Green’s functions and one-body quantum problems
G F ca n b e u se d to co n v e rt
(
1
2
 r  V ( r )   ) ( r )  0
2
in to a n in te g ra l e q u a tio n
a n d to co n v e rt in te g ra l o v e r so m e v o lu m e  to su rfa ce in te g ra ls o v e r b o u n d a ry S o f  .
Source
introduce G satisfying:
m u ltip ly
(
G (r',r)( 
m u ltip ly
(
 (r ) (
1
2
1
2
1
2
1
2
(
1
2
 r  V ( r )   )G ( r ', r )   ( r '  r )
2
 r  V ( r )   ) ( r )  0
2
by
G (r',r)
The delta not
yet invented at
Green’s time
 r  V ( r )   ) ( r )  0 .
2
 r  V ( r )   )G ( r ', r )   ( r '  r )
2
by
 r  V ( r )   )G ( r ', r )   ( r )  ( r '  r )
 (r )
2
1
G (r ',r)( 
1
 (r ) (
su btract :
 (r ) (
1
2
2
V
1
2
 r  V ( r )   ) ( r )  0 .
2
 r  V ( r )   )G ( r ', r )   ( r )  ( r '  r )
2
disappears
1
 r )G ( r ', r )  G (r ',r)(
2
 r ) ( r )   ( r )  ( r '  r )
2
2
in te g ra te o n r
1

2
dr [  ( r )  r G ( r ', r )  G (r',r)  r ( r )] 
2
2


dr ( r )  ( r '  r )   ( r ').

N ow interchan g e

 (r ) 
1
2

r
and
r'
dr '[G (r,r')  r ' ( r ')   ( r ')  r 'G ( r , r ')]
2
2

This is the sought integral equation: next, we convert volume integral ->
surface integral
2
 (r ) 
S tartin g fro m
1
2

dr '[G (r,r')  r ' ( r ')   ( r ')  r 'G ( r , r ')]
2
2

convert volume -> surface integral using the divergence theorem.
divergence
theorem :

divA dr 

A    
 .(        ) dr 



A .n dS
S
In se rt



(        ).n dS
S
(  ( r )  ( r )     ( r )) dr 
2
2


(  ( r )    ( r )  ).n dS
S
N o w r  r ', th e n  (r')  G ( r,r ' ). In th is w ay th e re su lt
 (r ) 
1

2
dr '[G (r,r')  r ' ( r ')   ( r ')  r 'G ( r , r ')], r '  
2
2

is co n v e n ie n tly re w ritte n as a su rface i n te g ral
 (r ) 
1
2

S
[G (r,r')  r ' ( r ')   ( r ')  r 'G ( r , r ')]n dS , r '  S .
3
Simple use of Green’s functions for Schroedinger’s equation
H  E ,
E 
1
G reen 's o p erato r G su ch th at ( H   )G  1
In tro d u ce
H  H 0  H 1 w ith easy H 0 .
(  H 0 )
1
H
1
H
1
H
operator identities
 (  H 0  H 1  H 1 )


1
  H0
1
  H0


1
  H0
1
H
1
H
H1
H1
 1 H1
1
H
1
H
1
  H0
Example: H1=V(r): then, in the coordinate representation
r
1
H
r'  r
1
  H0
r'  r
1
  H0
H1
1
H
r'
4
N ext introduce com plete set in
r
r
1
H
1
H
w here
r'  r
r'  r
1
  H0
1
  H0
r' 
r'  r

r ''
r
1
  H0
1
  H0
H1
1
r'
H
H 1 r '' r ''
1
H
r'
H 1 r ''   V ( r '') r ''
Lippmann-Schwinger equation
G  r , r '  G0  r , r ' 

d r '' G 0  r , r ''   V  r ''  G  r '', r ' 
3
The Lippmann-Schwinger equation is most convenient when the
perturbation is localized . Typical examples are impurity problems, in
metals. The alternative is embedding (Inglesfield 1981)
5
W e w ish to solve the one-body problem H ψ = ε ψ w ith H  
1
  V ( r ),
2
2
but V ( r ) is com plicat ed in som e regio n.
Suppose we can solve except inside the surface S. We can also solve
S
(  ( 
1
2
 r  V ( r )) g ( r ', r ,  )    ( r  r '), r and r ' outside
2
and wave functions outside are related to g by Green’s theorem
 (r ) 
1

2
[G ( r, r')  r 
( r ')   ( r ')  r 'G ( r , r ') ] n dS , r outside, r ' on S .
'
S
It is assumed that G outside S is not modified by the presence
of the localized perturbation.
W e can also solve (   ( 
1
2
 r  V ( r )) g ( r ', r ,  )    ( r  r ')
2
 r ' g ( r , r ')  0.
w ith the boundary condition on S
T hen for r outside S
 (r )  
1

2
g ( r , r ')   ( r ')·ndS , r ' on S .
S
6
T he relati on
 (r )  
1

2
g ( r , r ')   ( r ')·ndS ,
r outsi de S
S
also fixe s a relati on betw een  and   on S through g. Let us see ho w .
S uppose  ( r ) is a know n function for r  S .
S
W e call it  ( r ). T hen,
 (r )  
1

2
g ( r , r ')   ( r ')·ndS , for r , r '  S .
S
T his c an be considered as a m atrix relation and inve r t ed:
  ( r ')·n   2  d S g
1
( r , r ') ( r '),
r , r ' S
S
w it h g
1
( r , r ') a m atrix inverse, and so know ledge of  ( r )
determ ines   on the surface S .
 know ledge of  ( r ) determ ines
 outside S .
7

N ext, w e w rite
S
d x
3
2
as a surface integral using G re en's theorem .
ouside S
D ivide by  and m ultiply by  :
M ultiply H 
*
 H
*
ψ
=ε

*
ψ

+ |ψ |
H  ψ =ε  ψ +ψ  ε
H ψ =εψ 
V ary ε :
2
 ε
ψ ψ
*
2
S ubstituting,  H
H  =|ψ |
 

*
ouside S
*
ψ


d x
3
2
 ε
*
ψ

H


*
by
ψ

:
*
d x [ H
3
ouside S
*
ψ ψ
*
H ]
 
T h e term w ith V can cels an d w e can u se G reen 's th e o rem :

(        ) dr 
2
2

ouside S
(        ).n dS
S
w ith    *,  


d x
3
2

1
2
 [
S
*




, H  
 -
ψ

1
2
 ,   o u tsid e S
2
  ].ndS (sign change for ouside norm al)
*
8
W e can vary ε k eep in g  ( r )   ( r ) o n S

ψ


 0 o n S . T h is is clever b ecau se w e g et :
d x
3
ouside S
2

1
2

S
*


  .ndS
(sign change for ouside norm al)
1
R ecall : w e have seen that   ( r ')·n   2  d S g ( r , r ') ( r '), r , r '  S
S
w ith g
1
( r , r ') a m atrix inve rse, and so know ledge of  ( r )
determ ines   o n S , henc e also

d x
3
2
.
ouside S
John E. Inglesfield,
Emeritus Professor
,Cardiff
9
Inglesfield embedding
W e w ish to solve the one-body problem w ith H  
1
  V ( r ), bu t V is com plicated in som e region.
2
2
The Lippmann-Schwinger equation is less convenient when the perturbation
is extended. Typical examples are surface problems, when one wants to treat the
effects of surface creation, reconstruction, or contamination. In such cases one
can resort to slab models, but there are serious drawbacks in any attempt to
represent a bulk by a few atomic layers, with quantized normal momenta. The
only practical alternative is the method of embedding.
1) in an extended system, there is a localized perturbing potential. A surface S
divides the perturbed region I from the far region II where the potential is
negligible. 2) the problem in I alone can be solved 3) the extended unperturbed
system could be treated easily because it is highly symmetric 4) the wave
functions match those of the extended system on S.
II
S
I
 (r )


P roblem : to find  ( r )  
 (r)
 ( r )   ( r )

such that E 
 | H |  
 |  
rI
r  II
r  S;
is stationary (that is, H |    E |   )
10
II
S
I
 (r )


P roblem : to find  ( r )  
 (r )
 ( r )   ( r )

such that E 
 | H |  
 |  
rI
r  II
r  S;
is stationary (that is, H |    E |   )
S trategy: w rite E only in term s of  in I an d on S .
 |   

d x
3
2

I

d x
3
V
2

1
2

I
3
2
an d sin ce
II
ouside S
 |   
d x
d x
3
2

S

1
2

*


S
*
  .n d S ,


w h ere   ( r ')·n   2  d S g
  .n d S ,
1
( r , r ') ( r '), r , r '  S .
S
S o ,   |   is a fu n ctio n al o f  .
11
 (r )


P roblem : to find  ( r )  
 (r )
 ( r )   ( r )

II
S
such that E 
I
 | H |  
 |  
rI
r  II
r  S;
is stationary (that is, H |    E |   )
S trategy: w rite E only in term s of  in I an d on S .
N ext,w e w an t  | H |   

d x H   
3
*
I
w here


d x  
3
*

 H d x
*
3
V
II
extends to a sm all volum e around S .
V

W e have just seen how to w rite
d x   in term s of  :
3
*
II
V

o u sid e S
d x
3
2

1
2

S

*

  .n d S ,
w h ere   ( r ')·n   2  d S g
S
1
( r , r ') ( r '), r , r '  S .
12
C ontribution in  V

 H d x 
V
V

 H d x  
*
*
3

3
 (
*
1
  V ( r ))  d x , but the term in V vanishes w it h  V
2
3
V
2
1

  d x  
*
2
3
2 V
V
1

  d x
*
2
3
2 V
In each dS , w ith z  dS ,  
1
2

*

  dSdz  
2
1
2
V

*

div gra d  dSdz
V
S
By the divergence theorem

1
2

*

div grad  dSdz  
1
2
V
T o sum up,

*
    .n outer    .n inner  dS  
 H d x  
*
3
2
V
B ut recall:
1

1
2
*
        .n outer dS .
*
        .n outer dS .
S
  ( r ')·n   2  d S g
1
( r , r ') ( r '),
r, r ' S
S
so everything is written in terms of  and one obtains a localized variational
problem in region I equivalent to a Schrödinger equation with a potential  at S.
13
Absence of Magnetism in classical physics
Ampere Postulated
“molecular-ring”
currents to explain the
phenomenon of
magnetism
Bohr-Van Leeuwen Theorem (1911)
Partition function for a Classical electron in canonical ensemble
Z 

 px2  p y2  pz2
d x  d p ex p 
2 m K BT

3
3



Classical electron in magnetic field
eH

2
2
2
(
p

y
)

p

p
y
z
 x
3
3
c
Z   d x  d p exp 
2 m K BT







A change of variable removes the field.
 dp x

 d ( px 
eH
y)
c
Z  V  2 m K B T
3
2
14
magnetism cannot exist (catastrophe of classical physics)
Langevin paramagnetism
"gas" of ions w ith m agnetic m o m e nts   g  B J in fiel d B .
H am iltonian
H  g  B J .B
M 
m agnetization
N z
V olum e
Q uantum partition function
J
Z 

m J
 (2 J  1) g  B B 
sinh 

2
K
T
mg B B

B

exp[ 
]
K BT
 gBB 
sinh 

 2 K BT 
 z  K BT
M ean m o m en t
 ln ( Z )
B
 g  B JB J (
Paul Langevin
(1872-1946)
g  B BJ
),
K BT
w h ere B J ( x ) is k n o w n as B rillo u in fu n ctio n
BJ ( x) 
2J 1
2J
co th (
(2 J  1) x
2J
)
1
2J
co th (
x
2J
)
15
C ase J 
1
(electron), g  2 :
2
J
the partition function is Z 

m J

exp[ 
mg B B
]  2 cosh(
K BT
BB
)
K BT
 BH 
BH
M 
 B tanh 

H
igh
tem
perature
behaviour:

V
K
T
K BT
 B 
N
N B H
1
2
 M 
V
 C urie law (for param agn etis m )
K BT
16
Can we explain ferromagnetism as due to dipole-dipole
interactions?
Orders of magnitude estimate
e
Interaction energy of tw o dipoles  
(B ohr M agneton)
2mc
B
2
at distance a lattice  several B ohr radii : E dip 
B
2
3
aB
S ince
(
e
)
2mc
 
e
1
2



2 
 me 
2
2

c
1
3
e

2
2
2
4m c
3
m e
2
6
6

1
mc  ,
2
4
4
, m c  0.5 M eV , 1
2
3
a lattice
0
K  0.025 eV
137
B
2
if
a lattice  4 a B  E dip 
 dipole-dipole
 0 .1 K .
0
3
64aB
interaction far to o sm all to explain ferrom ag nets!
Heisenberg chain: a solvable 1d model of magnetism
N
H H eisen berg   J

n 1
S n .S n  1 ,
J0
ferrom agn etic
18
Hans Bethe solved the linear chain
Heisenberg model H  ,  ,   E ( ,  , 
1
2
N
1
2
N
)  1 , 2 ,
Born
July 2, 1906
Strassburg, Germany
Died
March 6, 2005
Ithaca, NY, USA
Residence
USA
Nationality
American
Field
Physicist
Institution
University of Tübingen
Cornell University
Alma mater
University of Frankfurt
University of Munich
Academic advisor
N
Arnold Sommerfeld
Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette
Zeitschrift für Physik A, Vol. 71, pp. 205-226 (1931)
19
 1 , 2 ,
N spins ½ on a ring with a configuration
, N
Symmetries
N
H H eisen berg   J

S n .S n  1
is in varian t
for
tran slation s of all sp in s
n 1
 th ere is a con served
w avevector
K
of all th e sy stem
N
H H eisen b erg   J

S n .S n  1
is in v a ria n t
fo r
sp in
ro ta tio n s
n1
(sin ce sca la r p ro d u cts a re in v a ria n t)
 c o n se rv e d
to ta l sp in S ,
c o n se rv e d
Sz 
N  N 
Separated problem s for each set K ,S , S z
2
.
20
N
H H eisen berg   J

S n .S n  1 can b e re w ritte n
n1
N
H H eisen berg  H 0  H X Y ,
H0  J

S
z
.S
n
z
, H XY  
n1
n 1
H XY  
J
N

2




(Sn Sn  1  Sn Sn  1 )
m oves
,
to
J
2
N





(S n S n  1  S n S n  1 )
n 1
rig h t
or
le ft
n1
Ground state for J>0: all spins up, no spin can be
raised, and so no shift can occur
H XY  
J
2
N





( S n S n  1  S n S n  1 )  H X Y    ...   0
n1
N
H H eisen berg    ...    J

N
S
z
.S
n
z
n1
   ...    J
n1
 g ro u n d state en erg y E

n1
( N 0 )
 J
1
4
   ... 
N
4
21
1 reversed spin in the chain
Consider for the moment no hopping, only H0
N
H0  J

S
z
.S
n
H 0     ...   E (     ...  )
z
n1
    ... 
n 1
E (     ...  )
E (     ...  )  E
w h e re
a facto r
e x tra
2
2
( J)
 E (     ...  )  2 * 2 *
( N 0 )
e n te rs b e cau se
b e cau se
2
of
 J
4
N
 E (     ...  )  J ,
,
4
p o sitiv e  n e g ativ e ,
th e
scalar p ro d u cts
are
re v e rse d .
A configuration with r=1 is conveniently denoted by |n>, n=position of down spin.
22
They are all degenerate.
Now introduce the hopping term.
H XY  
J
N

2




(Sn Sn  1  Sn Sn  1 )
m o v es

to
rig h t
or
left, so
n 1
( H H eisenberg  E
 N  0
) n  J n 
J
( n  1  n  1 ).
2
Introducing the eigenvect or 
E
,
H H eisenberg  E  E  E
Introduce 

E
E
n  f n
( H H eisenberg  E
 N  0
am plitude
) n  J 
of
E

n 
on
J
2
(
site
E
n in eigenvec tor of H .
n 1  
E
n 1 )
th at is ,
(E  E
 N  0
 f ( n  1)  f ( n  1)

) f n  J 
 f (n )
2


23
One- Magnon solution: spin wave
(E  E
 N  0
i s solved by
N
 f ( n  1)  f ( n  1)

( N 0 )
) f n  J 
 f (n )  , E
 J
,
2
4


f (n) 
e
ik j n
indeed f ( n  1)  f ( n  1) 
,
N
e
ik j n
(e
ik j
e
N
We must insert the periodic boundary condition for the N-spin chain
f n  N  
energy :
f (n)

kj 
 N 0 e
(E  E
)
2 j
N
ik j n
 J
N
 E  E
( N  0 )

e
E

1
N
N

e
ik j n
n
n 1
 e ik j  e  ik j

 1

2
N 

ik j n
 J 1  cos( k j ) 
24
 ik j
).
N
H H eisen berg   J

S n .S n  1 ,
J0
ferro m ag n etic
n1
One- Magnon solution: the spin wave is a boson (spin 1)
E E 0
E 
1
N
N

e
ik j n
n ,
kj 
n 1
2 j
2.0
N
1.5
J
1.0
E  E
( N  0 )
 J 1  cos( k j ) 
0.5
k
3
2
1
1
2
3
This magnon dispersion law looks like a free quasi-particle
dispersion for small k
Two- Magnon solution: scattering, bound states
25
Several reversed spins
N
L e t u s first c o n sid e r H 0   J

z
S n .S
z
n1
w ith o u t h o p p in g ,
n 1
w ith
r  N 
re v e rse d
H 0  1 , 2 ,
sp in s.
 N  E ( 1 ,  2 ,
 N )  1 , 2 ,
N .
A n y p atte rn o f sp in s w ill d o , b u t
E ( 1 ,  2 ,
 N ) d iffe rs if far o r c o n se c u tiv e sp in s are re v e rse d :
        ... 
h av e d iffe re n t E ( 1 ,  2 ,
E ( 1 ,  2 ,
N) E
N   n u m b e r
        ... 
an d
( N 0 )
of
2
 N ).
J
N  ,
4
re v e rse d
sc alar
p ro d u c ts . N o w in c lu d e H X Y .
Two Magnons
a configuration is denoted by |n1,n2>,
H XY  
J
N

2




(Sn Sn  1  Sn Sn  1 )
m oves
n1<n2 reversed spins

to
rig h t
or
le ft
n 1
provided that adiacent sites are up spins: if the reversed spins are close
the energy is different and this is an effective interaction.
W e in tro d u ce th e case n 2  n 1  1 by m ean s o f a bo u n d ary co n d itio n .
F irst, w e so lv e fo r n 2  n 1  1, w h en th e sp in s are 'far' an d m o v e in d ep en d en tly .
( H H eisen berg  E
( N 0 )
) n1 , n 2  2 J n1 , n 2  J
n 1  1, n 2  n 1  1, n 2  n 1 , n 2  1  n 1 , n 2  1
2
27
.
( H H eisen berg  E
J
L et
E
J
( N 0 )
) n1 , n 2  2 J n1 , n 2
n 1  1, n 2  n 1  1, n 2  n 1 , n 2  1  n 1 , n 2  1
2
f ( n1 , n 2 )  
E
n1 , n 2 ,
f  n1 , n 2   2 f  n1 , n 2  
T hus r.r. is
i ( k1 n1  k 2 n 2 )
O ne
the energy origin :
n 2  n1  1
EE
 N  0
 E
f ( n1  1, n 2 )  f ( n1  1, n 2 )  f ( n1 , n 2  1)  f ( n1 , n 2  1)
.
2
satisfied
f  n1 , n 2   e
R ecall
shift
,
,
n1  n2
for
E
J
M agnon
by
the
produc t
 (1  cos( k 1 ))  (1  cos  k 2  ).
S olution :
E
J
 1  cos( k j ).
Two independent magnons? NO! New k1 and k2 must be found since PBC
do not separately apply to n1 and n2 (you cannot translate one magnon
across the other). However there is
In v arian ce
u n d er
g lo bal
tran slatio n s o f bo t h rev ersed sp in s:
f  n1  N , n 2  N   f  n1 , n 2  .
28