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)