Introduction to Linear Response Propagators and Green’s Functions

Many-Body TheoryApplication to Electrons in Solids

Charles Patterson [email protected]

School of Physics, Trinity College Dublin

• 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

• 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– Wick’s Theorem for evaluating time-ordered products of operators

– Lehmann Representation to extract retarded functions

• Integral equations which arise in the new concepts– Dyson’s equation G = Go + GoG

– Bethe-Salpeter Equation = o + o v

– Effective Potential V = v + v o 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

Probability that system is in infinitesimal region of phase space d

Elements of position p and momenta q

Average value of variable A

Density in phase space

1),(d

),P(d

),P(),(

),P(d

),)P(,A(dA

...dddd

...dddd

ddd

)d,P(

N21

N21

qp

qp

qpqp

qp

qpqp

qqq q

ppp p

qp

qp

p1, q1

p2, q2


• Average value of variable A in NVT Canonical Ensemble

Probability depends on total energy of state

Canonical partition function normalises

Helmholtz free energy

Average value of variable A

E)/kT-(F

F/kT

)/kT,E(-

)/kT,E(-

)/kT,E(-

)/kT,E(-

)/kT,E(-

)/kT,E(-

AedA

ez

1

lnz kTF

edz

ed

)e,A(dA

ed

e),(

e),P(

qp

qp

qp

qp

qp

qp

qp

qp

qp


• Correspondence with Quantum Mechanics

Density operator

Variables represented as operators

Complete set of states

Expectation value of A

Trace of A

Sum of diagonal elements with probability weights

Pure state

Mixed state

0p 1p

0p 1p

a pa pA

...Tr

ˆATrAˆTrA

aA

pˆ

iji

iji

nnn

iji

ij

i

statesallnjiiin

nstatesall

n

i

ji

ii


• Linear Response in Classical Mechanics

Hamiltonian contains Ho and perturbation B

Average value of A in absence of perturbation B = 1/kT

<A> changes in presence of perturbation B

Defines the linear response

BH-

BH-

H-

H-

H-

H-

o

o

0

o

o

o

o

o

ed

AedA

A

ed

AedAAA

ed

AedA

BHH


• Linear Response in Classical Mechanics

Expansion of exponential with scalar exponent

Linear response function

ooo20

2

o

o

o

o

o

o

2

o

o

32

2

BAABV

UV'-VU'lim

BAABV

UV'-VU'

ed

e Bd

ed

eA d

ed

e ABd

V

UV'-VU'

e Bd V'

e ABd U'

...A3!

1A

2!

1A1e

V V'

U U'

V

UV'-VU'

V

U

B-H-

B-H-

B-H-

B-H-

B-H-

B-H-

B-H-

B-H-

A


• Example of application of Static Linear Response

Molecule in gas phase with permanent dipole moment

-p.E term in Hamiltonian

<px>o vanishes in absence of field

p is magnitude of permanent moment

P is polarisation

is molecular susceptibility of gas phase

kT3

p

EkT3

pE

V3

NpP

3

pp

Epp

EpppEppEppp

EpB-

pA

2

x

2

x

2

x

2

o

2x

xo

2xx

xoxoxo

2xoxxoxoxxxx

xx

x


• Time-dependent linear response

Relaxation after static perturbation in time domain

B(0)A(t)

B(0)A(t)A(t)

ed

e B(0)1A(t)d

ed

e A(t)dA(t)

Bt)(HH

0

o

o

o

H-

H-

H-

H-

• 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)

(-t)

t

1

0


• 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

Relaxation after static perturbation in time domain(t)(0)pp(t)

E(t)(0)pp(t)p

(t)(0)p(0)Ep(t)p

B(0)A(t)

xx

xxxx

xxxx

kT

B(0)A(t))(χ d)(χ dA(t)

0 t 0)f(t'

0 t)f(t'

t τ0 t' τ- t'dt'dτ

t'tτ

))f(t't'-(tχdt'A(t)

)t'-(tχ)t'(t,χ

t' t 0)t'(t,χ

))f(t't'(t,χdt'A(t)

t

AB

t

AB

t

-

AB

ABAB

AB

-

AB


• Time-dependent linear response

Define the response function AB

Causality requires this for t < t’

Response depends only on time difference

Introduce change of variable

f(t’) switches off at t’ = 0

Linear response in terms of response function


• Linear response function

Linear response function is correlation function

0 0)(χ

0 )(AB(0)-)(χ

(t)AB(0)B(0)A(t)dt

d

rule Leibniz' (t)χ)(χ ddt

d

B(0)A(t))(χ d

AB

.

AB

.

AB

t

AB

t

AB


• Example: Mobility of particle in a fluid

Phenomenological relation

Force acting on charged particle in uniform field

Linear response approach

Need to know velocity auto-correlation function

Mobility

0

xx

..

0

xxx

0

x

.

x

0

x

.

xx

0

xvx

t

-

xvxx

xx

xx

)((0)vv d

τ)(tBA(t)τ)(t)B(tA )((0)vv dF

0τ)A(t)B(tdt

d )((0)vx dF

)(vx(0) dF (t)v

)(χ dF)t'-(tχdt'F(t)v

kxρ(x)φ(x)HqEF

F (t)v

xx


• Example: Electric conductivity

Hamiltonian for charged particle in vector potential, A

Omit nonlinear A2 term

Define current density

Hamiltonian as Ho + perturbing part

t),(.t,dc

1HH

)'-((t)et),(c

1 U

2m

pH

AO.mc

eHH

UA2mc

e.

mc

e

2m

pH

Uc

e

2m

1H

o

.

pot.

2

o

2o

pot.2

2

22

pot.

2

rj)r A(r

rrvrj

AE

pA

pA

Ap

-


• Example: Electric conductivity of harmonic oscillator e.g. phonon

Equation of motion

Set driving force to zero

Equation of motion in absence of driving force

Solution in absence of driving force

A, depend on initial conditions

Required for EoM to be satisfied – defines 1

t/2

t/2

eδ)tsin(2

-δ)tcos(A(t)x

4-

δ)tsin(Aex(t)

0xxx

0 F(t)

F(t)x mxmxm

111

.

22o

21

1

2o

...

2o

...

)t'-(t )t'-G(tdt

d

dt

d 2o2

2



• 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.

G(t-t’) is the Green’s functionDirac delta function is a unit impulse function

Velocity of oscillator when given an unit impulse at t = 0

t/2et)sin(2

-t)cos(m

1(t)x 111

1

.

)F(t'

m

e )t'(tsindt'x(t)

e )t(tsin Am

1x(t)

e )t'(tsin2

)t'(tcosm

1(t)

e )t'(tsinm

1x(t)

m

1

m

)F(t'dt'

m

)F(t'lim(t)dtxlim)t'(x)t'(x

t

0 1

1

ii1

ii

1

1111

11

t'

t'0

t'

t'

..

0

..

)t'-(t2Γ-

)t-(t2Γ-

)t'-(t2Γ-.

x

)t'-(t

2Γ-




Unit impulse F(t’) = 1

t t+ t t+ t'

Position following unit impulseVelocity following unit impulsePosition following impulse seriesPosition following continuous force

)F(t')t't(m

e)t'(tsin

2-)t'(tcos dt'

)F(t')t'G(tdt'dt

d(t)x

)F(t')t'G(tdt'x(t)

)t'-(t )t'-G(tdt

d

dt

d

)t't(m

e )t'(tsin )t'-G(t

1111

t

t.

t

2o2

2

1

1

)t'-(t2Γ-

)t'-(t2Γ-




Green’s function

Defining relation for G

Position, from G for a given F

Velocity, from G for a given F




Change of variable

Example: impulse at t’=0

Recover original solution from impulse response function

)t(m

et)sin(

2-t)cos((t)x

)-(t)(m

e)sin(

2-)cos( d(t)x

)-(t)(t')F(t'

)-F(t)(m

e)sin(

2-)cos(ω d(t)x

0t t' - t'dt'd t't

)F(t')t't(mω

e)t'(tωsin

2-)t'(tωcosω dt'(t)x

1111

.

1111

0

.

1111

0

.

1111

t.

2Γt-

2Γ-

2Γ-

)t'-(t2Γ-

ii

i ii

i

ii

22o

22o

.

1111

0

.

tt

-t2Γ-

-tt'

e

dt

dRe

m

1e Re

m

1x

e )(m

e)sin(

2-)cos( dRe(t)x

eRe e Re)F(t'


Example: Periodic forcing



If we set the lower limit in the integral dt’ to 0 instead of – we obtain additional transient terms which depend on initial conditions


2Γt-

2Γt-

2Γ-

2)Γ(t-

x

x2Γt-

et)sin(2

-t)cos(m

kT(t)x(0)x

et)sin(2

-t)cos(1

m

kT(t)x

m

kT(0)x

2

(0)xm

2

kTE

e)sin(2

-)cos(2

e)(tsin2

-)(tcos

et)sin(2

-t)cos(dt)(t(t)vvdt

1111

..

1111

...

1111

111

111

00

xx


• Evaluate velocity auto-correlation function

T

t')t'(T

'

e f *f dT

ee )(t'f )t'-(Tf dt'dt)()g(g

dT dt t'-T tt'tT

e )(t'f dt'e (t)fdt )()g(g

e (t)fdt )(g

f*f)'(tf)t'-(tf dt'f*f

21

2121

2121

11

122121

i

ii

titi

ti



• Parseval’s Theorem

Convolution of f1 and f2

Fourier transform of f1

Product of Fourier transforms

Fourier transform of convolution



)().()( )(mV

e

e)(e

)sin(2

-)cos(dmV

e)G(

)-(tE)(e)sin(2

-)cos(m

kT

V

e)-(t).()((0)

V

e

)()((0).c

)-(td

V

e(t)

)((0))G( )'((t)et),(

))F(G()X( )-F(t)G( d)-F(t)G( d)F(t')t'G(tdt'x(t)

22o

2

1111

2

1111

22

0

.2

.

0

t

2Γ-

2Γ-

EJ

Evv

vvA

j

ABrrvrj

i

i

i



Same result as Drude when o tends to zero

22

2

22

2

2

2

22

Drude

22

2

oSHO

1

1

m

e1

1

m

e

1

1

m

e

1

1

m

e)(

m

e)0;(

ii

i

i

i


• 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

Introduction to Linear Response Propagators and Green’s Functions

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