the last chapter, in the next three chapters we demonstrate various applications of

the real-time formalism to the study of quantum dynamics.

6

Linear response theory

There exists a regime of overlap between the equilibrium and non-equilibrium be-

havior of a system, the non-equilibrium behavior of weakly perturbed states. When

a system is perturbed ever so slightly, its response will be linear in the perturba-

tion, say the current of the conduction electrons in a metal will be proportional to

the strength of the applied electric ¬eld. This regime is called the linear response

regime, and though the system is in a non-equilibrium state all its characteristics

can be inferred from the properties of its equilibrium state. In the next chapter we

shall go beyond the linear regime by showing how to obtain quantum kinetic equa-

tions. The kinetic-equation approach to transport is a general method, and allows

in principle nonlinear e¬ects to be considered. However, in many practical situa-

tions one is interested only in the linear response of a system to an external force.

The linear response limit is a tremendous simpli¬cation in comparison with general

non-equilibrium conditions, and is the subject matter of this chapter. In particular

the linear response of the density and current of an electron gas are discussed. The

symmetry properties of response functions, and the ¬‚uctuation“dissipation theorem

are established. Lastly we demonstrate how correlation functions can be measured in

scattering experiments, as illustrated by considering neutron scattering from matter.

Needless to say, in measurements of (say) the current in a macroscopic body, far less

information in the current correlation function is probed.

6.1 Linear response

In this section we consider the response of an arbitrary property of a system to a

general perturbation. The Hamiltonian consists of two parts:

H = H0 + H t , (6.1)

where H0 governs the dynamics in the absence of the perturbation Ht .

For the expectation value of a quantity A for a system in state ρ we have

A(t) = Tr(ρ(t) A) = Tr(U (t, t ) ρ(t ) U † (t, t ) A) . (6.2)

151

152 6. Linear response theory

Expanding the time-evolution operator to linear order in the applied perturbation

we get

t

i ¯†

U (t, t ) = U0 (t, t ) ’ U0 (t, tr ) dt HI (t) U0 (t , tr ) + O((Ht )2 ) ,

¯ (6.3)

t

where the perturbation is in the interaction picture with respect to H0

†

HI (t) = U0 (t, tr )Ht U0 (t, tr ) . (6.4)

For the statistical operator we thus have the perturbative expansion in terms of the

perturbation

ρ(t) = ρ0 (t) + ρ1 (t) + O((Ht )2 ) , (6.5)

where

† †

ρ0 (t) = U0 (t, ti ) ρi U0 (t, ti ) = U0 (t, tr ) ρ0 (tr ) U0 (t, tr ) (6.6)

and the linear correction in the applied potential is given by

t

i ¯†

¯

ρ1 (t) = U0 (t, ti ) ρi U0 (ti , tr ) dt HI (t) U0 (t, tr )

ti

t

i ¯† †

’ ¯

U0 (t, tr ) dt HI (t) U0 (ti , tr ) ρi U0 (t, ti ) . (6.7)

ti

We have assumed that prior to time ti , the applied ¬eld is absent, and the system is

in state ρi . For the expectation value we then get to linear order

t

i ¯ ¯

A(t) = Tr(ρ0 (t) A) + dt Tr(ρ0 (tr ) [HI (t), AI (t)]) . (6.8)

ti

So far the statistical operator at the reference time has been arbitrary; however,

typically we shall assume the state prior to the application of the perturbation is the

thermal equilibrium state of the system.

We ¬rst discuss the density response to an external scalar potential, and after-

wards the current response to a vector potential.

6.1.1 Density response

In this section we consider the density response to an applied external ¬eld. The

external ¬eld is represented by the potential V (x, t), and the Hamiltonian consists

of two parts:

H(t) = H + HV (t) , (6.9)

where H governs the dynamics in the absence of the applied potential, and the

applied potential couples to the density of the system as speci¬ed by the operator,

Eq. (2.28),

HV (t) = dx n(x) V (x, t) . (6.10)

6.1. Linear response 153

The density will adjust to the applied potential, and according to Eq. (6.8) the

deviation from equilibrium is to linear order

∞

δn(x, t) = n(x, t) ’ n0 (x, t) = dx dt χ(x, t; x , t ) V (x , t ) , (6.11)

ti

where

n0 (x, t) = Tr(ρ0 (t) n(x)) (6.12)

is the density in the absence of the potential, and the linear density response can be

speci¬ed in the various ways by the density“density response function:1

i

’ θ(t ’ t )Tr(ρ0 (tr )[n(x, t), n(x , t )])

χ(x, t; x , t ) =

i

≡ ’ θ(t ’ t ) [n(x, t), n(x , t )] 0

i

’ θ(t ’ t )Tr(ρ0 (tr )[δn(x, t), δn(x , t )])

=

≡ χR (x, t; x , t ) . (6.13)

The density operator is in the interaction picture with respect to H

n(x, t) = eiH(t’tr ) n(x) e’iH(t’tr ) (6.14)

and we have introduced the density deviation operator δn(x, t) ≡ n(x, t) ’ n0 (x, t).

The retarded density response function appears in Eq. (6.11) in respect of causality;

i.e. a change in the density at time t can occur only as a cause of the applied potential

prior to that time.

Before the external potential is applied we assume a stationary state with respect

to the unperturbed Hamiltonian H, and the initial state is described by a statistical

operator of the form

ρ» |» »|

ρi = ρi (H) = (6.15)

»

where the |» s are the eigenstates of H,

H |» |»

= (6.16)

»

and ρ» = ρi ( » ) is the probability for ¬nding the unperturbed system with energy

» . The unperturbed statistical operator is then time independent, ρ0 (t) = ρi , and

the equilibrium density pro¬le is time independent, n0 (x, t) = n0 (x) = Tr(ρi n(x)).

1A response function is a retarded Green™s function. Our preferred choice of Green™s functions, for

which we developed diagrammatic non-equilibrium perturbation theory, is thus the proper physical

choice.

154 6. Linear response theory

The response function will then only depend on the time di¬erence:

χ(x, t; x , t ) = χ(x, x ; t ’ t )

i »’ »

i

’ θ(t ’ t ) (ρ» ’ ρ» ) »|n(x)|» » |n(x )|» e ( )(t’t )

= . (6.17)

»»

In linear response, each Fourier component contributes additively, so without loss

of generality we just need to seek the response at one driving frequency, say ω,

V (x, t) = Vω (x) e’iωt . (6.18)

For any ω in the the upper half plane, m ω > 0, the applied potential vanishes in

the far past, V (t ’ ’∞) = 0, and the state of the system in the far past becomes

smoothly independent of the applied potential. For ω real we are thus interested

in the analytic continuation from the upper half plane of the frequency-dependent

response function.

Since we shall be interested in steady-state properties, the time integration in

Eq. (6.11) can be performed by letting the arbitrary initial time ti be taken in the

remote past. By letting ti approach minus in¬nity, transients are absent, and there

is then only a linear density response at the driving frequency

δn(x, t) = n(x, t) ’ n0 (x) = δn(x, ω) e’iωt . (6.19)

We obtain for the Fourier transform of the linear density response

δn(x, ω ) = δn(x, ω) δ(ω ’ ω ) , (6.20)

where

δn(x, ω) = dx χ(x, x ; ω) Vω (x ) (6.21)

and

ρ» ’ ρ»

» |n(x )|»

χ(x, x ; ω) = »|n(x)|» (6.22)

» ’ » + ω + i0

»»

is the Fourier transform of the time-dependent linear response function for a steady

state. The positive in¬nitesimal stems from the theta function; i.e. causality causes

the response function χω ≡ χR to be an analytic function in the upper half plane.

ω

If the Hamiltonian H describes the dynamics of independent particles, the linear

response function becomes

i »’ » — —

i

χ(x, x ; t ’ t ) = ’ θ(t ’ t ) (ρ» ’ ρ» )e ( )(t’t )

ψ» (x)ψ» (x)ψ» (x )ψ» (x )

»»

(6.23)

where ψ» (x) = x|» now denotes the energy eigenfunction of a particle correspond-

ing to the energy eigenvalue » , and ρ» the probability for its occupation. For the

Fourier transform we have

ρ» ’ ρ» — —

χ(x, x ; ω) = ψ» (x)ψ» (x)ψ» (x )ψ» (x ) . (6.24)

» ’ » + ω + i0

»»

6.1. Linear response 155

Looking ahead to Eq. (D.22), we can express the Fourier transform of the density

response function in terms of the single particle spectral function (see Appendix D)

∞ ∞

dE ρi (E ) ’ ρi (E)

dE

χ(x, x ; ω) = A(x, x ; E) A(x , x, E ). (6.25)

2π E ’ E + ω + i0

2π

’∞ ’∞

Introducing the propagators for a single particle instead of the spectral functions

= i GR (x, x , E) ’ GA (x, x , E)

A(x, x ; E) (6.26)

we have expressed the response function in terms of the single-particle propagators,

quantities we know how to handle well, as we have developed the diagrammatic

perturbation theory for them.2

6.1.2 Current response

In this section, we shall discuss the linear current response. We shall speci¬cally

discuss the electric current response to an applied time-dependent electric ¬eld rep-

resented by a vector potential A:3

‚A

E=’ , (6.27)

‚t

thereby in view of the preceding we have covered the general case of coupling to an

electromagnetic ¬eld.

Inserting the expression for the current density operator, Eq. (2.47), into the

linear response formula, Eq. (6.8), and recalling the perturbation, Eq. (2.51), the

average current density becomes to linear order

t

i ¯

j(x, t) = Tr(ρ0 (t) jA(t) (x, t)) = dt Tr(ρ0 (tr )[jp (x, t), HA(t) ]) , (6.28)

¯

ti

where jp (x, t) is just the paramagnetic part of the current density operator in the

interaction picture with respect to H.

To linear order in the external electric ¬eld we therefore see that the current

density

∞

p

j± (x, t) = Tr(ρ0 (t) j± (x)) + dx dt Q±β (x, t; x , t ) Aβ (x , t ) (6.29)

β ti

is determined by the current response function

e2 ρ0 (x, x, t)

Q±β (x, t; x , t ) = K±β (x, t; x , t ) ’ δ±β δ(x ’ x ) δ(t ’ t ) , (6.30)

m

2 Ifthe particles have coupling to other degrees of freedom the propagators are still operators

with respect to these, and a trace with respect to these degrees of freedom should be performed, as

discussed in Section 6.2.

3 The case of representing the electric ¬eld as the gradient of a scalar potential can be handled

with an equal amount of labor and the treatments are equivalent by gauge invariance.

156 6. Linear response theory

where we have introduced the current-current response function

i p

θ(t ’ t ) Tr(ρ0 (tr ) [j± (x, t), jβ (x , t )])

p

K±β (x, t; x , t ) =

i p

≡ θ(t ’ t ) [j± (x, t), jβ (x , t )]

p

(6.31)

0

p

and Tr(ρ0 (t)j± (x)) is a possible current density in the absence of the ¬eld. Here we

shall not consider superconductivity or magnetism, and can therefore in the following

assume that this term vanishes.

Assuming that we have a stationary state with respect to the unperturbed Hamil-

tonian before the external ¬eld is applied, the response function depends only on the

relative time

i »’ »

i

θ(t ’ t ) (ρ» ’ρ» ) »|j± (x)» » |jβ (x )|» e

p

p ( )(t’t )

K±β (x, t;x , t ) = .

»»

(6.32)

In linear response each frequency contributes additively so we just need to seek

the response at one driving frequency, say ω,

A(x, t) = A(x, ω) e’iωt . (6.33)

The time integration in Eq. (6.29) can then be performed by letting the arbitrary

initial time, ti , be taken in the remote past (letting ti approach minus in¬nity), and

we only get a current response at the driving frequency

j± (x, t) = j± (x, ω) e’iωt . (6.34)

For the Fourier transform of the current density we then have

dx Q±β (x, x ; ω)Aβ (x , ω) + O(E2 ) ,

j± (x, ω) = + (6.35)

β

where

ρ0 (x, x)e2

K±β (x, x ; ω) ’ δ±β δ(x ’ x )

Q±β (x, x ; ω) = (6.36)

m

and

ρ» ’ ρ» p

» |jβ (x )|» .

p

K±β (x, x ; ω) = »|j± (x)» (6.37)

’ » + ω + i0

»

»»

For the case of a single particle, the paramagnetic current density matrix element

is given by

’ ←

e ‚ ‚

—

’

»|jp (x)|» = ψ» (x) ψ» (x) , (6.38)

2im ‚x ‚x

6.1. Linear response 157

where the arrows indicate whether di¬erentiating to the left or right. For a system

of independent particles, the response function then becomes

∞ ∞

2

dE ρi (E ) ’ ρi (E)

e dE

K±β (x, x ; ω) =

2π E ’ E + ω + i0

m 2π

’∞ ’∞

— [GR (x, x ; E) ’ GA (x, x ; E)]

” ”

— ∇x± ∇xβ [GR (x , x, E ) ’ GA (x , x, E )] . (6.39)

We have introduced the abbreviated notation

’ ←

” 1 ‚ ‚

’

∇x = (6.40)

2 ‚x ‚x

for the di¬erential operator associated with the current vertex in the position repre-

sentation.

In the expression for the current response kernel we can perform one of the energy

integrations, and exploiting the analytical properties of the propagators half of the

terms are seen not to contribute, and we obtain for the current response function for

an electron gas (the factor of 2 accounts for spin)

∞

2

””

e dE

K±β (x, x , ω) = ’2 f0 (E) A(x, x ; E) ∇x±∇xβ GA (x , x; E ’ ω)

m 2π

’∞

” ”

GR (x, x ; E + ω) ∇x± ∇xβ A(x , x; E)

+ . (6.41)

Gauge invariance implies a useful expression for the longitudinal part of the cur-

rent response function, i.e. the current response to a longitudinal electric ¬eld,

∇ — E = 0, viz.4

e2 ρ0 (x, x, ω = 0)

δ±β δ(x ’ x )

K±β (x, x ; ω = 0) = (6.42)

m

and the longitudinal part of the current response function can be written in the form

Q±β (x, x ; ω) = K±β (x, x ; ω) ’ K±β (x, x ; ω = 0) . (6.43)

We can therefore express the longitudinal current density response solely in terms

of the paramagnetic response function

dx [K±β (x, x ; ω) ’ K±β (x, x ; ω = 0)] Aβ (x , ω) .

j± (x, ω) = (6.44)

β

4 For a detailed discussion see chapter 7 of reference [1], and for its relation to the causal and

dissipative character of linear response see appendix E of reference [1].

158 6. Linear response theory

6.1.3 Conductivity tensor

Expressing the current density in terms of the electric ¬eld

dx σ±β (x, x ; ω) Eβ (x , ω) + O(E2 )

j± (x, ω) = (6.45)

β

introduces the conductivity tensor,

Q±β (x, x ; ω)

σ±β (x, x ; ω) = (6.46)

iω

or, equivalently for the longitudinal part,

K±β (x, x , ω) ’ K±β (x, x , ω = 0)

σ±β (x, x , ω) = . (6.47)

iω

We note that the conductivity tensor is analytic in the upper half plane as causality

demands, and as a consequence the real and imaginary parts are related through

principal value integrals, Kramers“Kronig relations,

∞

1 m σ±β (x, x ; ω )

e σ±β (x, x , ω) = P dω (6.48)

ω ’ω

π

’∞

and

∞

1 e σ±β (x, x ; ω )

m σ±β (x, x , ω) = ’ P dω . (6.49)

ω ’ω

π

’∞

The time average of the response function, K±β (x, x ; ω = 0), is a real function,

and we have (for ω real)

’i 1

K±β (x, x ; ’ω)

e σ±β (x, x ; ω) = e = m K±β (x, x ; ω) . (6.50)

ω ω

The real part of the conductivity tensor for an electron gas is according to

Eq. (6.41) given by

∞

f0 (E) ’ f0 (E + ω)

1 e 2

e σ±β (x, x , ω) = dE

π m ω

’∞

— [GR (x, x ; E + ω) ’ GA (x, x ; E + ω)]

” ”

— ∇x± ∇xβ [G (x , x; E) ’ G (x , x; E)] .

R A

(6.51)

In the case where the electron gas in the absence of the applied ¬eld is in the thermal

state, only electrons occupying levels in the thermal layer around the Fermi surface

contribute to the real part of the longitudinal conductivity, as expected.

6.2. Linear response of Green™s functions 159

6.1.4 Conductance

Often we are interested only in the total average current through the system (S

denotes a cross-sectional surface through the system)

ds · j(x, ω)

I(ω) = (6.52)

S

and a proper description is in terms of the conductance, the inverse of the resistance.

Let us consider a hypercube of volume Ld , and choose the surface S perpendicular

to the direction of the current ¬‚ow, say, the ±-direction. In terms of the conductivity

we have (where ds± denotes the in¬nitesimal area on the surface S):

I± (ω) = ds± dx σ±β (x, x , ω) Eβ (x , ω) . (6.53)

βS

Since the current, by particle conservation, is independent of the position of the

cross section we get

I± (ω) = L’1 dx j(x, ω) = L’1 dx dx σ±β (x, x , ω) Eβ (x , ω) . (6.54)

β

For the case of a spatially homogeneous external ¬eld in the β-direction, E± (x, ω) =

δ±β E(ω), we have in terms of the applied voltage across the system, Vβ (ω) = E(ω) L,

I± (ω) = G±β (ω) Vβ (ω) (6.55)

where we have introduced the conductance tensor

L’2 dx dx σ±β (x, x , ω)

G±β (ω) = (6.56)

the inverse of the resistance tensor.

For a translational invariant state, the conductance and conductivity are related

according to

G±β (L) = Ld’2 σ±β (L) . (6.57)

6.2 Linear response of Green™s functions

The linear response of physical quantities can also, for a many-body system, con-

veniently be expressed in terms of the linear response of the single-particle Green™s

function as it speci¬es average quantities. For example, the average current density

can be expressed in terms of the kinetic component of the matrix Green™s function

(recall Eq. (3.83)), and we are therefore interested in its linear response. We repre-

sent the external electric ¬eld E by a time-dependent vector potential A according to

160 6. Linear response theory

Eq. (6.27), and not by a scalar potential; the two cases can be handled with an equal

amount of labor and are equivalent by gauge invariance. According to Eq. (2.51),

the linear coupling to the vector potential is through the coupling to the current

operator. The linear correction to the Green™s function for this perturbation is thus

represented by the diagram depicted in Figure 6.1.

1

2

Figure 6.1 Linear response diagram for a propagator.

The vertex in Figure 6.1 consists of the diagrams produced by inserting the vector

potential coupling into any electron line in any diagram for the Green™s function in

question. For our interest, the Green™s function is the kinetic or Keldysh propagator,

as the labels 1 and 2 allude to in Figure 6.1, referring to the triagonal representation

of the matrix Green™s function. The resulting propagator to linear order in the vector

potential is denoted by δGK .

To get the current density, say for electrons, we insert the kinetic Green™s function

into the current density formula, Eq. (3.83), and assuming that the current density

vanishes in the absence of the ¬eld we obtain

e2

e ‚ ‚

’ δGK (x, t, x , t) A(x, t) GK (x, t, x, t) . (6.58)

j(x, t) = +i

2m ‚x ‚x m

x =x

Recall, if the particles are coupled to other degrees of freedom, the propagators are

still operators with respect to these, and a trace with respect to these degrees of

freedom should be performed, resulting in the presence of vertex corrections corre-

sponding to insertion of the external vertex into all propagators.

According to the expression for the linear correction to the Green™s function, as

depicted in Figure 6.1, the propagator to linear order in the vector potential is

⎛ ⎞

⎜ ⎟

ie ‚ ‚

Tr ⎝„ 1 d2 A(2) · ’

δGK (1, 1 ) = ⎠ , (6.59)

G(1, 2 ) G(2, 1 )

2m ‚x2 ‚x2

2 =2

6.2. Linear response of Green™s functions 161

if in the trace Tr we include the trace over interactions as well as meaning trace over

Schwinger“Keldysh indices. The matrix „ 1 in Schwinger“Keldysh space insures that

the kinetic component is projected out at the measuring vertex.

For the conductivity tensor we then get in the triagonal matrix representation of

the Green™s functions

⎛ ⎞

∞

2 2

e ⎝„ 1 1 ⎠ ’ ne ,

σ±β (ω) = Tr dE p± pβ G(p+ , p+ , E + ω) G(p’ , p’ , E)

2πω V iωm

pp ’∞

(6.60)

where n is, for example, the density of electrons, and p± = p ± q and p± = p ± q, q

being the wave vector of the electric ¬eld. Here we have directly arrived at expressing

the response function, the conductivity, in terms of the single-particle propagators,

quantities we know how to handle well, as we have developed the diagrammatic

perturbation theory for them.5

Transport coe¬cients are thus represented by Feynman diagrams of the form

depicted in Figure 6.2, an in¬nite sum of diagrams captured in the conductivity

diagram.

Figure 6.2 Linear response or conductivity diagram.

Di¬erent types of linear response coe¬cients correspond to the action of di¬erent

single-particle operators at the excitation and measuring vertices. The excitation ver-

tex is proportional to the unit matrix in the dynamical or Schwinger“Keldysh index

in the triagonal representation and the measuring vertex is attributed in the dynam-

ical indices the ¬rst Pauli matrix in order that the trace over the dynamical indices

picks out the kinetic component. As we discuss next, for the case of fermions the

high-energy contribution from the propagator term exactly cancels the diamagnetic

term in Eq. (6.60).

Let us consider the case where the electrons only interact with a random potential,

5 Ifthe particles have coupling to other degrees of freedom the propagators are still operators

with respect to these, and a trace with respect to these degrees of freedom should be performed.

162 6. Linear response theory

in which case the conductivity becomes

⎛ ⎞

2 2

e ⎝„ 1 dE p · p G(p, p , E + ω) G(p , p, , E) ⎠ ’ ne , (6.61)

σ±β (ω) = Tr

2πω iωm

where the bracket means average with respect to the random potential. The quantity

to be impurity averaged is thus the product of two Green™s functions, as depicted in

Figure 6.3.

1

2

Figure 6.3 Propagator linear response diagram.

Denoting the ¬rst term on the right in Eq. (6.61) by K±β (q, ω), and unfolding the

trace in the dynamical indices it can explicitly be represented by the three terms6

RA RR AA

K±β (q, ω) = K±β (q, ω) + K±β (q, ω) + K±β (q, ω) , (6.62)

where

∞

i e 1

2

dE (f0 (E) ’ f0 (E + ω))

RA

K±β (q, ω) =

π m V ’∞

pp

— p± pβ GR (p+ , p+ ; E + ω) GA (p’ , p’ ; E) (6.63)

and

∞

i e 1

2

=’

RR

K±β (q, ω) p± pβ dE f0 (E)

π m V ’∞

pp

— GR (p+ , p+ ; E + ω) GR (p’ , p’ ; E) (6.64)

6 Actually,the terms should be kept together under the momentum summation for reasons of

convergence. For clarity of presentation, however, we write the three terms separately.

6.2. Linear response of Green™s functions 163

and

∞

i e 1

2

AA

K±β (q, ω) = p± pβ dE f0 (E)

π m V ’∞

pp

— GA (p+ , p+ ; E) GA (p’ , p’ ; E ’ ω) . (6.65)

In the ¬rst term integrations are limited to the Fermi surface and can be easily

performed. The two other terms are regular, having the poles of the product of

Green™s functions in the same half plane. The leading-order contribution from these

terms cancels exactly the density terms giving for a degenerate electron gas the

conductivity tensor

∞

f0 (E) ’ f0 (E + ω)

e2 vF

2

σ±β (q, ω) = dE

π ω

’∞

1

— p± pβ GR (p+ , p+ ; E + ω) GA (p’ , p’ ; E) .

ˆˆ (6.66)

V

pp

The apparent singular ω-dependence is thus canceled which is no accident but, as

noted earlier, a consequence of gauge invariance.7

We shall make use of this formula in Chapter 11, where we discuss weak localiza-

tion.

Exercise 6.1. The classical conductivity of a disordered conductor corresponds to

including only the ladder diagrams for the impurity averaged vertex function, i.e. all

the diagrams of order ( /pF l)0 ,

p+ E+ p + E+

p+ E+

p+ E+ R

= + .

qω qω qω

p’ E A

p’ E p ’E

p’ E

(6.67)

Analytically we have that the three-point vector vertex in the ladder approximation,

“L , satis¬es the equation

dp

|Vimp (p ’ p )|2 GR (p+ , E+ )GA (p’ , E) “L (p , q, ω).

“L (p, q, ω) = p + ni

E E

3

(2π )

(6.68)

7 For details of the calculations regarding this point we refer the reader to chapter 8 of reference

[1].

164 6. Linear response theory

Consider the normal skin e¬ect, where the wavelength of the electric ¬eld is much

1.8 We can therefore set the wave vector of the

larger than the mean free path, q l

electric ¬eld q equal to zero in the propagators, and thereby in the vertex function

as its scale of variation consequently is the Fermi wave vector kF = pF / .

Show that the classical conductivity is given by

ne2 „tr

σ0

σ(ω) = , σ0 = (6.69)

1 ’ iω„tr m

where „tr ≡ „tr ( F ) is the transport relaxation time

dˆ F

p

|Vimp (pF ’ pF )|2 (1 ’ pF · pF ) .

ˆˆ

= 2πni N0 (6.70)

„tr ( F ) 4π

In the following we shall assume that, prior to applying the perturbation, the

system is in its thermal equilibrium state. It is therefore of importance for the

relevance of linear response theory to verify that this is a stable state, i.e. weak ¬elds

do not perturb a system out of this state, as will be shown in Section 6.4. But ¬rst we

establish the general properties of response functions satis¬ed in thermal equilibrium

states.

6.3 Properties of response functions

Response functions must satisfy certain relationships. In order to be speci¬c, we

illustrate these relationships by considering the current response function. We have

already noted that causality causes the response function to be analytic in the upper

half ω-plane. The current response function therefore has the representation in terms

of the current spectral function, mK±β (the current response function vanishes in

the limit of large ω),

∞

dω mK±β (x, x ; ω )

K±β (x, x ; ω) = . (6.71)

ω ’ ω ’ i0

π

’∞

Since K±β (x, t; x , t ) contains a commutator of hermitian operators multiplied by

the imaginary unit, it is real, and we have the property of the response function (ω

real)

[K±β (x, x ; ω)]— = K±β (x, x ; ’ω) (6.72)

and the real part of the response function is even9

eK±β (x, x ; ’ω) = eK±β (x, x ; ω) (6.73)

8 For an applied ¬eld of wavelength much shorter than the mean free path, q 1/l, the corrections

to the bare vertex can be neglected.

9 In the presence of a magnetic ¬eld B, we must also reverse the direction of the ¬eld, for example,

mK±β (x, x ; ω, B) = ’ mK±β (x, x ; ’ω, ’B).

6.4. Stability of the thermal equilibrium state 165

and the imaginary part is odd

m K±β (x, x ; ’ω) = ’ mK±β (x, x ; ω) . (6.74)

From the spectral representation, Eq. (6.37), we have

ρ( » )| »| j± (x)|» |2 (δ( » ’ ’ ω) ’ δ( » ’

p

mK±± (x, x; ω) = π + ω)) .

» »

»»

(6.75)

For the thermal equilibrium state, where

»’ »

ρ( » ) = ρ( » ) e (6.76)

kT

we then obtain10

m K±± (x, x; ω) = π 1 ’ e’ ρ( » ) | »| j± (x)|» |2 δ( ’

ω/kT p

+ ω).

» »

»»

(6.77)

For a state where the probability distribution, ρ( » ), is a decreasing function of the

energy, such as in the case of the thermal equilibrium state, the imaginary part of

the diagonal response function is therefore positive for positive frequencies

m K±± (x, x; ω ≥ 0) ≥ 0 . (6.78)

For the imaginary part of the diagonal part of the response function K we therefore

have11

ω mK±± (x, x, ω) ≥ 0 . (6.79)

From the spectral representation, Eq. (6.71), we then ¬nd that the real part of the

response function at zero frequency is larger than zero, eK±± (x, x, ω = 0) > 0. The

diagonal part of the real part of the response function is therefore positive for small

frequencies. Since for large frequencies, the integral in Eq. (6.71) is controlled by

the singularity in the denominator, and as mK±± (x, x, ω) is a decaying function,

the real part of the response function is negative for large frequencies, eventually

approaching zero.

6.4 Stability of the thermal equilibrium state

In this section we shall show that the thermal equilibrium state is stable; i.e. manip-

ulating the system by coupling its physical properties to a weak classical ¬eld that

vanishes in the past and future can only increase the energy of the system. The

average energy of a system is

E(t) = H(t) = Tr(ρ(t)H(t)) . (6.80)

10 Note that the nature of the discussion is general; we already encountered the similar one for

the spectral weight function in Section 3.4.

11 We stress the important role played by the canonical ensemble.

166 6. Linear response theory

The rate of change of the expectation value for the energy (the term appearing when

di¬erentiating the statistical operator with respect to time vanishes, as seen by using

the von Neumann equation, Eq. (3.14), and the cyclic property of the trace),

dE dH ™

= ’ dx Tr(ρ(t) jt (x)) · A(x, t)

= Tr ρ(t) (6.81)

dt dt

has the perturbation expansion in the time-dependent external ¬eld, A,

t

’i

dE ™ p

p

= dx dx dt A± (x, t) [j± (x, t), jβ (x , t )] Aβ (x , t )

0

dt

±β ti

™

’ · A(x, t) + O(A3 )

dx jt (x) 0

∞

™

=’ dx dx dt A± (x, t) Q±β (x, t; x , t ) Aβ (x , t )

±β ti

™

’ · A(x, t) + O(A3 ) .

dx jt (x) (6.82)

0

The dot signi¬es di¬erentiation with respect to time. We recall that the equilibrium

current, jt (x) 0 , is in fact time independent.

An external ¬eld therefore performs, in the time span between ti and tf , the work

≡ E(tf ) ’ E(ti )

W

∞

tf

™

’

= dt dx dx dt A± (x, t) K±β (x, t; x , t ) Aβ (x , t )

±β t ti

i

∞

tf

dK±β (x, t; x , t )

= dt dx dx dt A± (x, t) Aβ (x , t )

dt

±β ti ti

O(A3 ) .

+ (6.83)

In the ¬rst equality we have noticed that the diamagnetic term in the response

function Q does not contribute. For the second equality we have assumed that the

vector potential vanishes in the past and in the future (i.e. the time average of the

electric ¬eld is zero), so that the boundary terms vanish, and we observe that in

6.4. Stability of the thermal equilibrium state 167

that case there is no linear contribution; to linear order the energy of the system is

unchanged.

For an isotropic system we have

K±β (x, x , ω) = K(x, ω) δ(x ’ x ) δ±β (6.84)

and we obtain, in view of Eq. (6.79), the result that the mean change in energy of

the system to second order is positive

∞

dω

ω m K(x, ω) A(x, ω) · A— (x, ’ω) ≥ 0 .

”E ≡ W = dx (6.85)

2π

’∞

Interacting weakly with the physical quantities of a system in thermal equilibrium

through a classical ¬eld, which vanishes in the past and in the future, can thus only

lead to an increase in the energy of the system; the energy never decreases. The

thermodynamic equilibrium state is thus a stable state.12

In the case of a monochromatic ¬eld

1

A(x, ω)e’iωt + A— (x, ω)eiωt = e A(x, ω) e’iωt

A(x, t) = (6.86)

2

we have for the mean rate of change of the energy to second order in the applied

¬eld, T ≡ 2π/ω,

T T t

T

’i 1

dEω 1 dE ™

≡ dt = dt dt dx dx A± (x, t)

dt T dt 4T

±β 0 ti

0

p

— p

[j± (x, t), jβ (x , t )] Aβ (x , t ) (6.87)

0

as the diamagnetic term averages in time to zero. Turning the ¬eld on in the far

past, ti ’ ’∞, we have in terms of the response function

T

’iω

dEω

dx dx A— (x, ω) (K±β (x, x , ω) ’ Kβ± (x , x, ’ω)) Aβ (x , ω)

= ±

dt 4

±β

ω

dx dx A— (x, ω) mK±β (x, x , ω) Aβ (x , ω) .

= (6.88)

±

2

±β

We can, according to Eq. (6.74), rewrite the average work performed by the external

¬eld in the form

T

dEω

ω ρ( » ) (P» ( ω) ’ P» (’ ω)) ,

= (6.89)

dt

»

12 Itis important to stress the crucial role of the canonical (or grand canonical) ensemble for the

validity of Eq. (6.79).

168 6. Linear response theory

where

2

2π 1

dx » | jp (x) · A(x, ω)|» ’

P» ( ω) = δ( + ω) (6.90)

» »

2

»

is Fermi™s Golden Rule expression for the probability for the transition from state » to

any state » in which the system absorbs the amount ω of energy from the external

¬eld, and P» (’ ω) is the transition probability for emission of the amount ω of

energy to the external ¬eld. The equation for the change in energy is thus a master

equation for the energy, and we infer that the energy exchange between a system and

a classical ¬eld oscillating at frequency ω takes place in lumps of magnitude ω.

At each frequency we have for the average work done on the system by the external

¬eld:

T

dEω 1 —

= dx dx E± (x, ω) eσ±β (x, x , ω) Eβ (x , ω) , (6.91)

dt 2

±β

where we have utilized Eq. (6.50) to introduce the real part of the conductivity tensor.

For a translational invariant system we have

1

σ±β (x, x , ω) = σ±β (x ’ x , ω) = eiq·(x’x ) σ±β (q, ω) (6.92)

V q

and we get for each wave vector

E± (x, ω) = E± (q, ω) eiq·x (6.93)

the contribution

T

dEqω V —

= E± (q, ω) eσ±β (q, ω) Eβ (q, ω) . (6.94)

dt 2

±β

Each harmonic contributes additively, and we get for the average energy absorption

for arbitrary spatial dependence of the electric ¬eld the expression

T

dEω V —

= E± (q, ω) eσ±β (q, ω) Eβ (q, ω) . (6.95)

dt 2 q

±β

For an isotropic system the conductivity tensor is diagonal

σ±β (x, x , ω) = δ±β σ(x ’ x , ω) (6.96)

and we have

T

dEω V

|E± (q, ω)|2 eσ±± (q, ω) .

= (6.97)

dt 2 ±

6.5. Fluctuation“dissipation theorem 169

For the spatially homogeneous ¬eld case, E± (q = 0, ω) = 0, we then obtain

T

dEω V —

|E± (q = 0, ω)|2

= e σ±± (q, ω) . (6.98)

dt 2 q

±

Since

1 1

mK±± (x, x, ω) ≥ 0

e σ±± (q, ω) = e σ±± (x, x, ω) = (6.99)

V ω

q

we obtain the result that, for a system in thermal equilibrium, the average change in

energy can only be increased by interaction with a weak periodic external ¬eld13

T

dEω

≥0. (6.100)

dt

The thermal state is stable against a weak periodic perturbation.14

Considering the isotropic d.c. case we get directly from Eq. (6.91) the familiar

Joule heating expression for the energy absorbed per unit time in a resistor biased

by voltage U

T

dE 1 1

G U2 = R I2 ,

= (6.101)

dt 2 2

where R is the resistance, the inverse conductance, R ≡ G’1 , and we have used the

fact that in the d.c. case the imaginary part of the conductance tensor vanishes.

The absorbed energy of a system in thermal equilibrium interacting with an ex-

ternal ¬eld is dissipated in the system, and we thus note that e σ, or equivalently

m K, describes the dissipation in the system.

6.5 Fluctuation“dissipation theorem

The most important hallmark of linear response is the relation between equilibrium

¬‚uctuations and dissipation. We shall illustrate this feature by again considering the

current response function; however, the argument is equivalent for any correlation

function. We introduce the current correlation function in the thermal equilibrium

state

1

˜ (j) p

K±β (x, t; x , t ) ≡ {δj± (x, t), δjβ (x , t )} 0 ,

p

(6.102)

2

where

δj± (x, t) ≡ j± (x, t) ’ j± (x, t) 0

p p p

(6.103)

is the deviation from a possible equilibrium current, jp (x, t) 0 , which in fact is inde-

pendent of time. However, for notational simplicity we assume in the following that

13 In fact, from the positivity of e σ±± (q, ω) for arbitrary wave vector we ¬nd that the conclusion

is valid for arbitrary spatially varying external ¬eld.

14 Since this result is valid at any frequency, we again obtain the result that a system in thermal

equilibrium is stable.

170 6. Linear response theory

the equilibrium current density vanishes. By taking the anti-commutator, we have

symmetrized the correlation function, and since the current operator is hermitian,

the correlation function is a real function.

Since the statistical average is with respect to the equilibrium state (for an arbi-

trary Hamiltonian H), we have on account of the cyclic property of the trace

Tr e’H/kT j± (x, t) jβ (x , t )

p p

K±β (x, t; x , t ) ≡ ≡

> p p

j± (x, t) jβ (x , t ) 0

Tr e’H/kT jβ (x , t ) j± (x, t + i /kT )

p p

=

<

= K±β (x, t + i /kT ; x , t ) (6.104)

as we de¬ne

K±β (x, t; x , t ) ≡ Tr e’H/kT jβ (x , t ) j± (x, t)

p p

≡

< p p

jβ (x , t ) j± (x, t) . (6.105)

0

We note the crucial role played by the assumption of a (grand) canonical ensemble.

We assume the canonical ensemble average exists for all real times t and t , and

consequently K < is an analytic function in the region 0 < m(t ’ t ) < /kT , and

K > is analytic in the region ’ /kT < m(t ’ t ) < 0. For the Fourier transforms

we therefore obtain the relation

K±β (x, x ; ω) = e’

> ω/kT <

K±β (x, x ; ω) . (6.106)

We observe the following relation of the commutator to the retarded and advanced

correlation functions

p

K±β (x, t; x , t ) ’ K±β (x, t; x , t ) =

> < p

[j± (x, t), jβ (x , t )] 0

’i K±β (x, t; x , t ) ’ K±β (x, t; x , t ) ,

R A

= (6.107)

where we have introduced the advanced correlation function

i p

K±β (x, t; x , t ) = ’ θ(t ’ t) [j± (x, t), jβ (x , t )]

A p

(6.108)

0

corresponding to the retarded one appearing in the current response, Eq. (6.31),

K±β (x, t; x , t ) ≡ K±β (x, t; x , t ) .

R

(6.109)

Since the response function involves the commutator of two hermitian operators

we immediately verify that (for ω real)

K±β (x, x , ’ω) = [K±β (x, x , ω)]— .

R(A) R(A)

(6.110)

Analogous to Eq. (6.105) we have for the correlation function, the anti-commutator,

p

{j± (x, t), jβ (x , t )}

p > <

= K±β (x, t; x , t ) + K±β (x, t; x , t ) . (6.111)

0

6.5. Fluctuation“dissipation theorem 171

Using Eq. (6.105) we can rewrite

1>

˜ (j) ω/kT

K±β (x, x , ω) = K±β (x, x , ω) (1 + e )

2

1 1

’ K±β (x, x , ω) ’ K±β (x, x , ω)

> < > <

= K±β (x, x , ω) + K±β (x, x , ω)

2 2

1

— ω/kT

(1 + e ) (6.112)

2

and thereby15

ω

˜ (j) K±β (x, x , ω) ’ K±β (x, x , ω)

R A

K±β (x, x , ω) = coth . (6.113)

2i 2kT

Using Eq. (6.113), and noting that for omega real (we establish this as a conse-

quence of time-reversal invariance in the next section)

K±β (x, x , ω) = [K±β (x, x , ω)]—

A R

(6.114)

we then get the relation between the correlation function and the imaginary part of

the response function

ω

˜ (j)

K±β (x, x , ω) = coth mK±β (x, x , ω) . (6.115)

2kT

We have established the relationship between the imaginary part of the linear re-

sponse function, governing according to Eq. (6.88) the dissipation in the system, and

the equilibrium ¬‚uctuations, the ¬‚uctuation“dissipation theorem.16

According to the ¬‚uctuation“dissipation theorem we can express the change in

energy of a system in an external ¬eld of frequency ω, Eq. (6.88), in terms of the

current ¬‚uctuations

T

dEω 1 — ˜j

= dx dx E± (x, ω) K±β (x, x , ω)Eβ (x , ω) . (6.116)

ω

dt 2 ω coth 2kT ±β

For the current ¬‚uctuations we have (recall Eq. (6.50))

1p ˜ (j)

p

{j± (x, ω), jβ (x , ’ω)} = K±β (x, x , ω)

2

15 If we introduced

i i ˜ (j)

p

K p

K±β (x, t; x , t ) = {δj± (x, t), δjβ (x , t )} = 2 K±β (x, t; x , t )

0

we would be in accordance with the standard notation of the book.

16 Formally the ¬‚uctuation“dissipation theorem expresses the relationship between a commu-

tator and anti-commutator canonical equilibrium average. The ¬‚uctuation“dissipation relation,

Eq. (6.115), is also readily established by comparing the spectral representation of the imaginary

part of the retarded current response function, Eq. (6.37), with that of K (j) . The ¬‚uctuation“

˜

dissipation relationship expresses that the system is in equilibrium and described by the canonical

ensemble.

172 6. Linear response theory

ω

= ω coth e σ±β (x, x , ω) (6.117)

2kT

and the equal-time current ¬‚uctuations are speci¬ed by

∞ ∞

dω ˜ (j) dω ω

˜ (j)

K±β (x, t; x , t) = K (x, x , ω) = coth mK±β (x, x , ω)

2π ±β 2π 2kT

’∞ ’∞

(6.118)

and Eq. (6.79) guarantees the positivity of the equal-time and space current density

¬‚uctuations.

In a macroscopic description we have a local relationship between ¬eld and current

density, Ohm™s law,

j± (x, ω) = σ±β (x, ω) Eβ (x, ω) (6.119)

or equivalently

σ±β (x, x , ω) = σ±β (x, ω) δ(x ’ x ) . (6.120)

The equilibrium current density ¬‚uctuations at point x are then speci¬ed by

1

≡ d(x ’ x ) {j± (x, ω), j± (x , ’ω)}

2 p p

j± xω 0

2V

1 ˜ (j) 1 ω

= K±± (x, ω) = ω coth e σ±± (x, ω) . (6.121)

V V 2kT

We note that the factor

ω ω 1

coth = ω n(ω) + (6.122)

2 2kT 2

is the average energy of a harmonic oscillator, with frequency ω, in the thermal state.

The average energy consists of a thermal contribution described by the Bose function,

and a zero-point quantum ¬‚uctuation contribution.

In the high-temperature limit where relevant frequencies are small compared to

the temperature, ω kT , we get for the current ¬‚uctuations in a homogeneous

conductor with conductivity σ, Johnson noise,

2kT σ

2

j± = (6.123)

ω

V

independent of the speci¬c nature of the conductor.

In the linear response treatment we have assumed the ¬eld ¬xed, and studied the

¬‚uctuations in the current density. However, ¬‚uctuations in the current (or charge)

density gives rise to ¬‚uctuations in the electromagnetic ¬eld as well. As an example

of using the ¬‚uctuation“dissipation theorem we therefore turn the point of view

around, and using Ohm™s law obtain that the (longitudinal) electric ¬eld ¬‚uctuations

are given by

1

2 2

E± xω = j± xω . (6.124)

|σ(x, ω)|2

6.6. Time-reversal symmetry 173

According to Eq. (6.121) we then obtain for the (longitudinal) electric ¬eld ¬‚uctua-

tions

1 ω eσ±± (x, ω)

2

E± xω = ω coth . (6.125)

2kT |σ±± (x, ω)|2

V

In the high-temperature limit, ω kT , we have for the (longitudinal) electric

¬eld ¬‚uctuations, Nyquist noise,

2kT

2

E± = . (6.126)

ω

σV

6.6 Time-reversal symmetry

Hermitian operators will by suitable phase choice have a de¬nite sign under time

reversal: position and electric ¬eld have positive sign, and velocity and magnetic

¬eld have negative sign.17 The following considerations can be performed for any

pair of operators (see Exercise 6.2), but we shall for de¬niteness consider the current

operator, and show that Eq. (6.114) is a consequence of time-reversal invariance.

In case the Hamiltonian is time-reversal invariant,

†

T [j± (x, t), jβ (x , t )] T † [T jβ (x , t ) T † , T j± (x, t) T † ]

p p

p p

=

p

’ [j± (x, ’t), jβ (x , ’t )]

p

= (6.127)

and

T ψ|T [j± (x, t), jβ (x , t )]† T † |T ψ ,

p p

p p

ψ|[j± (x, t), jβ (x , t )]|ψ = (6.128)

where |T ψ is the time-reversed state of |ψ . Consequently,

p p

Tr(ρ(H)[j± (x, ’t), jβ (x , ’t )]) = ’Tr(ρ(H)[j± (x, t), jβ (x , t )])

p p

(6.129)

and we therefore ¬nd that time-reversal invariance implies

—

R A

K±β (x, x ; ω) = K±β (x, x ; ω) (6.130)

i.e. we have established Eq. (6.114).

Exercise 6.2. Consider two physical quantities represented by the operators A1 (x, t)

and A2 (x, t), which transform under time reversal according to

T Ai (x, t) T † = si Ai (x, ’t) si = ±1, i = 1, 2.

, (6.131)

Show that when the Hamiltonian is invariant under time reversal, the response func-

tion

Aij (x, x , t ’ t ) ≡ Tr(ρ(H)[Ai (x, t), Aj (x , t )]) (6.132)

17 For a discussion of time-reversal symmetry we refer the reader to chapter 2 of reference [1].

174 6. Linear response theory

satis¬es the relations

Aij (x, x , t ’ t ) = ’si sj Aij (x, x , t ’ t) = si sj Aij (x , x, t ’ t ) (6.133)

and thereby18

Aij (x, x , ω) = ’si sj Aij (x, x , ’ω) = si sj Aij (x , x, ω) . (6.134)

6.7 Scattering and correlation functions

In measurements on macroscopic bodies only very crude information of the micro-

scopic state is revealed. For example, in a measurement of the current only the

conductance is revealed and not any of the complicated spatial structure of the con-

ductivity. To reveal the whole structure of a correlation function takes a more indi-

vidualized source than that provided by a battery. It takes a particle source such as

the one used in a scattering experiment, using for example neutrons from a spallation

source.

In this section we shall consider transport of particles (neutrons, photons, etc.)

through matter. To be speci¬c we consider the scattering of slow neutrons by a

piece of matter. A neutron interacts with the nuclei of the substance (all assumed

identical). The interaction potential is short ranged, and we take for the interaction

with the nucleus at position RN 19

V (rn ’ RN ) = a δ(rn ’ RN ) . (6.135)

We have thus neglected the spin of the nuclei (or consider the case of spin-less

bosons).20 For the interaction of a neutron with the nuclei of the substance we

then have

V (rn ’ RN ) = a δ(rn ’ RN ) .

V (rn ) = (6.136)

N N

The interaction is weak, and the scattering can be treated in the Born approximation.

For the transition rate between initial and ¬nal states we then have

2π ˆ

| f| V (ˆn ’ RN )|i |2 δ(Ef ’ Ei ) .

“¬ = (6.137)

r

N

For simplicity we assume that the states of the substance can be labeled solely by

their energy

(i) (i) (f) (f)

|i = |p , ES = |p |ES |f = |p, ES = |p |ES

, , (6.138)

18 Ifthe Hamiltonian contains a term coupling to a magnetic ¬eld, the symmetry of the correlation

function is Aij (x, x , ω, B) = ’si sj Aij (x, x , ’ω, ’B) = si sj Aji (x , x, ω, ’B).

19 We thus exclude the possibility of any nuclear reaction taking place.

20 However, it is precisely the magnetic moment of the neutron that makes it an ideal tool to

investigate the magnetic properties of matter. The subject of neutron scattering is thus vast, and

for a general reference we refer the reader to reference [21].

6.7. Scattering and correlation functions 175

where the initial and ¬nal energies are

p2 p2

(f) (i)

Ef = ES + , Ei = ES + (6.139)

2mn 2mn

and mn is the mass of the neutron. We introduce the energy transfer from the neutron

to the material

p2 p2 (f) (i)

’ = ES ’ ES

ω= (6.140)

2mn 2mn

and we have for the transition probability per unit time

2π (f) (i) (f) (i)

ˆ

| p, ES |a δ(ˆn ’ RN )|p , ES |2 δ(ES ’ ES ’ ω) .

“¬ = (6.141)

r

N

Since the interaction is inelastic, the di¬erential cross section of interest, d2 σ/dˆ d ,

p

is the fraction of incident neutrons with momentum p being scattered into a unit

solid angle dˆ with energy in the range between and + d . Noting that

p

”p = p2 dp dˆ = mn p d dˆ (6.142)

p p