has been introduced, and we shall also use the notation introduced in Eq. (12.14).

4 Stochasticdi¬erential equations should be approached with care, since di¬erent discretizations

can lead to di¬erent types of calculus.

5 Often in Langevin dynamics the over-damped case is the relevant one, i.e. in the present case

corresponding to the absence of the mass term, m = 0. In such cases it can be convenient to throw

in a mass term at intermediate calculations as a regularizer. In Section 12.1.6 we show that for the

over-damped case the Jacobian leads in diagrammatic terms to the absence of tadpole diagrams.

12.1. Field theory of stochastic dynamics 455

The quenched disorder is assumed described by a Gaussian distributed stochastic

potential with zero mean, V (x) = 0, and thus characterized by its correlation

function

ν(x ’ x ) = V (x)V (x ) , (12.30)

where now the brackets denote averaging with respect to the quenched disorder. The

interaction part is then

∞ ∞

‚2

i

xβ .

=’ x±

SV [˜ , x] dt dt ˜t ν(x) ˜t (12.31)

x

‚x± ‚xβ

2 ’∞ ’∞ x=xt ’xt

The above model thus describes a classical object subject to a viscous medium and

a random potential. In the case of two spatial dimensions it could be a particle on a

rough surface experiencing Ohmic dissipation, a case relevant to tribology. However,

the theory is applicable to any system where quenched disorder is of importance, say

such as when studying critical dynamics of spin-glasses.

The generating functional for the theory is thus

Dx D˜ eiS[˜ ,x]+i˜ ·F+ix·J

x x

Z[F, J] = (12.32)

x

where the action is S = S0 +SV with S0 and SV given by Eq. (12.25) and Eq. (12.31),

respectively. The normalization

Z[F, J = 0] = 1 . (12.33)

allows us to avoid the replica trick for performing the average over the quenched

disorder [123].

The generating functional generates the correlation functions of the theory

δ n Z[J]

xt1 · · · xtn Dx xt1 · · · xtn n

= Px [xt ] = (’i) . (12.34)

ξ

δJ(t1 ) · · · δJ(tn )

J=0

The retarded full Green™s function, GR , is seen to be the linear response function

±±

to the physical force F± , i.e. to linear order in the external force we have

∞

dt GR (t, t ) F± (t )

x± (t) = (12.35)

±±

’∞

and GK is the correlation function, both matrices in Cartesian indices as indicated.

±±

12.1.4 Dynamical index notation

It is useful to introduce compact matrix notation by introducing the dynamical index

notation. We collect the path and auxiliary ¬eld into the vector ¬eld

˜ φ1

x

φ= = (12.36)

φ2

x

456 12. Classical statistical dynamics

as well as the forces

F f1

f≡ ≡ . (12.37)

J f2

This corresponds to introducing the real-time dynamical index notation we used to

describe the non-equilibrium states of a quantum ¬eld theory. Here they appear as

the Schwinger“Keldysh indices in the classical limit of quantum mechanics.6 In this

notation the quadratic part of the action becomes

1

φD’1 φ ,

S0 [φ] = (12.38)

2

where

’1 ’1 ’1

DK DR i ξt ξt DR

’1

D = = (12.39)

’1 ’1

DA 0 DA 0

is the free inverse matrix propagator

D’1 D = δ(t ’ t )1 (12.40)

and

DA (t, t )

0

D(t, t ) = . (12.41)

R

DK (t, t )

D (t, t )

Exercise 12.1. Show by Fourier transformation of Eq. (12.27) that

1

R

Dω = (12.42)

(ω + i0)(mω + i·)

and thereby that the solution of Eq. (12.27) is

1

Dtt = ’ θ(t ’ t ) 1 ’ e’·(t’t )/m .

R

(12.43)

·

The generator for the free theory is

√

det iD e’ 2 f Df ,

i

Dφ eiS0 [φ]+iφf =

Z0 [f ] = (12.44)

where the matrix D is speci¬ed in Eq. (12.41). The diagonal component, the kinetic

component, is given by the equation

DK = ’DR i ξξ DA (12.45)

and its Fourier transform is therefore

Dω = ’2i·kB T Dω Dω .

K RA

(12.46)

The free correlation function of the particle positions

δ 2 Z0 [f ]

(’i)2 K

= = iDtt + xt xt (12.47)

xt xt

δf2 (t) δf2 (t ) f2 =0

6 See also the Feynman“Vernon theory, discussed in Appendix A.

12.1. Field theory of stochastic dynamics 457

has connected and disconnected parts.

The generating functional in the presence of disorder becomes

Dφ eiS[φ]+iφf ,

Z[f ] = (12.48)

where S = S0 + SV , and the action due to the quenched disorder is

∞ ∞

‚2

i

φβ (t )

’ dt φ± (t)

SV [φ] = dt ν(x) (12.49)

1 1

‚x± ‚xβ

2 ’∞ ’∞ x=xt ’xt

and the normalization condition becomes

Z[f1 , f2 = 0] = 1 . (12.50)

The generator generates the correlation functions, for example the two-point

Green™s function

δ2Z

φt φt = ’ . (12.51)

δft δft f2 =0

The generator of connected Green™s functions, iW [f ] = ln Z[f ], for example generates

the average ¬eld

1 δZ[f ] δZ[f ]

= ’i = ’i

φt . (12.52)

Z[f ] δft δft

f2 =0 f2 =0

12.1.5 Quenched disorder and diagrammatics

Let us investigate the structure of the diagrammatic perturbation expansion resulting

from the quenched disorder, i.e. the vertices originating from the quenched disorder.

The perturbative expansion of the generating functional in terms of the disorder

correlator is

1

’1

Dφ eiφD φ+if φ

(iSV [φ])2 + . . . .

Z[f ] = 1 + iSV [φ] + (12.53)

2!

The vertices in a diagrammatic depiction of the perturbation expansion are deter-

mined by SV , Eq. (12.31) and can be expressed as

i dk

ν(k)k · xt eik(xt ’xt ) k · xt .

˜ ˜

SV [φ] = dt dt (12.54)

2

2 (2π)

˜

The vertices of the theory thus have one auxiliary ¬eld, x, attached and an arbitrary

number of ¬elds x attached, and are depicted as a circle with the time in question

marked inside and a dash-dotted line to describe the attachment of an impurity

correlator

˜

x t . (12.55)

···

x

x x

458 12. Classical statistical dynamics

As any vertex contains attachment for the impurity correlator, vertices occur in pairs

ν(k)

˜ ˜

x x

t t (12.56)

·

·· x

···

x

x x

x

˜

resulting in vertices of second order in the auxiliary ¬eld x but of arbitrary order

in position of the particle, x. The diagrammatic representation of the perturbation

expansion in terms of the disorder is thus speci¬ed by this basic vertex, and the

propagators of the theory are in this classical limit of the real-time technique, the

propagators DR , DA and DK . Diagrams representing terms in the perturbation

expansion of the generating functional consist of the vertices described above and

connected to one another or to sources by lines representing retarded, advanced and

kinetic Green™s functions. An example of a typical such vacuum diagram of the

theory, containing two impurity correlators, is displayed in Figure 12.1.

Figure 12.1 Example of a vacuum diagram. The solid line represents the correlation

function or kinetic component, GK , of the matrix Green™s function. The retarded

Green™s function, GR , is depicted as a wiggly line ending up in a straight line, and

vice versa for the advanced Green™s function GA . A dashed line attached to circles

represents the impurity correlator. The cross in the ¬gure represents the external

force F.

As an application of the above Langevin dynamics in a random potential, we shall

study the dynamics of a vortex lattice. But before we discuss the phenomenology

of vortex dynamics, we consider the relation of the theory with a mass term to the

over-damped case.

12.1. Field theory of stochastic dynamics 459

12.1.6 Over-damped dynamics and the Jacobian

We have noted in Section 12.1.3 that the presence of the mass terms can be used as a

regularizer leaving the Jacobian for the transformation between paths and stochastic

force an irrelevant constant. However, many situations of interest are concerned with

over-damped dynamics and we shall therefore here deal with that situation explicitly.

We show in this section that the neglect of the mass term in the equation of motion

gives a Jacobian, which in diagrammatic terms leads to the cancellation of the tadpole

diagrams.

In the over-damped case the inverse retarded Green™s function, Eq. (12.26), be-

comes

’1

DR (t, t ) = ’· ‚t δ(t ’ t ) (12.57)

corresponding to setting the mass of the particle equal to zero. The Jacobian, J, is

for the considered situation the determinant

δξ t

J = det (12.58)

δxt

which by use of the equation of motion can be rewritten

δ∇V (xt ) δ∇V (xt )

J = ’(DR )’1 + = · ‚t δ(t ’ t ) + (12.59)

tt

δxt δxt

or equivalently

‚ 2 V (xt )

det ·‚t δ(t ’ t )δ δ(t ’ t )

±β

J = +

‚x± ‚xβ

t t

det ·‚t δ(t ’ t )δ ±β

=

2

˜ ‚ V (xt ) ˜

’1

’1

— det δ(t ’ t )δ , t) ± β δ(t ’ t )

±β

+· ‚t (t , (12.60)

‚x ‚x

˜

t

where the inverse time di¬erential operator is

’1

‚t (t1 , t2 ) = θ(t1 ’ t2 ) . (12.61)

Using the trace-log formula, ln det M = Tr ln M , the Jacobian then becomes

det ·‚t δ(t ’ t )δ ±β

J =

2

˜) ‚ V (xt ) δ(t ’ t )

’1

· ’1 ‚t (t

— exp Tr ln δ(t ’ t )δ ˜

±β

+ ,t

‚x± ‚xβ

det ·‚t δ(t ’ t )δ ±β

=

∞ 2

1 ˜) ‚ V (xt ) δ(t ’ t ))n

’1 ’1

— exp ’ Tr(’· ‚t (t , t . (12.62)

n ‚x‚x

n=1

The Jacobian adds a term to the action, and the diagrams generated by the Jacobian

are seen to be exactly the tadpole diagrams generated by the original action except

460 12. Classical statistical dynamics

for an overall minus sign, and the Jacobian can thus be neglected if we simultane-

ously omit all tadpole diagrams. This is equivalent to choosing the step function in

Eq. (12.62) to be de¬ned according to the prescription

t¤0

0

θ(t) = (12.63)

1 t>1

since then the ¬rst term of the Taylor expansion of the logarithm will be

Tr(‚ ’1 (t ’ t )V (xt )δ(t ’ t )) = dt θ(0)V (xt ) = 0 . (12.64)

The higher-order terms in the Taylor expansion are similarly shown to be zero. The

result we obtain for the Jacobian for this particular choice of the step function is

therefore independent of the disorder potential V

J = det (·‚t (t ’ t ) δ(t ’ t )) = const . (12.65)

The derivation of the self-consistent equations can therefore be carried out in the

same way as for the case of a nonzero mass when we have chosen this particular

de¬nition of the step or Heaviside function. The only di¬erence is that the following

form of the free retarded propagator is used:

1

DR (t, t ) = ’ θ(t ’ t ) . (12.66)

·

The equations obtained by setting the mass equal to zero in the previous equations

are then exactly the same as the ones obtained for the over-damped case.

12.2 Magnetic properties of type-II superconductors

The advent of high-temperature superconductors has led to a renewed interest in

vortex dynamics since high-temperature superconductors have large values of the

Ginzburg“Landau parameter and the magnetic ¬eld versus temperature (B“T ) phase

diagram is dominated by the vortex phase.7 In this section we consider the phe-

nomenology of type-II superconductors, in particular the forces on vortices and their

dynamics. Since vortex dynamics in the ¬‚ux ¬‚ow regime is Langevin dynamics with

quenched disorder, they provide a realization of the model discussed in the previous

sections.

12.2.1 Abrikosov vortex state

The essential feature of the magnetic properties of a type-II superconductors is the

existence of the Abrikosov ¬‚ux-line phase [124]. At low magnetic ¬eld strengths,

7 The Ginzburg“Landau parameter, κ = »/ξ, is the ratio between the penetration depth and

the superconducting coherence length. The magnetic ¬eld penetration depth was ¬rst introduced

in the phenomenological London equations, μ0 js = E/»2 and μ0 ∇ — js = ’B/»2 , the latter the

™

important relation between the magnetic ¬eld and a supercurrent describing the Meissner e¬ect of

¬‚ux expulsion as obtained employing the Maxwell equation to get B + »2 ∇ — ∇ — B = 0.

12.2. Magnetic properties of type-II superconductors 461

just as for a type-I superconductor, a type-II superconductor exhibits the Meiss-

ner e¬ect, magnetic ¬‚ux expulsion. A counter supercurrent on a sample™s surface

makes a superconductor exhibit perfect diamagnetism, giving it a magnetic moment

(which can provide magnetic levitation). Above a critical magnetic ¬eld, Hc1 , the

superconducting properties of a type-II superconductor weakens, say for example its

magnetic moment on increase of magnetic ¬eld, and the superconductor has entered

the Shubnikov phase (1937). In this state, magnetic ¬‚ux will penetrate a type-II

superconductor in the form of magnetic ¬‚ux lines, each carrying a magnetic ¬‚ux

quantum, φ0 = h/(2e), with associated vortices of supercurrents. This phase is the

Abrikosov lattice ¬‚ux-line phase, and persists up to an upper critical ¬eld, Hc2 , where

superconductivity breaks down, and the superconductor enters the normal state. The

supercurrents circling the vortex cores, where the order parameter is depressed and

vanishing at the center, screen the magnetic ¬eld throughout the bulk of the material.

The coupling of magnetic ¬eld and current results in a repulsive interaction between

vortices which for an isotropic superconductor leads to a stable lattice for the regular

triangular array, the Abrikosov ¬‚ux lattice.

The energetics of two vortices are governed by the magnetic ¬eld energy and the