Many-Body Theory
Application to Electrons in Solids
Charles Patterson
Charles.Patterson@tcd.ie
School of Physics, Trinity College Dublin
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Introduction to Linear Response
Propagators and Green’s Functions
Green’s Function for Schrödinger Equation
Functions of a Complex Variable
Contour Integrals in the Complex Plane
Schrödinger, Heisenberg, Interaction Pictures
Occupation Number Formalism
Field Operators
Wick’s Theorem
Many-Body Green’s Functions
Equation of Motion for the Green’s Function
Evaluation of the Single Loop Bubble
The Polarisation Propagator
The GW Approximation
The Bethe-Salpeter Equation
Numerical Aspects of Many-Body Theory
The GW Approximation
Hybrid Density functionals
Recommended Texts
• A Guide to Feynman Diagrams in the Many-Body Problem, 2nd Ed.
R. D. Mattuck, Dover (1992).
• Quantum Theory of Many-Particle Systems,
A. L. Fetter and J. D. Walecka, Dover (2003).
• Many-Body Theory of Solids: An Introduction,
J. C. Inkson, Plenum Press (1984).
• Green’s Functions and Condensed Matter,
G. Rickaysen, Academic Press (1991).
• Mathematical Methods for Physicists, 5th (Int’l) Ed.
G. B. Arfken and H. J. Weber, Academic Press (2001)
• Elements of Green’s Functions and Propagation,
G. Barton, Oxford (1989).
Overview
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Propagation of single particles or holes
Zero temperature formalism (c.f. finite temperature formalism)
Applicable in first principles, model Hamiltonian, … methods
Scattering of particles and holes from each other and external potentials
–
–
–
•
Renormalisation of particle or hole energies
Finite lifetimes for particles or hole excitations (quasiparticles)
Particle-hole bound states (excitons, plasmons, magnons, …)
New concepts such as
–
–
N-body Green’s functions particle, hole, particle-hole, particle-particle, …
propagators
Self energy (energy renormalisation and excitation lifetime)
Overview
•
Approximations to propagators derived from expansion in (Feynman)
diagrams or by functional derivative technique
–
–
•
Integral equations which arise in the new concepts
–
–
–
•
Wick’s Theorem for evaluating time-ordered products of operators
Lehmann Representation to extract retarded functions
Dyson’s equation
Bethe-Salpeter Equation
Effective Potential
G = Go + GoSG
P = Po + Po v P
V = v + v Po V
Applications of Concepts and Methods in Many-Particle Theory
–
–
–
Correlation Effects and the Total Energy (No-body propagator)
Self-Energies and the GW approximation (1-body propagator)
Collective Excitations and the Bethe-Salpeter Equation (2-body propagator)
Classical Statistical Mechanics
• Average value of variable A
• Probability distribution in phase space r
P(p, q)dG
Probability that system is in infinitesimal region of phase space dG
dG  d p dq
dp  dp1dp 2 ...dp N
dq  dq1dq 2 ...dq N
A
dGA(p, q)P(p, q)


 dGP(p, q)
r (p, q) 
P(p, q)
 dGP(p, q)
 dGr (p, q)  1
Elements of position p and momenta q
Average value of variable A
Density in phase space
p1, q1
p2, q2
Classical Statistical Mechanics
• Average value of variable A in NVT Canonical Ensemble
P(p, q)  e- E( p, q )/kT
r (p, q) 
A 
Probability depends on total energy of state
e- E( p, q )/kT
- E( p, q )/kT
d
G
e

- E( p, q )/kT
d
G
A(
p
,
q
)e

- E( p, q )/kT
d
G
e

z   dGe- E( p, q )/kT
Canonical partition function normalises r
F  kT lnz
Helmholtz free energy
1
 e  F/kT
z
A   dGAe (F - E)/kT
Average value of variable A
Classical Statistical Mechanics
• Correspondence with Quantum Mechanics
rˆ  p i  i  i
Density operator
  a ij  i  j
Variables represented as operators
 
i
   
  Tr rˆÂ  Tr Ârˆ

Tr 
n
...  n
Complete set of states
Expectation value of A
Trace of A
all states
 
p
i
a ij  n  i  i  i  j  n  p n a nn
all states
p i  1 p j i  0
p i  1 p j i  0
Sum of diagonal elements with probability weights
Pure state
Mixed state
Classical Statistical Mechanics
• Linear Response in Classical Mechanics
H  H o  B
A
o

Hamiltonian contains Ho and perturbation B
 dGAe
 dG e
- bH o
- bH o
A  A o  A 

A

Average value of A in absence of perturbation B
b = 1/kT
- bH
d
G
Ae

- bH
d
G
e

<A> changes in presence of perturbation B
Defines the linear response
 0
- b H   B 
o

  dGAe
A 

 dGe - bH o  B 

Classical Statistical Mechanics
• Linear Response in Classical Mechanics
  U  U' V - UV'
U
V
U' 
V' 
 
2
  V 
V


1
1
2
3
e  A  1  A  A   A   ...
2!
3!
U'  b  dGAB e
V'  b  dGB e
-b H o -B 
Expansion of exponential with scalar exponent
-b H o -B 
U' V - UV'
b
2
V
 dGAB e
 dGe
-b H o -B 
-b H o -B 
b
U' V - UV'
 b  AB  A B 
2
V
U' V - UV'
lim
 b AB o  A o B
2
 0
V

o

 dGA e
 dGe
-b H o -B 
-b H o -B 
 dGB e
 dGe
-b H o -B 
-b H o -B 
Linear response function
Classical Statistical Mechanics
• Example of application of Static Linear Response
A  px
Molecule in gas phase with permanent dipole moment
- B  p x E x
-p.E term in Hamiltonian


px  b pxpxEx

p x  b p 2x E x
 px
o
o
pxEx
o
 b  p
2
x o
 px
o
px
o
E
<px>o vanishes in absence of field
o
2
p
p 2x 
p is magnitude of permanent moment
o
3
Np2 b
rp 2
Px 
Ex 
E x P is polarisation
3V
3kT
rp 2
c
c is molecular susceptibility of gas phase
3kT
x
Classical Statistical Mechanics
• Time-dependent linear response
• Consider a system where a steady perturbation –B is applied from t→-
• The perturbation is switched off abruptly at t = 0
• To obtain <A(t)> we must use the perturbed system density at t = 0 (full H)
1 (-t)
H  H o   ( t) B
-b H
A(t)
dGA(t) e


-bH
d
G
e


A(t)

b B(0)A(t)
dGA(t) 1  B(0)  e


-bH
d
G
e

-b H o
t
0
o
 b B(0)A(t)
 0
Relaxation after static perturbation in time domain
Classical Statistical Mechanics
• Time-dependent linear response
• Onsager regression hypothesis
The way in which spontaneous fluctuations in a system relax back to
equilibrium is the same as the way in which a perturbed system relaxes
back to equilibrium, so long as the perturbation is small
b B(0)A(t)
p x (t)  b p x (0)E x (0)p x (t)
p x (t)  b p x (0)p x (t) E x
c (t) 
r
kT
p x (0)p x (t)
Relaxation after static perturbation in time domain
Classical Statistical Mechanics
• Time-dependent linear response

A(t)   dt' χ AB (t, t' )f(t' )
Define the response function cAB
-
χ AB (t, t' )  0 t  t'
Causality requires this for t < t’
χ AB (t, t' )  χ AB (t - t' )
Response depends only on time difference
t
A(t)   dt' χ AB (t - t' )f(t' )
-
τ  t  t'
dτ  dt' t'  - τ  
f(t' )   t  0
f(t' )  0 t  0
Introduce change of variable
t'  0
τt
f(t’) switches off at t’ = 0
t


t
A(t)    d χ AB ( )    d χ AB ( )  b B(0)A(t)
Linear response in terms of response function
Classical Statistical Mechanics
• Linear response function

 d χ
AB
( )  b B(0)A(t)
t
d
dt

 d χ
AB
( )   χ AB (t) Leibniz' rule
t
.
d
b B(0)A(t)  b B(0) A (t)
dt
.
χ AB ( )  -b B(0) A ( )
 0
χ AB ( )  0
 0
Linear response function is correlation function
Classical Statistical Mechanics
• Example: Mobility of particle in a fluid m
v x (t)  m Fx
Phenomenological relation
Fx  qE x  H  ρ(x)φ(x)  kx Force acting on charged particle in uniform field

t
v x (t)  Fx  dt' χ v x x (t - t' )  Fx  d χ v x x ( )
-
Linear response approach
0

.
v x (t)   b Fx  d x(0) v x ( )
0

.
  b Fx  d x (0)v x ( )
0

  b Fx  d v x (0)v x ( )
0

m  b  d v x (0)v x ( )
0
d
A(t)B(t  τ)  0
dt
.
.
A(t)B(t  τ)   A(t) B(t  τ)
Need to know velocity auto-correlation function
Mobility
Classical Statistical Mechanics
• Example: Electric conductivity s
2
1 
e 
H
 p  A   U pot.
2m 
c 
Hamiltonian for charged particle in vector potential, A
p2
e
e2
2
H

A.p 
A
 U pot.
2
2m mc
2mc
e
Omit nonlinear A2 term
2
H  Ho 
A.p  O A
mc
p2
1 .
Ho 
 U pot. E  - A
2m
c
Define current density
j(r, t)  ev(t) (r - r ' )
 
H  Ho 
1
dr A(r , t ).j(r, t)

c
Hamiltonian as Ho + perturbing part
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
..
.
m x  mG x  mwo2 x  F(t)
Equation of motion
F(t)  0
Set driving force to zero
..
.
x  G x  wo2 x  0
Equation of motion in absence of driving force
x(t)  Ae Gt/2sin( w1t  δ)
Solution in absence of driving force
A,  depend on initial conditions
2
G
2
2
w1  wo Required for EoM to be satisfied – defines w1
4
.
G


x (t)  A w1cos(w1t  δ) - sin( w1t  δ) e Gt/2
2


Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
We can view the continuous force applied to a mass on a spring as
a sequence of delta function impulses. If we know the response of
the system to a single impulse, provided the system is linear, we can
immediately write down the solution in terms of the impulse
response function.
 d2

d
 2  G  wo2 G(t - t' )   (t - t' )
dt
 dt

G(t-t’) is the Green’s function
Dirac delta function is a unit
impulse function
1 
G
 Gt/2 Velocity of oscillator when given an
x(t) 
w
cos(
w
t)
sin(
w
t)
1
1
1 e

unit impulse at t = 0
mw1 
2

.
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
.
.
x ( t' )  x ( t' )  lim
 0
t'  ..

t' 
x (t)dt  lim
 0
t'

t'
F(t' )
F(t' ) 1
dt' 

m
m
m
Γ
Unit impulse F(t’) = 1
- (t -t')
1
x(t) 
sin w1 (t  t' ) e 2
mw1
.
x (t) 
Γ
1 
G
 - 2 (t -t')
 w1cosw1 (t  t' )  sin w1 (t  t' ) e
mw1 
2

t'
t t+ t t+
Γ
- (t -t i )
1
x(t) 
A i sin w1 (t  t i ) e 2

mw1 i
t
x(t)   dt'
0
- Γ (t -t')
sin w1 (t  t' ) e 2


mw1
F(t' )
Position following unit impulse
Velocity following unit impulse
Position following impulse series
Position following continuous force
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
G(t - t' ) 
- Γ (t -t')
sin w1 (t  t' ) e 2


mw1
 ( t  t' )
 d2

d
 2  G  wo2 G(t - t' )   (t - t' )
dt
 dt

Green’s function
Defining relation for G
t
x(t)   dt' G(t  t' )F(t' )
Position, from G for a given F

t
d
x (t) 
dt' G(t  t' )F(t' )

dt 
.
Velocity, from G for a given F
- Γ (t -t')
e 2
G

  dt' w1cosw1 (t  t' ) - sin w1 (t  t' )
 ( t  t' )F(t' )
2

 mw1

t
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
- Γ (t -t')
e 2
G

x (t)   dt' ω1cosω1 (t  t' ) - sin ω1 (t  t' )
 ( t  t' )F(t' )
2

 mω1

  t  t' d  dt' t'  -   
t'  t   0
Change of variable
.
t

- Γ
e 2

- Γ
e 2
G

x (t)   d ω1cos(w1 ) - sin( w1 )
 ( )F(t -  )
2

 mw1
0
F(t' )   (t' )   (t -  )
Example: impulse at t’=0
.
G

x (t)   d w1cos(w1 ) - sin( w1 )
 ( ) (t -  )
2

 mw1
0
.
- Γt
e 2
G


x (t)  w1cos(w1t) - sin( w1t) 
 (t)
2

 mw1
.
Recover original solution from impulse
response function
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
 iw t -  
iwt'

F(t' )  Re e
  Re e
 

Example: Periodic forcing
- Γ
e 2
G
iw t - 

x (t)  Re  d w1cos(w1 ) - sin( w1 )
 ( ) e  
2

 mw1
0
.

  iw e iwt  1
d

1
e iwt
x  Re  2
  Re 

2
m  wo  w  iGw  m  dt wo2  w 2  iGw 
.
If we set the lower limit in the integral dt’ to 0 instead of – we obtain
additional transient terms which depend on initial conditions
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Evaluate velocity auto-correlation function

Γt

G

 -2
0 dt v x (t)v x (t   )  0 dt w1cos(w1t) - 2 sin( w1t) e x
Γ(t  )
G

 - 2
x w1cosw1 (t   ) - sin w1 (t   ) e
2


.
.
kT m x (0)
kT
E 

 x (0) 
2
2
m
Γ
Gw1 
G
 -2

w1cos(w1 ) - sin( w1 ) e

2 
2

Γt
kT 1 
G
 -2
x (t) 
w1cos(w1t) - sin( w1t)  e

m w1 
2

.
Γt
kT 
G
 -2
x (0) x (t) 
w1cos(w1t) - sin( w1t)  e

mw1 
2

.
.
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Parseval’s Theorem

f1 * f 2   dt' f1 (t - t' )f 2 (t ' )  f 2 * f1
Convolution of f1 and f2


g1 (w )   dt f1 (t) eiwt
Fourier transform of f1


g1 (w )g 2 (w )   dt f1 (t) e
iwt


iw t '
dt'
f
(t'
)
e
Product of Fourier transforms
 2

T  t  t' t  T - t' dt  dT




g1 (w )g 2 (w )   dt  dt' f1 (T - t' ) f 2 (t' ) eiw (T  t') eiwt'

  dT f1 * f 2 eiwT

Fourier transform of convolution
Classical Statistical Mechanics
• Example:
Electric conductivity
of harmonic
oscillator e.g. phonon
t


x(t)   dt' G(t  t' )F(t' )   d G( )F(t -  )   d G( )F(t -  ) X(w )  G(w )F(w )


0
.
j(r, t)  ev(t) (r  r ' ) G( )  b B(0) A( )

.
e b
A(t -  )
j(t)  
d
. v (0) v( )  ( )

V 0
c
2
Γ
e2 b
e 2 b kT 
G
 -2

v (0) v ( )  ( ).E(t -  ) 
w1cos(w1 ) - sin( w1 ) e  ( )E(t -  )

V
V mw1 
2

e2
G(w ) 
mV

- Γ
e 2
G

iw
d

w
cos(
w

)
sin(
w

)

(

)
e
1
1 
  1
2
 w1
e2
 iw

 s (w )
2
2
mV wo  w  iGw
J (w )  s (w ).E(w )
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
re 2 G  iw
s SHO (wo  0; w ) 
m G2  w 2
re 2
1
re 2 1  iw
s Drude (w ) 

m 1  iw
m 1  w 2
1

1


i
w


re 2  



m 2 1
m 1
2
2
  2 w 
 2 w 




   iw 
2
re   

Same result as Drude when wo tends to zero
Classical Statistical Mechanics
• Conclusions
• Linear response functions, e.g. transport coefficients, are derived
from correlation functions
• The correlation function is independent of the external stimulus
(Onsager)
• The reponse function contains the step function () to satisfy
causality