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)