the weak localization e¬ect is thus one of the few e¬ects that can be used as a reference for zero

magnetic ¬eld.

428 11. Disordered conductors

i.e. the typical size of a di¬usive loop of duration t is proportional to D0 t.

For the ¬‚uctuations we have

∞

≡

2

dS S 2 ft (S) = 8π 2 (D0 t)2

S (11.228)

t

’∞

and we can write

π 1

ft (S) = . (11.229)

cosh2 √ πS 2

2 S2 t

2S t

The probability distribution for di¬usive loops is thus a steadily decreasing function

of the area.

The weak localization e¬ect in cylinders and rings leads through the Aharonov“

Bohm e¬ect to an amazing manifestation of the quantum mechanical superposition

principle at the macroscopic level. Furthermore, the weak localization e¬ect can

be reversed to weak anti-localization if the impurities, such as is the case in heavy

compounds, give rise to spin-orbit scattering. Discussion of these e¬ects can be found

in chapter 11 of reference [1].

11.5 Coulomb interaction in a disordered conductor

The presence of impurities changes the e¬ective electron“electron interaction. We

shall study this e¬ect in the weak disorder limit, F „ , which is the common situ-

ation in conductors such as metals and semiconductors. The change from ballistic to

di¬usive motion leads to di¬usion enhancement of the electron“electron interaction.

This leads to interesting observable e¬ects such as the temperature dependence of

the conductivity of a three-dimensional sample being proportional to the square root

√

of the temperature [99], σ ∝ T , instead of the usually unnoticeable T 2 -term due to

Umklapp processes in a clean metal. For experimental evidence of the square root

temperature dependence see references [100, 101].

Let us assume that the inverse screening length is much smaller than the Fermi

wavelength; i.e. the range of the screened Coulomb potential, V , is much larger than

the spacing between the electrons. The exchange correction to the electron energy

» due to electron“electron interaction is then much larger than the direct or Hartree

term. We shall use the method of exact impurity eigenstates and, since diagonal

elements dominate, Σ» ≡ Σ»» , we have for the exchange self-energy

— —

Σex = ’ dx dx V (x ’ x ) ψ» (x) ψ» (x ) ψ» (x ) ψ» (x) , (11.230)

»

» occ.

where the summation is over all occupied states » , i.e. all the states below the Fermi

level since for the moment we assume zero temperature. We are interested in the

mean energy shift averaged over all states with energy ξ (measured from the Fermi

energy)

1

δ(ξ ’ ξ» ) Σex

Σex (ξ) = (11.231)

»

N0 V

»

11.5. Coulomb interaction in a disordered conductor 429

for which we obtain the expression, say ξ > 0,

0

1

’ dξ dx dx V (x ’ x )

Σex (ξ) =

N0 V

’∞

— —

— δ(ξ ’ ξ» ) δ(ξ ’ ξ» ) ψ» (x) ψ» (x ) ψ» (x ) ψ» (x) , (11.232)

»,»

where the prime on the summation sign indicates that the sum is only over states »

occupied and states » unoccupied. The impurity-averaged quantity is the product of

two spectral weight functions in the exact impurity eigenstate representation, except

for the restrictions on the summations. However, these are irrelevant as the main

contribution comes from ξ ξ. In the standard impurity averaging technique we

encounter in the weak-disorder limit, 1/kF l 1, the di¬usion ladder, and we obtain

∞

D0 q 2

1 dq

’

ex

Σ (ξ) = dω V (q) 2 . (11.233)

(2π)d ω + (D0 q 2 )2

2π

ξ/

In the above model of a static interaction the average change in energy is purely

real. The result obtained can be used to calculate the change in the density of states.

To lowest order in the electron“electron interaction we have for the change in density

of states due to the electron“electron interaction

‚Σex (ξ)

δN (ξ) ≡ N (ξ) ’ N0 (ξ) = ’N0 (ξ)

‚ξ

D0 q 2

N0 dq

= V (q) (11.234)

2

(2π)d

2π ξ

q 2 )2

+ (D0

as the change in the density of states due to disorder is negligible in the weak-disorder

limit.

Exercise 11.3. Verify that if V is a short-range potential, the change in the density

of states near the Fermi surface due to electron“electron interaction is in the weak-

disorder limit

|ξ|

δN3 (ξ) V (q = 0)

√

= (11.235)

4 2π 2 ( D0 )3/2

N3 (0)

in three dimensions and, in two dimensions,

|ξ|„

δN2 (ξ) V (q = 0)

= ln . (11.236)

(2π)2 D0

N2 (0)

430 11. Disordered conductors

The singularity in the density of states is due to the spatial correlation of the exact

impurity wave functions of almost equal energy, as described by the singular behavior

of the spectral correlation function. The singularity in the density of states gives rise

to the zero-bias anomaly, a dip in the conductivity of a tunnel junction at low voltages

[102].

Quite generally the propagator in the energy representation satis¬es, in the pres-

ence of disorder and electron“electron interaction, the equation

(0)R (0)R

GR (E) = G»» (E) + G»»1 (E) ΣR1 »1 (E) GR1 » (E) , (11.237)

»» » »

»1 »1

where the propagator in the absence of electron“electron interaction is diagonal,

(0)R (0)R

G»» (E) = G» (E) δ»» , and speci¬ed in terms of the exact impurity eigenstates

(here in the momentum representation)

—

ψ» (p) ψ» (p ) —

R(A) (0)R(A)

≡

G0 (p, p , E) = ψ» (p) ψ» (p ) G» (E) .

E ’ » (’) i0

+

» »

(11.238)

Since energy eigenstates are only spatially correlated if they have the same energy,

only the diagonal terms, ΣR (E) ≡ ΣR (E), contribute in Eq. (11.237), and we obtain

» »»

the result that the propagator is approximately diagonal and speci¬ed by

1

GR (E) = . (11.239)

E ’ » ’ ΣR (E)

»

»

The imaginary part of the self-energy describes the decay of an exact impurity eigen-

state due to electron“electron interaction. When calculating the inelastic decay rate,

we should only count processes starting with the same energy, and on the average in

the random potential we are therefore interested in the quantity

1

δ(E ’ ξ» ) ΣR (E) .

ΣR (E) = (11.240)

E »

N0 V

»

To lowest order in the electron“electron interaction we can set E equal to E in

Eq. (11.240) because their di¬erence is the real part of the self-energy, and we get

for the inelastic electron“electron collision rate

1

’2 m ΣR (E) = i ΣR (E) ’ ΣA (E)

= E E E

„e’e (E, T )

1 (0)R (0)A

’ ΣR (E) ’ ΣA (E) (E) ’ G»

= G» (E) ,

» »

2π N0 V

»

(11.241)

11.5. Coulomb interaction in a disordered conductor 431

where we have expressed the delta function in Eq. (11.240) in terms of the spec-

tral function. We thus have to impurity average a product of a self-energy and

a propagator, say the retarded self-energy and the advanced propagator, presently

both expressed in the exact impurity eigenstate representation. In the weak-disorder

limit, kF l 1, the contributions to the collision rate are therefore speci¬ed in terms

of the Di¬uson and the e¬ective electron“electron interaction as depicted in Figure

11.6. For the case of the product of the retarded self-energy and the advanced prop-

agator there are contributions from the two diagrams depicted in Figure 11.6. In the

case of the retarded interaction, the Di¬uson occurs only for the case where the ki-

netic Green™s function appears right at the emission vertex since impurity correlators

e¬ectively decouple momentum integrations (recall the similar analysis in connection

with Eq. (11.82)).

R K

q, ω q, ω

R K R R

p’ q p’ q p’ q p’ q

E’ ω E’ ω E’ ω E’ ω

D D

+

A A A A

p, E p ,E p, E p ,E

Figure 11.6 Lowest order interaction diagrams for the inelastic collision rate.

We then obtain for the inelastic collision rate or energy relaxation rate in terms

of the Di¬uson and the electron“electron interaction

1 1 dq dω R A

’ D(q, ω)(V (q, ω) ’ V (q, ω)) u4

= m

2 V2 (2π)3

„e’e (E, T ) 2π

— GR (E ’ ω, p ’ q) GA (E, p ) GR (E ’ ω, p ’ q) GA (E, p)

pp

E’ ω ω

— tanh + coth . (11.242)

2kT 2kT

Here we have used that the e¬ective Coulomb interaction has similar statistics prop-

erties as bosons, and in arriving at Eq. (11.242) we have in fact used the ¬‚uctuation“

432 11. Disordered conductors

dissipation relation that relates the kinetic component of the e¬ective Coulomb in-

teraction to the spectral component

ω

K R A

V (q, ω) ’ V (q, ω) coth

V (q, ω) = (11.243)

2kT

accounting for the second term arising from the second diagram in Figure 11.6.64

At this point, we bene¬t in interpretation from an important feature of the de-

veloped real-time non-equilibrium diagram technique, viz. that for the choice of

propagators we have made, the quantum statistics of fermions and bosons manifest

itself in a distinct way in diagrams as noted in Section 5.4. In the ¬rst diagram in

Figure 11.6, where the retarded interaction appears, it leads (according to the dia-

grammatic rules of Section 5.4) to the appearance of the quantum statistics of the

fermions, accounting for the ¬rst term in Eq. (11.242). It is important that this term

occurs in combination with the term containing the boson statistical properties of

the e¬ective Coulomb interaction, and that the boson kinetic component couples to

the electrons as a classical external ¬eld. This feature is generic, and leads in the

present case to the physical feature that zero-point ¬‚uctuations do not cause dissipa-

tive e¬ects. In the present context it corresponds to the fact that the imaginary part

of the self-energy, the inelastic collision rate, for an electron on the Fermi surface,

E = 0, vanishes at zero temperature. Or equivalently, that in accordance with the

exclusion principle the lifetime of an electron on the Fermi surface, E = 0, at zero

temperature is in¬nite.65

The momentum integrals over the impurity-averaged propagators are immediately

performed and we obtain

E’ ω

dq dω ω

R

= mV (q, ω) eD(q, ω) tanh + coth

(2π)3

„e’e (E, T ) 2π 2kT 2kT

(11.244)

from which we can calculate the collision rate.

The e¬ective electron“electron interaction itself, specifying the electron self-energy,

is also changed owing to the presence of impurities. It is thus the dynamically

screened electron“electron interaction in the presence of impurities, as expressed by

the dielectric function, (q, ω),

V (q)

R

V (q, ω) = , (11.245)

(q, ω)

64 Inthe calculation in Section 11.3.2 of the weak localization phase-breaking rate due to electron“

K

electron interaction with small energy transfers, only the kinetic component of the interaction, V ,

was included, but this is justi¬ed by the presence of its quantum statistics factor making it the

dominant component in the low frequency regime. This is the reason for the success of the single-

particle description used for the calculation, where the electron“electron interaction is represented

by a Gaussian distributed classical stochastic potential since it has identical properties with respect

to the dynamical indices as the kinetic component.

65 Such spurious zero-point ¬‚uctuation e¬ects are with frequency conjectured in the literature for

various physical quantities. For an early rebuttal in the context of weak localization see reference

[91].

11.5. Coulomb interaction in a disordered conductor 433

which appears in Eq. (11.244), and not the bare Coulomb interaction, V (q). The

basic excitation of the bare Coulomb potential in an electron gas is the particle“

hole excitation, and it will lead to screening of the interaction. It is su¬cient to

use the random phase approximation where additional interaction decorations by

electron“electron interactions are negligible since the disorder e¬ects are driven by

the long ranged Di¬uson.66 Before averaging with respect to the random impu-

rity potential we thus have the diagrammatic matrix representation of the e¬ective

electron“electron interaction

p+ E+

= +

qω q q q

p’ E’

p+ E+ p+ E+

+ + ...

q q q

p’ E’ p’ E’

p+ E+

= + , (11.246)

q q qω

p’ E’

where the thick wiggly line represents the e¬ective Coulomb interaction, i.e. in the

triagonal representation the matrix

R K

V (q, ω) V (q, ω)

= (11.247)

A

qω 0 V (q, ω)

and similarly for the thin line representing the bare Coulomb interaction for which we

note V K (q) = 0. Analytically the Dyson equation for the matrix Coulomb propagator

has the form

V (q, ω) = V (q) + V (q) Π(q, ω) V (q, ω) , (11.248)

where the polarization, Π, in the triagonal representation has the form

ΠR (q, ω) ΠK (q, ω)

Π(q, ω) = (11.249)

ΠA (q, ω)

0

66 The random phase approximation can also be stated as the linearized mean-¬eld approximation

as discussed for example in chapter 10 of reference [1].

434 11. Disordered conductors

and in the random phase approximation speci¬ed in terms of the dynamical indices

according to

Πkk = ’2i˜ii Gi j Gji γj j .

γk k

(11.250)

Solving the Dyson equation for the e¬ective interaction in the random phase

approximation gives

1

= . (11.251)

qω p+ E+

’1 ’

q qω qω

p’ E’

According to our universal rules for boson“fermion coupling in the dynamical

indices, Eq. (5.51) and Eq. (5.52), the retarded polarization bubble is given by

∞

dp dE

ΠR (q, ω) = ’i GR (p) GK (p ’ q) ’ GK (p) GA (p ’ q) , (11.252)

(2π)3 2π

’∞

where p = (E, p) and q = (ω, q). In the diagrammatic expansion of the e¬ective

electron“electron interaction, we must then impurity average the electron“hole or

polarization bubble diagram. To lowest order in the disorder parameter 1/kF l, we

should insert the impurity ladder into the bubble diagram; i.e. we encounter the

diagrams of the type

R R

. (11.253)

qω qω

A A

The impurity-averaged bubble diagram is evaluated using the standard impurity

Green™s function technique, and we thus have in the di¬usive limit, ql, ω„ 1 (in

the three-dimensional case), for the dielectric function, ql, ω„ 1,

e2 2N0 D0 q 2 D 0 κ2

s

(q, ω) = 1 + = 1+ , (11.254)

2 ’iω + D q 2 ’iω + D0 q 2

0q 0

relating the bare Coulomb interaction to the e¬ective interaction.67 Inserting into

Eq. (11.244), we can calculate the inelastic collision rate.

67 The calculation is equivalent to the calculation of the density“density response function of a

disorder conductor giving the expression

2N0 D0 q 2

χ(q, ω) = .

’iω + D0 q 2

This is understandable since we note that a ¬‚uctuation in the density of electrons creates an electric

potential, which in turn is felt by an electron. Fluctuations in the density or current of the elec-

trons give rise to ¬‚uctuations in an electromagnetic ¬eld inside the electron gas, as discussed quite

generally in Section 6.5 in connection with the ¬‚uctuation“dissipation relations of linear response.

11.5. Coulomb interaction in a disordered conductor 435

We could also calculate the inelastic collision rate or energy relaxation rate in

the dirty limit by solving the Boltzmann equation with the two-particle interaction

modi¬ed by the impurity scattering

∞ ∞

‚f ( ) d

= 2π dω P (ω) R( , , ω) , (11.255)

‚t ’∞2π

’∞

where

f ( ) f ( ’ ω) (1 ’ f ( ’ ω)) (1 ’ f ( ))

R( , , ω) =

’ f ( ’ ω) f ( ) (1 ’ f ( )) (1 ’ f ( ’ ω)) (11.256)

and

2

2N0 „ 2 (D0 q)2

dq V (q)

P (ω) = (11.257)

| (q, ω)| ω 2 + (D0 q)2

(2π)3

π

is analogous to Eliashberg function, ±2 F , for the electron“phonon case. We notice

that we can rewrite

„ dq ζ(q, ω)

V R (q, ω)

P (ω) = m , (11.258)

1 ’ ζ(q, ω)

(2π)3

πω

where ζ is the insertion Eq. (11.20) (here the relevant case is the particle“hole channel,

but the result is identical to that of the particle“particle channel) and given (in two

and three dimensions) by

i ql + ω„ + i

q ≡ |q|

ζ(q, ω) = ln , (11.259)

’ql + ω„ + i

2ql

with the limiting behavior

§ π

⎨ ql > ω„, ql > 1

2ql

ζ(q, ω) = (11.260)

©

1 + iω„ ’ D0 „ q 2 ql, ω„ < 1 .

In the three-dimensional case we have, ω„ < 1,

ω ’1/2

√

P (ω) = . (11.261)

3/2

8 2π 2 N0 D0

We therefore get for an electron on the Fermi surface in a dirty metal the electron“

electron collision rate at temperatures kT < /„ 68

∞

„ 1/2 (kT )3/2

1 2ω

√

= dω P (ω) =c , (11.262)

ω

„e’e (T ) kF l

sinh kT F„

0

The dielectric function and the density and current response functions are thus all related

e2

iσ(q, ω)

(q, ω) = 1 + = 1+ χ(q, ω) .

2

ω0 0q

For a discussion we refer the reader to chapter 10 of reference [1].

68 From the region of large ω and q we get the clean limit rate, Eq. (7.206), which dominates at

temperatures kT /„ .

436 11. Disordered conductors

where c is a constant of order unity (ζ(3/2) 2.612)

√

√

3 3π

ζ(3/2)( 8 ’ 1) .

c= (11.263)

16

For an electron in energy state ξ, ξ < /„ , we get analogously in the dirty limit

for the electron“electron collision rate at zero temperature69

√

„ 1/2

1 6

ξ 3/2 .

= (11.264)

3/2 (k l)2

„e’e (ξ) 4 F

The scattering rate due to electron“electron interaction is thus enhanced in a

dirty metal compared with the clean case [103, 104, 105], di¬usion enhanced electron“

electron interaction.70 Equivalently, the screening is weakened owing to the di¬usive

motion of the electrons. The interpretation of this enhancement can be given in terms

of the previous phase space argument of Exercise 7.10 on page 214 for the relaxation

time and the breaking of translational invariance due to the presence of disorder.

The violation of momentum conservation in the virtual scattering processes due to

impurities gives more phase space for ¬nal states. Alternatively, viewing the collisions

in real space, owing to the motion being di¬usive instead of ballistic the electrons

spend more time close together where the interaction is strong, or, wave functions

of equal energy in a random potential are spatially correlated thereby leading to an

enhanced electron“electron interaction. The scattering process now includes quantum

interference between the elastic and inelastic processes as signi¬ed by the collision

rate /„e’e being dependent on .

We note that the expression for the energy relaxation rate in two dimensions

diverges in the infrared for a dirty metal in the above lowest-order perturbative

calculation. For the Coulomb potential for electrons constricted to movement in two

dimensions the bare Coulomb potential is

2πe2

V (q) = (11.265)

|q|

and for ω„ < 1

1 1

P2 (ω) = (11.266)

8 F„ ω

69 At temperatures and energies kT, ξ > /„ , the expressions for relaxation rates are those of the

clean limit, recall Exercise 7.10 on page 214.

70 In the case of electron“phonon interaction, local charge neutrality forces the electrons to follow

adiabatically the thermal motion of the ions, and because of the coherent motion with the lattice of

the ¬xed impurities, the interaction with the longitudinal phonons is in fact decreased owing to this

compensation mechanism. The imaginary part of the electron self-energy will therefore be given by

the results obtained in Section 11.3.1 for the phase-breaking rate. As shown there, the interaction

with transverse phonons are either enhanced or diminished depending on the temperature regime.

The in¬‚uence of impurities will not be universal for the case of interaction with phonons as will be

the case for the di¬usion enhanced electron“electron interaction.

11.6. Mesoscopic ¬‚uctuations 437

giving the divergent expression for the relaxation rate, kT < /„ ,71

∞

1 1 1

= dω . (11.267)

ω

„e’e (T ) 2kF l sinh kT

0

However, this is not alarming since we do not expect the relaxation rate to be the

relevant measurable quantity, as in this quantity scattering at all energies is weighted

equally. We do not expect such divergences in physically measurable rates, and indeed

the phase relaxation rate of the electronic wave function in a dirty two-dimensional

metallic ¬lm does not diverge because of collisions with small energy transfer, as

discussed in Section 11.3.2. There we made use of the expression for the e¬ective

electron“electron interaction at low energies and momenta in a dirty metal for which,

according to Eq. (11.243), we have72

’4ie2 kT

K

V (q, ω) = = . (11.268)

σ0 |q|2

The low frequency electron“electron interaction in a disordered conductor is thus

identical to the Nyquist noise in the electromagnetic ¬eld ¬‚uctuations, the correlator

we used in Section 11.3.2 (here represented by the scalar potential),

2kT

φ(q, ω)φ(’q, ’ω) = . (11.269)

σ0 q 2

We observe the generality of the result of Section 6.5.

11.6 Mesoscopic ¬‚uctuations

In the following we shall show that when the size of a sample becomes comparable to

the phase coherence length, L ∼ L• , the individuality of the sample will be manifest

in its transport properties. Such a sample is said to be mesoscopic. Characteristically

the conductance will exhibit sample-speci¬c, noise-like but reproduceable, aperiodic

oscillations as a function of, say, magnetic ¬eld or chemical potential (i.e. density

of electrons). The sample behavior is thus no longer characterized by its average

characteristics, such as the average conductance, i.e. the average impurity concen-

tration. The statistical assumption of phase-incoherent and therefore independent

subsystems, allowing for such an average description, is no longer valid when the

transport takes place quantum mechanically coherently throughout the whole sam-

ple. As a consequence, a mesoscopic sample does not possess the property of being

self-averaging; i.e. the relative ¬‚uctuations in the conductance do not vanish in a

71 Wenote that the relaxation rate due to processes with energy transfers of the order of the

temperature is

1 kT

∼ .

„T mD0

72 The factor of ’2i between Eq. (11.268) and Eq. (11.269) simply re¬‚ects our choice of Feynman

rules.

438 11. Disordered conductors

central limit fashion inversely proportional to the volume in the large-volume limit.

To describe the ¬‚uctuations from the average value we need to study the higher

moments of the conductance distribution such as the variance ”G±β,γδ . We shall

¬rst study the ¬‚uctuations in the conductance at zero temperature, and consider the

variance

”G±β,γδ = (G±β ’ G±β )(Gγδ ’ Gγδ ) . (11.270)

For the conductance ¬‚uctuations we have the expression

= (L’2 )2 dx2 dx2 dx1 dx1 σ±β (x2 , x2 ) σγδ (x1 , x1 ) .

G±β Gγδ (11.271)

The diagrams for the variance of the conductance ¬‚uctuations can still be managed

within the standard impurity diagram technique in the weak disorder limit, F „ ,

and a typical conductance ¬‚uctuation diagram is depicted in Figure 11.7 (here the

box denotes the Di¬uson).73

r1 r1

p p

R

R

Ap p A

± γ δ β

p p

R R

p p

A A

r r

Figure 11.7 Conductance ¬‚uctuation diagram.

The construction of the conductance ¬‚uctuation diagrams follows from impurity

averaging two conductivity diagrams. Draw two conductivity bubble diagrams, where

the propagators include the impurity scattering. Treating the impurity scattering

perturbatively, we get impurity vertices that we, upon impurity averaging as usual

have to pair in all possible ways. Since we subtract the squared average conductance

in forming the variance, ”G, the diagrams for the variance consist only of diagrams

where the two conductance loops are connected by impurity lines. As already noted

in the discussion of weak localization, the dominant contributions to such loop-type

diagrams are from the infrared and long-wavelength divergence of the Cooperon, and

here additionally from the Di¬uson.

73 The diagram is in the position representation, and the momentum labels should presently be

ignored, but will be explained shortly.

11.6. Mesoscopic ¬‚uctuations 439

To calculate the contribution to the variance from the Di¬uson diagram depicted

in Figure 11.7, we write the corresponding expression down in the spatial represen-

tation in accordance with the usual Feynman rules for conductivity diagrams. Let

us consider a hypercube of size L. If we assume that the sample size is bigger than

the impurity mean free path, L > l, the spatial extension of the integration over the

external, excitation and measuring, vertices can be extended to in¬nity, since the

propagators have the spatial extension of the mean free path. We can therefore in-

troduce the Fourier transform for the propagators since no reference to the ¬niteness

of the system is necessary for such local quantities. Furthermore, since the spatial

extension of the Di¬uson is long range compared with the mean free path, we can

set the spatial labels of the Di¬usons equal to each other, i.e. r1 = r and r 1 = r .

All the spatial integrations, except the ones determined by the Di¬uson, can then be

performed, leading to the momentum labels for the propagators as depicted in Figure

11.7 Let us study the ¬‚uctuations in the d.c. conductance, so that the frequency, ω,

of the external ¬eld is zero. The energy labels have for visual clarity been deleted

from Figure 11.7, since we only have elastic scattering and therefore one label, say

, for the outer ring and one for the inner, . According to the Feynman rules, we

obtain for the Di¬uson diagram the following analytical expression:

∞ ∞

2

e2 2 u 2 ‚f ( ) ‚f ( ) dp dp

’4

G±β Gγδ =L d d

D

4πm2 (2π )3 (2π )3

’∞ ‚ ‚

’∞

— GR (p )GA (p )GA (p )GR (p )GR (p)GA (p)GA (p)GR (p)

— p± pγ pδ pβ dr dr |D(r, r , ’ )|2 . (11.272)

In order to obtain the above expression we have noted that

’ ) = [D(r, r , ’ )]— ,

D(r, r , (11.273)

which follows from the relationship between the retarded and advanced propagators.

At zero temperature, the Fermi functions set the energy variables in the propagators

in the conductance loops to the Fermi energy, and the Di¬uson frequency to zero. At

zero temperature we therefore get for the considered Di¬uson diagram the following

analytical expression, D(r, r ) ≡ D(r, r , 0),

2

e2 2 u 2 dp dp

’4

G±β Gγδ = L p± pγ pδ pβ

D

4πm2 (2π )3 (2π )3

— [GR (p)GA (p)GR (p )GA (p )]2 dr dr |D(r, r )|2 . (11.274)

F F F F

440 11. Disordered conductors

It is important to note that the same Di¬uson appears twice. This is the leading

singularity we need to keep track of. If we try to construct variance diagrams con-

taining, say, three Di¬usons, we will observe that they cannot carry the same wave

vector, and will give a contribution smaller by the factor / F „ . The momentum

integrations at the current vertices can easily be performed by the residue method

(recall Eq. (11.27))

4π p2 N0 3

dp

p± pγ [GR (p)GA (p)]2 = F

j±γ = „ δ±γ (11.275)

3 3

(2π ) 3

F F

and for the considered Di¬uson diagram we obtain the expression

2

e2 D0 „

’4

δ±γ δδβ dr dr |D(r, r )|2 .

G±β Gγδ =L (11.276)

D

2π

To calculate the Di¬uson integrals we need to address the ¬nite size of the sample

and its attachment to the current leads, since the Di¬uson has no inherent length

scale cut-o¬. At the surface where the sample is attached to the leads, the Di¬uson

vanishes

D(r, r ) = 0 or r on lead surfaces (11.277)

r

in accordance with the assumption that once an electron reaches the lead it never

returns to the disordered region phase coherently. On the other surfaces the current

vanishes; i.e. the normal derivative of the Di¬uson must vanish (recall Eq. (7.96) and

Eq. (7.97))

‚D(r, r )

=0 r or r on non-lead surfaces with surface normal n .

‚n

(11.278)

74

We assume that the leads have the same size as the sample surface. Therefore by

solving the di¬usion equation for the Di¬uson, with the above mixed (Dirichlet“von

Neumann) boundary condition, we obtain the expression

2

1/„

dr dr |D(r, r )|2 = , (11.279)

2

D0 qn

n

where n ≡ (nx , ny , nz ) is the eigenvalue index in the three-dimensional case

π

qn± = n± n± = nx , ny , nz (11.280)

L

where

nx = 1, 2, ..., ny,z = 0, 1, 2, ... (11.281)

74 This “thick lead” assumption is not of importance. Because of the relationship between the

¬‚uctuations in the density of states and the time scale for di¬using out of the sample, the result

will be the same for any kind of lead attachment [106].

11.6. Mesoscopic ¬‚uctuations 441

and we have assumed that the current leads are along the x-axis. Less than three

dimensions corresponds to neglecting the ny and nz . We therefore obtain from the

considered Di¬uson diagram the contribution to the conductance ¬‚uctuations75

2

e2

G±β Gγδ = cd δ±,γ δδ,β , (11.282)

D

4π

where the constant cd depends on the sample dimension. The summation in Eq.

(11.279) should, in accordance with the validity of the di¬usion regime, be restricted

to values satisfying n2 + n2 + n2 ¤ N , where N is of the order of (L/l)2 . However,

x y z

the sum converges rapidly and the constants cd are seen to be of order unity. The

dimensionality criterion is essentially the same as in the theory of weak localization,

as we shall show in the discussion below of the physical origin of the ¬‚uctuation

e¬ects. The important thing to notice is that the long-range nature of the Di¬uson

provides the L4 factor that makes the variance, average of the squared conductance,

independent of sample size (recall Eq. (6.57)). The diagram depicted in Figure 11.7

is only one of the two possible pairings of the current vertices, and we obtain an

additional contribution from the diagram where, say, current vertices γ and δ are

interchanged.

In addition to the contribution from the diagram in Figure 11.7 there is the other

possible singular Di¬uson contribution to the variance from the diagram depicted in

Figure 11.8.

Rp

r r1

r r1

p p

R R

Ap

A A

p p

± γ δ β

p

R

A

p

Figure 11.8 The other possible conductance ¬‚uctuation diagram.

75 Because of these inherent mesoscopic ¬‚uctuations, we realize that the conductance discussed in

the scaling theory of localization is the average conductance.

442 11. Disordered conductors

This diagram contributes the same amount as the one in Figure 11.7, but with a

di¬erent pairing of the current vertices. We note that the diagram in Figure 11.8

allows for only one assignment of current vertices.76

Reversing the direction in one of the loops gives rise to similar diagrams, but

now with the Cooperon appearing instead of the Di¬uson. Because the boundary

conditions on the Cooperon are the same as for the Di¬uson, in the absence of a

magnetic ¬eld, the Cooperon contributes an equal amount. For the total contribution

to the variance of the conductance, we therefore have (allowing for the spin degree

of freedom of the electron would quadruple the value) at zero temperature

2

e2

”G±β,γδ = cd (δ±γ δδβ + δ±δ δγβ + δ±,β δγ,δ ) . (11.283)

2π

The variance of the conductance at zero temperature, and for the chosen geometry

of a hypercube, is seen to be independent of size and dimension of the sample and

degree of disorder, and the conductance ¬‚uctuations appear in the metallic regime

described above to be universal.77

Since the average classical conductance is proportional to Ld’2 , Ohm™s law, we

¬nd that the relative variance, ”G G ’2 , is proportional to L4’2d . This result should

be contrasted with the behavior L’2d of thermodynamic ¬‚uctuations, compared with

which the quantum-interference-induced mesoscopic ¬‚uctuations are huge, re¬‚ecting

the absence of self-averaging.

The dominating role of the lowest eigenvalue in Eq. (11.279) indicates that meso-

scopic ¬‚uctuations, studied in situations with less-invasive probes than the current

leads necessary for studying conductance ¬‚uctuations, can be enhanced compared

to the universal value. In the case of the conductance ¬‚uctuations, the necessary

connection of the disordered region to the leads, which cut o¬ the singularity in the

Di¬uson by the lowest eigenvalue, nx = 1, re¬‚ecting the fact that because of the

physical boundary conditions at the interface between sample and leads, the max-

imal time for quantum interference processes to occur uninterrupted is the time it

takes the electron to di¬use across the sample, L2 /D0 . When considering other ways

of observing mesoscopic ¬‚uctuations, the way of observation will in turn introduce

the destruction of phase coherence necessary for rendering the ¬‚uctuations ¬nite.

In order to understand the origin of the conductance ¬‚uctuations, we note that,

just as the conductance essentially is given by the probability for di¬using between

points in a sample, the variance is likewise the product of two such probabilities.

When we perform the impurity average, certain of the quantum interference terms

will not be averaged away, since certain pairs of paths are coherent. This is similar to

the case of coherence involved in the weak-localization e¬ect, but in the present case

of the variance of quite a di¬erent nature. For example, the quantum interference

76 Thecontribution from the diagram in Figure 11.7 can, through the Einstein relation, be ascribed

to ¬‚uctuations in the di¬usion constant, whereas the diagram in Figure 11.8 gives the contribution

from the ¬‚uctuations in the density of states, the two types of ¬‚uctuation being independent [107].

77 However, for a non-cubic sample, the variance will be geometry dependent [108, 109].

11.6. Mesoscopic ¬‚uctuations 443

process described by the diagram in Figure 11.7 is depicted in Figure 11.9, where the

solid line corresponds to the outer conductance loop, and the dashed line corresponds

to the inner conductance loop. The wavy portion of the lines corresponds to the long-

range di¬usion process.

R

R—

R

R—

Figure 11.9 Statistical correlation described by the diagram in Figure 11.7.

When one takes the impurity average of the variance, the quantum interference terms

can pair up for each di¬usive path in the random potential, but now they correspond

to amplitudes for propagation in di¬erent samples. The diagrams for the variance,

therefore, do not describe any physical quantum interference process, since we are not

describing a probability but a product of probabilities. The variance gives the statis-

tical correlation between amplitudes in di¬erent samples. The interference process

corresponding to the diagram in Figure 11.8 is likewise depicted in Figure 11.10.

444 11. Disordered conductors

R

R—

R—

R

Figure 11.10 Statistical correlation described by the diagram in Figure 11.8.

When a speci¬c mesoscopic sample is considered, no impurity average is e¬ec-

tively performed as in the macroscopic case. The quantum interference terms in the

conductance, which for a macroscopic sample average to zero if we neglect the weak-

localization e¬ect, are therefore responsible for the mesoscopic ¬‚uctuations. In the

weak-disorder regime the conductivity (or equivalently the di¬usivity by Einstein™s

relation) is speci¬ed by the probability for the particle to propagate between points

in space. According to Eq. (11.95)

|Ac Ac | cos(φc ’ φc )

P = Pcl + 2 (11.284)

c,c

as

1

Ac = |Ac | eiφc , φc = S[xc (t)] , (11.285)

where |Ac | speci¬es the probability for the classical path c, and its phase is speci¬ed

by the action. When the points in space in questions are farther apart than the mean

free path, the ensemble average of the quantum interference term in the probability

vanishes. The weak localization can be neglected because for random phases we

have cos(φc ’ φc ) imp = 0. However, for the mean square of the probability, we

encounter cos2 (φc ’ φc ) imp = 1/2, and obtain

|Ac | |Ac | .

P2 2

= P +2 (11.286)

imp imp

c,c

Because of quantum interference there is thus a di¬erence between P 2 and

imp

11.6. Mesoscopic ¬‚uctuations 445

P 2 resulting in mesoscopic ¬‚uctuations. Since the e¬ect is determined by the

imp

phases of paths, it is nonlocal.

The result in Eq. (11.283) is valid in the metallic regime, where the average

conductance is larger than e2 / . To go beyond the metallic regime would neces-

sitate introducing the quantum corrections to di¬usion, the ¬rst of which is the

weak-localization type, which diagrammatically corresponds to inserting Cooperons

in between Di¬usons. Such an analysis is necessary for a study of the ¬‚uctuations in

the strongly disordered regime, as performed in reference [84].

The Di¬uson and Cooperon in the conductance ¬‚uctuation diagrams do not de-

scribe di¬usion and return probability, respectively, in a given sample, but quantum-

statistical correlations between motion in di¬erent samples, i.e. di¬erent impurity

con¬gurations, as each conductance loop in the Figures 11.7 and 11.8 corresponds

to di¬erent samples. In order to stress this important distinction, we shall in the

following mark with a tilde the Di¬usons and Cooperons appearing in ¬‚uctuation

diagrams.

We now assess the e¬ects of ¬nite temperature on the conductance ¬‚uctuations.

Besides the explicit temperature dependence due to the Fermi functions appearing

in Eq. (11.272), the ladder diagrams will be modi¬ed by interaction e¬ects. The

presence of the Fermi functions corresponds to an energy average over the thermal

layer near the Fermi surface, and through the energy dependence of the Di¬uson

and Cooperon introduces the temperature-dependent length scale LT = D0 /kT .

Since the loops in the ¬‚uctuation diagrams correspond to di¬erent conductivity mea-

surements, i.e. di¬erent samples, interaction lines (for example caused by electron“

phonon or electron“electron interaction) are not allowed to connect the loops in a

¬‚uctuation diagram. The di¬usion pole of the Di¬uson appearing in a ¬‚uctuation

diagram is therefore not immune to interaction e¬ects. This was only the case when

the Di¬uson describes di¬usion within a sample, since then the di¬usion pole is a con-

sequence of particle conservation and therefore una¬ected by interaction e¬ects. The

consequence is that, just as in the case for the Cooperon, inelastic scattering will lead

to a cut-o¬ given by the phase-breaking rate 1/„• . In short, the temperature e¬ects

will therefore ensure that up to the length scale of the order of the phase-coherence

length, the conductance ¬‚uctuations are determined by the zero-temperature expres-

sion, and beyond this scale the conductance of such phase-incoherent volumes add

as in the classical case.78 A sample is therefore said to be mesoscopic when its size

is in between the microscopic scale, set by the mean free path, and the macroscopic

scale, set by the phase-coherence length, l < L < L• . A sample is therefore self-

averaging only with respect to the impurity scattering for samples of size larger than

the phase-coherence length.79 A sample will therefore exhibit the weak-localization

e¬ect only when its size is much larger than the phase-coherence length but much

smaller than the localization length L• < L < ξ.

An important way to reveal the conductance ¬‚uctuations experimentally is to

measure the magneto-resistance of a mesoscopic sample. To study the ¬‚uctuation

e¬ects in magnetic ¬elds, we must study the dependence of the variance on the

78 Forexample for a wire we have g(L) = g(L• ) L/L• .

79 Theconductance entering the scaling theory of localization is thus assumed averaged over phase-

incoherent volumes.

446 11. Disordered conductors

magnetic ¬elds ”G±β (B+ , B’ ) , where B+ is the sum and B’ is the di¬erence in

the magnetic ¬elds in¬‚uencing the outer and inner loops. Since the conductance

loops can correspond to samples placed in di¬erent ¬eld strengths, the di¬usion pole

appearing in a ¬‚uctuation diagram will not be immune to the presence of magnetic

¬elds, as in the case when the Di¬uson describes di¬usion within a given sample,

since particle conservation is, of course, una¬ected by the presence of a magnetic ¬eld.

According to the low-¬eld prescription for inclusion of magnetic ¬elds, Eq. (11.105),

we get for the Di¬uson

e 1

˜

(’i∇x ’ δ(x ’ x ) ,

A’ (x))2 + 1/„•

D0 D(x, x ) = (11.287)

„

where A’ is the vector potential corresponding to the di¬erence in magnetic ¬elds,

B’ = ∇x — A’ , and we have introduced the phase-breaking rate in view of the

above consideration. In the case of the Di¬uson, the magnetic ¬eld induced phases

subtract, accounting for the appearance of the di¬erence of the vector potentials A’ .

For the case of the Cooperon, the two phases add, and we obtain

e 1

˜

(’i∇x ’ δ(x ’ x ) ,

A+ (x))2 + 1/„•

D0 C(x, x ) = (11.288)

„

where A+ is the vector potential corresponding to the sum of the ¬elds, B+ =

∇ — A+ .

The magneto-¬ngerprint of a given sample, i.e. the dependence of its conductance

on an external magnetic will show an erratic pattern with a given peak to valley ratio

and a correlation ¬eld strength Bc . This, however, is not immediately the information

we obtain by calculating the variance

[G±β (B1 ) ’ G±β (B1 ) ][Gγδ (B2 ) ’ Gγδ (B2 ) ] , (11.289)

”G±β,γδ (B+ , B’ ) =

where B1 is the ¬eld in, say, the inner loop, B1 = (B+ + B’ )/2, and B2 is the

¬eld in the outer loop, B2 = (B+ ’ B’ )/2. In the variance, the magnetic ¬elds are

¬xed in the two samples, and we are averaging over di¬erent impurity con¬gurations,

thus describing a situation in which the actual impurity con¬guration is changed, a

hardly controllable endeavor from an experimental point of view. However, if the

magneto-conductance of a given sample, G(B), varies randomly with magnetic ¬eld,

the two types of average “ one with respect to magnetic ¬eld and the other with

respect to impurity con¬guration “ are equivalent, and the characteristics of the

magneto-¬ngerprint can be extracted from the correlation function in Eq. (11.289).

The physical reason for the validity of such an ergodic hypothesis [110, 111], that

changing magnetic ¬eld is equivalent to changing impurity con¬guration, is that

since the electronic motion in the sample is quantum mechanically coherent the wave

function pattern is sensitive to the position of all the impurities in the sample, just

as the presence of the magnetic ¬eld is felt throughout the sample by the electron.80

The extreme sensitivity to impurity con¬guration is also witnessed by the fact that

changing the position of one impurity by an atomic distance, 1/kF , is equivalent to

shifting all the impurities by arbitrary amounts, i.e. to create a completely di¬erent

sample [113, 114].

80 The validity of the ergodic hypothesis has been substantiated in reference [112].

11.6. Mesoscopic ¬‚uctuations 447

The ergodic hypothesis can be elucidated by the following consideration. In the

mean square of the probability for propagating between two points in space we en-

counter the correlation function

cos(φc (B1 ) ’ φc (B1 )) cos(φc (B2 ) ’ φc (B2 )) (11.290)

imp

where (φc (B) ’ φc (B)) depends on the phases picked up due to the magnetic ¬eld,

i.e. the ¬‚ux through the area enclosed by the trajectories c and c . When the

magnetic ¬eld B1 changes its value to B2 (where the correlation function equals one

half), the phase factor changes by 2π times the ¬‚ux through the area enclosed by

the trajectories c and c in units of the ¬‚ux quantum. This change, however, is

equivalent to what happens when changing to a di¬erent impurity con¬guration for

¬xed magnetic ¬eld, i.e. the quantity we calculate.81

In order to calculate the variance in Eq. (11.289) we must solve Eq. (11.287) and

Eq. (11.288) with the appropriate mixed boundary value conditions in the presence of

magnetic ¬elds, and insert the solutions into contributions like that in Eq. (11.276).

However, determination of the characteristic correlations of the aperiodic magneto-

conductance ¬‚uctuations can be done by inspection of Eq. (11.287) and Eq. (11.288).

The correlation ¬eld Bc is determined by the sample-to-sample change in the mag-

netic ¬eld, i.e. B’ . According to Eq. (11.287) and Eq. (11.288), this ¬eld is deter-

mined either by the sample size, through the gradient term, or the phase coherence

length. When the phase-coherence length is longer than the sample size, the cor-

relation ¬eld is therefore of order of the ¬‚ux quantum divided by the sample area,

Bc ∼ φ0 /L2 , where φ0 is the normal ¬‚ux quantum φ0 = 2π /|e|, since the typical

di¬usion loops, like those depicted in Figures 11.9 and 11.10, enclose an area of the

order of the sample, L2 . We note that in magnetic ¬elds exceeding max{φ0 /L2 ,

φ0 /L2 }, the Cooperon no longer contributes to the ¬eld dependence of the conduc-

•

tance ¬‚uctuations, because its dependence on magnetic ¬eld is suppressed according

to the weak-localization analysis.82

We note that the weak-localization and mesoscopic ¬‚uctuation phenomena are a

general feature of wave propagation in a random media, be the wave nature classical,

such as sound and light,83 or of quantum origin such as for the motion of electrons.

The weak-localization e¬ect was in fact originally envisaged for the multiple scat-

tering of electromagnetic waves [81].84 The coherent backscattering e¬ect has been

studied experimentally for light waves (for a review on classical wave propagation in

random media, see reference [116]). For the wealth of interesting weak-localization

and mesoscopic ¬‚uctuation e¬ects, we refer the reader to reference [1], and to the

references to review articles cited therein.

81 Another way of revealing the mesoscopic ¬‚uctuations is to change the Fermi energy (i.e. the

density of conduction electron as is feasible in an inversion layer). The typical energy scale Ec for

these ¬‚uctuations is analogously determined by the typical time „trav it takes an electron to traverse

the sample according to Ec ∼ /„trav . In the di¬usive regime we have „trav ∼ L2 /D0 .

82 For an account of the experimental discovery of conductance ¬‚uctuations, see reference [115].

83 Here we refer to conditions described by Maxwell™s equations.

84 It is telling that it took the application of Feynman diagrams in the context of electronic motion

in disordered conductors to understand the properties of classical waves in random media.

448 11. Disordered conductors

11.7 Summary

Quantum e¬ects on transport coe¬cients have been studied in this chapter, espe-

cially the weak localization e¬ect, which is the most important for practical diag-

nostics in material science as it is revealed at such small magnetic ¬elds where the

di¬usion enhancement of the electron“electron interaction is una¬ected and classical

magneto-resistance e¬ects absent. Though the weak localization e¬ect is a quantum

interference e¬ect, the kinetics of the involved trajectories were the classical ones,

be they Boltzmannian or Brownian, and we could therefore make ample use of the

quasi-classical Green™s function technique developed in Chapters 7 and 8. We calcu-

lated the phase breaking rates due to interactions, the phase relaxation of the wave

function measured in magneto-resistance measurements, thereby opening the oppor-

tunity to probe the inelastic interactions experienced by electrons. We studied how

the interactions are changed as a result of disorder. In the case of Coulomb interac-

tion a universal di¬usion enhancement or weakening of screening resulted, whereas

for the case of electron“phonon interaction, the longitudinal interaction was weak-

ened owing to the compensation mechanism of the vibrating impurities, whereas the

interaction with transverse phonons could be enhanced or weakened depending on

the temperature regime. Finally, we discussed the phenomena that sets in when the

electronic motion is coherent in the sample and the signature of mesoscopic ¬‚uctu-

ations are present in transport coe¬cients, such as the quantum ¬‚uctuations in the

conductance, the universal conductance ¬‚uctuations.

12

Classical statistical dynamics

The methods of quantum ¬eld theory, originally designed to study quantum ¬‚uctua-

tions, are also the tool for studying the thermal ¬‚uctuations of statistical physics, for

example in connection with understanding critical phenomena. In fact, the methods

and formalism of quantum ¬elds are the universal language of ¬‚uctuations. In this

chapter we shall capitalize on the universality of the methods of ¬eld theory as intro-

duced in Chapters 9 and 10, and use them to study non-equilibrium phenomena in

classical statistical physics where the ¬‚uctuations are those of a classical stochastic

variable. We shall show that the developed non-equilibrium real-time formalism in

the classical limit provides the theory of classical stochastic dynamics.

Newton™s law, which governs the motion of the heavenly bodies, is not the law

that seems to govern earthly ones. They sadly seem to lack inertia, get stuck and

feebly ramble around according to Brownian dynamics as described by the Langevin

equation. Their dynamics show transient e¬ects, but if they are on short time scale

too fast to observe, dissipative dynamics is typically speci¬ed by the equation v ∝ F

where the proportionality constant could be called the friction coe¬cient. This is

Aristotelian dynamics, average velocity proportional to force, believed to be correct

before Galileo came along and did thorough experimentation. If a sponge is dropped

from the tower of Pisa, it will almost instantly reach its saturation ¬nal velocity. If

a heavier sponge is dropped simultaneously, it will fall faster reaching the ground

¬rst. If on the other hand an apple is dropped and when reaching the ground is

given its opposite velocity it will according to Newton™s equation spike back up to

the position it was dropped from, before repeating its trip to the ground. If a sponge

has its impact velocity at the ground reversed, it will ¬zzle immediately back to the

ground. Unlike Newtonian mechanics, which is time reversal symmetric, sponge or

dissipative dynamics chooses a direction of time.

We now turn to consider dissipative dynamics, in particular Langevin dynamics.

In this chapter we will study systems with the additional feature of quenched disorder,

in particular vortex dynamics in disordered superconductors. The ¬eld-theoretic

formulation of the problem will allow the disorder average to be performed exactly.

The functional methods will allow construction of a self-consistent theory for the

e¬ective action describing the in¬‚uence of thermal ¬‚uctuations and quenched disorder

449

450 12. Classical statistical dynamics

on vortex motion. This will allow the determination of the vortex response to external

forces, the vortex ¬‚uctuations, and the pinning of vortices due to quenched disorder,

and allow to consider the dynamic melting of vortex lattices.

12.1 Field theory of stochastic dynamics

In this section we shall map the stochastic problem, formulated ¬rst in terms of a

stochastic di¬erential equation, onto a path integral formulation, and obtain the ¬eld

theoretic formulation of classical statistical dynamics. We show that the resulting

formalism is equivalent to that of a quantum ¬eld theory. In particular we shall

consider quenched disorder and the resulting diagrammatics. The ¬eld theoretic

formulation will allow us to perform the average over the quenched disorder exactly.

12.1.1 Langevin dynamics

A heavy particle interacting with a gas of light particles, say a pollen dust particle

submerged in water, will viewed under a microscope execute erratic or Brownian

motion. Or in general, when a particle interacting with a heat bath, i.e. weakly with

a multitude of degrees of freedom in its environment (of high enough temperature

so that quantum e¬ects are absent), will exhibit dynamics governed by the Langevin

equation

m¨ t = F(xt , t) ’ · xt + ξt ,

™ (12.1)

x

where m is the mass of the particle, F is a possible external force, · is the viscosity

or the friction coe¬cient, and ξ t is the ¬‚uctuating force describing the thermal agi-

tation of the particle due to the interaction with the environment, the thermal noise,

or some other relevant source of noise. For a system interacting with a classical envi-

ronment assumed in thermal equilibrium at a temperature T , the ¬‚uctuating force is

a Gaussian stochastic process described by the correlation function for its Cartesian

components1

(±) (β)

= 2·kB T δ(t ’ t ) δ±β

ξt ξt (12.2)

relating friction and ¬‚uctuations according to the ¬‚uctuation“dissipation theorem,

as proper for linear response.

Being the dissipative dynamics for a system coupled to a heat bath, Langevin

dynamics is relevant for describing a vast range of phenomena, and of course not just

that of a particle as considered above. For example, randomly stirred ¬‚uids in which

case the relevant equation would be the Navier“Stokes equation with proper noise

term [117]. The ¬eld theoretic formulation of the following section runs identical for

all such cases. Also, the coordinate above need not literally be that of a particle,

but could for example describe the position of a vortex in a type-II superconductor,

as discussed in Section 12.2. However, we shall in the following keep referring to the

degree of freedom as that of a particle.

1 The quantum case and the classical limit are discussed in Appendix A.

12.1. Field theory of stochastic dynamics 451

12.1.2 Fluctuating linear oscillator

For a given realization of the ¬‚uctuating force, ξ t , there is a solution to the Langevin

equation, Eq. (12.1), specifying the realization of the corresponding motion of the

particle xt . In other words, xt is a functional of ξ t , xt = xt [ξt ], and vice versa

¯

ξt = ξt [xt ]. The properties of the ¬‚uctuating force is described by its probability

¯

distribution, Pξ [ξ t ], which is assumed to be Gaussian

’1

Pξ [ξ t ] = Dξ t e’ 2 dt1 dt2 ξt 1 Kt 1 , t 2 ξt 2

1

, (12.3)

’1

where Kt1 ,t2 is the inverse of the correlator of the stochastic force

Kt,t = ξ t ξ t (12.4)

and we have used dyadic notation to express the matrix structure of the force corre-

lations in Cartesian space. This structure is, however, irrelevant as the quantity is

diagonal.

Using the one-to-one map between the ¬‚uctuating force and the particle path,

xt ←’ ξ t , the probability of given paths, xt s, equals that of the corresponding

forces, ξ t s,

Px [xt ] Dxt = Pξ [ξt ] Dξ t . (12.5)

In general, this does not allow us to state proportionality between the two probability

distributions, since the volume change in the transformation from Dξ t to Dxt must

be taken into account, the change in measure described by the Jacobian. Only if the

Langevin equation, Eq. (12.1), is linear is this a trivial matter, restricting the force

in Eq. (12.1) to that of a harmonic oscillator, F(xt , t) = ’mω0 xt (and a possible

2

external space-independent force, F(t), which we suppress in the following), i.e. the

equation of motion is that of a harmonic oscillator in the presence of a ¬‚uctuating

force

’1

’DR x = ξ (12.6)

where we have introduced the retarded Green™s function for the damped harmonic

oscillator

’1

DRd. h. o. (t, t ) = ’(m‚t + ·‚t + mω0 ) δ(t ’ t )

2 2

(12.7)

and suppressed the time variable and used matrix multiplication notation in the time

variable.2 In the considered linear case the Jacobian is a constant and

Px [xt ] ∝ Pξ [ξ t ] = Pξ [m¨ t + · xt + mω0 xt ] ,

2

™ (12.8)

x

where the last equality is obtained by using the equation of motion, Eq. (12.6). Using

the fact that the ¬‚uctuations are Gaussian, gives for the probability distribution of

paths,

’1

e’ 2 dt1 dt2 ξt 1 [xt 1 ] Kt 1 , t 2 ξt 2 [xt 2 ]

1

Px [xt ] ∝ ¯ ¯

’1

e’ 2 dt1 dt2 (m¨ t 1 +· xt 1 +mω0 xt 1 )Kt 1 , t 2 (m¨ t 2 +· xt 2 +mω0 xt 2 )

x x

2 2

1

™ ™

= (12.9)

2 It could of course be considered a matrix in Cartesian coordinates, but since it would be diagonal

it is super¬‚uous.

452 12. Classical statistical dynamics

for the case of a harmonic oscillator coupled to a heat bath.

Completing the square in the following Gaussian path integral

∞˜∞ ˜ ∞˜

D˜ t e’ 2 xt Kt , t xt ’i dt xt ·(m¨ t +· xt +mω0 xt )

x˜ 2

P [xt ] = N ’1

1

˜˜ ˜ ˜ ˜˜ ˜˜ ™˜

dt ’∞dt ˜

(12.10)

x ’∞ ’∞

gives the previous expression, and the path integral representation for the probabil-

ity distribution of paths has been obtained. The proportionality factor is ¬xed by

normalization of the probability distribution.

We can also arrive at the expression for the probability distribution of paths in

the following way. For a given realization of the ¬‚uctuating force, ξ t , the probability

distribution for the particle path corresponds with certainty to the one ful¬lling the

equation of motion as expressed by the delta functional

P [xt ] = N ’1 δ[m¨ t + · xt + mω0 xt ’ ξ t ] ,

2

™ (12.11)

x

where N ’1 is the constant resulting from the Jacobian. Introducing the functional

integral representation of the delta functional we get

∞

D˜ e’i dt x·(m¨ +· x+mω0 x’ξ)

x 2

P [x] = N ’1 ˜ ™

. (12.12)

x ’∞

The average over the thermal noise, being Gaussian, can now be performed and we

obtain

∞ ˜∞

D˜ e’i dt ’∞dt x·(m¨ +· x+mω0 x’˜ i

x x ξξ x)

2

= N ’1 ˜ ™ ˜

Px [xt ] = P [x] . (12.13)

x ’∞

ξ

Realizing that the correlation function, Eq. (12.2), is the high-temperature classi-

cal limit of the inverse of the correlation function for a harmonic quantum oscillator

coupled to a heat bath (see Appendix A), i.e. the kinetic component of the real-time

matrix Green™s function, we introduce the notation

’1

’iDK (t, t ) = K ’1 (t, t ) = ξt ξt = 2·kB T δ(t ’ t ) (12.14)

where reference to the irrelevant Cartesian coordinates is left out, i.e. K ’1 now

denotes the scalar part in Eq. (12.2).

In addition the advanced inverse Green™s function is introduced

’1 ’1

DA (t, t ) = DR (t , t) (12.15)

and both functions are diagonal matrices in Cartesian space and will therefore be

treated as scalars. We can then rewrite for the path probability distribution (rein-

troducing an external force F(x, t))

D˜ eiS0 [˜ ,x]+i˜ ·F+ix·j ,

x x

Px [xt ] = (12.16)

x

where

1

x ’1 ’1

˜ ˜ ’1 ˜

S0 [˜ , x] = (˜ DR x + xDA x + xDK x) (12.17)

x

2

12.1. Field theory of stochastic dynamics 453

and we have absorbed the normalization factor in the path integral notation, it is

¬xed by the normalization of the probability distribution

Dxt Px [xt ] = 1 . (12.18)

We have in addition to the physical external force F(t) introduced a source, J(t),

and have the generating functional

D˜ eiS0 [˜ ,x]+i˜ ·F+ix·J

x x

Dx

Z[F, J] = (12.19)

x

with the normalization

Z[F, J = 0] = 1 . (12.20)

The source is introduced in order to generate the correlation functions of interest,

for example

∞

δZ[J]

Dx xt Px [xt ] = ’i =’ ¯ ¯ ¯

dt DR (t, t) F(t) ,

= (12.21)

xt ξ

δJ(t) ’∞

J=0

where the last equality follows from the equation of motion, and the retarded prop-

agator DR is thus the linear response function (for the considered linear oscillator,

the linear response is the exact response).

Because of the normalization condition, Eq. (12.20), all correlation functions of

˜

the auxiliary ¬eld x (generated by di¬erentiation with respect to the physical external

force F), vanish when the source J vanishes.

We now realize that the above theory is equivalent to the celebrated Martin“

Siggia“Rose formulation of classical statistical dynamics [118], here in its path inte-

gral formulation [119, 120, 121], albeit for the moment only for the case of a damped

harmonic oscillator.3 This restriction was of course self-in¬‚icted and the formalism

has numerous general applications, such as to critical dynamics for example studying

critical relaxation [122].

We note that whereas equilibrium quantum statistical physics is described by Eu-

clidean ¬eld theory (recall Section 1.1 and see Exercise A.1 on page 506 in Appendix

A), non-equilibrium classical stochastic phenomena are described by a ¬eld theory

formally equivalent to real-time quantum ¬eld theory.

We hasten to consider a nontrivial situation, viz. that of the presence of quenched

disorder. The corresponding ¬eld theory will be of the most complicated form, in

diagrammatic terms it will have vertices of arbitrarily high connectivity.

3 Inother words, the Martin“Siggia“Rose formalism is simply the classical limit of the real-time

technique for non-equilibrium states, where the doubling of the degrees of freedom necessary to

describe non-equilibrium situations is provided by the dynamics of the system. We note that the

presented ¬eld theoretic formulation of the Langevin dynamics is the classical limit of Schwinger™s

closed time path formulation of quantum statistical mechanics of a particle coupled linearly to, for

the considered type of damping, an Ohmic environment. Equivalently, it is the classical limit of

the Feynman“Vernon path integral formulation of a particle coupled linearly to a heat bath, as

discussed in Appendix A.