Many-body Green’s Functions
•
•
•
•
•
Propagating electron or hole interacts with other e-/h+
Interactions modify (renormalize) electron or hole energies
Interactions produce finite lifetimes for electrons/holes (quasi-particles)
Spectral function consists of quasi-particle peaks plus ‘background’
Quasi-particles well defined close to Fermi energy
• MBGF defined by


iG(r, t, r ' , t' )  o T ψ̂ H (r, t)ψ̂ H (r ' , t' ) o
i.e. correlatio n function of field operator averaged over
exact Heisenberg ground state, o
Many-body Green’s Functions
•
•
•
•
•
Space-time interpretation of Green’s function
(x,y) are space-time coordinates for the endpoints of the Green’s function
Green’s function drawn as a solid, directed line from y to x
Non-interacting Green’s function Go represented by a single line
Interacting Green’s Function G represented by a double or thick single line
y
x
Add particle
Go(x,y)
Remove particle
t’
t > t’
y,t’
time
o ψ̂ H (x, t)ψ̂ (y, t' ) o  (t  t' )

H
x
y
Remove particle
G(x,y)
y,t’
Add particle
t
t’ > t
x,t
time
o ψ̂ H (y, t' )ψ̂ H (x, t) o  (t' t)
x,t
Many-body Green’s Functions
• Lehmann Representation (F 72 M 372) physical significance of G


iG(r, t, r ' , t' )  o T ψ̂ H (r, t)ψ̂ H (r ' , t' ) o
exact Heisenberg ground state, o
exact Heisenberg state, n , any particle number
n n  1 unit operator in occupation number formalism
iG(r, t, r ' , t' )  o ψ̂ H (r, t)ψ̂ H (r ' , t' ) o  (t  t' ) - o ψ̂ H (r ' , t' )ψ̂ H (r, t) o  (t'  t)
o ψ̂ H (r, t)ψ̂ H (r ' , t' ) o  o ψ̂ H (r, t) n n ψ̂ H (r ' , t' ) o
 o e  iĤt ψ̂S (r )e -iĤt n n e  iĤt'ψ̂S (r ' )e -iĤt' o
-i(E n - E o )(t - t')
o ψ̂S (r ) n n ψ̂S (r ' ) o
-iE n t
 iE t
o e  iĤt  e o o
e
e-iĤt n  e
n
Many-body Green’s Functions
• Lehmann Representation (physical significance of G)


iG(r, t, r ' , t' )  o T ψ̂ H (r, t)ψ̂ H (r ' , t' ) o
iG(r, t, r ' , t' )  o ψ̂S (r ) n n ψ̂S (r ' ) o e
 o ψ̂S (r ' ) n n ψ̂S (r ) o e
-i(E n - E o )(t - t')
 (t  t' ) -
 i(E n - E o )(t - t')
 (t'  t)

iG(r, r ' ,  )  i  d(t - t' )G( r, t, r ' , t' )e i (t  t')


o ψ̂S n n ψ̂S o
ε  ( E n  E o )  iδ
n̂   dr ψ̂S (r )ψ̂S (r )

o ψ̂S n n ψ̂S o
ε  ( E n  E o )  iδ
n̂ψ̂S o  ( N  1)ψ̂S o
ψ̂S o reduces particle number in o by one
Many-body Green’s Functions
• Lehmann Representation (physical significance of G)

S

S
o ψ̂S n n ψ̂ o  n ψ̂ o
2
connects N and N  1 particle states
E n ( N  1)  E o ( N)  E n ( N  1)  E o ( N  1)  E o ( N  1)  E o ( N)
E n ( N  1)  E o ( N)  E n ( N  1)  E o ( N  1)  

S
o ψ̂ n n ψ̂S o  o ψ̂S n
2
connects N and N  1 particle states
E n ( N  1)  E o ( N)  E n ( N  1)  E o ( N  1)  E o ( N  1)  E o ( N)
E n ( N  1)  E o ( N)  E n ( N  1)  E o ( N  1)  
Many-body Green’s Functions
•
•
•
•
Lehmann Representation (physical significance of G)
Poles occur at exact N+1 and N-1 particle energies
Ionisation potentials and electron affinities of the N particle system
Plus excitation energies of N+1 and N-1 particle systems
• Connection to single-particle Green’s function
0 is the single - particle (non - interactin g) ground state
iG o (r, t, r ' , t' )  0 ψ̂ H (r, t)ψ̂ H (r ' , t' ) 0  (t  t' )
  ψ m (r )ψ*n (r ' ) 0 ĉ m (t) ĉ n (t' ) 0  (t  t' )
m, n

unocc
 ψ n (r)ψ*n (r' )e
-i n (t -t')
 (t  t' )
0 ĉ m ĉ n 0   mn n  unoccupied
n
Sum limited to unoccupied states as ĉ n 0  0 for states below  F
Many-body Green’s Functions
• Gell-Mann and Low Theorem (F 61, 83)
• Expectation value of Heisenberg operator over exact ground state
expressed in terms of evolution operators and the operator in question in
interaction picture and ground state of non-interacting system o
o Ô H (t) o

o o
 i 
n t
Û I (t,0) 
n!
o Û I (, t)Ô I (t) Û I (t,- ) o
o Û I (,-) o
t
t

0
0
0
Many - Body Green' s Function
iG(r, t, r ' , t' ) 

 dt1  dt 2 ... dt nT Ĥ I (t1 )Ĥ I (t 2 )...Ĥ I (t n )  Te


o T ψ̂ H (r, t)ψ̂ H (r ' , t' ) o
o | o
t
i  dt'Ĥ I (t')
0
F 57
Many-body Green’s Functions
• Perturbative Expansion of Green’s Function (F 83)
iG( x, y ) 

1
o Û I  ,  o

o Û I  ,  o  
n 0
•
•
•
•
•
•
•

 i n dt
n 0
 i n dt
n!
n!

-


-
-
1
-

1


 dt ...  dt
2
-



dt
...
dt

T
Ĥ
(t
)
Ĥ
(t
)...
Ĥ
(t
)
ψ̂
(
x
)
ψ̂
( y ) o
2
n
o
I
1
I
2
I
n


n


o T Ĥ I (t 1 )Ĥ I (t 2 )...Ĥ I (t n ) o
-
Expansion of the numerator and denominator carried out separately
Each is evaluated using Wick’s Theorem
Denominator is a factor of the numerator
Only certain classes of (connected) contractions of the numerator survive
Overall sign of contraction determined by number of neighbour permutations
n = 0 term is just Go(x,y)
x, y are compound space and time coordinates i.e. x ≡ (x, y, z, tx)
Many-body Green’s Functions
• Fetter and Walecka notation for field operators (F 88)
ψ̂(x)  ψ̂ (  ) (x)  ψ̂ (-) (x)  â  b̂ 
()
(-)
ψ̂  (x)  ψ̂  (x)  ψ̂  (x)  â   b̂
ψ̂ (  ) (x)ψ̂  (  ) (y )  iG o (x, y ) t x  t y   ââ 
0
x
ψ̂ (  ) (x)ψ̂  (  ) (y )  0
x
 iG o (x, y )

t
t
t

x
 ty 
 ty 
 t y  - b̂b̂ 
ψ̂  ψ̂   ψ̂  (  )  ψ̂  (  ) ψ̂  (  )  ψ̂  (-)


â

 b̂

â

 b̂


â  â   â  b̂  b̂â   b̂b̂
â  â   â  b̂  b̂â   b̂b̂  0
ψ̂  ψ̂   0 similarly
ψ̂ψ̂  0
Many-body Green’s Functions
• Nonzero contractions in numerator of MBGF


















ψ̂ (r )ψ̂ (r ' )ψ̂(r ' )ψ̂(r )ψ̂(x)ψ̂ (y ) (1)
ψ̂ (r )ψ̂ (r ' )ψ̂(r ' )ψ̂(r )ψ̂(x)ψ̂ (y ) (2)
ψ̂ (r )ψ̂ (r ' )ψ̂(r ' )ψ̂(r )ψ̂(x)ψ̂ (y ) (3)
ψ̂ (r )ψ̂ (r ' )ψ̂(r ' )ψ̂(r )ψ̂(x)ψ̂ (y ) (4)
ψ̂ (r )ψ̂ (r ' )ψ̂(r ' )ψ̂(r )ψ̂(x)ψ̂ (y ) (5)
ψ̂ (r )ψ̂ (r ' )ψ̂(r ' )ψ̂(r )ψ̂(x)ψ̂ (y ) (6)
(-1)3 (i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y)
(-1)4(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y)
(-1)5(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y)
(-1)4(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y)
(-1)6(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y)
(-1)7(i)3v(r,r’)Go(r,r) Go(x,r’) Go(r’,y)
Many-body Green’s Functions
x
• Nonzero contractions
-(i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y)
(1)
r
+(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y)
(2)
y
x
(1)
y
(2)
x
-(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y)
x
(3)
r’
r
+(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y)
(3)
y
x
x
(4)
(5)
(6)
(5)
y
r
r’
r
r’
-(i)3v(r,r’)Go(r,r) Go(x,r’) Go(r’,y)
r’
r
(4)
y
+(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y)
r’
r
r’
y
(6)
Many-body Green’s Functions
• Nonzero contractions in denominator of MBGF
• Disconnected diagrams are common factor in numerator and denominator
ψ̂  (r )ψ̂  (r ' )ψ̂(r ' )ψ̂(r ) (7)
ψ̂  (r )ψ̂  (r ' )ψ̂(r ' )ψ̂(r ) (8)
r
(-1)3(i)2v(r,r’)Go(r’,r) Go(r,r’)
(-1)4(i)2v(r,r’)Go(r,r) Go(r’,r’)
r’
r
r’
(7)
(8)
Numerator = [ 1 +
+
+…]x[
Denominator = 1 +
+
+…
+
+
+…]
Many-body Green’s Functions
• Expansion in connected diagrams


(i)

iG( x, y )  
dt
...
dt

T
[
Ĥ
(t
)
...
Ĥ
(t
)
ψ̂
(
x
)
ψ̂
(y )] o
1
m
o
1 1
1 m


m 0 m! 


iG(x, y) =
•
•
•
•
•
+
+
connected
+…
Some diagrams differ in interchange of dummy variables
These appear m! ways so m! term cancels
Terms with simple closed loop contain time ordered product with equal times
These arise from contraction of Hamiltonian where adjoint operator is on left
Terms interpreted as


iG o (x, x) t 'limt  o T ψ̂(x, t)ψ̂  (x, t' ) o
  o ψ̂  (x)ψ̂(x) o  ρ o (x) non - interactin g charge density
Many-body Green’s Functions
• Rules for generating Feynman diagrams in real space and time (F 97)
• (a) Draw all topologically distinct connected diagrams with m interaction
lines and 2m+1 directed Green’s functions. Fermion lines run continuously
from y to x or close on themselves (Fermion loops)
• (b) Label each vertex with a space-time point x = (r,t)
• (c) Each line represents a Green’s function, Go(x,y), running from y to x
• (d) Each wavy line represents an unretarded Coulomb interaction
• (e) Integrate internal variables over all space and time
• (f) Overall sign determined as (-1)F where F is the number of Fermion loops
• (g) Assign a factor (i)m to each mth order term
• (h) Green’s functions with equal time arguments should be interpreted as
G(r,r’,t,t+) where t+ is infinitesimally ahead of t
• Exercise: Find the 10 second order diagrams using these rules
Many-body Green’s Functions
• Feynman diagrams in reciprocal space
•
•
•
•
•
For periodic systems it is convenient to work in momentum space
Choose a translationally invariant system (homogeneous electron gas)
Green’s function depends on x-y, not x,y
G(x,y) and the Coulomb potential, V, are written as Fourier transforms
4-momentum is conserved at vertices
q2
q3
d 4k
ik .( xy )
G( x, y )  
G(k )e
4
q1
2 
v( q)   d(r - r ' ) v( r, r ' )e -iq.(r r ')
d 4k  d 3kd
k.x  k.x - t
Fourier Transforms
4
d
 xe
iq1.x -iq2 .x -iq3.x
e
e
 2 
4
 q1  q 2  q3 
4-momentum Conservation
Many-body Green’s Functions
• Rules for generating Feynman diagrams in reciprocal space
• (a) Draw all topologically distinct connected diagrams with m interaction
lines and 2m+1 directed Green’s functions. Fermion lines run continuously
from y to x or close on themselves (Fermion loops)
• (b) Assign a direction to each interaction
• (c) Assign a directed 4-momentum to each line
• (d) Conserve 4-momentum at each vertex
• (e) Each interaction corresponds to a factor v(q)
• (f) Integrate over the m internal 4-momenta
• (g) Affix a factor (i)m/(2)4m(-1)F
• (h) A closed loop or a line that is linked by a single interaction is assigned a
factor ei Go(k,)
Equation of Motion for the Green’s Function
• Equation of Motion for Field Operators (from Lecture 2)
iG(r, t, r ' , t' ) 
i

o T ψ̂ H (r, t)ψ̂ H (r ' , t' )o
o o


ψ̂ H  ψ̂ H , Ĥ H  ĥ(r )ψ̂ H (r )
t
for Ĥ H   dr1ψ̂ H (r1 )ĥ(r1 )ψ̂ H (r1 )



1

i ψ̂ H  ψ̂ H , Ĥ H   dr2 ψ̂ H (r2 )
ψ̂ H (r2 )ψ̂ H (r )
t
r  r2
for Ĥ H 
1
2


d
r
d
r
ψ̂
(
r
)
ψ̂
1
2
H
1
H (r2 )

1
ψ̂ H (r2 )ψ̂ H (r1 )
r1  r2
Equation of Motion for the Green’s Function
• Equation of Motion for Field Operators
i





ψ̂ H (r, t)  ψ̂ H (r, t), Ĥ H (r, t)  e  iĤt ψ̂S , Ĥ S e  iĤt
t
 e  iĤt ĥ(r )ψ̂(r ) e  iĤt  e  iĤt  dr2 ψ̂  (r2 )
 ĥ(r, t)ψ̂ H (r, t)   dr2 ψ̂ H (r2 , t)
1
ψ̂(r2 )ψ̂(r ) e  iĤt
r  r2
1
ψ̂ H (r2 , t)ψ̂ H (r, t)
r  r2
1
 


i

ĥ
(
r
,
t)
ψ̂
(
r
,
t)

d
r
ψ̂
(
r
,
t)
ψ̂ H (r2 , t)ψ̂ H (r, t)
2 H 2

 t
 H
r  r2


Equation of Motion for the Green’s Function
• Differentiate G wrt first time argument


iG( x, t, y , t' )  o T ψ̂ H (x, t), ψ̂ H (y , t' ) o
i



G( x, t, y , t' )  o ψ̂ H (x, t)ψ̂ H (y , t' ) (t - t' ) - ψ̂ H (y , t' ) ψ̂ H (x, t) (t'-t) o
t
t
t
 o
ψ̂ H (x, t) 
ψ̂ (x, t)
ψ̂ H (y , t' ) (t - t' ) - ψ̂ H (y , t' ) H
 (t'-t) o 
t
t
 o ψ̂ H (x, t)ψ̂ H (y , t' ) (t - t' ) - -ψ̂ H (y , t' )ψ̂ H (x, t) (t - t' ) o
 o
ψ̂ H (x, t) 
ψ̂ (x, t)
ψ̂ H (y , t' ) (t - t' ) - ψ̂ H (y , t' ) H
 (t'-t) o 
t
t

 ψ̂

(y , t' )
o ψ̂ H (x, t), ψ̂ H (y , t' )  o  (t - t' )
o
H
(x, t), ψ̂ H

o  (t - t' )  o | o  (x - y ) (t - t' )
Equation of Motion for the Green’s Function
• Differentiate G wrt first time argument

i G( x, t, y , t' )  iĥ o ψ̂ H (x, t)ψ̂ H (y , t' ) (t - t' ) - ψ̂ H (y , t' )ψ̂ H (x, t) (t'-t) o 
t
1
 i  dr1
o ψ̂ H (r1 , t)ψ̂ H (r1 , t)ψ̂ H (x, t)ψ̂ H (y , t' ) (t - t' ) o 
x  r1
1
 i  dr1
o - ψ̂ H (y , t' )ψ̂ H (r1 , t)ψ̂ H (r1 , t)ψ̂ H (x, t) (t'-t) o
x  r1
  (x - y ) (t - t' )


1
 iĥ iG( x, y )  i  dr1
o T ψ̂ H (r1 , t)ψ̂ H (r1 , t)ψ̂ H (x, t)ψ̂ H (y , t' ) o
r  r1
  (x - y ) (t - t' )


1
 



i

ĥ
G(
x
,
t,
y
,
t'
)

i
d
r

T
ψ̂
(
r
,
t)
ψ̂
(
r
,
t)
ψ̂
(
x
,
t)
ψ̂


1
o
H
1
H
1
H
H ( y , t' ) o

x  r1
 t

  (x - y ) (t - t' )
Equation of Motion for the Green’s Function
• Evaluate the T product using Wick’s Theorem
 dr1


1
o T ψ̂ H (r1 , t)ψ̂ H (r1 , t)ψ̂ H (x, t)ψ̂ H (y, t' ) o
connected
x  r1
• Lowest order terms
ψ̂ H (r1 , t)ψ̂ H (r1 , t)ψ̂ H (x, t)ψ̂ H (y, t' )
x
r1
(i)2v(x,r1)Go(x,r1) Go(r1,y)
y (9)
x
ψ̂ H (r1 , t)ψ̂ H (r1 , t)ψ̂ H (x, t)ψ̂ H (y, t' )
•
•
•
•
r1
(i)2v(x,r1)Go(r1,r1) Go(x,y)
Diagram (9) is the Hartree-Fock exchange potential x Go(r1,y)
Diagram (10) is the Hartree potential x Go(x,y)
y
Diagram (9) is conventionally the first term in the self-energy
Diagram (10) is included in Ho in condensed matter physics
(10)
Equation of Motion for the Green’s Function
• One of the next order terms in the T product
 d1d2dr1
1
1
ψ̂ H (1)ψ̂ H (2)ψ̂ H (2)ψ̂ H (1)ψ̂ H (r1 )ψ̂ H (r1 )ψ̂ H (x)ψ̂ H (y)
1 - 2 x - r1
x
(i)3v(1,2) v(x,r1)Go(1,x) Go(r1,2) Go(2,r1) Go(1,y)
• The full expansion of the T product can be written exactly as
 dx' S(x, x' )G
o
(x' , y ) S is the self - energy
r1
S(x,1)
1
2
Go(1,y)
y
(11)
x' is a dummy variable (1 in this diagram)
At higher orders some diagrams are repeated and others are unique
Unique diagrams cannot be cut into two by cutting a single Go line
This distinctio n divides higher order diagrams into proper and
improper diagrams. The latter are generated by iterating proper diagrams
Equation of Motion for the Green’s Function
•
•
•
•
•
The proper self-energy S* (F 105, M 181)
The self-energy has two arguments and hence two ‘external ends’
All other arguments are integrated out
Proper self-energy terms cannot be cut in two by cutting a single Go
First order proper self-energy terms S*(1)
x
x’
x
(9)
• Hartree-Fock exchange term
x’
(10) r
1
Hartree (Coulomb) term
Exercise: Find all proper self-energy terms at second order S*(2)
Equation of Motion for the Green’s Function
• Equation of Motion for G and the Self Energy
1
i  dr1
o T ψ̂ H (r1 )ψ̂ H (r1 )ψ̂ H (x)ψ̂ H (y ) o  i  dx' (x, x' )G o (x' , y )
x  r1


 (1)   (1) (direct)   (1) (exchange )
 (1) (direct)( x, x' )   dr1
1
 (x  x' )G o (r1 , r1 )  VH (x, x' )
x'r1
Convention in condensed matter physics is to put  (1) (direct) in Ĥ o
(x, x' )  (x, x' )  VH (x, x' )
 

 i  ĥ  VH G( x, y )  i  dx' (x, x' )G o (x' , y )   (x - y )
 t

x, y , r1 time dependence suppressed here
(x, x' ) is the exchange - correlatio n potential
Equation of Motion for the Green’s Function
• Dyson’s Equation and the Self Energy
 

 i  ĥ  VH G( x, y )  i  dx' (x, x' )G o (x' , y )   (x - y )
 t

Equation of Motion for G for interactin g system
 

 i  ĥ  VH G o (x, y )   (x - y )
 t

Equation of Motion for Go for non - interactin g system ( Ĥ  Ĥ o incl. VH )
G( x, y )  G o (x, y )    dx'dx' ' G o (x, x' ) (x' , x' ' )G o (x' ' , y )
Dyson' s Equation
Equation of Motion for the Green’s Function
• Integral Equation for the Self Energy
The self - energy S and the proper self energy S* are related by
S(x, x' )  S* (x, x' )   dx' 'dx' ' ' S* (x, x' ' )G o (x' ' , x' ' ' )S(x' ' ' , x' )
i.e. improper (repeated) terms in the self energy
generated by iterating the proper self energy
S  S*  S*G o S*  S*G o S*G o S*  ...
Compare  dx' S(x, x' )G o (x' , y ) and
*
d
x
'
S
 (x, x' )G( x' , y )
using G  G o  G o SG o and S  S*  S*G o S
SG o  S * G o  S * G o S * G o  S * G o S * G o S * G o 
S *G  S *G o  S *G o S *G o  S *G o S *G o S *G o 
Hence we may replace SG o by S*G in Dyson' s equation
Equation of Motion for the Green’s Function
• Dyson’s Equation (F 106)
G( x, y )  G o (x, y )    dx'dx' ' G o (x, x' )S* (x' , x' ' )G( x' ' , y )
G( x, y )  G o (x, y )    dx'dx' ' G o (x, x' )S(x' , x' ' )G o (x' ' , y )
G(x,y) =
S(x’,x’’)=
•
•
•
•
•
=
+
+
+
+…
+…
In general, S* is energy-dependent and non-Hermitian
Both first order terms in S are energy-independent
Quantum Chemistry: first order self energy terms included in Ho
Condensed matter physics: only ‘direct’ first order term is in Ho
Single-particle band gap in solids strongly dependent on ‘exchange’ term
Evaluation of the Single Loop Bubble
• One of the 10 second order diagrams for the self energy
• The first energy dependent term in the self-energy
• Evaluate for homogeneous electron gas (M 170)
, q
, k-q
, ℓ
, q
d 3q d
2
i
G
(
k

q
,



)
(

i
V(
q
))
x
, ℓ+q  
o
3 
2  2
d 3  d
x (-1).2.
iG o (  ,  ) iG o (   q ,    )
3 
2  2
, ℓ
, ℓ+q  i o (q,  )
 o  iG o G o Wick' s Theorem
 i o   i  G o G o  iG oiG o
2
Evaluation of the Single Loop Bubble
• Polarisation bubble: frequency integral over 
d
 2 iG o (,  ) iG o (  q,   )
i
i
iG o (,  ) 
iG o (  q,    ) 
     i
      q  i
• Integrand has poles at  =  ℓ - i and  = - +  ℓ+q + i
• The polarisation bubble depends on q and 
• There are four possibilities for ℓ and q
  kF
  q  kF
  kF
  q  kF
  kF
  q  kF
  kF
  q  kF
y
      q  i
x
     i
  kF
  q  kF
Evaluation of the Single Loop Bubble
• Integral may be evaluated in either half of complex plane


Anti clockwise
Upper half plane

-

semicircle in upper half plane
 2i  residues
 
lim
d
d rei i
i
1


y
 2 r   2 rei rei r  0
      q  i
1
x
f(z) 
     i
z  a z  b 
1
  kF
  q  kF
residue f(z) at z  a 
a  b 


i
i
residue 
 for pole at       q  i
      i       q  i 

i 2
     q  i     i 

i 2
     q     i
Evaluation of the Single Loop Bubble
• From Residue Theorem
d
2i
1
i
G
(

,

)
i
G
(


q
,



)

o
 2 o
2      q     i
i

     q     i
• Exercise: Obtain this result by closing the contour in the lower half plane
Evaluation of the Single Loop Bubble
• Polarisation bubble: continued
d
iG o (,  ) iG o (  q,    )
• For 
2
  kF
  q  kF
  kF
  q  kF
• Both poles in same half plane
• Close contour in other half plane to obtain zero in each case
• Exercise: For
• Show that
  kF
  q  kF
d
i
i
G
(

,

)
i
G
(


q
,



)

o
 2 o
   q     i
d 3
2i
d 3
2i
 i o (q,  )  

2 3    q     i  2 3    q     i
• And that
 i oA
  kF
  q  kF
 i oB
  kF
  q  kF
Evaluation of the Single Loop Bubble
, q
• Self Energy , k-q
, ℓ , ℓ+q
, q
d 3  d
d 3q d
2
iG o (  ,  ) iG o (   q ,    )
iG o (k  q,    )(iV(q)) 
S  -2
3 
3 
2  2
2  2
i
d 3q d
B
A
2
(q,  )

i

)

,
q
(

i

))
q
V(
i

(

o
o
3 
2  2     ε k q  i k q

i
d 3q d
2
A

i

)
q
V(
S 
o (q,  )
3 
    ε k  q  i k  q
2  2

A
d 3q

2 3

  kF
2i
i
d 3  d
2
 2 3  2 V(q)     ε k q  i k q   ε q  ε   i
poles at     ε k q  i k q and   ε  q  ε   i
k  q must be  k F otherwise both poles in lower half plane

  q  kF
Evaluation of the Single Loop Bubble
• Self Energy: continued


2i
i
residue 
 at     ε k q  i
     ε k q  i   ε  q  ε   i 
2
  kF ,   q  kF , k -q  kF

  ε   ε   q  ε k  q  i
d 3q
 i S  2i 
2 3
A
1
d 3
2
 2 3 V(q)   ε   ε q  ε k q  i
  kF ,   q  kF , k  q  kF
d 3q
 iS  2i 
2 3
B
1
d 3
2
 2 3 V(q)   ε   ε q  ε k q  i
  kF ,   q  kF , k  q  kF
Self energy is energy and wave vector dependent
Evaluation of the Single Loop Bubble
• Real and Imaginary Parts
1
a  i

a  i a   2
a
1
 1 
Re 

P

2
a
 a  i  a  

 1 
Im
   (a )

2
a 
 a  i 
 ( x ) lim0
 /
from lecture 1
2
2
x 
• Quasiparticle lifetime t diverges as energies approach the Fermi surface
d 3q
Re( S )  2 P 
2 3
A
d 3q
Im( S )  2 
2 3
A
d 3
1
2
V(
q
)
 2 3
  ε   ε   q  ε k q
d 3
2
V(
q
)
   ε   ε  q  ε k q 
 2 3
t 1   Im( S A )     ε F 2