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

’∞