .
Itisknow nfrom thegeneraltheory ofthem echanicsofdeform ablesolidslthat
such a medium can supportboth transverse (shear)and longitudinal(compressional)waves. Onlythclongitudinalmodegivesriseto thechangesindensity
thatare necessary to m odify the coulom b interaction between the electrons and
the background. Thusforthe presentpurposes we furthersim plifythe m odel
assum ingthatthebackground hasnoshearstrengthand iscom pletelydeterm ined
bytheadiabatic bulk m odulus
B*-Fê
P#
Prjs
(44.j;
justasifitwereauniform fluid. W eshallassumeBtobegiven,althoughitis
in fact determ ined by the potential energy of the lattice.which depends selfconsistently on 170th the interatom ic and electrostatic interactionsbetween the
ionsand on the interaction w'
ith the electrons.
The large ionic m assleadsto an importantsim plihcatiol'thatallows usto
treatthe problem in two distinctsteps. First.the smallam plitude ofthe ions'
motion meansthattheionsm aybeconsideredfixedattheirequilibrium positions
in calculating the behaviorofthe electrons. Second,the low-frequency ionic
m otionisthen determ ined from thechangein energy associated with asequence
of such stationary ionic consgurations,assum ing that thu electrons have the
wavefunction appropriateto theinstantaneousionicconhguration. To agood
approximation,it follows thatthe electronic system and the ionic system are
decoupled.z A sa sim ple exam ple ofthis separation,we recallthecalculation
ofB foradegenerateelectrongasina uniform positivebackground (Sec.3). In
thatcase,theground-stateenergy E(r,)iscalculated asafunction ofthedensity
ofthe background;the bulk m odulusattheequilibrium density isthen propor-
tionaltothecurvatureofE(rs)at(r,)mi.= 4.83,andwe5nd
e2
(#)mi.= 4.5x 10-52 = 0.66x 101odyne/cm2
/1
The discussion ofSec. 16 showsthatthe variationsin mass density 3pm
obeytheequation ofm otion
1 :23
? Dt2 pm= V2&pm
(44.2)
'G.Joos,t*-rheoreticalPhysics,''2d ed.,chap.VIII,HafnerPublishingCompany, New York,
1950.
2ThisistheBorn-oppenheimerapproximation EM .BornandJ.R.Oppenheimer.Ann.'/l
'.çfà,
.y
% :457(1927)).whichisdiscussed,forexample.inL.1.Schifll.touantum M echanics,''3ded.,
p.446,McGraw-l-lillBookCompany.New York,1968.
PH O NO N S A N D ELECTRO NS
391
where
2
B
B
(
y zzcP
...
-un
:lki.
n.
v
?n(
)c
isthelongitudinalspeed ofsound and
/7moK M /70
(44.
4)
is the m assdensity ofthe background. The solutions to this wave equation
describe the longitudinalsound waves. W e m ay again use oursim ple m odelof
a degenerate electron gas at the equilibrium density to 5nd the theoretical
expression
2 c2 2 (0.916)2
c = jtzo9,: ktj-é--f)
.
For M = 2?m p,uzhich is appropriate for sodium . this expression gives ctj,=
l.Ix 105cm/sec.in reasonable agreement with cexp= 2.3 x l05 cm/sec,w'
hich
wasevaluated withtheobserved valuesforsodium 1# = 5.2 x l0l0dyne/cmzand
pmo= 0.97 g,
.
'
cm 3. Sineethe theoreticalcurve Elrxj(thelirsttwo termsof Eq.
(3.37)J is a variationalestimate that represents an upper bound on the true
ground-state energy,ourcalculation presum ably gives too sm alla curvature at
the m inim um and thustoo sm alla value forB and c.
LAG RANG IAN A N D HA M ILTONIAN
A com plete description ofsound wavesisreadily obtained from thelagrangian
forthesystem . Foreachpointinthem edium wefirstintroducethedisplacement
vectord(x)thatcharacterizes the displacementfrom the equilibrium position.
The change in volum e of the elem ent dxdîbdz under deformation then can be
com puted to Iowestorderinf/xlxlzfrom
(
3#- PJ. Pd.
dF'=#x'd)'dx'=#xJy'Jz(l+o.
x
A'
i
-kj,+0z-+'''
,
1
orto firstorderin derivativesofd
#l'''= JU'(1-c.V .d)
To the sam e order,the change in density is
bL,n= 3
,
-- = --V .
d
-
9mQ
S0
(44.5)
and the wave equation can therefore be w'ritten in term sofd as
l 02d
'
V.gzjj.
y-V2dj=(
)
(44.
6)
'esAmerican Institute ofPhysicsHandbook,''2d ed.,pp.2-21and 3-89,M cGraw-ldillBook
Com pany,New York.I963.
z:z
Appuscga joNs To pHvslcAu SVSJ EM S
In an elastic m edium w'
ithoutshearstrengthand vorticitl'. d satisfiesthe general
condition
lanypaIt
:
which m eans
V x d==0
ThusNse conclude
1t
i2d
'
'
VX((
.2o/c--V'd)-0
1
-.t
,-.
-r -- l'
L
(44.l1a)
'
(/jedi ?t( è(I0='
!
'.
i
'tï3.
V$p,,
(
,('??
yot--Xav,àl-y-y1
'
.
(44.llbs
where repeated latin indices are sum m ed from 1 to 3. The Euler-Lagrange
equati(
717s),ield Eq.(44.9).5$hiie Lq.(44.10)plalsthe ro1e ofa subsidiary con.
dition guaranteeing longittldinalU a:es. The usualcanonicalprocedLlre alIow s
us to derive :
1 ham iltonian.and NNe find
ld
<tx.J1s
t
z:/7,'()g
-r
'
Ht
)--!(tl3x(p,
-u)=2- #(V .d)2)
whereEq.(44.l0)hasbeen substittlted illto thesecond term in Eq.(44.1Ibband
the result integrated tw ice by parts. Introduce the llorm al m ode expansions
(we w'
ork in a large box ofvoIume 1,'and use periodic boundao'conditions)
Pd
j
=(x.?)=pa,
t
ljjE!
E.
--(.
%Itl
q
j) (h
gcs
ovjlg
k(ck6
4
ik.x-i.,
j
.
t.(,
t.j
k
.
,-fk.x-ic
t
?
k,)
.
k î.
(44,l4)
lForadiscussion ofcontinuunnluechanics. seeH.Goldstein.''Classi
calMechanics,-'chap.l1,
A cldison-W esley Publishlng Colllpanà.lnc.,Reading,M ass.. l953.
PHONONS AND
393
(44.15)
coy= ck
(44.16)
which now explicitly incorporate the subsidiary condition V x d= 0. After a
little algebra,the ham iltonian becomes
Hv= .
!.jklâcoytcf
kck-1
-t'
kC'1)
(44.17)
whïch represents a system of uncoupled lm rm onic oscillalors.' A lthough we
can im posecom m utation relationson = and d,the subsidiary condition makes
this procedure quite intricate'
,instead,it is m uch m ore convenient to consider
ckandtrlasOurindependentcanonicalvariablesbecausethesubsidiarycondition
is then explicitly included. Thus we shall quantize by using the canonical
com m utation relations
Cf-k,cl'1= Jkk'
(G .l8)
DEBYE THEO RY OF TH E SPECIFIC HEAT
Equation (44.l7) provides a basis fbr investigating the thermodynamics and
statistical m echanics of the free phonon system . W e Erst observe that the
chem icalpotentialofthe phonons vanishes
(44.l9)
which m ay be seen in the following way. C onsider the H elm holtz free energy
computed foralixed numberofphononsF(F,V,Nps). Sincethereisno restriction onthe num berofphonons,theequilibrium stateofthe assem bly atfxed F
and Iz'is obtained by m inim izing the H elm holtz free energy
èP
N/ph-j
.(;
,!rpr
(44.x)
Thisexpressionisjustthechemicalpotential(Eq.(4.6)1,therebyverifying Eq.
(44.19). Thethermodynamicpotentialforthiscollection ofbosonsisgiven by
(compareEq.(5.8)1
D
/ 4,.
o=-kBTlnTrexpj-y
j
.
sw
k
'j
t
ao-k.vJ/tlng1-exp(-k
jo;)1+zh
k'
:k
w)
(44.21*
(44.21:)
where theextra term com esfrom thezero-pointenergy in theham iltonian
lîo= Jk(hulk(clck+ I.
J
lThisisthereasonforchoosingtheparticularcoemcientsappearinginEqs.(44,14)and(44.15).
APPLICATIONS TO PHYSICAL SYSTEM S
Sincerzp:= 0,weim m ediatelyderive
oe--z'p'-E-vs-E+r(.
P
p
fr
l
?j,
E--rzjP
j.(Py)F
,-zkj/
jt
-,
.exptj
tB
oa
w
k)-l-'+pt
wj
(44.
22)
(44.23)
Thesum can beconverted to an integralintheusualfashion
Zk --*J#a'#(t,a)
(44.244)
#(tDl= (2R2)-1ZC-3œ2
(44.24:)
(44.24c)
ln a uniform medium sthe frequtncy u)has no upperlim it. ln a realcrystal,
however,it is clear that the wavenum ber of propagation cannotexceed the
reciprocal of the interparticle spacing. W e can determ ine the m axim um
frequency u)o byobservingthatthetotalnum berofdegreesoffreedom in areal
crystalis?N ,whereN isthenumbtrofions. Thus
?N=j9
'
E
'
Dgtoallf
.
z
y
(44.25)
which gives
g(fx?)dœ =
9N o)llf
x)
a
(44.26)
Defining the dimensionlessvariable u= âtza/ksF and the Debye temperature
# huln
= ks
(44.27)
wefindl
y 3 #/r x3du
gxksp
E-gxksz'
(j)j,vu.j+ s
c.-(D
PT
f)y
,-9xks(F
.
j-)'j(
'
)
'T(e
&4'
#'d
1)
&2
.
(44.28)
(44.29)
which istheD ebyetheoryofthespeciicheat.2 Equati
on(44.29)hasthefollowinglim itingvalues
Ck = ?Nkg
lNotethattheonlyquantitydependingon thevolumein theseexpressionsisœo.
:P.Debye,Ann.Physik,39:789(1912).
(44.30/)
PHONONS AND ELECTRONS
395
p 3 . x4eut/x
Cz= 9Nks #
12=4
=
z
() (
c'- 1)
F 3
5 Nka .
j.
(44.30:)
Thefirstisjusttheresultoftheclassicalequipartitionofenergyandthesecond
is the fam ous D ebye 7*3 law. The D ebye theory provides an excellent oneparameterdescriptionofthespecischeatsofcrystalsasisevidentfrom Fig.44.1.1
6
4
C;z
Pb
e= 90.3
Ag
#= 2l3
AI
2
0
1
p=389
C (di
amond)
e= 1,890
3
2
logjoT
Fig.44.1 Theheatcapacity'in caloriesperdegreeperm olefor
severalsolid elements. ThecurvesaretheDebyefunction with
the # values given. (From N, Davidson. '*statistical
M echanics.''p.359,M cGraw-ilillBook Company,New York,
1962,afterG .N .Lewisand M .Randall,AtTherm odynamics,''
2ded.,revised byK.S.Pitzerand L.Brewer,p,56.McGraw-Hill
Book Company,New'York,1961. Reprinted by permission.)
In Table 44.lwe com pare som evaluesof$ determ ined by fitting speciflc heats
with the valuescalculated from the elastic constantsofthe m aterial.z
Table 44.1 Values of Debye tem perature $ in 2K obtained from therm al
and elastic m easurem ents
r
j Fe
i
(
Thermalvalue
1
Elasticvalue(room temp.
) I
El
asticvalue (absolutezero) 1
1
398
402
488
315
332
344
215
2l4
235
Source:G.H.W annier,û*statisticalPhysics,''p.277,JohnW ileyand Sons,Inc.,
New Y ork.1966.
IM etalshaveanadditionalelçctronicspecificheatproporti
onalto T(seeEq.(29.31)),which
becomesimportantatvery low temperatures(;
4:1 K).
2Asw'e havepointed out.an elasticsolid with shearstrengthcan supportthreetypesofsound
waves:two transverse m odesand one longitudlnalmode ot-the type consldered here. Tht
onlycflkctontheabov'
eanalysisistoreplace1(-3inEq.(44.24:)by(2c)=-1c?bH 3.ds, This
merelychangestherelationot
-cootothedensityandspeedofsound (Eqs.(44.24:)and(44.26)
);
Eq.(44.26)andthesubsequentanalysisremain unchanged.
396
APPLICATIONS TO PHYSICAL SYSTEM S
45rTHE ELECTRO N-PHONON INTERACTIO N
Oursimplemodel(theDebyemodel)forthedynamicsoftheuniform background
allowsusto investigate the interaction between theelectronsand the collective
m odesofthebackground. Theinteraction ham iltonian isthatofSec. 3
HeI-:- J#3xJ3x'pel
,(x')
jt
xx)px?
j
(45.1)
-
wherep,= zen isthebackgroundchargedensity and zisthevalenceoftheions.
Withthedefinitions(seeEq.(44.5)J
Pb= Po+ bPb
(45.2)
(45.3)
3p,= zebn = '
-zenoT -d
Eq.(45.1)becomes
Hel-:- Hz
el + ftf3xt/3x'fzd(x)3p.(x')
b
Ix- x,I
-
Thesrstterm hasalreadybeenincluded inthehamiltonianN el-gasOfthe electron
gasin an inertuniformlycharged background (Eq.(3.19)J;the second isthe
electron-phonon interaction. To exam ine this quantity we use Eqs. (2.8)and
(45.3)andsubstitutetheieldexpansionsintheSchrödingerpicture
Wx)= 1)
P'-+edk*xynckz
kA
(45.5/)
1(x)-
(45.5:)
-
i
/i lk
(%ru)1 k 2f,pkP' I(ckeik*x-clc-ik@x)ptf
x)s- (
.
oyl
.
wherefooistheDebyefrequency. (NotethatEq.(44.15)isin theinteraction
picturewithrespectto.
?%,andthattheDebyemodelcutsofrthephononspectrum
ata wavenumbercoojc.j Thisprocedureyields
Xel-p,-Jd3xdqx'XI(
X)3%(x')
jx
x/!
(45.6/)
.
zel n 1
/.
).
.2
el-ph= c .
M
.
kA q
hœ 14,
v
q q
,o.q. Aav, zo
2P'
-j-#((so- f.sqj(uk
+ J1zJkyq.ACq
f1 (45.6:)
The electron-phonon coupling isproportionalto thecharacteristiccoulom b
interaction
&J(q)= 4r+2q-2
(45.7)
Ourinvestigationsin Secs.12 and 14 showed, however,thattheeflkctiveinteraction between two chargesin the m edium governed by A el-gasism odised bythe
dielectricconstantand becomesUclq)= &J(q)/s(t
?). In theapproximation of
PHONONS AND ELECTRONS
397
sum m ing the ring diagram s with repeated coulom b interactions between the
electrons(Fig.12.4),theqfRctil'
estaticcoulombinteractionbecomeslgseeEq.
(12.65)J
L'f
rlq)u='z4=e2-1-
ptl?<<
:1'z-
(45.8)
q '
V qvF
where the Thomas-Fermiwave numberisgiven by Eqs.(12.67)and (14.13)as
4=
u
--
vs
'9rr 1 77.
2
=
'
rrzaf
j
(.
y1,
.
,â
-j
s-'
k,
(45.9)
Itisclearfrom thisrelationthatqv
-lisoftheorderofthelatticespacing;since
thecutofTolnlcin Eq.(45.6:)isofthissameorder,wemaytake
&c
r(q)Q;4nelq.
t.i
-
(45.10)
foralm osta11phonon wavenum bersofinterest. W ith thesubstitution
4=
4=
4=
u.-.+ --j-. j- Q:.j
9
Q H'0Tr CTF
theelectron-phonon interaetion in Eq.(45.6)may berewritten as
4er-p,- yJJ3.
v'
;1(x)'
fxtx)(
/.
(x)
(45.1l'
)
where the following defnitionshave been introduced
1..- u 'Y .vli2.zv2
ZP.2 /'
FH fIQ$ un: n /l(j
g & -g g
.ys..
yy-) u
=s
''y).
-sh'X
(45.12)
(45.13)
N ote that the coupling constant y is independent of the ion m ass J.
/ and has
dimensionsof(energys volumeli'. Thenew phonon field ;(x)can bew'ritten
in the Schrödinger picture
/hco '1
f
/'
(x)= j.
jyZ
.j((
kFi
k*x-(
7
)p-l
kexj#t(
z)
s-(
skj
k
%
(45.14)
The density variations ofthe baekground also m odify its own coulom b
energy according to
-#3xd'
)x.
,P?s
(X)
X'
Hv= J.
$
--?-N(p
;)
(45.154)
IX - X !
1Although thisequationisstrictlyvalidonll'athigh densitiesfr,-m 1).anqoregeneraltreatment
merelyreplacest
/1.
,-- 312
')b#'(1p
2t.
s
'7.u'
heresistheexactspeedofsoundinthemodelconP1t
sidered in Sec.3 (D.Plnesand P.Nozi
ères.''TheTheorl'ofQuantum Liquids.'':70).I.-secs.
3.land3.4.NV.A.Benjamin,lnc.,Ncw York.l966).
398
APPLICATIONS TO PHYSICAL SYSTEM S
3a(x)bnlx')
J/'
,=H3+!.z2e2J#3x#3x' lx x,(45.l$b)
l
wherethesecondresultfollowsfrom J(/3x3n(x)= 0. Thefirstterm ontheright
.
-
side ofEq.(45.15:)hasalready been included in Hel-gzs. Furthermore,the
coulom b interactionin theintegrand ofthesecond term isagainshielded atlarge
distances,and its eflkcts atsm alldistances have already been included in B,
which isassum ed to begiven. Thisterm thereforewillbeneglected.
Inthisway,wearriveatthe(approximate)hamiltonian forthecoupled
electron-phonon system l
X =Pel-gas+Pp,+)?J#3.
'
t'#â(x)('a(x)t/(x)
(45.16)
whereXel-zaswaSdiscussed in Sec.3 and Xp,in Sec.44. Sincelîphdescribes
physicalphononswiththeexperim entallong-wavelengthdispersion relation,the
interaction term clearly representsthecoupling between electronsand physical
phonons.z Itm ust be rem em bered,however,that the bare electron-phonon
hamiltonianwith UJ(q)IEq.(45.6/9)hasbeenscreenedbysummingtheelectron
ringdiagrams,asshown in Fig.45.1. The resulting eflkctiveinteraction U,
f(q)
k+ q '
q
.
k
+ .- - + . .
- ..
&0
t(q)
k+ q
q
=
-% - --
k
tl
G --- +
&tr(q)
k+q
c
tl
--+ - -
k
Nx& r
t(0)
Fig.48.1 D iagramssummedin passingfrom :hebareinteraction &j(q)(wavyline)
to theshielded electron-phonon interaction t7;(0).
1F.Bloch,Z.#/?.y.
î,
'
/C.52:555 (1928);H,Fröhlich,Phys.Ret'
.,79:845 (1950);H.Fröhlich.
Proc.Roy.Soc.(London),215)291 (1952). The approach in thissection essentially follows
the latterpaper.
1Asan alternative approach,the ions and electronsare often considered a gas of charged
parti
cles. The low densityoftheions(h2)
'M e2<.
:ai
-ljthen l
eadsto theformation ofaW igner
lattice,whosebarelongitudinalphononsoscillateattheionicplasmafrequencyl4vnoz2elt
'
M lk.
Theinclusion ofring diagramsscreensthe singularcotllomb interactions,however,and yields
theobserved phononspectrum. Thismodelalso allowsan improved calculation oftheelastic
constants,which agree quite wtllwith experimentalvalues. See,for example,J.Bardeen
and D.Pines.Phys.Aerl..99:1140 (1955);T.D,Schultz,Keouantum Field Theory and the
Many-Body Problem,''chap.IV,Gordon and Breach,Science Publishcrs,New York,1964;
J.R.Schriefer,t'Theory ofSuperconductivity,''sec.6.
2,W .A.Benjamin,Inc.,New York,
1964.
PHONO NS AND ELECTRONS
399
is then approximated by the constant UC
r(0),thus yielding the renormalized
electron-phonon hamiltonian (45.11). W hen computing with Eq.(45.16),w'
e
m ust,ofcourse,om itthose diagram s already included in this renorm alization
procedure.
46E7TH E CO U PLED -FIELD TH EO RY
To sim plify theensuingdiscussion,thecoulom b interactionsbetween theelectrons
willbe treated in the H artree-Fock approxim ation,and weshallexam ine
VmOdelEEk21
1 CkYk/tV/- k21
( Ckdll/Yk2V k<)
( (f'1Ck'
V1')hl
Dk
> kr
< ks
u?o/c
+ y j-J3xka
ttx)'
;alxl(7
.
(x)
where t,kare now the Hartree-Fock single-particle energies ofthe electron-gas.l
The present treatm entconcentrates on the zero-tem perature properties ofthe
system ,and the extension to snite temperatures isleftforthe problem s. W e
now w ish to exam ine the G reen's functions for the coupled problem specilied
in Eq.(46.1). The motivation isthesameasbefore;theGreenesfunctionsgive
theexpectationvalueofanyone-bodyoperator,thepolesoftheG reen'sfunctions
givetheexactexcitationenergiesofthesystem,and theground-stateenergyshift
ofthe interacting system can be com puted from the electron G reen'sfunction
according to Eq.(7.30)
/.
r.
-.Evs
.
=-j'
vd
, im
-X I
,/4î (hq,,
y -eq)iGtilq)eiqo'l
o'
/' n.
->
o- (2*)
-
where Ev is the ground-state energy ofthe uncoupled electron-phonon system .
(ThereisnofactorofJ in thisresultsincetheinteraction ishereproportionalto
l
/'f?J.) ln addition,the Green'sfunctionsdescribe the linearresponseofthe
system to an externalperturbation.
FEYNM AN RULES FO R
Thegeneralfield theory ofChap.3appliesjustasbefore. In theinteraction
picture W'ick's theorem allow'
s a decom position into Feynm an diagram s. and
onlyconnecteddiagram sneedbeconsidered. W eshallsimplystatetheFeynm an
rulesin m om entum space foran arbitrary process.z
lThese energies mustbe appropriately smoothed atthe Fermisurfaceto take into account
theefrectofsumming theringdiagrams(seeProb. 8.2).
2Notethatthese Feynm an rules, because oftheirgenerality.aregiven in a slightl
y diflkrent
form from those in Secs.9 and 25. To construct the Green's functions iGxnlqjor iD(q),
an explicit Qctor (2=)48t4'(4 - q'
) must be inserted forone of the end lines(See Fig.9.10
and accompanying discussionl; for example, the lowest-order contribution to iDlqj is
Jd*q'?
'
(2rr)-4D0(#')(2=)4:t4)(t
?- q2)= /
.p044
/).
400
APPLICATIONS TO PHYSICAL SYSTEM S
1.Construct a1l topologically distinct Feynman diagram s,the basic vertices
being theem issionand absorption ofaphonon byanelectronasillustratedin
Fig.46.1.
p- q
p
Fig.46.1 Basicelectron-phonon vertex.
A ssign afactor--iyt'
h foreach orderin perturbation theory.
lncludeafactori(2=)'-4G0
a/$(q)foreach electronlinewhere
G0 (q.(/o)= 3.o ..p-(
.qf- ks)
-....
xb
.
. .-
0(/,
-s- q.)
.
,
,
t
:
/0- 6q.
h -.jyy qv--sq,,
Jj.
- iy
and take the spin m atrix productsalong the electron lines.
4. Incltlde:
1factor/420-)-4Dofk
qjforeachphonon line. To com putethephonon
PCOPa,
gatOr.V'
P r11tlStCXalT)ine
(46.4)
(46.5)
(46.6)
(2,,)4r
$f''(V
'
x,
'(1ij
6. I1
1tegratc()ver:1llintcrn:
111ines ((14q.
7,lncltldeafactor(-..1)f'wherc J-isthe nunnberofclosed ferm ion loops.
8. D iscard alldiagran1s that haN'c subunits connected to the rest ofthe graph
3no111i1)e asin Fig.46.2. T his resultfollowsfrom m om entum
b.y o11) onc ph(
pHosoNs AND ELECTRONS
401
conservation. which im plies that the phonon line m ust have q = 0,and Eq.
(46.6) shows that DQ(q= 0)EED0(0)v
-zO for any hnite y. M ore generally,
wenotethat.
D0(q= 0,ç0).
=0becausethefundamentalheld(
;isproportional
toV .(
1'
,hencetheintegraloftfoveral1spacevanishesidentically.
A ny connected
subunit
p
q=0
z
/
D0(0)= 0
p
Fig.46.2 Generaltadpolediagram,whichvanishes.
THE EGU IVALENT ELECTRON-ELECTRO N INTERACTION
These rules for phonon exchange allow us to put a1l the phonon-exchange
Feynm an diagram s for the electron G reen's function in one-to-one correspondence with the Feynm an diagram s ofan equivalentspin-independent potential
(Fig.46.3),which isgiven by1
L'
)(q.(
/n)EEE--ih-1y2iD0(q)= ,,2h-lDoLq)
(46.75)
(46.7:)
This potentialis now frequency dependent,even in lowestorder. In the static
limit,Eq.(46.7:)reducesto
Since ulw'c is com parablew ith kr,the upper cutofl-can be neglected in alm ost
a1lcasesofinterest. lfweassume &o(q,0)= -y2forallq,then theequivalent
interelectron potentialbecomesan attractit'
edeltafunction.
lfeqtx)= -y28(x)
(46.9)
'The phonon-exchange potentialbetween electronsw'assrstdiscussed by H .Fröhlich,Phvs.
Ret..,loc.ci:.
4c2
APPLICATIONS TO PHYSICAL SYSTEM S
The consequences of an attractive interaction between particles close to the
Ferm isurface have been considered in Secs.36 and 37.and they are explored
indetailinChap.13. ltisalsoclearfrom Eq.(46.7:)thatl.%(q.t?c)willceaseto
be attractive when the energy transferqo satisses I
çoh> trq. Since opq< œns
we therefore have
or()
tqstyolu-0
if1
t?(, o.u)o
(46.10)
ln the noninteracting ground state w here the electrons form a slled Ferm isea
&o(q,t/o)can be attractiveonly forthose electronslying within an energy shell
of thickness hujo below the Ferm isurface,because only those electrons can be
excited to unoccupied levels with energy transfer less than hulo. Sim ilarly,
(zltq,ç(
)lcan beattractiveonly w'
hen a pairisexcited to unoccupied stateslying
within an energyshellofthicknesshuloabovetheFerm isurface.
VERTEX PARTS AN D DYSO N'S EQUATIO NS
The graphicalstructure ofthe coupled-held theory can be elegantly sum m arized
in a setofequationshrstderived by D ysonlin connection with quantum electrodynam ics.z These nonlinear integralequations involve the exact propagators
and vertices. W 'hen iterated consistently to any given order in the coupling
constant,they reproduce the Feynm an-Dyson perturbation theory. N evertheless, Dyson's equations are m ore than a convenient sum m ary of perturbation
theory,and theirgenerality is often used to seek nonperturbative solutions.
The proper electron self-energy hasalready been discussed in Sec.9. The
properphonon self-energy Il* isintroduced in an exactly analogousfashion.and
leadsto equationsthatare form ally identicalw'
ith those oftheeflkctive interaction
propagator ofSec.9
G = G0+ G0l:*G
D = D0+.,
:,2h-1DQ11*D
(46.11)
(46.12)
where 11* iscom puted with the effective potentialofEq.(46.7). There isone
im portant new elem entahowever,known as a t'ertex part,defined to be a part
connected to the restofthe diagram by two ferm ion lines and one phonon line.
Two exam ples are show'
n in Fig.46.4. M om entum conservation im pliesthata
1F.J.Dyson,Phys.Reu'..75:486(1949);75:1736(1949).
2ln fact,the Feynman-D yson perturbation theory ofquantum electrodynam ics isform ally
identical with that Of the electron-phonon problem .the m ain dipkrence being the explicit
form Ofthe propagators. In addition.quantum electrodynam ics has no diagram s with an
Oddnum berOfphotonlinesconnectedtoaclosed ferm ion Ioop'
.suchprocessesvanishbycharge
Conjugation.inaccordanceWithFurryfstheorem (W .Furry,Phys.Sf'
r..51:125(1937)). F0r
qtlite a diflkrentreason (See rule8 Ofthe Feynman rul
esin thissection),theelectron-phonon
System haS no closed electron loops connected to the restofa diagram by one phonon line.
Consequently.neitherfeld theoryhastadpolediagram s.
PHONONS AND ELECTRONS
403
Proper
(a)
1n3proper
(b3
Fig.46.4
vertex partdepends on two independent m om enta.and allspin indices nlal be
suppressed because the interaction is spin independent. A proper l'
tar/cx par:
is desned to have no self-energieson the externallegs ofthe N'
ertex. Thus the
second diagram in Fig,46.4 isim proper,forithasself-energy insertions on both
theferm ion and phononlegs.
To obtain Dyson'sequations.wenow attem ptto writethe integralequations
for the G reen's functions in term s of the proper vertex and proper self-energy
Take an arbitrary Feynm an diagram with an)
. num ber of externallegs
The rem ainderis known asan irreducible or5'
/&'c/t'/(p??(liagratït. The irreducible
diagram s for the proper self-energies and vertex parts 'llt,l'lsL'il'es are defined to
be those diagram sremaining afterthisprocess has been carried outl?p to the
pointwherea furtherreduction would lead to a sim ple lineorpoint.respectively'.
Some exam ples are shown in Fig.46.5. The lowest-order fernnion self-energy.
phonon polarization part,and vertex correction are thusdetined to be theirou 17
skeleton diagram s. W'e now claim the fo1Iowing:
l. Forevery graph there is a ulliquc skeleton.
2. AIlgraphs can be builtup by inserting the ekactG reen's function G.phonon
propagator D.and propervertex part F in aIlthe skeleton diagram s.
404
APPLICATIONS TO PHYSICAL SYSTEM S
D iagram s:
hh
Skeletons:
Fig.46.5 ReductionofFe)nnlan diagram sto theirskeletolls.
).
S*(t?) - x
x
11*(t?) - x
x
X
X
X
Ia ëux .
X
=z
+
4.
X
X
X
X
+
x
X
X
+
+
xx
x
Fig.46.6 Dyson'sequationsforthe electron-phonon system .
F'iioNo Ns AN D ELECTRONS
405
il
-orexam ple.both )-:*(q )and Il*((1)has'
e on13.
,one skeleton.thatshou 1
) i1)
Fig.46.l
qa and b.and aIIcontributions to Y*(f/)and Ii*(t
y)are obtained b),'
inserting theexactG.D.and 1-in these skeletonsasindicated i11Fig.46,6. N ote
that the exact proper vertex m ust bc inserted aton1),' on e end of these s'
keleton
diagram s.
'otherwise some diagram s u i11be counted tw ice. Forexam ple.there
is only one graph of the type shou 1 ln Fig.46.5/?. u hereas this graph B ou1d
appear twice ifue were to insertthe exactN'
ertex i11both ends in Fig.46.6.
T he rennaining problem isto u rite an equation for the exactproper Nertex.
U nfortunatel).'.this aim ca11notbe achie:ed in closed forn1 bccause :here is an
intinite 11um ber of irreducib1e sk'eleton :ertex diagram î
i. Nve m ust therefore
resortto a seriesexpansion and haNe shou n lhe firstfeu term sin Fig.46.6.1 The
h-.
NW >
Fig.46.7 The rroperNertex l-toordern5,
1
)
2EEE'
)z(1 ..J'f2)-- )
I'N(4)-'-.
To iterate these equations through order y3 in the interaction strength. the
propagators and vertex correction com puted to order :,2 m ust be inserted in
the second diagranl.uhile the lou'
est-order resultm aq'be inserted in the last
three diagram s because they are alreadj explicitly of order y'. This iteration
procedure yields the expansion sho::1
: in F'ig. 46.7.NNhich contains aII proper
vertex parts through order :.
z5. h 'e can now com bine these results to obtain
7/equations shown in Fig.46,6. A lthough
the nonlinear coupled D b'.
%ollp
'
?7lcgr
, (
we have not derived them .we claim it is plausible thata t't?l?.
5'
l'
.
5
'/t
;'
r7;iteratioll
q/- the D 1.j'
t?,7 intezral t
a/
?l/tz?I
'
f?l?.
s lo tz?7J' gitoetl order //7 y reprodtlces J/?(a
ftzI.
????7t7??-D 1.5//?perturbatiollr/pt>t
'
)r1..
4û6
APPLICATIONS TO PHYSICAL SYSTEM S
The com ponents of D yson's equations are the exact G reen's functions G
and D and theexactpropervertex r. ltisthereforepossibletoattem ptadirect
solution of these equations without resorting to perturbation theory. For
electrons in a norm alm etalinteracting through phonon exchange,such an
approach can be greatly sim plified by the rem arkable theorem derived by
M igdalt
r=yl+OV
mjl
(46.j4j
wherem(M isthe ratio oftheelectron massto the ion mass. To an excellent
approximation,Eq.(46.14)allowsustoreplacetheL?prr:x by thepointtw/l/èy,
and the problem is thusreduced to the coupled equationsforG and D . The
detailed discussion ofthe corresponding solutions isbeyond the scope ofthis
book,and thereaderisreferred to theliteraturefordetails.z W eshall,however,
conclude thischapterwith a discussion ofM igdal'stheorem .
47J M IG D A L'S TH EO REM 3
M igdal's theorem states thatthe exactvertex in the eleetron-phonon system
satissesEq.(46.14),wherem(%Iistheratio oftheelectron massto theionmass.
W e shallnot prove thisresultto aIlorders butm erely show itto be true for the
hrst vertex correction. This calculation illustrates the physicalideas involved
in the theorem ,and the extension ofthe proofto higher ordersrequiresno new
principles.4
p+q
p + q- l
I
p- l
#
Fig. 47.1 Second-ordervertexcorrection.
Thesrstvertexcorrection,whichisdenoted Pt2)inEq.(46.13)andisshown
in Fig.47.1,can be w ritten with the Feynm an rules
/
Pt2'(p+
2.k
..). #4/
(
.oj
.4.
q,p4= ..
y
--17
(2=)kIk- (-(
,&T)c
lA . B.Migdal,Sov.Phys.-JET1L7:996(1958).
2See,forexample, J.R.Schrieflkr.Ioc.cit.:G .M .Eliashberg,Sot'.Phys.-JETP,16:780(1963).
3A .B.M igdal.Ioc.cit.
4ln fact,M igdaliscontentto assert'*Itcan beshown thatthisestimate isnotchanged w'hen
diagramsofa higherorderaretaken into account.
'' (A.B.M igdal,Ioc.cl
'
J.)
pHoNcNs AND ELECTRONS
407
Here the integralisnow dim ensionless,allwavenum bersbeing m easured in units
ofksand al1frequenciesin unitsofeF
o/
'h= hky
lf
lkm ,and
5-(q)EEEf''-l
IqJ> l
t-l (q1< 1
In these units the phonon frequencies are proportionalto the factor(mlakljk
glpy7.i B i
a,.....
-
(k
$
j)(,,
-al?
lp
and themaximum wavenum berQaxEHStt
?s cisapurenum berfsee Eqs.(44.24:)
and (44.26))
/maX= (6/
'z)i
(47.
4)
N ote that 1-Xt2)depends on the ion mass only through the factor coland thus
through the sound velocity. ln contrast,the bulk m odulus B depends only on
the potential-energy surface for the crystal and is independent of M . The
phonon propagatorhasafactorofo)lin thenumerator,andintegrationsover
frequenciescanreducethisatmostbyonepower(thatis.(2=s
')-lj-dIot,?/g/à((z?,- /z7)2)-1= -!.
uj),
.consequently,the remaining expression for the vertex
correction isstilllinearin (z)land hencein (m(1
%1)1. Thisobservation provides
the basis for M igdal's theorem .
W e now verify thisresultexplicitly for1M62)by j
lrstcarrying outthe integral
overQ,closingthecontourineithertheupperorlowerhalfoftheQplane. After
som e familiar algebra,the resultcan be written in the form
-
'
pn+qzs----- (--- i'sbsq-v+ q- 1)1)t4?.5'
-
-
lfthequantityinbracesisdenotedby/tf
.
,?ll,thisexpressioncanberewritten
(47.6)
Providedthattheintegralcontaining/to)converges,theIimitingexpressionfor
small (rrl/.
#/')1 can be obtained by neglecting the second term containing
408
APPLICATIONS TO PHYSICAL SYSTEM S
/'((s))- /-(0). The proof is therefore reduced to showing that the limiting
.
.
expression
.3jay?y'.
i./ s !- j
'
v d3/
l-'
(2'(p -+.
-q.p);4rV2/k
v
ac
-Fj-jj
y
jjj.
j(
'
-j-z/j
-.
3/8(Iî
nax-/)
(;sy .
.
iswelldefined,fortheexplicitfactor(,77.31)1'infrontofthedimensionlessintegral
then givesthe desired result.
The expression of Eq.(47.7).howeNer. is as u'
elldetined as any integral
over ferm ion G reen-s functions thatappears in the r = 0 theory. The D ebye
cutof'
flimits the m om entum integralto a finite region.thusrem oving any divergences at large /. In addition.realcrystals haN'
e a sm ooth ctltol
'l-at the upper
end ofthe phonon spectrunè so thatally logarithm ic singularitiescaused by the
sharp Debye cutoil-are spurious. Ftlrtherm ore. the infrared divergence of
Eq,(47.7)atqv= q = pv= p = 0 gwhich mal'give rise to term sproportionalto
(/??,'.$J')1ln(/3?,
.51)1jisirreleN'
ant because thc weak phonon interaction is only
importantforelectronsnearthe Ferm isurface where ;p :
4;l.
The only rem aining source of dipicultl'arises from the singularitiesin the
integrand in Eq.(47.7). Although each one indlvidualll'gives rise to a énite
integral,the resulting expression m ay diverge u'hen the tw'o singularities com e
together. W e m ay isolatc the contribution of this regton with the following
seriesofsteps.
l. The inénitesim al il'
naginary term s in the denom inators are relevant only
wheretherealpartN'anishes. Thusu'emay replace5'(p .-1)and 5-(p = q- 1)
w'ith .
p(pv)and .
v(/?(
).,-qù)sw'here.
,(p())isdet
ined by
(
-l
lrçjqEEE'
t-i
1
o
.-
Here 6,.< ep- ,and b
'
?ev.
'è.p:isassum ed positive.
2. Theradialpartot'theintegrandin(47.7)containsthefactorl3#/,whichweights
the regioll/;
47Quxm ostheaN.ily. Asa sim pleapproximation wereplace one
factoroflbyitsNalueat/;?
:uxandthentransform tothenew'integrationvariable
tHEp - 1. Since the rem aining integralover t converges if e:o:t1at large t,
we extend the tintegration over allspace,
3. The energy 6t..q in the second denom inator is expanded abot1t the vaIue 6,
409
2k)
'
-x
t2d;
ds -i
r'
s
(2)
(,.q,y?
),
ej
V.
n
.
--e
1
1O
-,
*qPJ/opn-'
-:
-?
'.
iT
-vfpo
-bqoj'j
.
--!
e-+-j.dE.,/
'
?.
+-i
xlalJye--qyl
-.
%,8x lnC?.
- - .
-
-
:,- q)c;/J/+.lyulpv... )
4. To study the region where the singularitîes coalesce,w'
e evaluate the slowly
varying functions tz and Pef?
/olatthe position ofthe tirst singularity,e?= pf
j.
W ith thenew variable (EEEE'
f- p0,Eq.(47.8)reducesto
1-32)(
ck3 .t
.
?(Fj-2 ag,
-.
r
-P .
p-q,p):
4;-j
tmyC
sy(jp'),
r.s,
d(
t
.
-ys(
..
).
(j)
, .po
j,
.,
.
--
.
.
(--q,- q(?6r.
'J?l6-.po- l
'T.
;lpù- qzq
)
a')j)(
-------.
-q(-J.
7---..
--<.j-.----.
q
o--v..
ef,'83/.)
,.'
l- pv
i'
r
z
o-l
pç
t
?-cu)
-
Thesingularity at(= iyvlpv)isimportantonly forpv> eo,w'
hen theintegral
fncludestheorigin. Inthiscase,thestructureofthesingularitiesisunchanged
when the integral is extended to - v: and evaluated with contour methods.
lf.;(p0)and Jlpv-.-t
y0)havethesamesign.theintegralcanelearlybeclosed in
thathalfplanecontainingno singularitiesand vanishes.
5.ThusJ(,(h)and atpo-.
-qo4musthaveopposite signsto yield a nonzero value.
The integral can then be closed in that half plane containing the pole at
(= l
'.
r).
)lpolwiththesnalresultforpv> 6etreintroducingdimensionalunits)
Thisexpression isfiniteeverywhereexceptat
hqn=z
kq(a
os
t
yj
y
,,.sp,
where it develops a logarithm ic singularity. 5Ve note the following points.
however.
(J) A logarithmicsingularity isgenerally integrable(forexample.when we
substitute r intotheskeleton diagramsforl1* and Y*).
4:0
APPLICATIO NS T0 PHYSICAL SYSTEM S
(/?) Iftheelectronsareontheenergyshelland elosetotheFermisurface,then
hpl
j;4;esand 1*k21isonlysingularw'hen i
c/tli;
4;qt'p. Sincec.c
t&s,thissingu.
Iarity isalso unim portantboth forrealphononsw'
ith 1% , = qcand forvirtual
phononsw'
ith I
ço < qc < t
ro.
Thus Eq.(47.7) is well defined in most regions of interest,and we have a
dem onstration ofM igdal'stheorem to ordery3.
M igdal'stheorem also can be understood qualitatively by observing that
the dimensionless ratio of-the displacement ofthe lattice d(x) in Eq.(44.15)
to theinteratomicspacing(;4:qw).)dependsontheion massasLm,
1M )i. This
,
qtlantity is smallforheavy ions,explicitly verifying ourrem arksatthe beginning
ofthischapter. (Infact,thesmallvalueof-theratio(?è?,
'.
W )1'isjustthecriterion
for the validity ofthe Born-oppenheimer approximation.1) The ratio enters
twice in evaluating a second-order correction as in Fig. 47.1, so that
r =ytli-Og(rn,
/
',
%1)1)). ItisalsotruethatY
--*= Okltnj'k
W )1)forthesamereason;
as seen in Secs. 36 and 37,however,a weak interaction can still have drastic
eflkcts nearthe Ferm isurface.
PRO BLEM S
12.1. Proke thatypy= 0 in the interactl
hg electron-phonon system .
12.2. Theexactphonon G reen'sfunetion isdefined in theH eisenberg picture
as
iDqx- .
x')= .
,
xO1Fg(/s(x)(
/s(x'))iO)
where kO) is thc exact normalized Heisenberg ground state of the coupled
electron-phonon system. Hencederivethe Lehmann representationforiD(q).
12.3. Using the generalrelation 0O,
'èt= lijhj(4 ,t))forHeisenberg operators,
show thatthe Heisenberg phonon 5eld forthe problem deined in Eq.(45.16)
satisfies the following relations
lt/'
fJ(X),t
f'ff(A'')1?-:'= 0
(u(x4,P
t
/'
r')
2 23(x- x')
-è
Vt(
./'
.r = â
--c
i.Vx
. ,,
in the lim itcoo .
-.
>.z'
z. Thus derive the following equation ofm otion
V2
1 92
- p g j ;s(x)- -yVx
2zfl
salx)#,,atxll
H ow are these relations m odified forsnite f
z?s?
1L.1.Schifllop.cit..p,447.
PHONONS AND ELFCTRONS
l é?2
411
(Vx
2-(y
.
jjy.
j1iD(a---x).
-h
,
.Vx
a(
5(x.
-x,)(
5(?--t,)
,
--
yVa
2..O Fgjlf
'
t
faj.
v)q?jya(a')(
/jj(A-'))O
N ote thatthe lastexpression isone lim ltotthe generalvertex function. W'hatfs
the corresponding resultfor snite u;o'
?
12.5. (J) Evaluatethelow'
est-ordercontribtttion to 11*(q)shoun in Fig.46.6.
(b) Compute the corresponding G(q)and discussthe rezulting expression:for
the single-particle energy'and lifetim e.
(c) How arethechemicalpotential.thespeciécheat.and tlneground-stateenerg);
aflkcted ?
12.6. (t7) Evaluatethelouest-ordercontrlbution to 11*(61)shown in Fig,46.6.
(??) Compute the corresponding phonon propagator and deri&e the fol1oNNing
expressions for the renorm alized phono1)frequentzy fzjq and in%erse llfe!inne èq
(neglectcorrectionsoforder c'r7s):
(12
:=(
sjgl-2x
&(0):2gjtF11
c
f
.t)z
a
2
Jq= '
-n
-?
u
hY
.A'(0)ptayrs- q)
qkr
- ..
-
Hereg(x)isgiven in Eq.(14.8)and JV(0)= pnl's.277.2/12isthedensityofstatesfor
one spin projection atthe Fermisurface. Note thatJt).
:èq becomesinflnite
atq = 2/t'r(theKohn eflkctt)and thatzbql'cistheultrasonicattenuationconstant
xnin the pure norm alm etal.
12.7. W rite outD yson'sequations explicitly in m om entum space to the order
indicated in Fig.46.6 using G(q),D(q)and F(t?l,t?c).
12.8. Show thattheenergy shiftin Eq.(46.2)also can bew'
ritten
*y s/ / *
.
E-Eo-Jj0.'
)
(
?jd3xx
-OI
v'?
/'
a(x)y
#
lx(
.
x)(
;(x)i
O'
)y,
.
= -
*y
a
,j.
a
.
i Jy'fd3.:.lzim,x l
i
m
.
O
F
l
y
'
s
a
t
.
x
'
l
y
'
u
z
(
>
'
)
(
;
'
s(
2
'
)
)
O
y.
y,--,x -
'
u 0
.
-.
where the m atrix elem ent m ust be calculated for aIl0 < y'< y. A nalyze the
rightside ofthis expression in term s of Feynm an diagram sand use M igdal's
::W .KohnsPhys.Ret'.Letters,2:393(1959).
412
theorem to show'thatin the norm alm etal
12.9-
12.11.
12.12.
J.Jex= (
'cl34-î
-'
ex(x :)jgitx)
.
=
((/3&-)'cx(xt)(
j
.(x)
13
S upercond uctivity
ll1 tIRis (2hap.ter.U e stud)
. the rem arkab1e phc11onlenon (
7f
.
-superconductiA.1t)
..l
A lthough the basic experinlental fat?è> are easi1.3 stated.the co11s!ructic)ll of a
satisfactory m icrostzopic descr1ptio1
1 pro'
.ed e'
&ceedingIl
k d1Picult. a11d earl),
theorie> yNere essential1),'phenom enological. N evertheltss.these theories 5.
$ere
surpri>i11gl)- accurate and profound1) 1ntltlt
'nced the prese11t rn:1n),-bodq theor)
ot'superco11ductor$. Fo1-th1s reason. $
,
5e k)o1'
1sider 1t e>$ent1d1 to disk'
zuss t1
)e
Phenom enolog1ca!the0r1
'
e> in deta1l.both tor the1r intr1l:>1c I!1terestand a> an
i1
,
1troduct1on to thk
lm icrt
'
3
.scop1capp1'o'
tcI
R. T hu-1atter(B C S)theor(
h7 represents
hGood introductory refcrencesare F.Londtan.''Surertlu1ds.'' NvAl.l.L7oNer Publicatlens.
Inc..Ne% N'ork.196l.D .Shoenberg.''Surerconduc:lNk:2..''7d ed .Can)bridge Ln1Nersll'.
Press.Canlbr1dge. 19(aj'
.N1.-1'!nkhan:.''Suyacrck.
nndttc:2Nl:.'
s.'' G o:ktuan and Breach.Sc1ence
Ptlblishers.N eyî'
bJ:'k.i9(af.
2J.Bardeen.L.N .Coope!'.andJ.R.SJhrlel-l-e1*.Pt'::'k.Stl'..108:11-f(l9f-).
414
APPLICATIONS TO PHYSICAL SYSTEM S
one of the m ost successfulapplications of m any-body techniques,
'in addition
to itsnew predictions,italsojustitiesthe earlier descriptionsand allowsan
evaluation ofthe phenom enologicalconstants.
48JFU N DA M ENTA L PRO PERTIES O F SU PERCO N D UCTO RS
W e start by sum m arizing the simplestexperim ent.alobservations. Atthe very
Ieast,any theory ofsuperconductivity m ustaccountfor these facts.
BAslc EXPERIM ENTAL FACTS
1.lnhnite('
t//7t/l/c/l
'
l'//y':W henanyoneofalargeclassofmetallicelements
orcom poundsiscooled to within a few'degreesofabsolutezero.itabruptly loses
a1ltrace ofelectricalresistivity at a dehnite criticaltem perature Fc.1 Since the
transition is not accom panied by any change in structure or property of the
crystallattice,itisinterpreted asan electronic transition,in which the conduction
electronsenteran ordered state. A sa hrstapproxim ation,we assum e the usual
constitutiveequation (Ohm 'slaw)
(48.2)
thenim pliesthatthefluxdensityB rem ainsconstantforanym edium with inhnite
conductivity because E vanishes inside the m aterial. In particular,consider a
superconductorthatiscooled below rcin zero m agnetic field. The above result
showsthatB remainszeroevenifafieldissubsequentlyapplied(Fig.48.1).
2. M eissner p/
ft
oc/:Althoughtheinsniteconductixityisthemostobvious
-
characteristic.the true nature ofthe superconducting state appears m ore clearly
in its magnetic eflkcts. Consider a norm alm etalin a uniform m agnetic field
(Fig.48.1). W hen thesample iscooled and becomessuperconducting.experim entshrstperform ed by M eissnerand Ochsenfeld zdem onstratethatal1m agnetic
flux isexpelled from the interior. N ote thatthis result does notcontradictthe
previousconclusion ofconstantB in thesuperconducting state;ratheritindicates
thatthe constantvalue m ustalways be taken aszero.
3. CriticalhkldïTheM eissnereflkctoccursonlyforsumcientlylow magnetic
felds. For sim plicity,we considera long cylinder ofpure superconductorin a
parallelapplied field H ,where thereare no dem agnetizing eflkcts. Ifthesam ple
is superconducting at tem perature F in zero held,there is a unique criticaltield
Hc(T)abovewhich the samplebecomesnormal. Thistransition isreversible,
lH .K.Onnes,Comtnun.#/?
y'
.
3
'.Lab.L%I
k..Leiden.Suppl..M b (1913).
2W .M eissnerand R.OchsenfeldaNatltrx'
iss..21:787 (1933).
SUPERCONDUCTIVITY
416
Sam plrcx led in
zero fleld
thensubjcttoan
Fig.48.1 M eissnereFectinsuperconductors. No
malicmç
n
mar
gnet
flt
ea
ll
di
t
hen
V(
Sc)œ led below
applie field
forsuperconductivityreappearsassoonasH isreducedbelow Nc(F). Experimentson puresuperconductorsshowlthatthecurve.
Sc(F)isroughly parabolic
(Fig.48.2)
,c(r)-,c(o)g1-(j)2q empi
rical
H
(48.3)
Hc(r)
Normal
Fig.48.2 Phase diagram in S-F plane.showing
superconductingandnormalregions,andthecritical
curve Hc(T4orFc(S )between them.
Superconducting
)
T
4.Persistent currents and .,#?
.
zx quantization: As a diflkrent example of
magnetic behavior,considera normalmetallic ring placed in a magnetic held
perpendicularto itsplane(Fig.48.3). W hen the temperatureislowered,the
m etalbecom essuperconductingand expelstheflux. Supposetheexternalseld
lltmustbe mentioned thatmany superconducting alloysexhibita çemixed state,''in which
theresistivityrem ainszero yetflux penetratesthesam ple. Althoughwe brieoyreturn to this
question in Sec.50,the presentchapter islargely restricted to pure superconductors.where
Sc(0)isusuallya few hundred oersteds.
APPLICATIONS TO PHYSICAL SYSTEM S
416
N
NormalIingin
magneticfleld
Cooled below Fr,
'
magnetic field ihen
remove
Fig. 48.3 Flux trapping in a superconducting ring.
isthen rem oved;no flux can passthrough the superconducting m etal,and the
totaltrapped fluxm ustrem ain constant,being m aintained by circulating supercurrentsin theringitself. Suchpersistentcurrentshavebeenobservedoverlong
periods.l Furtherm ore,the flux trapped in sum ciently thick ringsisquantized
in unitsof2
970= hc
j-i= 2.07 x 10 7 gausscmc
-
(48.4)
5.Spec@cheat:ln addition to its magnetic behavior,a typicalsuperconductoralso hasdistinctive thermalproperties. For zero applied held,the
transition isot-second order,which im pliesa discontinuousspeciticheatbutno
latentheat'(Fig.48.4). Asdiscussed in Secs.5 and 29,theelectronic specifc
Fig.48.4 Schcmatic diagram ofspecitic heatin
asuperconductor.
heatCnin thenorm alstate varieslinearly with the tem perature. In the super-
conducting state,however,the specific heat Csinitially exceeds Cnfor TL Fr,
butthendropsbelow Cnand vanishesexponentiallyasF -.
>.0.1
- ào
Csr
c expk
aT
lTypicalrecentmeasurementsarethose ofJ.File and R.G.M ills,Phys.Ael''.Letters,10:93
(1963),who 5nd lifetimesoforder105years.
2B.S.Deaver,Jr.andW .M .Fairbank,Phys.Rev.Letters.7:43(1961);R.Dolland M .Nâbauer,
Phys.Rev.Letters,7:51(1961).
den,221e:(1932).
3W . H.Keesom and J.A.Kok,Commun.Phys.Lab.Univ.Lei
1W .S.Corak,B.B.Goodman.C.B.SatterthwaiteandA.W exl
er.Phys.Ret's,96:1442(1954).
sgpEncoNoucTlvlTY
417
Thisdependenceindicatestheexistenceofa gap intheenergy spectrum ,separating
theexcited statesfrom the ground statebyan energy à().t Formostsuperconducting elem ents,Ao is som ewhatlessthan 2ksFc.
6.Isotopecfcc/:A finaldistinctive property ofsuperconductorsis the
isotopeeflkct. W enoted thatthecrystallographicpropertiesofthenorm aland
superconducting phases are identical. N evertheless, careful studies of isotopically pure sam ples show thatthe ionic lattice plays an im portantrole in
superconductivity,forthetransition tem peraturetypically varieswith theionic
ITIaSSl
FCa:M -*
(48.6)
Thisresultindicatestheim portanceoftheattractiveelectron-phononinteraction,
whichprovidesamechanism fortheformationofboundCooperpairs(compare
Secs.36,37,and46).
THERM O DYNA M IC RELATIONS
Forcom pleteness,wefrstreview thebasicequationsofelectrodynam ics. O n a
microscopic(atom ic)scale,thereareonlytwo fundamentalseldseand b,dehned
by the equations
dive = 4=p
divb= 0
(48.7/)
(48.7:)
ob
curle= -c-1&
pe 4=
curlb= c-1'
)J+ -c pv
(48.7/)
(48.7:)
H erep isthe totalchargedensity and v isthe m icroscopic velocity Eeld. The
macroscopicseldsaredesned asspatialaverages
E= (elvol
B = (blvol
(48.8)
overa volum e appropriateto theparticularproblem in question;they obey the
conventionalM axwellequations
divE = 4rtplvol
divB = 0
(48.94)
(48.9:)
PB
CUI'IE = -C-1
(48.9c)
1Althoughtheenergygapisatypicalfeatureofsuperconductivity,itisbynomemnsnecessaryy
asshown by theexistenceof''gapless''superconductorswith zero dcresistivity.
!H.Fröhlich,Phys.Rev..79:845(1950);E.M axwell,Phys.#er.,78:477(1950);C.A.Reynolds,
B.Serin,W .H.W right,andL.B.Nesbitt,Phys.Re?
J.&7*:487(1950).
'
4!g
Appulcv loNs TO PHYSICAL SYSTEM S
OE
4=
curlB = c-lW + - '
lp#lvol
( c
(48.9J)
Although these equations are formally complete,it is customary to separate
(plvolintoa polarizltiondensity.
-divp and afree(externallyspecised)charge
densitypz. Equatim (48.9/)thenbecomes
div(E+ 4'
rrP)2
- divD = 4rpy
(4810)
.
where P and D are known asthe polarization and displacem ent,respectively.
In a similar way,the totalcurrent is separated into a magnetization current
ccurlM .a polarizatipn currentPP/J/,and a free(again externally specised)
currentjzassociated with themotion offreeeharges. Ifthemagneticfield is
desnedasH * B - 4>M ,thenEq.(48.9#)gives
4=
lPD
curlH = -c jg+ c èt
(48.ll)
Thelastterm isgenerally negligible in low-frequency phenom ena,and w'e m ay
interpret curlH as arising solely from the free currents. W hen the various
m agneticquantitiesarechanged by sm allam ounts,thew orkdoneonthesystem
isgivenby(474-1J#3a.H .dB,(so thatthechangeintheHelmholtzfree-energy
density becom es
dF = .
-sdT+ (4.
* -1H.#B
(48.12)
w ith thecorresponding diserentialrelations
,- -
td
Jwjs u-.(d
aF
s)v
(48.13)
Hereweassumethevolumeisheld constant,and sistheentropydensity.
Thefluxexpulsion associated with the M eissnereflkctindicatesthata bulk
superconductorin an externalm agneticheld H isuniquely characterized bythe
conditionB = 0,independentofthewaythestateisreached (Fig.48.1). W e
therefore infer thatthe superconductor isin true therm odynam ic equilibrium
and accordingly apply the techniques of macroscopic thermodynamics. For
m ostexperim ents,however,it isim possible to m anipulate the ;ux density B
directly;instead.theexternalcurrents(ina solenoid,forexample)controlthe
m agnetic seld H ,and we preferto m ake a Legendre transform ation from the
HelmholtzfunctionF(F,B)toanew (Gibbs)free-energydensity
G(r,H)- F - (4.
* -1B-H
(48.14)
1Thisresultcan bederivedverysimply bysurrounding thesystem witha surface.
/
1in free
space. Poynting's theorem showsthatthe energy Cowing in througb ,
4 in a shorttime dtig
given bydWim= -dt(cI4=4Jxds.E x H. Thedivergencetheorem and Maxwell'sequations
(48.
9c) and (48.11) then yield JHz'
em=(c/4=)fy.d% (c-tH .#B + c-bE.JD +(4zr/c)E.Jz#J1.
which verihestheassertion.
SUPERCONDUCTIVITY
419
with theeorresponding diflkrentialrelations
JG = -sdT - (4=)-lB.#H
s- -
tj
aG
zjHs--4=(
bt
j
'
j
n/
jT
(48.l5)
(48.16)
Consider a long superconducting cylinder in a parallel m agnetic Eeld.
Iftheheld H = Hâisincreased atconstanttemperature,Eq.(48.16)gives
G(F,S)-G(F,
0)=-(4r
8-1j0
HB(H'
)dH'
(48.17)
To agood approxim ation,thenorm alstateofm ostsuperconductingelem entsis
nonmagnetic(B = H),and wehnd
Gn(T,H)- Ga(F,0)- -(8,
n.
)-1H 2
(48.l8)
In contrast,B vanishesinthesuperconductor,which yields
G,(F,S)= Gs(F,0)
(48.19)
Thetwo phasesarein therm odynam icequilfbrium atthecriticalfield H c. This
condition m aybeexpressed by theequation
GXF,1fc)= Ga(F,Sc)
(48.20)
andacombinationofEqs.(48.18)-(48.20)immediatelygives
Gs(F,0)= Gk(F,0)- (8=)-lHj
(48.21)
F,(r,0)= Fa(F,0)- (8z8-1Hl
c
(48.22)
These equations show thata negative condensation energy -Hk(b= per unit
volumeaccompaniesthe formation ofthesuperconducting state. In addition,
a sim plerearrangementleadsto thegeneralresult
Gs(T,H4- Ga(F,S)=(8r4-l(S2- H6j
(48.23)
so thatthesuperconducting phaseistheequilibrium stateforallH < Hc(T).
ThederivativeofEq.(48.23)withrespecttotemperatureyieldstheentropy
diflkrencebetween thetwophases
J,(F,S )- sn(T,H )- (4,* -'Sc(F)y
o
-sc
#F
(48.24)
Figure 48.2 shows thatthe rightside is negative,so thatthe superconducting
phase haslowerentropy than the normalphase. Note that Eq.(48.24)is
independentofthe applied field,which also followsfrom the therm odynam ic
M axwellrelation derived from Eq.(48.15). The latent heatassociated with
the transition is Tlss- .
%),which vanishes at r = 0 and atFc(see Fig.48.2).
420
APPLICATIONS TO PHYSICAL SYSTEM S
Finally,thetherm odynam icidentity
cs-v(p
q)H
(48.25)
givesthedit
lkrence in thtelectronicspeciticheatsatconstantEeld
F JS 2
dlH
c'H- cnn= y,;; dpc + Sc(v 2F.
lnparticular,thejumpinthespecihcheatatTcbecomes
(
T dH
2
Cs- Cntvc= 4
-5
=- dTC v
C
(48.26)
(48.27)
and thisrelation between m easurable quantitiesiswellsatissed in practice.
O EJLO NDO N-PIPPARD PHENOM ENOLOGICAL THEORY
TheLondonequationslprovided thesrsttheoreticaldescriptionoftheM eissner
eFect. Although these equationsare a pair ofphenomenologicalconstitutive
relationsdescribing theresponseofthesupercurrentjto applied electricand
m agneticûelds,theym ay also bederived from thefollowingsim plem odt1.2
DERIVATION OF LONDON EQUATIONS
Ifthe superelectronsareconsidered an incompressible nonviscouscharged :uid
withvelocityseldv(x?),thenthesupercurrentisgivenby
j(x/)= -natw(x/)
(49.1)
wherensisthesuperelectron num berdensityand.
-eisthechargeonan electron.
Thecontinuityequationand Newton'ssecond law give
divj= diNv= 0
Jv
7/=
e
1
m E+cYxh
- -
(49.2)
(49.3)
wheredyldtisthetotal(hydrodynamic)derivativeand h(x)isthesne-grained
average ofbtx)overa microscopicvolumeofdimensionslargecompared to
atom ic size but sm all com pared to the penetration depth. In a11subsequent
discussion we shallreferto h(x)asthe microscopic held.3 The leftside ofEq.
lF.London and H.Londora,Proc.Roy. Soc.(London),A147:71(1935).
2F.London,op.cl't.,sec. 8.
3W ehere follow F.London,op.cl?.ssec. 3,although theseld in question isreally B.dehned
in Eq.(48.8). To beveryexplicit,themean flux density(the coarse-grained averageoverthe
sample)isherecalled J instead ofLondon'sB.
SUPERCO NDUCTIVITY
421
(49.3)m aybe rewrittenwith standard vectoridentities
A
?v
êv
a
1 = yt+ (v.V)v= -gj+ V(!.?
p)- vx curl>'
(49.4)
and we5nd
Pv
E
+ :-
'
ji m+v(!.
?2)-vx(curlv-m
eh
c)
(49.5)
Thecurlofthisequationmaybecombinedwith M axwell'sequation(48.9c)to
give
PoQt= curl(vx Q)
(49.6)
where
elk
Q H curlv- mc
-
(49.7)
Consider a bulk superconductor in zero feld, when Q = 0. Equation
(49.6)thenimpliesthatQ remainszeroevenwhenaEeldissubsequentlyapplied.
Since the M eissnereflkctshowsthata superconductorin a magnetic held isin
therm odynam ic equilibrium ,independentofhow the 5nalstate isreached,we
m aktthefundam entalassum ption thattheequation
Q - eurlv eh = 0
mc
-
(49.8)
-
correctly describesa superconductor underallcircumstances. Substitution of
Eq.(49.8)intoEq.(49,5)gives
êv
z
eE
W + V('
l+ )= --m
(49.9)
Equations(49.8)and(49.9)togetherconstitutetheLondonequations.
so LuTlo N F0 q HALFSPACE AND SLAB
The implications ofthese phenomenologicalequations are m ost easily under-
stoodbyrewritingEq.(49.8)as
h=
-
nmseczcurlj
(49.10)
A combinationwiththecurlofMaxwell'sequation(48.9#)forstaticseldsthen
gives
mc
m cl
m cl
h= -a, e2curlj= -4=nsegcurlcurlh= 4=nsecV2h
(49.11)
wherethe lastequality followsfrom Eq. (48.
90. Fordeâniteness,westudya
APPLICATIONS TO PHYSICAL SYSTEM S
r 2'
'
'
%f.
. .
1'''
,
'
.
.
'
., ..-
X
5
1
g x
1 '
1'''
.
'
'
S()
(&)
(DJ
Fig.49.1 Geometry ofsupercoqducting (J)halfspace,(bjslab in a
parallelapplied held Hz.
semi-inEnite superconductor (z > 0) in an applied field Ho= Hzk parallelto
the surface (Fig.49.1J). The microscopic seld h(z)= hlzl.
k in the interior
satisfesEq.(49.11)withtheacceptablesolution
hlz)= Hoe-'/1'.
(49.12)
where
c2
+
As= n= ns'
7
(49.13)
isknow n astheLondonpenetration depth. Thusthem agneticseld isconsned
to a surface layerofthickness ;
k;Asand vanishes exponentially for z> As. If
Gistakenasthetotalelectrondensity,thenX istypicallyafew hundredangstroms
(see Table49.1).1 Experimentsindicatethatthepenetration depth increases
Table49.1 Characteristic lengthsforsuperconductors
'Az,
(0),âl
&,A
As(0)/& A(0)t',.Aj Molexp,A
I
A1
1 1*
Sn ' 340
Pb
370
16
.(X*
2
3*
830
,
0.010
0.16
0.45
530
510
440
490.515
5l0
390
1Az.
(0)iscalculatedbyassuminga,=natF= OOK .
jA(0)tN isthe penetration depth atzero temm rature,calculated
with the BCS theory for diffuse scattering ofelectrons atthe
boundary.
Source..R.M eserveyandB.B.Schwartz,Equilibrium Prox rties:
ComparisonofExm rimentalResultswithPredictionsoftheBCS
Theory,inR.D .Parks(ed.
),lçsux rconductivity,''vol.1,p.174.
M arœ lDekker,lnc.,New York,1969.
!Thissmallsize makesprecise measurementsquitedimcult;typicalexperimentalprocedure
arediscussed in F.London,op.cit.,sec.5.
SUPERCONDUCTIVITY
423
with increasingtem perature,and thefunction
&(
A F)= 1- wF 4-+ em pirical
(49.14)
zX0)
1c
providesagood:tltothedatafbrF/Fck;!.. Sincensistheonlyvariablequantity
inEq.(49.13),weinfer
NXF)= 1 - F 4
G(0) (y1 empi
rical
(49.15)
Asa second example,considera slab ofthickness2# in an applied seld
Hok paralleltoitssurface(Fig.49.10. IftheoriginofcoordinatesisIocatedin
them iddleoftheslab,then the appropriatesolution ofEq. (49.11)is
hlz)= cosh(z/As)
Hzcosh(#/,V)
(49.16)
Them ean fluxdensity.
# isthespatialaverageofthemicroscopicheld
J 1 d
HoAs
d
-s
J.,dzhlz)- dtanhjj
(49.17)
In thelim itofathick slab,Eq. (49.17)becomes
S oAs
J;k; #
dy.As
(49.18)
andthesampleexhibitsaMeissnerefl
kct;intheoppositelimit(#.c
.
:As)themean
.
flux densityisessentiallyH u.
CONSERVATION AND QUANTIZATIO N O F FLUXOID
The London equations im ply a striking conservation law. Forsim plicity,we
study only the linearized equations
1 êj
(49.l9)
nse'
à
and Eq.(49.8).buta more generaltreatmentcan also begiven (Prob.13.1).
-
eE = 0Y=
m Ji
-
-
C onsidera surface S bounded by a hxed closed curve C thatlieswholly in the
superconducting material (Fig.49.2). Independent ofwhether S also lies
entirely in the superconductor(Fig.49.2/ orb),we may integrate Maxwell's
equation(48.9$ to obtain
J#S-êh
g=-cJJs-curlE=-cjc#l.E
(49.20)
1Fora comparison ofthe experimentswith Eq. (49.14)and the theoreticalprediction ofthe
BCS theory.see,forexample,A.L. Schawlow and G.E.Devlin,Phys.At,t).,113:120 (1959).
424
APPLICATIONS TO PHYSICAL SYSTEM S
6C
4'
.i
.,
.
'.
t.)
.u
.
r.2
u$
.
(.. y
yA,.i..14w,
..
. '
::
w
'
.k
.z
'q!
'v
..,
.,
.f..:.. ;z... a.
..
,:
.
e
s
,
,
a.
. r y.
Holein su rconductor
(J)
(A)
Fig. 49.2 Integration contour for evaluation of
quxoid:(J)simplyconnected;(b4multiplyconnected.
where therightside has been transformed w ith Stokes'theorem . Since C lies
inthesuperconductor,Eq.(49.19)appliesateverypoint.giving
&
0(ds.h+nm
sc,jc(
81.j-O
(49.21)
weseethatthe/uxofffldehnedas
mc
(I)s:j#s-h+M
-s ea c #l.j
(49.22)
rem ainsconstantforalltim e. Itisclearthat(1)diflkrsfrom the m agnetic flux
by an additionalcontributionarising from theinduced supercurrent.
W ith added assum ptions,itispossibleto derive m orespecific results.
1.IfC issuëcientlyQrfrom theboundaries,thenjisexponentiallysmall,and
(I)reducesto them agneticflux.
2.lfthe interior ofC is wholly superconducting (Fig,49.25),then the other
Londonequation(49.8)immediatelyimpliesthat* vanishes.
3. Asacorollary ofthepreviousconclusion,* isthesame foranypath C 'that
can bedeform ed continuously into C,alwaysrem aining in thesuperconductor.
F.London also observed thatEq.(49.8)can bewritten in termsofthe
vectorpotentialA as follows
eA
curl mv - - = 0
c
(49.23)
where
h = curlA
(49.24)
Thecanonicalm om entum isgiven by
eA
P = nIM - c
'F.London,op.cit..sec.6.
(49.25)
SUPERCONDUCTIVITY
42:
and Eq.(49.23)thusbecomes
curlp = 0
(49.26)
which m ay be considered a generalized condition of irrotational flow. In
addition,theCuxoid m aybewritten as
C
(Il= -e-
c
#l.p
(49.27)
Thisequation isrem iniscentoftheBohr-som merfeld quantization relation,and,
indeed,London suggested thattheCuxoid isquantized in unitsofhcle.î As
noted in Sec.48,thisprediction wassubsequently consrm ed,butthe observed
quantum unitishcllegEq.(48.4)J.
PIPPA RD'S GENERA LIZED EQ UATION
Itisnow convenienttochooseaparticulargauge(Londongauge)forthevector
potential
divA = 0
onboundaries
which allowsustorewritethestaticLondonequation(49.10)as
M
(49.284)
(49.28:)
2
J(X)= - se A(X)
mc
(49.29)
Thischoiceofgaugeisalwayspossible,becauseEq.(49.28:)can besatissedby
adding the gradient of an appropriate solution ofLaplace's equation. N ote
thattheLondongaugeensuresthatdivj= 0andthatJ.H= 0ontheboundaries.
Equation (49.29)showsthatthe London theory assumesj(x)proportionalto
A(x)atthesamepoint. Furthermore,thetheorypredictsthatthepenetration
depth dependsonly on fundamentalconstantsand na(Eq.(49.13)). To test
these predictions, Pippard studied the properties of superconducting Sn-ln
a1loys.2 Although thethermodynamicpropertiessuch asSc(F)and Frwere
unaltered byratherlargeconcentrationsoftheIn impurities(f3X),hefound
that the penetration depth increased by nearly a factorof2. Such behavior
cannotbe understood in the London picture,becausean increased penetration
depth w ould im ply a reduced value ofnsand a corresponding m odihcation in
thefreeenergyandothertherm odynam icproperties. Forthisand otherreasons,
Pippard proposed a nonlocalgeneralization ofEq.(49.29),in which j(x)is
determ ined as a spatialaverage ofA throughout som e neighboring region of
dim ension ro. For heavily doped alloys,rz iscom parable with the electronic
mean freepath Iin the norm alm etal;forpurem etals,however,rcisnotinfnite,
IF.London,op.cf?.,p.152.
2A.B.Pippard,Proc.Ro)'
.Soc.(London),A216:547(1953).
426
APPLICATIONS TO PHYSICAL SYSTEM S
butinsteadtendsto acharacteristiclengthL,knownasthe Pippard (orBCS)
coherencelength. Pippard thusm ade theparticularchoice
l
1
1
-
rz= J'
:+ j
(49.3g)
and assumed
n,e2 3
;,
X(x.A(x')) .
-x/r,
j(x)= - mc4,
n5: d x X4 e
(ys.?j)
where X uzux - x'. A n experim ental fit to the m easured penetration depths
yieldedl
Jll;z0.15k
r
.hvFc
(49.32)
s
Forcomparison,theBCS theoryofpuresamplesleadsto averysimilarnonlocal
relationandidentises(zas
fn= 0.180kâlls = -hv
m
a Tc
(49.33)
rr2:n
'
wherezxtlistheenergygap atzero tem perature. Typi
calvaluesofL areshown
in Table49.1.
ThePippardequationrelatestheinducedsupercurrentjtothetotalvector
potentialA. In the presenceofan applied magnetic ield,however, A contains
contributionsfrom theexternalcurrentsaswellasfrom jitself,therebyrequiring
asimultaneoussolutionofEq.(49.31)and Maxwell'sequationdeterminingthe
totalmagneticheld. Nevertheless,itispossibletoextracttheimportantphysical
features by noting the existence of two characteristic lengths. The vector
potential varies w ith the self-consistent temperature-dependent penetration
lengthA,whichneednotbethesameastheLondonpenetrationlengthAs(defined
inEq.(49.13)),whiletheintegralkernelhasatemperature-independentrangera,
whichisapproximatelythesmallerof% and/. Ifrz< A,thenthevectorpotential
variesslowly and can beevaluated atx'= x. In thisway,we obtain
n ,2
3
jklxs= -mc
'ln,1l(x)s
n ez
s
e-l/re
dbX X.m x4
=-rqcf:4!(x)b.tj=
0Jye-x/
re
j(x)-
-
z
n'e2l A(x)
mcll+ &)
ro.xA
!T.E.Fa% rand A.B.Pippard,Proc.Roy.Soc. lZtladt/al,A231:336(1955),
(49.34)
su PERCONDUCTIVITY
427
Any sample thatsatisses this localcondition (ro<b
LA)is known as a London
l'r//pcrc'o/ipl/c/t
pr,because L-q.(49.31)then reproduces thc form ofthe London
equation (49.29)butwith thecoeflicientreduced byafactor(I : (tvI) î. Comparison with Eqs.(49.l0) and (49.1l) im mediatcly giNes the corresponding
penetration depth atzero tem perature
J- Jo 1
l(0)-.
'
=z
ls(0)( j ) lOCall
imit:r(
)-,
èZ
J
z
Foraplfrc Lolldon superconductor(#0<,'/).F.q.(49.35)reducesto thepre:ious
London expression.
lnpractice(seeTable49.1),mostsuperconductingelementsatlow temperature violate the condition for a pure London stlperconductor.u hich requires
ro;
:
k:Jt
l<.
z
tA. Hence London superconductorsare usual1.
).hea:i1) dopcd allo)s,
where thelength rvisdetermined by /instead ofJ't
).and thet'
ollou illg inequality
holds:rv;
4;/-s.A. ln thiscase,Eq.(49.35)explainstheobser:ed increasei1 the
penetration depth fordirty alloys&$here/.<-(!(
). lfa sam ple isa Londol)superconduetorat low temperature.Eq,(49.I4)shov:s thati( remains onc forall
F ..
c rc. Sillce the penetration depth ofans superconducting m aterialinureases
rapidly as r -.
' Fc,how ever-aII.
îl?p&r('t
???t/s?c'J(?r.
5
'bel-ollll,1.o/pti//?sltpL'rcoltthlctors
s'lk-/
/i'
c-àt.
antIî'c/(?A't
?ïo Tc.
Itisim portantto em phasizethatLq>.(49.34)and (49.35)areolllyeorrect
forrvk
tz
'
t-a1
1d we !1oïNeonsiderthe nlore t)picalno1
)local11m it(?'().
,.z
ï),N.
'
N.hen
the m aterialiskno% i
)asa Pippartl.
y'
l/1t'/'t'
t)/?t/Ift'/ö?
-. A lthough a generalso1ution
isverydit
licuIt.wecanevaluatejandheom pietel)foraninfi11itesuperconductor
-,
t
$(z)lying in thex1'plane. The curre1tsheetis
surrounding a currentsheetjv:
.
asourceofmagneticfleld.?hich in turI
1Induces:1supcrcurrentjrx). The total
current is the sun'
lofthest
'
ltïso contributions.a11d 51axUeli'> equatio11(
9J)
.48.
beconles
This equation m ust be solved along with Pippard-s equation.u hich provides
anotherrelation betweenjand A.
Thetransiationalinvariancem akesitusefuIto introducethree-dim ensional
Fouriertransform s
jtx)- (2=1-3 4
-(13:(afq-Xjtq)
A(x)= (2:7.
)-31
'J?qtpfq-xA(q)
(49.37/)
and thel-ondon gaugeimplies
q'A(q)= 0
(49.38)
42g
AppulcATloss To pHyslcAL SYSTEM S
ItisstraightforwardtoverifythattheFouriertransform ofEq.(49.31)takestht
form
J(q)- -sc A-(ç)A(q)
(4939)
.
where
3ase2
X(V)= -mczït4*
=
*dx
o
1P sinqx
-àt-1C-XF'Oq
(? qx
.
6r- ,e2 arctanqra
1
c
mcafnq (çro)k (1+ (qrv)j- q%
(49.40)
and thegaugecondition (49.38)hasbeenused to simplifythevectorcomponents.
Inaddition,theFouriertransform ofEq.(49.36)isgivenby
-
qx qx A(q)= 4rrc-1fd?xE'-iQ*X(.j()#3(z)+jtxlj
=
4=c-3E(2=)2.&#3(#x)3(çx)+j(q))
(49.41)
andtheltftsidamaybesimplifedwithEq.(49.38):
qX q>.A(q)=qlA(q)- q(q.A(q))= q2A(q)
IfEq.(49.39)isused to expressj(q)in termsofA(q),the resulting algebraic
-
equation isreudilysolved to yield
A( 4,
p.(2,
r)2jo#3(çx)3((n)
q)= -c qz+ Kg)
(49.42)
Thecorresponding spatialquantitiesbecom e
77 * dt?. J'
kt7i?T
A(x)M -4(z)#= 4
a:
c -x2=q + K%)
h(x)u >(z)9- -4
v =d
pakiqei4z
-.
X '
c -.2=:2+ K(q)
.-
(49.43t8
(49.435)
which providethecom pletesolution.
Itisinteresting to examint the asymptotic form ofhfzjasz-+.œ. If
ql+ Kfqjhasno zeroson thertalqaxis,then theintegration contourcan be
dtformedintotheupper-halfqplane,showingthathlz)vanishesfasterthanany
algebraicfunction ofz. Sincethisbehaviorisexactlythatfound intheM eissner
elect,weobtain thegeneralcriterion
qn+ K(q4# 0
forqreal:Meissnereflkct
(49.44)
Inpractice,Eq.(49.44)isnotvery convenient,becauseK(q)isacomplicated
function ofq,and we m ustinstead rely on graphicalm ethods. The Pippard
kernel(Eq.(49.40))isplotted in Fig.49.3. Ithastwo characteristicfeatures:
SUPERCONDUCTIVITY
429
1.0
K (<)
A-(0)O.j
Fig.49.3 Pippard kernelKlqj(Eq.(49.40)1.
00
1.0
2,0
3.0
q%
:
K(q4ispositiveforallq and K(q)decreasesmonotonically. Forany kernel
with thefirstproperty,wemay simplify Eq.(49.44)to theform
&(0)# 0 Meissnereflkct
(49.45)
&(0)= 0 noM eissnereflkct
.
.
which servesasausefuland sim plecriterion.l
TheforegoingdiscussionconsidersonlytheasymptoticbehaviorofA(z),
which generallydiflkrsfrom a pureexponential. Hencewe m ustgeneralizethe
conceptofa penetration depth,and a naturalchoice is2
A=
x$
hlz)
,dtz= 0)
jvdz/
yz=o=yqz-()
Theintegralin Eq.(49.43$)convergesonly ror fzt# 0,and h(z= 0)mustbe
determ ined by otherm eans. Thispresentsno problem ,however,forA m père's
1aw im plies
h(z= 0)= 2=c-1h
and we5nd
A= -2 * ty
x
1
jvq(
yz.j
.g((
y)
(49.
46)
Therathercomplicated form ofthePippardkernel2Eq.(49.40))precludes
ananalyticevaluationofAforallvaluesofIandS. Wenote,however,thatthe
importantvaluesofq areoforderA-!. ln thelocallimit(A> roj,Fig.49.3
showsthatwemayapproximateK(q)by .
&(0),and a simpleintegration yields
thepreviousexpressionEq.(49.35). lntheopposite(nonlocal)limit,weapproximateK(q)byitsasymptoticform asq-->.cs
Kqq)x
3=2n e2
j-?-mc #JII
q..
xw
lM .R.Schafroth,Helv.Phys.Acta-,24:645(1951).
2A.B.Pippard,Ioc.cit.
(49.47)
4ac
Appulcv loNs To PHYSICAL SYSTEMS
and a straightforward calculation givesthezero-tem peratureresultl
A(0) s V'
J1
=;
(z.
y)(Az
,
(0)2Jo)1 nonlocalli
mit:à<
:
trf
t
(49.
48)
Thisexpression isindependentofthemeanfreepath /becausethespatialintegra-
tioninEq.(49.31)islimitedbythepenetrationdepthAandnotby%. Equations
(49.35)and(49.48)constituteacentralresultofthePippardtheory. Inaddition
to explaining the increased penetration depth in alloys,the theory clarisesthe
discrepancy ofa factor œ 2-3 between the m easured penetration depth in pure
PippardmaterialsandtheLondonvalue(seeTable49.1)because
A(0) 8 VJfo 1
=I
As(0) l2=A,(0)1
(49.
49)
A lthough the original Pippard theory does not determ ine the tem perature
dependenceofA(F),theempirical1aw (Eq.(49.14))isgenerallyassumedforboth
localand nonlocallim its.
SX GINZBURG-LANDAU PHENOM ENOLO GICAL THEORY
Theoriginalform ofthe London theory hasoneseriousflawsforitapparently
predictsthatthe M eissnerstatein a seld H < H cwillbreak up into alternating
norm aland superconducting layersofthicknessdn.:
4ds.l Thisresultisreadily
verihed: If #a<K #s,then the condensation-energy density remains essentially
unchangedat-S J/8zr,whileifJs.
c
4A,thenthefeldpenetrationlowerstheGibbs
free-energy density by -S2/8gr. F.London noted thatthisargumentignores
the possibility of a positive surface energy associated with a norm al-superconducting interface,and he used theobserved stabilityoftheM eissnerstateto
estimate thissurface-energy contribution. A sa m orefundam entalrem edy for
this defect, Ginzburg and Landau3 proposed a diflkrent phenomenological
description ofa superconductor,which accounts for the surface energy in a
naturalway.
EXPANSION O F THE FREE ENERGY
The phase transition atFcsignalsthe appearance ofan ordered state in which
the electrons are partially condensed into a frictionlesssuperquid. Ginzburg
lAnequivalentprocedureisto rewrite Eq.(49.46)indimensionlessform
1=2=-1J0
Odt(
/2+(A3/A)J:)I-1
F(lr(
j
/à)j-l
whereF(x)=1.I(x-2+1)arctanx-.
x-lj. Thelimiting valuesofFltrolh)forro/à..
+0and
ulh-.
y.* reproduceEqs.(49.35)and(49.
48),respectively.
:F.London,op.cI'
?.,pp.l25-130.
3V.L.Ginzburgand L.D .lnndau:Zh.Eksp.Teor.Fiz,20:1064 (1950). An English translation mayix foundin4'M enofPbysics:L.D.Landaus''vol.1,p.138,PergamonPress,Oxford,
1965.
SUPERCONDUCTIVITY
431
and Landau describe the condensate with a complex order parameter 1F(x).
The observed second-order transition im plies that Vl''vanishesforF > rc, and
thatitincreasesin m agnitudewith increasingrc - r > 0. NearFc,thequantity
r'Frissmall,andthemicroscopicfree-energydensityFsoofthesuperconducting
state in zero Eeld isassumed to have an expansion oftheform
Fsz- Fno+ at
'l'
-12+ q.b1.1,
-t4+ '''
where Fnois the free-energy density ofthe norm alstate in zero m agneticfield
and wherea and b are realtem perature-dependentphenom enologicalconstants.
Sincetheorderparam eterisuniform in the absence ofexternalfelds, G inzburg
and Landau added a term proportionalto 1V'F(25tending to suppressspatial
variationsinYP. ln analogywith theSchrödingerequation, thisterm iswritten
as
(2-*)-'r--fâV:l'(2
where m * isan egkctive m ass. W hen m agneticfields are present, this term is
assum ed to take the gauge-invariantform
(2
1
esA
2
,,+)-11
(-ihT.
,.-cj'
1,:(
.
where-e*issomeintegralmultipleofthechargeon an electron and
curlA(x)= h(x)
(50.1)
determ ines the m icroscopic m agnetic l
ield. In this way,the totalfree-energy
density ofthe superconductingstatein am agneticfield becom es
Fs -
i
Fnv+ aIY15I2+ jb1YF14+ (2rn*)-11
I-l'
hT +
c*A
c
2 h2
4'j
!+ @-#
(f0.2)
where the lastterm represents the energy density ofthe m agnetic :eld.
conventionaltochooseaparticularnormalization forV1S:
1
'
t
F12> ns
*
(503)
.
where ns
* desnesan eflkctivesuperelectron density. Thischoice fixestheratio
(t7/:)2. Experiments:indicatethat
ra*= 21q
(50.4)
inagreementwiththepairing hypothesisofSecs. 36and 37;wethereforeidentify
ns
*with thedensityofpaired electronsand detinens by theequation
'ns
*- lws
(50.5)
!B.S.Deaver.Jr.andW .M .Fairbank.Ioc. cl't.;R.DollandM .Nâbauer,Ioc.cit.;J.E.Zimmerman andJ.E.M ercereau,Phys.Rev.Letters,14:887(1965).
4a2
APPLICATIONS TO PHYSICAL SYSTEMS
The free energy ofthe sample isobtained by integrating Eq.(50.2)over
the totalvolum e Pr. In a uniform externalfield H ,however,therelevanttherm o-
dynamicpotentialistheGibbsfreeenergy(compareSec.48),andwemustconsider
f#3xgF,- (4,
7
$-1h.H).
= fJ3xGs
.
(50.6)
where Gs is the m icroscopic G ibbs free-energy density. The condition of
therm odynamic stability requiresthatEq.(50.6) be stationary with respect to
arbitraryvariationsoftheorderparameterand vectorpotentialsubjectto the
constraint of Eq.(50.1). A straightforward variationalcalculation gives the
follow ing Geld equations
+A'
.2
(2-*)-1-l
'
hv--C
-C
--j'
1
*+t
/l
'
-+bI
Vl
'
-.
'
21e=0
(50.7J)
c
r
*Ji.(à1,**VYI'
e*)2 (kl7';2A
k'
p/curlh= -éf
--YT-VYI-*I- (
,n L
m.c
(50.7:)
whiletherem ainingsurfaceintegralsvanish ifA andY*satisfytheboundaryconditions
,
.
ov
bx y,
l
i*(-!/
XV c 1a=(
;
(50.8J)
zix (h - H)= 0
(50.8:)
Equations(50.7)show thatY'obeysanonlineartçschrödinger''equation,while
the m agnetic held iqdeterm ined by the supercurrent
j=
--
*h (t)*ï&Vl.
(c*)2ty12A
lem*
i- l.a- tI.%V&I.*+)- -&/-cj
''
(50.9)
TheboundaryconditiononV1-isverydiflkrentfrom thatoftheusualSchrödinger
theory,however,and may be understood asguaranteeing thatjeH vanishesat
the surface ofthe sample. Equation (50.8:)impliesthatthe tangentialcomponentofthe m agnetic seld iscontinuousacrossthis surface.
so bu-rlo N IN SIM PLE CASES
Although a com plete solution of these coupled equations cannot be obtained,
wecan discoverqualitative featuresby exam ining limiting cases. ln the absence
of-a magneticlield (A = 0).Eq.(50.7J)hasthespatially uniform solutions
.y .a
.
a
1l = -J
The first solution clearly represents the norm al state because F, then equals
Fnv. The second solution is physically acceptable only if a(b is negative;it
SUPERCON DUCTIVITY
433
represents the superconducting state with a corresponding free-energy density
lal
Fso= Fno- jy
zerolield
(50.11)
and comparison with Eq.(48.22)yields
al= -H (F)2
i
Ei.
-.
-=
(50.12)
Equation (50.12)showsthatb mustbe positive,which also ensuresthatFxis
bounded from below (see Eq.(50.2)1. Following Ginzburg and Landau,we
assume thatb is independentoftem perature,while
a(T)= (F - FclJ'
(50.13)
is negative forF < Fc,vanishing linearly at Fc. Thischoice correctly :ts the
linearslopeof.
Jfc(F)nearFcand also predictsthatns
*(T)a:(rc- F)gcompare
Eq.(49.15)1.
A s a second sim ple case,consider a one-dim ensionalgeom etry,where Y1'
variesbuth vanishes:
hl d21.
1,
%
-Lm %-t?1,
*=0
z-s +.tzVl,+ bj'Fj2A.
(50.14)
W eassum ethatt1,
-isrealand introduce adim ensionlessorderparam eter
/( 'F(z)
z)- 1Y1.x r
(f0.15)
where
1
'
1.
.1E
s(1
&!
b
jl
(50.16)
isthe m agnitudeof'F deep in the sam ple. A combinationofEqs.(50.l4)and
(50.15)gives
-
z J2
y
lm â
,y
-rJ)zzc-f+f3= 0
(50.17)
whichdefnesanaturalscaleoflengthforspatialvariationsoftheorderparam eter
ft''
hl
1
h
- è'--tt
zt
-f-) - Ez,'
,?-tt - r)u'1+
t50'1''
Thislengthisknownasthe(Ginzburg-l-andau)coherencelength. Itbecomes
large as F .
-->Fc and is thus very dipkrentfrom the tem perature-independent
parameters(0orraintroducedinSec.49.
In certainphysicalsituations(seeFig.50.1J)'1Pessentiallyvanishesatthe
boundary ofthe superconducting region (z= 0). Equation (50.17)can then
434
APPLICATIONS TO PHYSICAL SYSTEM S
Norm al
Fig.50.1 Surfaceregion betweennormalandsuperconductingmaterialfor(c)A<:
<J(v.
c
:l)
and (/
p)z
ï?>#(,:'
s
.
-1).
beusedtostudyhow !/ approachesitsasymptoticvalue1at:c. Animmediate
hrstintegralisgiven by
(50.l9)
If/increases
which iseasily integrated to yield
(50.20)
Justasin theelectromagneticpenetration.thespatialvariation of/ isconsned
to a region !z $
k:J',becausel- If ;vanishesexponentiallyforizr
.y.J.
,
Fora inalexam ple,consideran applied m agnetic held with an essentially
uniform orderparameter%'
-= I
YFXI(see Fig.50.1:). The supercurrent (Eq.
(50.9))thenreducesto
jt
(c*)2n*
e2n
x)= - FNY.C'A(x)= - m c'A(x)
(50.21)
whichtakespreciselytheform oftheLondonequation(49.29). Thepenetration
depth form agneticseldsfollowsim m ediatelyas
m * c2
+
A(r)-jyxp
,,
Nwl(pk
jjl
A(F)= 4
m *clb
=(/V (rc---'lv
(50.22)
1
and isproportionalto(Fc- F)-1 forF -->Fc(compare Eq.(49.1444.
SLIPCRCONDUCTIVITY
435
1:isimportanttoemphasizethatthecoherencelength ((T)andpenetration
lengthA(F)areboth phenomenologicalquantitiesdehnedintermsoftheconstants
aand b. ltisconventionalto introducethe Ginzburg-l-andauparam eter
(
KEEA
-ï
(F)
/'
)
(50.23)
w'hich is independent oftem perature near Fc. W ith the preceding deénitions,
a sim ple calculation yields
v'l'g,
<= -hc ,/c(r)A(r)2
p;
4rpc
l?1*f /7 1
(f0.24(z)
(50.24:)
&= We-b i,
-r
each of which is usefulin applications. ln particular,Eq.(50.24/)relates /t
'
to them easurablequantitiesH cand A;itm ay also berew ritten as
H - -.-11-C 2rrN.2JA
(jo.a5)
where(asshown in thefollowingdiscussion)
(50.26)
is the flux quantum .
FLUX QUA NTIZATION
TheGinzburg-l-andauexpression gEq.(50.9))forthesupercurrentallowsusto
verify London's prediction of a quantized tluxoid in a superconductor. The
order param eterm ay be written as1
j e.h -a
(ta*)2I
V1'
el2A
-- m#1% (V+- m .c
(50.28)
A + (j??c?c. =
P*)2i
V2
(50.29)
-
hcVv
e*
1Thisrepresentation ofVl-*isparticularly usefulin describing the Josephson eflkct.w'herethe
phase ofthe orderparameterisofdirectphysicalinterest(see,forexample,B.D .Josephson,
Advan.Phys.,14:4l9(1965)).
436
APPLICATIONS TO PHYSICAL SYSTEM S
lntegratethisequationaround aclosedpath C lyingwhollyinthesuperconductor
(Fig.49.2)
(c JI.A .-z-(?7
7*-c
#
1
.
j
hvc cJI.V(p
j ty,
kj
-k
,a= -g
I
ta,yj
.
(50.30)
Thet
irstterm ontheleftm ayberewrittenw'
ith Stokes'theorem ;ifVl'isassum ed
to beslnglevalued,theintegralon therightm ustbean integralm ultiplen ofl.
z:
,7121c J1.i
hc
'$*JS .h + .
(g-ï').j yj.*
.a= L
zznx
(50.31)
c, :
e
Comparison with Eq.(49.22) shows thatthe left side is London's fuxoid *
generalized to nonunil
-orm system s,and weconclude that(1)isquantized in units
of(pv= /7c.'
t?*. The presentderivation appliesonly nearFc,butitisplausibleto
expectthe sam e quantization foral1F < Fc.
SURFACE EN ERGY
The significance ofthe param eterK ism osteasily understood by studying the
energy associated with a surfaceseparating norm aland supereonducting m aterial
(see Fig.50.1). Theboundary ismechanically stable only ifthenormalregion
has an applied tield H c parallelto the surface,since the G ibbs free energy deep
in the norm alregion
H2
G(.
z-+-:
x)= G'
,,
()- y7c
F
(50.32)
then equalsthatdeep in thesuperconducting region
G(z--+.:c-)= Gbv
(50.33)
(com pareEq.(48.2l)1. Thepossibilityofasurfaceenergyarisesinthefollowing
wal'from theoccurrenceofthetwolengthsAand J. Ifthesamplewereentirely
norm alorentirely superconducting,theG ibbsfreeenergy perunitareaw ould be
fc
J.dz(-S2
cy'
8=). lnthesurfaceregion,however,thefluxisexpelled forzk;A,
while the condensation energy buildsup forzk.(. Hencethe true Gibbsfree
energy perunitarea isgiven approxim ately asthe sum oftu'o term s
-A
-JI1 -. - .s 2
;
4;(-..(lzV,
;rE'=-(jJz-p,v,--c
.
By dehnition.the surface energy ansis the diflkrence between the actualGibbs
free energy per unit area and the value that w ould occur if the sam ple were
uniform ly norm alor superconducting
H 2 .-A
c.
c.
c,,;4:--jgc((..dz4-jgdz-j..Jz)
(50.34)
stlpEncosDuc-rlvi'ry
437
The G inzburg-l-andau theory perm its us to study the surface energy in
greater detail. Consider a one-dimensional geometry (Fig. 50.I) w'
ith the
magnetic seld h(z)= /?(z).
f and vectorpotentialA(z)= ,
4(z).
f. ln the present
problem ,'1'm ay be chosen real in an appropriate gauge.i and the G inzburgLandau equations become
hl
.
,
42Vlp'
..
Sm Jkj.
av. j
u'
j
).
j%.yj/jq3.j.(jy+j2l.
.2J.c
..
r?
2. ()
-
f.
'
- . h'=
4
n.
(8*)2
11'
02W
- ---.-
(j:.t
j
.
yjuj
(50.355)
ln c
(50.35c)
w here the primes denote diflkrentiation w ith respect to z. The corresponding
boundary conditions
YF
h -0 jz >.- :c
- -
-
Hcj
A1'
-= '1
Y.
l'-:*1j
,z.
-,.+:
x.
h= 0 J
(50.36)
guaranteethatEqs.(50.32)and (50.33)are satisfied.z W ith the samedesnition
ofthe surface energy,we l
ind
Here the second and third linesareobtained with Eqs.(50.6)and (50.2),respectively. lfEq.(50.35/)ismultiplied byYF* and tntegrated overallz,a simple
integration by parts yields
j-xt
fzjh
vk
z+d
?1
.
l.
-k
4+(2p7+)-1;
l
t-mv+t
'
*c
A
-),
.
I,(
i
r
2j-()
!V.L.GinzburgandL.D.Landau,loc.cit. IfI
Y*I2dependsonlyonz,theorderparameter
takes the form ciWï;
r.F'àF(z). ln additions M axwell's equation implies thatj(z)=.
/(z)#.
Equation (50.9)then showsthatVF(z)mustberealwhileP(
p/
'P.
xvanishes. Theremainingphase
factor+(y)mustbe a linearfunction ofy and maybe absorbed with thetrivialgaugetransformation z4 ->.z4- (hcle*jPg)/Py'.
2TheseboundaryconditionsviolateEq.(50.8:)atz= +*,owing to ouridealized one-dimensionalgeometry. A morephysicalconiguration isaIongcylinderplaced in a solenoid.with
am acroscopicsuperconducting cortsurrounded by anorm alsheath.
438
APPLICATIONS TO PHYSICAL SYSTEMS
SubstitutionintoEq.(50.37)givesthefinalfdrm
x
4 (#c- hj2
c.s=
j-c
odz-'
Wl
k
l'l+ g.
(50.
38)
BothEqs.(50.37)and(50.38)areexact,theformerisalsovariationallycorrect
while the latter,which m akes use ofthe exactheld equations, is considerably
simpler,
Itisconventionalto characterizethesurfaceenergy witha length 3:
M2
cp&= - cb
8=
where
(59.39)
ss.je
-œJzg
r-j
,
l
o
Xwo!
T
.
1
4+(l-pj)2j
(50.40)
hasbeenrewrittenwiththeaidofEqs.(50.12)and(f0.16). Althoughacomputer
solutionofEqs.(50.35)and(50.36)isneededtoevaluate3forarbitrary<,special
lim iting caseshavebeen studied analytically. The num ericaldetailsarerather
tedious,however,and we only state the results'
4v'jJ
3 ;
k;1.89/
b=
(50.4lJ)
0
.-
1
)(v'
'1- 1)A;
:
z-1.l0A
a.= x,é
(50.41:)
Kà>1
(50.41c)
inagreementwithourqualitativeestimateofEq.(50.34).
The surface energy isim portantin determ ining the behavior ofa superconductorinan applied m agneticseld,and am aterialisconventionallyclassised
astype Iortype 11according asn sispositive ornegative. Com parison with
Eq.(50.41)yieldsthefollowingcriterion
T
1
ypeI: K< O
l
Type11: K > O
J(F)> V1'A(F)
ans> 0
(50.42)
((T4< vYA(r)
Thepositivesurfaceenergy oftype-lm aterialskeepsthesam plespatially hom ogeneous,and itexhibitsacomplete M eissnereflkctforallH < Hc. In contrast,
type-ll m aterialstend to break up into m icroscopic domains as soon as the
magneticseld exceedsalowercriticalseldHcj(seeProb.13.5),which isalways
lessthan H c. ForH > H cL,m agnetic flux penetratesthesam ple in the form of
!V.L.Ginzburg and L.D.Landau,loc.cït.;D .Saint-lames,G.Sarma,and E.J.Thomas,
tt-l-ype-llSuperconductivity,''chap.II,Pergamon Press,London,1969.
SUPERCONDUCTIVITY
439
quantized flux lines,and the sam ple is said to be in the m ixed state. Thisstate
Persistsup to an uppercriticalfield Hc;= V'IKHc(see Prob.l3.6)abovewhich
the sam ple becom es norm al. The possibility oftype-ll superconduetivity w as
:rstsuggested by A brikosov,lwho used theG inzburg-Landau theory to study
themixed statein detail. W'
e regretthatthisvastsubjectcannotbeincluded
here,and we m ustreferthe readerto other sources.z
6IEM ICROSCOPIC (BCS) THEORY
Forthe rem ainderofthischapter.we considerthe m icroscopicm odelintroduced
by Bardeen, Cooper, and Schrieffer (BCS) in 1957.3 This model has had
astonishing successin correlating and explaining the properties ofsim ple superconductors in term sofa few experim entalparam eters. The ground state ofthe
m odelhasalready been determ ined in Sec.37 with a canonicaltransform ation.
Such an approach can beextended to hnitetem peratures,butG orkov4hasshown
thatitis pretkrable to reform ulate the theory in term softem perature G reen's
functions. A sdiscussed in the following paragraphs,this approxim ate description represents a naturalgeneralization ofthe Hartree-Fock theory ofSec.27.
GENERAL FO RM ULATION
The theory starts from the following m odelgrand canonical/7(7??7?
'
/r()r2ï(7?7for an
electron gas in a m agnetic field
-
' )
..#.
,
..'
F
,.
a
J.
q
jrpii'.
'
t-?'a(x)l
j's(x)?
,'slx)L'
a(x)
whereA(x)isthevectorpotentialand-eisthechargeonanelectron. Asshoun
in Chap.l2.the exchange of phonons leads to an effective attraction between
electronsclose to the Ferm isurface. Thisinterparticlepotentialhashere been
approximated by an attractive delta function with strength g nw0 (com pare Eq.
'A.A.Abrikosov,Soc.Phys.-JETP%5:1174(1957).
1See.forexample,P.G .deGennes.*wsuperconductivity ofM etalsand A1lo)'s.''chaps.3.6.
W .A.Benjamin.lnc.,New'York.1966'
.A.L.Eetterand P.C.Hohenberg,TheoryofT/'
pe-ll
Superconductors.and B.Serin.T3'pe-11Superconductors:Experiments.in R.D.Parks(ed.),
.Ksuperconductivityy''vol.ll,chaps.14,l5.MarcelDekker,lnc..New'York,1969,
3W ehavefoundthefollowingbooksparticularlyuset-ul:P.G.deGennes./tlc.cI'J.;G .Rickalbzen,
Sk-fheory ofSuperconductivitys''John W'ileyand Sons.lnc..Neu'York,1965.J.R.Schriefer,
Sû-rheory ofSuperconductivityv''W .A.Benjamin,lnc.,New York,1964. Seealso.A.A.
Abrikosov,L.P.Gorkov.and 1.E.Dzl'aloshinskii,**Methods ofQuantum Field Theory in
StatisticalPhysics,'qchap.7.Prentice-Hall.lnc.eEnglewood Clifl-s.N.J..1963.
4L.P.Gorkov,Sot'
.Phys.-JETP,7:505(1958).
l4.tl
Appulcv loNs 'ro pHyslcAu sys-rEM s
(46.9)1.1 Thephilosophyofthemicroscopictheoryisthento solvethismodel
ham iltonian ascom pletely as possible.z
As an introduction to our subsequentdevelopment,we first review the
H artree-Fock theory,which m ay be obtained by approxim ating theexactinter-
action operatorf'in Eq.(51.l)asa bilinearform
P;z rr'fsa -g fd?xgxl
J.
f(x)fatxllss#j
ltx)'
fp(x)
.
-
q-'
f'a
f(x)fjlxllz?s#)(x)fatxll
Heretheangularbracketsdenoteanensem bleaveragewith thedensity operator
e.- jxss
)zfs=Tr
- e-jk-s
',
(51.3J)
and
Xss= ko+ Pzfs
(51.3:)
Inthisapproach,Xffsisused to define atinite-temperature Heisenbergoperator
('
xytxz)= ekssT'A'
t
;y(x)e-*HFr/h3withtheequationofmotion
hPI
X (x'
r)
l'
eA 2
.
y2
(
---=-yt(-j
Jj
vh-y.
-j-sfxytxr)
'-g-,
;J(x)fztxl-,ssf'
xytxv.
)--g(#â(x).
ll,
/xllp
.ss4xztxr)
The corresponding single-particle G reen'sfunction
%x/1(xr,x'r')= -t
$
x
'rzg;.
'Ku(x'
r)ylx
fp(x'm')))us
satistiesthe sam e self-consistentequation ofm otion asthatderived in Prob. 8.3
and istherefore identicalw'ith the H artree-FockG reen'sfunction ofSec. 27(see
Prob.l3.7). Theself-consistency here appearsthrough f'
ss,which both deter-
minesandisdetermined byEq.(51.3).
The precisestructure of P'sscan be understood by seeking a linearapproxi-
mation to the exactequationsofmotion. Since the commutator (P,fk(x)J
contains three tield operators, it m ust be approxim ated by a linear form
(P,f'
?.(x)J-.
>./.
'#?
J#(x)wherethe/ajarec-numbercoemcients. Thisreplacement
hastheconsequencethat((P,'
t
Ja(x))s?/j
f(y))o /xjftx- r). lnfact,theleftside
ofthisrelation isstillquadratic in thefield operators. W e therefore replace it
by itsensem ble averageto obtain the linearized theory,which providesa prtscrip-
tionfordetermining/a#. Theapproximateform bhyinEq.(5l,2)ischosento
reprodueethecorrespondinglinearequations.
1Thesingularnatureofthispotentialoccasionally leadstospuriousdivergentintegrals.which
wlllbecutof:ratthe Debye frequency inaccordancewiththediscussion in Chap.12.
2Superconductingsolutionsoftheelectron-phononhamiltonian(Eq,(46.1)Jhavebeenstudied
by N.N .Bogoliubov.Sov.Phys.-JETPn7:41(1958)and byG .M .Eliashberg,Sov.Phys.-JETP,
11:696 (1960)and 12:1000 (1961). This approach is essentialfor strong-coupling superconductorssuch asPb (Fig.51.2 and Table 51.l).which do notfitthe usualBCS theory Isee,
forexample.J.R.Schrieler.tlp.cil..chap.7,andW .t-.M eM illan,Phys.Sep.,163:331(1968)).
SUPEnCONDUCTCVITY
441
W e m ay also recallthe alternative derivation ofthe H artree-Fock theory
studied in Probs.4.4 and 8.3. ln analogy w ith W ick's theorem ,the ensem ble
average offourfkrm ion held operators is factorized asfollows:
'Frg1/')a(1)1Jlj(2)'
?
Jxy(3)?
fxô(4)))zfs= '
:(Fz('
(4a(1)1Jxô(4)))zfs
x (FzE1
J1:(2)fxy(3)1)ps.
-'Fz('
?
/'la(l)?
Jsy(3)1'
zHF(rz(1;l.
j(2)'
(,xôt4lllzfs
?.
(51.4)
This procedure im mediately yields the H artree-Fock equationsfor F '
.it also
provides a generalm ethod fordealing with the typicalexpressions occurring in
thetheory ofIinearresponse(see,fbrexample,Eqs.(32.7)and (32.11)).
The foregoing discussion m ustnow begeneralized to includetheone essentially new feature ofa superconductor.namely the possibility thattwo electrons
ofopposite spins can form a self-bound Cooper pair(Secs.36 and 37). As a
m odel for this phenom enon,we add tw'o extra terms representing the pairing
amplitudeto Eq.(51.2):
f-,
.z:fHF-l.g.
6*#3xgktfattx)?
J;(x)k.?
Jj(x)'
t
/atx)
1
-#'J(x)?
/)(x)'
t'
;,(x)'
t
/'atxlhq (5l.5)
-
Thisapproxim ation correspondsto adding a term cçzjfj
ftx)to the linearform
/zj!
Jjtxl.l Sincetheresultinglinearequationsofmotionfor.
?
Jand 9%arenow
coupled,the theory no longer conservesthe num ber ofparticles. Indeed,the
condensed Cooperpairsmay beconsidered a particle bath,in close G?1(7lO#>'With
the Bose condensation studied in Chap.6 and Sec.35. FOr consistency,the
factorization procedure of Eq.(51.4)mustalso be generalized to include the
pairing am plitudesz
trzEfx
fa(l)'
t
;)j(2)'
#xy(3)fxô(4)J)= (Fz(?
J1
ka(1),
?
Jx,44)1)ktrzll/1:(2)1Jxy,
(3)1)
't
rglé)a(1)?
Jxz(3)1)'
trmlfx
fj(2)'
?
Jxô(4)))
+ f'Fr(fla(1)'
t
ix
tj(2)1)k'Fz(?
;xy(3)('
x:(4))) (51.6)
-
Although allthetermsinEq.(51.5)canberetained,itismuch simplerto
omit the usual Hartree-Fock contribution f'ss entirely,thereby treating the
norm alstateasa freeelectron gas. Thisapproxim ation isbased on theassum p-
tion thatPss isthesamein both normaland superconducting phasesand does
notafectthecomparison between thetwo states(seethetreatmentofSec.37).
SincetheCooperpairhasspinzero,theindicesaandjinEq.(5l.5)mustreferto
oppositespinprojections,and thetotaleflkctivehamiltonianbecomes
#.,,-Xo-gJ#3xE--f1tx)#1
!(x))f?(x)?
;)(x)
+'
(hltxlf!
$(x)t4,(x).
#t(x))J (51.7)
!P.G.deGennes,op.cl
'
/,.p.141showsthatthepresentchoiceoffxnandgxgminimizesthe
freeenergy.
2L.P.G orkov,Ioc.cà/.
442
APPLICATIONS TO PHYSICAL SYSTEM S
whichform sthebasisfortheBCS theory. Thetheoryisselflconsistent,because
theangularbracketsareinterpreted asan ensembleaveragcevaluated with Xefr;
in particular quantitiessuch as
y.
j.jo-jkeff't
'
;t(x)yC!t
jx)j
(
1'
j-,)t
1(x)yaf
?(x)-,=
..y.
-.. /
u.jtk.
r
-r..-!
--.
re
.
donotvanish,becausegxeff,x'l+ 0.
W enow introduceH eisenbergoperators
k1,
'
K!(x'
r)= cxeffT'h1l
3
.r(x)t7-XrffT'h
ê*)
X'
t = --1 ihV + ex l- s y?,x
'
tt--g q'?t
yy
?) ;,xy
h J&r
..
.
.
lm
c,
A sthc linalstep in our form ulation,we define a single-particle G reen's function
(51.10)
where a particularchoiceofspin indices hasbeen m ade to sim plify the notation,
Diflkrentiate$ with respectto r. Thederivativeactsboth onthet
ield operator
and on the step functionsim plicitin the 'itim e''ordering,whieh yields
il
:,
? ,
4
q zz'
.
hjfTtx'
r,x m)= -/it
'
i('
r- T).
'(yx:(xr),1
.
/x:(x r))/'
T
(5l.ll)
(5l.l2a)
(51.12b)
--
â3tx - x')ôt'
r- m')
443
SUPERCON DUCTIVITY
In the usualcase ofa time-independenthamiltonian,the functionsF,.
W ,and
+ %depend only on the difference '
r- r'-and itisconvenientto introduce the
abbreviation
tlp
.
xt
xz)-
'
vx:(xr)
?/
-'
t
xt(XT)
(51.18)
and a 2 >'2 m atrix G reen-s function
=
$(xT.x'r')
#-f(x'
r.x,r,)
(5l.19)
,.
1W e herefollow theconvention ofA.A .Abrikosov,L.P.Gorkov,and I.E.Dzyaloshinskii,
op.cI'/.ysec.34,butseveralauthors(forexanApleaP.G.de Gennes,op.('f'
/..p.I43)introduce
an additionalminussign.
2Y.Nambu,Phvs.RE't'.s117:648(l960).
444
Appt-lcv loNs TO PHYSICAL SYSTEM S
The corresponding equation ofm otion becom es
(51.20)
l&(x)
à*(x)
-
t
?.
'
:jz
l
CA 2
+y,
-,(ov+yj
,
(5l.21)
This m atrix form ulation has been used extensively in studies of the electronphonon interaction in superconductors.l lt also provides a convenient basis
fora gauge-invarianttreatmentofelectrom agnetic flelds.2 U nfortunately,itis
notpossibletoconsiderthesequestionshere,and wegenerallyrelyon theoriginal
Gorkovequations(51.l5)and(5l.17).
sotu-rloN Fo R UNIFO RM M EDIUM
ln alm ost all cascs of interest the hamiltonian is tim e independent, and the
corresponding G reen's functions depend only on .
r- r'. lt is then useful to
introduce a Fourierrepresentation
Ftxr.x'r/)= ()h)-1N'e-iLe
'D(T-m'F(x,x',(
s,,)
(51.22J)
(51.22:)
where the choice t.
,)?
,= (1114-lj=,
'lh guarantees the proper Fermistatistics.
Thecorresponding equationsofm otion are
1
t
4A 2
ihu)n- 2m .
-ihT + -c-- + y.
z t#(x,x'.ts,,
)+ ,â(x).X 1(x.x',(s.)
=
1
-
hblx- x') (51.23/)
ex 2
ihcon- A
-N ih% + -C- -hp,.
W t(x,x',f
w )- .
'
X*(x)F(x,x',(sa)= 0
-'l
(5l.23:)
whichm ustbesolved along with theself-consisteney condition
(51.24)
lJ.R.Schrieflkr,op.cit..chap.7.
2J.R .Schrieokr.op.cit.,secs.8-5 and 8-6.
s UPEaCONDUCTIVITY
446
In the m ost general situation,these coupled equations m ust be augm ented by
M axwell's equation relating the m icroscopic m agnetic tield h = curlA to the
supercurrent j and any externalcurrents used to generate the applied held.
It is clear that a com plete analytic solution is im possible,so thatwe m ust turn
to Iimiting cases. Therem ainderofthissectionisdevoted to thetherm odynam ics
ofaninhnitebulksuperconductorin zero seld. W ethen exam inetwoexam ples
where the existence of a sm allparam eter pernnits a rather com plete solutîon:
the linearresponse to a weak magneticfleld (Sec.52)and the behaviornearFc,
wher: the gap ftlnction A(x)is small(Sec.53). These cases are particularly
interesting.forthey providea mlcroscopiejustitication oftheLondon-pippard
and G inzburg-l-andau theories.respectively.
In the absence ofa m agnetic seld,the vector potentialcan be setequalto
zero.and Eqs.(51.23)and(51.24)assumeasimpletranslationally invariantform
(51.25J)
(51.25:)
&. -. .l&h
g-.conn,xttx- 0,(
,?u)
?1
W ith the usualFouriertransform s
lklx.(unl= (2:71-3(d?keik*x@lk,conj
(5l.27)
we obtain a pairofalgebraic equations
(-iho,n- (kj=Wflks(wl- z
X*Ftk,f
zlnl=0
(51.28J)
(5l.28:)
where EcompareEq.(37.24))
h2k2
fkEE lm -- p.
(51.29)
These equationsare readily solved to give
F(k
hlilta)n+ J7
k)
a?a)= jcuj.
s+ jk
2-j.j-j-ri
n
(51.30J)
Y ttk,
hA*
(snl= haco2
u-t-Ja
k-i--s
!--'i
(5l.30:)
,
.-
-
446
APPLICATIONS TO PHYSICAL SYSTEM S
In theabsenceofan appliedfield,theparam eter,
.
A m aybetaken asrealwith no
lossofgenerality,thereby ensuring that
W ftk,tzln)= W tk,tzlnl
(5l31)
which follow'
smostsimplyfrom Eq.(51.16)forauniform medium .Furthermore,
Eq.(51.30)may bewritten in theform
.
(5l.32J)
(51.32:)
(51.33)
and
X
.
Nk!hk= jyk
!
7
k l- uv,,-jl I êkb
'
,
-j
2 ,.-
-
(51.34)
..
arepreciselythequantitiesintroducedinEqs.(37.32)and(37.34). Comparison
with Eqs.(2l.9)and(2l.l0)showsthecloserelation between superconductors
and condensed Bose systems.
Forthe presentuniform medium ,the self-consistentequation (5l.26)for
J
îbecomes
.
A
.=b#b
-
#3/t
-
h.
.s
-it
>7j(#Jk-)a+-yj
k
(5j.35)
-
where theconvergence forlarge nJallowsusto sety = 0. Theseries m ay be
summed directlyw'
ith a contourintegral(compareProb.7.6)orwith thepartialfraction decomposition ofEq.(51.32:);canceling the'
common factor.
l,we5nd
l=g(2=)-3(d?k(2Fk)-1tanh(JjFk)
(5l.36)
W e now introduce the approxim ation
which willbe used consistently to evaluate integralsthat are peaked nearthe
Ferm isurface. The resulting gap equation isaflnite-tem perature generalization
ofEq.(37.45). Itmustbecutofl
-at (k(,
=hcoo<:
<esinexactlythesameway,
and the sym m etry ofthe integrand allow sus to write
xj)ytanhgz
-jt.
l=gX(0)(&c
oo(tj-s
dt
:
1
y2
.)
-
.f0
.
s
(51.37)
SUPERCONDUCTIVITY
M7
whichdeterminesthetemperature-dependentfunctionA(r). Since
( h2k2
= lm - p
in the presentm odel,the density ofstatesbecom es
#(0)= mkF
(51.38)
2.
=zha
whereJ.
thasbeen expressed in termsoftheFermiwavevectorks(compareEq.
(37.47)),andweneglectthesmalldiserenceinthefreeenergybetweenthenormal
and superconducting states.
DETERM INATION OF THE GAP FUNCTION Z (F)
ItisevidentthatEq.(51.37)reducestoEq.(37.48)asT-+.0and givesthesame
solutionàll- 2â(z)oe-17N(0)g In theoppositelimit(r-+.Fc),thegap vanishes
identically,and we 5nd
1
lcopd(
(
= g#(0) o -f tanhlkBTc
(51.39)
Tosolvethisim plicitequationforFc.itisconvenienttointroduceadim ensionless
integration variable
l
X(0)#=
zd
z
4) z
tanhz
(5l.40)
wherethecutofl-z = âoao/zksrcisessentialbecausetheintegraldivergeslogarithm ically forZ -->.:s. Integrate by parts2
1
z
A(0)g (
1nztanhz)
I-jodzlnzsechzz
=
(51.41)
In a11casesofinterest,the upperlimitisvery large(seeTable 51.l)and Eq.
(51.41)maybeapproximatedas
1
.
N(0)# ln(1k*
sr
Bc)-j0
'
*dzlnzsechzz
-
-
(51.
42)
where the dennite integralis given in A ppendix A . A sim ple rearrallgem ent
yields
(51.43)
W eseethatrcandthezero-temperaturegap é(F= 0)- tâtlEEq.(37.49))both
dependsensitivelyon theparameterA(0)g;thisdependencecancelsinforming
APPLICATIONS TO PHYSICAL SYSTEM S
theirratio
Atl
k ='
n.
e- yQJj.ys
s Fc
(51.44)
whichisauniversalconstantindependentoftheparticularm aterial.
Table 61.1 Therm odynam ic properties of typicalsuperconductors
1
.
1
.
â: H-loj,oei
z'
s
-c(
;
)
t
ril,ca-c.j
, k.zi '
1
i ct . c. ,.-
l
j
rc.oK e-hœolkz,oK N(0)g:
'
Bcs
cd
0.56
A
1.
snl 3
.2
75
Pb
7.22
164
1
I
i ,76 1
4
1
-.----
; 0.168 1
1.43
i1
0.18 ;
l.6 I 29 :'i 0.l77
' 13-2.1 I 1o6 ' l 0.171
32-1.4o
.
3
75 !0
1
8I
.
4
5
195
0..
25
' .
l6
I 305 i
!0.161-.
0.164l
!l
l.
60
.
96 t0.39
2.2 j 805 t
l 0.134 1
; 2.71
5
1N(0)#iscalculatedfrom Eq.(51.43).
Source:R.Meserveyand B.B.Schwartz.Equilibrium Properties:ComparisonofExperimental
Resultswith Predictionsofthe BCS Theory,in R.D.Parks(ed.),tesuperconductivity,''vol.1.
pp.122.141.165.M arcelDekker,lnc-,New York,1969;D.Shoenberg,*lsuperconductivity,''
2d ed.,p.226,Cambridge University Press,Cambridge,1952.
A sseenin Table51.1,thisrelation isquitewellsatissed in practice. Fora
givenelement.Eq.(51.43)showsthatFcisproportionaltothecutosœo x M -%,
which therefore yieldstheisotope eflkct,discussed in Eqs.(36.2)and (48.6).
The ratio #/Fc= hœolkaFcalso determinesthe eFective interaction #(0)g>
N(0)y2,whereyistheelectron-phononcouplingconstantinEq.(45.12). For
ionswithchargeze,acombinationwithEq.(51.38)leadsto
#(0)y2= zl hlnj =2
2(3=2)1 m McI
(5l.
45)
and the observed values'for Cd,which has the sm allest valence of those in
Table51.1(z= 2,ncA.
/= 8.6g/cm3,cl= 2.t8 x 105cm/sec),giveA(0)y2= 0.54.
The approxim ate agreem entwith the value in Table 51.1 is evidence for the
im portance ofphonon exchangein superconductivity.z
The tem perature dependence ofthe gap param eterm ay be derived from
Eq.(51.37). Sinceitssolutionrequiresnumericalmethods,weshallnotattempt
1See,forexample,C.Kittel,tklntroduction to Solid State Physics,''2d ed.,pp.1* and 259,
John W iley and Sons,Inc.,N ew York.1956.
2The comparison between theory and experimentis complicated by the repulsive coulomb
i
nteraction(see,forexample,G .Rickayzen,op.cit.,sec.5.7andW .L.McM illan,Ioc.cit.t.
SLIPERCONDUCTIVITY
449
a detailed analysisand instead merely state the limiting behavior(see Prob.
13.9)
A(F);
:
zAo- (2=tXaksF)+e-%1kBT
(5l.46J)
atwla,k.r-,rg,/(3)j'(l-j)*
z' 1
cs3.06:sFc 1- p
Tc- F< Fc
(51.46:)
'œC
Thecom plete functionisshown in Fig.51.1.
à(T)
k
T
S
1.5
oTIN33.5Mc
* TIN 54Mc
1.0
N
%
- BCS THEORY
%
Fig. 61.1 Variation of A(F)/ksFc in Sn,
measuredfrom therelativeultrasonicattenuation in superconducting and normalstates
(see Prob.13.19).(From R.W .M orseand
H.V.Bohm,Phys.Rev.,10*:1094 (1957).
Reprinted by perm ission oftheauthorsand
0.5
0
0
0.2
0.4
theAmericanInstituteofPhysics.)
0.6
0.3
1.0
F/Fc
TH ERM O DYNA M IC FUN CTIONS
Thethermodynamicpotentialmaybedeterminedfrom thebasicequation(23.21).
Sinceourmodelneglectsthe usualHartree-Fock termsin both thenormaland
superconducting state,the thermodynamic potentialDain the normalstate is
justthatofafreeFermigasDn,and we5nd
('
h'
:-taa-J0
't
czz-ltz/ll
=-j
1j'
0d
#
',
'j#3x#,
t4J(x)fj
f(x)4j(x)4.(x))
=-j,
'#g'(g
1,
)2jt
isx?
.
A(x)l
2
(51.
47)
whereEqs.(51.1),(51-6),and(51.14)havebeenusedinthesecondandthirdlines.
Notethat(AXI)difersfrom theensembleaverageofthelastterm inEq.(51.7)
by afactor!.. Thisfactoralwaysoccursinany Hartree-Fock theory andmay
beseenexplicitlyin Eq.(27.20).
4B0
APPLICATIONS TO PHYSICAL SYSTEMS
In the presentuniform system ,the spatialintegration m erely introducesa
factor P''
,a sim ple change ofvariablesthen gives
(1
g
l2
,-
D.=-F(odg',
.--;.
tX2
-I
,
'j-o
gdg,dj&o
l#,
fy
,.
').
,
:
Xc
a
-v)-v
fs,
1,o,)a #v(
/g
nl
,)
(5l.48)
Thislastform isparticularly convenientbecauseEq.(51.37)expresses1/g asa
functionofz
V FI.anddirectsubstitutionleadsto
t
kp'fk x(())jo
''
.n(
/jr(o'
&dh.(a
,,
)zp.P
.
anh(
j.
?E'
)
â't
E'
.
â-'z'
-x(0)jof
yg'
â
gztanhtl.
js)-2jc
sJa'v
'
l,
'tanl
atèjz-'
lj(51.
49)
wherethesecondlineisobtainedwithpartialintegration. Thehrstterm isjust
therightsideofthegap equation (51.37),andthesecondcan besimplified by
changingvariablesfrom z
'
X'toF'= ((é')2+ J2j1. Inthisway,we5nd
ha)o
Jogy(j
.
js
co.
s
k
h
à(
t
jè)
jE
tj)
4N (0) *h*l1
+ # j, d(ln(1+e-if)(5l.5û)
Sincehœo>>ksF,thelastintegralontherightm ay beextended to insnity. A n
easy calculation then shows that it equals the srst tem perature-dependent
correction to the thermodynamic potentialin the normalstate (compare Eq.
(5.53)):
4N(0)U' a'
.
-
#
j,(
dl'l
ntl+e-bt)-.
jx(0)p'
xzt1.
sr)2;
k:-((k(r)-(1a(0)j
(.
5l.5l)
In addition,itisreadily verised that
'- D
-2x(o)jo #f(e--f)a.-x(0)(
/
.
1
.
a2+a2Inz
-/.
i
,
t
j
ooj
(5l.52)
SUPERCON DUCTIVITY
451
*hf.tpo
-
4Ar(0)kaTJ(0 dlln(1+e-bE)+.
j,
rr2.
N(0)t/('sF)2 (51.53)
where the inequalities F ,
:(Fc<:
.
tP <<:rs have been assum ed.
Thisexpression correctly reducesto Eq.(37.53)atF = 0. ltalso allows
us to evaluate the leading Iow-tem perature correction, which arisessolely from
the norm al-statecontribution becauseA - AoN'anishesexponentially forF --+.0
(Eq.(51.46J)J. A directcalculation with Eqs.(23.9)and (5l.32J)shows that
N .s ;
k:N,,foralIrc.
crcgcompareEqs.(37.51)and(37.52)1,and wemaytherefore
reinterpretEq.(51.53)as
(51.54)
A com bination with Eqs.(48.22)and (51.44)givesthe low-temperature critical
tield
(,2? 'T '2
s-(w)-l4x,
vtel.
spl
xgl--j(.
y;)j
.u-(.)g,-I.
e6(yrf
'
j2j
.
where
(5l.55)
Hc(Q)= (4=2V(0).Xà)i
is the criticaltield at F = 0. Since 2N'(0)determines the normal-state speciflc
heat(compareEqs.(5.59)and(51.38)q
C,,. . 2x2
.
X'
j x(()jkjv
(5l.58)
which isindependentofthe m aterialin question. Each ofthese parametersis
measurable,and experimentalconhrmation issatisfactory (Table 51.l).
Itisalsointerestingtoevaluatej'
k itself,whichmaybeobtainedfrom Eqs.
(5l.5l)and (51.53)
f
kz, DyjF= 0)
y.((jg
s=--.
j
y-..-jN((j
).
yz(jm2jnAg
jj-4,
pjho
a
,syj
.
rju(js.g-,s)
g
(51.59)
4s2
APPLICATIONS TO PHYSICAL SYSTEM S
Inthtpresentlow-tem peraturelim it,itisperm issibleto evaluate theintegralby
settinghœo -.
>.(
x,and /.
ï QJétl;an approximatecalculationyields
-*
lcdtln(1+expE-j(f2+u
&à)1))a;.
I.d-/.
ln(2=1(j#-1)1
whichcanbecombinedwithEqs.(5l.46J)and(51.59)togive
f
l
-.-.,.
p'P-,(Fp
--0)-lxt()ag-2x(()(2z
p
rA
qo)1e-bho
(51.60)
Theelectronicspecischeatin thesuperconducting statecan then beobtained by
di/erentiation
c
k
xo ê
t
-- J;
k;2A(0)(
'
X(
)/cs(2zr)1 ksr e-L*/kBT T-->.0
(5l.61)
p'
Thisexpression clearly exhibitsthe energy gap mentioned in Eqs.(36.1)and
(48.5).
The preceding calculations have considered only the low-tem perature
behavior,where.
â - tâtlisexponentiallysmall. Although a generalevaluation
ofEq.(5l.53)foralIF < Fcrequiresnumericalanalysis,itispossibletoderive
explicitexpressionsnearFc,where #,
..
â
ctlprovidesa smallparameter. W e
startfrom the gap equation (51.35),which may be expanded in powersof z
'
X:
1- 2Jf(0) h*n
g
2x(O) i
-h'
.o .v f
1
.
g' j,,Jyntj.so(,y-s
-
l
.
:2
=-p J(
) dtzzLîytzpn)-i+yz-((:f
.
sa)2+J2)2+'
Thederivativewith respectto z
'
X iseasilyevaluated,and acom bination with Eq.
(51.48)gives
f1,- f'
)a=
p'
-
z
V(0).
54 hcoo#f
j
()
1 au
a
'
E
-(
-/u
',n) + f 1
(51.62)
Sincehœo> kzFc,theintegralmaybeextended to insnity:
f1,- Da
X (0)Z4 = 1
s =- y é jjjms.
j
--!.
x(0).
â(1
.)2éX(zn+
l.j)3.-7
..
4:
.
/1xto142
#.2c
8 N(0)(
7(
(3) rksrclzj
l(I-.
v
Tc
,
)2
(51.
63)
4
a=0
SUPERCONDUCTIVITY
4B3
where thesym metry ofthe sum m and hasbeen used in the second line and the
explicitform ofé(F)in Eq.(5l.46b)hasbeen used in thelastIine.1
Equation (48.22)againdeterminesthecriticalheld,and wehnd
y -8
- -
Hc(T)=zJc(0)e7((j)'jl-y
rcj
cs1.
74.
s-(0)I-zrC
-j r.
-sv,
(51.
64)
where Sc(0) has been taken from Eq.(51.56), Note that Eqs. (51.55) and
(51.64)areverysimilarto thephenomenologicalrelation (48.3). Nevertheless
there are smallbut distinct diflkrences (Fig.51.2),and the BCS predictions
provideabetterfltformostsimplesuperconductors. SinceS1/8=istheactual
diflkrencein free-energy density between thenorm aland superconductingstates,
theexcellentagreementbetweentheoryand experimentjustihesourassumption
thatthe Hartree-Fock energy is the sam e in both states. Thisassum ption is
extremely diëcultto justifya prioribecausethe condensation energy (= 10-7
ev/particle) isso much smaller than the Hartree-Fock energy (;
k;l to 10 eV/
particle). Equation(51.63)alsoallowsustoevaluatethejump inthespecihc
heatatFcgcompareEq.(48.27)1
l
8
'
ptc'
x- Cn)iwc-yttjjY(0)n.2/fjFc
(5l,65)
0.02
S((F)
,-(()) -
I7
Pb
gl-(z))j
0
02
0.4
0.6
0.8
(F/7))2
-
0.02
A1
-
Sn
Bcs
0.04
Fig. 61.2 Difrerence between actual critica! Neld
/f(F)/Sc(0)and theempiricalcurve1-(F/rc)2. (From
I.B.M .JournalofResearchandDevelopment,vol.6,no.1,
Jan.l962,frontcover, Reprinted by permission.
)
'An argumentsimiiarto thatin Eq.(37.52)showsthatFs(N)- Fn(N )= D,(Jz.)- DntJzal,apart
from corrections of order tp.
x- p.
alz//
zz
a. W e can therefore use Eq,(51.63) to determine
y.
s- p,
nnearTc. A simplecalculation with Eqs.(51.38)and (51.44)gives
N (p.
s- H.nb/(Fs- Fa)='
à-f2(Plnz&u,
'êlnN )Fc/
.
'
(Fc- F).
APPLICATIONS TO PHYSICAL SYSTEMS
464
andacombinationwithEq.(51.57)yields
C,- Cn
C
=
12
a;1.43
7((3)
inreasonableagreementwithexperiment(Table51.1).
n
rc
(51.66)
SZDLINEAR RESPONSE TO A W EA K M AGN ETIC FIELD
Asa furtherapplieation ofthemicroscopic theory,we now turn to theelectrodynamic behavior of a superconductor,which is probably its most striking
feature. In principle,allsuch eFectsare contained in the Gorkov equations
(51.15)and (5l.17),butthey are quite intractable in theirmostgeneralform.
Forthisreason,itis simplerto evaluate the electrodynamicsofa bulk superconductorwiththegentraltheoryoflinearresponsefrom Sec.32.1 lnparticular,
weshallevaluate the transversecurrentj(xJ)induced by an applied magnetic
fieldspecifedbyatransversevectorpotentialA(x?). Althoughitispossibleto
maintain fullgaugc invariance throughoutthe calculation,zthe detailsbecom e
quite com plicated and tend to obscure the sim ple physicalresults. Hence the
presentcalculationw illbecarried outintheLondon gauge
divA(x/)= 0
q-A(q.tz))= 0
(52.1)
Moregenerally,thevectort
ield A(q,t,))alwayscan beseparated intoitslongitudinaland transverse parts
A'(q.oa)= t
iE#.A(q.t,a)1
z4'(q.t,a)= A(q,oJ)- tjE#.A(q,f
a))q
.
(52.24)
where,by dehnition
qx Al(q,œ)= 0
q.A'(q.(s)=0
(52.2:)
A generalgaugetransformation ofthevectorpotentialA and scalarpotentialg)
takestheform
A(q,œ)->'A(q,œ)+ l'qA(q,(z,)
m(q,u,)-+e(q,*)+ iœc-,A(q,œ)
(52.3)
whereA isan arbitraryscalarfunction. ltisevidentthatthegaugetransforma-
tion afects only the longitudinalpart(Al-+.At+ iqA),so thatAîisgauge
invariant.
lThe same results can l
x derived by a perturbation expansion ofthe Gorkov equations.as
shown in A.A.Abrikosov,L.P.Gorkov,and 1.E.Dzyaloshinskii,op.cit.,sec.37.
:Carefultreatments ofthisquestion may be found in P.W .Anderson.Phys.Rev..110:827
(1958)and 112:19X (1958);D.Pinesand P.Nozières.e'The Theory ofQuantum Liquids.''
vol.1,sec.4.7,W .A.Benjamin,lnc.,New York.1966;G.Rickayzen,op.cit.,chap.6;J.R.
SchrieFer,op.cit..chap.8.
SUPERCONDUCTIVITY
4BB
DERIVATION O F THE GENERA L KERNEL
In the presence ofa vector potential, thetotalhamiltonian operatorXtisthat
given in Eq.(5l.1):
4,= X + Xx
(52.4)
whereW isthehamiltonianinzeroseld
ff
c3x,
iltxl(-âl
2m
:72)4-(x)-!.
gj(
F3xf)(x),
o,(x),
;,(x)4.(x) (52.5)
:.- .
.
and 4x istheperturbation
z?x-J#3x(a,
E
,,
.
*
-l
C
,(
.
,
/
,
)(x)vf-(x)-(v,
c(x))4-(x)j..(x)
+2mcr
.(x)1
.
2,
J)(x),
J-(x))(,a.
6)
A s noted previously,e isa positive quantity and the charge on an electron is
t'2
-- -
c
e.
-
For any operator ö(x/),theensemble average (0(x/))x in the presence ofthe
vectorpotentialisgiventolowestorderby(compareEq.(32.2))
wheretheunlabeled bracketsdenoteanaverageintheunperturbedbutinteracting
ensem ble. Since we shall deal w ith operators that conserve the num ber of
particles,itispermissibleto replacethe Heisenberg picture by (compare Eq.
(32.6))
Jx(x?)- cifl/âJ(x)e--ikbl
't
(52.8)
detined in termsofk = X - JzX.
W e now specialize Eq. (52.7)to thetotalel
ectromagneticcurrentjin the
presence ofA . Thisoperatorhasthe intuitive form
j- -'
l'
e'E'
?
;'lv';'a+ lv#zll';'al
where gcompare Eq.(49.25)) A= p1-l(-f/iV + :A,/c).
Ieads to
j=
-
eh ,
.
el
gml.(yl;Vy'
,x-(Vy*'
a
T)y'a)- N1C,A'
;)a
1y'?(
z
(529)
.
simple calculation
(52.10)
anditcanbevèrisedfrom theheldequationsforb?and ('%thatthesecondterm
isessentialto guaranteeconservation ofcurrent. A combination ofEqs.(52.7)
and (52.10)yields
(52.ll)
4se
APPLICATIONS TO PHYSICAL SYSTEM S
Thelinearresponseisobtained byexpandingtohrstorderinA ;sincetheexpecta-
tionvalueofjvanishesinzeroseld.wefind
J(x/)- tj(x?))x
(52.12)
where
eh
jl(x?)- -js,,.l#la(x?)V#xx(x?)- (Vf:a(x?))#xa(x?))
(52.13)
isthe current operator for A = 0. lt is convenient to rewrite this equation as
anintegralrelation between thevectorpotentialA and theinduced supercurrent
j(compareEq.(49.3l)1
C
jklxt)= -j.
j #.
4x,#/,Kktlxt,x'/')zll(x,t,)
..
.
(52.14)
wherethe repeated lower-case latin subscriptsreferto vectorcom ponentsand
are sum m ed over the three spatialcoordinates. Tbe specifcproperliesofthe
m edium are contained in the kernel
and theproblem isthusreduced to theevaluation ofaretarded currentcom m utatorin the exactunperturbed system
Pklçxtnx't')- u-iqk.
ltklxtl,jttlx't-.
)1),b(t- t')
(52.16)
Asdiscussed in Chaps.5 and 9.the retarded function isinconvenientfor
perturbationanalysis,and weinstead introduce atem peraturefunction
Mktlxr,x'T')M -(F,.(.
/1k(xT).
/kl(x'T')1)
dehned in termsofthe çtlleisenberg''picture
OX(x,
r)= ekrlh0(x)e-krlh
(52.18)
The relation between Eqs.(52.16) and (52.17) is readily established with the
Lehm ann representation,which gives
#J(X,x ',œ)= * daœ'
p
')
.
rr(
skI
-(x,
(
.
0x'
,,
+u,
j
,y
(52.19)
o'%k.
ekl(x,x'.v.)- = dl.
'(x'x''œ')
(52.20)
,
-x 2.
n Ip.a- ttlr
wherethe Fouriertransformsare desned asin Eqs.(32.12)and (32.13),and
vn= ln=lph.
SUPERCONDUCTIVITY
467
Thefunction@ktmayberewrittenasaspatialdiflkrentialoperatorapplied
to a ittim e''-ordered productofseld operators
(52.21)
wheretheargument1isanabbreviationforthevariables(xj,'
r1). Thisequation
is exact,butit is now necessary to introduce approxim ations. Schafrothl has
show n to allorders in perturbation theory that an expansion in the electron
phonon interaction can never yield the M eissnerefl
kct. W etherefore rely on
Gorkov'sself-consistentfactorization fEq.(5l.6))to write
'FmE?
/4(
x(2)#x.(1)f4j(2')kxjtl')))
'
;
k;4C#(12)F(l'2')- 2F(12')F(1'2)+ 2.W(11').Ff(2'2) (52.22)
where@ and.W'aretheG orkovfunctions,and thespinsum shavebeenevaluated
explicitly. A combinationofEqs.(52.21)and(52.22)gives
ltis now convenientto specializeto a uniform tim e-independentsystem ,
where@ktlxw,x'.
r')-'
@ktlx- x',r- r')maybeexpressedinaFourierexpansion
@kl(x- x','
r- r')= (2=)-3f#3geiq*(x-x')(jâ)-1j
je-ivnç'
r-r'b.
@kl(q'p,
n)
n
-
.
(52-24)
The corresponding Fouriercoeëcientiseasily evaluated with Eqs.(51.22)and
(51.27). Thefrstterm involving F(l2)F(1'2')isjusttheproduct-(X(x))x
(7/(x')). A simplecalculation showsthatitvanishesidentically,conhrming
the previousassertion thatthere isno supercurrentin zero feld,and we obtain
'
eh 2
#3p l
J?
Ql
(q,'
za)-2(.)J(a.)3yjF
.(p+ 'l'q):(p+ 'l-qll
,
x lf#tp+ q,f
.
,?,+ L)f#'
lp.t.
u,ll+ ,
* (p+ q,t,
'
al+ u).
* (p,r
.
zal)) (52.25)
lM .R.Scharroth,Ioc.cit.
468
APPLICATIONS TO PHYSICAL SYSTEM S
Here F(p,t,)I)and .Wtp,(t)jlaregiven in Eqs.(51.30)and (5I.31);since they
depend only on lp(,a change ofvariablesin Eq.(52.25)zp-.
>.-p- q,f
aal--+
f
,.
,l- valshowsthat.
@ktisanevenfunctionofra,exactlyasinEq.(30.l8):
#'kllq.wl- J/kltq,-%)
(52.26)
'
..
The evaluation ofthe frequency sum islengthy butnotdim cult, because
tht partial-fraction representation in Eq.(5l.32)reduces allthe variousterms
to theform studied in Eq.(30.8). Hencewemerely state theGnalresult
ekl
(q,
u)-(E
,J12J(sj2
r)
jpkp:((r
/+u-+t'
-r
,
-)2L.
t'
(E-)-.
/'
(s-))
,
x
l
i
l,w --#u (
..---/.
-j
E/.
.-----sr..1- lu-s+.àuj-(/-
+ (u.r-- t,.u-)2@l-f(E+)-f(F-))
1
l
t
ivn- /-1(/.+ # ) ivn,à'Q(#+e.
L
E'-jJ
.
-
where the sym m etry in va has been m ade explicitand p has been replaced by
p- !.
q. Here the subscripts + denote the arguments p:i:l.
q (that is.Ex=
fpàlq,etc.),and
f(E)> (p9E+ 1)-1= Jg1- tanh(JjF))
(52.28)
isa modised Fermifunction. Notethat/(F)diflkrsfrom theusualdistribution
function even in thelimitz
'
â--.
>0 (normalmetal),because E then reducesto
Ifi= re- p.r. Equation(52.27)isnow intheproperspectralform,and#!(q,(,p)
is obtained with the substitution ivn->.u)+ lh. The factors involving u and v
maybeevaluatedwith Eq.(51.34),and wethereforeobtaintheFouriertransform
oftheintegralkernel(Eq.(52.15))
4=nelj
Xkîtq,(
z
)l-mcz.1o4.
a.j
'
.
m
câ
.
c
-42)ta
#,3
r
g
lj/Npl(j
)(js.(+s
J.
-.
s
+-,
12)
.
-E.
,.
(s-)-.
,.
(s-)!gà-.,
.
,-1(s--e.,
(52.29)
SUPERCONDUCTIVITY
459
which describesthe generalresponse ofa uniform superconductorattem perature
F to a ueak applied (transverse)vectorpotenlialA
c
(52.30)
J'
ktq-tz?l.
'
--- 4= A'k/tq-t
.
z?l.
1Itq-tz?l
Unfortunatel),thisexpression iscom plicated and unwieldy,and we henceforth
consideronlya statict
ield (r
.
,?.
-.0). A sseellbelow,even thissim plercaseleads
to rather long calculatiollssand the generallzation to finite frequency doesnot
involve any new qucstionsofprinciple.
M Elss N ER E FFECT
TheM eissnereflkctcannow bedem onstrated byexam iningthelim itingbehavior
ofA-(q.0)as(1..
-'.0 (com pare Eq.(49,45)1. ltismostcon:enienttoevaluatethe
induced supcrcurrentilself
?L,2
1
) ch 2 .
* J3?
j(q)--??cAtq)-t(
.nt-j j(2=)/3pg
p.A(q)j
-
.
s
.
(--.X2 f(E.)-/'(E )
(.-t-.
â2 1-.
(F-)-.
g(l.
-#-v
.v j-j.
-.s.è-(l-J.
ly
..jy.j-fy
..g).lE-jj
(1flE-)- /(F-)
..
- --.
F .-.E-
ê/-(Ev)
:z:2
?J.p
and ue find
jt
nel
q)$
4;- s
2 eh l '
* (1'p
èj'qk-)
')
(.A(q)-c
'(,
?
y)!(2.))plp-A(q)1'-ègv'
=
-
.
??c2
hl *aD
A(q)t;;c
. 1-,-?.,s ajjjj
-
èfqE j
!(
;d;)p4-yyp
'
.
= -
.
7
r)t,
A(q),
.
-f(-2
n2c
hz xaz
ns(T)H /7- j4cjt,.
(52.32)
4;-qs )
)(
jp4dp--jyP
C
(52,
33)
4en
APPLICATIONS TO PHYSICAL SYSTEM S
dehnesthe superelectron density in the BCS theory.t ltslimiting behavior is
easilydeterm ined with thetechniquesdeveloped in Sec.51:
n,(F) I-(a
,
r
.
rk
y
r
,.jle-a,/.sr
a a(j-y
vC.
j
(52.344)
œ
(52.34:)
and thecompletefuactionisshowninFig.52.1,alongwiththeem piricalfunction
g.
è.
J.
q.
1.q21,o
à(
r)
0.8
O.6
0.
4
0.2
O
0
smpirjcajk.
t-(y,,yl4)
.
BCS(nonlocal)
BcS(locall-
O.2 0.j 0.6 0.8 1.0
TlT
Fig.62.1 Temperaturedependenceofthepenetrationdepthin
apuresuperconductor. ThecurvesBCS (local)and BCS (t
l0nlocal)were plotted from Eqs.(52,33)and (52,
44),respectively,
using1helablescomputedbyB.Muhlschlegel.Z.Physik,155:313
(1959). NotethatthecurveslabeledBCS(local)andempirical
alsorepresentthecorrespondingratiosnslTlln(Eqs.(49.15)and
(52.35)1.
'
in Eq.(49.15). Sincethe form ofEq.(52.32)agreeswith thatofthe London
theory gEq.(49.29)J,we immediately conclude thatthe penetration depth fora
pure London(local)superconductorisgiven by
lA(0
r)j2.j/
As
(z
0'
)j2.n
.'
i
L
l
n
'
iz
b joeajjjmi
t
(52.35)
wheretheonlyeffectofthemicroscopictheoryistheuseofEq.(52.33)onthe
rightside. Notethatz7s(F)isdefined through the strength ofthe response to a
long-wavelength perturbation and reduces to the totaldensity n as F -.
>.0.
1The BCS single-particle excitation spectrum hasa gap A. Asshown in Sec.54,a simple
quasiparticle modtlthen yields both Eq.(52.33)and a criticalvelocity l
?c$
kSé/âks for tbt
destruction ofsupercurrents. Thisexplanation offrictionless flow mustbe psed with care,
however,for there are gapless superconductors,which have a finite order parameter A(x)
butno gapin theexcitationspectrum (A.A.AbrikosovandL.P.Gorkov,Sov.Phys.-JEF#,
12:1243 (196l);M .A.W oolfand F.Reif,Phys.Rtrl
,
'.,137A :557 (1965)),whereasinsulating
crystals also have a gap in theexcitation sptctrum. The correctprocedure, ofcourse.isto
aska physicalquestion.suchasthe Iinearresponseto atransversevectorpotential.
461
SUPERCON DUCTIVITY
PENETRATION DEPTH IN PIPPARD (NON LOCAL) LIM IT
Electrom agnetic eflkcts involve wavelengths com parable with the penetration
depthA(F),so thatthepreviouslimitq(oEaqlht,s/vr-kv)<:1maybeinterpreted
asdescribing the behaviorofa London(local)superconductor. W enow evaluatethekernelintheoppositelimitq% y.1,whichwillthenallow ustodetermine
the penetration depth ofa pure Pippard superconductor. In addition to the
aboverestriction,qm ustalsosatisfytheconditionq<:
tkF,becausethepenetration
depth isalway'sm uch larger than the interatom icspacing.
Itisconvenientto evaluate Eq.(52.29)in sphericalpolarcoordinateswith
(asthepolaraxis. TheLondongauge(Eq.(49.38)1restrictsptotheequatorial
plane,and theazimuthalintegrationshowsthatjtakestheform ofEq.(49.39.
),
where
A'tiq,0)H Klqt 4.7./7t.:2 4=62h
r4 *rsin3#tzp
w2 *= JJ7
-x/
x
11lC2
?z?V 2 c(2z
r)z2 e
.
>:l(l-r#->
.
E
).
-E
+-i-2j.
fJfz
7
')
.-f
EbE
.
-A
--(l-l.
tj+
r-j
t-/
xzj1.u
fE
fE
-t
.
+
t.
tT
tf-j
(52.36)
ThisexpressionmayberewrittenwithEqs.(5l.29),(52.28),and(3.29)as
J+(-+..:2 tanh(Jjfyj.
- tanh(Jjf.s)
L(l+ s-s-) s.-s#-+ ,
)+ tanh(Jjf-) j,
--( (.s
-,--.â2)tanh(Jjf+e.
,-s- 1)(gg)
.
.
..
'
j-
.
where
(t- f+ j'
hqt,
rz
neglectingtermsoforderqllkl.
(52.38)
W e shallsrstconsiderthe responsein thenormalstate,which isobtained
bysetting.
(X= 0and Et- ffz(inEq.(52.37). Anexaminationofthediflkrent
possiblesignsof#+and(-yieldsthesimpleform
4nnelg a rx a, j
-l a ,! . c%tanh(à.jJ+)- tanh(èjJ-)
Knlqb-
mczjl-.
j.
p--ag.
,-,aztIz, 3.-t- 1
4=ne2
l
g . t
anh(!.jf-)
jl-.
pj-,dz(1-z).
(-.y tanh(JjJ+y)g-v,tz
1
.
.
-
m c,
The(integrationcanbeevaluatedexactlyandgives
4=nel
l
-,
y gl-t.
(.jJz(1-z2)1=0
Knlq)= -tn
(52.39)
*ea
APPLICATIONS TO PHYSICAL SYSTEM S
This resultholds forallq<:
.ks,thusconhrming thatthe normalstate is nonm agnetic,apartfrom theweak Pauliparam agnetism and Landaudiam agnetism ,
both ofwhich areneglected in thepresentapproximation. (SeeProbs.13.17
and 13.18.)
The evaluation ofthe superconducting kernelmay be sim plised by con-
sidering thedifl
krence between Eqs.(52.37)and (52.39). Aftercollectingtht
factorsoftanhtè#fz)separately,wefindi
?nnel *
l
a
J-.#fJ-!dZ(1-z)
-(#,
4é2#tgtanhs
(J.jf+)-tanhs
(!
.
#f-)1
+(+21-jyojtanhf
(!+
.
jF+).tanhf
(!
+.
#J+)j
yjtanhS
(!-#f-).tanh1
(è
-jf-ljjj (jr.
4o
KlqtH Klqj- Knlqt= - 4m c2
Since f+- (-= hqvyz,tht lastttrm (in square brackets)containsthe factor
.
f+
ja(jjy..
J-xdfE+tanhq?E+)-tan
j
along with anotherterm with thesubscript*ç-'' Itisevidentthatthisintegral
convergesabsolutelyforlargelf!;itisthenpermissibletochangevariablesfrom
(tof+,andtheoddsymmetryoftheintegrandshowsthattheintegralvanishes.
lnthisway.Eq.(52.40)reducesto
?nmel12 1
Klq)= - lmccâçl?s
l- z2
dz
-1
z
Jç
*-j
f
.
tL
gtanhE
(!+
.
#f+)-tanhE
(J-#F-()
j
52.41)
Eachterm oftheintegrandseparatelydivergesnear(œ 0,z= 0,and thisregion
dom inates the integral. W e may therefore approxim ate the slowly varying
numerators (1- zzltanhtljfz)by theirlimiting valuestanhtl#l) at(= 0,
z= 0,which yields
Kl
anh(J#Z) 1 Jz * X 1 1
q,ar-3nmekz12
m
-ctz
xp
s J.,zJ-.y(zt-r()
=-L
qzrnel82tanh(J#A)g* #(g*dl
mczhqvp
J-.T jaY
'
x1:
(f+.
1.
0+a
-
'
) (52.
42)
-..- :(;- !.()c+ a-!.
l1.M .KhalatnikovandA.A.Abrikosov.Advan.Phys.,8:45(1959).
SUPERCONDUCTIVITY
463
w'here(= hqt'
pzand theconditionq(o=qhvptlvr-ïv>>lallow'
susto extend the
limitsto(= +.aa. Theremainingintegralscanbereducedto(-4'A).
f;(sinh-vl-lxffx
whichiseasilyevaluated(seeAppendixA)togive-=l,
Ih,and Eq.(52.42)becomes
KL 3=2nelA(F)
gj= mC
...g#ço
y ks
tanhgj/jztFjj
(52.43)
..a:
,
where Eq.(49.33)has been used to identify #o. Thisexpression should be
comparedwiththecorrespondingnonlocallimitlqt(
jy-l)ofthePippard kernel
(Eq.(49.47)j. TheremainingcalculationofthepenetrationdepthinthePippard
limitproceedsexactlyasbefore,and weobtain (Ay-J'
:)
A(w)-z(0)g'
lu
9
:
r
o
-'tanhy
tt
s
r/)j-1
v/
j)
1(
where40)- 8j(j.
#oA/
.
(())
1
(52.45)
is the com m on zero-tem perature lim it of b0th theories. N ote thatthe present
microscopic calculation predicts a definite temperature dependence for A(F)
in the nonlocallim it;thisfunction isshown in Fig. 52.lalong with the em pirical
form Eq.(49.14) and the theoreticalcurve predicted for the locallimit gEq
(52.33)4.
.
NoN LOCA L INTEGRAL RELATIO N
A s a 5na1 conhrm ation of the Pippard phenom enological theory, w'e shall
transform Eq.(52.31)to coordinatespace. Forsimplicity,the calculation NNi11
be restricted to r = 0,butthe sam e approach applies for a1lr < rc.l W ith the
change ofvariables p+ l.q -.
+k.p - Jq ..
--1,the inverse Fourier transform at
F = 0 may be NNritten as
(52.46)
where
F(f',ï')- )J
t
X
L'
'
'E-V,(.
#!+-EEE
-)-'
,
.
because/tf')vanishesasF -->.0.
To putEq.(52.46)in theproperform,weconcentrateonthesecond term ,
w hich containstheintegralkernel
464
APPLICATIONS TO PHYSICAL SYSTEM S
This expression is m ost easily evaluated by introducing a generalized kernel
Sjj(X,X')EH(2Jr)-6fJ3V#3J(k-t-l)j(k+ l)ycftk-ll*xei(k+I)@x'y'
((yg(,)
w'hich clearly reducesto 5'
f,(x)asx'-+.0. Thevectorcomponentscan now be
replaced by spatialgradientswith respectto x'
where./
.
(,(z)= (si1)z)'
z. Thefunction F((v.(t4issharplypeaked neark ;
4:/;4$ks,
while thc relevantspatialdistancesarc m uch largerthan the interparticle spacing
$
kJkkl. Consequently. the trigononnetric functions oscillate rapidly,and the
variation ofthedenominatorof'
h maybeneglected inevaluating thederivatives.
A straightforNvard calctllation gives
r x . *b
'
E'
*:
f'k J/.'/dl
'
%,.
(x)---$'
ë,(x,0)--8â.,
2..si.4J
.
%
-
j -(j.
a)
y--y((g.(t)cos((k-/)xq
0 v.
l0
The convergence of the integralallou s us to introduce the integration variables
(kand (t.w'hichy'ields
A--Yj/7?2j.2
'
*c. '
*:
x
s .
s'
fj(x)--1
'
rk.
s'
rk,4j
.
. -. . -.
,
,
.
-Y
,
(i
(#ïF(#
.
:
.t
-)cost.
c-()hL
-'y
The rem aining double integralm ay be evaluated with the change of variablesl
(= At
lsi1h('
r- m')
('= ét
lsinh(r- m')
2..
:,.
vsx-j/??
2k/ j
'2
:
-
*x' dv#'
r'cos((2.Xox)
IhL'y)sinhT'cosh'
r)sinh2T'
The r'integration iseasily perform ed w ith the substitution z = sinh r'
772kF
l-vfx.j 'co dr
72.
.: x cos
Siy(x)==- -..î.0/à
0 ..-- h z'
..x
4 - . (2O.S.
n'=.h
'
= --.
Y.,
=
h T 2=3 -- .h
!'.
-
-
rrexp
2:X()xcosh T
àt'p
466
SUPERCONDUCTIVITY
and the rem aining expression m ay be rewritten as
mk 2x xl
a
(52.49)
s'
fytx)- '
rrz; .
X
'4l-z'
,mr,.'
3(.
x)+=-'
&oJ(x)1
HerethefunctionJ(x)isdefined by
(52.50)
and Kv is the Bessel function of im aginary argum ent desned in A ppendix A .
Substitution of Eq.(52.49)into Eq.(52.46) show'
s that the one-dimensional
deltafunction exactlycancelsthefirstterm ofEq.(52.46),1andw'
ehnallyobtain
j(
?nelvrk'
xg 3 ,X(X.A(x'))
x)..
--4= m c yvs # '
r -yi--. J(X)
whereX = x - x'. Thisexpressionisform allyidenticalwith Pippard'sequation
(49.31)forapuresuperconductor(r()= ïo)atF= 0,apartfrom theappearance
ofJ(X)inplaceofe-X/'
f'. ItiseasilyseenthatthefunctionJIX)isverysimilar
to thePippard kernelforallX (Fig.52.2),and,in particular(seeAppendix A),
/(0)= l
.
co
jv#.
j'
,
.
/(.
y'
)=w
h.
v
,jF
,
.
1.0 l
0.
8j
-
exp(-A'/f(
j)
O.
6j
O.4
0.2 J(X )
Fig.52.2 ComparisonoftheBCS kernel(52.50)
with the Pippard kernele-A'
'C0 for a pure superconductoratzero tem perature,
O
1.0
2.0
3.0
4.0
#/f()
which allowsusto identifythePippard coherencelength
fo = hït
s
zrtïtl
!NotethatJr ô(x)#x- !..
(52.52)
466
APPLICATIONS TO PHYSICAL SYSTEM S
in term s of the param eters of the m icroscopic theory. The above m icroscopic
calculations have been extended to snite tem peratures and frequencies and to
alloys w ith finite m ean free path. ln allcases,the resulting supercurrent m ay
becastin theform ofPippard'soriginalexpression,therebyjustifying theearlier
phenom enologicaltheory in detail.l
S3L M IC RO SCO PIC D ERIVA TIO N O F
G INZBU RG -LA N DA U EQ U ATIO N S
The com plicated and delicate self-consistency condition of the m icroscopic
theory m akes direct study of spatially inhom ogeneous system s very dim cult.
ln m arked contrast, the G inzburg-l-andau theory is readily applied to nonuniform superconductors.which is probably its single m ostim portant feature.
The sim plicity arises from the diflkrentialstructure that allows full use of the
analogy with the one-body Schrödinger equation. A sdiscussed in Sec.50,the
original G inzburg-l-andau equations were purely phenomenological with the
various constants fixed by experiment. For our finaltopic in this chapter,we
now presentG orkov-sm icroscopicderivation ofthe Ginzburg-l-andau equations.z
This calculation determ ines the phenom enologicalconstants directly in term s
ofthem icroscopicparam eters;italso clarisestherangeofvalidityoftheequations
and allowsdirectextensionsto m ore com plicated system ssuch as superconducting alloys. Indeed, m any m icroscopic calculations now proceed by deriving
approximate Ginzburg-l-andau equations,.whose s3lution is considerably
simplerthan thatc;ftheoriginalequatidns.
W estartfrom thepairofcoupled equations(51.23)for'ff and .
W f. Here
we are interested in the eflkctofarbitrary magnetic fields (unlike Sec.52),so
thatA cannotbe considered sm all. lnstead,the calculation is restricted to the
immediatevicinity ol
-Fc.wherethegapfunction.
l(x)itselfprovidesthenecessary
smallparameterI.
l(x)//y rc!.
'
.
v1. ltisconvenienttointroduceanew temperature Green's function r
#0(x,x'.(sn)that describes the normalstate in //?& same
lnagneticheld. ltsatishestheequations
ihu)n+ lhl V tpA(
+-?'
-x)
- 24-p, #0(x,x',fw )= hblx - x')
m
hc
ihu)n+ l:2 V - ?'
cA(
-x)
- 2+.p.V0(x',x,t
w )= â3(x- x')
m
hc
obtained from Eq.(51.23/)and its analog forF(x',x,(
w )with .
â= 0. This
auxiliary function enables usto rewrite Eqs.(5l.23)asthefollowing pairof
$D.C.M attisand J.Bardeen,Phys.Ret'..1l1:412 (1958);A.A.Abrikosov,L.P.Gorkov,
and 1.E.Dzyaloshinskii,op.cl'?..secs.37and39;G.Rickayzen.op.cit., chap.7;J.R.Schrieflkr,
op.cit,.sec.8.4.
2L.P,G orkov.Sot'.Phys.-JETP, 9:1364 (1959);A.A.AbrikosovsL.P,Gorkov.and 1.E,
Dzyaloshinskii,op.cit.,sec.38.
supEncoNDucT1vITY
467
coupled integralequations
Ftx5x',f.
z
.
ln)= #0(x.x',(zp:1)- h-l.(J3.).r#0(xqj'
,u)n ).:(
,),
.
k
.z'
1(y.x',(wl (53.25)
.)
'
oFf
tx5x's(
.
t?,l= h-l1
'(/3).t
,
:
40(y'x.-(,.
)u)..
:*(y)F(y,x'.r
.
s,,)
(53,2:)
.
..
w hich are easily veritied by direct substitution into the origir)aldiflkrentia!
equations. N ote carefully the rather complicated arguments in Eq.(53.2:);
theyarenecessaryto reproducethestructureofEq.(5l.23:). A sïmplemanipulation ofEqs.(53.2)yields
'
V(xx',t
.
u )= V0(xx',(.
t?)-h-2(d3)'d3zV0(xyq(snlA()r)
;
< C#0(z,y,.
--t,a,).
l*(z)F(z,x',(z?u)
(xNX's(.
t)n)= h-1.fJ3.J'tf40(y.x,-.
-f
xprl)Z*(y)1
'
VQ(y'x's(.
wl- /i-2..fJ3yd3,
2
.
wvf
x'
#0(y.x,-c,
.
,n).
&d'(y)%
.0(y,zA(,?,)A(z)..
/-t(z,x'.(su) (j3.3J,)
''
.
which areexactintegralequationsforfT andu'
/-fseparately.
Further progress depends on the assum ption of sl
nall -X,and B'e first
concentrate on Eq.(53.3A). The second term on the rightbecomes a small
perturbation i1
4 this Iim it.and an expallsiollgiNes
:
/-f(x5x'5(,?n.)--h-1.fd3.$'4
J0(y-x,-t.
,
?n)..
ï.
*(y):#0(y.x'.u,,).
- li-3.1
-J3J.J3z
g rï3w /
.
40(y x.-(z?,)..
ï.dc().)'
.t0().,z.Lun).
V
l(z)
'
<'
u
L
W0(wqz:-(,-)rl).
îi'(w)'
,
4D(w!x'.(
.
ur!)- .. .
,.
.
..
.
'
.
W hen Eq.(53.4)iscombined w'ith the self-consistentgap condition (5l.24),we
obtain an integralequation forthegap function itself
(53.f)
W e see thatthe assum ption ofsm all '
y Ieadsto a nonlinearin/egralequa,.-X onl
tion. The simpler d#erentialstructure ofthe Ginzburg-l-andau equations
requires the additionaland separate assum ption rc- r <x rc,since .X* and A
then vary slowly with respectto the range ofthe kernels Q and R. Thesetwo
conditions are physically quite distinct,for a sum ciently strong m agnetic lield
can render.X* small,even atr = 0.
46s
Appt-lcATtoNs To PHYSICAL SYSTEM S
ltisnow necessaryto examinethe kernelsQ and A. AsaErststep,we
evaluatethenormal-stateGreen'sfunetionf#0in//?eabsenceofamagneticjeld.
Thisfunction dependsonly on x - x'and isprecisely thatstudied in Sec.23
V0(x,a)n)= /i(2=)-3fJ3i-eîkwxlt
hoy- y()-1
(j3.8)
A lthough the com plete spatialdependence is rather com plicated,the relevant
lengthsin Eqs.(53.5)to (53.7)area11m uch longerthan interatomicdimensions,
and itistherefore perm issible to assum e kpx )
.
> l. In addition,ifthe discrete
frequency satisfies the restriction I/it
z?a;<:
:/
.t, then the dom inant contribution
arisesfrom the vicinity ofthe Ferm isurface,and we flnd
F0(
- dl
x,
(
sn);
4:/
jAr
(0)jjjoswiy'
oqkk.
l
.
â
j'
/
v
t.
t%s
Y.j
-jj.
-dnqyj
fexpg,
.jz-y
.+j))xj-expg-/(/
cs+.(v
;)xj)
.
=/kia
V(
'
0)
rco
,
= - -.- exp I
'
kFx sgn (.
un- xl
-n1
(53.9)
Thislastrestriction (E/it.
z
.
;p,'-.
tkpt)isfullyjustifled inpractice,becausethe terms
omitted makeanegligiblecontributionto thesum overninEqs.(53.6)and(53.7).
Themagnetictield in Eq.(f3.l)isthatin the superconducting state,which
z--Y
rs
varies with the natural length A(F). In contrast,$Q oscillates with a much
shorter length k;l,so that A can be considered locally constant over m any
wavelengths. Gorkov thusmakesan eikonal(phase-integral)approximation,
assum ing thatthe dom inantefrectofthe m agnetic field can be included in a
slowly varying ens'elope function f
p
-
t.#. ()tx,
z
x'nt
,
z?,
l)= t?i(J)(x.x')t:.
J.a'0(.
x - x,stonl
(53.10)
Theeontribution ofA isnegligibleforx ;
k;x';hence+ isehosen to satisfy
(
F(X,X)= 0
Directcalculation with Eq.(53.10)gives
(v+i
j
e
j
y
A
-j2L
#è-ci
wj
y7.
2f
<'
0+lit'
vw+4
eA
c
.
-v g 0
.
,
ivzv + i
-
+
i$
%c
.
'A
-)-(vv.
j
-j
,A)2goj(53.l2)
c
Herethetermsaregrouped inapproximateascendingpowersofeAjhckybecause
F 0varieswith thzcharacteristic length /s
-s
-l. Since,'
fisoforderhH ,thisparam eter m ay be rewritten as hebli
lhcks'which is sm allfbr a1lm agnetic seldsof
Given A(x).thissrst-orderdiflkrentialequation can alual'
sbeintegrated.
W e now return to the kernelç)(x.y), A com bination of Eqs.(53.6)and
(53.10)gives
(53.l4)
whichdefinesthe kernel/0in the absence ofa magneticfield. Thisfunction is
ou):
readilyevaluated w'ith Eq.(53.9)andtherelation)é/(con')==2 V f((
n
Ncb'
Q0(x)-.jcj
t
v
F
/x
o
-ljz.
1
/
jnexpg-2
'
.
:
J
.
?
i
.
con.
s
'
j
=h'(0)2
1
ksx jsini'
t(é=x.
'j/7l's)
NearFc.the kernelQQvanishesexponentially for.
vw âl's'ksrcQ:0(ï0),and it
thus has a range comparable with the (temperature-independent) Pippard
coherencelength. Since(vismuchshorterthanthescaleofvariationsofeither
the vector potentialor the gap function,it is perm issible to treat A and .X* as
slowly varying functions. ln particular.Eq.(53.13)can beintegrated explicitly
e
'
,
+(x,x')- - jjC(A(x.
)+ A(x)).(x- x )
(53.16)
wherethesym metrized form representsacom prom ise between thetwo form sof
Eq.(53.l). Furthermore,A isoforderSc(F)A(F)cc(Fc- F)1;therestricted
rangeof:0then meansthat+(x,x')itselfbecomessmallasF ..
->.Fc,permitting
an expansion ofeiT in powersof%.
The above conditions allow us to evaluate the firstterm on the rightside
ofEq.(53.5). W iththedetinitionzHëEy- xwehave
(53.17)
470
APPLICATIONS TO PHYSICAL SYSTEM S
TheshortrangeofQ0requiresthatz% #:,andtheremainingfunctionsmaybe
expanded ina Taylorseriesaboutz= 0. Retainingtheleadingcorrection term ,
we5nd
.
f-,3,c(x,y)a.(y)a.a+(x)fv
lszvn(z)+)gv-2't
'
,
A-(x)jza-tx)
.
x J#3zz2;0(z) (53.18)
where V denotes the gradient with respectto x and acts only on the vector
potentialA(x)andthegapfunctionA*(x). Notethatwehavenow reducedthe
originalintegraloperatorto a simplerdiflkrentialone.
The properties of the norm al m etal appear only in the num erical co-
emcients of Eq. (53.18), and we flrst consider j#3z:0(z), which diverges
logarithm ically atthe origin. This singular behavior reflects the unphysical
approximation ofa short-range potentialin Eq.(51.1). lt must therefore be
cutofl-inmomentum spaceatIttp= halo:
J#3z;0(z)=(jâ2)-1(2:7.
)-3j(
/3/:jFlyg0(k,(
w)F0(-k,-(.
on)
=
j-1(2rr)-3fd3k IDg(hlct
)j+.42)-1
*ljwo
v(0)j0 #t:#-1tanhllj#)
- .
Thisintegralisjustthatconsidered in Eq.(5l.39),and comparison withEqs.
(51.42)and(51.43)gives
J#3zQ0(z)= .
N(0)lntzâtt)oqeY=-1)
-xtelln(F
w)+g-t.x(0)(1-z
()+g-1
C
.--
(53.
19)
The other integralin Eq.(53.l8)can be evaluated directly in coordinate space
usingEq.(53.l5). Sincethisterm isalreadythecoeëcientofasmallcorrection,
we setF = Fcand obtain
Jdqzzzco(z).)cg'
rv
kF
tojzjy3zcschjl
cx
yv
z,
=$N(0)(#c
=
-1
&f)2Jo
-dysin
A'
h
2y
=8
7(
13)x((j)('
rk
hs
t
'
r
Fc,
12
whereEq.(51.38)hasbeen used.
(53.20)
SUPERCONDUCTIVITY
471
Theonlyothercalculation isthesmallnonlinearcorrection inEq.(53.5),
which maybeevaluated inlowestorderby setting #0$
k;f#'
0and taking al1the
factorsof7
X atthesame point
.
J#3y#3z#3u'S(x,y,z,w)u
.
&*(y)A(z)aX*(w)
;4J.
1*(x)1u&(x)(2j
-d?))J3z#3wA(x,y,z,w)
A straightforward calculationin m om entum spacegives
J3)(=kswc)-z
- -x (0)TJ8
(53.21)
w'
hereF hasagain been setequalto rc.
Thefinalequation forA*(x)isobtained bycombining Eqs.(53.5),(53.18),
and (53.19)to (53.21). Aftersome rearrangement,theterm g-lA*(x)cancels
identically,and we find
hl
9A(# 2
(/:Fc)2 F -F :
)X*(x)r
j
gv.
-2-ycj,
â+(x)u-6-=2y(
tlj
cyjfy.
:y(x)-7((3j
j
zusy,â;z
xlI
2j
.
(53.22)
Therelationwith theGinzburg-l-andauequationcanbemadeexplicitbydehning
avb'
avefunc:ion
(3)
kl.-(x)e j74.
-n.-1.
l(x)= 7-((3v)1,l
,(
-x-)
u-n1.
l=ksTc)<
yrrz #sI'
c
(53.23)
thatsatishesthefollowingequation(notetheeomplexconjugation)
(53.24)
Heren isthe totalelectron density.and comparison gith Eq.(50.7J)identises
the phenom enologicalparameters
6=2(ksrc)2
r'
cp = 2:
6=2(/N
' rc)2
a---.
.
))(?,,
,y(l-w-j b-.
jt(
-j)r
.
yn-
(53 25)
.
472
AppulcATloNs To pHyslcAL SYSTEM S
Furthermore,thecharacteristiclengthsA(F)and((T)canbeevaluatedinterms
ofthemicroscopicparametersAs(0)and ,
.
10rEqs.(50.18)and(50.22))
l
X(r)= 1s10)(1-y
Xj-i
(53.26/)
'
J(w)-ï'
o=e-vg?)s
??1'(1-z
rC
-v
)-'
T j-1
c
r0.
7395(1-j.
-)
(53.26:)
,
while the G inzburg-l-andau parameter becom es
r)
A O)
KEBA(
ttFl;
4:0.957.g.
t'a. r.-+Fc
-
..
The penetration depth agrees w ith that obtained with the w'
eak-l
ield response
nearrclcompare Eqs.(52.34:)and (52.35)1,butthe coherencelength can only
be determ ined by ineluding spatialvariation of the gap funetion. ln addition,
the Ginzburg-l-andau expressions for nsll-j= 2/?s
*(F) and Sc(F) derived with
Eqs.(50.12),(50.l6),and (53.25)agreeu.ith Eqs.(52.34:)and (51.64). Thelast
calculation requiresthe expression
ES
c
3=3(/csF )2tle-lY
(0)12= .
-- --& --- - -
6/
obtained by combining Eqs.(5l.38),(51.44).and (51.56).
The preceding derivation shows how the srst G inzburg-l-andau equation
em erges as an expansion of the self-consistent gap equation near Fc. W e now
consider the supercurrent, which can be related to spatial derivatives of the
single-particleGreen'sfunctionr# tcompareEq.(52.10))
eh
:2
jtx)= - Ftx'r,x'r?)lx,-x- 2-r#'(x'
r,xr+)
m i(V -V')'
lncA(x)'
(53.28)
Here the factor 2 arises from the spin sums. If Eq.(53.3/)ïs expanded as
F(x,x',fw )= V0(x,x',f
w )+ 3F(x,x',t
w)
with
473
SUPERCONDUCTIVITY
In this way,the totalsupercurrentreducesto
2e2
e
j(x)= --p
m j
RV - V')3F(x,x',(s.)Jx,-x- m cj
-jA(x).
'
j
x,x,f
w)
aj3F(
The remaining calculation depends on theexplicitform of8@ ,and sub-
stitutionofEq.(53.29)gives
e
j(x)+ -2,2 A(x)â 3g(x,x.(w)- i
lâ,
mcph
n
m
J3ydazA(y)sqv(z)
#n(z,y,-a?n)E#0(z,x,/
w )Vx#0(x,y,(,an)- #0(x,y,t
.
oa)Vxf#0(z,x,(sn))
(53.30)
Thespatialderivativescan beevaluated with Eqs.(53.10)and(53.16),and the
x
resultcan besim plised with theslow variation ofA
el A(
j(x)+m2c
bh x)j
a)3g(x,x,f
w)-à-j
.
--pXj(
e
$
s3yt
/3zatyla..
tz)
x
A(x)#(
)(x,y,tz)n)go@,X,t.'n)j.ef@(x,x):2+(z.x)
#0(z,y,-r
a?nl -2f:â
c
..
x(g0(z-x,(
snlvxF0(x-y,f
,
.
anl-f
#otx-y,oanlvx@o(z-x,(
sn)!'
)
TheErstterm on therightsidenow cancelsthe second term on the left. W ith
the same approximationsas in Eq.(53.17),the supercurrentnear Fcbecomes
j(x)- e
#3y#3z@ulz- y,-u,n)(Fotz- x,aulvxC#otx- y,(s'nl
mishc
lie
F0(x- y,ulnlTxF0(z- x,f
xlnll(1A(x)k2- Ià(x)12hc
-
x A(x).(z- y)+ A*(x)(y- x).VA(x)+ à(x)(z- x).Và*(x))
(53.31)
Severaltermsvanish identicallyowingto thesphericalsymmetry:
j(x)= eyz
c('- ys-f
.
vnlvxgotx- y,tol
d3yd3zp,
-t
)al(gntz..x,(
mi$ n
-
x
g0(x:-y,fwlvx@nlz- x,(sa))
(.
-2;I
u
xlxli
zA(x).(z-y)+ltxlvxu
q+txl-z+l*(x)vx.
l((
x).y1
53.32)
474
APPLICATIONS TO PHYSICAL SYSTEMS
Therem aining integration ism osteasily perform ed w'
ith the Fourierrepresenta-
tion ofr#0from Eq.(53.8),and a lengthy butstraightforward calculation givcs
W ith thewave function in Eq.(53.2.
3),wefinallyobtain
2
t2I
ij/l.*(x)Vkl.(x)-4ja(xjv).
j.y:(xjj- .
2C x(xj:
jtx)= - j'
tj%(x)(2
m
lAlf'
..
(53.34)
in complete agreementwith Eq.(50.9).
In sum mary,we have derived theGinzburg-l-andau equations(50.7J)and
(50.9)from theGorkovequationsunderthefollowing setof-assumptions:
l.TheorderparameterA(x)and thevectorpotentialA(x)aresmall.
2.Therange ofthe kernelsin Eqs.(53.5)and (53.30)issmallcompared to the
characteristic length forspatialvariationsof.
l(x)(i.e.,thecoherencelength
#)andA(x)(i.e.,thepenetrationIengthA).
3.The eikonalapproximation also requireshkFy.1 (which isgenerally valid).
Forany supercondtlctor,thesecriteria are alwayssatisfied sum ciently closeto
the transition tem perature Fc.
PRO BLEM S
13.1. Show that Eq. (49.6) can be rewritten :Q/ê/+ (#.V)Q = (Q .VIV.
Hence prove the conservation ofthe flux moEH2
(ds.Q through any closed
surfacebounded byacurvethatrnoré'.
sx'ith//76./u1#. Asacorollary,conclude
thattheguxoid (1)isa rigorousconstantofthem otion.
13.2. lfFs(F,0)isthefree-energy density ofa superconduetorin zero field,
show that Eq.(49.11)is the Euler-l-agrange equation ofthe totalHelmholtz
free energy UFs(r,0)+ J#3x((8rr)-1h2+ jrqpyvzj for arbitrary variationsOfh
subjectto curlh= 4=jjc. How doesthisderivation failfora normalmetal?
13.3, (tz) Evaluate the mean Helmholtz free-energy dcnsity gsee Prob. 13.2
andEq.(49.16)JforaslabinaparallelappliedfieldHvk. UseEqs.(48.13)and
(49.17)to verifythatH = Hvk throughoutthesample,andfindthemagnetization
(.
: - S )/4=.
(:) From the corresponding Gibbs free energy,show thatthe sample remains
superconducting up to acriticalfield //c
*determined by theequation (H c
*/fc)2=
(1- tAs/#ltanh(#/As)1-1,whereScisthebulkcriticalseld. Discussthelimiting
cases#<.
:htand # à>As.
13.4. Considerasem i-infinitesuperconductor(z > 0)w'ithanexternalmagnetic
seld H parallelto the surface. Use the Ginzburg-l-andau theory to 5nd Y-(z)
SUPERCONDUCTIVITY
475
and >(z)for >cy>1. Show thatthe jield-dependent penetration depth A(/.
f)is
given by (A(S)/A(0))2= 1Hc,
l
'LHc+(Hc
l- S2)1j. Sketch the spatialvariation
of/7and Vl-forvarious values ofH . W'hat happens forH > Hc? W hy is it
permissibleto violatetheboundaryconditionJ'l
''l
/#z - 0?
13.5. (J) Asa modelforan extremetype-llsuperconductor(2 t>().solve the
Londonequation(49.ll)intheexteriorofacylindricalholeofradius((seeFig.
50.1b)containing one quantum ofPuxoid (
;)(
j= /?c/2c. Discussthe spatialform
ofhandj.
(:) Evaluate the Gibbs free energy as in Prob.13.3 and show.that the lower
critiealtield fort
luxpenetration isgiven by Scl= ((
ro/'4'
mA2)ln(A/#).
13.6. Consideran intinite m etalin a uniform applied tield H thatissufliciently
strong to m ake the sam ple normal. Solve the linearized G inzburg-l-andau
equations.and show thata superconducting solution becom es possible below'
an uppercritiealéeld Hcz(T)= '
$ 2&Sc(r).
13.7. Usetheprocedureoutlined below Eq.(51.3)toderivethe Hartree-Fock
theoryatfinitetemperaturefora potential-g3(x - x').
13.8. RetaintheHartree-Fock termsin Eq.(5l.5)andderiveasetofgeneralized
equations for C#'and ..
'
F *. Solke these equations for a uniform system and
compare with the calculations Ieading to Eqs.(37.33)to (37.35). Specialize
the equationsto a norm alsystem and rederive the results ofSec.27.
13.9. (a4In the weak-coupling limit (.
A()-s:Atzpsl,show that the BCS gap
equation (5l.37)m ay bewritten
and hence proveEq.(51.46:).
13.10.
476
APPLICATIONS TO PHYSICAL SYSTEM S
to solve Eq.(37.35)given the interparticle potentialgrecallthe convention otEq.(37.6)thatg> 0 foranattractivepotential)
(k- k(I,
z'2k'- k')-glU-1blhu)o- 1ù r)hho)o- 1Jk'l
.)
g1P'-iblhûpt- IL 1)d(âf1,,- IL,1)
-
Hence derive l= N(Q4g.fflntzâ(,ao/z
'
Xll where geff= ,
:1-g2(1+ N(Q)g2x
lntDpr/tz)sll-l(insteadofgl- g2asonemightnaivelyexpect).
(b)lfksrc= l.k?hœne-l/NtolgerrandtssL<LM -%,showthat-#lnFc/#lnM < .
i.,
which may accountforthereduced isotopeeflkctin transitionmetalsal
13.11. Use Eqs.(51.50)and (51.51)to show thattheentropyin thesuperconducting statecan be written
Xs= -2PWs)
(;f(Ekjln/(Fk)+ E1-f(Ek)jln(1-.
/'
(ék)))
k
where/tf')- (ebE+ 1)-1. ComparewithProb.2.1andinterprettheresult.
13.12. A s a m odel of a supercondacting fllm ,consider an inflnite superconductorcarrying a uniform current,where the gap function takesthe form
Aczfqexwith.
:
breal(seeEqs.(53.23)and(53.34)1andq<:
.
:kr.
(tz) SolvetheGorkovequationsto :nd F and .F %and show thatthesupercurrent
isgivenbyj$k;-twn,(F),wherev= âq/m gseeEq.(52.33)4.
(y) Find the self-consistency condition for z'
X. At F = 0 show that .
â is in.
dependentofq forq < qc;4;k
âohvr. N earFcexpand the gap equation to hnd
gk
Z.V
V
Fcj2x78
('
J
(
r
3
21jj.X
F,
j.i
2jmhk
2B
kT
scj2ga
and determinethe criticalvalueofqthatmakes2X vanish.û
13.13. Carryouttheanalyticcontinuationtoobtainareal-tim etherm odynam ic
G reen'sfunctioncorrespondingtothephysicalsituationinProb.13.12. D iscuss
theexcitation spectrum forsm allq.
13.14. (J) Use the analogy with Bose systems(Sec.20)to generalize Dyson's
equationsfortheelectron-phonon system to asuperconductor.
(b4 Approximate the electron self-energies by the lowest-order contributions
expressedin termsofF,.
X ,.
F %,and 9 ,andwriteoutthecoupled self-consistent
equationsforF and+ %in m om entum space.!t
tN.N.Bogoliubov,V.V.Tolmachevand D .V.Shirkov.$1A New M ethod in theTheoryof
Superconductivity,''chap.6.ConsultantsBureau,New York,1959.
jIn fact,the system makes a hrst-order transition to the normalstate at a criti
calvalue
qc;4r1.23(.
Ac/ât
,
.s)I1- (r/Fc))#,whi
ch issmallerbya factor1/V3. See,forexample,P.G.
de Gennes.i'Superconductivity ofMetalsand All
oys,''pp,182-184,W .A.Benjamin,Inc..
N ew York,1966.
I)
'G.M .Eliashberg,Sov.Ph.vs.-JETP,11:696 (1960);12:1000 (1960).
SUPERCONDUCTIVITY
477
13.15. Assume thatthetwo functionsl/(x)and ?
Xx)satisfy thecoupled eigenvalue equations
t(2-)-lf-ihT + c-1eA(x))2- pt)lgtxl- é(x)r(x)= Eulx)
(-(2-)-i(//i
V + c-it,A(x)12+ p,
)!,
(x)- zâ*(x)l
?(x)= Fp(x)
known astheBogoliubovequations.l If?.
(+1(x)and ?'
y
t-'(x)isasolutionwith
positiveenergy E i
,show thatthereisalwaysasecondsolutionI,
(-'(x)= -!
,J
(.
+'(x)*,
.
o(-'(x)- lzy
t+'(x)+ with energy .
-EJ. Let U be a two-component vector with
elementsu and ?.. with the usualdehnition (U,U)= j#3.
'
r(2;/22+ )è,J2)show
that (U.
f
/2E)7Uj,
.r'l= bJ.
j,, while (U(
y:
-',I.
-Jj
(F')- 0. Construct the eigenfunction
expansion ofW(x,x',(z?u)2Eq.(51.19)1and henceexpressrl(x)and A(x)interms
ofthese eigenfunctions. W hy isEyasingle-particleexcitation energy ? Specialize to a uniform m edium and rederive the resultsofSec.51.
13.16. Prove Eqs.(52.345)and (52.34:).
13.17. (J) lfK(q)in Eq.(49.39)vanishes like K(q);
k;xqlasq ->.0,use Eq.
(49.43:)toshow thatthediamagneticsusceptibilityisgivenbyy;= -a/4zr(1.
c-a)
;
4r-x(4n ifa<'
t l.
(&) Derivea generalexpressionforKnlq)byevaluatingTkI(q,u)(Eq.(52.25))
in a norm al m etal. Hence show that the Landau diam agnetic susceptibility
à'aofa noninteracting electron gasisgiven ata1ltemperaturesby ya= -Jxp
where zpisthe corresponding paramagneticsusceptibility (Prob.7.5). (Note
p.t
l= ehjlmcfbranelectron.)
13.18. (J) Usethetheoryoflinearresponsetoshow thatthespinsusceptibility
y ofa superconductorisgiven by
;
t.- 2
y
p 3=2e
n
7j'
(
Opzoj-P/3t
)
E
Gplj
-
-
whereypisthePaulisusceptibilityofanormalmetal(Prob.7.5).
(b) VerifythefollowingIimits:y= 0atF= 0and z= ypatT= Fc. Interpret
these results.
13.19. (J) Repeat Prob. 12.6 for a superconductor at fnite temperature.
Show thattheapproximatephonon propagatorm aybewritten
glq,vn)=-ât1)ltpï+(z):
2+(tj,
y2Wtgypalj-lolcop- uyj
1Theseequationsarediscussed in detailinP.G.deGennes,SesuperconductivityofM etalsand
Alloysy''chap.5,W .A.Benjamin.lnc.,New York,1966.
478
APPLICATIONS TO PHYSICAL SYSTEM S
W here
d3k (
@(q,
v.)=jjsyj(u+u--t
?..,-)2Ey.
(s+)-y(s-)!
X -
1
- .-
1
ihvn- (E-- E+) ihvn+ (E-- E+)
+ (lk+r-+ u-n+)2g1-f(E+)-f(E.)j
X
1
-
1
ihvn- (f7+ E-) ihvn+ (E++ F-)
and thenotationisthatofEq.(52.27).1
(!))Derivethe retarded phonon Green'sfunction DRLq,t
.
o). Ifhul.:
t2é,:nd
the ultrasonic attenuation coelcient t
x,in a superconductor and prove that
as/t
xa- 2T(,
'
X),whichallowsadirectmeasurementofà(F)(seeFig.51.1).
13.20. Derive Eq.(53.33).
tForadetailed comparison ofultrasonicand electromagneticabsorption,seeJ.R.Schrieflkr,
S'TheoryofSuperconductivity,''chap.3.W .A,Benjamin,Inc.,New York.1964.
14
Superfluid H elium
TheelementHehastwo stableisotopes,He3(fermion)and He4(boson),which
diflkr only by the addition ofone neutron in the nucleus. l A tlow pressure,
bothisotopesrem ainliquiddowntoabsolutezero,wheretheysolidifyonlyunder
an applied pressureof;>30atm (He3)or QJ25atm (He4)(Fig.54.1). Sincea
classical system w ill always crystallize at suë ciently 1ow tem perature, these
substancesare known asquantum liquids. Theirunique behaviorarisesfrom a
combinationoftheweakinteratomicattraction(owingto theclosed 1Jelectron
shell)and the smallatomic mass,which produceslarge zero-pointoscillations.
tGeneraldescriptionsoftbe low-temperaturepropertiesofHemay lx found in K . R.Atkins,
:el-iquid Helium,''Cambridge University Press, Cambridge,1959:R.J.Donnelly,'iExlxrimentalSupercuidityy''UniversityofChicago Press, Chicago,1967;F.London,tssu- ldluidss''
vol.II&DoverPublications,Inc.,New York,1964;J. W ilks.:srhe Properties ofLiquid and
Solid Helium,''Oxford University Press.Oxford, 1967.
*79
48c
AppulcATloNs To Pl-lyslcAu SYSTEM S
A sa corollary,both liquidshave low density:p):
4:0.081g,/'cm 3 and p4:k:0.l45
g/Cm3.
D espite their superficialsim ilarities,the two isotopes diPkr profoundly
becauseoftheirquantum statistics. ThePauliprincipletendsto keep lermions
apart. and the interparticle potential m ixes in only unperturbed states with
k >ky. Thisbehaviorleadstoa smallhealinglength,and justifiestheuseof
theindependent-pairapproxim ation in com puting both theground-stateenergy
E
G
Q
(a)
Fig.54-1 (a4Comparison ofphasediagramsofHe3and He4in 'F plane. (Frol
l)J.DeBoer.
Excitation Modelfor Liquid Helium II,Course XXI,Liquid HeliumsProceedings of''The
International Schoolof Physics *Enrico f.-erm i'.** p. 3.Academ ic Press, New,York. l963.
Reprinted by permission.) (b4Enlarged portion ofmeltingcurveofHe3. (From D.P.Kell
y
and W .J.H aubachs'kcom parativePropertiesofHelium -3and Heliunl-4---A EC Research and
DevelopmentReportM LM -I161.p.18. Reprinted-by permission.)
and excitation spectrum of norm al Ferm isystems such as nuclear m atter and
liquid He3. Thesituation isvery diflkrentforbosonsbecausethe particlestend
to occupy thesam esingle-particlestate. In addition,theinterparticlepotential
preferentially m ixes in the unperturbed states thatare already occupied,thus
enhancingthem any-bodyeflkcts. Forexam ple,thesuccessoftheindependentparticle m odel shows that the low-lying states of a norm al Fermi system are
generally sim ilar to those of a free Ferm igas. This correspondence clearly
fails for interacting bosons with snite m ass,where the low-lying excitations
usually havea collectivephonon character.l Thediflkrencebetween bosonsand
ferm ionsalso appearsin the expansionsforthe dilute hard-core gas. Although
this system serves as a usefulm odelfbr nuclear m atter and H e3,the correction
term in Eq.(22.19)becomes 1j2j8(nt
z3/,
rr)1;zr;.4 when evaluated for He4where
'R.P.Feynman.Phys.Ret..,91:1301(1953);94:262 (1954).
su PER FLCJlD H CLlU M
481
(at73)à;
:
zr0.5.1: Asahnaldiference,wenotethateventhe/recBosegasexhibits
acom plicated phasetransition (compare Figs.5.2and 5.4).
54E7FU N DA M ENTA L PRO PERTIES O F He 11
W efirstreview som eim portantfeaturesofsuperiuid H e4.
BAslc EXPERIM ENTA L FACTS
l, h-point: W hen liquid H e4 in contact with its vapor is cooled below
FA= 2.17CK ,it enters a new phase know'
n as He II. The transition is m arked
byapeakinthespecificheat,whichbehaveslikelnIr- FAIonbothsidesofthe
transition.l Thissingularity isgenerallythoughtto representthe onsetofBose
condensation,butno theory hasyetprovided a satisfactory description ofthe
phase transition.
2.Superyuidily(p7#//?:twouthlidmodelLHe11hasremarkablehydrodynamic
properties. lt can flow through fine channels with no pressure drop, which
seem inglyim pliesthattheviscosityiszero. O ntheotherhand,adirectmeasure-
mentoftheviscosity (forexample,with a rotating cylinderviscometer)yieldsa
value com parable with the viscosity of He I above FA. The apparent paradox
wasexplained byTisza2and byLandau3w'itha two-fluidm odel,in which He 11is
considered a m ixture oftwo interpenetrating com ponents:the superiuid with
density psand velocity vs.and the normalfluid with densitypnand velocity % .
The superquid is assum ed to be nonviscous,which accountsfortherapid flow
through 5ne channels,and irrotational
curlv,= 0
(54.1)
The norm al fluid behaves like a classical viscous Puid, which provides the
measured viscosityofHel1. Thistwo-iuid picturewasstrikinglyconfirmed by
Andronikashvili4with a torsion pendulum ofclosely spaced diskssuspended in
He lI. Theoscillatingdisksdragged thenorm alCuid andthusallowed a dired
measurementofpntr).which variesas7-4forF.:
.
CFA(Fig.54.2).
3.Second soundLThe presence oftwo com ponentsgives H e 11an extra
degree of freedom ,and Landau predicted that He 11 would support two independent oscillation m odes that differ in the relative phase of vx and va. If
L and vxm ovetogether,the wake transm itsvariationsin density and pressure
(atconstanttem peratureand entropy)and isknown asérstorordinary sound.
tW e follow N.M .Hugenhoitz and D.Pines,Phys.ReL?..116:489 (1959),p.505 in taking
a= 2.2X. SeealsoF.London,op.cit., p.22.
$M ,J.Buckingham andW .M .Fairbank. TheNature of:heA-Transition in Liquid Helium , in
C.J,Gorter (ed.),''Progress in Low-Temperature Physicsy''vol,111,p. 80,North-ilolland
Publishing CompanysAmsterdam,1961.
CL. Tisza,Nature,14l:913(1938).
3L.D .Landau, 7.Phys.(USSRq,5:71(1941))11:91(1947).
''E..L.Andronikashvili.J. Phys.(USWR),10:201(1946).
*82
Appt-lcATloNs To PHYSICAL sYsTeM s
If'v. and v,.are opposite, however,the m ode transm its periodic variations in
p,/paand hence representsatem peratureand entropy wave(atconstantdensity
and pressure) known assecond sound. The calculation is a straightforward
generalization ofthatleading to Eq.(16.19),and wemerelynotethatthevelocity
ofsecond sound isproportionalto (p,/'
s )1,which showsthe importanceofthe
two com ponents.l Second sound was srst Observed by Peshkov;2 the correspondingvaluesofpsandpnagreeverywellwith thoseobtained with theoscillat-
ingdisk(Fig.54.2).
l.
0
0.8
0.6
%
0.4
Fig.54.2 Tem perature dependenceof p..,''p.
The dots are obtained from the velocity of
second sound;the crosses are obtained from
0.2
0
1.J
l.4 l.6 l,3 2.
0 7.2
F (OK )
the oscillating-disk method. IFrom V. P.
Peshkov. J. Phys. (f
75'
5'
S), 10;389 (1946).
fig.6. Reprinted by permission.)
4.Criticalvelocities:Superiuid flow cannotpersistup to arbitrarily high
velocities'
,instead,itbecom esdissipativeatacriticalvelocity t,
cthatdependson
the width d ofthe channelbut is independent of tem perature exceptvery near
FA. Although theprecisefunctionalfbrm ofvcld)isuncertain,manyexperimentsare approximately described by a relation w = h/md,wherem isthe mass
ofahelium atom .3
5. Rotating H e11and tporfïceî4:Ifvsism uch lessthanthespeed ofsound,
itispermissible to treatthe superiuid as incompressible (divv,= 0). Asa
result.vsisderivable from a velocity potentialthatsatisses Lapiace'sequation
h
vs=m-V+
(54.2)
V2p = 0
(54.3)
hL.D .Landau.Ioc.cit.
2V.P.Peshkov,J.Phys.(USSR4,10:389(1946).
3See. however,W .M .Van Alphen,G.J.Van Haasteren,R.De Bruyn Ouboter.and K.W .
Taconis,Phys.Letters,26:474 (1966),who find t?c= Cff-+,whereC isaconstantoforderunity
in cgsunits.
*ThissubjecthaslxenreviewedbyE.L.Andronikashviliand Yu.G.M amaladze.Rev.Mod.
.,3#:567(1966)and by A.L.Fetter.Theory ofVorticesand Rotating Helium.in K.T.
J%.)u'
Mahanthappa and W .E.Brittin (eds.
),'sl-ecturesin TheoreticalPhysics,''vol.Xl-B,p.321,
Gordon andBreach,SciencePublishers,New York.1969.
SUPERFLUID HELIUM
483
C onsidera sim ply connected cylinderrotating about itsaxis with an angular
velocity u). The boundary condition o%jon= 0 ensures that+ is constant so
that&'
,vanishesforany co. ThusEq.(54.1)impliesthatthe superiuid cannot
participate in the rotation,and the shape of the free liquid surface should be
given by the equation
k 2r 2
z- kt
j- y
(54.4j
# P
whereg istheacceleration dueto gravity. Thiseflkctshould bereadilyobserv-
able because pn/p becomes very smallbelow I.SOK. Equation (54.4)was first
tested by Osborne,lwho found insteadthatthefreesurfacewasthatofaclassical
Puid
2r 2
zc,= -j#
(54.5)
independent of tem perature. To explain this discrepancy,we reexam ine the
cylindricallysymmetricsolutionsofEq.(54.1),which necessarilytaketheform
Y,(r)=const
--. .1
r
(54.6)
wherer= (r,#)isavectorin thexJ'plane. lfcurl#,= 0everywhere(including
theorigin),theconstantmustbezero. Sincethispredictioncontradicts(54.5),
Feynm an 2 suggested a less stringent condition, allowing curlv, to becom e
singular atisolated points in the quid. The strength ofthe singularity m ay be
characterized by thecirculation
K= j(/1.v
(54.7)
and thecorresponding sym m etricsolution becom es
J'
Ystr)= 2K
=r
(548)
.
which is precisely the velocity ofa classicalrectilinearvortex parallelto the z
axis. Feynm an further postulated thatrotating H e 11contains a uniform array
ofparallelvorticesdistributed with a density2f
.
,:/v perunitarea. On am acroscopic scale, the resulting superfluid velocity field ls indistinguishable from a
uniform rotation v = to x rbecauseboth flow patternsim ply the sam e circulation
about any contour large compared with the spacing between vortices (Prob.
14.2). In addition,the superiuid now contributes to the depression of the
lD.V.Osborne,Proc.Phys.Soc.,A63:909(1950).
2R,P.Feynman,Application ofQuantum Mechanicsto Liquid Helium,in C.J.Gorter(ed.),
''Progress in Low--femperature Physics,''vol.1,p.l7,North-ilolland N blishing Company,
Am sterdam ,19f5.
4'
g*
APPLICATIONS TO PHYSICAL SYSTEM S
meniscus, thereby reconciling O sborne's experiment with the generalized
condition thatcurlvx= 0 alm osteverywhere.
6.Quantizedcirculation:MultiplyEq.(54.7)bym '
,theresultingequation
mK= jdtwmy= j#l.p
isagain reminiscentofthe Bohr-sommerfeld condition (compareEqs.(49.26)
and (49.27)1.and Onsagerland Feynman2independently suggested thatthe
circulation in H e11wasquantized in unitsof
K
h
0.997 x 10 3 cnncysec
-
= -- =
m
(54.10)
This prediction has been confrm ed 170th for irrotational flow in m ultiply
connected regions3and for vortex rings.4 Itis now possible to flnd the energy
perunitlength ofvortex line
2
2 PsK2 dr
Ev= f# rjpsrs= 4.n. -r
Ps,2 R
QJ4= InJ
tj4.)1)
H ereR isan uppercutoF thatm aybeinterpreted astheradiusofthecylindrical
container,whilefisalowercutoF representing theradiusofthe vortexcore.
Experimentssindicattthat#QJ1â,sothatEv;z1.3x 105ev/cm forR ;
kJ1cm.
LAN DAU'S QUASIPARTICLE M O DEL
Atlow temperatures(F<tL ),thespecischeatofHe11variesasF3. Toexplain
thisobservation,Landau6interpreted the low-lying excited statesofH e 11asa
weakly interacting gasofphononswith theusualdispersion relation
phonons
(54.12)
wherec= 238m/secisthespeedof(srst)sound. M oregenerally,hesuggested
thattheenergyspectrum hastheform shown in Fig.54.3,with alinear(phonon)
region neark = 0and adip near/o ,where ekbehaveslike
ek= à +
hllk- kz)2
2
rotons
àw
!L.Onsager,Nuovo Cimento,6,Suppl.2. 249(1949).
2R.P.Feynman,ttprogressin Low-Tem perature Phyeics,''loc.cit.
3W .F.Vinen,Proc.Roy.Soc.(f-t7?lJt)a).A2* :218 (1961).
4G.W .Rayseld and F.Reif,Phys.Rev-,136:A 1l94 (1964).
5G .W .Rayseld and F.Reif, Ioc.cit.
t'L.D.Landau,loc.cit.
(54.13)
sulPERFLU ID H ELIU M
485
Excitations with this latter dispersion relation are known as rotons. Landau
originally determ ined the roton parameters by titting the specihc heat above
;4:0.6 K . M ore recently.howr
ever,the theoreticalcurve in Fig.54.3 has been
contirmed in greatdetailby inelastic neutron scatteringl(compare Prob.f.13),
ek
Fig.54.3 D ispersion curve for liquid kB (oK.)
helium . The experim ental points are
the neutron-scattering mcasurements.
(From D. G . Henshavv and A . D. B.
W oods, Phr.$. Sel'.. 121:l266 (I961).
5g.4. Reprinted by perm ission ofthe
authors and the A merlcan Institute of
Physics.)
w'hich yields an independent and m ore accurate measurement of the sam e
parameters(Table 54.1).
Table 54.1 Roton param eters
Landatl(l947))
9.6 K
A'
etl/rolls'l'
attering (l959)j
8.6'K
1.9f,
4-l
1.91.
i-'
0.77vlse
0.16 hhlj-lr
tSotlrce:L.D.Landau,J.Pll
.
vb'
.(USSR),11:91fl947).
jSource.
'D .G .Henshaw and A .D .B.W'oods.Phys.Ret'.,
l21:1266(1961).
In Landau's m odelofH e I1.the totalfree energy arisesfrom the thermally
excited quasiparticles.U hich aretreated asan idealBose gas.2 Sincethe num ber
ofquasiparticles a
Tqv isnotconserved,itm ustbe determ ined by m inim izing the
'
free energy
l'JjF
,;.-0
'qvl
-
(54.l4)
ED .G .HenshavNand A .D .B.l 'oods.#/?y'â.Rek'.,l21:1266(196l).
7An equivatentapproach isto describe He Ilb/ the approxlnlate quaslparticle ham iltonian
P4:- N
'ek;.
'
>
:.here6kIStheexcitatlonspectrurl:ln Fig.54.3.
.
K wk. 5
k
486
APPLICATIONS TO PHYSICAL SYSTEM S
andcomparisonwithEq.(4.6)showsthatthecorrespondingchemicalpotential
iszero (seethediscussion ofEqs.(44.19)and (44.20)). Thethermodynamic
potential(1thenreducestotheHelmholtzfreeenergyF,andEqs.(5.8)and(5.9)
give
F(r,F)=ksFF(2=)-3J#3/
cln(1-e-ifà)
Nqp(F'F)= F(2rr)-3J#3*(eie:- 1)-1
(54.15)
(54.16)
whereekistheexactdispersionrelationfrom Fig.54.3andj = (/fsF)-1. Atlow
tem perature,however,only the phonon and roton portions contribute signiicantly,and we m ay separate the integraloverk into two independentparts
F = Fp,+ Fp
(54.17)
(54.l8)
AsF-.
>.0,wemayassumethatEqs.(54.12)and(54.13)holdseparatelyfora1lk.
N qp- Nph+ Nr
A straightforwardcalculation yields
Np.-1=
(3
,)p-jkâ
sc
rj'
Fph=-ibP'
ksTLkh
*c
Z)'
(54.194)
='
(54.19:)
Nr = â2(2PT3G rkaF)*e-hjg.v
(54.20/)
Fr= -ksTNr
(54.20:)
rr)1
Thecorresponding entropy and heatcapacity are then obtained with the usual
relations
s--(ê
az
'.
). cv-rtp
as
w).
(54.21)
ItisclearthatCvpsvariesasF3,whileCvrvanishesexponentially;both ofthese
predictions are verifed by experim ents.
In addition to the foregoing therm odynam ic functions,Landau's quasi-
particledescriptionjustisesthetwo-iuid modelandallowsacalculation ofthe
normal-iuid density. Consider a situation where the quasiparticles have a
m ean driftvelocity v with respect to the rest fram e of the superquid. Their
equilibrium distribution as seen in that rest framel isflek- âk.v),where
1Thisresultismosteasilyderived in them icrocanonicalensemble,wherethesystem ofquasi-
particleshasasxed energy E and momentum P. Theadditionalconstraint(tixed P)necessitates an additionalLagrange multiplierv.and the maximization ofthe entropy immediately
verihestht aboveassertion. See,forexample,F.London,op.cl'J.,pp.95-96,and Prob.14.4.
487
SUPERFLUID HELIUM
/-(:)= tt,/e- 1)-1isthe usualBose distribution function. In the sanne franne
.
the totalm om entum carried by the quasiparticlesisgiven by
P = 1z'(2zr)-3fd?kâk/-tek- âk.v)
-
a:p-(2=).
31
'd'khkg.
/(.
k)-âk..P'
/
e
'
t
e6
kk)+...1
Vh2 -.
oflek)
% 6=avj0dkkk- 8ek
(54.22)
Thusthe flow ofquasiparticles is accom panied by a netm omentum iux;the
coeëcientofproportionalitydehnesthenormal-iuid massdensity
pa(F)H6h
.
v
lcj
J
*o
*u
a'
sa
x
s4j
g.8.
/
a
'
(
6
s
k)j
(54.23)
whilethe superiuid m assdensity becom es
ps(F)= p- p2F)
(54.24)
Thisresultisverysim ilartothatforthesuperelectron densityin superconductors
discussed in Sec.52. In fact,acomparisonofEqs.(52.33)and (54.23)shows
thatthey diFeronly in thestatisticsand spindegeneracy.
W hen Eq.(54.23) is separated into phonon and roton contributions,an
elem entary calculation gfves
2=2h #, r 4
p,,p, - 4jc .....
(#c
--)
hzk.jN,
pnr= 3ks F U
(54.25/)
(54.25:)
whose sum ;ts the observations within experimental error. This prediction
ofpa(F)wasmadebeforeany experimentshad been carried out,thusproviding
a particularly striking confrm ation ofthequasiparticle m odel.
Landau's description also gives a qualitative explanation of the critical
velocity in He II. Considera macroscopic objectof mass M moving with
velocity >'through stationary He 11 at zero tem perature. ln this casc, the
m omentum and energy are
P = My E =!.A,
/:2
(54.26)
Iftheobjectexcitesa quasiparticlewith momentum p and energy e,the new
velocityv'isdeterm ined bym om entum conservation
V'
P
= V- M
(54.27)
48:
APPLICATIONS TO PHYSICAL SYSTEM S
Correspondingly,the new energy E 'becom es
(54.28)
w here the recoil energy pl)
'l&l has been neglected because u
k/ is large.
suë ciently low veloeity,theprocessviolatesenergyconservation
E - E'= e
(54.29)
As 1,increases.however,we eventually reach a criticalvelocity l'
c,where e - p.vc
firstvanishes. ltisevidentthatp willbe parallelto vc,and a sim ple rearrangem entyieldsthe Landau criticalvclocity
(54.30)
where t'istaken from Fig.54.3. lf6werea purephonon spectrum sthe m inim um
value of E/'
p would be the speed ofsound ck 238 m 'sec. W hen rotons are
included, however,t.c drops to ;
kh 'likt= 60 m z'
sec,and such roton-lim ited
criticalvelocitieshave been observed with ionsin He11underpressure.l
A sim ilartheoreticaldescription appliesto flow in channels,foragalilean
transform ation to the rest fram e of the super:uid reproduces the previous
situation.withthewallsast'hemacroscopicobject. Unfortunately,thepredicted
criticalxelocity of ;
4:60 nn,sec is m uch too large to explain the observed breakdown of superiuid flow in channels.w here l'c is instead thought to signalthe
creation ofquantized vortices.z
SSQ W EA K LY INTERA CTIN G BO SE G A S
The m icroscopic description ofChap.6 and Sec.3.
5wasrestricted to a stationary
Bose system at F = 0.and we now generalize the theory to include spatially
nonuniform system s at l
inite tem perature. A lthough such a problem can be
treated verygenerallys3thepresentdiscussionisrestricted toa weaklyinteracting
Bosegas.wherethesmalldepletionofthecondensateallow sustocarrythrough
the calculations in detail. As noted previously.this m odel is quite unrealistic
because the depletion in theground state ofHe 11isbelieved to be large(,
.
'
4r90
percentl.4 Furthermore,the modelhasa complicated behaviornearthe phase
transition-sso that%'
zshallstudyonlythelow-tem peratureproperties. N ever!G.W .Rayfield.Pll.t's.Ret'.l(,l/é'rA',16:934(1966).
2See,forexanlple,R.P.Feynman,i-progressin Low--fkmperature Physicss''Ioc.cJ?.
3See,forexalrple,P.C.Hohenbergand P.C.M artin,Ann.Phys. (N.F.),34:291(1965).
40.Penroseand L.Onsager,Phys.Re'
!'.,104:576(1956);NV.L.M cM illan,Pll
5'
s. Ret'.,138:442
(1965).
5K.Huang,C.N,Yang,and J.M .Luttinger.Phys.St7l'.p105:776 ç1957).
SUPERFLUID HELIUM
489
theless.an im perfect Bose gas contains m any features of the Landau quasiparticlem odelandclearly illustratesthespecialdimcultiesinherentincondensed
Bose system s.
G ENERA L FORM ULATIO N
Consider an assembly of bosons with the grand-canonical ham iltonian
X = f#3x(?f(x)gF(x)- p,jf(x)+ JJJ3x#3x'<f(x)1J1'
(x'l
x Utx- x')#(x')#(x) (55.1)
where F(x)= -â2V2/2/:is the kinetic energy. Since Eq.(55.1) iscompletely
general,an exactsolution is clearly im possible,and we m ustintroduce approxim ations appropriate to a condensed Bose system . To motivate our treatment,
recallthe Bogoliubov approximation ,
3,(x)-->.(0+ /(x) used to analyze the
ground state1O)ofauniform stationaryBosesystem (Secs.18and 19). M omen-
tum conservation furtherimplied that(0 i(
/?(x)1O)- 0 (Eq.(19.9)1,and the
quantityL could thereforebeinterpretedastheground-stateexpectationvalue
oftheseld operator
)O $'
f(x)lO.
.
,- fo
.
uniform system
W e now generalizetheconceptofBose condensation to fknite tem perature and
nonuniform system s,assum ingthatan assem blyofbosonsiscondensedwhenever
theensembleaverage t#lxl) remainslinitein the thermodynamiclimit. Itis
convenientto introducethenotation
'F(x)H t#(x))
and the deviation operator
/(x)ë/#(x)- #f'
(x))- ,
;(x)- '.
l-(x)
(55.4)
Thec-numberfunctionkF(x)isfrequentlyknownasthecondensatewavefunction;
itiscloselyanalogoustothegapfunctionA(x)(Eq.(51.14))inasuperconductor,
and,indeed,thepreseneeofsuch flniteanom alousam plitudesservesasageneral
criterion fora condensed quantum quid.l Note thatEqs.(55.3)and (51.14)
are rnerely general defnitions. In any practical calculation, they m ust be
evaluated withsom eapproxim ation schem ethatdependsonthepreciseassem bly
considered.
In the presentweakly interacting system ,alm ostallthe particlesare in the
condensate. Consequently,theoperatort
/)maybeconsideredasmallcorrection
!Thisconcept,which firstappearedin V.L.Ginzburg andL. D.Landau,Zh.Eksp.Teor.Fiz..
20:1064(1950)and in 0.Penroseand L.Onsager,Ioc.cit.,isa morepreciseformulationof
London's:slong-rangeorder''IF.London,op.cit.,vol,1,secs.24-26andvol.Il,sec.221. See
also.C.N .Yang.Rev.M od.#/l)zJ'.,34:694(1962).
49c
Appulcv loss To PHYSICAL SYSTEM S
toT',andweexpandk inpowersofj'
handC,retainingonlythelinearand
quadraticterm s:
k = A'
()+ X + X'
where
Kv= J#3xT*(x)(F(x)- JzJT1x)+ !.J#3x#3x'Ftx- x')l
1F(x)!2
x !'
.
le(x')12 (55.6)
kt=J#3x<#(x)(F(x)-p,+jd3x'Ftx- x')1
T(x')I2)T(x)
+JJ3x((F(x)-p,+ JJ3x'F(x-x')IN'
-(x')I2J11'*(x))(
/(x) (55.7)
#'= J#3xt/#(x)(F(x)- JZJ(/(x)+ J#3x#3x'U(x- x')
x E(
'1'(x')t2t/3(x)t
/(x)+ kl'
e*(x)Y*(x')t/#(x')#(x)
+ èkl'*(x)'I''*(x')4(x'):(x)+ à;1(x):1(x')'F(x')kI'(x)l (55.8)
Theresultinghamiltoniancan besimplihedbychoosingT to satisfythefollowing
nonlinearseld equation
2F(x)-p,)kF(x)+J#3x'Ftx-x')t
T(x')l2àF(x)=0
(55.9)
which m aybeinterpretedasaself-consistentH artreeequation forthecondensate
wavefunction.l ln thisway,tht linearterm .
:1vanishesidentically,giving an
eflkctivequadraticham iltonian
# çç= Kz+ X'
(55.10)
Thetheory can now be m adeself-consistentbyintroducing astatisticaloperator
e- jx.,,
).ff= Tre-jx./f
(55.11)
and acorresponding ensem bleaverage
;.'')= Tr().ff'
(55.12)
Sincea
orfdoesnotcommutewithX,thecondensatewavefunction
X1'(x)= Tr(âerr#(x)1
(55.13)
m ay besnite,in contrastto thesituation fora norm alsystem .
Therem ainingform ulation followsin directanalogyw ith thetreatm entof
superconductorsgiven in Sec.51. W eintroduceHeisenbergoperators
/xtx'
r)= eX.'f'/5:(x)e-Ref'T75
(55.14)
41
x(x,
r)= eAeffv'/â4#(x)e-*.n'1%
hE.P.Gross.Ann.Phys.(N.y.
),4:57(1958)andJ.M ath.Phys.,4:195(1963).
SUPERFLUID HELIUM
491
which satisfy linearfeld equations
t-:
'?
hy (/xtxml= -(F(x)- p.+ J#3x,U(x- x,)1Y.(x,)1z)fxlxrj
-
jdbx'P'tx- x')X%*(x')Qxtx'm)+ 4)(x''
r)àF(x!))VF(x) (55.15J)
'
0-
3
hjz (
/ltxml= (F(x)- Jz+ jd x' Utx- x#)1t1'(x')12)(/))(xr)
(55.15:)
These field operators can be used to Uesne a single-particle G reen's function
(compareEq.(19.4))
Ftxm,x'r')= -(Fz((,x(x'
r)fllx'r'')1)
= -
YF(x)Y1P*(x')+ F'txr,x'm')
(55.16)
w here
g'(xr,x''r')EB-((Fz(;x(xr);)(x'r')))
(55.17)
m easures the deviation from the equilibrium values. The two term s of Eq.
(55.16) refer to condensate and noncondensate,respectively,while the cross
terms(linearin (
/)vanish identically owingto Eq.(55.4). Asacorollary,the
m ean density becom es
n(x)= n()
(x)+ n'(x)
(55.18)
w here
?1a(x)= j
T-txl12
n'(x)= -F'(xm,xr+)
(55.19)
(55.20)
Theequationfor@'can beconstructed from Eq.(55.15):
h 0 @ ,(
o'r
xr,x'm')= -â3('r- r')((t
/x(xm),#)(x'r')))
-
jv.jo*K
oç
v
xrbç
4(x-zljk(55.21)
Equation (55,4)showsthattheequal-timecommutatoron the rightsidereduces
to a delta function,and we readily obtain
t-âjo
.
;-F(x)+p.-.
f#3x'Ftx-x?
')$
k
l'
Xx*)r
2)F'
(xm,x''
r'
)
-
fdsx'Ftx- xelà1'
etxl(T *(x'
')F'(x,r,x'm')+%P(x&)FJl(x>'
r,x'.
r'))
=
hbl.
r- '
r')3tx- x') (55.22)
492
APPLICATIONS TO PHYSICAL SYSTEM S
whereFa'lisananomalousGreen'sfunction(compareEqs.(20.i4:)and(51.120)
F(l(x'
r,x?'
r')>-(Fz#)(xm);)(x''
r')j)
(55.23)
A very sim ilarcalculation givestheotherequation
hy;- r(x)+ p.- Jd?x Ftx- x )l
kF(x )I Fzjtx'
r,x r)
Jd?x'Ftx- x'
')T'*(x)(tlP(x'
')Fc'1(x'
'm,x''
r')
-
+t1P*(x*)g'(x'
'r1x'v'))= 0 (55.24)
whichcom pletelydeterm inestheproblem . N otethatH'
'appearsasacoe/ cient
in Eqs.(55.22)and(55.24). Thissituationisanalogoustotheappearanceof.
â
'
in Eqs.(51.15)and (51.17),buttheself-consistency hereissimplerbecauseY'
satissesanunc8upledequation(55.9)insteadoftheself-consistentgapequation
relatingk
â and$-(Eq.(51.14)).
Intheusualcaseofatime-independentassembly,@'andFc'lhav- Fourier
representation
M'lx.
r5x!m')= (#â)-ljje-foaatm-m''@'(x'x'*u)n)
(55.25)
F2
'ltxm.
,x',
r:)= (jâ)-1z
N.
,
-e-îconçr-r't@2'l(x,x'
.'
q
f.
sn)
whereu)n= l'
Trn/lhisappropriateforbosons. Thecorresponding equationsof
m otion becom e
kfâtt?,r+ (2,'
n)-1hlV2+ y - f#3x'
'Ftx- x'
'))
VF(x'
')l2jF'(x,x',(z)a)
fd3x//I.
'tx-.x')11.*(x)g11**(x')(#''(x'5x',f.
tln)
-
.
(55.26/)
-
f#3x''lz'tx - x')YF*(x)('
t1-(x'
')M2'l(x'
',x'.f
.
on)
+ 'tI
.-*(x'
')F'(x'
',x'.(s,,)J- 0 (55.26:)
.
A lthough these coupled equationscan,in principle.be solved for any particular
choiceofAIPthatsatishesEq.(55.9),onlya few simplecaseshavebeenstudied
in detail.
U NIFOnM CO NDENSATE
A sa srstexam ple.weconsidera stationary assem bly ofbosonsatfinitetem perature,when the condensate wave function is a tem perature-dependentconstant
kF(x)H /o(F))1
(55.27)
suPER FLU ID H EL1U M
493
thatdeûnesno(F). Directsubstitutioninto Eq.(55.9)givesthefollowingrelation
between thechemicalpotentialand thecondensatedensity
J.
t= nvlr)1,(0)
(55.28)
where I
z,40)EEEP'(k ='0). Furthermore.a Fouriertransfbrm solvesthecoupled
integrodiflkrentialequations(55.26)becausethecoeëcientsarespatialconstants,
W 'ith the usualdehnitions
'
V'(X - X'.(e
t)nl= (277.)--3.fd/k (
2îk*(X-'X''F'(k*(.
t)n)
'
V2'l(X - X',u?rl)= (2*)-3fd'
bkfpik*tx-x')/
,
F2'1(k,(.
snj
.
(55.29)
.
a simplecalculation yields(compareEq.(5l.28)J
(ihu)n- .-2-nvU(1)JF'tk-t,pnl-noP'(#)FJItk.cs'al= h
(55.30)
w'here eo
k= hlL.z2p7and Eq.(55.28)hasbeen used. Thesealgebraic equations
are easily solved:
?
n.62-n
e-.
1,
'(1')) :.-..u
19'(k,fapsjcn.u.h(ih.(,
l
.
'
j
.
2-Ek
(h
-.
.
-.
.
-w
.jEk
....
-.
.
.
--.
,hc.
o-n)=-.
.
Ez
k ---..= ltz
anIh - c
lc
t
ln..
+.
- .
.
.
7k
v?j
)J.'(k)
1
1
'
h.'z
'j(k.f
x):)= (
-h
-u-).
--= -us
.
---Ey
.,
-i- :
.o--.
n.
I
Z----.
El
v
G!'s
aj
l.
co-n-/J
lc
n-,-Ek
/â
(55.32)
wheregseeEqs.(21.7)to(21.10))
(55.3.
3)
(55.34/)
(55.34:)
The presenttheory isvery sim ilarto thattreated in Secs.21 and 35,and
m ostofthe same rem arksapply. ln particularsthe usualanalytic continuation
to realfrequency identises Ekas the single-particle excitation energy,and its
long-wavelength behavior
k -. ()
(55.35)
(55.36)
c= g/7e(F)I'(0)?77-1)+
shows that 1,
'(0)mustbe positive. lf.in addition.Ekis positive forallk # 0,
thentheLandaucriticalvelocityist
inite;thusanim perfectBosegasissupersuid,
whereas an idealBosegas isnotbecause c and !'
evanish identically. This is
4:4
Appt-lcAyloss To pHyslcAt- SYSTEM S
one ofthe m ost im portantconclusions of Bogoliubov's originalcalculations'
foritdem onstrateshow thepresenceofrepulsiveinteractionsqualitativelyalters
the spectrum,leading to a Iineardispersion relation ask -.
>.0.
Itis interesting to study the temperature variation ofthe parametersnn
anda'. A combinationofEqs.(55.20)and(55.31)gives
n'(F)=
-
k)(jâ)-1
Q - t?
j
(J3
2=)
r
t
iœn- .
Ek
-/
-j icon+ Ekjh eiwn'
a
j(2
#=
3:
)3jebEk
Q-l.j
.l-r
e
?
I
-bEk
.j
(55.37)
where the sum has been evaluated with Eq. (25.
38)and theparametersEk,uk.
and ykdependontemperaturethrough?u(F). Forlow temperatureand weak
interactions, however,we may set ao(F);
kJaa(0)in Eq.(55.37). A sim ple
rearrangementyields
n,(r) n,(0);:z d3kj(Iz2
k;
,+
jk.!?
vjl!v-o
-
(2=) e
. .
m
;
k;12âac (ksF)2
j
F-.
>.0
where n,
(0) is the noncondensate density (compare Eq.(21.15)J and c=
(zu(0)F(0)/-)1'isthespeedofsound,bothatF= 0. Sincethetotaldensityis
independentoftem perature,thecondensate m ustbe depleted according to the
relation.z
,
m
c
?u(O = a0(0)- 12âac (k.r)
(55.38)
This equation m ay also be expressed in terms ofthe macroscopic condensate
wavefunction
1- 'l'(F)= mlkzF)2
T(0) l4h3c?u(0)
(55.39)
showing thatthe temptrature-dependentpartofklpvaries as r2. In addition,
wehaveseen that11F12isalwayslessthan abecausetheinterparticlepotential
introduceshigherFouriercom ponentseven atF= 0.
1N.N.Bogoliubov,J. Phys.(&SfR),11:23(1947).
2Ifthesecondterm ontherightismultiplied byafactorzu(0)/a,whichis ;
<$1inthepresent
modelIseeEqs.(21.15)and(22.14)),theresultingequation correctlydescriixsaBosesystem
with an arbitrary short-rangerepulsivepotential. aslong ascisinterpretedasthe actualsrwxed
ofsoundatr=0IK.Kehr.Z.Physik,;21:291(1969)).
SUPERFLUID FIELSUM
495
ltisinterestingtocompareLkl'-(r)(2withthesuperouiddensityps(F),which
can be dehned through the linearresponse ofthe system to an externalperturba-
tion. TheLandauquasiparticlemodelpredicts(seeEq.(54.25J))
l hçXè 2R2h /fd!F 4
p--- yjc
.g-
(uc
--)
.
(55.40)
so that p,(F = 0)= p,with temperature-dependent corrections :z:7-4.1 It is
clearthatthere isno directconnection between p,(F)and 'Xl'
(F)I2because these
physicalquantities have diPkrent tem perature dependence and difI
krent values
atr = 0 (seealso Prob.14.9).2 W ehavealreadydiscussed the measurementof
ps(F) in Sec.54. The eorresponding experimentaldetermination of?7f
;(r) is
m uch m ore dië cult: nevertheless it m ay be possible to obserke anom alous
quasielastic scattering (see Sec.17)ofenergetic neutrons(;4:1 eV)from He 11
below FA.3
NoN klN IFo R M CO N DEN SATE
W enow turn to spatially nonuniform systems,whereVl'
-(x)mustbedetermined
from the nonlinear integrodiflkrentialequation (55.9). lt isoften convenient
to write
t1'(x)= F(x)ei*ixb
(55.4l)
where F and ç;,are real.
relation
i(x)= 2h-.E(V - V,)Ftxm,x'r*))x,-x
ml
-
(55.42)
thecontributionjctx)ofthecondensatebecomes
h
av
.
s
jo(x)- y -jl'l- (x)V'1'(x)- ,
.(x)vkj.
xvjxlj
h
= -
m
c
h
(F(x)j V+(x)= rlo(x)-V(p(x)
m
wheren:H /-2isthecondensatedensity. Thisequation identiqestheeondensate
velocityas(compareEq.(54.2))
h
votx)= .
-V+(x)
m
(55.44)
!Equation (55.
40)and the phonon specil
ic heat(a:F3)obtained from Eq.(54.19)have also
been derived from the microscopic theory to a11orders in perturbation theory by K . Kehr.
Physica,33:620 (1967)and by W .Götzeand H.W agner,Physica,31:475(1965),respectively'
.
2In the sptcialcase ofan idea!Bose gas, these two quantitiescoincide;see Probs.14.6 and
14.l1.
3P.C.i'lohenbergandP. Nl..Platzn-kan,Phys.Rer..,152:1.98(1966).
4g6
APPLICATIONS TO PHYSICAL SYSTEM S
N'otethat&'
(j(x)isautonlaticallyirrotational(Eq.(54.I))whenever V(/,'isbounded.
because v: isproportionalto the gradientofthe phase of:1-. If:1-*is assum ed to
besingle 5,alued.then the 1ine integralofEq.(55.44)around aclosed path lying
55holl),
'i11 the condensate inlnlediatell giNes the quantization of circulation in
tlnitsof11???gEq.(54.l0)1.justasi11Eq.(50..
31).
For si171plicity'. ut'
t approxinlate the interparticle potential by a repulsive
delta ftllctio!
1
l'(x)--v
tri
s(x)
(55.46)
ilto (f5.461.the im aginar/ and real parts can be
(55.47)
jc-z$r ;12 z
-.u - ' .-- .7 a .Tr /- -.ya&.c()
lJl -221-/
T1
-let,e eqtlatio1)s are rk
lad1ly retlogl
'
l1zed :
ts the conti1:uity'equation for the condellsate alpd ïtqtlal
lttl11'
tCtllalog ot-Berloulli-sequation for steadl'flou .
.h
.s aI1 eNttl
'
tlpluto1-:111t3!ltl1
)1fornl s)steni.considera stationarq'condensate
co1l
illetl to a t
seIlli-i11ti1)ltt
l dk4nla11) (.
2' '0). The boundary colld1tion Nvi11 be
takeI
A:).
>î1'-.0 at- 0 >11
-1ce Eq.(:
55.46)1ss1n)i1arto a o1:e-part1cle Schrödinger
eqtlcttio11. I1)th1sone-d1Ille11s1
'onalgeonletry,11'takesthe fbrm
(55.50)
= llvg
(55.51)
1This equation is ./
(?l
'
'?)7J//I'identi
calwilh a Ginzburg-l-andau tl'pe offield equation forpl
valid nearrc(V.L.Ginzburg and L.P.Pitaesskil.St
?t'.Ph)'
s.-JETP,7:858 (1958)),butthe
physicalinterpretation is dit/krent. In the presentcontext.Elq.(55.46)B'astirststudied by
E.P.G ross,N 2?t?ë'f.?(.-illl(>'lîç).20:4j4 (l96l)alld by l-.P.Pilaes'skii.S(?1..Pll.b,s.-JETP,13:451
(l96I).
SU F'ERFLU lD H EL.IU M
497
justasinEq.(55.28). Thenaturalscalelengthforspatialvariationin:1'
-isgiven
by gcompare Eq.(50.l8))
(
7-c
zj
(.
j
r
t
j
q
j
t
y
,
-g
.j
(55.52)
whilethepropersolutionofEq.(55.50)hasalreadybeenobtainedinEq.(50.20)
(55.53)
Thusthecondensatedensity vanishesatthewalland rises to itsasym ptotic value
nvinacharacteristiclengthJ. ltisinterestingthatthissamelengthalsooccurs
in a uniform sy'
stem with a delta-function potential (see Eq.(21.14)),for the
excitationspectrum Ekchangesfrom lineartoquadraticatthevaluek;
4:#-1.
To explore the implicationsofEq.(55.53).we evaluate the energy a per
unit area associated w ith the formation of the surface layer. The num ber of
condensate particles per unitarea contained in a large region 0 -'
z-z < *-'- given
by Eq.(55.19)1
OL
*
u$'
()=?70j0dztanh2--,=
.
$7J
.
:znoL - ?7o-$L(
(f5.54)
where1 isassumed to be much larger than (. Sinlilarlq'
.the totalcondensate
energyperunitareabecomesgseeEq.(55.6)1
-L
h2V 2
'
E0= j0 Jz -X1R*..
---.11't-i
wgkl-.4
lnl
.
*L
.
-
û'
1
-r?0
2g I,Jz 1- sech4-g
.y j'
:
4:J,70
2gl-- J
3-/70
2gl
-4 2
(55.55)
M ostofthis energy arisesfrom the interparticle repulsion and also occursfor a
uniform system . B 'e therefore desne the surface energy o'as the diflkrence
between the totalenergy Eç
j and the energj'of the sam e num ber of particles
'ThelengthL ishereconsidered am athemalica!culofl-uith no phqrsicalsigniticance.butthe
samecalculationalsodescribesaphyrsicalchannelofuidth21.>>fsincetheresultingH-(x)is
sym m etric.
APPLICATIONS TO PHYSICAL SYSTEM S
498
uniform ly distributed
a.= E0- !.N:
2gl--1
x.
'é/12n9 pvc
;
k:-V n,l
.
- ;4r3.
,
(55.56)
where we have used Eq.(55.54)for Nvand setl'
rlrlo= mn= p in the lastline.
This surface energy contains170th the (positive)kinetic energy associated with
the curvature of the wave function and the (positive) compressional work
involved in m oving particlesfrom the surface to the bulk ofthe iuid,where
the density is slightly increased. A lthough the present m odel applies only if
the surfacelayerismuch thickerthan theinterparticle spacing (nf3yp.l),the
last form of Eq.(55.56)remains welldeûned even for a strongly interacting
system wherer,/3$4:l. W ith the numericalvaluesappropriateforHe II,we
5nd c;
kJ0.36 erg/cmz,in good agreement with the observed low-temperature
surfacetension ;
k;0.34dyne/cm Ea0.34erg/cmz.
As a second exam ple of a nonuniform system ,consider an unbounded
condensateoftheform l
n-(x)= rlleipytr)
where(r,û)areplane-polarcoordinatesand/tr)isreal,approaching1asr.
-+.cc.
Equation (55.44)immediately gives
h
vatx)= -.
1
(55.58)
mr $
so thatEq.(55.57)representsa singly'quantized vortex with circulation h(m
gseeEq.(54.8)1. TheGross-pitaevskiiequationreducesto
:2 '1 # #
1
% (,j;ry?
.-pf+pf-no,
:/-3=0
y
(55.
59)
whose asymptotic behavior again leadsto Eq.(55.51). Itisconvenientto
introducethedimensionlessvariable(= r/f:
J
Vk-+ (
1,d
ft- (
à(
y-1
-if-vf-fs=0
(55.60)
Forsmall(,thecentrifugalbarrierdominatesthesolution,whichtakestheform
T(()= C(
(55.61)
where C is a num ericalconstant. This resultshowsthatthe particle density
fallstozeroinaregionofradius(.whichmaybeinterpretedasthevortexcore.
:V.L.Ginzburg and L.P.Pitaevskii,Ioc.cit.'
,E,P.Gross,Nuovo Cimento.Ioc.cit.,L.P.
Pitaevskii.Ioc.cit.
sUPERFLtliD HELlU M
499
To clarify thisresult,wecombineEqs.(55.36)and (55.52)to gike
K
C
'--
2,rx jy
showingthatthecoreradiusJmaybeinterpretedasthepointwherethecirculating super:uid velocity becomes comparable with the speed ofsound (see Eq.
(54.8)). Thiscriterionisindependentofthestrengthofthepotential:forHe1I,
Eq.(55.62)givesJ;
k;!.A.inrough agreementNNith theobservedvalue ;
k;lX.1
Forlarge (,anexpansion in powersof(-2yields
f(()- 1- (2(2)-1
(55.63)
.
N ote that the density perturbation here vanishes algebraically rather than
exponentially as was the case for a sem i-intinite dom ain; this ditlkrence arises
from the long-range circulating velocity held around the vortex. The exact
solution/tt)mustbeobtainednumerically(Fig.55.1),w'
hilethecorresponding
Fig. 65.1 Radial wave function for a
singly quantized vortex. Tbe curve was
plotted from the numericalsolution to Eq.
(55.60) of M . P. Kawatra and R. K .
Pathria.Ph.b'.î.Re('..151:132(1966).
O
num ericalcalculation oftheenergyEvperunitlength associated with thevortex
givesz
E
=hlnfln -1.46R a;j!
c-2ln l,46A
-
- - - - --
.
j
4.
-.y -.
H ereR isacutofl-atlargedistancesthatm aybeinterpreted asthe radiusofthe
container,and weagaintakeno = ??inthelastform . Thisexpression isanalogous
to that for a classicalvortex (54.ll).butthe core region is now welldesned
w'ithout recourse to specialm odels. The quantity Et.determ ines the critical
angular velocity for the form ation of quantized vortices in a rotating cylinder
ofHe11(Prob.14.3). ForR ;z;lmm.experiments3consrm thatln(R 'J)Q:16
.
PRO BLEM S
14.1- Use the Clausius-clapeyron equation to discussthe melting curvesfor
H e3and H e4in Fig. 54.1. D evise a theory ofthem inim um in the melting curve
'G.W .Rayfield andF. Reif,Ioc.cit.
2V . L.Ginzburgand L.P.Pitaevskii. Ioc.cit.;L. P.PitaevskiiaIoc.cil.
3G . B.H essand -W'.M . Fairbank,Phys.Aeè'.Letters,19:216 (1967).
scc
Appulcv loss 'ro pHyslcAu SYSTEM S
of He3 based on the observation thatCurie's1aw describes the m agnetic susceptibility ofsolid He3down to a few m illidegrees. W hathappensto them elting
curve as F --.
>.0 ?
14.2. (t
z) If #(r)= o).
'
fx r, prove that curlv= 2f
.
,)J. Hence deduce that
jctd1.Y= 2a).
d.where,
4= j
-#S.JistheareaenclosedbythecontourC.
(b) lfvtr)= (x/2rrr)J,prove thatcurlv= sJ3(r)where risa two-dimensional
vectorinthexyplane. Henceprovethatjc#l.v= KNcwhereNcisthenumber
ofvorticesenclosedinC,andverifythediscussionbelow Eq.(54.8).
14.3. Conskdera system rotating atangularvelocity at.
(J)Prove thatthe equilibrium statesare determined by minimizing the'tsfree
energy''F - (
,
0L,whereL is theangular m om entum and F isthe H elm holtz free
energy.
(&) For a superfuid in a rotating cylinder of radius R show thatthe critical
angularvelocitytzlclforthecreationofasinglevortexisgivenby(sc!= (a'
/2=A2)x
ln(#/f)where (:is the radiusofthe core. Compare your ealculation with
Prob. 13.5. W hy can a quantized flux line in a superconductor be interpreted
asa N'
ortex with circulation bt'ltne?
14.4. A system ofN classicalparticleshasa totalenergy E and a totalcenter-
of-massmomentum P. Maximizetheentropysubjectto theseconstraintsand
derive the eqtlilibrium distribution function /'
(e- âk.v).wherewf(e)istheusual
.
Boltzm ann distribution function and v isa Lagrange m ultiplier. Com pute the
ensembleaverageofP6/Pk and henceidentify:,.1:
14.5. Derivethedensity ofphononsand rotonsgEqs.(54.19J)and (54.20J)J,
and find the corresponding contributions to the free energy, the specifc heat,
andthenormal-nuiddensitypngEqs.(54.25)).
14.6. UseEq,(54.23)toshow thatpn= pforanynoninteractingnonrelativistie
gas, except for an ideal Bose gas below its transition tem perature Fo, w'here
Pa= P(F/F0? .
14.7. Consider a dense charged Bose gas in a uniform background where the
excitation spectrum is ek= ((/i(1pI)2-i
-(h2k22p7s)2j'
# (Probs. 6.5 and 10.2).
U se the Landau quasiparticle m odelto show that the criticalvelocity and the
norm al-iuid density atlow tem perature are given by
e2
l,c= 1.32r-Q
sh
PFQJ0 3608 la0kBT 1exp '-3.46r-*
.
,
s
-
P
e,
el
ytuk.v
wheren= 3/4*r)av
b,ru= hljmselistheBohrradius,and es/h isthevelocityin
the lowest Bohr orbit. D iscuss the num erical values involved if m uQ:rzue.
1V.Lo13dorl.i%St
1l
7CFfluidS.3'VOl.1I,pp.95-96,DoverPublications.lnc.,New York,1964.
sUPERFLU ID HELIU M
501
14.8. (tz) Show thatthesingle-particleGreen'sfunction foraperfectBosegas
below its transition temperattlre isgiven by f#'
0(k,(s.)= -42=)3lhnoj(k)f
s/
u(
)(i(
.o?3
1
!- 62,
/
'h)-l'
(/?) Use Ekao.(25.25)and (26.10)to rederivetheresultsofSec,5.
14.9. Consider a weakly interacting Bose gas with a uniform condensate that
m ovesNvith velocity v.
((z)Show that11'(x)= ntt/i''
v*x.handso1veEq.(55.9)forrz.
(b) Using thisexpression fortl'
(x).solveEqs.(55.26)for1#,and r#a'). Com pare
with the solution obtained in Prob.6.6.
(c.
) Provethatn'isunalteredatF= 0 untilè'exceedstheLandaucriticalvelocity
forthe Bogoliubov excitation spectrum Fk. A ttinite tem peratures.show that
the second term on the rightside ofE'q.(f5.38)ismultiplied by (l- t'1,,.c2)-1.
(#) Expresstbe total-m omentum densitj'l'-1,P in terms of $''. Expand to
firstorderin v.and verify that)'-lxP --gp - p,,
(F)j&'.wherep,,isgiven by the
Landau expression Eq.(54.23). Evaluate pnqF)for r -'
*0 and compare N&ith
Eq.tf4.255).
(g) RepeatProb.13.13 forthissystem ,
14.10. (t
z) Derive the remaining equationsofmotion (compare Eqs.(20.l8),
(55.22).and(55.24)1forthematrixGreen-st-unction'
i
9''txr.x'r')EE-.Frllbstx'
r)x
4)K
1(x'r')1N'in aweak1),
.interacting Bosegas.
(??) Repeat the calculations of Prob. 13.15 for this systel'
n and compare ),eur
resultswith thosefora supcrconductor. The norm alization condition m tl$t1)ow
be written 1
-J!v(l
?/4
..
L12- /'
tï
z12)= s-1 N.
Nhere zo refers to the signot-theellergy
eigenvalue.
14 .11. U se the m ethods de:eloped i11Sec..
52 to sttid)
'the 1inearre>po1
:se ot-a
condensed nonintcracll
blg charged Bose gas to a transserse v'
ector pote11tial
A(x/). ln thelong-wa:elength statitr1inlit.shou'thattheindtlcedcurrentobeys
the London equation (49.29)u'ith atem perature-dependentcoefèicient/?s(F)'
ll==
l- (F'Fo)1(com pare Prob.14.6).and henceexhibittheNleissneref:
fect1
.
14.12. (t7)GeneralizeProb.l4.11to adenseilltcracting charged Bosegasin a
uniform background. Usetheana1og ofEq.(.
51.6)to deriveam oditied London
equation B ith a tem peraturc-dependentcoefhcient
?7
(T4
,(F)
zs
-(-F)
- = qs
--.
.
-- = 1 - #,
-.
.
11
P
P
1
wherepn(T4.
,
'
p isgiven in Prob.14.7.
(b4 Athighdensity'and zero tem peratureshow thattheelectrodl'namicsislocal,
apartfrom correctionsoforder?-s
î.
tN.
'.L.G inzburg.C'
sp.Fiz.,
%'
alIk.48:25 (l952)(Gernlan translation:Forts('/;,Phy'
.
$
'..1:101
(1953)J:51.R .Schafroth.Phys.ReL,.100:463(195.
6).
Kz
Appt-lcArloss To pHyslcAu SYSTEM S
14.13. SolveEq.(55.46)fora stationarycondensateinachannelofwidth #,
anddiscussthetransitionbetweenthelimitingcases#y,(andd.
q (. Compare
the resulting density prohle with thatforan idealBose gas. Assum e Ng;
k;A s
which neglectsthçnoncondensate,and takeN cc#.
14.14. Consider a semi-inhnite dom ain (z > 0),where thecondensate moves
uniform ly with velocity v= t?k. Assum ethattheasym ptotic density atinsnity
isgivtnbytheresultsofProb.14.9foranintinitemedium,andsolveEq.(55.46)
fork1P(z). Discusshow ac(z)varieswithv.
14.15. Show thatEq.(55.46)foràlpcanbeobtainedfrom avariationalprinciple
fortheenergy keepingthenum berofcondensateparticlessxed. Asaparticular
exam ple,consider a hard sphere ofradiusR in an insnite m edium ,where the
condensate wave function takes the form 1F(x)= ntflrj withf(R)= 0 and
/((
x))= 1. Usethetrialfunction/tr)= 1- e-tr-Rl/tanddeterminetheelective
surfacethickness/. ln thelimitR -->.oashow thatl= (6â2/lLmnvgjkand that
the corresponding surface energy becomes g= 0.479/12,1()/m4 (compart Eq.
(55.56)1.
14.16. Usethevariationalprinciplefrom Prob.14.15 to study a vortex in the
condensate. Assume a radialfunction ofthe form /(r)= r(r2+ /2)-+ and
show that/2= 2J2isthe bestchoice. Hence derive the approximate result
Ev= (rr/j2ac,
/rz?)ln(1.497A/J)andcomparewithEq.(55.64).
15
A pp licatio ns to Finile System s:
T he A tom ic N ucleus
A hnite interacting assem bly ism uch m ore com plicated than a uniform m edium .
lndeed,itisamajortaskjusttogeneratetheHartree-Fockwavefknctionsfora
hnite system .whereas in a uniform mediunl,translational invariance requires
thesewavef-unctionsto besim ply planewaves(see Sec.10).
Sincethesingle-particlestatespresentsuch a dim cultproblem .an5 practical
application ofm any-body techniquesis necessarill'm uch less sophisticated than
those discussed so far. Nevertheless. ue shallsee that second quantization,
canonicaltransformations.G reen'sfunctions.and the independent-pairapproxim ation provide powerful tools for discussing the properties of a tinite manj'
particle assem bly.
Fordehniteness.we concentrate on atom ic nuclei.butthe sam e technlques
also apply to atom s or molecules. Hence we defertbe speciéc problem sarislng
so3
so4
from the singtllar nature of the nucleon-nucleon force tlntilthe last sectio1
.i1'
1
thischapter.
For a finite assem bly,the anguIar m omentum pIays a t?rtlciaIrole.and a
briefre&iew ofthissubjectisgi&.en in AppendixB.l Therotationalinvariance
ot
-the totaIham iItonian im plies thatalIthe eigenstatescan be labeled by J.
î/ .2
l11 addition.the eigellstates are labeled b) their parity .
7
z because thttillxariallce
of the strol
lg i1teractions tlnder i1&ersiol
ls im plies that parit)'is also a good
quantum ntlm ber.
S6CG EN ERA L CA N O N ICA L TRA NS FO R M A TIO N
TO PA RTIC LES A N D H O LES
W estartbypertbrnlingageneralcanonicaltransform ationtk'
lparticlesand holes.
In thefirstapproxinlation.theground $tateorthecore(see Fig,.
56,l)isassum ed
to be a setofkzonppleteià tilled single-particte levels.% hose exat!t natttre NSi11be
specified bhortly. For a spherically sjm m etric systenl. these single-particle
statescan alwaj's becharacterized bs the qtlanttlnl nunlbers3 a o
.n
E 1115)
j11:y yNith
.
.
5-.J
.and /--./:J
.. The parity ofthcse statesis(.
-I)d. Itiscollvenlentto
'
-
m ake the /??dependellu.
e explicitb)'tlsing the notation
a jz a .-.-/))y
-.
uhere .
ta) de1)otes the qtIa11tum num bers :llls)'J. Asstlme the single-particle
quantum num bersto be ordered u ilh F a nunlberthatlies betueen the lastj
-ully
occupied and first tlnocctlpied (or partially occtlpied) state. as illustrated i11
Fig.56.1.
*..- j
Fig.56,1
'W e follow the angular-lllolllentulll notation ofA .R.Edlptonds.''Angular N1olllellttllll in
Quantun,Mechanics.''Princeton L'nlversltj Prcss.Princeton,N.J..l957.
?The exacteigenstatcscan be characterized b/ the center-of-nlass Illollqentulll Pt,,,and the
angularnlom entunl '
.1 in thecenter-of-nlassfrallle,butthe dependenccon P(,,;wI11notbe
.J%
tuade explicit(see,howes'er.Prob.I5.l8). W c shalIgenerally study the behav'torof$.alence
nucleons nloving abouta heavy lnerlcore. To a good approxilppation.the totalcenlerof
lllassthen colncldes%Ith thatofthecol'e,
71-orclarity.weassuIltelnltlaIlythattlleshstenl1scotplposed of1ustonetyptloffernl1on. The
1sotopic spin ofthe ntlcleon leadsto sktghtIllotllticat1ons.NKh)ch are dlsctlsseklatthe ent!t7f
this section.
APPLICATIONS TO FINITE SYSTEM S :THE ATOM IC NUCLEUS
B06
lfthe generalferm ion creation and destruction operatorsaredenoted by
(?)and ('a,then theparticleand holeoperatorscan bedefined by therelations
(56.2/)
(56.2:)
:.:./S-xt
b% :
.
--a
where we adoptthe phase convention
S * (- lljl-'nx
Thisisclearly a canonicaltransform ation,foritleavestheanticom m utation rules
unaltered
tux,uf
a-)- (à)a,d?'
),)- 3za.
A11otheranticom m utators= 0
Thev-dependentphaseinEq.(56.2:)guaranteesthattheoperatorblcreatesa
holewith angularmomentum 1jxmul'.which may beproved by shouing that
/)1isan irreducible tensoroperatorofrankh and componentmx. ltishrst
necessary to constructthe angular-m om entum operatorfor the system
+
z
N.i ('
'r?'
l1J.!jm')(- 1)/-mbn,./,-m(-1)J-m'b%
.J
ntj.-m,
nll?nrn < F
(56.5)
where the second line follows because the single-particle m atrix elem ents of J
arediagonalin(>?/.
j)andindependentofnand/. Thelasttwooperatorscanbe
written (-l)p-'1bnti,-m(-l)?-'
n':llj,-m,= (-1)'.-'
n'(3mm,- ?(lj,-m.bntj,-m),andthe
tirstterm in parenthesesmakesno contribution to Eq.(56.5)because
V
z-
(m lJl
i
m)= J=
Y m =0
&1
m
Furthermore,the W igner-Eckarttheorem showsthatthe m atrix elementofJ3q
satissesi ;jm @
JIVljm'j= (-1)m'-'
n+l.
tj,-m''
f
-zj:/j,-zn)
,.. With the change of
sum mation variables (mtm'->.-rn',-?'
n), the angular momentum operator
linally becom es
J=
:W eusethegeneraltensornotation TxQ discussed in AppendixB.
s06
APPLICATIONS TO PHYSICAL SYSTEM S
Theproofthatbjisatensoroperatornow follow'simmediatelyfrom therelations
(J-??),,
a)- 2)'
C/
-llzty)(>/
f
?b,,,>1J
py
cy-- pt,
)z- F)av+.t?(F - y)u
s'
.
yé/-y
(56.8/)
(56.8:)
and the nexttask is to write the totalham iltonian
f'
?-W-k-cœ&
t.v.'r i)'scj+ à.aV'
Vöc'
z
lc'
t
x)'
iIz
'
-.y3>c.,cy
j-.
.
(56.9)
.
in norm al-ordered form with respect to the new particle and hole operators.
Jttst as in Sec. 37. W ick*s theorem can sim plify the calculation. W e w rite
,N'(cx
tc#).#-c.
a
f'cb
.EEnc
'x
fcj
3d
(56.l0)
I
where A'now stands for the norm alordering with 'respectto the new'operators.
Take matrixelementsofEq.(56.10)w'ith respectto the new vacuum 10''
.dehned
by
It follow's that
(56.l2)
Cj
.'
'
#
'.= Q.C)
.= ()
= C
j
(56.l3)
(56.l4)
t'1c;
1c:cy,
- (Cayt
sja- (
5zôt
sjy)#(.F- a)0(F- ô')+ 23ay0(F- a)z
V(c'j
Ac:)
23J
?(F- a)Ntcj
fc)?)a-A'tcJt'
j
'
tcst'
s) (56.15)
7st
wherethesymmetrykajpP'iy3.= u'$1
x.1z''3y)'hasbeen used tocombineterms
-
.
thatgivethesamecontributionwhensummedovertheindicesa#y(
à.
W hen Eqs.(56.14)and (56.15)are substituted into Eq.(56.9),the hamiltonian separates into a c-ntlmber part.a partcontaining N (c1t-).and a part
containing .
Y(('ltXcc). Although thisresultis valid forany setofsingle-particle
APPLICATIONS TO FlNlTE SYSTEM S : TH E ATOM lC N UCLEUS
B07
states, we shall now choose the particular set that diagonalizes the quadratic
partofthe transformed ham iltonian. A sim ple calculation yieldsthe eondition
which isjustthe energy-eigenvalue equation derived from the Hartree-Fock
single-particle equatfons of Sec. 10. Consequently, if the single-particle wave
functions (
J)satisfyl
then the totalham iltonian takes the form
X = Ho+ X 1-t
-Xa
H0= N
J (ra s-)
4'
.-.
-.
a)
x<F
(56.20)
(56.21)
(56.22)
(56.24)
Thisgeneralresultisnotrestricted to spheriealsystem s.
'itappliesw henever
ta)denotesasetofsingle-particlequantum numbersspecifyingthesolutionsto
theHartree-Fockequations(56.l7)and(56.22). lnal1thecasesconsideredhere,
.
however,the rotationalinvariance of the Hartree-Fock core im plies that the
corresponding single-particle energies are independentof?'
zz,nam ely'Fz= L,
Jz
'7 = 1z
'
aband 6X = ea.
.
Equations (56.18)to (56.21)are an exacttransformation ofthe original
hamiltonian.Eq.(56.94. For alm ost a1Ipurposes.Ho- X :providesa better
starting pointform any-body calculationsbecause italready includesthe average
interaction of a particle w'ith the particles in the core. The new ham iltonian
hasthefurtheradvantagethatHv-,-X,isexplicitlydiagonalandthatx0!
.Vc10)
vanishes.
1Note that the solutions corresponding to difrerentenergies are orthogonal,whereas tht
degeneratestatesalwayscan beorthogonalized.
50:
Appt-lcA-rloNs To PF#YSyCAL SYSTEM S
Thepresentdiscussion appliesdirectly to atom s,wherethecoulom b interactionwiththenucleuscanbeincorporated in theone-bodyoperatorF. Forthe
nuclearproblem ,however,thepreceding treatmentm ustbe generalized slightly
because the nucleus consists of 170th protons and neutrons. ln this case,the
setofquantum numberstalsimplycanbeexpandedtoincludetheisotopicspin
ofa nucleon,asdiscussed in Sec.40. Forsphericalnuclei, wethereforedehne
!a )> i/?/J.
jmj.#z?r'
.
(56.25)
with mt= -t-1-(-1')fOra Proton(neutron). ThephaseSzin Eq.(56.3)mustalso
.
be generalized to
sya (-j)Ja-ma(-j)y-mt.
(j6.26)
so thatblisalsoa tensoroperatorofrank1.with respectto isotopicspin(the
prooffollow'
s exactly as w'ith angular momentum ). W ith this enlarged basis,
Eqs.(56.18)to (56.24)applyequallywellto tinitenuclei.
57/THE SIN G LE-PA RTIC LE SH ELL M O D EL
The solution of the Hartree-ll-ock equations for a snite system is extrem ely
dië cult. Fortunately,m ostpropertiesofsnite nucleican be understood with
approxim ate,solvablessingle-particlepotentials.and wenow discusstw'
o specific
exam ples.
A PPROXIM ATE HARTREE-FOCK W AVE FUNCTIO NS AN D LEVEL O RDERINGS
IN A CENTRAL POTENTIAL
Considerfirstthe sim plestsingle-particlepotential,an infinite square w'ellshown
Fig.67.1 The square-welland harmonic-oscillator approxim ationstothesingle-particleH artree-Fock potential.
in Fig. 57.I. The solutions to the Schrödinger equation that are snite at the
origin are
n'
'
*lm = Nn,./l(#r))$.,
(f)r)
with energy
h2k2
6= -S-?-- G
APPLICATfONS TO FINITE SYSTEM S : THE ATOM IC NUCLEUS
509
j(kR4
,=,0 giving theeigenvalue
The uave function m ustN'
anish atthe boundar).l
spectrum
k,1lR .uu-j'
l1I
u here Xn;is:he ??lh zero ofthe /th sphericalBesselftlnction.excluding the origin
and including the zero atthe boundary. The ordering of thebe zeros isshow 1)
oI) the right side of Fig. 57.2. Thd nornpalization integral can be evaluated
explicitly to givel
Nk'
re u rite
)tjarllplw.
lfultr) -rl(j.
jr.
j
c-. j.1?
.
t*
(57.6)
Ns'
hosesolutionsareLaguerrepolynomialsz
unêl
q)= z
%rntql.lfa...1.:2Ll
'
,v
-.#
l(q2)
f
l-(J + p +.l)ez#J'
c(J)SE-.
>
t'(p -.-1)-- (zJ-J't
7--z)
zo-.
Jzp
,
hz
Act?EH - j
mf
lThisresultfollowsfrom the relations in L.I.Sehiff'
.''Quantum slechanics.''3d ed..r.8t7.
M cGraw-i-lillBook Companl',New s'ork.1968.
2P.M .M orse and H .Feshbach.-'M ethodsofTheoreticalPhlrsics.''rr.-34.1667.slcG rauHillBook Company.New'York.1953.
3A iitto m ean-squarenuclearradiiyleldstheroughesîim alehts';k:41NleN'A1.
61B
APPLICATIONS TO PBYSICAL SYSTEMS
A sinthesquarewell,ndenotesthenum berofnodesin theradialwavefunction
includingthe oneatintinityand takesthevalues
n = 1,2, .. .,x
(57.11)
f- 60
f- fp
Isotropic harmonicoscillatorlevels
Square-well
.levels
z
(inhnite walls)
(1681z/..-'y(56
1
1381
..-..
)(1/.lg,3#,4s) 6h(
.
o- :
,
S..c
.
L
jp (6)
j
(138)
.
>'> . l
.
L
q
p
(
1
1
2
J
'
'
A'
L.
.
1
3
2
1
.(26)
..w (
e
e
,
A
lI
.
.
'
2
/
(
1
06
3
.
.
.
.
'
n(
)
6
)
2
/
(14)
.
A
(
(1121 a
l..-.
-...
.42)(lh,2/.3r) 5hu) E'
-...1/1(921
w
N
x
3.
:('70) ' * *x > x t9aj y.y(2)
(701 .-.
---- 2:(681
(681 zh(21)
-
(30)(lg,2(
t 3s)
4hu)
;&N-..h- -...
N
-- -- -
2d(10)
lgIj8) x.
N.
f40)
(20)(1/)2p)
z ..
--.2p(4c) ...
m.
. N
L
jhu;
.
1
z (201
ht.o
jj134)
.w N. . . m
1hub
N '..h.
w-'
--(34) 1.
/(14)
zstzl))
j
.
w
..Q--. ..
.h
.wN ld(181 ..-...w-..,
.- (2()j z (g)
.
(6)(1#)
(58)
lg(18)
j
4
(
)
j
-lp(6)
xx
l8J
w w
.. jjyy jy
;jyo
-.,
..Nw jpjg)
>'>.
ohul
!21
-- --
1si2)
----
jg(6j
t2)
1:(2)
Fig.57.2 Levelsystem ofthe three-dimensionalisotropic harmonicoscillator(on Ieft)and thesquarewellw'
ith intinitelyhigh walls(on right). The
dtgeneracyofeach state isshown in parentheses,and the totalnumberof
l
evels up through a givtn state is in brackets. (Il
-rom M .G.M aycrand
J.H.D .Jensen,e'Elementary Theory ofNuclearShellStructureq''p.53s
Jghn W ileyand Sonsalnc..New Yorks1955. Reprinted by pernnission.)
Thesefunctionscan benorm alized using standard form ulasl
N 2=
-
2(
n- .l
)!
.-- -.
n' Jtt%
-(l,+.l-.
4
-!.)j3
APPLICATIONS TO FINITE SYSTEM S :THE ATOM IC NUCLEUS
:11
The resulting eigenvalue spectrum of the three-dim ensionaloscillator is well
known
e.t= hoA(N + !)- P'
o
(57.13J)
N = 2(a- 1)+ l= 0,1,2,.
(57.135)
and isplotttd on theleft-hand sideofFig.57.2. Thesedegeneratelevelscontain
statesofdiflkrentland are spaced a distancehœ apart. W e shallreferto these
astheoscillatorshells. In comparingthetwo modelpotentials(Fig.57.1),the
valueof l'chasbeen chosen so thatthe 1.
:levelshavethesameenergy.
Thetrue nuclearsingle-particle wellhasa snite depth. Nevehheless,the
low-lyinglevelswithlargebindingenergiesareessentiallyunchangedbyextending
the wallsofthepotentialtoinfnity. Forthesestates,atleast,therealH artreeFock potential presumably has a shape intermediate between the square well
and the oscillator,and we can easily interpolate between the two cases. 80th
n and Iremain good quantum numbersasthebottom edgeofthesquare wellis
roundedorthebottom oftheoscillatorisiattened out. ThehighestJstateshave
the largestprobability ofbeingneartheedgeofthepotential,and asindicated in
Fig.57.2.theirenergy willbe Iowered in going from theoscillatorto the square
well.
SPIN-OBBIT SPLIU ING
It is know n experim entally that certain groups of pucleons possess sm cial
stability corresponding to the cl4sed shells ofaY mic physics. These Sfmlgic
numbers''are2.8,20,28s50,82,126,. .. . ClearlyFig.57.2correctlypredjcts
onlythesrstthreeofthesemajor-shellclosures. MayerandJensenltherefore
suggested that the nuclear Hartree-Fock single-particle potentialalso has an
attractivesingle-particle spin-orbitterm 2
H'= -l(r)l.s
(57.14)
In the presenceofthisinteraction the single-particle eigenstatescan becharac-
terizedby bnlj.jml),andtheoperator1.shastheeigenvalues
l.s(al!..
j??n)= #12- 12- shknljjmjj
jfjlj+ 1)- l(I+ 1)- s(s+ l)JI
a/!./z?n)
(57.15)
Theççstretched''case(j= /+ !.)leadstoI.s= //2.whilethetjack-knifed''case
=
'See M .G.Mayer and J.H . D .Jensen,bbElementary Theory ofNuclear SbellStructureq''
John W iley and Sons.lnc.,New York,1955.
:Tbereissometheoreticalevidence thatthisterm ari
sesfrom thespin-orbitforceG tweentwo
nucleons. See,forexample.K.A.Brueckner.A.M .Lockett.and M .RotenG rg.Phys.Ret'.,
111:255(1961)and also B.R.Barrett,Phys.Rev..1M :955(1967).
512
APPLICATIONS TO PHYSICAL SYSTEMS
(j= /- !.
)leadsto I.s= -(I+ 1)/2. Thusthe srst-ordersplitting isgiven by
61-1- el.1.= xnl1(2/+ 1)
(57.16)
xntMJ&1(r)Vr)dr
(57.17)
Forhxed Iand.j,the(lj+ 1)degeneratemlstatesaresaidtoform ajshell(see
Fig.57.3).
Foranattractivespin-orbitforce(aa1>0),thestretched state,orstateof
higherj fora given ( lieslowerin energy. In addition,the state ofhighestl
in any oscillatorshellispushed down when the bottom ofthe wellisiattened,
as illustrated in Fig.57.2. Itis therefore possible thatthe state ofhighestj
and lfrom oneoscillatorshellm ay actually bepushed down into thenextlower
oscillator shellas indicated in Fig.57.3,thus explaining the observed m agic
numbers. Notethatthepushed-down statehastheoppositeparity(-1)1from
the otherstates in the oscillator shelland has a highj. Thisobservation has
manyinterestingconsequences;inparticular,itexplainstheislandsofisomerism
orgroupingsofnucleiwith low-lying excitationsthatdecay by y transitionsto
theground stateonlyvery slowly.
SING LE-PARTICLE M ATRIX ELEM ENTS
A s our srst description ofnuclear structure,we considerthe extrem e singleparticle shellm odel,which appliesto nucleiwith an odd num berofnucleons.
In thispicture,thelevelsin Fig.57.3areflled insequence,and alltheproperties
ofthe nucleus are assum ed to arise from the lastodd nucleon. Thism odelis
extrem ely successfulin predicting the angular m om entum and parity ofsuch
nuclei. lt also allow s a calculation of other nuclear properties such as the
electrom agnetic m om ents.
Them agneticdipoleoperatorsfora proton and neutron are
1* = Jl,(1+ 2le#
lAn= Ixsllhns)
where
h
p= +2.793
Aa= -1.913
(57.18f8
(57.18:)
eh
p,s= 2mp c
(57.19)
The magnetic momentof a nucleus with angular momentum j is desned byl
t:j.
il0Ijljj)
rzH ($ m -JI>lcl./,m -.
/)- (zj+ j)1 (.
/l
IIzi
1.
/)
(57.20)
In the extrem e single-particle shellm odel,thisquantity iscom puted from the
matrixelementsforthelastnucleon,which are oftheform (1èj111/è.
/) and
.
lThroughoutthis chapterwe generally suppress a1Iquantum numbers thatare notdirc tly
relevantto thediscussion.
APPLICATIONS TO FINITE SYSTEM S:THE ATOM IC NUCLEUS
513
Ij1(i
sl
f/.
ljX. Thesearejustreduced matrixelementsofatensoroperatorin a
coupled scheme(seeAppendix B),and wehnd
X =
I
l,Ls
2(./+ 1)((.
/('
j+ 1)+ /(/.
#-l)-.
5'(.
5
,+ l))+ 2à('j(j+ l)
+,
î(!+ 1)- I(/+ 1))) (.57.21)
N
N
N- I
-
6At,,
even
-
4s
3d
2#
!7,
.j
z
3:3
4
-.u'-- 4s)'
?
x a'
lg7z
>''
' 1i1'
/2
t'
;J
Jys
z> N
z
2#%
N
.
li
.. .
(16)
(4)
(2)
(8)
(12)
(6)
(10)
(184)
184
(14) (1261
(2)
t4j
126
N
N
N
N.
-jp
5Aco
odd
-
1f
<.-- - - 3,/1
--ypjt
.-.2./-71
<.
-.,
.
2)y2
Z
lj'
E
h
(6)
1hy:
(8) (1œ )
(jc)
z
-
1h
'N
N
4h
N
-
3:
N
---?s72
--
-
2#
c:w
t.
o
even
.-
-z
1:
1N
N
N.
even t- IJ
Ihol
odd
-
1g
82
(641
1y'7
/2
xx.
,
.
-
lf
/?/z
(4) (20)
(2)
(6j
(16)
(j4)
20
(2)
(4)
I8!
(6)
8
C
1g%
(821
*'
'
f>zN
2:(
,, f-2.
î
:dj/l
(12)
(2)
(4)
(6)
(gj
(1()).
--.(50)
(2) (401
(6) (38)
(4)
(8) 128)
...-
3hco - lp
odd -1/'
ldh
lgyn
AV
-
1/1'1
/2
2/1
$
2Rx
1.f%
t:- '
h--251
/2
-'
1t
:
/j
/2
..-..-'-1,'
/?
-- 1p3
/
'
?
50
.
28
0 -15.
---- l&/2
(2) !21
2
Fig.57.3 Schematic diagram ofnuclear levelsystems with (right)and
without(left)a spin-orbitcoupling H '= -a!.s. (From M .G,Mayerand
J.H.D.Jensen.S'Elementary Theory'of-Nuclear ShellStructure,''p.58,
John W iley and Sons,lnc., New York,l955. Reprinted by permission.)
*
:!4
AppLlcv loNs To PHYSICAL SYSTEMS
withs= !.. Forneutrons,onlythesecondterm (proportionaltoA)contributes.
Equation (57.21)isimmediately recognized asthatobtained from thesimple
yectormodelforthe addition ofangular momenta. The two casesj= /+ !.
becom e
j- !.+ A
A j
p.
,. .
/+j+ j(!'- A)
(57.22/)
.
1- l- '
1'
(57.22:)
which are thesingle-particle Schm idtlinesforthe magneticm oments.l These
results are independentofany radialwave functions and depend only on the
angular-m om entum coupling. Outof 137 odd-,
d nuclear m oments,only five
lie outside the Schm idtlines. Itisan interesting factthatallbut 10 ofthese
moments1iebetween thevaluegiven by the fullmomentofEq.(57.19)and the
fullyquenched(pureDirac)valuehp= l,Aa= 0.2
The electric quadrupole operatorfor a single proton is
Qzft= 3z2- r2= 2r2cc;
(57.23)
where cxoH (4=/(2# + 1))1Fxo,and the quadrupole momentofthe nucleus
isdehned by
!c r
Q-(j,m-.
/1Qzzl
j,m-.
/)-('-jy.2()jj)Ijr
2gyy
(j,
y
c4)
.
In the single-particle shellm odel,the relevant m atrix elem ent is again thatfor
thelastparticleta/è/jj:cl
jn/.
!.J'). Evaluatingthematrixelementofthistensor
operatorinacoupledschemeyields(seeEq.(B.34J)j
2J.- 1
Q = -(r2),aj27+u
(57.25)
O dd-proton nucleiwith lessthan half-filled shellstend to havenegativequadrupole mom ents.as predicted by this m odel. Ifthe shells are more than half
filled, however,the experimental quadrupole m om ents tend to be positive.3
This showsthe need fora consistentm any-body treatment,which isgiven in
Sec.58.
In the single-particle shell model, the quadrupole m om ent satisses the
relation
lUp< .
R2
(57.26)
where R is the nuclear radius. ln m ostcases, (he experim entalquadrupole
1T.Schmidt.Z.#/l.)u'
l
'
/(.14)6:358(1937).
2M .A.Preston,*Kphysics oftheNucleus.''pp.70-75,Addison-Wesley Publishing Company,
Inc.,R eading.
,M ass-,1962.
3 M . A.Preston,op.cff..p.68.
APPLICATSONS TO FINITE SYSTEM S : THE ATOM IC NUCLEUS
B15
m om ents are m uch larger,particularly away from closed shells. Forexam ple,i
Q(7lLul75),
?(1.2.
1'
.
iF)2= +13. Odd-neutron nucleishould have no quadrupole
m om entsin the single-particle shellm odel. Experim entally,they tend to behave
as ifthe lastneutron were really a proton.and the quadrupole m om entisagain
m uch Iarger than the m odelwould predicteven for a proton. Forexam ple,z
;(Ertt7),
/(1.2,
d1F)2;
4r4-23. The explanation for these large values is that
thecoreisnotinert,butcanactuallybedeformedbythepresenceofthevalence
nucleons. Since the core hasa charge Z,a sm alldeform ation hasa large eflkct
on @.t Unfortunately.adescription ofdeformed nucleiwould takeusoutside
ofthe scope ofthe presenttext.3 and the rem ainderofthis chapterconcentrates
on sphericalnuelei,nam ely those in the vicinity ofclosed shells.
58U M A NY PA RTIC LES IN A SH ELP
The previoussection investigated an extrem e single-particle m odelofthe nucleus.
W e now take the nextlogicalstep and consider the problem of slling a g'shell
with interacting valence particles. Throughout this discussion, the shells
already tilled w illbe assum ed to form an inertcore.
Tw o VALENCE PARTICLES: GENERAL INTEBACTION AND 3(x) FORCE
W ithjustoneextraparticleina level,thenucleushastheangularmomentum
and parity of that extra nucleon. W hat happens when two identiealnucleons
areplaced inthesam eg'shell? lfweconhneourselvestothesubspaceofagiven
j shell.wecan usethe shorthand notation fnl
jmj-->.m. Thestatesofdesnite
totalJ fortwo identicalparticlestherefore can be written
ij2JM A= 1
jmjjmz!JjJM )t/mjfcar0)
q.
,jmlM : (.
(58.1)
where the superscript2 denotes the num ber ofparticles in the shell. Since the
fermion operatorsanticommute,the symmetry property(B.l9b)oftheClebschG ordan coeë cientunderthe interchange m k= m 1,together with a change of
dummy variables,yields II'IJM )= (-1)27-1+1jj2J.
M j. We conclude thatJ
m ustbeevenorthestatevanishes'
,two identicalvalencenucleonsthereforehave
the allowed states
Jr= 0+92+54*5*'@!(2.
1- l)+
'M .G .M ayerand J.H.D.Jensen,op.cl'/.,pp.79-80.
2lbid.
1J,Rainwater,Phys.Rev.,79:432(l950),
3Forathoroughdiscussion ofthistopicseeA.BohrandB. R.M ottelson,seNuclearStructure,''
W .A.Benjamin,Inc.,New York,vol.I(1969)andvols.11and1ll(tolx published).
*An excellentdiscussion ofthe many-particle shellmodelis contained in A.de-shalitand
1.Talmi.e*NuclearShellTheorys''Academic Press.New York,1963.
s1e
AppLlcAn oxs To pHYslcAu SYSTEM S
In thesingle-particleshellm odel,these statesaredegenerate because the coreis
spherically sym m etricand thevalenceparticlesdo notinteract.
Forrealnuclei,thisdegeneracyislifted,and itisan experimentalfactthat
alleven-even nucleihaveground stateswithJn= 0+. Furtherm ore,theproperties of odd-W nucleican be understood by assum ing that the even group of
nucleonsiscoupled to form Jn= 0+in the ground state. Thequestion arises,
can these facts be understood within the fram ework ofthe many-particle shell
m odel? The degeneracy ofthe statesformed from the valence nucleons can
be rem oved only by adding an interaction between these nucleons. A s an
introduction,we shalluseilst-orderperturbation theoryand com pute thelevel
shiftsfrom thehrst-quantizedmatrixelements(j2JM IV3j2JM ). Thesematrix
elementsarediagonalinJ and M and independentofM becausethetwo-particle
potentialisinvariantunderrotations. They stilldepend on Z however,owing
to thediflkrenttw o-particle densities.
To investigate the levelordering oftwo interacting particlesin aj shell.
weirstm ake a m ultipole expansion ofthe generalrotationally invariantinter-
action Ftrl,n,cos#jz)
U'
(rl,r2,C0s#lz)= Z/xlrl,rcl#xtcos#12)
This relation ismerely an expansion in a complete setoffunctionsofcos#Ic,
and itcan beinverted with the orthogonalityofthe Legendre polynom ials
Tx-
IK + 1 1
z
-
#x@)P'(r!.rz.x)#.
Y
(58.4)
l
Equation(58.3)canberewrittenwiththeexpansioninsphericalharmonics
4,
r
'xtcosp!z)- ZK j.j
.
Fx-tf1llFx-(f1c)- ct-d
.
M
which gives
P'(rl,n,x)- Z/x(rl,r2)t't-1
(58.6)
For generality we evaluate the expectation value of F for two arbitrary
single-particlestatespnjljylmytllandt;
uzlz,zm,(2)coupledtoform desniteJand M.
Thismatrix elementfollowsfrom therelation(B.33)ofAppendix B
(.
/1jZJM lP'l.
/l./2JM )
/1 h J
7
lcxl
l-/l)(A1
1cx!
1A) (58.7)
x2Fx(-1)#.+#a+J h ./l K ;jLI
.
wheretheradialm atrixelem entisdefned by
FxH Jl
dIlj(rl)dzl,(r2)./k(r1,r2)drj#r2
(58.8)
A11the dependence on the totalangular momentum J is contained in the 6j
symbol,and allthe specihc nuclear properties are contained in the multipole
APPLICATIONS T0 FINITE SYSTEM S : THE ATOM IC NUCLEUS
517
strengthsFK. Sincel
0
K ,4
jj xuven
(58.9)
K odd
Eq.(58.7)can be w'
ritten
L
,/.1./.
I
.
jy,
s
2Jiu :
.p.I).
j./.
2J-;. ev,
e
.
n,Ksx(x-j)y-1J
js
Jf
.
2
.
.
x
jjl() ..
yljjj2 ''
1
.
Z
x
'
t
j
,
j (58.1O)
'j2Jj
r
%L
flJz
'ëjlJ,
/
/, Jv l
.
/' K;
&z
J),=ev(
j) Fg(-1)J-tJ
j
'
;.
j
'(2 .
k.1)z(
en K
l/ .
/ n )
. /
hj (
.
,
.
lfa1lthe FK are negative,corresponding to an attractive two-nucleon fbrce,then
adetailed examination ofEq.(58.l1)showsthatthespectrum willbeasin Fig.
58.1a. This fbllows'since the largest 6j symbols are generally those with
J = 0.1
Equation (58.10)derived abovealso can beapplied to anodd-odd nucleus
with an extra proton and neutron in theshellsylandh. In thiscasethelowest
stateforan attractiveinteractionw'
illhaveJ = jbv-g'
a andthefirstexcited state
turns out to haveJ = /y-jz(Fig.58.1:). ltiseasy to seethe reason forthese
results. The m atrix elem ent .
:
.jj
.
jzJ,V P'jL/27,
%/ essentially measures the
.
.
@
@
@
@
@
*
-.
4*
....
()jM-
'
Jl
-.
.0*
j.
j)- j:j
Two identicalparticles
Two particles in diflkrent
in same shell
shellsjbandh
(J)
(*)
Fig.58.1 Two-particle spectrum w'ith an attractive interaction.
!M ,Rotenberg.R.Bivens.N . Metropolis.and J.K.W ooten.Jr..--The ?jand 6jSymbols.*'
p.6.TheTechnologyPress,M assachusettslnstitute ofTechnology.Cambridge.Mass.,1959.
t M .Rotenbergetal..op.c,
'
J.
m:
APPLICATIONS TO PHYSICAL SYSTEMS
ovelap ofthetwo single-particlewavefunctions. The bestoverlap alwayswill
% obtained by opposing theangularmomenta and thenextbestby liningthem
up. Forexample,ifwewrite L = 11+ lzand take 11.Iaasameasureofoverlap
(---Fig.58.2),then
1
/1h
L = 11+ h
1.l
a=ilf,
(Z+1)-Jj
t/j+1)-lz(/z+1))=tujj.
j.j)tg g.jj.jav()
(58.12)
and thisoverlap islargerifthe angularmomenta areopposed. Two identical
particlescannotbe lined up becauseofthePauliprinciple.
lj
lz
Fig.**.2 Qualitativepictureoftheoverlapoforbits.
Theseresultscan a seen m oreclearlywith asim plemodeloftheattractive
interaction potential. A ssumethat
P(12)= -g3(xI- xa)= -#ô(1- cosejzj3/1- '2)
,
'zr
rjrz
(58.13)
with g > 0. Thisyields
f
A (2x + 1)3(rI--rz4
vlvg
rl,n)= -4=
xt
(58.14)
Fx= -21gflK + 1)
(58.15)
where
1
dr
1- j-# l4l,,(r)l
zt,,(r)p
(58.16)
Theenern shiftsoftwo identicalparticlesin the.jshellcan now bedetermined
by explicitly evaluating the sum on K in Eq.(58.l1).1 Foreven J,we 5nd
72 (2K + 1) j .
/ J l K l2 1j j J 2
j .
/ K !' 0 --i = - j !' --1 ()
.v..x
(58.17)
1M .Roten% rgetal.,op.cit.,Eq.(2.19). Note thatthesum oneven K can tx converted toa
sum ona1lK byinxrtingafactor111+ (-1)X1.
APPLSCATIONS TO FINITE SYSTEM S:THE ATOM SC NUCLEUS
0
-
2
2
(3/2)2
(5/2)2
2+
(7/2)2
4
+
2+
2
-4
< j JIF1
p.J >
619
6
4)
a+
0+
#1
-
-
6
-
s
0+
()9
l0
Fiq.68.3 Slxctrum (Eq.(58.18))fortwoidenticalparticlesinthe/shell
withaninteraction P'
(xl- xz)= .
-gôtxl- xz).
Itfollowsthat
(j2JMjP'I
jlJM)=-1g(2j+1)2(j
1 J
0)l
(58.18)
The resulting spectrum isindicated forsomesimplecasesin Fig.58.3. W esee
thatthe0+stateindeed lieslowestand issplitos from theexcited states,which
arenearly degenerate with thisdelta-function potential,byapairing energy
(./2(X)tFl.
/2(K))=-(2.j+ 1)Ig
(58.19)
Notethatthisenergy shiftisproportionalto lj + 1so thatitmay sometimesbe
energetically favorable to promotea pairofidenticalparticlesfrom theoriginal
jshellinto ahighery'shellif-/'-jislargeenough. Inthislowest-ordercalculation (i.e.,takingjusttheexpectation value of F(1,2)),the relativeposition of
these levelsisadirectm easureofthe two-body interaction G tween the valence
nucleons. Actualnuclearspectra show many ofthe featuresillustrated in Fig.
58.3.
SEVERAL PARTICLES: NORM AL COUPLING
Fora short-rangeattractiveinteraction,wehaveshown explicitlythatthelowest
energystateofa pairofidenticalparticlesin thesam e.jshellwillbe
IJ'2œ)=V10)
(58.20)
wherewehavedehned a new two-particleo> ratorby
11st- çé
l
M j-a
'lim,.
/'nzpijJMSc:,da
.
(58.21)
52Q
APPLSCATSONS TO PHYSICAL SYSTEM S
with(
a(
ytH (
NJ
y-v.sf-v. Thisargumentsuggeststhatthe lowestenergy state for ,%.
identicalparticlesina.jshellisobtained byaddingthemaximum numberofJ= 0
pairs consistent with the Pauli prineiple. These states (unnorm alized) are
3-particlestate= a,
'
t.J
*f
1
)10N
,,
4-particlestate= (
A.
(
yl
j
Ax
s
yi
x
o
) Normal-coupling shell-m odelground states
.
s
.k.
particlestate= t/lmJ'j(
)
Jcjjixz
.
-
,
etc.
(58.22)
ltisafundamentalassumptiont#'theshellmodelthatthesenormal-couplingstates
form the ground states of the multiply occupied j-shellnuclei.z This model
correctly predicts the ground-state angular m om entum and parity for m ost
nuclei. Weshallattemptatheoreticaljustiscationofthisassumption shortly.
butletustirstbrieqy exam inesomeofitsadditionalconsequences.
Probably the m ost usefulresult is thatthe ground-state expectation value
ofanarbitrarymultipoleoperatorlkonow canbecomputedexplicitly. Foran
oddgroup ofnucleons,3henormal-coupling schem eyieldsl
K odd
K even
K#0
(58.234)
(58.23:)
whereN isthe(odd)numberofparticles(forevenAthemodelhas.
/=0). These
relations express the many-particle m atrix elem ents in the norm al-coupling
schem edirectlyin term softhesingle-particle m atrix elem ents, which in general
stilldepend on the quantum numbersn and /. Forodd moments,such asthe
magneticdipole,one hndsjustthe single-particlevalue. Foreven moments.
such astheelectricquadrupole,thereisareductionfactor(2j+ l- lN)((lj- 1).
Thisreduction factorvanishesathalf-lilled shells,N = J(2/+ 1),and changes
signupongoingfrom N particlesinashelltoN holesin ashell, N -+lj+ 1- N.
Thesign changeagreeswith theexperimentalobservations(seethediscussion
followingEq.(57.25)1.
ThederivationofEqs.(58.23)illustratesthepowerofsecondquantization,
sothatweshallnow provetheseimportantresults. The Q =0 componentof
an arbitrary tensoroperatorin the subspaceofthe m ultiply occupied jshellcan
.
.
.
bewritten
L
tkz- 7)c'
mt-lrxolrn)am
m
-
/
./A'O)C711Q 11.
/)',
i
Z
(-1)i--(imj- rnI.
(2# + 1)+ m
m
.
'M .G.M ayerand J.H.D.Jensen,op.cit-,p.241.
2A.de-shalitand 1.Talmi.op.cit.,p.53l.
(58.24)
APPLICATIONS TO FfNITE SYSTEM S : THE ATOM IC NUCLEUS
521
where the W igner-Eckart theorem has been used to sim plify the single-particle
matrix element. The many-body matrix elementsof?xoalso can be rewritten
w ith this theorem
/'
(c/-'
V.JJIt..&)).yr
.
uN/h
.= y (-j).
f-mqz
g.-m gp.'
KQ)(.
y.Ny.m ;tgv(y.N.
/v) (58.25)
(LX + 1)
,
n
ThusacombinationofEqs.(58.22),(58.24),and(58.25)gives
.
--
..
'u./Nj!
1!T' I
1jNj)
, ,.
z; àJp.
) .- = y (-1)j-,n(-1)J-p(jmj- tnjyjKoyx
,qypj- pl
j.
)jK0),
.--
''
J11
.FxIj)
x
,,,p
,.
,..
# ,..j.
A.t
.
rfo '''mjf
o
0) (j:.2,6)
xr
-01&.x'.'lc-amHst
,rr
gt
.;
()1
y
-
.
'
:0lfo ''.foaqaqJ'o '''(z.,
wherethefactors(
h andloappear(N - 1)/2timesandthenormalizationfactor
in the denom inatoris independent ofq by rotationalinvariance. Using the
pair ofidentities
(Hp,&)1= bpmYp
(58.27)
,..
2
..
Zp (-l)J-#(jp.j-p!
,JJ
/A-0)(Hp.J1)-(2J.
-jkJ'lc
+-)-
(58.28)
togetherwithEJ
-)05#
'
7Mq-0foraIlJandM,wenndthat
SinceJ
*'*
)0> 0forodd K (seetheargumentfollowing Eq.(58.l)).Eq.(58.23/)is
proved. For even K,we use the identity amaL= l-fctz.j= l-Hm. The
term 1willnotcontributein the num eratorifK # 0,and the-Hm can be m oved
to theleft,exactly asbefore. Thedenom inétorisindependentofc/.so thatthe
average
l
2j+7
j.
q<
a & = l-
X
2/+ l- (N - 1)
-> J
lj+ l
2J'+ l
can be taken. ln this way,we obtain
2/+ l
Lk'+2l--.
N (js.30)
APPLSCATIONS TO PHYSICAL SYSTEM S
s22
W ith therelations
(l:.lâ)-lj+
zjl-24
(58.31)
Elo,?1nJI0)-O
(58.32)
j.j
.
K+0
itisnow straightforward to show that(seeProb.15.4)
(0Ilxolo...(
*'
0J
'
*'
o
1..'l0
#lx
1oI0) 2 2j+ l-(N - 1)
(0lf
,
..
()lc'''
-x
J'
()f
.(
1'''f
x(
.
tQj!0) x - j )j+ j-j.)
.
-
(58.33)
ThisimmediatelyyieldsEq.(58.23*.
Thenorm al-couplingm odelcanbeextended toexcitedstates;lforexam ple,
itisassumed thatthelow-lying statesareobtained by breaking one oftheJ = 0
pairs.thusreplacing1(
%
)in Eq.(58.22)byIJsf. Thisassumptionallowsusto
:nd the energy spectrum of the many-particle system. W e observe that the
om rator describing a generaltwo-particle potentialcan be written within the
llshellsubspaceas
P-!.MRIJ,;t'1,f'll?,'nlP')/,ç)aqap
(58.34)
where the two-particle matrix elementhas been discussed previously and only
them dependenceisexhibited. Thetwo-particlewavefunctionscan alwaysbe
expanded in a basis with given totalJ and M by therelation
+p(1)97q(2)OTJ'
p(1)7./4(2)= JM
X (jpjql.
#JA'
/)*J?z>f(1.2)
(58.3j)
lnaddition,thepotentialF isascalar.sothatitsgeneralm atrixelem entsbecom e
(jlJ'$f'!P'1LIJM L= b.
u,&Mu.hjll6V(jlJ)
(58.36)
with the remaining factor independent of M . Thus we obtain the general
expression,valid within a.jshell,
P-JIM t'
IMLL-iIJIP'l./2J)f
-J.
(58.37)
wheretheoperatorlJMisdetinedinEq.(58.21). NotethatQuvanishesunltss
J iseven. W em ay now add and subtracttheexpression
'*#
Zt
zul.
I20(p'(./20)lJM- ;jz0!p'Ij20)!.mp
)jam
kcp
'
tapa,.
.
JM
=
(j201P')j20)JXIX - 1)
(58.38)
and theinteraction potentialin the.jshellbecomes
P- JM
):l,
%ML(jlJIP'1.
/2J)- t.
/20lP-..
/20))lJu+s,./201P'!.
/20)4.4(4- 1)
(58.39)
1A. de-shalitand 1.Talmi,op.cfJ..secs.27 and 30.
APPLICATIONS TO FINITE SYSTEM S: THE ATO M IC NUCLEUS
523
W hen the m any-particle interaetion energy iscalculated asthe expectation value
ofthis operator,the lastterm givesa constantdepending only on the num berof
particles in the shell. Thustheenergy dl
yerencesbetween the many-particle
levelscan bedirectly relatedtotheenergydl
yerencesbetween thetwo-particle
levels. The techniques of second quantization provide a very elegant and
straightforward way ofsndingtheserelations,lw hich form the basisfora great
deplofbeautifulworkcorrelating and predictingthelevelsofneighboring nuclei
corresponding to thejNconsguration.z
THE PAIRING -FORCE PRO BLEM
W enow returntotheproblem ofjustifyingthemany-particleshellmodel. W e
confine ourattention to the subspaceofagivenjshelland ass'umethatthereis
an attractive.short-range interaction between the particlesin the shell. In this
case.it is evident from Fig.58.3 thatthe dom inanttwo-particle m atrix elem ent
ofthe potentialoccurswhen the pairiscoupled to form J = 0. Thisobservation
suggeststhatthepotentialin Eq.(58.37)can bewrittento agood approximation
as
P';
k'f
*'Jt/20IP'lj20'
)I()HE-G(
,
*'jl0
.
(58.40)
Such an interaction iscalled a purepairingforce. Ithasthegreatadvantage
thattheresulting problem can besolvedexactly-h
W e wish to l
ind thespectrum oftheham iltonian
X = 60X + f'
(58.41)
whereeoisthesingle-particle Hartree-Fock energy oftheparticularshelland f'
is given in Eq.(58.40). The calculation is most easily performed with the
auxiliaryoperators
X+O (82/+1)J1f
V
''
X M (1'(2/+ 1))1f
0
.
(58.40)
.
(58.42:)
% -t(2X'-(2.
j+ 1)J
(58.42/)
-
.
Equation(58.31)and therelation
(X,ê11- 2f
*1
show thatv
Ltand.93obeythefamiliarcommutationrelations
Ef+,,9-)- 2X3
(58.43)
(58.44/)
'Thecoeëcientsin theserelationsareknownasthecoes cientsofJbactionalparentage (sec
Prob.15.8).
2Forageneralsurveysee1.Talmiand 1,Unna,Ann.Rev.Nucl.Sci..10:353(19* ).
3G.Racah,Phys.Rc&.,63:367 (1943). The presentsolution is due to A.K .Kerman,Ann.
Phys.(N.F.).12:3* (1961). Thesamepseudospintechniquewasused earlierbyP.W .Anderson,Phys.#er',,112:19* (1958),in a discussion ofsuperconductivity.
APPLICATIONS TO PHYSICAL SYSTEM S
s24
(X+,X31= -X+
Iu
f-,uf3J- uf-
(58.44:)
(58.44c)
which arejustthose ofthe angular-momentum operators. The quantity :2
can bewrittenas
92=J'
J+.
à./+.
f-+xf-u
f..)- .
L,(<
% - 1)+ .f+u
#-
(58.45)
and a rearrangem entyields
f+.
L--92- &(.
9z- 1)
.
(58.46)
In term s of these pseudospin operators,the interaction ham iltonian becom es
(jg4y)
b
7- -l2G ,f.,
j-- - 2G (gc- g3(g3- j;j
j+ 1
2j+ 1
so thatP isexpressed entirelyin termsof:2and Sb. Sinc.
e the spectrum of
theangular-momentum operators,12and7,followssolelyfrom thecommutation
.
relations,we can im mediately deducetheeigenvaluespectrum ofthe operators
:2and Sj,which thussolvestheproblem .
Theeigenvaluesof:z'
areoftheform N45*+ 1).whereSisintegralorhalfintegral. Italso followsfrom the theory ofangularmomentum thatS > 5-31
,.
and % ishxedfrom Eq.(58.42c)ifthe numberofparticlesN < 2./+ 1isgiven.
Theabsolutemaximum possible valueof.
S isclearly 1,S Imax= (2.
/+ 1)/4,and
thegeneraltheory ofangularm omentum also requiresthat.
S'< 1
,5
'
9gmax. Thus
forfxed N ,theeigenvaluesS liein therange
jjzj+ 1)> .
&> '
t(2A - (lj+ l)!
(58.48)
and mustbeintegralorhalf-integraldependingonwhether5-;= (2.
N - (2j+ 1)1/4
is integralor half-integral. ln eithercase,the allowed values of S diflkr by
integers,suggesting thedehnition
S-.
j(2.
/+ 1- 2c)
(58.49)
where c isan integerknown astheseniority., Itfollowsfrom Eqs.(58.48)and
(58.49)thatthepermissiblevaluesoftzforsxedN are
0,2,4,...,N (X even) IN < (2.j+ 1)
a = 1, 3,5, .. .,N
(58.504)
(.
N'odd) partjcjes
,4,...,2j+ 1- N (A even) 2N >
c= 0,
1, 2
jj0(l
j:js+ 1)
(f8.50:)
3,5....,lj+ 1- N (N odd)
Thehrstcasecorrespondsto a shelllessthan halffilled,orparticlesin theshell,
whereasthe second refersto a shellm orethan halffilled,orholes in the shell.
'G.Racah,loc.cI'/.
APPLICATIONS TO FINITE SYSTEM S : THE ATOM IC NUCLEUS
B25
Theexaetspectrum ofX isobtainedbycombiningEqs.(58.41),(58.42*,(58.47),
and (58.49)
N - a 2/+ 1 +.2 - N - a
Eg= S'6()- G --.
j-.
-Z- j.jj- j.
--.
-.
-.
ge:-G
j(I-a
'
Y
.
/y
-o
--2.
)j+.G
'
-scj'I-j
t
'
yyu
2)
(58.51)
o,
withtheallowedvaluesoftrgiveninEqs.(58.50).1
W e now explicitly construct the lowest few eigenstates of 4 and srst
consider an even num ber of particles N. The lowest energy state clearly has
o.= 0. We shallshow thatthisstate isjustthe normal-coupling shell-model
ground state
1N&J=0;c.
=0z,=J
V .,.(
*(
$
)
'J
'
71
0.
)
(58.52)
wherethereareN(2factors11. ToprovethatEq.(53.52)isthecorrecteigenstate
ofX,usethecommutationrelationsofEq.(58.31),replacingX bytheappropriate
eigenvalue ateach step
P%(
A.
(
A
)'..
2
+1 ,
..#
#x
+ 2.-/
u
).'.'
f.(
)1
0.
,
/-+ l f(
.
N 2j+ 1- N + 2 ,
..j.
xt
G j -.--2/.
+
.--l
...--.-.(v ...(ok
(/.
),
.- .-.
whichistherequiredresult. Itisevidentfrom E).(58.51)thatthefirstexcited
stateshave(z= 2. Thesestatesareobtained by breakingoneoftheJ = 0 pairs.
1N..
.
tM ,tz 2x
?=kj...,
;
-.
(
1
))g
#x1oy
p J= 2,,4,6,...,2./- 1 (58.53)
Theproofgoesjustasbefore.exceptatthelaststep,whereIf
%SlJM210)-0for
.
=
J # 0,
'thereforetheeigenvalue ofP is
N.
--Glj(2
-.
/+
-L1
.j
+
-.
N
l+2)-Ij--cN
-.2
-22.
12
+.
/1.
+
-lN
asclaim ed. The seniority c isclearly the numberofunpaired particlesin these
states.
tW edonotdiscussthedegeneracyoftheei
genstates. Theenumerationoftheantisymmetric
statesGftotalJ in thej shellisastraightforward problem,and theanswercan thefound.for
example,in M .G .M ayerandJ.H.D.Jensen.op.cl'/.,p.64.
qa:
APPLICATIONS TO PHYSICAL SYSIEMS
For an odd nume r ofvalence nucleons,the lowestenergy state can be
written
tN.J=j,m;c- 1)=11...l#
t,t/>jû)
(58.54)
whichisprovedjustasaboveusing(la,a,
#,,JJ0)-0. Thefirstexcitedstateswith
c- 3and J# ./canl
x written(steProb.15.7)
qN,J#.
/,Af';c- 3)= )
()Lknjmkkj-lMsl(
1...llat4priol
-#
(58.55)
Up throughthe7/2shell,thereisonlyonestatewithJ=j,and thestateswith
J#/areuniqueand thusindem ndentofkinEq.(58.55).1
W ecan now discussthe s- ctrum in the-/& confgurationusingthepairing
interaction V'=-G!1l:. Foreven N,tbe many-particle ground state has
Jn= 0+,and thesrstexcitedstatealwaysliesadistanceG abovetheground state.
One can thusthink ofG astheenergy gap,orpairing energy. In thismodel,
a11the hrstexcited statesaredegenerate and consistofeven angularmomenta,
corresponding to the possible statesofa broken pair. Forodd N.the normalcoupling state with c = 1 is always the ground state,and tht degeneratefirst
excitedstatesareseparatedfrom thegroundstatebyadistanceGllj- l)/(2.j+ 1).
Thuswehavetheimgortantresultthattheshell-modelcoupling rulesholdfora
potentialoftheform F=-G111c. Totheextentthatanyshort-rangeattractive
potentialcan also lx written in thisform,we have a more generaltheoretical
justiEcationofthemany-particltshellmodel.
The modelis quite good atpredicting the ground state ofnucleiand the
energy gap to theârstexcited states,butthe predicted degeneracy oftheexcited
statesislessrealistic.
THE BO%ON APPROXIM ATION
Theform oftheexactcomm utator
(l:,lJ1= 1-224
/+ l
(58.56)
.
suggestsa very sim ple approximation in thecasethatthegiven.jshellhasonly a
few particles with N 4 2j+ 1. W e shalltherefore assume the approximate
commutation relations
!lJM.l1,M')r'
:'JJ'3--.
(58.57)
which hold in thislimit. Theexactspectrum ofthegeneralpotential
P=>!
Zz11/,(1)P')A)L.
$M .G.M ayerand J.H.D.Jensen,op.cit.,p.64.
(58.5%
APPLICATIONS TO FINITE SYSTEM S : THE ATOM IC NUCLEUS
527
forsxed N can then beobtained becauseEq.(58.57)isjustthecommutation
relationforbosonsand P dependsonlyon theboson numberoperator
P- â (AlP'IA).
,
-#*zs
(58.59)
AM
Ah.>t1.L.
(58.60)
.
Thecorresponding eigenjtatesofP'foreven N areobtained by applying N(2
bosoncreationoperators(1vtothevacuum. Inparticular,if
f'--Glâlo-,-G-JL
(58.61)
thentheenergydependsonlyontheeigenvalueot-.z#% (i.e.,onthenumberof
pairscoupled to form J = 0),and the spectrum isshown in Fig.58.4. If we
0
-
2+4
#+9**'
*(2g-1)+
G
0+
2+4+ (2./-1)+
0+
-
2G
-
3G
2+4+ (2g- 1)+
0+
Fig.58.4 Pairing-forcespectrum in the boson approxi-
mation fortheconfiguration/f.
identify N - 2..
+% - c,then Eq.(58.61)reproducestheexactspectrum ofEq.
(58.51)inthelimit2.
,
3/
% < N -:
t2j+.1. ThisresultjustifestheuseoftheapproximatecommutationrelationsofEq.(58.57)inthissamelimit.
THE Bo Gol-luBov TRANSFO RM ATION !
W e have seen thatpairing playsa very im portantrole in nuclearspectroscopy.
ltisthereforeofinterestto includethepairingefectsinamodisedXt,which
should then providea betterstarting pointfordiscussingnuclear spectra. This
objectivecanbeachievedwiththeBogoliubovcanonicaltransformation. Since
theresultingXoandX - Powillnotseparatelyconservethenumberofparticles,
Xowillbea sensibleapproximate hamiltonian only ifthe numberofparticles
underconsideration islargeand thecorrespondingquctuations((X 2)- (X )2)/
lS.T.Beliaev.ApplicationofCanonicalTransformation M ethod to Nuclei.in C.DeW itt(ed.),
t'The M any-Body Problem,''p.377,John W ileyandSons.Inc..New York,1959. Foramore
detailed discussion of this approach to nuclear slxctroscopy see M .Baranger,Phys.Re?J.,
120:957(19K )andA.M .Lane,t.Nucle-ar'
fheory,''W .A.Benjamin.lnc.,New York,1964.
saa
'
AppulcAzloNs To pHvslcAu sys-rEM s
X)2 negligibly small.l These conditions are best satissed in heavy nuclei,
and we here consider only sphericalsystem s.
W e shall generalize our previous discussion of spectroscopy in a single
jshellbyworkingwith asetpfstatesthatarecompletelyspecised bythequantum
numbers bl
jm),,the radialquantum numbern then being fully determined by
the originalchoice of shells. This set can be as large as two neighboring har-
monic-oscillatorshells(seetheleftsideofFig.57.2). Notethattheparityofa
given state is(-1)1. W eagain usethe notation
jt
x)-.ljmj).EH lJ,z?tal
l-,
.
x)- ha,-rna).
(58.62J)
(58.62:)
wherethequantum numbers((z)denote(n(j;,withn redundantasalreadydiscussed. Fortherem ainderofthissection,we assum e thatthe shellsare ûlled
withjustonekind ofnucleon,forexample,anucleuswithclosedprotonshells
and partiallyilled neutron shells.
Thekineticenergyisascalaroperatorand hencediagonalin =, la,
iFla.
')=
3az.Ta. Consequently the therlnodynam ic potentialat zero tem perature can
be written
(58.63)
wheretheinteractionisassumedattractive(P'> 0)exactlyasinEq.(37.6). We
now carry out the following canonicaltransform ation on this therm odynam ic
potential
Wl
%- uJc%
? - va Sac-a
d Es uacu - vaSuc'
- .
(58.64)
.
wherethephase.
% = (-1)Jœ-'*ehasagain beenintrodueed to makez41'an irreducible tensor operator. The coeë cients ua and vg are taken to be realand norm alized
ul
a + n2
a= 1
so thatthe transform ation is indeed canonical
lWz,Wt'l= &x.'
(58.65)
1A resnementoftheapproachdiscussedinthissectionistoprojectfrom theeigenstatesof4:
thatpartcorresponding to a dehnite numberofparticles. See,forexample,A.K.Kerman,
R.D.Lawson.and M .H.Macfarl
ane,Phys.Rev.,124:162(1961).
APPLICATIONS TO F!N ITE SYSTEM S : THE ATOM IC NUCLEUS
529
Equations(58.64)can beinverted to give
(- --.uaA x -#-r'
a.
b'
xz.
l1
- ,
x
(yX
t -.,uazzl(
1
'-;-!.,a.Sazd-
(58.66)
::
The nexttask is to w'
rite k in normal-ordered form with respect to the
operators .
,
1 and .
,
1I. Thiscalculation is readily done w ith W ick-stheorem ,just
asin Sec.37,and the only nonzero contractions are
The
enables us to rewrite the
(58.68)
(f8.69)
'a/3iP'!y't
i'= N'
'
'z ??7a.y'
j??2j.jay'jJ.îf , 7':zl1ly.js??7y /y./'
x- .g
,.I.
b1
JM
.
(58.70)
.
abJ :l.
';cdJ
Q/.
9!p'.y3.:
v(ca
1(-)csc.,)
5:*
APPLICATIONS TO PHYSICAL SYSTEM S
and the other terms are those remaining in Eqs.(j8.63). W ith the identities
(seeEq.(B.20>))
i (ju'z7,jbmbA .
5
VJM S(./amx.
ibmbjh 7,JM )- l
and
=
N
.
u
ut
x
'.
l'
m.
1'- m tjj'00'
.
)L.j/ng'- m !
1gg
'JM )= bJ(j8M()
lA1
thec-numbertermsin Eqs.(58.63)can bewritten
&= )
(?
'
J,
2(L - y.+ !.L )- !.j
(valq'
îa
œ
Q
(58.74)
wherethesingle-particlepotentialenergyhasbeen desned by1
(58.76)
Thetermsproportionalto Nlctc),which givebilineartermsin .d and ,
41,
can be sim plified with the identity
V =Y tju?
V
Na/jmî1A .
ji?JiV )(.
lumx.
/:ms1A AJM L
/
.
?,1a
=
2J + 1
2
3, jsjpsms
%+ 1 #
.
Furthermore,P'(1,2)isinvariantunderspatialreiections,and conservation of
parityimpliesthat/j= Isinthematrixelement(ab.l1VIadJ$. (Thepossibility
h = Is:iu2isruledoutby-/j=h.4 Withinoursubspaceoftwoadjacentoscillator
shellswe can therefore write
lIn order to emphasize that L and .ï. are independent of mx. we use the abbrevi
ation
J'
aEëjx.etc.
APPLICATIONS TO FINITE SYSTEM S :THE ATOM IC NUCLEUS
B31
Byexactly the sameargument,thetermsproportionalto Nlcc)and h'lctc1)can
besim plised with therespectivereplacements(seeEq.(B.20>))
Svfvuoj
x
t.X ??
'
1y-&ms!jyh 00.
.
'--+(j.
l-- j:
c-h 1)
6a,:'jamxJ'jmj1jajj00h
>.(Sx
,-.
jj -.j-j
)1.
.
a+
ln thisway,we obtain
X l= Jgltu2
t
,- l?j)(eu- y,
)+ 2uavakâajj.
tzdt
'
lzla+.adla ad-a)
-
wherea new single-particleenergy hasbeen dehned by
6u> Ta+ L
(58.78)
Theanalysisoftheseequationsnow proceedsexactlyasin Sec.37. Defining the energy w ith respectto the chem icalpotential
fsO îa- p,
and choosing uand t)to eliminateXcleadsto
(58.79)
k-Kc-!.xby
J(& fta#jp'ry,3)Nlcx
%4cscy)
(58.80)
where
kne U + jg(Jj+ u
/
Xjll.dJ,1a
(58.81)
and
l
âo- j
z
ljc+ 1 1
c
jy s-f (t=0IP'lc.co'
)(.
f7
.
Ac
1
âi+ fl)
(58.82)
W e note in passing thatthe normalsolution to the gap equation
uc= #(c- F)
l'c= f(F- c)
(58.83)
reproducesthe resultsofSec.56. W ithin oursubspace,howeversweneed not
assutne thatJ/7esingle-particle wwrtpfunctionssatisfvthe Hartree-Fock equations.
Itisonly necessary thatthe statesbe com pletely characterized by thequantum
numbersfljmji.z
lThe presentdiscussion considersonly the interaction between particlesin arestricted setof
basisstates,butitiseasilygeneralized to includean additionalinertcore. Onestartswith the
Hartree-Fockw'
avefunctionsdetermined bytheinteraction withthecore(seeEqs.(56.16)to
(56.
24))andthencarriesputtheBogoliubovtransformationontheadditionalvalenceparticles.
lfT.isaugmented bytheHartree-Fockenergy ofinteraction withthecoreT..
-.
>.F.+ P'lï
're5
then theresultsofthissection correctlydescribethepropertiesofthesevalenceparticles.
sa2
APPLICATIONS TO PHYSSCAL SYSTEM S
The operatorXoin Eq.(58.8l)isnow diagonal, and itsground state 10)
satisses
zlx10)- 0
(58.84)
Ifthe exactground stateofX isapproximated by theground stateofko, then
z
thetherm odynam ie potentialofthesystem isgiven by
fltr= 0,P'
,rt);
4ù/O1
.é !O)- (OIX:1O)= &(l
z',Jz)
(58.85)
The expectation value ofthe num ber operator
X -(
j
gca
fcz= t'
Y gttfz
.- t'2
u).
,4Jz
4x+ uao xatv
dJ.
,4&a +..,4 a .,da).y?yz
uj (j8.g6)
z
.
inthestate 1O)is
N = 5- f.a = -l 1 - -s--w (
-a k-j
-&- & X 2 ('
.
'
&j+ (j)
.
(58.87)
so thatthe occupation numberisagain continuousat6u= t
a.. Thisequation can
be used to elim inate /.
tin term s ofN.
Thequantity ofdirectinterestin nuclearphysicsisthe energy athxed N.
Fortheground stateofXo,thisenergy canbe obtained from therelation
E(N)= s.O 1k + P.X rO)'= :Xo)O)+ P.
N = U +y,N
(58.88)
-0 !
where y,
(N)isdetermined from Eq.(58.87). Theexcited statesofthe system
are obtained by adding quasiparticles to the ground state;they take the fbrm
jnlna ...)= t-dlla.tz.
lllna..'10)withaiquasiparticlesinthesingle-particle
state l,nzquasiparticlesin thesingle-particlestate2,and so on. ltisclearthat
ni= ()or 1. For fixed p.
,each ofthese excited states has a slightly diflkrent
expectation value of-X. W e therefore consider a collection ofassemblies at
slightlydiflkrentchemicalpotentialsJzexsuchthat(nlnz ''.IX 1nln2 ...)= N
ineach case. lnthepresentapproxim ation,theexcitationenergyofthesestates
isgiven by
LE*(N)EEE(nInz
IXo+ rtexX InIn2 ''')- (OIXO+ P,
X jO)
=
Z1 (f,
2+ éa
2))x?7a+ (&(/.
zex)- 1
.
7(/2,
)+ (Mex- /z)X)
(58.89)
Fora smallnumberofquasiparticles,Eqs.(18.3% and (58.85)imply thatthe
term in brackets vanishes;in addition,the coeëcientofnx can be evaluated
usingthe/.
zobtained forthestate 10). Thusifj)nuissumciently small,we
obtain the im portantresultthat
Zf*(X)= )((C + ,
âjllaa
(
where(j.
(N4isdeterminedbyEq.(58.87).
x
(58.90)
AppulcA-rloNs To FINITE SYSTEM S : THE ATOM IC NUCLEUS
533
Theseideascan now beapplied to even and odd nuclei. The ground state
ofk(
)isidentihedwith theground stateofaneven-evenJ== 0-nucleus.t Since
thestate ilxHE,1)1O.
rhascomponentswiththenumberofparticleschanged by
one and hasodd half-integralJ values,itthen refers to an odd nucleus. To get
excitationsin theoriginaleven'nucleus,we m ustcreateatleasttw'
o quasiparticles.
and the spectrum forthese excitations(Eq.(58.90)1starts atan energy greater
than zlmin(theminimum valueof2.Xa,2-XN,etc.)abovetheground state. Thtls
thepresentmodelIead.
%toanenergl'gap tk/'zâminin taL'e/?nuclei.
For an odd nucleus,we can repeatallthe previous argum ents.taking the
state11'
,',
.
-z41!
0..asthegroundstate. lnthiscase
(58.9l)
To getthe excited statesofan odd nucleus,w'
e can again add pairsofquasiparticlesin the-jsshell. Thereis,however,anotherclassofexcited'
states,w'hich
are obtained by simply promoting the single quasiparticle from the originalja
shellto a new'h shell. Theexcitation energyofthesestatesisgiven by
(58.92)
and we conclude that otld /?l/c.JcI
'hat'
e A?()energl'gap ?
'
??lhis/7t
?t# /.2 There is
some evidence that this m odel indeed describes hea:),
'nuclei,3forexam ple. the
set of Sn or Pb isotopes.4
The m atrix elem ents of an arbitrary' m ultipole operator between these
low-lying (single-quasiparticle)states ofan odd nucleuscan be evaluated immediately. TheQ = 0componentofageneraltensoroperatorcan bewr
ritten
=
N7 f
a
')jFA:0l
'
,
.gN(cj
h(aa).
y-t.
v
l3jaj
Ia.
-.
aj
-
(58.93)
lOnecan show (see Prob.15.14)thatthestate O doesindeed haveJ== 0-.
2Typicalexcitation energiesforsphericalnucleiare aboutan M e%'foreNen .4 and a feu
hundred keV fbrodd ,4.
3A.Bohr.B.R .M ottelsonsand D ,Pines.Phys.Retx.,110:936(1958).
4L.S.Kisslingerand R.A.Sorenson.Kgl.Dt
?l7x/k
'e k
''
idenskub.Selskab.,
V/t?r.-/#5'
.A
v edd'32,
No.9(1960).
wa
sz4
Appulcv loss To pHyslcAu svs-rEu s
TheexpectationvalueofT'
xnisgivenby
(lllko1* = (œlrxclœ)1:1- r-œlFxol
-œ)rl
= r:œlrxcIa)E&a
2- (-I)&pj)
K #0
and we therefore 5nd
tœlTknl* -
fœtrxolœ)
L
(lî+Al)+'
Ctzlrxnlœl
K odd
A even
K#0
(58.90 )
(58.94:)
The odd m ultipolesare unchanged by pairing,whereasthe even m ultipolesare
againreducedbypairingand becomenegativefore< p,(i.e.,forholes). These
resultsshouldbecomparedwithEqs.(58.23). A similarresultholdsfortransition multipolesbetween diflkrentjshells:
.
(/lTknIœ)= (#lFxoI=)ubua- t-=IFxo1-/)vav.5'-aS-ô
(58.95)
(i:1(fxl(L)- (Al
!FxI
l.&)uslza+ (-1)#.+J.+X(.&(lrxlIA)w t?
a
The presentapproach enables usto include the pairing interaction in the
starting approximation. To understand the validity and implications of this
approximation,consider a singlej shellwith N particlesinteracting through a
pureattractivepairingforce(G >0)
(.
IlJIU3jlJ)= G'
ôJ:
(58.96)
Thisispreciselytheproblem solved in theprevioussection,with theexactspec-
trum ofEq.(58.51). Sincewenow haveonly onesingle-particleleveland p,
u,A.and( arealready independentofm,allsubscriptscan bedropped. The
sum in Eq.(58.87)then becomestrivial;thusthe chemicalpotentialyjN4is
determ ined from therelation
x-(v+1)sz-(2y+1)j
lgl-(yz+3az).j
(58.97)
which hxesthecoeëcientsuand &:
v-
taytjl* u-(1-zj)j)'
withthesignschosenasinSec.37. Similarly.thegapequation(58.82)reduces
to thesimplecondition
(é2+ J2)+= I.
G
(58.98)
APPLICATIONS TO FINITE SYSTEM S :THE ATOM IC NUCLEUS
637
g(2/-+-1- 2N )/(2J*+ 1)J2a:r1. Thisapproximationyields
G
(58.107)
.
XE'(N):
k;2q(l-2/n+-1
n
.
Forcomparisontheexactresult(Eq.(58.51)1isgiven by
LE *(?V)- G nq
(58.108)
i-(1-X
'
kj
?+
-2
l1
,
w here we have again identitied o'E2a(,. These two expressions now'agree up
through thefirstcorrectionterm in thethermodynamicIimit:Q'- I-v s'.'
aith
nq/lzj+ 1)snite.
The validity of the Bogoliubov approach to nuclear spectra can also be
tested by exam ining theQuctuation in .
N'(com pare Prob.l0.6)
nX O
which leadsto thegeneralresult
2
k:
V'-2).- L..N''Ax..
fz
'
2
.N '
.)
z
2Y
Ill.
2
. l
a a ff
x'1.2 2
(58.1091
a
ln the case ofa single.jshellwitha purepairingforce.Eq.(f8.109)can besimplihed to give
fk 2'
x.- ,.
4 z,
2 a
N
t
Ck
q)2
' -'
:,l-lj+11
.
.
t58.1I0)
The theory m akes sense only ifthisexpression issm allswhich istrue provided t
%I
.
islarge'
enough.
ln sum m ary,weconclude thatthediagonalJ/?tpr/zzt/ti
/b'
atzzps'
rpotenlial
#o= & -;
-N'(J24..
â2)1,
.
zIâ.4x
(58.1ll)
/J/?apureptzàr/
k,
ginteracliongEq.(58.96))
correctly describes a ./N'conhgurationu'
in the lim it
S' sxed
lj+ i
---
U + l-->.oz
(58.ll2)
538
APPLICATIONS TO PHYSICAL SYSTEM S
S9LEXCITED STATES : LIN EA RIZATIO N O F
TH E EQ UATIO N S O F M O TIO N I
Afterthisexcursion into the theoreticalfoundationsofthe m any-particle shell
m odel.we return to the m ore generalproblem ofcollective excitations builton
the Hartree-Fock ground state ofa Gnite interacting assem bly. In the present
section.the ground state is assum ed to contain only closed shells. Ifthe system
is excited throtlgh the single application of a pne-body num ber-conserving
operator.then the resulting states m ust have a single particle prom oted from
theground state to a highershell.orequivalently,m ustcontain a particle-hole
pair. H encewe m ay hopeto describe the strongly exeited collectiveoscillations
assom egenerallinearcom bination ofparticle-/olestates. Thisprocedure will
now becarried outintwostagesofapproxim ation.
TAM M -DANCOFF APPROXIM ATION (TDA)
(59.2)
where ,
11'.(
),istheexactground state ofthe hamiltonian X in E'q.(56.l8)and
.t
1'?!) is an exactexcited state. The lef'
tside can be rewritten by evaluating the
com m utatorexplicitly:
-''
'
E//),(xtjl= ()
(59.3tz)
,
(59.3:)
(59.3c)
u'
here the norm alordering in the last line again refers to the particle and hole
operators
(59.4)
The firsttwo com m utators are sim ple.but the last one is generally very com plicated.
1More delailed discussions of the application of these llpanl'-body techniques to nuclear
spectroscopy can be found in A .M .Lane,Ioc.cit.,M ,Baranger,loc.cit.,and G .E.Brown,
-'Unihed TheoryofNuclearModels.'qNorth-Holland Publishing Company.Amsterdam.1964.
APPLICATIONS TO FINITE SYSTEM S:THE ATOM IC NUCLEUS
B39
W e startby evaluating Eq.(59.3c)in the Tamm-Dancofr approximation
(TDA),1where the allowed statesin Eq.(59.2)are restricted to the closed-shell
Hartree-Fock ground state
ilI-0 ,= 1
.0 .
!
ax 0'
'-.bar0s.a.O
.z
and a linear com bination ofparticle-hole states
;tj*. vc aln)+1t.j
o.
,1 = 7/+'aj q
azj v',
ln thiscase,the commutatorin Eq.(59.3t-)can be evaluated by retaining only
those term s that yield a'
fbt because aIlthe other operators vanish between the
statesapproximated asin Eqs.(59,5)and (59.6). Thesetermsbecome
where the sym m etry'.pa'P'lzT' =
isreadily evaluated to give
1, p'
/.
t has been used. The com m utator
(/?2-(
Qa
#j1=L
c)t.% t.t.
S'-/,(A- /
îl
z'--sz - A--)
z
.?r'x-t
-
-à
I.i
.Aj*n!ta
'
tj'tj*()..a Vlz
(,
1
p)
. .
TheeigenvaluesEnareobtained by settingthedeterm inantofthecoeë cientsequal
to zero,and thecorresponding solutionsgivethe eigenvectorcoeëcientsb
lljn'.
ln thisexpression,the indices(aj)and (Apt)denote particle-hole pairs,while
the m atrix elem ent?'a,;As represents theparticle-hole interactionpotential.related
totheparticle-particleinteractionby Eq.(59.10). Thequantityeu- c-jisjust
the unperturbed Hartree-Fock excitation energy fora particle-hole pair. Since
the Hartree-Fock energy e. isthe energy ofinteraction with thefilled ground
11.Tamm.J.Phys.(USSR'.9:449(I945)andS.M .Dancofr.Ph5'
s.Re!
'..78:382(1950). These
authors were concerned with lmeson-nucleon tield theory and aolved this problem within a
truncated basis. The approach isa standard one,lpowesrer.and w'as used earlier in many'
othercontexts.
540
APPLICATIONS TO PHYSICAL SYSTEMS
state (Eqs.(56.22)to (56.24)),the role of the particle-hole interaction gEq.
(59.10))istosubtracttl
f theinteractionwiththeunoccupiedstate.
Theexcited statevectorshavealreadybeen given in Eq. (59.6),and wecan
now usetheidentity
.
:
t01L,:t1j10)- 3,,bb.
(59.11)
to evaluate their innerproduct
:
'
tA.
1p
-n,jvyn)= pj/f
apj')4t
a,
n/
k*= j''n,
r
(j9.ja)
xb
Theorthogonality followsbecause (iç
xn)isthenth eigenvectorofa hermitian
matrix. Furthermore, Eq. (59.9) does not determine the normalization of
4t
xnj
',and Eq.(59.12)providestheproperprescription.
Considernextthe problem ofcomputing the transition m atrix elem entof
some m ultipole operator
t- p(c1.
tœIF1/.
?)cb
(59.13)
xb
between oneofthe collective excitationsand theground state. W ithin theTD A.
itagain followsthatonly theterms(/>tin ? contribute to .
?:1*n1t1:1-0):and we
.
.
find
(59.l4)
(59.l5)
Thus the transition m atrix element is a sum of single-particle m atrix elem ents
weightedwiththecoeëcientsp
/zt
a//determinedabove.
W eshalldiscusstheTDA inmoredetailbelow (includingsomeapplications).
butt
irstwe generalize the sam e approach.
RANDOM -PHASE APPROXIMATION (RPA)
A n im proved approxim ation can be obtained by working w ith a m ore general
system of states than that considered in the TD A. In particular.we treat the
ground state and the excited states m ore sym m etrically,allow ing both to have
particle-hole pairs. W ithin this expanded basis,the equations ofm otion are
stillIinearized. Usingthesam edefinitions
t'
a
AjH tzx
fbj
%
lxj= bîax
(59.16)
we retain allm atrix elements ofthe form
'/
Ja
(n;
). yvjc,jt'
x
.
kbjv()j
tpa
fnj'- t.!zn!t'
xj1..
1.
,0)
.
(jq.j.
;a)
(59.I7,)
ApPLICATIO NS TO FINITE SYSTEM S : THE ATOM IC NUCLEUS
541
in the equations ofm otion. This condition meansthatthe excited statescan
be reached either by creating or destroying particle-hole pairs in the ground
state. For historicalreasons this prescription is known as the random -phase
approximation (RPAI.I
The next section presents a physicalpicture of this approxim ation. For
thepresent,however.wejustexaminethematrixelements
,,.
x
1-?)((*.IA
:
zj
)2p'l.
-())-(E1
,-Ev)/a
t'/
'
3
'
(59.18a)
IVl.
-,1LlI,(
''
'l.
-t)'
).= (En- f'
0)(
pt
''
abJi
'
,
xb
(59.18:)
.
ThecommutatorsontheleftsidearewritteninEqs.(59.3). (Notethat(P,Lj)
isobtainedfrom thenegativeadjointofEqs.(59.3).1 lncontrastto theTDA,
we now retain a11terms in Eq.(59.3:
-)thatl'ield eitheraq'bk orba,since this
procedurestillgiveslinearequationsofmotion. Thehrstcontribution(.
:ct/à)'
F)
wasevaluated in the discussion ofthe TD A .the second requires only the single
term
(59.20)
uxbkàvO .
S-bS-s(-;-p.l'aA - -;?-Jzl'A:
z1
and theequationsofmotion(59.18)become
(:E0- (.'u- e-,))- En'
L4a
('
1
j'- D
N'(l.aj;.
ppz,
lk'
?
u
'- 1/aj;vbu(
;''
a'
,a
'J- 0
t(.
E'
o- (ex- 6-,)1- f'
,)(
px
t'b
''- ;
u
.
C(1*x*niàv(rzt'#','-i
-ua*j:A,,4t
&.'1- 0
,
.
(59.22:)
.
which again form a linear hom ogeneous set of algebraic equations. lf the
com plete set of H artree-Fock single-particle wave functions is taken as the
bound statesplusthecontinuum stateswith standing-waveboundaryconditions.
then aIlthe m atrix elements in thisexpression are real2
L'xj;Av=S'
,
1;:AJ,= C'A/
xkaj
(59.23J)
uak:zs= &Apz:aj= l?a
*j:àbz
(59.23:)
,
.D.Bohm and D.Pines.Phvs.:,1...82:625(19jl)(secthecolmnnentsinSec.12). Thcapproach
discussed herew'asoriginallh'developed rortheelectron gasb)rK.Sawada.Phvs.Ret'.-106:3--h
(1957).
2ln ourapplications,theHartree-Fockua&efunctionsareapproximated bythe (bound$stales
ofthe harmonic oscillator. Fora discussion oftransitionsto the continuunnstalessee.t-oT
exam ple,F.Villars and M .S.W'eiss.Phys.f-erpcrs',11:318 (1964)and J.Friar,Phbns.Sc:.,
C1:40 (19703.
5*2
APPUCATtONS TO PHYStCAL SYSTEMS
In this case, the solutions corrèsponding to two diflkrent eigenvalues with
En- Eu> 0 satisfy the following orthogonality relation
; j4l
Xnq'l#4(
(
Z:
f)- p@
XF
')*pt
7n/lj= 3XX,
œj
(j9.g4)
Equation(59.24)alsoimpliesaparticularnormalization;thejustihcationofthis
choiceisrathersubtle,however,and now willbetreated in detail.l
WcfrstnotethatEqs,(59.22)and(59.24)côrrectlyreducetoEqs.(59.9)
and(59.12)asfpç
xn;->.0. A morecompellingargumentforEq.(59.24)canbe
found within theframework ofthequasiboson approxim ation.l So farwehave
approxim ated onlythematrix elem ents.butthenorm alization ofthecoeë cients
requiresfurtherrestrictions. D efnethe operator
Q-â- )(('
/,a
tn,')1,-vu
çn)Ljl
ub
(59.25)
ThenthecommutatorwithX yields
E4-p-ll'
-'-(En-e())Q1
(59.26)
where wehave used the linearization prescription thatthe only term s retained
ontheleftsi4eshouldbeproportionaltolAort'(Eqs.(59.3)and(59.20))along
withtheequationsof-motion(59.22)and(59.23). Assumenow thatEq.(59.26)
holdsasanoperatoridentity,andthattheinteractingground statecanbedesned
by the condition
Q-1(
k1'())= 0
alln
(59.27)
Thecollectiveexcitationscan beconstructed as
l'Fal- ()11Y-(
))
(59.2%
which isaneigenstateofP becauseofEq.(59.26). These eigenstatesm ustbe
orthonorm al:
.
r
t.l,
-,,'1.I.
',
-)-3,-,.
-t.1'
-(,r(:-,,p1)l'I,
-(
'
))
(59.29)
SubstitutionofthedehningrelationofEq.(59.25)thenleadstothenormalization
condition ofEq.(59.24)provided thatthereplacement
(Lj,1'
1p,1='
=3.Abbb,
(59.30)
1M .Baranger,loc.cit.
2Notethatthequasibosonsdiscussed here areformed from particle-holepairs;in Eq.(58.57),
theboson operatorscontain pairsofparticles.
Appl-lcATloNs To FINITE SYSTEM S : THE ATOM IC NUCLEUS
KA.Q
is perm issible. A direct evaluation of the com m utator yields the following
necessary conditionsforthequasiboson description
IYIS'IIlt
/yay1Y'o)-t 1
(kl'ck?/Yb'
/'1:1P0).
c
.
:1
(59.31J)
(59.31:)
Tointerprettheseconditions,weassumethatthetruegroundstatetV%)consists
mainlyoftheparticle-holevacuum k0)plussmalladmixturesofotherparticlehole states with Jn = 0+. Since the num ber ofparticles m ustequalthe num ber
ofholes,Eq.(59.31)reallyassumesthatNKIN <:
<1whereN isthetotalnum berof
ElledcorestatesandNhisthemeannumberofholesin lN%),forthentheprobability ofsnding a hole in the state y is
(k1% b%b 1:F );k$Nb< l
y? 0 N
(59.32)
--
Thus ifthe ground state doesnotcontain too m any holes,the norm alization
condition can be w ritten consistently as
t'l'cI(Qa,
::-n
1)I
1.F0,
h- )(g4t
an
j''*$ç
(
:n
jt--pt
a'j
'''*wt
apj
'')-3nz.
(59.33)
The transition matrix element of a multipole operator from one ofthe
excited statesto the ground state can be im mediately evaluated. W ithin the
RPA.onlythosetermsof?containingt
/Aforbawillcontributeto(H'al/(
kl%),
and we find
t-.
-I
('-arF(-,)5--j)1,+'t-,rFl=)s-bL,1
xb
(59.34)
(1l'
-nI1-1
'i-0)- 7)5'-,E(alF1-#)l
/za
t/j+(-#1Fla)tpt
zaj')
(59.35)
.
Equation(59.35)isagainasum ofsingle-particlematrixelementsweighted with
thecoemcients(/t
œn
j),wt
an
jl)determinedabove.
RED UCTION O F THE BASIS
The dimension of-the matrix equations(59.22)can be reduced considerably by
noting thatJ is a good quantum num ber. W e use therotationalinvariance of-
thesystem toassumethattal= (nljmj;anddesne
3%abJM4- Z (&mz-ibrnj1.
&.&JM )Dj
@l@mn
(59.36)
w hich isan irreducibletensoroperatorofrankJ.
Incontrast,t(abJM 4isnot
.
anirreducibletensoroperator,althoughwehavealreadyseen that(-1)JY-mœJ-z
EAœ
APPLICATIONS T0 PHYSICAL SYSTEM S
and (-1)J#-''
:5-paretensoroperators. Thisdefectiseasily remedied with the
identity
Hxmx-àmbkhiôlsis- (-1)Jœ+*-J(A - mx-ib- mbl.hibl- Af) (59.37)
m u + m b= M
whichshowsthat%((abJ-M )isatensoroperatorofrankJwhere
SJM (-l)J-M
(59.38)
Thuswefurtherdefne
hnblabqe ('l3MIêXeJM)l'l%)
Z iixmxjnmpljxà-lhf?s/a
t$
mlmp
+!?'(Jd9H&('l3-Il(J1J-M)l
'l%)
'
-
(59.39/)
% Z (-/amx.
/,mbIh .
&.
/- M )'Pç
xnb
'
(59.39:)
Mœmn
which are independentofM by the W igner-Eckarttheorem . In thisway,the
generalmultipole operatorin Eq.(59.34)can be written with the aid ofthe
W igner-Eckartthcorem as
% M 'zk
1
a
u )tujjsjy)nj
fs-bqlxFAla.%vjl.
/a.
&.7'
(u .
j.j)1.Y
a#
+ su5'-#r.
&-'nj.&,--mxïjbhlMs'
:?':
(r,Lf')Ljl
-'-
(u .1 l)+X ttJl
I
F,l
I
:)t,(ayJu l
a.
and itfollowsthat
'
::131
!T'
,11
:1%)- Z f(cIF,l
1O 'hnblabt+ (-1P -J*-'(:1IrgI!
c)Jqnblabjk (59.41)
Thebasisforthelinearequations(59.22)cannow bereduced bysumming
withtheClebsch-Gordancoeëcientsin Eqs.(59.39). Forasphericalsystem,
e.- e-j= ea- evand thehrsttermsofEq.(59.22)immediately yield (Gnblab)
andçijnblabj. Theinteractiontermsaremorecomplicated. Sincetheinteraction
potentialisinvariantunderrotations(compareEq.(58.70)),wemaywrite
x L;lbJ'@Fl?ncJ')- (-l)J,+J--J'(IbJ'!P'1/vJ')) (59.42/)
AppulcATloNs To FINITE SYSTEM S : THE ATOM IC NUCLEUS
r846
(
1( X-jS-vk
'
,X -r#lsjs-#A7j;j(
z/j7'M ')(
tl'
zmàjxrz7.(/Ajx-l''M ')
.
.
(59.42:)
In these expressions (ab) and (Im) denote the remaining quantum numbers
identifying the particle-hole pair, for example (abjEe(nalajasnvlbjvq. W hen
ë..j;Ayissum med withtheClebsch-Gordan coemcientin Eq. (59.39J),weusethe
relatlons
-
$' k'./Xmœ.
& mbIjxJ'
j-/.
lz;t.
/AmnJ'
j-p;jlhy-;'A:
f')
x-M '
mqm#
,
(-jj1'
-J
'
l-Jbj'g.jy
.
.
.
j.
yy
'ty
.
s
j j.
xqgs-m..
lxmxp.
/pJu ?)-j
.
=
(59.43)
(59.44)
j- J'
'
/1 J L(IDJ';IzramJ')
.
-
(-j)jo+J'
m'
:J'k
(jylrjFtt'
nalrjj (59.45)
The overallm inussign com es from interchanging the order ofthe two matrix
elementsin Eq.(59.42/).
The sum oftlf
ybkàv with the Clebsch-G ordan coeë cientin Eq. (59.39/)is
im m edi
ately carried outwith the help ofthe following identity obtained from
Eq.(59.43)
Z (JkmxX mb.
1
X 7,.
/* )Ih P?Ah Mlz.1./,4jxJ'M ')
mamp M '
Com biningthephaseswiththefactorS- v in Eq.(59.42:),we have
SAS-!
.
I(-j)Ja+#l+J'= (-j)#a+#I+J'(-jIJ-M(-1)Jm-Jt-J
:46
APPUICATIONS TO PHYSICAL SYSTEM S
Therequired sum istherefore given by
(59.47)
where
ul
ab;jaja (-.j)lm-.
/1-Jj.
Jtpi?F1l
(j9.
4g)
N ote the interchange ofthe indicesIand ?)?on the rightside ofthis relation.
The recoupling relations(59.44)and (59.47)now makeitpossible to rewrite
thelinearEqs.(59.22)as
lEfc+ (Ea- e,)1- fnl'
lqntlabbv Z
:LJ,:l,,b
llqnbllm)+ uibilmvqntllm)?-0
lm
(59.494)
(59.49:)
Thiscalculation explicitly decouples the states of diflkrentJ and dem onstrates
that the eigenvalues and eigenvectors are independent of ,
j.
I. The TD A is
recovered from Eqs.(59.41)and(59.49)by setting(ptn'= 0.
The equations to this pointare entirely generaland apply to atom saswell
as nuclei.l In the nuclearcase,however,the invariance ofthe interaction under
rotations in isotopic-spin space can be used to reduce the basis further. W ith
ttxl=fnljmj;Jrnr)itisclearthatwemustmerelyincludeanisotopic-spincoupling coeë cientand phasefactoralong with every angular-m om entcoupling co-
eëcientand phasefactor. From Eq.(59.41)we immediately obtain
r'
t
tliTiitgviEk1.
-0)
= 5
-tft
't
zE!TJviib)s/!J.
/(J/?)+ (-1)1-1-r(-1)#.-Ja-JLbiiTJ.
yEEaj(
ylj
yn
ylttzyll
J;
(59.50)
wherethe sym bol EEdenotesa reduced m atrixelem entwith respectto both angu-
larmomentum andisotopicspin. Thecoemcientskqn
gj,(
pj
r
n:
/inthisrelationsatisfy
thesame setoflinearequations(59.49)with
vl
r;,m
a,
- -
Jjm
5- (2J'+ 1)(2F'+ 1)jy,
Jx
,
J't''
l' !'
Jjj j
uJT
i'-F f
J*;lm = /
N- 1
'1
I1-'
%- 1
*L
JJm-JI-J ,
t.JT
ab$/?y1
lForanapplicationtoatomsseeP.L.AltickandA.E.Glassgold,Phys.Ret'..133:,*632(1964).
s47
sot-u-rloN FOR THE (ISI-DIM ENSIONAL SUPERM ULTIPLET W ITH A 3(x) FORCE
Jn this section the TDA and RPA equations are solved for a specifc m odel
n here theinterparticle forceistaken to be independentofspin and isotopicspin.
The ground state of the nucleus is assumed to consist ofclosed shells and to
have the quantum num bers .
% = L = F = (). This m odel illustrates som e very
interesting sjstem atic features of the particle-hole calculations. The Hartree-
Fock single-particleenergiesnow depend only on (n//,and thesingle-particle
levels can be characterized by
because b0th m :and n:sare good quantum num bersforspin-independentforces.
Thegeneralresultsderived abovearewritten ln a-/-jcoupling schemeappropriate
t'
oractualnucleiwherethesingle-particlespin-orbitforcetsimportant. Although
we could explicitll'transform to an L-S coupling schem e appropriate for the
sim plespin-independentm odel.itism uch m ore convenientto repeatthe previous
analysisfrom the beginning.starting with a sllghtly diferentcanonicaltransform ation
(59.53)
(59.55)
Since the interaction and single-particle energies are assum ed independent of
spin and isotopic spin,the basis can be reduced by introducing states of given
L.S.and F,forexam ple.
blLn
àvlab)H N' .
'Ia?A7k./,?37,;
!'
.IaIvL.
'
S:
JJ. .Jrn,zj.
msî!.!.5'-$'
f,
(alkrrl's?
x
l'
lziral'
v'r,'
i'l'rk/r..
,'
)ç
xn) (59.56)
Furtherm ore.the spin and isotopic-spin dependence ofthe interaction m atrix
elements factors.z Svhen Eq.(j9,55) is summed with the Clebsch-Gordan
coemcientsin Eq.(55.56).thespinand isospincoeëcientsmaybetakenthrough
the second term ofthe interaction and applied directly to the w'
ave function.
'Note thatBt
orj(ï - lx - !.and Bt
orja =lœ - .
i.so thatthe!wo operators
œ = b%
X f
tz= -b%
q f
merely diflkrb5anJ-dependentphase.
2Thisistheadvantage ofthecanonicaltransformation in Eqs.(59.53)and (59.54).
s48
APPLICATIONS TO PHYSICAL SYSTEM S
Inthehrstterm oftheinteraction,thespindependence(andsimilarlytheisospin
dependence)isgivenby
3mzA.-mJp 3*1:#.-mzl(-1)1-?
Nœ(-1)1-'NA
=('
vZ112(1.
ra,zj.
msjt!.!.00)(!.
??:,A!.
rnsj
,I!.!.00)
which contributes only for S = r = 0. The /dependence of the interaction
m atrix elem entsistreated exactly asin theprevioussection. Thusthe reduced
TDA equationsbecome(con,pareEq.(59.45))
(Ea- eb- en)SLn.
lrlabs+ Z L'
)l;
Q I'Lnîrçlm)= 0
fm
(59.57/)
Im It f,
,
I , (($lbL IP'1/almL )
b a L
vu
Tl
qbi
m = - N=.;x
tgL'+ j) /
L'
-
43.s: 3w(
j(-1)lb+l'-l-'(Ib/1L'lIZ.
1/t,/mf,'/
') (59.57:)
where the order ofthe two potentialmatrix elements in Eq.(59.55)has been
interchanged. In these equations,the labels (ab) and (lm) again denote the
remaining particle-holequantum numbers,forexample,(ab)H (ngla,n,/,).
Thequantum numbers.5'andFinEq.(59.57)arisefrom couplingthespins
and isotopic spinsofa particle-hole pairand can take the valueszero and one.
W e shallsrstconcentrate on the stateswhere eitherS or r diflkrsfrom zero;
theparticle-holeinteraction(59.57:)isthen particularlysimplebecausethelast
Table 69.1 Degenerate states of the
(16)-dimensional spin-isotopic spin
superm ultiplets
N
l
I
I
)
L
L - 1,LsL + 1
l
term proportional to 3sot
s
iwc vanishes. As the remaining interaction is in-
dependentof S orF,weconcludethatthe l5 spin and isotopic spin statesin
Table 59.1alllie atthesam eenergy. Thesestatestherefore form a degenerate
supermultiplet(the(15)supermultiplet)and mustbecombinedwiththenstates
ofgiven L obtained by diagonalizing the matrix in Eq.(59.57/). Thevaluesof
J in each degenerate supermultiplet(obtained by coupling L and 5') are also
shown in Table59.l. Thesinglestatewith .
&= F = 0foreachn and L isclearly
APPLICATIO NS TO FINITE SYSTEM S : THE ATOM IC NUCLEUS
649
splitofl-from theotherstatesbytheinteraction in Eq.(59.57:),and we shall
return to itatthe end ofthis section.l
Theproblem can be furtherreduced by assum ingthattheparticle-particle
interactionpotentialisshortrangeandattractive(1 >0)
P'= -g3txl- xc)
(59.58)
In thiscase,the integration overdbxzin the m atrix elem entcan be perform ed
im m ediately. Furtherm ore,the identityz
Z (I3r#71/2m2!
./1/2ZM ) Yl,,?
,,(f2)Fl:ma(D)
m l*t2
*q2lj+ lj
)é2j+l)1'II Iz L y- (j.
=(-l)g,-tz(
j;
m
.
..
-- .-
j((
)(
;()z.
sf
leads to the result
N
Fnumb?Fl!mm
((r?JlIbrn:IIt&L'M ''
)(
LlavlaIm?.
?k,
a;IgfmL'M ')
x
(0
b0
/
,0
f-1
/y
(0
la0
lmZ
0)
The recoupling relation3(note thateach ofthe ?-j sym bolsvanishesunlessthe
sum oftheJ'siseven)
é.
u
((21,sj)t
f/
l,
ml
/a
,z
L.
,)q
'
b O
0
1',
j(.
I
0
a
=(-1) (0
/
,0
'
Z/
0
,jq0
b 0
1.
,)
'
l,+z,.t.
thenallowsustorewritetheparticle-holeinteractionfortheglsl-supermultiplet
inthevery sim pleform
LV!.ii= fLfyL'k
(59.59/)
LjyH(-l)l
.((2/a+.1)(2/
p+1))10
C V
0j
(59.59:)
.
lIftheforcesareindependentofspinand isotopicspin, thehamiltonianisinvariantunderthe
symmetrygroupS&(4),whi
ch isW igner'ssupermultipl
ettheory(E.P.W igner,Phys.Reth..
51:106 (1937)). The degenerate particle-hole states then form bases fbr the irreducible
representations4(
& i= I1)(
.
9 (15).whichisexplicitlyillustratedinthepresentcalculations.
7A.R.Edmonds,op.cit..eq. (4.
6.5).
3A , R .Edmonds.op.cl
'
?..eq.(6.
2.6).
APPLICATIONS TO PHYSICAL SYSTEM S
t?.
% = (6l
Icz.l
I/,)
(59.59$
where
f- # *un,1,unst.un t.un dr
4.
n
o
g
m
l,ar
.
-i'
(59.60)
Inthisexpressionunllristheradialwavefunction (see.forexample,Eq.(57.6)).
Notethatthet'
jyarejustthereducedmatrixelementsofthesphericalharmonics
cuM = L4=I(lL + 1)1+Yuu. Now the radialwave functionsare peaked atthe
surfaceofthenucleusforparticlesinthesrstfew unoccupied shellsandforholes
inthelastfew flledshells,andtheoverlapintegral(willnotchangeappreciably
from oneparticle-holepairtothenext.l Weshallthereforeassumethatfisa
positive,state-independentconstant. In thiscase the particle-hole interaction
isseparable. TheresultingTDA equations(here,%b= e.- e,)
(e.,- 6n)/(
(:))z.(J& -1-ftl, Z
?k /t
(:)2z-(/r?;) - 0
lal
(59.6l)
canthenbemultipliedbyva
Lvlleav- en)andsummedoverabtoyieldtheeigenvalue
equation
1
(t,1,.
)2
f J: 'n- ea.
(59.62)
Thesolutionsto thisequation foragivenL areindicated graphicallyinFig.59.l.
Ifn particle-hole statesare included,then n - lofthe eigenvalues lie between
l
f
Fig. 59.1 Pl
ot of )((&'
jJ2(6- 6.,)-1 as a
*ab
function of e with the eigenvalues ea of Eq.
(59.62) indicated bycrosses.
adjacentunperturbed energieseav,and oneeigenvalue eyopispushed up to an
arbitrarilyhighenergydependingonthequantity l/f.
Toinvestigatefurtherthepropertiesofthesestates,wem akethesim plifying
assum ption thatalltheparticle-holestatesareinitially degenerate
fa,= fa - k'b X C0
lSeeH.Noya,A.Arima.and H .Horie,Suppl.Prog.Theor.Phys.çhr
yotols8:33(1959),Table2,
foran explicitevaluation ofthese matrix elements. They are.in fact,remarkably'constant.
APPLICATIONS TO FINITE SYSTEM S:THE ATOM IC NUCLEUS
651
In thiscase,theenergyofthetop superm ultipletbecom es
(59.63)
t'top= tb+ fZ
(X22
ab
while Eq.(59.61)showsthat/l7)(J:)ccpj,forthese states. Thenormalized
w avefunction istherefore
/l1l (JO 1s
r). +
Z (XJ2
(59-64)
Fortheothern - 1eigenvalues,Fig.59.1showsthat
n -o = 0
othern- 1eigenvalues
(59.65)
and itthereforefollowsforeachofthesesuperm ultipletsthat
)
(!7f,
a/j1))z,
(/ra)=0
1/$
othern- 1supermultiplets
(59.66)
W e m ay use these resultsto evaluate the m atrix elem entsofthe various
m ultipolesofthechargedensity
A
QL.
M - J=l
Z r.
r-cz--(f1,)!.(1+ r3(./)1
(59.67)
between the ground state with F = S = L = 0 and these excited m odes. From
Eq.(59.14),thetransitionmatrixelementsofageneralirreducibletensoroperator
aregiven by
('l'a1/I'l%)- x7b2(œIF1-/)+-b'
lt
zs
(59.68)
ThespinmatrixelementsagainfactorbecauseQuM isspinindependent,andthe
m ssum m ation
Z 3,,
u.,-,,
s(-1)++'N:/2'ms
o- v'1$L.
n
-bz
@I&amsp
eliminatesallbuttheS = 0excitedstates. '
W ithin the(151supermultiplet,the
stateswith S= 0musthaveT = 1(seeTable59.1),and onlythe'r)term in Eq.
(59.67)contributesto thetransition matrix element. Since (!.1l
!.T1lè)= !.V6,
itfollowsthat(compareEq.(59.50))
(klczlslz-EEQslE'l%l-A/'J3sn7
)(&lIcz.E
l/,)/t
(:):z-(JO (rLfab
ab
.
(59.69)
Ifweagain assum ethattheradialm atrixelem entsareindependentofthestawzs
(ab),thereducedrhatrixelementcanbesimplisedto
(klczlsqz-i!P-sIE:F())'- VJ&sulr'.s )
JL'
),/t
(:))s(JO
ab
(59.70)
Oa
APPLICATIONS TO PHYSICAL SYSTEMS
In thespecialcasethattheinitialparticle-holestatesaredegenerate,thisexpres-
sionbecomes(seeEqs.(59.64)and(59.66))
+
()jz(
1o
ls
p)z..
:jg'
-L.
::ljr()j= V'g
-jsejrLj y tsab
z.42
(59.71/)
(VI'(
''
151Li*
:QL:
.'
:%'0)=0
(59.71:)
..
u,
othern- 1supermultiplets
Foreach Z,we 5nd the following rem arkable result. The one superm ultiplet
that is pushed up in energy carriesthe entire transition strength, whereas the
rem ainingn - 1degeneratesuperm ultipletshaveno transition strength whateyer.
Itis an experimentalfact that low-energy photoabsorption in nucleiis
dom inated by the giant resonance. This state is a ftw M tV wide, and its
excitation energy decreases systematically from about 25 M eV in the lightest
nucleito 10 M eV in the heaviestnuclei. Thedom inantm ultipoleforphotons
ofthisenergyiselectricdipole,and the giantresonance also systematically ex-
hauststhedipolesum rule(seeProb.15.17). Theelectricdipoleoperatoris1
z:
QlM = #=l
Z (3/4=)1xIM(./)JT3(.
j)
Thusthe excited state m usthave S = 0,F = 1,and Ln= Jn= 1- ifthe ground
statehasS = F = 0 andLn= Jn= 0+.
The sim plest picture of the giant dipole resonance, due to Goldhaber
and Teller,2isthat'theprotonsoscillateagainsttheneutrons. Them oresophisticated m odelpresented herewasoriginally proposed by Brown and Bolsterli.(
j
The observed giantdipole resonance m ay be identihed with the T = 1, S = 0
state ofthe top L = 1supermultipletand doesindeed exhibitmany featuresof
thism odel. Itappearsatanenergy higherthan the unperturbed consguration
energieseavobtained from neighboring nucleiand carriesallthedipoletransition
strength.
ForL = 1,the presentschematic modelpredictsa supermultlpletofthese
giant resonance states. The m odelalso predicts other giant-resonance superm ultiplets,one foreach allowed Z. In He4.the simplestclosed-shellnucleus
!Notethattheirst(isoscalar)term inEq.(59.67)fortheelK tricdipoleoperatorisproportional
to
A
â x(7)
Jal
which isjustthepositionofthecenterofmass. Hencethisterm cannotgiverisetointernal
excitation ofthesystem.
2M .GoldhaG rand E.Teller,Phys.Rev.,74:1046 (1948).
3G .E.BrownandM .Bolsterli,Phys. Rev.Letters,3:472(1959). (Theseauthorswereprimarily
concerned withtheF = l.J'
'= 1-excitedstatesofr = 0,J'
'
t= 0+nuclei.) %xealso G.E.Brown,
op.cfl.,p.29.
APPLICATIONS TO FINITE SYSTEM S:THE ATOM IC NUCLEUS
5B3
withfournucleonsin thelsoscillatorshell.theflrstnegative-parityexcited states
are expected to belong to the (1p)(1:)-.particle-holeconfguration with L = I.
A11the states ofthe (15)supermultiplet builton thisconhguration have been
experimentallyidentified.l In addition,thereisevidencefrom inelasticelectron
scattering thatthe F= 1,S= l,Jn= l-and 2-componentsoftheZ = 1,(15J
superm ultipletof
-giantresonancesarealso presentinbotb C i2and O '62
Beforeconsideringthe.
S'= 0,r = 0excitedstates,weinvestigatethesesam e
(15)supermultipletswithintheframeworkoftheRPA. Thelinearizedequations
aregiven by Eqs.(59.22)and(59.23),and thebasiscanagainbereducedbyintroducingstatesofdesniteF,S,andL. Justasbefore,theûrstterm inEq.(59.21)will
not contribute if either S or F diFers from zero, and the C lebsch-G ordan
coeë cientsforspinand isotopicspin maybetakenthroughthesecondinteraction
term ontothewavefunction. ThereductionoftheIpartsofthem atrixelements
proceedsexactlyasin thepreviousdiscussion. Thustheinteractioninthe(15)
supermultipletbecomes(compareEq.(59.48))
r'
sl
,
l
.li- Jr),h'k
&a
(:
15
;l
)m
L= (t%c
L,
15
:,
)
?
,
Ll(-j)lm-l!-L(-j).
%+T
(59.72/)
(-j)L+S+F(va
Lbt
yz
hl
Lm
(59.72:)
=
W ith theassumption o1-aconstantJ,thepotentialisagain separable,and the
RPA equations can be written3
(Ec,- E,,
)4t
(:)9z.
(c&)+ fra
''
, 5-tl
t'k k't
J)!,,
(/?'
n)+ (-1)L+:+''cfmfpt:s
'lst/?''ll - 0
(59.73/)
(%,+ %)g'g
t:s
'lt-lab)+ êra
'.
b 5-E'.
'
k J'
f
çks
bjllms-i
-(-1)L+1'+''1.
p,,/(
1
t7s
'3,.(/??'
))J - 0
(59.73:)
Theeigenvalueequation followson multiplying by ru
f'
t
p/tea,.
-F5ea)and summing
over(ab)
1
- =
uc
l
j- aYb (L,) 6n-
1
eu,- en+ %b
(59.74)
This equation isexplicitly sym m etric in en,and the excitation energiesare the
solutionsforpositiveen(indicated graphically in Fig.59.2). Justasin theTDA,
1W . E.MeyerhofandT.A.Tombrello,Nucl.Phys.,A109:1(1968).
2T.deForestand J.D.W alecka.Electron Scattering and NuclearStructure,in Advan.Phys.,
15:1(1966).
3The explicitphase dependenceon S and F can beeliminated by considering the amplitudes
4and(-l)1'+S+T'@.whichdeterminethephysicalquantitiesofinterest(Eqs.(59.74)and(59.78)J.
554
APPLICATIONS TO PHYSICAL SYSTEM S
Z
f
f
fa,
Fig.59.2 Plotof)g (pjb)2((e- :.)-1- (6+ :jo)-1)asa function of e
with thepositiveeigenvaluesqnindicated by crosses.
n- 1eigenvaluesaretrapped betweentheadjacentvalueseu,andoneispushed
up,although notasfarasin theTD A .
Thesolutionsforpositive enagain sim plifyconsiderably in thedegenerate
casewhere%.= o . Thehighesteigenvalueisgiven by
13
1
top==6: 1-r-ee
-- (ps.,)z
6
(59.75)
Forthissupermultipletitfollowsfrom theRPA equationsthat/at,and +avare
both proportionalto sa
Lb,while (Jav/çpat,= (-1)L+:e'T(e()+ e)/(o - e). W ith the
normalizationofEq.(59.24),thewavefunctionbecomes
bsa
tb
t'o - e'top
+l73s(J:)- (-1)1,+.
5+'
r-'$ (L 2 . 2(..lOp,(,)i
LJ
/1ljhLlab)-
(59.76J)
E,f-
ab foY-ftop
Z ( y ztftop
'eo).
;
L122
ab
(59.76:)
.
The othern - 1eigenvalues are unaltered,and itfollowsthat
e.= o
(59.77/)
+j:)jz.
(c??)= 0
othern- 1supermultiplets
Z
r
l
p
/t
(
1
)
1
L
l
a
bj
=
0
ab
(59.77:)
(59.77c)
Thetransition m ultipole m atrix elem entsfrom theground stateareobtained
from Eq.(59.35)with S-;replacedby R -j. ForthechargemultipolesofEq.
(59.67),itisagaintruethatonlytheS= 0,F = 1excited statescontribute,and
we:nd (compareEq.(59.50)1
tkl'zlslz.i!QuEiY%)= V'j3s0a)b('
Lrtb
,a,IL'):/l1)jtlab)+(-l)L+&+'
r
'
>:ra
'7+l:)),,
(tz?')) (59.78)
APPLICATIONS TO FINITE SYSTEM S :THE ATOM IC NUCLEUS
555
ln thecase ofdegenerate unperturbed consguration energiesand radialm atrix
elementsindependentof(ab),thisexpression becomes
e
-
1
'
t
êtyE
tl
os
plL.
:.
:QL::tyo). V'?jsoj?.Lj et0
y
(j)
t.)2
ab
op
ab
1
. ..
tVl''
(
'
l5)L'
:'
:QL'
:'
:Vl'0.
b-0
k
(59.79)
othern- lsupermultiplets
Hencethe transition probabilityisreduced bya factor(ec/e)1 from the TDA
resultfor the corresponding states. Thisrepresents an im provementbecause
the single particle-hole TDA calculation generally puts too m uch strength in
the top superm ultiplet-l
Finally,wereturntotheF= 0,S= 0states(the(1)supermultiplets),where
theparticle-holeinteractionhasadditionalterms. ltisclearfrom Eq.(59.57:)
and thecalculationleadingupto Eq.(59.59J)thattheparticle-holeinteraction
is again separable for a delta-function potential. In fact,the relevant m atrix
elem entsare given by
r5111,,= -3LQ!
.lm
(59.80J)
l/5l,
!,r
,,- -3/5
1!
.1m
(59.80:)
andthereforechangeslknforthe5-= F= 0states. Allthecalculationsproceed
exactlyasbefore,zbutwith a new J'thatisnegative,('= -3J. Figures59.1
and 59.2 im ply thatthe collective levelis now pushed down in energy and is
pusïjzdfurtherintheRPA thanintheTDA. Inaddition,theS= F= 0collectivestate hasa Iargertransition m atrix elem entto theground state in the RPA
because(o/e)1> 1,and thusbecomesmore collectivethan in theTDA.3 Both
ofthesefeaturesoftheR PA generallyim provethecom parison withexperim ents.
AN A PPLICATION TO N UCLEI: O1:
To illustratetheapplicationofthistheoryto arealphysicalsystem ,wetum m arize
acalculation intheTDA oftheF= 1negativeparitystatesof016.1 Therelatively strong spin-orbit force in nuclei m akes it appropriate to return to the
single-particlestateslabeled with(n#). Theground stateofO16isassumedto
form aclosedp shell,andallnegativeparityconsgurationsobtainedbyprom oting
lThe RPA can beshownto preservecertainenergy-weighted sum rules;seeProb.15.16.
2Forthe(1Jsupermultiplet,onlytheisoscalarpartofthechargemultipolescontributetothe
transition matrix elemenls,and therightside ofEqs.(59.69)to (59.71)and (59.78)and (59.79)
mustbemultipliedby1/$.
'
,
/1.
3Foran illustration ofthesepoints,seeV.Gilletand M .A.M elkanofl Phys.Rev.,133:B1190
(1964).
tThesrstcalculationsofthistypeforO16werecarriedoutbyJ.P.ElliottandB.H.Flowers,
Proc.Roy.Soc.(Londonq,A242:57(1957)and G.E.Brown,L.Castillejo,and J.A.Evans,
Nucl.Phys-,22:1(1961). The presentcalculation isdueto T.W .Donnellyand G.E.W alker,
Ann.Phys.(N.F.),(to be published).
:se
AppujcArloss To PHYSICAL SYSTEMS
particlesfrom the 1p oscillatorshellto thenext2,-1#oscillatorshellareretained
(see Fig.57.3). The resulting particle-hole states are listed in Table 59.2,
togetherwith thepossiblevaluesofJ'. Table59.2 also givestheH artree-Fock
Table 69.2 Neqative-parity partiole-hole eonfiguration: In O16
Conhguratîons o - e.,MeV
States
'
(2a.
1.
)(1.
:.
/-1
(1d1)(1#J-'
(1d+)(1#:)-1
(2.
h)(1#J-:
(1Jp (1pJ-'
(lt
%)(1.
:.
/-.
18.55
17.68
22.76
12.39
11.52
16.*
1-#21-'2-;3->40-#1-'2-# 30- 12-931-,2.
Source:T.W .Donnelly and G.E.W alker, Ann.Phys.
(N.K ),(tobepublished).
confkuration energiesea- % ofthesestates(seeEqs.(56.22)to(56.24)),which
are obtained from neighboringnucleiwith an extra neutron particleorhole.
The particle-hole interaction is computed from Eq.(59.51J). In this
calculation the nucleon-nucleon potential F is taken to be ofa nonsingular
: in M eV
-
gj3'
2'
1'
'
2
j-
1.#..2..
. >.
1,3,3.4
()-:1-91-&3
1% :
A
10
29
*)-
Fig.69.3 T = 1 spectrum of O!6 computed as
descri% d in the text. Also shown is the supermultipletstructure obtained when thespin-dexn-
Calcuiate
Sum rpultiplet
spectrum
approxlm atlon
dentforcesareturned ofr. (Theauthorswish to
thankG .E.W alkerforpreparing thisfigure.)
APPLICATIONS TO FINITE SYSTEM S:THE ATOM IC NUCLEUS
s:7
Y ukawa form with Serber exchange;it is lit to low-energy nucleon-nucleon
scattering:
F(1,2)= (1F(rlz)1# + 3F(rjz)3#)j.(j+ /u(1,2))
3P = $(1- 11.qz) 3P = 13+ 11.ec)
e-Ht'lz
1'(rl2)= Farzrla
lN = -46.87M eV
3L = -52.13MeV
V
-6
(59.81)
1/:,= 0.8547F'l
3y.= 0.7261F-1
e= 135*
ki= 124 MeV
4.
l 6x10
t;
T
3-
r
f
q<
Q
22
2-
a-
3A/i/@é
t
# jj
4 '2i
1- j
#*<: .:il/sA:j
3 i
j- 2- #k.%9:: z
j2- 1. j#i
>
30
28
26
24
22
ao
18
16
14
12
lp
e(M eV)
Fig.69.4 Exm rimentalslxctrum ofelKtronswith incidentenergyet= 224 MeV
inelasticaily scattered at6= 1359asafunctionofnuclearexcitation energy k.The
calculated spedrum using the states of Fig.59.3 isalso shown (with arbitrary
overallnormalization;the integrated areasforthe variouscomplexesagree with
the theory to approximately afactorof2). Thesolid lineisa calculation ofthe
nonresonantbackground above the threshold for nucleon emission. IFrom
1.Sick,E.B.Hughes,T.W .Donnelly,J.D.W alcka,andG.E.W alker,Phys.Aep.
Letters,23:1117(1969)andT.W .DonnellyandG.E.W alker,Ann.Phys.(N.F.),
(to be published). Reprinted by xrmission oftheauthorsand the Amerieqn
InstituteofPhysics.)
Theharmonic-oscillatorwavefunctionsofEq.(57.6)areused asapproximate
Hartree-Fock wavefunctionsin computing matrix elem ents,and the oscillator
parameterb = 1.77 x 10-13cm isdetermined bysttingelasticelectron scattering.
The calculated spectrum isshown in Fig.59.3. These collective statesmay be
seen,for example,through transitionsexcited by inelastic electron scattering,
and Fig.59.4 com paresan experimentalspectrum with the predicted location
andrelativeintensitiesofthepeaks. Thetransitionmatrixelementsarecomputed
usingthecoeëcients#ù*?(* )obtainedintheTDA calculation EseeEq.(59.50(1.
Thecalculated m ak locationsareconsistently aboutan M eV too high;furthermore,the experimentallevelsarequitebroad aboveparticle-em issionthreshold
:%
AppulcA-rloNs To piiyslcAu sysTEv s
(;
>19 MeV inthepresen'
tcase)so thattheuseofsimple bound states isinadequate. Nevertheless,thegeneralagreementisverygood.
In accord with our previous modelcalculation,wc also investigate the
supermultiplet structure of these states. The appropriate particle-hole con-
hgurations(aa&)(ru/J-lare(2J)(1,)-land(1#)(1X-iwithL = 1(twice),2and
3. Thesuperm ultipletswillbedegenerateonlyforspin-independentforces. To
uncoverthisstructure,wehrstçkturn ofl-''thesingle-particlespin-orbitforce by
assumingthattheobserved (n#)stateshavebeen shifted from an originalfnl)
conhguration accordingto Eqs.(57.14)to (57.16).and wethtn retainonlythe
spin-independentpartoftheinteraction in Eq.(59.81). The resulting F= l
membersofthe(15)supermultipletsareshowninFig.59.3. Thesupermultiplet
approxim ation clearly provides only a qualitative description of the actual
spectrum .
6X EXCITED STATES : G REEN'S FUNCTIO N M ETHO DSI
In thissection,we again study the collective excitations ofa fnite interacting
assem bly,butwenow apply the m oregeneraland powerfultechniquesofquantum Eeld theory.
THE POLARIZATION PROPAGATOR
Asdiscussed in Sec.l3,theresponse ofthesystem to a perturbation oftheform
4ex(/)=J##(x)F*'(x?)#(x)d3x
(60.1)
is governed by the density-correlation function, or polarization propagator.
In thepresentcontextofa hnitesystem,the poiarization propagatorisdehned
by2
fl1à;z:./l- J')M(V%l.
F(t1r,(?)cnnlttt12J')cuôlt')11:1'
-0)
(60.2)
wheret51'
%)istheexact(normalized)Heisenberggroundstateofthehamiltonian
inEq.(56.18). TheGreeksubscriptsnow referto thesingle-particleHartree-
Fockstates,whilees.(/)and t1.(/)areHeisenbergoperators,related to the
particleand holeoperatorsby
c.- #(œ- F)n + d(F - x)S.bt
-.
(60.3)
Justasin Sec.59,weagainconsideronly closed-shellground states.
'Theapproachinthissœtionislargelybasedon D.j.Thouless,Rep.Prog.Phys.,27:53(196*.
> alsoW .Czy:,ActaPhys.Polon..R :737(1961)andG.E.Brown.CourseXxlll-Nuclear
Physics,in eeproc.Int.SchoolofPhysics:Enrico Ferm i,'*9p.99,AcademicPress,New York,
1963.
'Notethatheretheorderoftheindiceson11isslightlydiSerentfrom thatin Eq.(9.39)(compare
Fio .9.18 and * .1). The presentchoice is more convenientin studying particle-hole inter-
actionsandallowsustowritetheimportantequations1(* .20).forexamplelinaparticularly
transpxwmtform .
APPLICATIONS TO FINITE SYSTSM S : THE ATOM IC NUCLEUS
569
ThegeneraldiagrammaticstructureofH(/- t')isshown in Fig.(60.1).
A particle-holepairinthestatestt
xjliscreated attimet'. ltthenpropagates
to time /where itis destroyed in the states(Ap.). Note thatthe desnition of
Eq.(60.2)isjusta piece ofthe fullpolarization propagatorfortheassembly
defned by
ïI1(x,x')= ('!'
-(,hFif,f(x).
(,(x)f1(x'),
;(x'))ykF())
l'I1(x,x')- )( wp(txlf+A(x)+z(x')1@,(x')ïHAs;.j(/- r')
a;28
A
pt
x
p
(60.4J)
(60.4:)
Fig.60.1 StructureofFeynmandiagrams
contributing to 11à#z;
x#?- J'
)inEq.(K .2):
(f8 general.(b)lowestorder.
where(@)denotes the Hartree-Fock w'
avefunctions (Eq.(56,17)1. l'
lzsiaj is
easilyshownto beafunction oft- t'by themethodsofChap. 3,and thustakes
theform
I1As;aj(/- t'j= (2=)-1Jdt
.
oFlAs;aj((,>)e-ïetf-l'l
(60.5)
Furthtrmore,theLehmannrtpresentationisderivedjustasinSec.7
1-I
IXF'
()lrp
fcaI
'
'Fa)('l''nIcf
acj!'l%)
Av;a/t
.
&')=h
âf,a- (En- fcl+ i.rl
-
t'l'ohrt
1c,1'Fa)'
.
t'F,?ctfîc,
à11,
'
-n) (60.6)
l
.
o+ (En- Eo)- l
h
Notethattheintermediatestates1
'Fa)refertoanassemblyofpreciselyAparticles.
The poles ofthe polarization propagator evidently determine the energies of
theexcited statesthatcanbereached by aperturbation oftheform ofEq. (60.1).
To express17 in termsofFeynman diagrams, wesrstgo to theinteraction
representation and write
ïl-lApz;a,(?'- /')
a)
=
n=0
(-il'
hjn
''n'!' #/.
j...dtn(01T'j/ct/j)....
J'
.
h(K)cllt)cA(/)
x c'f
t?')cjt/'))p0)connecteu
z
s6c
APPLICATIONS TO PHYSICAL SYSTEM S
whcreX2istheinteractioninEq.(56.21). Thestate10)isnow theHartree-Fock
vacuum orgroundstateillustratedin Fig.56.1. W henthisexpressionisanalyzed
With Wick'stheorem,thereareno contractionswithin agivenXzbecauseXz
isalready norm alordered. Thusthe presenttheory hasno Feynm an diagram s
ofthe type shown in Fig.10.1;allthese term s are explicitly sum m ed by the
canonicaltransformation ofSec.56. Thepolarization propagator(Eq.(60.7))
stillcontainsdisjointgraphswiththestructureshowninFig.60.2. Suchterms
A
)
,
#t
Fig.60.2 Disjointgraphscontributing to Eq.
(K .7). They contribute only at ts= 0 and are
thereforeomitted from the subsequentanalysis.
. ,
areindependentoft- t',however,and contributeonlyto theu)= 0 com ponent
of17((t)). They willtherefore be omitted entirely because we here consne
ourselvesto those com ponentswith (
.
o'
# 0. Alternatively,we could considera
polarization propagatordefined in term softheCuctuation densities
3Ecf
sz(r')c,,;(/')1- c#
,,xtr')cublt'b- (%-olc1
s.(?')csj(?-)i
.
4'
-c)
(60.8)
aswasdonein Secs.12and 32. Thelastterm isa c num berand can cause no
transitions;itmerely servesto removethen = 0 termsin Eq.(60.6)so thatthe
moditled H((.
o) has no singularity at (
.
o= 0. ln coordinate space the correspondingc-numbertermsjustcancelallFeynmandiagramsofthetypeshownin
Fig.60.2.
Thefreesingle-particlepropagatorisdefned by
(0r(0(/)(-t(/')1(0)E,I
'Gya(?- t'
)
t01r(cA(/)c1
a(/'))I
0)N(2,4-1j#(z)dGqz(r
.
,))e-ïf
a'tf-l'l
(60.9)
(60.10)
where the c'sare now in theinteraction representation. A familiarcalculation
leadsto
Gyat/aal- 3za b(-=- #) + p(F- t
x)
u)- ts. + i.
q (,
o - o)u - iyl
(60.11)
Theonelowest-ordercontributionton isshown inFig.60.1:,and an application
ofW ick'stheorem gives
/H0
Av:xj(?- ?')- .
-(2,4-2.
fdo),fdu),ft4.(r.
z
al)i'
Gb
0pz(ta,2)E--'t>'t,-'''e't-'
ztt-'''
(60.12)
APPLICATIONS TO FINITE SYSTEM S : THE ATOM IC NUCLEUS
561
in coordinate space and
(60.13a)
ïHksiajt(zal= -à2(2=)-1fdœkGjzttsj)Gj
osttsj- (
x;)
(60.1?b)
ifl2
$a- F)0(F-p4 0qF- a.lp(j-F)
4
,
;
a
,
(
f
.
,
?
)
i
b
n
x
3
,
,
g
.
--(.a--.
-.j)s.,
j-..
s
jto-j
.-.a)-j,
j (60.13c)
in m om entum space.
It is now possible to derive the hrst-ordercorrections to the polarization
propagator. Theinteraction ham iltonian is
4cl?- /')- !. )( (pg.;p'1n.
i
z?N Ecp
fl/)cJ(??)cvlt')c,?(?)13(?- ?')
'
-
P@TP
(60.14)
and the delta function can be written
H ence we find
jjj'
(
jl
.);xb(t- t:)- -1
.
y-11*)*dt3#?l
,.l.+
x. f,
jj,
zj'r)p.
x
;j(gxj-l
pc.
.
PrPt7
xexplj'
(
.
t)jt/l- t('
))Jf
x?j,
'0!TLN tc1
pt/j)cl(/()G,
(?j
')c'
pt/'j)1
x cl
/)cA(t)c1(t')cj(t'))k
0)
, (60.15)
st
W hen this expression is analyzed w ith W ick's theorem , there are tw o sets of
contributions disering only by the interchange of dum m y variables pk::
â(z,
r)k
'
,
:er.andtjk::
âtt
'. W emaykeepjustthefirstsetandcanceltheoverallfactor!..
Conservation offrequency ateach vertex m eansthatthe variable t- t'alw ays
occurs through a factor c-ftœt-a'
z'tt-f'' as in Eq. (60.12). Thus a factor
2=3(/
.
5- f
zll+ (.
,)al must be included in the Fourier transform to accountfor
theend points(compareEq.(60.13ë). A combination oftheseresultsyields
-
JH t
AI
p1
tiaj(ts)= -yji
, . .. Jat
x?j
ttpc!
.Pr.
lnp,
)
p #a(s
=
Tr
TNJ'EI
*
(60.16)
andthesrst-orderFeynm andiagram sareshowninFig.60.3. The(.
:,5integration
is the sam e in 170th term s.uhile changing dum m y variables p ccc
âo.in the second
562
APPLICAM ONS TO PHYSICAL SYSTEM S
term andusingEq.(60.13)gives
âllkstz/all= TVXPG il10
As;pv(t,))Etpcl#'1
,r?'
y'),- (cplP-l,?r)JfI10
,?g;.j((s) (60.17)
Thisexpressioncan beclarihed bywriting
llt
A'
H';a,(a')-
Z Hjl/
x:pvt/
.
?
al
.XttlttzgppincHo
nc,:ajtf.
,?l
TpP@
âA'
4ltt,alpvinc- ''
-tpo'lP'l'
nv)+ tcplF'ln'
z)
à
ttll
#
Y)
pz
%
fxl3
(z)j
ttlj
à
7 +
CF
Y?
P (.t)5
f.tl;
f.tl2
(s2
(60.18)
(60.19)
rz
%
@
F
Fig. 60.3 Feynman diagrams contributing to
a
#
x
#
lI,ï
t
l'a,(t'
.
/)in Eq.(60.16).
si
wherethetwo matrixelementsinA7)comefrom thefirstand seconddiagrams
in Fig. 60.3, respectively. N ote that the lowest-order proper particle-hole
scatteringkernelA(
t)isindependentofu). Furthermorethesimpleexpression
(60.18)arisesonly becausethe interaction potentialitselfisindependentot-the
frequency(z)j,which isactuallydiflkrentin thetwodif
lkrentgraphs.
w enextwriteaBethe-salpeterequation fbrparticle-holescattering. This
equation iteratesK * and allowsa calculation ofthepolarization propagator11
to allorders. Ittakesthe form
l-lzs:ajltz?l= l-l0
As;aj(tz;)+4'1
%I1V;pv(t,a)A'*tœlpringl.
lne;zj(œ)
'P@
(60.20)
corresponding to Fig.60.4. W e have here m ade the simplifying assum ption
thatthe kernelK * dependsonly on o),as in Eq.(60.19). The mostgeneral
h
tt
P s r .'1' '' I'
. .:
.â
.
..f'
<.
yv
(
z
#
Fig.60.4 Bethe-salpeterequation (60.20)for l1As:t
o(tt?).
APPLICATIONS TO FSNITE SYSTEM S : THE ATO M IC N UCLEUS
563
particle-holeBethe-salpeterequationwillnotQctorasin Eq.(60.20)butinstead
becom esan integralequation involving an integraloverfrequency.l
Equation (60.20)is now a simple matrix equation,which can be written
in the form
H4(s)= I10(f.
s))- r10((s)K*(ts)H((s)
(60.21)
The inverse ofthis m atrix relation yields
(60.22)
Hence H(ts)isa hermitian matrix for realo) and can be diagonalized with a
unitary transform ation
UH((z?)U-1- HD((o)
(60.24)
The inverse ofthisrelation
UI1(t.
&?)-lU'-1= H D(t.
t?)-l
showsthatthetransformationU alsodiagonalizesH ((
.
t)4-l. Someofthediagonal
matrixelementsofI1D(t
.
,?)haveapoleattheexactexcitationenergyoftheassem bly
l'
1co = En-- Ev,which means thatthe corresponding diagonalm atrix elem ents of
H D((o)-lhave a zero atthe same point. Since these diagonalmatrix elements
arejusttheeigensr
aluesof114t
.
u,)-l.Uearriveatthegeneralprinciplethatthezero
eizent'
alues of H((s)-lconsidered asa J'
lkr/c/ïtp/?qfco correspond rt
p the c'
o//ccràè'
c
eltergl'/t?!'t7/çof'r/?casseînbt
q'. ltistherefore necessary to solve thesetoflinear
equations
Xta?)-l= 0
(60.27)
N'g140((s)-1- K *(cojjqs;ajCyj= 0
(60.28)
xb
!NV . Czyà,loc.cfr.
s64
Appl-lcv loNs To PHYSICAL SYSTEM S
RAN DO M -PHASE APPROXIM ATION
Asafirstapproximationin Eq.(60.284,wetakeK*Q;K)),which expressesthe
polarization propagatoras a sum ofFeynm an diagram s ofthe type shown in
Fig.60.5. Thesediagram stontain asa subseta1ltheringdiagram sincluded in
's ):
'
-
.
/
t
Fig.60,5 Feynmandiagramsforn summedintheapproximationK*;
krK!k).
ourdiscussion oftheelectrongas,land theapproxim ation here,asin thatcase,
iscommonlyreferredtoastherandom-phaseapproximation. Equation(60.28)
maybewrittenoutindetailusingEqs.(60.l3c)and(60.l9)
(60.29)
Thissetoflinear,hom ogeneous,algebraic equationspossesses solutions only for
certain eigenvalues 6n.
W enow separatetheindicesin Eq.(60.29)intotheregionsaboveand below
E andchangedummyindicessothataandAalwaysrefertoparticles(œ,A> F),
andjand J.
talwaysrefertoholestjsp,< F)
+ 'tap.rP'I#A)1G a)- 0
a> F,j < F
a >F,# < F
A >F,p,< F (60.30:)
(60.31t8
(60.31:)
:Thelargersetofdiagramsshownin Fig,60.
5isjustthatconsidered inProb.5.8onzero sound.
APPLICATIONS TO FINITE SYSTEMS:THE ATUMIC NUCLEUS
566
XA l''-3-p.X!(p'
z
*j)= 0 (60.32/)
s-) r'xA
-.
-
.
-,
3-/
.
tr'aA'
,jt
)1j
a- y,1,-jA,)(;'1jJ= 0 (60.3lb)
Ifwerecallthedefinitionsin Eqs.(59.10)and (59.21),weseethattheseequations
can be rewritten as
(eA-p-e)/A
*.-,-oN'(!'
a
*.ia,/a
*;-u,
1.;ajç;1?
T)-0
(60.334)
(eA-.,+ 6)+A
*p-'
-(
1l(t'A,
x;z,%'
1b-uv
jvkuî'
)a
*;)-0
(60.33:)
,
un
Nvhich areexactly theRPA equations(59.22)analyzed in detailin thelastsection.
Itisinstructive to com pare thisresultw'
ith thatofthe Tam m-D ancos approxim ation.
TA M M -DA NCO FF APPROXIM ATIO N
To the extentthatthe positive eigensalues e = Enofthe collectlve excitationslie
in the vicinitl.ot
-the unperturbed particle-hole energies(namely,en:4:k:x- 6j).
the lowest-order polarization propagator (Eq.(60,13c)1 can be replaced by
(60.34)
(60.35)
whicharejusttheTDA equations.
W ecannow seetherelationbetweentheR PA andTDA . TaketheFourier
transform (Eq.(60.5)Joftheapproximatezero-orderparticle-holepropagator
(Eq.(60.34)1used in IheTDA. ltisevidentthatthisquantity only propagates
forward in tim e. In contrast,the originalzero-orderparticle-hole propagator
in Eq.(60.13c) retains the fullsymmetry in u)and propagates in both time
directions. Consequently the RPA sum s all possible iterations of the two
lowest-orderdiagramsforK)I)in thesenseofFeynman diagrams(Fig.60.5),
whereastheTDA sumsjustthatsubsetofdiagramswhereallparticle-holepairs
propagate forward in time. Thus the TD A has one and only oneparticle-hole
566
APPLICATIONS TO PHYSICAL SYSTEMS
pairpresentatanyinstantoftime,whereastheRPApermitsanynumberofparticleholepairstobepresentsimultaneously.
Itisclearfrom Eqs.(60.33)thattheTDA isthelimitingcaseoftheRPA
when them atrixeltm entsofthtpotentialartsm allcom paredtotheunperturbed
energies,orm oreprecisely when
4U)
<.
:j
en+ (eA- e-s)
-
-
(60.36)
In thiscaseEqs.(60.33)uncouple,and Eq.(60.33/)becomesjustEq.(60.35).
An equivalent condition is that the true eigenvalues should lie close to the
unperturbed conhguration energies,nam ely,enQJeA- e-s. H ence.exeited states
thatarestronglyshiftedfrom theunperturbed particle-holeenergies(particularly
thosestatesshifted to lowerenergiesforwhicho ;k;0)should betreated in the
RPA ratherthan theTDA ;thesearegenerallythemostcollectivestates.
CONSTRUCTION OF n (u;) IN THE RPA
Thepresentresultscan becombined with thoseofSec.59 to constructnt(,)l
explicitlyintheRPA. Equations(60.31)and(59.17)togethergive
Ca;=Sbt
/4-,-'
&('l'
%l?'-jJz1Ye.)
œ>F,/<F
(60.37J)
Cuî=Sx+;-.-'
Sxt'l'nlJl
jàl.zI'L) #>F,œ<F
(60.37:)
Theseexpressionscan becom bined w ith theparticle-holetransform ation ofEq.
(59.4)
c)cj='
=p(x- F)0(F - #)d >ljSj+ p(j- F)0(F- a)>-zabS.
RPA
(60.38)
and wethereforeidentify
(tl%IcJcj(
kL)=Cj
tl' RPA
(60.39)
AsaresultthepolarizationpropagatorintheRPA canbewritten(seeEq.(60.6))
IIRPA (
Aàz;xjoa)=
Cà
(a.)C'
(
(ax
)C'
(
)v
aa!)# Cb
paz
w
(
.
0- f
w + IT (z)+ oö - i.
q
(60.40)
Thisexpressioncharacterizesthelinearresponsein RPA toanexternalperturba-
tionoftheform ofEq.(60.1). ltthereforepermitsustostudytheexcitationof
nuclear states obtained by electron scattering,photon processes.weak inter-
actions,and even high-energy nucleon scattering (to theextentthatthe Born
approximationappliestothecxcitation process).
APPLICATIONS TO FINITE SYSTEM S:THE ATOM IC NUCLEUS
667
6IQREALISTIC NUCLEAR FORCESI
Sofar,thediscussion ofhnite system shasbeen castin rathergeneralterm s,and
the techniques apply to atom s and m olecules,as wellas to atom ic nuclei. In
this section we brieiy discuss som e ofthe specihc problem s arising from the
singular nature of the nucleon-nucleon force. The hard core (and strong
attractionjustoutsidethecore)meansthatthematrixelements(ajlF1àp,
)must
bereplacedbyasum ofladderdiagrams,asintheGalitskiiorBetàe-Goldstone
approaches. W ehereusethelatterfram eworkbecauseofitsrelativesim plicity.
'
Tw o NuctEoNs O UTSIDE CLOSED SH ELLS: THE INDEPEN DENT-PAIR
A PPROXIM ATION
Inprinciple,theErststep intreatingasnitesystem with singulartwo-bodyforces
isto constructthe H artree-Fock single-particle wave functionsforan eJective
two-body interaction obtained by sum m ing the ladder diagram s. This sum ,
in turn,dependson the choice of single-particle wave functions. Such a selfconsistentproblem isclearly very dië cult. Asan illustration,weshallinstead
assum ethatthesingle-particlewavefunctionsand energiesareknown from our
discussion ofthe shell-m odeland considera single interacting valence pairof
neutronsorprotonsoutsideadoubly magiccorewith closed majorshells(e.g.,
He6,O18ACa42' etc.). Thissimplised problem servesasa prototypeforany
study ofa snite system with singularforces. In addition,italso allowsa direct
com parison with experim ent,forweknow from Sec.58thatthem atrixelem ents
(jZJM (Iz-jJ'ZJM Ldeterminethespectrum ofsuch a nucleusiftheinteraction
between the valence nucleons is nonsingular. For a singularinteraction,we
m erelycom putethecorresponding m atrix elem entsintheladderapproxim ation.
Fig. 61.1 Two nucleons outside closed
shells. '#'- indicates tbe degenerate valence
levelsand 'ë'denotes the closed-shellcore.
lThe sim plified discussion in thissection follows thatofJ.F.Dawson,1.Talmi.and J.D.
W alecka,Ann.Phys.(N.F.),18:339 (1962); 19:350 (1962). For a thorough treatmentof
nuclearspectroscopywith realisticforces,see B,H.Brandow,Rev.M od.Phys.,39:771(1967)
and M .H.Macfarlane,CourseXL,NuclearStructureand NuclearReactions.in Proceedings
oft:-fheInternationalSchoolofPhysicstEnrico Fermi,'''p.457,AcademicPress,New York,
1969.
su
AppulcAyjoxs To pHvslcAt. sys-rEM s
The present situation is illustrated irt Fig.61.1. Two idtntival xalenee
nucleonsoccupythe(degenerate)setofstates'
#'-,whilethecore@rconsistsof
doubly magic closed shells. Letthe totalbinding energy ofthe nucleuswith
4 nucleons,Z protons,and N neutronsbedenoted by zB1= zE1- ZmpclNmncz. Theground-statebindingenergyBE(2.
N)ofthevalencepairofneutrons,
.,
forexam ple,canthen bedefned by
BE(2N)= zB1- z#4-l- 2(z#4-l- z#ï.
-J1'
(6l.l)
Thebasicproblem isto com putethistw o-particlebinding energy, aswellasthe
excitation spectrum ofthe nucleus. To achieve thisaim ,we assum e thatthe
inertcore servesonly to providea self-consistentHartree-Fock potentialwhose
levels@Garealreadyslled. W ethen workintheindependent-pairapproxim ation
and write a Bethe-G oldstone equation for the valence pair thatallows virtual
excitationtoany statesexceptthosealreadyoccupied. Sincethisl'
ntegralequation
dependsonlyonthecombination F/,itssolutioniswelldesnedevenforsingular
potentials.
BETHE-GO LDSTO NE EQ UATION
The Bethe-G oldstone equation for the system illustrated in Fig. 61.1 follows
directly from thediscussion in Sec.36.1
(61.2J)
(61.2:)
Here the index n denotesthe single-particle levelsfnlmtJ-s)available to the
valence pair,and we assume initially thatthe unperturbed single-particle spectrum hastheform shownontheleftsideofFig.57.2. Thecorrespondingsingle-
particlewavefunction is+n,and fJ,paistheinitialunperturbed energy ofthe
pair. The perturbation HLconsistsofthe(singular)two-particle interaction
andanyresidualone-bodyinteractionnotincludedinthestartingapproxim ation.
In Eq.(6l.2t#,the sum X'
'runs over aII unoccupied single-particle states as
indicated in Fig.61.1.
Theseequationsareexactwithin thepresentframework,butEq.(61.2c)
is not very usefulasitstands,for)Z'
'includes states thatare degenerate with
@a,az. SincethecorrespondingexacteigenvalueEnjaz isclose to E.%l,,a,theenergy
denom inator for these degenerate states becom es very sm all. This dië culty
can be elim inated by starting overand choosing a m odised setofinitialstates
forthevalence particlesin '
#'
-. ln particular,weconsidera linearcom bination
ITosimplifythesetquations,wesuppresstheantisymmetrization ofthewavefunctionsforthe
identicalvalencepair.
569
APPLICATIONS TO FINITE SYSTEM S : THE ATO MIC NUCLEUS
ofthese degenerate states
à'a(.1,2)i
K
'
V
-
njna (
72 1
C ,C
'la J?':(l)(?
.,,a(2)
)u
T,
requiring only thatthe transform ation be unitarj
x'
c?(
3/);!11;:
.u...
Ja
j?)
Jr*
r cIt
1:
jr)
/;;),= jll;ll!,
Q
kl.-a (1,2)- xa(l.aj.
:z r J
2
) ?''.
J.
/!
I
%+,
.(
--1.,
.
.
-.
-X1.*
'
.a..
E.x-.&.
o
- E,,
-
'%,!'(1)%,,
z(2)#?;?l2'.
f/1tI'a (6j.juj
E 1'0,
Z.
I ?lij
.
?1 .
.
..- ..... . ... .
.
:z
?:1 ?!;j
Ex- Eî = à'aH3'11':
'
ln theseexpressionstheeigenvaluesarenow labeled b) fa.and Ek'denotes
the unperturbed energ) ofa pair i11 '
/'. The sum in Eq.(6!,fa1ha8 been àplët
into a part com ing fronl the other degenerate unperturbed stateà in y 'and a
rem ainderZ 'containing at least one excltation to a highershelltsee Fig.61.11.
Since the sm allenergy denom inatorsoccur only in the firstsum .ue shalltrl
k to
choosethe coem cientC)1
xb
/
11sothatthentlmeratorsin thetirstsum in Eq.(61.f5)
N'anish identically
,
.rya,sj:j.j..tsa-.sy.jjaj
ln thiscase.only the second sum in Eq.(61.fa)rem ains:furtherm ore.allthe
statesin 12'involvean excltation to highershellsso thatthe num eratorsNNillin
generalbe sm allerthan.orcom parable u ith.the denonlinators.
This reform ulation ofthe originalequations(61.2)is sti11Ner) difl
icult to
solve becausethe exacteigenvaluesappearin lhedenonpinatorsin Eq.(6l..
qa..
W 'e m ay now'
, however.observe that )
Z'runs over allstates U here the Nalence
particles are prom oted to excited states (in the harm onic-osciliator nlodel.
valence particles m ust be m oved an even num berofoscillatorspacingsto m ix in
statesofthesameparity). Asa result.a1ltheenergy denominatorsin thesum
Y'are large(atleast2/io.):
k:81M ev .
,1îin theoscillatormodel)compared to the
energy shiftsf'
a- Eî tobeealculated below (a few MeV).and wemaytherefore replace Ex bJ'Fy
o.in these term s. This approxim ation allows usto wrlte
k1'
-:
Z(1.2)= n lêl.2N'
L-'
0
2)
....
tf
1a
Il'
l2c
'$
?;:#;2#1.
C*
.7
*%
nj&.2(l*3
vNj(l)v,
?2(2)-)= q
--'r
/'?
,(l)(
/-n:(2) l?1
-?72
-f.
flub0 u
- ' -- -.-O..
----j.-.--- -0
--1
-S.
.
z.
,.: ?,2
t;y.- E,1l ,;l
570
APPLICATIONS TO PHYSICAL SYSTEM S
with a know n kernel. The addi
tionalcondition on the matrix elements (Eq.
(61.6))becomes
where Eq.(6l.4) has been used. This set of linear hom ogeneous algebraic
equationsdeterminesthecoet
TicientsCj1
uj2.andtheNanishingofthedeterminant
gives the eigenvaluesFa. ltisclearthatEqs.(6l.7)and (6l.8)reproduce Eq.
(61.5J)becausetheconditionofEq.(6l.6)eliminatesthefirstsum in Eq.(6l.5c).
andFaœ fk.inthesecond sum within thepresentapproximation. Equations
(61.8)and (61.9) togetherform the (approximate) Bethe-Goldstone equations
fora finite system B ith initialdegenerate states.1
HARM ONIC-OSCILLATO R A PPROXIM ATION
The discussion can be greatll'sim plit
ied if we approxim ate the H artree-Fock
$ingle-particle u as'e functions r
/,,and energies En
Qin2 by those of the harm onic
oscillator. ln this case,the perturbation becomes
/.
/I= 11l(1)-2.11l(2).
-.l'(I,2)
where lz'(1,2)is the two-nucleon potential.and /?jis a single-particle operator
thatrepresentsthe dl
j/ercncv between the true Hartree-Fock potentialand that
ofthe harmonic osclllator. (To be s'
ery explicit.11l= .îP'(r)- al.s,where the
tirst term is the change in the potentialwellleading from the spectrum on the
leftside ofFig.57.2 to thatol7the leftside of Fig.57.3,and the second term is
the spin-orbitintcraction.) W e shallbe contentto treathbin tirst-order per-
turbation theory,and Eq.(6l.9)becomes
Nx
- g,x??l??aI
'1,'40
:
7!.?
l:, - .A),nzt/?l(1).c./?I(2);
'
n(nj'
x
.
,l'nLn-c y
h(E'
0
,.- f.
zl8,
,,,,,, 3na.,,,!Cj1
ald
ni
),- 0
-
Attheendofthecalculationwemayintroducethesingle-particlestates(nljj-j)
.
that diagonalize hj. and the resulting single-particle consguration energies
E 0 - ./3j'
v- :,/?2.- c: - 6a can be determ ined em pirically from the energies of
neighboring odd nuclei.
The lirststep in obtaining the energy levels Eu and coem cients C,J
œ
I'
2
u2isto
solve the approximate Bethe-Goldstone eqtlation (6l.8) with HjEH 1z'. The
resultingwavefunction J,
0
,lhendeterm inesthem atrixelementsofP'appearing
zl1 t
in Eq.(6l.1l). Thechoiceofharmonic-oscillatorwavefunctionsJ?nnow allows
1See in thisconnectlon H .A .Bethc.Ph.b,
.b'.Sct'..103:l353(1956)and C.BlochandJ. H orow itz,
.:
/1/('/.Plo'.b..8:9l(1958). The basisforthesetofEqs.(6I.9)canagain bereduced in theusual
nlanner by introducing eigenstates ofthe totalangular molmentum . W'e Ieave this as an
excrciseforthededicatcd reader.
APPLICATIONS T0 F'INITE SYSTEM S:THE ATOM IC NUCLEUS
671
a verl'im portant sim plification.for the tu'
e-body wave function ç'nkç'nz is an
eigenstate ofthe ham iltonian
P2
,
'
oi .,,:t.,)2x2- 4.n1r
x
Hv= .$. =. - -+- 1
.
I . ,.72xî
.
..
.
n: 2,?)
X =.
1.(xl-.xz)
P = p:- p:
x = xa- xI
p= ;
(p:- p:l
-.
(6l.12)
Hç
jcan be reu'
ritten as
p2
P2
H o wss-.- - nlcuzx 2- . -- 4,
z.,24
.
,cxz
4nt
l,7
which separates the center-of-m asb m otion of the Nalence palr from the relall
'
k'
fv
m otion. The elgenstates of Ho as written in Eq. (61.14) take the form
'
%s(X)+,,
(x),where the quantum num bersare .Y Es$
,.
h'.:
#'.
'
-//.
s
y!for the center of
.
.
m ass and n HE.
t
lml.
!.
lA?5!Jr/?sy1
.tor the reiatlve NNa:e functlon. Each of thepe
,n
statescan beexpanded asa 11nearcom bination ofthe orlginalstates>-n(l)q'-;(2h
with the sam e total tq o-partiele energy'(and hence sam e elgen:alue of .
/Y(,)
because the latter form a eom piete set ofeigenstates of the same ham iltonlan
H o. Thuswe haNe
%..
x(X)+ntx)= s uY- ?kI??zNnzC
/n,(1)c
#'na(2)
!s s
u'here5'
2clenote.
îalIpossible lu'
(?-//tzr?/c
Vcuflates#t
x cl?t'
rtz?c 't'
?'
J/?tllc initialJ'
/tz/tp.
These orthogonal transform atlon bracketsl 11jnz.Y?? have been extensisely
tabulated.z Itfollowsfrom Eqs.(6l.15)and (61.8)that
..
.
??:l:z'.'
l.z '
l;0n,.,,z-'-=
NJ
j..
.
,
b-171-'i
.
/
.
/0v.n'
-1.
1l11:.-N'17 '
h''??t11l'nz' -'
(N ote thatthe sum on njnzin Eq,(6l.lf)can be applied directly'to the wave
function $0
n,naon the rightsideofEq.(6l,8Jsince the quantit)'Ey
o.isthe sam e
foral1thedegeneratestatesin fz'y,)
Theexplicitcalculation ofkc
t.uiscomplicatedby'therestrictedsum Z'in
Eq.(61.8)-uhich eouplesthecenterofmassand relativeuave functionsofthe
N'alence pair, Itis therefore conv'enlen:lo reu rite the sum as
N-'
N'
N'
n*1na n1n77-J2y. alor'
n
-2
'='
<
!l,Ta1mi.Helk'.Phys..,4cta..25:185(1952).
2T,A.Brod) and N1,M oshlnsk).''Tables ofTransformatlon Brackets.''M onogratias De1
lnstituto deFislca.slexlco Clts.1960.
s72
Thefirstternlsun-lsoveralI.
5'/tz/tp.
q
'bs'
illlenerg)'dijh'
lren:/àt????1-y
0.and thesecolld
term rem ovesthosestatesthatviolate the PauIiprinciple. W e neglectthesecond
term (lhe Pauliprinciplecorreclion)forthepresentand return to itlaterin this
section. Ifonly thetirstterm isretained,the corresponding wave function 4-0
satisfiesthe equation
/-%
0.
?1
(X-x)- (
px(X)fr,,(x),-
%s.(X)%,?.
(x,
à't%''/l'(f
z-t'
Jt.
.
t,,
é-; - é'y$
j,
(ojjyj
.
4-0
:
.
k?
l(x*x)..(/-s(xJ'
.
JO
x)
tlI
'
W ithi1thedegeleratesetof-statesQ.ysthematrixelement A'
)?'lz'1
/-N0?1,,)
,inEq.
.
.
.
.
(6l.l6)isalso diagonalin the relativequantum num bers,land Eqs.(61.l6)and
(61.!8)takethetbrm
Y
'
b
I t
'tlj/7:t)
,U'
:4,j,1.,,2. = xx
e' nj/7a.
N./7..C
'
N61:.,
'/?tj11zr..tî!
,l
,
-.:/7
,'
.
x
%11
(6l.20J)
.
/-0
x)'
-r
?,
'
,,
(x)z'
,,(
r,1 .'à1
Y'I'CX) 11'.I''
l
J
0,
'
.u
----(, -(?
--
s,,- é.,
Theenergydenom inatorsi1 Eq.(61.20/))areagai1
)atleastaslargeaszhot;
to the same degree of approxim ation as before. we ean therefore rewrite this
equation as
E,
1--1-,
0aE'nIpzI
1
;J1'
(6l*21b)
Theproblem isnow solved,forEqs.(6l.21)arejustarewritingoftheSchrödinger
equation
-
:2
-
2.v..
/?J(s2.
:.
2u-p'(x) /u(x)= Enjs(x)
N15-
(61.22)
Hencethematrixelementsrequired inEq.(6l.20J)can beobtained directly with
Eq.(61.21bqrrom the discrete eigenvalues ofthe diflkrentialequation (61.22).
Theunperturbed (U'-=0)solutionsto Eq.(6l.22)have already been given
in Eqs.(,
57.7)to (57.I2).
'they are plotted in Fig.6l.2 fordiflerentvaluesof/.
1W eassunnethat I''isdiagonalin /.
573
APPLICATIONS TO FINITE SYSTEM S : THE ATOM IC NUCLEUS
1.
0
Fig. 61.2 Square of the unperturbed
solutionsuîttoEq.(61.22)togetherwiththe
square of the txactJ-wave solution /z:sto
1u15I
2
0.9
0.8
07
06
05
Eq.(61.22)in thepresenceofthe potential
(61.23).Thewavefunctionsarenormalized 04
O3
to Jll
z!I12 dr= 1. Also shown are the
0.2
interactions in Eq.(61.23). IFrom j.F.
0
x 7
N7
ranges of the hard-core and attractive 0.1
Dawson.1.Talmi,and J.D.W alecka.Ann.
Phys.(N.F,).18:339 (1962). Reprinted
bypermission.)
-
x t.
4
r
()2 0.6 1.
0 14 l8 22 2.
6 30 bNT
i 1
+1/M
6.1
Forcomparison,wealsogivetheexacttnumerical)J'
-wavesolutionforapotentia?
X
pr=
-
e-vr
P'o
(61.23)
y.r
Uo= 434 M eV
y,= 1.45F-l
c= 0.4F
with the parameters determined from a fit to the l,
S nucleon-nucleon phase
shift. Throughout these calculations the oscillator parameter is taken as
b= lhlmotl'
k= l.70F,appropriatetoOïS. ltisclearthatonlytheJwavesare
m uch aflkcted by the potentialand thatthe correlated and uncorrelated wave
functions are very similar, the correlations being most important at small
distances.
Thistheory has been applied to 018,wherethe initialunperturbed confguration has two valence neutrons in the 2J-1# oscillator shell. W ith the
Brueckner-Gammel-Thaler nucleon-nucleon potential,Eqs.(61.11),(61.20*,
(61.21:),and (61.22)givethe spectrum shown in Fig.61.3.1 Thissgurealso
7+
-
Fig.61.3 Experimentaland theoreticaltwoneutron binding energy and low-lying excitation spectrum of018. Thetheoreticalresults
werecomputedusing(a)1hesi
ngularBrueckner-
-
2++
0
4
Gammel-rhalernucleon-nucleon potential,(:)
the nonsingular Ser* r-Yukawa potentialof
Eq.(59.81). (From J.F.Dawson,1.Talmi,
and J.D.W alecka,Ann.Phys.(N.F.).18:339
(1962);19:350 (1962).-Reprinted by permission. Theauthorswish to thankG.E.W alker
forpreparingpart(d8.)
lJ.F.Dawson etal.,Ioc.cl't.
Theory
4
0*
:74
Appucv toNs To puyskchu svsu M s
shows the result obtained by using thenonsingular force given in Eq. (59.81)
and replacing /k.n,--,.+s.+.,in Eq.(61.16). In both casesthesingle-particle
consguration energiesweretaken from O17:e(l#.
j)x 0,E(2.
u)= +0.871MeV,
and e(1#!.)= +5.()B M eV.
PAU LI PRINCIPLE CO RRECTIO N
To includetheeflkctsofthe Pauliprinciple,wedesneprojectionoperatorsP
and Q representing the srstand second restricted'sum on therightside ofEq.
(6l.17). lfEq.(61.8)with Hjreplaced by Prismultiplied by P',itcan be rewritten asan iterated operatorequation
G= P'+ P'iPt---Hz
%.p'+ r'fo
P-p
-P'UPQ F+ .
--Hv
-Hv
.
-
(61.24)
where theoperatorG isdehned by
Gvn,t
pnz= P'/o
n,az
(6l.25)
Equation(61.24)can berearranged to give
(61.26/)
'
(61.2648
A comparison of-Eqs.(6l.26:)and (61.18)showsthatG0corresponds to the
wavefunctionwehavejustdiscussed
G0(
2?n1pnzH P'/Rln:
)
(61.27)
ThusthefirstPauliprinciplecorrection isgiven by
()
.Gvp= -Gû/-Q-..
N
tioG
.
0-
(61.2g)
The m atrix elem ents ofthisrelation are now snite even with singular potentials.
Furthermore,thedenominatorsareagain>2hu)inmagnitudesothatEq.(6l.28)
representsa smallcorrection to theenergy levels(<0.5M eV). Thiseflkcthas
been included in Fig.61.3a.
EXTENSIONS A ND CALCU LATIO NS O F OTHER Q UA NTITIES
Thesubjectofnuclearspectroscopy with realistic forces has beendeveloped
extensively.l The m ostcom pletecalculation isthatof K uo and Brown,zwho
also include a core-polarization term (Fig.61.4)in the etlkctive interaction.
1B.H.Brandow,Ioc.cit.;M .H.Macfarlane,Ioc.cit.
2T.T.S.Kuo and G.E.Brown.Nucl.Phys.,85:40 (1966);seealso B.R.Barrettand M .W .
Kirson,Phys.Letters.27B:544(1968).
APPLICATIONS TO FINITE SYSTEM S:THE ATOM IC NUCLEUS
57:
Itisclearthattheindependent-pairapproach also can beused to 5nd the
ground-statepropertiesofnuclei. Edenetal.havecomputedthebindingenergy
and density ofO l6 using the harmonic-oscillator framework.l A somewhat
diferent procedure is to use the G matrix calculated fornuclear matter asan
eflkctivepotentialin a Hartree-FocklorThom as-Ferm itheory3ofhnitenuclei.
=
Fig. 61.4 Tbe effective interaction of
Kuo and Brown.
F@fr
>
+
Fxre- polawzation
A com pletely consistentcalculation w ith the G m atrix for a snite nucleus has
neverbeencarried out.
PRO BLEM S
15.1. Provethe assertion thatthe solutionsto Eq.(56.17)corresponding to
diflkrenteigenvaluesare orthogonaland thatthe degenerate statescan always
beorthogonalized.
15.2. Derive theSchmidtresults (Eq.(57.22))fortheshell-modelmagnetic
m omentsstartingfrom Eq.(57.20).
15.3. Derive Eq.(57.25) for the quadrupole moments in the single-particle
shell-m odel.
15.:. ProveEq.(58.33).
15.6. Using thesecond-quantization techniquesofSec. 58,show thatforeven
N
1 1(7*0,.c= 0j
I',I/*J;c- 2)12
2J+ i'
.
.
IJ+
11jrx2jl
+jl-lNj(2j+
lljtyjpgjy)j
a)
In them any-particleshellm odel,observethatthistransitionstrengthisenhanced
with increasing N and reachesam axim um fbrahalflhlled shell.
1 R . J.Eden.V.J.Emery.anu S.Sampanthar.'roc. Roy.Soc.lLon#onl,A153:177 (19591)
A253:186(1959);seealso H.S.Kèhl
erand R.J.M ccarthy,Nucl.Phys.. % :611(1966).
2K.A.Brueckner.A.M .Lockett.and M . Rotenberg,loc.c/r.,whichcontainsotherreferences.
Fora review ofthis approach.see M .Baranger, 'sproceedings ofthe InternationalNuclear
Physics Conference.Gatlinburg.Tennessee.''p.659.Academic Press.New York,1967:and
CourseXL.NuclearStructure and NuclearReactions. in Proc.of''The lnternationalSchool
ofPhysics'Enrico Ferm ig.''p.51l. AcademicPress,New York,1969.
3H .A. Bethe,Phys.Ret'.,167:879 (1968).
s7e
AppulcATloNs To pHvslcAu sys-rEu s
1s.6. Consideranodd-odd nucleuswithp protonsinthe-/lshellandn neutrons
in thejz shell. Ifboth the protonsand neutronsare in the normal-coupling
shell-model ground state. prove that the energy splittings (in lowest-order
perturbation theory)dueto a neutron-proton potentialofthetypein Eq.(58.6)
can be com puted with the replacement
(ik'j,.
2y,+ 1- lpljz+ 1- 2n
/1.
/271PbjLjk/:.h ./) -.
>'- zjl j
zj,- j 't./1.
hJ1F1.
/1.
/27)
.
-
O bservethatifeithershellishalfslled,thisquantityvanishesand al1thesplitting
comesfrom thespin-dependentpartoftheinteraction.
1s.7. Prove thatthe statein Eq.(58.55)hasthe quantum numbersindicated.
1s.8. (c)Show that the matrix elements ofthe two-body potentialofEq.
(58.37)inthethree-particlestateofEq.(58.55)aregiven by
(.I3JM 1P!/3.fM )= 3)(1(.
/3.
/t).
/2(A).
V )I2(.j2AjP'1./2A)
.
A
wherethecoeëcientsoffractionalparentagearedehnedbytherelation
l)j2
J
1k
1'
)
.
(Thesymbolbljb.
/2.j3)meansthattheangularmomenta mustadd up,thatis,
1.
/1-./2l<.
/3<71+.
/2).
(>)Provethatthesamerelationholdsforthestatein Eq.(58.54)(takek= 0).
15.9. Assume the 0+ (ground state).2+(2.* M eV),and 4+ (3.55 M eV)states
in O18belongto the (#4.)2consguration.
(J)Prove thatthe only possibleJnvaluesforthree identicalparticlesin the
(#1)3conhgurationare!+,.
)+,and!+. (SeeM.G.MayerandJ.H.D.Jensen,
op.cit.,p.64.)
(:) UseEq.(58.39)andtheresultsofProb.15.8to predictthelocation ofthe
statesofthe(#!.)3consguration. Comparewith theexperimentalresultsfor
O19:.
j+(groundstate),!+(.096MeV),and!.+(2.77MeV).
15.10. Com putethem atrixelem entin Prob.1:.5 in theboson approxim ation.
15.11. Computethespectrum ofthejNconhguration(Aeven)foranattractive
delta-functionpotential(Eq.(58.18))inthebosonapproximation.
15.12. Extend theboson approxim ationto an alm ostslled shell.
APPLICATIONS TO FINITE SYSTEM S : THE ATOM IC NUCLEUS
677
15.13. ProveEq.(58.69)forapotentialoftheform F = P'
()(.
rlc)+ P'l(xIc)11.nz.
Assume realradialwave functions.
15.14. (J) W ritetheangularmomentum J in termsofthetransformed operatorsof-Eq.(58.64).
(>)Provethatthestate!O)hasJ= 0.
(c) Provethatthestate1O)hasevenparity.
15.15. Verify Eqs.(59.80).
15.16. LetD bea sum ofherm itian single-particleoperators.
(t
z) Provetheidentity
ïV'CrEé,EX,:11l'Fo)- 2X (En- fo)p('1'aIX$
'1'c)I2
'
FI
(>) Iftherightsideisevaluated in the RPA,show thatthe resultisthesameas
evaluating the leftsidein the H artree-Fock shell-m odelground state. W henever
thedoublecommutatorisacnumber(seeProb.15.17).theRPA thuspreserves
thisenergy-weighted sum rule.t
15.17. (J) Show that the electric dipole moment of a nucleus measured
relative to the center ofm assisgiven by
o-J$lx(J.
)gi.
-3(.
/)-4
'-r3j
(b)Iftheinteractionpotentialisoftheform F(xjy),derivethedipolesum rule
for N = Z .
4X (fn - Q )I'
2-,
4
SY'.fà1'
1e0)2= 3h
lm
n
(c) ProvethatifN # Z,therightsideismultiplied by4NZ/AI.
15.18. DiscusstheimplicationsoftheresultsofProb.l5.16iftheoperator X ?
commutes with X. (Forexample.the totalmomentum ê,the totalangular
m omentum j. Note thatthese operatorsgenerate translations and rotations
oftheentiresystem .lj
15.19. (J)Use the resultofEq.(60.40)forI1ls
P)a;((z))to discussthe linear
responseofa hnite system to aperturbation oftheform Eq.(60.1).
(>) Computethecrosssection forinelasticelectron scattering asin Sec.l7.
15.20. D erive an expression forthecore-polarization potentialshow n in Fig.
61.4.
tD .J.Thouless,Nucl.Phys.,22:78 (1961).
jIbl
d.
A P P EN D IX ES
ArQD EFIN ITE INTEG RA LSI
The gamma function 1n(z)and the Riemann zeta function ((z)are defined as
follows:
rtz)-j-o
Rdtt--'tz-l
((z)H:zN-p .z
/,-7.
These desnitionsconverge only in thespecihed regionsofthe com plexz plane.
butthefunctionscan be analytically continued with thegeneralrelations
I.(.c)1a(I- c)-= 7F
.
sin rrz
2'-;
'I>(z)((z)cos(at.
n.
z)= n.
z((1- z)
lW e here follow the notation of E. T.W hittakerand G .N.Watson. -*Modern Analysisp''
4thed..CambridgeUniversityPress. Cambrldge.!962,wheremostoftheserelaîionsareproved
.
579
s80
APPENDIXES
Thegammafunction also satissesthefunctionalequation
rtz+ l)= zPlzl
and ittherefore reducesto thefactorialfunction forintegralvalues
r(n+ 1)= n!
nintegral
Thedigammafunction/(z)isthelogarithmicderivativeofthegammafunction
/( d
1 #r(z)
z)= k-jl0gF(Z)= r(z) dz
A tz= l,itreducesto
/(1)=jf
*
)dte-llntM-y
w herey isEuler'sconstant. Usefulnum ericalvaluesarelisted below
r(!.)= Vzr;s1.772
r(l)= 0!= l
(40)= -!.
((1.
)Q:2.612
r(û);
kû0.9064
r(J)=Jv'=;e0.8862
r(:
));
k;0.9191
((2)- =2/6a;1.645
(0)= 1.341
((3):k:1.202
((4)= ,.
,4/90;
k;1.082
y œ 0.5772
l,
(o=(#(
dz
(z)jz.0o-jjn(a.)
e?a;1.781
N umerousseriesand dehniteintegralsm aybeexpressedinterm softhepreceding
functions:t
77
*(2,+1ljh-(1-2--'tt-'
p-0
Jc
xdxex
.A-l1- FtX)tt'')
jzdxerx+!1=C1-21-R)U(n)((NJ
-
x
*
z,-l
.
Jodxx,-lcschx-jn
*dxsinhx
-
2(1- 2-n)r(n)((n)
iAllbutthelastintegralinthislistaretakenfronlH .B.Dwi
ght,e'TablesofIntegralsandOther
M athematicalData,''4th ed.,chap.12,The Macmillan Company,New York,1964.
APPENDIXES
681
jo
=t
sxn-lsechx=(c
oJxco
xm
sj
-j
l
x
*
2r(a)p=0
jl(-I)p(2p+ 1)-a
=
jo
cdxx.
a-jsechzx=jo
xdxoo
rsj
-a
lx
=
l
22-,41- 22-/)P(a)((n- 1)
n> 0
l
jodxxz
n-(l-x)a-1=jj
xdxjjX
-j'
V/
NIoF9)
yT
.x
vw =ptm
n)
J-zyx lnxsechzx L
rv
Ja
-zgo-yxa--'sechcxaaq-l --In4xe''
.
-
TheBesselfunction#:(x)maybedtûnedbytheintegralrepresentation
A'
o(x)-j0
*dte-'
=@'h'
and we5nd
J*ogxA'(x)-jo
=dtsech?
o
BQREVIEW O F THE THEORY OF ANGU LAR M O M ENTU M t
BAslc COM M UTATION RELATIO NS
Thestarting pointisthefundamentalsetofcommutation relations
LL,L?= ieijklk
(B.1)
where eéikisthe totally antisym metric unittensorand a11angularm omenta are
m easured in unitsofh. A lthoughJ could beahrstquantized operator
L
l
ê
?
x= '
/ y'j.
j- zjy
-
etc.
asecond quantized operator
Q - J,;t(xll' P
P
yjys-zj
yk(x)d)x etc.
'Thisdiscussion is very brief, Fora moredetailed analysisseeany basi
c texton quantum
mechanicsandespeciallytreatiseson thetheoryofangularmomentum such asA . R.Edmonds,
**AngularM omentum in Quantum M echanics,''PrincetonUniversity Press. Princeton,N .J.,
1957 orM ,E.Rose.%sM ultipole Fields.''John W iley and Sons,Inc.. New York,1955. The
treatmtntherefollowsthatofEdmondsveryclosely.
*2
APPENDIXES
ora Paulimatrix
J= l.
l
Eq.(B.1)isindem ndentofany particularrepresentation,and allthe results
depend only on the :ftWc commutation relations. W e defne the raising and
lowering operators
Jk=Jx+ iJy
(B.2)
and thesquareofthe totalangularmomentum operator
J2-Jl+ J2
x+Jk=J;+ jLJ+J-+J-J+à
(B.3)
ltisthen easy to verify thefollowingcommutation relations
(J2,Jz)= (J2,
A )= 0
iJz.Jz1= :
iVz.
(B.4)
The om rators.
12and h can therefore be diagonalized simultaneously,and the
eigenstateswillbe labeled by 1./rn). From thecommutation relations(B.1),
itisreadilyproved thatl
Jk1./zn)- Aç-i.+,*1.
/- + 1)
(B.5)
where
Allmt= R.
/+ mtfl- m + 1)11
(B.6)
desnesa particularchoiceofphases,and
J21./-)=jlj+ 1)Ijm)
(B.7)
Z l.imâ= m 1jms
IngeneralwedealwithstatesIy)A)whereydenotestheremainingsetofobservablesand
El>,J2)= EP,.6)= 0
coupLlNG OF Tw O ANGULAR M OM ENTA: CLEBSCH-GORDAN
CO EFFICIENTS
Considerthe problem ofadding two com m uting angularm om enta
J= Jl+ A (Jl,Jc1= 0
(B.8)
whereJâand Jzcan referto diflkrentparticlesoroperateindiserentspaces(for
example,J= L + S). Wecanimmediatelyverifythat
(AJI!- (J,Jl)- (J,J2)- 0
(B.9)
tSee,forexample.L.1.Schifr,leouantum Mechanics.''3ded.,p.2œ ,M cG raw-llillBook
Company,New York.1968.
AppENnlxEs
Onecomplete setofco.
.
-ng operatorsisthereforede*ned by
r',J1,J'
lz,Jl,&x
A notherpossiblechoiceis
r:J2
1,J2
2:J2:JJ
Eitherdescription iscom plete,butthe set2 ism oreconvenientG causeoverall
rotationalinvariance impliesthatj and m are constantsofthe motion. Since
these basesare equivalent,they mustbe connected by a unitarytransformation
Iy-/lhjmb- J: (ij,:11h -21.
/,h jmsl
y-/lm,)'z-z)
(B.10)
*1lM z
ThesetofnumericalcoeëcientsareknownasClebsch-Gordan(CG)coeëcients;
they areindependentofy because thetransformation involvesonly the angular
partsofthewavefunction (thisresultwillbeshownbyexplicitconstruction;see
Eq.(B.l7)). Theorthogonality and completenessofthestatespy7jh jm)and
ly.
/lmb.
/2M2)implytherelations
ty-/l?Fl1h mz1y71mt
'jz?.
n2
')= 3mlml'3mamz,
(y-/lJ'z./rnl
y./ljzl'rn/)= 'JJ'3-,
.,
(B.l1)
)(Iy/lhims(y/lhjmz- Z Iy7l,:21h ?nz)ty.
/lm,.
/zmzi
J-
- 1mz
and itfollowsthat
7J (J,Jzl m 1J1m,J2rnaz4J,mtJa'nall,Jz-lms- JJ' mm,
?NlMz
(B.12)
'
(
E2'
:Jlm 'j/2mz'1J-IJzJ'n)(J-IJzJm 1J-,mlJ-zmz$- 3,,,,,
.,1'3-z-z'
.
Jm
The CG coeëcientscan befound by diagonalizing theom rators
Jz=Jls+ Jz,
(B.13J)
J2= J(+ Jl+ 2JI.Jz
(B.l3:)
intheEnitebasisIy/lml./arrl2). SincetheoperatorsJlandJIarealreadydiagonal,itisonly necessary to diagonalizeh and Jl.J2where
2Jl.Jc=Jt+A -+Jt-h++ u tzhs
(B.14)
Forthehrstoperator,wew rite
Jz1y71.
12.
/z>l)- &lly.
/lh .
/zn)
(B.15)
and use Eqs.(B.10)and (B.13J)to conclude that(jL-1hmzIjbjz.
/v)= 0 if
rnt+ m z# m . Forthesecond operator,wecom bine therelation
2Jl-J2ly/lh .
/r?
i)= n Iyylh .
/rn)
(B.
l6)
ux
AppENnjxEs
obtained from Eq.(B.l35)with Eqs.(B.l4)and (B.5)to 5nd a setoflinear
homogeneousequationsfortheCG coeëcients
(2,n1mz- aJ$(.
/1?.
n1.
/2mzi./lh jm$+ ,4(.
/1,m1+ 1).
,1(/c,-zn2+ 1)
x (.
/1mt+ l.
/zmz- l(.
/1.
/2jm3+ AljL.-?.
n1+ 1)
x -4(.
/2,n12+ 1)Ljkp1l- l./zl'
nc+ lh.
/lhjmb= (1
(B.17)
.
ThissetofeigenvalueequationsforaJhasa solutiononly if
Ijt-A I<j*.
/1+A
(B.l8)
with each valueoîjoccurringonceandonlyonce. TheresultingCG coeëcients
aredetermined onlyupto overallrelativephases. Thusacompletespecisdation
oftheCG coeëcientsrequiresanadditionalsetoîphaseconventions;thestandard
choicelisthattheCG coeëcientsarerealand satisfy
(B.19J)
(B.l9:)
I..jb?
N1h mz1./1hlmï
b= (-l)7'+#:-J(.
hmz.
/1mql)1)'
tjms
2.1+ l 1
(
jLM1h mlI./1J1jmb= (-1)72+M2 ljL+ i (jz-mzjm j/cjjj-1) (B.19$
.
.
TwoparticularCG coem cientsarevery useful:z
(L-ij/A71* 1./10./1m,)= 1
(j3&l1jj- -1r.
/l./1(X))= (-1F1-M1(ljj+ 1)-1
(B.20J)
(B.20:)
A more symmetric form ofthese coeëcients isgiven by the W igner ?-j
sym bol
(M
.
/1
l M
lq
3N(-jj.
il-Jz-maLljj.
j.j)-.
y(yjsyjjgyj
ajyjjgjj.m.
y)
which hasthe following properties:
mk+ mz+ mj= 0
71 .
12 -/3
-
r?1, -.rz;z - r?:3
(8.22(z)
=
(-j),.j./,+.y: m./1b
.
h
-3
(j;.zzy)
3.Any even perm utation ofthe columns leavesthe 3c/coeëcient unchanged.
4.Anyoddpermutationofthecolumnsgivesafactor(-1)7l+7a+J3.
Extensive tabulationsofthesecoeë cientsappearinthe literature.3
1A.R.Edmonds,op.cfJ.,pp.41-42.
2Notethatforhalf-integral/:(-1)7-m= (-1)'
n-1= -(-1)J+m.
3M .RotenY rg,R.Bivens.N.M etropolis,and J.K.W ooten.Jr.,K*f'
he qjand 6J'Symbols,''
TheTechnologyPress,M assachusettsInstituteofTechnology,Cambridge,Mass., 1959.
APPENDIXES
B85
COUPLING OF THREE ANGULAR M OM ENTA:THE 6-j COEFFICIENTS
Considerthreecommutingangularmomenta(theindexywillnow besuppressed).
A basisofdehnitetotalangularm om entum can be form ed by coupling the hrst
two to form adehniteylcandthencouplingylcand /3to form j
-
.
)<(.
/12mj2./3mql./i2jj.
/?n), (B.23)
orby coupling the second and third to form jzjand thencouplingjjandg'z3to
form /
x (/im lJ'
23=23Ijt.
/23jm?s (B.24)
Either ofthese schem es gives a com plete orthonorm albasis,for the properties
ofthebasisstates1./1znlh m2jjrna)andtheCG coeëcientsimply
kx(
J'l.
/2)jt
'z./3./'m'(./1/2).
/12J'
3jm3= 3jk,k,zbj.jbm.m
Z p(.
/l.
/2)./12jj.
/rn)((J'1hljj2j?jm i
jlajm
.
-
(8.25(7)
Z (.
/1,,71jz'nz.
/amq?'r:.
/,mb.
/cmzh r,731 (B.25:)
p1jm aFa:
with sim ilarrelationsforthesecond basis. Asaresult,thetransform ation from
one basisto the other is again unitary;the 6-j symbols are desned in termsof
thesetransform ation coeë cientsby
(B.26)
Thesetransform ationcoeë cientscanbeprovedindependentofm inthefollowing
way. Consider
(B.27)
*6
APPENDIXES
Now operateon both sidesofthisequationwithJ+lA(j,-m). Thisraisesthe
m valueofthestateswithoutchanging the coeë cients.w hich stillhavean index
m . By defnition.however,the expansion coeë cients ofthe new states have
an index m + l,and thusthecoeë cientsareindependentofm.
From the deinition,it is seen thatthe 6-j symbols are invariant under
(l)anypermutation ofthecolumnsand under(2)theinterchangeoftheupper
and lowerargum entsineachofanytw ocolum ns. Anothervery usefulrelation
thatfollowsfrom Eqs.(B.23)to(B.27)is
The6-jcoeëcientsalso havebeen extensivelytabulated.l
IRREDUCIBLE TENSO R OPEBATO RS AND THE W IG NER-ECKART THEO REM
A n irreducible tensor operator ofrank K isdefned to be the set ofIK + 1
operatorsTIKQ)(-A'< Q < A-)satisfyingtheequations
EJz,F(X'Q))- -4(& :
FQ)T(& Q :iu1)
(/z,F(#Q)l= QTIKQS
(B.29)
Someexamplesoftensoroptratorsare(in Nrstquantization):
T(r)YtM(0,+
(-Z < M <Z):tensoroperatorofrankL
2. rq
(-1< q< 1):tensoroperatorofrank l
where
1
rz1H :1Z v (.
X+ iy) roO z
vj
Jq (-l< q< 1):tensoroperatorofrank 1
where
J
1
tl* ::
F , (Jx+ iJ,)
vj
E'
N
1
otethatJzj= ZFVé.&.)
w e now introduce the W igner-Eckart theorem.which states thatthe m
dependence of the matrix elem ent of an irredueible tensor operator can be
1M .Rotenlxrgetal.,loc.cit.
A pPEN D 1.'xEs
687
extracted explicitly interm sofaClebsch-G ordancoeë cient:
'j'tn'Ir(K(?).yjm-= (-l)K-J-j''Koj
m--SK
j
'??
'X
'
.-.
.jj
--7
-.qxy'g''pjrx8
:yjh
'
i'.
(j).
;>.
).
;'
.
.
(B.30)
Here the reduced rntzrrïa-elemen: y'j'(Fxl
'y/ is independent qf the m t'
alues.
Thispowerfuland uset
-ultheorem can be proved asfollows:
l. Consider
The commutation relations (B.29) between the angular momentum and
TIKQ)(using J2= KJ-J-vJ-J-j-,
-Jzlgjand the setof linearequations
dehningtheCG coemcients(B.17)show that
,2!
.
Y!*jm),=jlj-..l)j
N.-jm)
,.
Jz1Y1*jm'
)= m i
klqjm''
,
.
2.W ecanthereforeexpand J
Vl->).inthebasisstatesJyjm'
Itjâjm(.z.
crP
N (ty'jm jk1,
*Jm).
.
j
y'jn)'h
and the coem cients kye-/
rrll
klLm.
'
)are independentofm
asproved following
Eq.(B.27),
3.Theorthogonality oftheCG coemcientsim pliesthat
TIKQbh'
lz
-/lm l)'- )'
('
.
L.
/1m lKQ i.
/lKjm)I
.
'1'
-km)
jm
4.Theinnerproductwith thestate '
k.y'jzmj'gives
.
fy'./2m2l
TIKQ4Iy-/)m tb= v../1/3:1KQ !.
/1KJ'zmz'
)'
t
.v'h 1
'
k1-;'
whieh isjusttheW igner-Eckarttheorem,with thereduced matrix element
dehnedintermsofky'jz1:1-jz''throughm-independentfactorsandsigns.
TENSO R O PERATO RS IN COU PLED SCHEM ES
ConsidertwotensoroperatorsF(A'j@l)and U(KzQz),which weassumerefer
to diserentparticlesordiflkrentspaces. ltiseasilyverised from thedehnition
Eq.(B.29)and from thedehning setoflinearequationsfortheCG coemcients
(B.17)thatthequantity
A-(A'Q)ëBE N' sKj(?jKj:21A'1K:KQ$F(A-IQ I)L'(K?;c)
(B.31)
OIO1
isagain an irreducibletensoroperatorofrank K. A specialcaseofthisresult
isthe scalarproduc:oftwo tensor operators
(2S v 1)1'(-1)K.
1,-(0)EEF(S).L'(A')= N-(-1)0TIKQ,&(A'- Q)
,
V
(B.32)
s88
AppENolxEs
which is a tensor operatorofrank zero and therefo
inserting a com plete setot-states
re com m utes with J. By
, usi
ng the W igner-Eckarttheorem , and using
thedeEnitio
noftht 6-jsymbol,wtcan provethat
(y'1Ih'j''n'Ir(A'
)-&(&)4y/.h ./.n)
.
J;'
lk
!Z (vzJ.I
I,
y,.y.
j.)y,
'jgtj
yy(g.jg
t
'
yy'
zj
/2J
v,
' ,J)y.(kj;
,
-
(B.33)
Inexactly thesamefashion, them atrixelem entofa singleoperatorin a
schem eis
coupled
/z
.
,,
K 't)y /l1F(A-)1y:
/1-,
,
(B.34J)
K
/')tz-/c
'l
jrtx
l
h
z
/
c
)
(B.34:)
.
In theserelations, F(#)and U(K)actonthefirstandsecondpartsof
-thewave
function,respectively.
IN D EX I
Addition theorem for spherical harmonics,
516
Adiabaticprocess,187
Adiabaticelswitchingon.''59-61,289
Analytic continuation,1l7p,297-298, 302303,493
Angularmomentum,344,504,571p
review of,581-588
Angular-m om entum operator.505
Anomalousamplitudes,489
Anomalousdiagrams,289
Bogoliubov replacement,2* ,201,203,315,
489
Bogoliubov transformatiom 316a, 326-336,
527-337
Bohrradius,25
Bohr-sommerfeld quantization relation,425,
484
Boltzmanndistribution,39,279
Boltzm ann'sconstanty36
Born approximation,188,197/,219:259,345,
368
AnomalousGreen'sfunctions(bosons),213
Born-oppenheimerapproximation,390a,410
Anticommutatitm relations,16
Bose-Einsteincondensation,44,198-2* ,211,
Atoms,116,121,168/.195/,503,508,546,567
481
rigorousderivationof,41a
andsuperconductors,441,446,476#
BCScoherencelength.426.465,469
Boson approximation in shellmodel,526-
BCSgapequation(seeGapequation)
BCStheory(seeSuperconductor)
Bernoulli'sequation,496
Betadecay,350-351
Bethe-Goldstoneequatiom 322,358-366,567
anomalouseigenvalue,324
withdegeneratestates,569-570
eflkctive-massapproximation.359
energyshiftofpair,371
andGalitskiiequations,377-383
ground-state energy of hard-sphere gms
371-374
J-waveinteractions,371-374
p-waveinteractions,374,386/
power-seriesexpansion,373-374
hard-core,solutionfor,363-366
partial-wavedecomposition,386/
J'-wave,360
self-consistent,358-360,377-378
square-well,solutionfor,360-363
Bethe-salpeter equation,131-139,219,562565
(SeealsoGalitskii'sequations)
Blochwavefunctions.5
Bogoliubov equations for superconductor,
477#
527,576p
Bosons,7,198-223,314-319,479-499
Bethe-salpeterequation,219
charged,223/,336/,5(f#,501#
chemicalpotential,202,206,216,493
condensate,2X ,493
attinitetemperature,492-495
moving,223#,336/,501#
nonuniform,495-499
densitycorrelationfunction,ll3p
distributionfunction.37,218,317
Dyson'sequations,211-214,223#
Feynmanrules:
incoordinatespace,208-209
inmomentum space,209-210.223#
fieldoxrator,2*
Greengsfunctions,203-215
ground-stateenergy,31#.201,207,318
hamiltonian,2X ,315
Heisene rgpicture,204
Hugenholtz-pines relation,216,220,222,
ll3p
interactionpicture,207-208
kineticenergy.205
Lehmannrepresentations214-215
momentum om rator.204
t'
FheletternafterapagenumG rdenotesafootnoteandp,aproblem.
09
œ en
INDEX
Condenute.33,2* ,220,317.491
inachannel,502,
nume ro- rator,201-202
idealBOK gas,42
m rturbationtheory,199,207-210
measurementof,495
potentialenergy,7tm,205-206
moving,502/
prom rselfenerr ,211,215,219
andsux rfluiddensity,495
temm ratureG- n'sfunction,491
surfaœ energy,497-498
thermodm amicpotential,37,38,202,207
wavefunction,489,492
vacuum state,201
boundarycondition,496
W ick'stheorem ,203,223#
atfinitetemx rature.494-49,
5
Hartre equationfor,490
fseealsoHard-sphereBOR gas1Interacti
ng
spatialvariationofy497
Bose> )
Brum kner-Goldstone theory and thermoCondensationenergy ofsum monductor,419,
dynamicpotential,288-289
453
Bruœkner'stbeory,116,357-377,382-383
Cov gurationspace,7
Bulkmodulus,30,390,407
Cormcteddiagrams,113,301-302
Bosons(cont'J):
noninteracting(seeIdealBosegasl
Bulkprom rtyofmatter,2A 349
(SeealsoDisconncteddiarams)
Constantofthemotion,59
Continuitym uation.183,420,496
Canonicale- mble,33
Contractions.87-89,238,327-329
Canonicalmomentum .424-425
Coom rpairs.320-326.359-3* .417,441
Canonical t- nmformation to particles and
bindingenergyy3zs,336
holes,70-71,l18#,332,5@$-508,547
bound-statewavefunction,336#
Charge-densityom rator,188.353
Com-polarizationpotential,574,$77p
Clwv dN x gas,223p,336/,5X #,501/
Correlation energy.29.155, 163-166,169#,
G emiOlpotential.34.* ,75.1B7,327,52828* 287
532,535
anddielxtricfunction,154
Y sons,202,206,216,493
andpolarizationpropagator,152
hard-spheregas,220-222
Correlations:
ideal.39-41,43
innuclei,362-363,365-366,572-573
clmssicallimit.39-40
two-pm icle.191-192
diFerenœ in sux rconducting and normal Coulombenergyofnuclei,35û
state,335.453a
Coulombinteraction,22,188
electrongas,278,284-285
(SeealsoScreeninginaneledrongasl
Coupling-constant integration,70, 231-232,
fermions)
hard-spbere- ,174
280,379
ideal,45-48,75.284-285
Creationox rators,12
Criticalangularvelocity,499,5(f#
phonons,393,410#,485-486
andprom rself-energy.1X
Criticalcurrent.476p
Criticalield,415,451,453,474#
Cimulation,483.498
f7lnttsills-clalxvoneqtlatioh.4gg-sY p
CriticaltemrratureofBosegms,40,259-261
Clebsch-Gordancœ mcients,544,582-584
Criticalvelmaty,4K a,482.487,5*,
Cloe fermionloops,98,1û3
Crosssection,scattering,189,191,314-315
Ce mcientsoffractionalparentage,523a,516p Crystallattice.21,30,333,389.390,394-396
Cohemnœ lengtb:
(SeealsoPhonons)
Curie'slaw,254#,309,,5*,
BCS,426,465,469
Curmntox rator,455
Ginsburga>ndautheory,z33,472
sulx= nductory4zz,426.433.472
Cyclicprox rtyoftrace.229
Collc tivemodes,171,183,193-194,538
Collisiontime,184
Damping,81,119.p,181,195#,308.309#,310#
Commutationrelations,12,19
nebyefrequency,333,394,395n,196,4ztcn
Compr- ibility.30.1X ,3:7J,,30 -391
s91
INDEX
Debye-Hikkeltheory,278-281,290,
Debyeshieldinglength,279,30* 308
Debyetemperature,394-395,448
Debyetheoryofsolids,389,393-395
Deformednuclei,515
Degeneracyfactor,38,45
Deltafunction,101,246
Density:
offreeFermigas.2* 27
ofHe3andHe4#480
ofnuclearmatter,348-352
ofstates,38,166-267,333,447
Density correlation function, 151, 174,217
3(10-303,558
analyticprom rties,181n,302-303
forbosons,2l?p
forfermions,194#,302.309,
atfinitetemperature,3(*-.302
Letlmann representation,3(*-301
perturbationexpansion,301-302
andpolariz-ation,153-154.302
relationtopolarizationpropagator,153,3û2
retarded,173,194#,3(0-301,307
time-ordered,174.175
D ensityiuctuationoperator,117#,189
Depletion,221,317,488
Destructionoperators,12
Deuteron,343
Deviationoperator:
forbosons,489
fordenstty,151,173,3K ,560
Diagrams(seeFeynmandiagrams)
Diamagneticsusceptibility,477/
Dielectricfunctions111,154,184,396
ringapproximation,1,
56,163.175,180
Digammafunction,580
Dipolesum rule.552,577#
Direct-productstate,13.17
Aisconnecteddiagrams.94-96,111,301,560
factorizationof,96
Dispersionrelationls):
forplasmaoscillations,181-182,310#
forpropagators,79,191,294-295
forzerosound,183-184
Distributionfunction:
forbosons,37,218,317
forfermions,38.46,333-334
formovingsystems486,5* ,
Dyson'sequations:
forbosons,211-214,223p
forelectron-phononsystem,402-406,411/
atfmitetemperature.250-253,412p
Dyson'sequations(coht'dl:
forGreen'sfunction,105-111
Hartree-Fockapproximation,122-123
forpolarization,110-111,119#,252,271
forsuperconductors,476#
Effectiveinteraction,155.166-167,252-253
Efrectivemass:
electrongas,169#,310/
ofimperfectBosegas,260
ofimperfectFermigas,148&168p,266
innuclearmatter,356,369-370
insuperconductor,431
Efective-massagproximation,359,3#4
E/ectiverange.342-343,386,
Eikonalapproximation,468-469,474
Electricquadrupoleoperator,514
Electron gas:
adiabaticbulkmodulus,390
chemicalpotential,278,284-285
classicallimit,275-281,290#
correlation energy,29,155,163-166,169/,
28* 287
couplingtobackgrounds389,396.
.406
degenerate,21-31,151-167,281-289
dielectricconstant,154
dimensionlessvariablesfor,25
eflkctiveinteraction,155,16* 167
efl-ectïvemass,169#,310/
electricalneutrality,25
ground-stateenergy,?2p,151-154,281-289
hamiltonianfor,21-25
Hartree-Fockapproximation,289p
heatcapacity,289-290/
Helmholtzfreeenergy,280.284-285
linearresponse,175-183,303-308
plasmaoscillations,l80-183,307-308
polarized,32p
properself-energy,169/,268-271
screening,175-180,195/,303-307
single-particleexcitations,310p
thermodynamic potential, 268, 273-275,
278.284
zero-temperaturelîmit,281-289
Electron-phononsystem :
chemicalpotentialofphonons,410p
coupl
ed-heldtheory,399-406
Dyson'sequations,402-406:411p
Feynman-Dyson perturbation theory,
399-4û6
propereleclronself-energy,402,411,
B92
Electron-phononsystem (cont'dj:
properphononself-energy,402,411p
vertexpart,402-406
equivalentelectron-electron interaction,
401-402
FeynmanrulesforT = 0,399-401
ieldexpansions,396-397
snite-tem peratureproperties,412/
ground-stateenergyshift,399,411p
hamiltonian,398
interaction,320,39* 399.417
linearresponse,412,
Migdal'stheorem,406-410
phononselds410/
phononGreen'sfunction,4* ,402,411p
screenedcoulom binteraction,397
superconductingsolutions,440/,476p
Electron scattering,171, l88-194,348-349,
557,566
Electronicmeanfreepath,425
Electronicspecischeat,395a
Energygap:
innuclearmatter,360,383-385,388/
innuclei,385,526,533
insuperconductors,320,330,417,447-449
Ensem bleaverage,36
Entropy,34-35
ofH eII,486
ofidealFerm igas,48
ofanidealquantum gas,49p
ofimperfectFermigas.265-266
ofsuperconductor,419,476/
Equationofstate,187
ofelectrongas,30,278-281
ofidealBosegas,39,42
ofidealclassicalgas.39
ofidealFermigas,45,46,47-48
ofultrarelativisticidealgas,49p
Equations ofmotion,linearization,440-441,
538-543
Equilibrium thermodynamicsand temperatureGreen'sfunction,227,229-232
Equipartitionofenergy.394-395
Euler'sconstant,580
Exchangeenergy,29,94,126-127,168,,354
Excitationspectrum ,81
interactingBosegas.217,317
innorm alstate,334
insuperconductors,334.334a
Exclusion principle(seePauliexclusion
principle)
Extensivevariables,29,35
INDEX
External perturbation, 118/,122, 172, 173,
253,,298,303
Factorizationofensembleaverages,441,457
Fadeevequations,377
Fermigas:
interacting (see Hard-sphere Fermi gas;
InteractingFermigas)
noninteracting(JecldealFermigas)
Fermi-gasmodelfornuclearmatter,352-357
Ferm im omentum ,26
Fermimotion,193
Fermisea,28,71
Fermisurface,179-180,306,334
Ferm ivelocity,185
Fermiwavenumber,27,46
Ferm ions,15-19
Fermi'seiGolden Rule,''189
Ferromagnetism,32p
Feynmandiagrams,96,378.399-401,559-562
incoordinatespace,92-1*
atfinitetem perature,241-250
inmomentum space,100-105
Feynman-Dyson perturbation theory, 1121l3,115,399-406
Feynm anrules:
forbosons,208-210,223p
forelectron-phononsystem,399-401
forfermions,97-99,102-103
athnitetemperature,242-243,24* 248
Fieldoperators,19,65,71
forbosons.200
commutationrelationsofs19
creation part,86
destructionpart,86
equationofmotionfor,68,230
Finite-temperature formalism, F = 0 limit,
288-289,293,296,308/
Firstquantization,relationtosecondquantization,15
Firstsound.184,481
Fission,351
Fluctuations.2œ ,337#,527-528
Fluxquantization,415-416,425.435-436
Fluxquantum ,416,435
Fluxoid,423-425,474#
Forwardscattering.133
Freeenergy:
Gibbs,34
ofmagneticsystems,418
ofsum rconductor,432
INDEX
Freeenergy Lcont'dj:
Helmholtz.34
ofclassicalelectrongas,280
ofm agneticsystem s,418
ofnoninteractingphonons,393
ofquasiparticlesinHeII,485
ofsuperconductor,43l,451,453,474,
FreesurfaceinrotalingH elI,483
Frequencysums,248-250,263.281
Friedeloscillations,179
Galitskii'sequations,139-146,358,370,373
376,567
andBethe-Goldstoneequations,377-383
Gammafunction,579-580
G apequation,331.492
insnitenuclei.531,535
normalsolutions,332,531
innuclearm atter,383-385
superconducting solutions, 333, 446-449,
475p
Gapfunction.443,466-474,489
Gaplesssuperconductors.417n,460n
Gaugeinvariance,444,454
Gell-slafm and Low theorem, 61-64, 1l3
208,
G iantdipoleresonance,552
Ginzburg-tandauparameter,435,472
Ginzburg-Landautheory,430-439,474
boundaryconditions,432
coherencelength,433,472
determinationofparameters,471-472
fieldequations,432,496a
fluxquantization.435-.436
microscopicderivationof,466-474
inonedimension,437
penetration length,434,472,475/
supercurrent,432
surfaceenergy,436-438
wavefknction,471
Goldstonedi
agrams.1l2.l18p,354, 376,381
Goldstone'stheorem,111-1l6.387/
G orkovequations,444,466
Grandcanonicalensemble.33,228
Grandcanonicalhamiltonian,228,2.56
Grandpartitionfunction.36.228,308/
Green'sfunctionsatzero temperature, 64-65,
2û5,213,228,292,4%
advanced,77
analyticproperties,76-79
asymptoticbehavior,79,297
593
Green'sfunctionsatzerotemperatuzetcanl'#):
bosons,203,215
anomalous.213
hard-sphereBosegas,220
idealBosegas,208
matrixGreen'sfunction,213-314.501p
dlagrammaticanalysisin perturbation
theory,92-111
forelectron-phononsystem,399-406
atequaltimes,94
equationofm otion,117,,411p
Feynmanrulesfor:
incoordinatespace.97-99
inm omentum space.102-103
atsnitetemperature(seeTemperature
Green'sfunctions)
frequencydependenceof',75
Hartree-Fockapproximation,124
foridealFermigas,70-72
forinteractingFerm igas,145
ininteractionpicture.85
m atrixstructure.75-76
perturbationtheoryfor,83-85.96
forphonons,400.402....404,410p,41jp
physicalinterpretation,79-82
real-time,at finite temperature (see RealtimeGreen'
sfunctions)
relationto observables,66-70
retarded.77
aszero-tem perature lim itofreal-tim e
Green'sfunction,293,296,308/
G ross-pitaevskiiequation,496
Groundstateinquantum-feldtheory,61
G round-stateenergy;
fbrbosons,31,,201,207,318
hard-spherebosegas.221-222
ofelectron gas,25-26,32/,151-J54, 28l289
electron-phononsystem.399,411p
andGreen'
sfbnctions,68
of-hard-sphere Fermigas.132, 135, 148149,319,373-374,387,
Hartree-Fockapproximation,126
ofidealFermigas,27,46
ofnuclearmatter,353-355,366-377
andproperself-energy,109
shiftof,70.109.l11
superconductingandnormalstates,335
andthermodynamicpotential,289
time-independentperturbation theory,31r.
l12.118p
Groupvelocity,183
694
Hamiltonian:
tirst-quantized,4
m odelsforphysicalsystem s:
bosons,200,315
electrongas,21-25
electron-phonon,333,391-393,396-398
pairingforce,infinitenuclei,523
superconductors,439-441
second-quantized,15,18
Hard-sphereBosegas,218-223,317-3l9,480481
chem icalpotentialy220,221-222
depletion,221,317
Green'sfunction,220
ground-stateenergy,221-222,318-319
otherphysicalproperties,222
properself-energies,219
Hard-sphereFerm igas:
chemicalpotential,147
efl
kctivemass,148.169/,370-371
eflkctivetwo-bodyinteraction,136-137
efectivetwo-bodywavefunction.137-139
ground-state energy, 135, 148-149, 169/.
319,373-374,387,,480
heatcapacity,148
properself-energy,136,142-146.168/
single-particleexcitations.146-148
zerosound.195,.196p
H arm onic oscillator,12.393,509-511.569571
Hartreeequations,127,490
H artree-Fockpotential,355-357,511,568
Hartree-Fock theory. 121-127, 167/. 168/,
399,475,,504-508.575
forbosons,259-261
equations,126-127,257-258,507
solution foruniform medium,127,258259
offinitenuclei,575
at snite tem perature, 255-259, 262-267,
308,,475/
Green's functions, 124, 168/, 257. 308/,
440-441,476p
ground-stateproperties,l26-127,332
properself-energy,12l-122,125.255-258
relationtoBCStheory,439-441
self-consistency,121-122,127.258-259,265
single-particle energy, 127.258,330,507,
510,513,539,556
Hartree-Fock wave functions,352-353,503,
508-511,541,558-559,567
Stl-lealingdistance,''366,376
INDEX
He4,liquidVeeHeII)
He3,liquid,49,116,128,480
Brueckner'stheory,150
heatcapacity,148
zerosound,187
HeII,44,481-488
criticalvelocity,482,488
entropy,486
heatcapacity.:l.4.484,486
phasetransitionof,44,481
quantizedvortices.482-484,488
quasiparticlemodel,484-488
phonons,484-488
rotons,484-488
surfacetension.498
two-iuidmodel,481
Heatcapacity:
Debyetheoryofsolids.393-395
ofelectrongas,269,289-290/
ofhard-sphereFermigas,l48
ofHe11.M ,484.486
ofidealBosegas,42.43
ofidealFerm igms.48
ofimperfectFermigas,261-267
ofmetals:
Hartree-Fockapproximation,269,289/
norm alstate,295n
superconducting state, 320, 416, 420,
451-454
Heisenbergpicture.58-59,73,173,189
forbosons.204
groundstate,65,558
modihed,forhnitetemperatures,228,234
operators,65,115,213,292
relationtointeractionpicture,83-85
High-energy nucleon-nucleusscattering,566
Hole-holescattering.149-150,381
Holes,70-71,504-508,514,520,524,538543,558-566
Homogeneous(uniform)system,69,190,214,
292.321
Hugeltholtz-pinesrelation,216,220,222,ll3p
Hydrostaticequilibrium,50p,177.195/,386#
IdealBosegas:
chemicalpotential,39-41,43
criticaltemperature,40
equationofstate,39,42
G reen'sfunction,208
heatcapacity,42,43
num berdensity,39
595
IN DEX
ldealBosegas(cont'
ô:
lnteractingBosegas(cont'
d):
nearFcv259-2.61,493
occupationnum ber,37
phasetransition.44
tuWcalsoHard-sphereBosegas)
statisticalm echanics.37.-44
lnteractingFermigas,l28-150,261-267,326superfluiuzty',493
336
temperatureGreen'sfunction,232-234,
distributionfunction.333-334
245-246,501p
effectivem ass,167/,266
therm odynam icpotential,37a38
entropy,265-266
ground-stateenergy,27,118#,168/,319
two-dim ensional,49p
heatcapacity.261-267
IdealFerm igas:
magnetfzatfon,32p,169p,?10p
chemicalpotential,45,48,75,284-285
properpolarization,169#,196/
density,26-27,45-47,352
entropy,48,266-267
zerosound,183-187,196/
(SeealsoHard-sphereFermïgas)
equationofstate,45,46,47..-48
Ferm ientrgy,46
Interactionpicture,54-58
ground-stateenergy,26,46
forbosonsa207-208
forfinitetemperature,234-236
heatcapacity.48,26* 267
Internalenergy,34,247,251
occupationnumber,38
param agneticsusceptibility,49.p,254.p,309/
Hartree-Fock,atEnitetemperature,258
statisticalm echanics,45-49
ofideaîquantum gas,39,46,49p
tem peratureGreen'sfunction,232-234.
and temperature Green's function,
247,2f2
245-246
tllermodynamicpotential,38,278,285
lntem articie spacing,25,27,349,366, 389,
two-particlecorrelations,192
394.397
lmaginary-timeoperator,228
lrreducibfedfagram,403-405
lmperfect Bose gas (see Hard-sphere Bose lrreducibietensoroperator,505,508.543,586
gas;lnteractingBosegas)
lrrotationalflow,425.481
ImpedkctFermigas(see Hard-sphere Fermi lsotopeeffect,320,417,448,476/
gas;lnteractingFermigas)
Isotopicspins353,508,546
Im pulsiveperturbation,180,184,307
Independent-pairapproxim ation,357-377,
480
Josephsoneflkct,435n
ground-stateenergy,368
justiscation,376
-,,.zoj.2,:9
K ineticenerg3,.4.a?,(,
self-consistency,368
Kohnejyec!. 4jkp
single-particlepotential,368
Kroneckerdelra. ,,23
lndependent-particle model of tht nucleus.
352-357.366
justification,376
Integralkem elforsuperconductor.456,458
inPippardIimit.463
Inlegrals,desnite,579-581
lntensivevariables,35
InteractingBosegas,215-218,219.314-319
chemicalpotential.216,336,
depletion.218.317
excitationspectrum,217,218,317
ground-stateenergy.318,336,
movingcondensate,223/,336,,501/
properself-energies.215
soundvelocity,217,317
supe/ uidity,493
Ladderdiagrams,131-139.358,378-379.567
Lagrangem ultiplier,203,486/,500/
Laguerrepolynomials.509
Lam bdapolnta481
Landaueriticalvelocity,488,493
Landaudamping.308
Landaudiamagnetism,462,477/
Landau'sFermiliquidtheoo',187
Landau-squasiparticlemodel.484-488
Legendrepolynomials,516
Legendretransformation,34,336,
Lehmannrepresentation:
forbosons.214-215
forcorrelationfunctionss299.300-301,456
596
INDEX
N eutronscattering,171,194,196,,485
forGreen'sfunctions atzero temperature, N eutronstars,49
66,72-79.107
Newton'ssecondlaw,183,186,420
forpolarizationpropagator,117p,174,300- Noncondensate,491s494-495
Nonlocalpotentials,322
301,559
forreal-timeGreen'sfunction.293-294
Nonuniform Bosesystem,488-492,495-499
fortemperatureGreen'sfunction,297
Normal-fluiddensity,481,486-487,5(X1,
Lifetimeofexcitations.81-82,119#.14* 147, Norm al-orderedproduct,87.327
N uclearm agneticresonance,179
291,308,309,.310p
Linearresponse.172-175
Nuclearm atters116,128,348-352.480
bindingenergyofA particlein,387/
ofchargedBosegas,501/
ofelectrongas,175-183,303-308
bindingenergy/parti
cl
e,352,353-355,371electronscattering,188-194
377
infnitenuclei,566,577/
Brueckner'stheory,150,357-377,382-383
atsnitetemperature,298-303
compressibility,387/
neutronscattering,196-197/
correlationsin,362-363,365-366
ofsuperconductor,454-.
466
density,348-352
efrectivem ass,356,369-370
toweak magneticfield,309/7,454-466.477/
zerosound.183-187
energygap,330,360,383-385,388/
Linearization of equations of m otion.440Fermiwavenumber,352
healingdistance,366
441.538-543
London equations,420-423,425,434,459independent-pairapproximation,357-377
independent-particle model,352-357,366,
460,47411,475/
Londongauge,425,427,454,461
376
many-bodyforces.377
Londonpenetrationdepth,422
Londonsuperconductor,427
pairing,351,383,385
Long-rangeorder,489a
with tsrealistic''nucleon-nucleon potential,
366-377
reference-spectrum method,377
M acroscopicoccupation.198,218
saturation,355.357,375
M agneticseld,309/.418,420
single-particlepotential,355-357.381
stability',355
thermodynamics,418-420
M agneticimpurities,l79
sym m etryenergy,386/
M agnetic susceptibility.l74,254/,309/,
tensorforce,efrectof,367,375,386/
three-bodyclusters,376-377
310/
Magnetization,169/,309/
N uclearreactions,17l
M ajoranaspace-exchangeoperator.346,354 Nucleon-nucleon interaction. 341-348, 354,
367,504,567
M any-bodyforces,377
M axwell'sequations,417....418
nucleon-nucleonscattering,342-347
Meissnereflkct,414,421,423,457,459-460
phenomenologicalpotentials,348.361,367,
557,573
fbrBosegas,501/
summaryofproperties,347-348
criterionfors429
MeltingcurvesforHe,480,499-500,
Nucleus,49,5Qp,121,188,503
Metallicfilms,194,476p
Bogoliubovtransrormation,316/,326-336,
M etals,21.49,121.180,188,333,389
527-537
energyatfixedN.532
M icrocanonicalensemble,486/,500/
forevenandoddnuclei,533-534
M igdal'stheorem,406-410
M ixedstateofsuperconductors,415/,439
iuctuationsofR,527-528,537
M olecules,503,567
forpairingforceinthe/shell,534-537
M om entum ,24,74,204,332
restrictedbasis,528
M ultipoleexpansion oftwo-bodyinteraction,
single-quasiparticlem atrixelements,533516
534
Lehmannrepresentation(cont'dj:
697
lNDEX
Nucleus(cont'd):
chargedistribution,348-349
deformed,515
energygapa385,526,533
excitedstates:
applicationtoO l6555-558
constructionofH (co)inR PA,566
with3(x)force,547-555
the(15Jsupermultiplet,548
Green'sfunctionmethods,558-566
Hartree-Fockexcitationenergy,539
Nucleus(cont'd):
two-particlebindingenergy,568
single-particlematrix elements,512-515
m agneticm om ents,514
m ultipolesofthechargedensity.514-515,
551
singleparticleshellmodel,508-515
spin-orbitsplitting,511-512,513
sum ruless577/
two-bodypotential:
3(x)force,518-519
generalm atrixelem ents-51* 518
particle-holeinteraction,539
m ultipoleexpansion.516
quM ibosonapproximation,542,543
N um berdensity,20.66,229,247,251
random-phase approximation (RPA),
comparisonofsuperconductingandnormal
540-543,564-565
state,334,451
reductionofbasis,543-546
ofelectrongas,284
relation between RPA and TDA, 565H artree-Fockapproximation,124,257
566
ofidealBosegas,39
Tamm-DancroF approximation (TDA), ofidealFerm igas,45
538-540.565-566
ofquasiparticlesin HelIs486
transitionm atrixelements,540,543
Numberoperator,12,17,20,73,201-202.315
giantdipoleresonance,552
Hartree-Fock ground state,506,538-539,
560
magicnumbers,511
Occupation-numberHilbertspace.12,37,313
many-particleshellmodel:
Occupationnum bers,7,37.38
bosonapproximation,526-527
Odd-oddnuclei,350.517,576p
coemcientsoffractionalparentage,523a, O perator,one-body,20,66,229,512-515
576/
Opticalpotential,135,357
normal-coupling excited states,522-523, O rderparameter,431
576p
Oscillatorspacingsinnuclei,509/,569
normal-couplinggroundstates,520
one-body operator in norm alcoupling,
520-522
pairing-forceproblem,523-526
Pairing,326,337,,351,431,519
seniority,524
Pairingforce,523-526
theoreticaljustification,526
Parity,344,504,577p
two-bodypotentialinjshell,522-523
Particle-holeinteractions,192,539,562-563
twovalenceparticles,515-519
Pauliexclusion principle,15,26,47,127,134,
odd-oddnuclei.350,517,576,
184,193.322,344,357.480,520,572,574
pairing,351,383,385,519,523-537
fornucleons,353
realistic forces for two nucleons outside Paulimatrices,75-76,104,119/,196/,343,
closedshells,567-575
353
applicationto0 18 573-574
Pauli param agnetism ,49/,254,, 309,,443,
Bethe-Goldstoneequation,568-570
462,477p
harm onic-oscillatorapproxim ation, 57> Penetrationlength:
575
Ginzburg-tandautheory,434,472.47.$p
independent-pairapproximation,567superconductor(seeSuperconductor:
575
penetrationl
ength)
Pauliprinciplecorrection,574
Periodicboundaryconditions,21,352,392
relativewavefunction,572-573
Persistentcurrents,415-416
the(11supermultiplet,555
698
Perturbationtheory:
forbosons,199,207-210
fordensitycorrelationfunction,301
diagrammaticanalysis,92-116
forsnitetemm ratures,234-250
forscatteringamplitude,132-133
fortime-developmentoperator,5* 58
Phaseintegral,468-469
Phaseshiftforhardsphere,129
Phasetransition,44,259-261,431,481
Phononexchangeandsuperconductivity,320,
439,448
Phonons:
Green'sfunction,4œ ,407
Lehmannrepresentation,410,
forsuperconductor,477-478/
inHeII,480,484-488
interaction with electrons, 320, 396-399,
417
noninteracting,390-395
chemicalpotential,393
Debyetheory,393-395
displacementvector,391-393
normal-modeexpansion,392
quantizations393
Photonprocesses,566
Pictures:
Heisenberg,58-59
interaction,54-58
Schrödinger,53-54
Pippardcoherencelength,426,465,469
Pippardequation,425-430,465
Pippardkernel,428-429
Pippardsuperconductor,427,461-463
Plane-wavestates.21&127,258,352,392
Plasmadispersionfunction,305
Plmsmafrequency,180,182,223#,307
Plasmaoscillations,21,180-183,194,307-308
comparedtozerosound,186
damping,181,195/.308,310,
dispersionrelation.181-182,307-308,310p
Poisson'sequation,177,183,279
Polarizationpropagator,110,152,190
analyticcontinuation,302-303
dispersionrelationfor,191,3*
insnitesystems,558,563
constructioninRPA.566
athnitetemperature,252,271
in lowest order, 158-163,272, 275, 282,
3(*-305,561
relation todensitycorrelationfunction,153.
302
INDEX
Polarizmionpropagator(cont'dj:
inringapproximation.193,307
(SeealsoProperpolarization)
Positronannihilationinmetal
s,171
Potential energy. 4, 67-68, 2X , 205-206,
230
Potentials:
core-polarization,574,577p
nonlocal,322
separable,322
sho!'trange vs.long rangey127,167-168/,
186
spin-independent,1G4-110
symmetryproperties,328,529
Poynting'stheorem,418/
Pressure,30.34-35,222,278
Propagationoftheenergyshell,130,382
Propagator (see Green's functions at zero
temperature;Polari
zationpropagator)
Proper polarization,110,154,252-253,302303,402-405
atsnitetemperature,252
forimperfectFermigms,169p.196,
inIowestorder,158-163,272.275,282,304305,561
Properself-energy.105-106,355,402
forbosons,211,215,219
forel
ectrongas,169#,268-271,273,402
athnitetem> rature,250-251,2M ,309,
hard-sphereFermigas,142-146
in Hartree-Fock approximation, 121-122,
125,256,308#
forphonons,402
Prom rvertexpart,403
Pseudopotential,196/.3l4
Pseudospinoperators,524
Quantizedcirculation,484,496
Quantized iux,415-416,425,435.
-436,438439
Quantizedvortex,488,498-499
Quantum huid,479,489
Quantum statistics,6
Qumsibosonapproximation,542-543
Quasielasticm ak,193-194,196p,495
Quasiparticles,147.316,317,327,487,532537
inHeII,484-488
ininteractingFermigas,332,4* a
weightfunction,309,
INDEX
Random-phaseapproximation;
elKtrongms,l56
innuclei,540-543.564-565
Real-time Grœ n's functions at snite temperature,292-297
dispersionrelations,294-295
Lehmannrepresentation,293-294
fornoninteractingsystem,298
relation to temperature Green's functions,
297-298
retardedandadvanced,294-297
relationtotime-ordered,295-296
time-ordered,292-293
zero-temperaturelimit,293.296,308/
Reducedmass,129,259
Retardedcorrelationfunction,174,299
RiemannRtafunction,579-580
Ringdiarams,154-157,271-273.281,564
RotatingHelI.482-484,5* /
Rotons,484-488
Ryde rg,27
<ottering:
opticaltheorem,131
phaseshifts,128-129
Scattering amplitude,128-130,143-146.314
Bornseries,132,135
Scatteringcrosssection,189,191,314-315
Scattering l
ength,132,143,218,314.342-343.
48ln
Bol'napproximation,135,219
Scattering theory in momentum space. 130131
Scm teringwavefunction,129.138-139,380
Schrödi
ngerequation,4,54,509,572
inmomentum space,130-131
insecondquantization,15,18
two-particle,129,320-322
Schrödingerpicture,53-54.172
Screening in an electron gas.32#,167,175180,195/.303-307,310/.397
Secondquantization.4-21,353
Secondsound,481-482
Self-consistentapproximations,120,358-360,
442.-446,492
Self-energy,104,107-108.250
proper(seeProperself-energy)
Semiempiricalmassformula,349-352
Seniority,524,536
599
Separablepotential,322
Serberforce,346,355
Shellmodelofnucleus,508-515
bosonapproximation in,52* 527,576#
many-particle (see Nucleus:many-particle
shellmodel)
singl
e-particl
e (Jee Nucleus:single-particle
shellmodel)
Single-particleexcitations,147-148,171,309#,
310/,399,508-515
Single-particle Green's function (see Green's
functionsatzero tem perature;Tem pera-
tureGreen'sfunction)
Single-particle operator, 20, 66, 229, 512515
Single-particle potential in nuclear matter.
355-357,381
G/symbol,585-586
Skeletondiagram.K 3-K s
Slaterdeterminants,16
Sodium ,30,391
SolidihcationofHe,479
Soundvelodty,187,391,407,484
ininteractingBosegas,217,222,317,494
Soundwaves:
classicaltheory,186-187
(SeealsoPhonons;Zerosound)
Specihcheat(seeHeatcapacity)
Spindensity,67,196/,229,309/
Spin-orbitinteraction,511-512,513
Spinsums,9#,104.189
Spinwaves,196,
Square-well potential, 360-361, 386/, 508510
Stabilityagainstcollapse,31/,355
Statisticalmechanics,review of,34-49
Statisticaloperator,36.228
Stepfunction,27,63,72
Structurefactor,189n
Sum ruless191-192,196#,296,577/
Sumsoverstatesreplaced byintegrals,26,38,
394
Superconductor:
alloys,415a,425,427
Bogoliubovequations,477p
and Bose-Einstein condensation,441,446,
476p
chemicalpotential,334-335,453a
coberencelength,422,426,433,472
condensationenergyof,419,453
Cooperpairs,320-326,417,441
criticalcurrent,460a,476p
INDEX
SuperconductorLcont'd):
criticalseld,415,451,453,474p
lowerandupper,438,439,475#
efectiveinteraction,448,476#
efrectivemassandcharge,431
and electron-phonon interaction,444,448,
476#
Dyson'sequation,476/
phononpropagator,477....478p
energygap,330,417,447-449
entropy,419,476p
excitationspectrum.334
experimentalfacts,414-417
films.474p,476p
fluxquantization,415-416,425
gapequation,333.446--449
gapfunction,443,466-474,489
gapless,417/,460a
gaugeinvariance,tt'!,454
SuperconductorLcont'dj:
thermodynamicpotential,449-454
typelandtypell,438-439,475/
ultrasonicattenuation,449,478p
uniform medium,444-454
variationalcakulationofgroundstate,336,
337#
Supercurrent,432,472-474,416p
Superelectrondensity,423,431,459-460
Superquiddensity,4*1-482,487,495
Supermultiplzts,548,549/,558
Surfaceenergy:
inBosesystem,497-498,502#
ofnuclei,35G
insuperconductors,430,436-438
Susceptibili
ty,174,254p,309#,310,
Symmetryenergyofnuclei,350,386#
Ginzburg-laandau theory (see Ginzburg- Tadpolediagram,108,154
Landautheory)
Tamm-DancoFapproximation.565-566
Gorkovequations,4zl4,466
ground-statecorrelationfunction,337,
ground-stateenergy,335
heatcapacity,320,416,420,451-454
Helmholtzfreeenergy,431,451,453,474p
isotopeeeect,320,417,448,476#
local,427
London,427
matrixformulation,443-444
M eissner eflect,414.421,423,457,459460
mixedstate,415a,439
modelhamiltonian,441
nonlocal,427
numericalvalues,tables,422,448
orderparameter,431
penetrationlength,427,434,472,475J
generaldesnition,429
locallimit,427,429.460
nonlocallimit,429-430:460,461-463
table,422
persistentcurrents,415-416
phasetransition,431
Pippard,427,461-463
relation to Hartree-Focktheory,439-441
self-consistencycondition,zltt,446
spinsusceptibility,471p
stabilityofM eissnerstate.430
strong-coupling.4K a
surfaceenergy,43û,436-438
temperatureGreen'sfunction,442-444
Temm rature,34
Temperaturecorrelationfunction,3*
Temm ratureGrœn'sfunction,228,262
analytic continuation to real-time Green's
function,297-298
forbosons,491
conservationofdiscretefrequency,246
equationofmotion,253,
Feynmanrule.s:
incoordinatespace,242-243
inmomentum space,244-248
Fourierseriesfor,244-245
Hartree-Fockapproximation,257
ininteractionpicture,235-236
Lehmarm representatiom 297
for noninteracting system ,232-234,245246,298
fornormalstate,468
periodicityof,236-237,244-245
andproperself-energy,251
relation to obsewables,247,252,261-262
forsuperconductors,442-444
weightfunction,29* 297,309/
Tensor force in nuclear matter, 367, 375.
386p
Tensoroperator,343
Thermalwavelength,277,304,306
Thermionicemission,49#
Thermodynamiclimit,22,75,78,199,489
Thermodynamic pottntial, 34-35, 268-269,
274,290,,327
01
IN DEX
Thermodynamicpotential(cont'
dl:
forbosons,37,38,202,207
coupling-constantintegrationfor,231-232
for electron gas,268,273-275,278, 284,
l%ûp
forferm ions,38,329-332
insnitenuclei,528-537
forphonons,393
relation:
tolruxkner-Goldstonetheory,288-289
to temperature Green's function, 232.
247,252
ringcontribution.274-275,281-286
ofsuperconductor,449-454
Thermodynamics:
ofmagneticsystems,418
review of,34
Thomas-Fermitheory,177-178.195,,386#,
575
Thomms-Fermi wavenum- r, 167,176, 178,
182,397
Vacuum state,13,201
Variationalprinciple,29,336.337/,333,502#
Vectorpotential,424,425-428,431-433,435437,454-456,459:465,468
Velocitypotential,482.495
Vertexparts,402-406.411#
VorticesinHeII.482-484,5*#,502/
energy/unitlength,484,499
vortexcore,484,498-499
W avefunçtions:
for condensate (see Condensate: wave
function)
Dirac,188
forGinzburg-l-andautheory,471
many-body.5-8,16
scattering.129,138-139,380
singleparticle,5
Hartree-Fock, 352-353, 503, 508-511,
541,558-559,567
spinandisospin.21,353
3/symbol,584
W eakinteractions,566
Time-developmentoperator,5&.58
Time-ordered density correlation function, W hite-dwarfstars,49,50p
W ick's theorem,83-92,327, 399,441,506,
174,175
579,560
Time-orderedproductofoperators,58s65,86forbosons,203,222p
87
atsnitetem perature,234-241,441
Transitionmatrixelements,540,543
Transition temperature of inttracting Bose W igner-Eckarttheorem,505,521,544,586587
gas,259-261,493
W ignerforce.354
Translationalinvariance,73-74
W ignerlattice,31,398/
Transversepartofvectorseld,454
W igner'ssuperm ultigiettheory,549/
Two-quidmodel,481
Twœparticlecorrelations,191-192
Two-parti
cl
eGreen'sfunction,116p,253,
Zero-pointenergy,41.393
Zerosound,183-187,195,.196/
comparedtoplmsmaoscillations,186
damping.187,195,.310/
dis> rsionrelation,183-184
Ultlasonicattenuation,411/,449,478#
inliquid He3.187
Uniform rotation,483-484,5* #
spin-waveanalog,196/
Uniform system,69,190,214,292,321
velocityof.185-187