12.1.3 Quenched disorder

We now return to the in general nonlinear classical stochastic problem speci¬ed by

the Langevin equation

m¨ t + · xt = ’∇V (xt ) + Ft + ξ t ,

™ (12.22)

x

where V eventually will be taken to describe quenched disorder. Owing to the pres-

ence of the nonlinear term V (xt ), the argument for the Jacobian being a constant is

less trivial. However, by using forward discretization,4 one obtains the result that,

owing to the presence of a ¬nite mass term, the Jacobian can be chosen as a constant

and the analysis of the previous section can be taken over giving5

∞

D˜ e’i dt x·(m¨ +· x+∇V

x ’F’ξ)

P [x] = N ’1 ˜ ™

. (12.23)

x ’∞

The averages over the thermal noise and the disorder can now be performed and

we obtain the following expression for the path probability density

= N ’1 D˜ eiS[˜ ,x]+i˜ ·F ,

x x

Px [xt ] = P [x] (12.24)

x

where the action, S = S0 + SV , is a sum of a part owing to the quenched disorder

and the quadratic part

1

x ’1 ’1

(˜ DR x + xDA x + xi ξξ x) ,

˜˜ ˜

S0 [˜ , x] = (12.25)

x

2

where we have introduced the inverse propagator for the problem in the absence of

the disorder

’1

DR (t, t ) = ’(m‚t + ·‚t )δ(t ’ t ) ,

2

(12.26)

i.e. the retarded free propagator satis¬es

’(m‚t + ·‚t )Dtt = δ(t ’ t )

2 R

(12.27)

with the boundary condition

R

Dtt = 0, t < t . (12.28)

The corresponding inverse advanced Green™s function

’1 ’1

DA (t, t ) = DR (t , t) (12.29)

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

kinetic energy of the supercurrent, and as governed by the London equation give a

repulsive force, assuming the same sign of vorticity, on each vortex of strength

F = φ0 js , (12.67)

where js is the supercurrent density associated with one vortex at the position of the

other vortex. In the presence of a transport current, j, through the superconductor

the vortices will therefore per unit length be subject to a Lorentz force of magnitude

FL = φ0 j , (12.68)

where j is the transport current density, and the direction of the force is speci¬ed by

j — B. Even a small transport current will give rise to motion of the vortex lattice

perpendicular to the current in a pure type-II superconductor in the Abrikosov“

Shubnikov phase. This motion causes dissipative processes due to the normal currents

in the core, which phenomenologically can be described, at low velocities, by a friction

force (per unit length) opposing the motion of a vortex with velocity v

Ff = ’· v . (12.69)

The friction coe¬cient is given by8

φ2

0

·= (12.70)

2πa2 ρn

where ρn is the normal resistance of the metal, and a is the size of the normal core

(approximately equal to the superconducting coherence length).

8 For a phenomenological justi¬cation of the friction term we refer to the Bardeen“Stephen model

[125], or analysis based on the time dependent Ginzburg“Landau equation [126, 127, 128]. As

proclaimed, we describe only the phenomenology of the relevant forces, no derivation based on the

microscopic theory will be done, instead we refer the reader in general to reference [129].

462 12. Classical statistical dynamics

In addition, there can also be a Hall force

FH = ± v — n

ˆ (12.71)

acting on the vortex [130].

In a real superconductor there are always imperfections, referred to as impurities,

causing the vortices to have energetically preferred positions. The pinning force is

caused by defects such as twinning or grain boundaries, or dislocation lines. These

can pin a vortex, which would otherwise move in the presence of a transport current.9

At low enough temperatures and below a critical value of the transport supercurrent,

the vortex lattice is pinned and the current carrying state dissipationless. At larger

currents or higher temperatures, the motion of the vortices occur by thermal excita-

tion of (bundles of) vortices hopping between pinning centers, the state of ¬‚ux creep.

In the regime where the pinning force, Fp , is weak compared with the driving force,

the motion of the vortex lattice is steady, characterized by a velocity, v, the super-

conductor is in the dreaded ¬‚ux ¬‚ow regime. The moving magnetic ¬eld structure

associated with the vortices, leads by induction to the presence of an electric ¬eld,

E = ’v — B. The electric ¬eld has, as a result of the friction force, a component

parallel to the current, and the work, E·j, performed by the electric ¬eld is dissipated

by the friction force. The resistance is of the order of the normal state resistance,

and the dissipation will drive the superconductor to its normal state.

There is also interaction between the vortices as discussed previously. We shall be

interested in the case where the deformation of the Abrikosov lattice is weak, leading

to a harmonic interaction between the vortices described by continuum elasticity

theory.

12.2.2 Vortex lattice dynamics

We now turn to the case of interest, the dynamics of the Abrikosov vortex lattice in

the ¬‚ux ¬‚ow regime. The formalism is identical to the previously considered case of

one particle, except the occurrence of the whole lattice of vortices with the additional

feature of their interaction.

We consider a two-dimensional (2D) description of the vortices, since we have

in mind a thin superconducting ¬lm, or a three-dimensional (3D) layered supercon-

ductor with uncorrelated disorder between the layers. We shall be interested in the

in¬‚uence of quenched disorder on the vortex dynamics in the ¬‚ux ¬‚ow regime. The

description of the vortex dynamics is, according to the previous section, described

by the Langevin equation of the form

¦RR uR t = F + ±uRt — z ’ ∇V (R + uRt ) + ξRt , (12.72)

™ ™ ˆ

m¨ Rt + · uRt +

u

R

ˆ

where uRt is the two-dimensional displacement, normal to z, at time t of the vortex

(or bundle of vortices), which initially has equilibrium position R, · is the friction

9 The existence of the Abrikosov vortex state and the pinning of vortices is, from the point

of applications using superconducting coils as magnets, the most important property. They can

produce magnetic ¬elds in the excess of tens of Tesla. Usual copper coils can not produce the stable

¬eld produced by the supercurrent, not to mention its mess of water-cooling.

12.2. Magnetic properties of type-II superconductors 463

coe¬cient, and m is a possible mass of the vortex (both per unit length). The mass

of a vortex is small and will eventually be set to zero. The interaction between the

vortices is treated in the harmonic approximation and described by the dynamic

matrix ¦RR whose relevant elasticity moduli is discussed in Section 12.6. The

force (per unit length) on the right-hand side of Eq. (12.72) consists of the Lorentz

force, F = φ0 j — ˆ, due to the transport current density j, which we eventually

z

assume constant, and the second term on the right-hand side is a possible Hall force,

characterized by the parameter ±, and V is the pinning potential due to the quenched

disorder. The pinning is 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 ) . (12.73)

The thermal noise, ξ, is the white noise stochastic process with zero mean and

correlation function speci¬ed according to the ¬‚uctuation“dissipation theorem (where

the brackets now denote averaging with respect to the thermal noise)

= 2·T δ(t ’ t ) δRR δ±±

± ±

ξRt ξR t (12.74)

and, since the forces are per unit length, the temperature T has the dimension of

energy per unit length.

Upon averaging with respect to the thermal noise and the quenched disorder, the

average restoring force of the lattice vanishes

’ ¦RR =0 (12.75)

uR t

R

since the average displacement is the same for all vortices, and a rigid translation

of the vortex lattice does not change its elastic energy, leaving the dynamic matrix

with the symmetry property

¦RR = 0 . (12.76)

R

™

Owing to dissipation, the vortex lattice reaches a steady state velocity v = uRt ,

corresponding to the average force on any vortex vanishes

F + Ff + FH + Fp = 0 , (12.77)

i.e. there will be a balance between the Lorentz force, F, the average friction force,

Ff = ’·v, the average Hall force, FH = ±v — ˆ, and the pinning force

z

Fp = ’ ∇V (R + uRt ) . (12.78)

The pinning force is determined by the relative positions of the vortices with respect

to the pinning centers and is invariant with respect to the change of the sign of ±.

The average velocity, v, is the only vector characterizing the vortex motion which

is invariant with respect to the change of the sign of ±, and the pinning force is

therefore antiparallel to the velocity. Thus, the pinning yields a renormalization of

the friction coe¬cient

’·v + Fp = ’·e¬ v . (12.79)

464 12. Classical statistical dynamics

The e¬ective friction coe¬cient depends on the average velocity of the lattice, the

disorder, the temperature, the interaction between the vortices, the Hall force, and a

possible mass of the vortex. In the absence of disorder, the e¬ective friction coe¬cient

reduces to the bare friction coe¬cient ·.

The pinning problem has no simple analytical solution. One way of attacking the

problem is a perturbation calculation in powers of the disorder potential. A second-

order perturbation calculation works well for high velocities, as we show in Section

12.5.1.10 At low enough velocities the higher-order contributions in the disorder

become important. We shall employ the self-consistent e¬ective action method of

Cornwall et al. [53] to sum up an in¬nite subset of the contributions in V . Such self-

consistent methods are uncontrolled but many times they yield surprisingly good

results. In order to apply the ¬eld theoretic methods of Cornwall et al. we need

to reformulate the stochastic problem in terms of a generating functional, which is

achieved by the ¬eld theoretical formulation of classical statistical dynamics.

In the following the in¬‚uence of pinning on vortex dynamics in type-II supercon-

ductors is investigated. The vortex dynamics is described by the Langevin equation,

and we shall employ a ¬eld-theoretic formulation of the pinning problem which al-

lows the average over the quenched disorder to be performed exactly. By using the

diagrammatic functional method for this classical statistical dynamic ¬eld theory, we

can, from the e¬ective action discussed in the previous chapter, obtain an expression

for the pinning force in terms of the Green™s function describing the motion of the

vortices.

12.3 Field theory of pinning

The average vortex motion is conveniently described by reformulating the stochastic

problem in terms of the ¬eld theory of classical statistical dynamics introduced in

Section 12.1. The probability functional for a realization {uRt }R of the motion of

the vortex lattice is expressed as a functional integral over a set of auxiliary variables

{˜ Rt }R , and we are led to consider the generating functional11

u

u

Z[F, J] = DuRt D˜ Rt J eiS[u,˜ ] , (12.80)

u

R R

where in the action

˜ ’1

u(DR u + F ’ ∇V + ξ) + Ju

˜

S[u, u] = (12.81)

the inverse free retarded Green™s function is speci¬ed by

’1

’DR uRt = m¨ Rt + · uRt + ¦RR uR t + ±ˆ — uRt ,

™ z™ (12.82)

u

R

10 Vortex pinning in the ¬‚ux ¬‚ow regime was originally considered treating the disorder in lowest

order perturbation theory [131, 132], and later by applying ¬eld theoretical methods [133, 134].

11 In the following we essentially follow reference [134].

12.3. Field theory of pinning 465

i.e.

’1

DR (R, t; R , t ) = ’¦RR δ(t ’ t ) ’ (m‚t + ·‚t )1 ’ i±σy ‚t δR,R δ(t ’ t ) ,

2

(12.83)

where matrix notation is used for its Cartesian components, i.e. 1 and σ y denote

the unit matrix (occasionally suppressed for convenience) and the Pauli matrix in

Cartesian space, respectively. The Fourier transform of the inverse free retarded

Green™s function is therefore the two by two matrix in Cartesian space given by the

expression

’i±ω

mω 2 + i·ω

’1

’ ¦q .

DR (q, ω) = (12.84)

mω 2 + i·ω

i±ω

In Eq. (12.81) we have introduced matrix notation in order to suppress the integra-

tions over time and summations over vortex positions and Cartesian indices. Thus,

˜ ’1

for example, uDR u denotes the expression

∞ ∞

˜ ’1 ’1±±

uDR u = dt dt u± (R, t) DR

˜ (R, t; R , t ) u± (R , t ) . (12.85)

RR ’∞ ’∞

±,± =x,y

The Jacobian, J = |δξRt /δ uR t |, guaranteeing the normalization of the generating

˜

functional

Z[F, J = 0] = 1 (12.86)

is given by

⎡ ¤

∞

‚ 2 V (R + uRt ) ¦

J ∝ exp ⎣’ R±±

dtDRt;Rt , (12.87)

‚x± ‚x±

R±± ’∞

where the proportionality constant is the determinant of the inverse free retarded

’1

Green™s function, |(DR )±± t |. As discussed in Section 12.1.6, in the case of a

Rt,R

nonzero mass, m = 0, the Jacobian is an irrelevant constant; and in the case of

zero mass, dropping the Jacobian from the integrand is equivalent to de¬ning the

R

retarded free Green™s function to vanish at equal times, Dtt = 0, which in turn leads

to the full retarded Green™s function satisfying the same initial condition. In terms of

diagrams, the contribution from the Jacobian exactly cancels the tadpole diagrams

as discussed in Section 12.1.6.

The average with respect to both the thermal noise and the disorder is imme-

diately performed, and we obtain the averaged functional, dropping the irrelevant

Jacobian,

Z Dφ eiS[φ]+if φ .

Z[f ] = = (12.88)

We have employed the compact notation for the ¬elds

φRt = (˜ Rt , uRt ) = (φ1 (R, t), φ2 (R, t)) (12.89)

u

466 12. Classical statistical dynamics

and for the external force and an introduced source, J(R, t),

f (R, t) = (F(R, t), J(R, t)) . (12.90)

The action obtained upon averaging, which we also denote by S, consists of two

terms

S[φ] = S0 [φ] + SV [φ] . (12.91)

The ¬rst term is quadratic in the ¬eld

1

φD’1 φ ,

S0 [φ] = (12.92)

2

where the matrix notation now in addition includes the dynamical indices, i.e. φD’1 φ

denotes the expression

∞ ∞

’1

φD’1 φ = i dt dt φ± (R, t) Dij ±± (R, t; R , t ) φ± (R , t ) . (12.93)

i j

RR ’∞ ’∞

±± ij

The inverse free matrix Green™s function in dynamical index space

’1

’1 ’1 2i·T δ(t ’ t ) δ±± δRR DR (R, t; R , t )

D11 D12

’1

D = = (12.94)

’1 ’1 ’1

D21 D22 DA (R, t; R , t ) 0

is a symmetric matrix in all indices and variables, since the inverse free advanced

Green™s function is obtained by interchanging Cartesian indices as well as position

and time variables

’1± ’1±±

DA ± (R , t ; R, t) = DR (R, t; R , t ) . (12.95)

The interaction term originating from the disorder is

∞∞

ν(uRt ’ uR t ) ±

2

i ‚

’ u±

SV [φ] = dt dt ˜Rt uR t .

˜ (12.96)

‚u± ‚u±

2 Rt Rt

’∞ ’∞

RR

±±

The source term introduced in Eq. (12.80)

∞

dt J(R, t) · u(R, t) ,

Ju = (12.97)

’∞

R

where the source, J(R, t), couples to the vortex positions, u(R, t), is added to the

action in order to generate the vortex correlation functions. For example, we have

for the average position

δZ

uRt = ’i (12.98)

δJRt J=0

12.3. Field theory of pinning 467

and the two-point unconnected Green™s function

δ2 Z

=’ . (12.99)

uRt uR t

δJRt δJR t J=0

Here and in the following we use dyadic notation, i.e. uRt uR t is the Cartesian

matrix with the components u± (R, t) u± (R , t ).

12.3.1 E¬ective action

In order to obtain self-consistent equations involving the two-point Green™s function

in a two-particle irreducible fashion, we add a two-particle source term K to the

action in the generating functional (recall Section 10.5.1)

i

Dφ exp iS[φ] + if φ + φKφ .

Z[f, K] = (12.100)

2

The generator of connected Green™s functions

iW [f, K] = ln Z[f, K] (12.101)

has accordingly derivatives

δW ±

= φi (R, t) (12.102)

δfi± (R, t)

and

δW 1± i

±

= φi (R, t) φi (R , t ) + G±± (R, t; R , t ) , (12.103)

2 ii

±± 2

δKii (R, t; R , t )

where φ is the average ¬eld, with respect to the action S[φ] + f φ + φKφ/2,

i

±

Dφ φ± (R, t) exp iS[φ] + if φ + φKφ

φi (R, t) = (12.104)

i

2

and G is the full connected two-point matrix Green™s function

⎛ ⎞

δ˜± δ˜± t δ˜± δu± t

uRt uR uRt R

δ2W

= ’i ⎝ ⎠,

Gij = ’ (12.105)

δfi δfj δu± δ˜± δu± δu±

uRt Rt Rt Rt

where

δuRt = uRt ’ uRt δ uRt = uRt ’ uRt .

˜ ˜ ˜

, (12.106)

In the physical problem of interest, the sources K and J vanish, K = 0 and

J = 0, and the full matrix Green™s function has, owing to the normalization of the

generating functional

Z[F, J = 0, K = 0] = 1 , (12.107)

468 12. Classical statistical dynamics

the structure in the dynamical index space

⎛ ⎞

u± u± t

0 ˜Rt R

Gij = ’i ⎝ ⎠

u± u± t δu± δu± t

Rt ˜R Rt R

GA (R, t; R , t )

0 ±±

= , (12.108)

GR (R, t; R , t ) GK (R, t; R , t )

±± ±±

where we observe that the connected and unconnected retarded (or advanced) Green™s

functions are equal. Similarly, in the absence of sources the expectation value of the

auxiliary ¬eld vanishes, and the average ¬eld is therefore given by

¯ ˜

φRt = ( uRt , uRt ) = (0, vt) , (12.109)

where v is the average velocity of the vortex lattice.

The retarded Green™s function GR yields the linear response to the force F± ,

±±

i.e. to linear order in the external force we have

∞

dt GR (R, t; R , t ) F± (R , t ) ,

u± (R, t) = (12.110)

±±

R ’∞

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

±±

The matrix Green™s function in dynamical index space, Eq. (12.108), has only two

independent components, since the advanced Green™s function is given by

GA (R, t; R , t ) = GR ± (R , t ; R, t) . (12.111)

±± ±

Pursuing an equation for the pinning force, we introduce the e¬ective action, “,

the generator of two-particle irreducible vertex functions, i.e. the double Legendre

transform of the generator of connected Green™s functions, W (recall Section 10.5.1),

1 i

“[φ, G] = W [f, K] ’ f φ ’ φKφ ’ TrGK , (12.112)

2 2

where Tr denotes the trace over all variables and indices, i.e. TrGK denotes the

expression

∞∞

dt dt G±± (R, t; R , t ) Ki i ± (R , t ; R, t) .

±

TrGK = (12.113)

ii

’∞ ’∞

R,R

±,± =x,y

i,i =1,2

The e¬ective action satis¬es the equations

δ“

= ’f ’ Kφ (12.114)

δφ

12.4. Self-consistent theory of vortex dynamics 469

and

δ“ i

= ’ K. (12.115)

δG 2

The e¬ective action was shown in Section 10.5.1 to have the form

i i i

’1

= S[φ] + TrDS G ’ Tr ln D’1 G ’ Tr1

¯ ¯

“[φ, G]

2 2 2

¯

’ i ln eiSint [φ,ψ] 2PI

, (12.116)

G

’1

where the quantity DS is the second derivative of the action at the average ¬eld

¯

δ 2 S[φ]

’1

DS [φ](t, t ) = (12.117)

δφt δφt

¯

and Sint [φ, ψ] is the part of the action S[φ + ψ] that is higher than second order in

ψ in an expansion around the average ¬eld. The superscript “2PI” on the last term

indicates that only the two-particle irreducible vacuum diagrams should be included

in the interaction part of the e¬ective action, the last term in Eq. (12.116), and the

subscript that propagator lines represent G, i.e. the brackets with subscript G denote

the average

’1

= (det iG)’1/2

¯ ¯

i

Dψ e 2 ψG

eiSint [φ,ψ] ψ

eiSint [φ,ψ] . (12.118)

G

The ¬rst dynamical index component of Eq. (12.114) together with the equa-

tion for the average motion Eq. (12.77) provide an expression for the pinning force,

Eq. (12.78), in term of the dynamical matrix propagator of the theory. The general

expression is still intractable, and in the next section we shall introduce the main

approximation.

12.4 Self-consistent theory of vortex dynamics

Because of the disorder, the equation of motion describing the vortex dynamics has no

simple analytical solution. The employed ¬eld theoretical formulation of the pinning

problem will therefore be used in combination with a self-consistent approximation

for the e¬ective action for studying vortex motion in type-II superconductors. Since

we have constructed the two-particle irreducible e¬ective action, we expect that its

lowest-order approximation contains the main in¬‚uence of the quenched disorder

on the vortex dynamics. The validity of the self-consistent theory is ascertained by

comparing with numerical simulations of the Langevin equation. The e¬ective action

method will be used to study the dynamics of single vortices and vortex lattices,

and yields results for the pinning force, ¬‚uctuations in position and velocity, etc.

The dependence of the pinning force on vortex velocity, temperature and disorder

strength is calculated for independent vortices as well as for a vortex lattice, and both

analytical and numerical results for the pinning of vortices in the ¬‚ux ¬‚ow regime

are obtained. Finally, the in¬‚uence of pinning on the dynamic melting of a vortex

lattice is studied in Section 12.7.

470 12. Classical statistical dynamics

12.4.1 Hartree approximation

In order to obtain a closed expression for the self-energy in terms of the two-point

Green™s function, we expand the exponential and keep only the lowest-order term

¯ ¯ ¯

’i ln eiSint [φ,ψ] ’i ln 1 + iSint [φ, ψ]

2PI 2PI

Sint [φ, ψ] , (12.119)

G

G G

i.e. we consider the Hartree approximation, which in diagrammatic terms corresponds

to neglecting diagrams where di¬erent impurity correlators are connected by Green™s

functions.

Figure 12.2 Typical vacuum diagram not included in the Hartree approximation

for the e¬ective action. 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 . The curly line ending up on the dot represents the ¬rst kinetic

component of the average ¬eld. A dashed line attached to circles represents the

impurity correlator and the additional dependence on the second component of the

average ¬eld as explicitly speci¬ed in Eq. (12.120).

A typical vacuum diagram not included in the Hartree approximation for the

e¬ective action is shown in Figure 12.2, and represents the expression

2 2

i 1 dk1 dk2 ¯

k2 · φ1 (R2 , t2 )

(2π)2 (2π)2

2 4!

—(k2 GR (R2 , t2 ; R1 , t1 )k1 )(k1 GR (R1 , t1 ; R1 , t1 )k1 )

—(k1 GR (R1 , t1 ; R2 , t2 )k2 )(k2 GK (R2 , t2 ; R2 , t2 )k2 )

—ν(k1 )eik1 ·(R1 ’R1 +v(t1 ’t1 )) ν(k2 )eik2 ·(R2 ’R2 +v(t2 ’t2 )) , (12.120)

where integrations over time and summations over vortex positions are implied, and

we have introduced the notation

kGR (R, t; R , t )k = k± GR (R, t; R , t ) k± (12.121)

±±

±±

12.4. Self-consistent theory of vortex dynamics 471

for Cartesian scalars.

In the Hartree approximation, Eq. (12.119), we drop the superscript “2PI” since

¯

the action Sint [φ, ψ] only generates two-particle-irreducible vacuum diagrams, due to

the appearance of only one impurity correlator. The Hartree approximation can be

expressed as a Gaussian ¬‚uctuation corrected saddle-point approximation [135].

The e¬ective action can in the Hartree approximation be rewritten on the form

i i i

“[φ, G] = S0 [φ] ’ Tr ln D’1 G + TrD’1 G ’ Tr1 + SV [φ + ψ]

¯ ¯ ¯ (12.122)

G

2 2 2

since

∞ ∞ ¯

δ 2 SV [φ]

i

¯ ¯ ¯

G ’ SV [φ] ’

Sint [φ, ψ] = SV [φ + ψ] Tr dt dt ¯ ¯ Gt t , (12.123)

G

2 ’∞ ’∞ δ φt δ φt

where the trace in the time variable has been written explicitly for clarity.

In the physical situation of interest the two-particle source, K, vanishes, and since

“ is two-particle-irreducible, Eq. (12.115) therefore becomes the Dyson equation

G’1 = D’1 ’ Σ , (12.124)

where the self-energy in the Hartree approximation is the matrix in dynamical index

space

¯

ΣK ΣR δ SV [φ + ψ] G

Σij = = 2i . (12.125)

ΣA 0 δGij K=0, J=0

The Dyson equation, Eq. (12.124), and the self-energy expression, Eq. (12.125), and

the equation relating the e¬ective action to the external force, Eq. (12.114), constitute

a set of self-consistent equations for the Green™s functions, the self-energies, and the

average ¬eld, in this non-equilibrium theory the latter speci¬es the velocity of the

vortex lattice.

The matrix self-energy in dynamical index space has only two independent com-

ponents since

ΣA (R, t; R , t ) = ΣR ± (R , t ; R, t) , (12.126)

±± ±

a simple consequence of Eq. (12.111) and the Dyson equation. From Eq. (12.125) we

obtain for a vortex lattice having a unit cell of area a2 and consisting of N vortices,

the self-energy components (each a matrix in Cartesian space)

i

ν(k) kk e’•(R,t;R ,t ;k;v)

ΣK (R, t; R , t ) = ’ ˜

(12.127)

N a2

k

and

∞

˜˜

ΣR (R, t; R , t ) = σ R (R, t; R , t ) ’ δRR δ(t ’ t ) ˜

dt σ R (R, t; R, t) ,

˜

R ’∞

(12.128)

472 12. Classical statistical dynamics

where

1

ν(k) kk (kGR (R, t; R , t )k) = e’•(R,t;R ,t ;k;v) .

σ R (R, t; R , t ) = ˜

N a2

k

(12.129)

We use dyadic notation, i.e. kk denotes the matrix with the Cartesian components

k± k± . The in¬‚uence of thermal and disorder-induced ¬‚uctuations are described by

the ¬‚uctuation or damping exponent

•k (R, t; R , t ) = ik GK(R, t; R, t) ’ GK(R, t; R , t ) k (12.130)

contained in

•(R, t; R , t ; k; v) = ’ik · (R ’ R + v(t ’ t )) + •k (R, t; R , t ) . (12.131)

˜

The pinning force on a vortex, Eq. (12.78), is determined by the averaged equation of

motion, Eq. (12.77), and the ¬rst dynamical index component of Eq. (12.114), which

in the Hartree approximation yields

∞

δ SV [φ + ψ]

’1±± G

’ ±

dt DR (R, t; R , t ) v± t = FR + (12.132)

±

δφ1 (R, t)

’∞

R ± φRt =(0,vt)

resulting in the expression for the pinning force

∞

dk

k ν(k)(kGR t k)e’•(R,t;R ,t ;k;v) .

˜

Fp = i dt (12.133)

RtR

2

(2π)

R ’∞

The self-consistent theory in the Hartree approximation is still intractable to

analytical treatment, except in the limiting cases considered in the following, but it

is manageable numerically.12 In the following we shall study numerically the vortex

dynamics in the Hartree approximation. The results obtained from the self-consistent

theory will then be compared with analytical results obtained in perturbation theory,

and with simulations of the vortex dynamics.

12.5 Single vortex

In order to study the essential features of the model and the self-consistent method,

we ¬rst consider the case of a single vortex, since this example will allow the important

test of comparing the results of the self-consistent theory with simulations. The case

of non-interacting vortices is appropriate for low magnetic ¬elds, where the vortices

are so widely separated that the interaction between them can be neglected. The

dynamics of a single vortex is described by the Langevin equation

m¨ t + · xt = ’∇V (xt ) + Ft + ξ t ,

™ (12.134)

x

12 In the rest of this chapter we follow reference [134].

12.5. Single vortex 473

where xt is the vortex position at time t. We defer the discussion of the Hall force

to Section 12.5.5.

When presenting analytical and numerical results obtained from the self-consistent

theory, we shall always choose the vortex mass (per unit length) to be small, in fact

3√

· 2 rp / ν0 , that the case of zero mass only deviates slightly from the

so small, m

presented results, i.e. at most a few percent.

In the analytical and numerical calculations, the correlator of the pinning poten-

tial shall be taken as the Gaussian function with range rp and strength ν0

ν0 ’(x’x )2 /2rp

ν(k) = ν0 e’rp k .

2 22

ν(x ’ x ) = e , (12.135)

2

2πrp

12.5.1 Perturbation theory

At high velocities, the pinning force can be obtained from lowest-order perturbation

theory in the disorder, since the pinning force then is small compared with the friction

force, and makes, according to Eq. (12.77), only a small contribution to the total force

on the vortex. We ¬rst consider the case of zero temperature, where we obtain the

following set of equations by collecting terms of equal powers in the pinning potential

∞

’1 (0)

’ dt DR (t, t ) xt = Ft (12.136)

’∞

and

∞

’1 (1) (0)

’ = ’∇V (xt )

dt DR (t, t ) xt (12.137)

’∞

and

∞

’1 (2) (1) (0)

’ = ’∇ xt · ∇V (xt ) .

dt DR (t, t ) xt (12.138)

’∞

Assuming that the external force is independent of time, the average vortex ve-

locity will be constant in time, and in the absence of disorder the average vortex

position is

Ft

(0)

= vt = , (12.139)

xt

·

i.e. the friction force balances the external force, ·v = F. The ¬rst-order contribution

to the vortex position vanishes upon averaging with respect to the pinning potential,

and the second-order contribution to the average vortex velocity becomes, according

to Eqs. (12.136) to (12.138),

∞

i dk

k k 2 ν0 e’k rp +ik·v(t’t )

22

(2)

’ R

™ = dt Dtt

xt

(2π)2

· ’∞

∞ 2 2

ν0 vt vt ’ vt

’

R

= dt Dt0 e . (12.140)

2r p

5·

4πrp 0 rp 2rp

474 12. Classical statistical dynamics

The second-order contribution is immediately calculated, and for example for the

3√

· 2 rp / ν0 , we obtain

case of a vanishing mass, m

ν0

(2)

=’

™ v. (12.141)

xt

4πrp · 2 v 2

4

The pinning force is then, according to Eq. (12.77), to lowest order in the disorder

strength, ν0 , given by

ν0

Fp = ’ v, (12.142)

4πrp ·v 2

4

i.e. the magnitude of the pinning force is inversely proportional to the magnitude

of the velocity. The perturbation result is therefore valid for large velocities, v

√ √

2 2

ν0 /·rp , i.e. when the friction force is much larger than the average force, ν0 /rp ,

owing to the disorder.

12.5.2 Self-consistent theory

The self-energy equations for a single vortex reduces in the Hartree approximation

to

⎡ ¤

∞

dk ⎣ R

σk (t, t ) ’ δ(t ’ t ) dt σk (t, t)¦

¯R ¯

ΣR (t, t ) = (12.143)

2

(2π)

’∞

and

σk (t, t ) = ν(k) kk (kGR (t, t )k) eik·v(t’t ) ’ •k (t,t )

R

(12.144)

and

dk

ν(k) kk eik·v(t’t ) ’ •k (t,t )

ΣK (t, t ) = ’i (12.145)

2

(2π)

with the ¬‚uctuation exponent

•k (t, t ) = ik GK (t, t) ’ GK (t, t ) k . (12.146)

Writing out the components of the matrix Dyson equation in the dynamical indices,

Eq. (12.124), we obtain the Cartesian matrix Green™s functions

GK (ω) = GR (ω) ΣK (ω) ’ 2i·T 1 GA (ω) (12.147)

and

1 ’ vv

ˆˆ ˆˆ

vv

GR (ω) = + , (12.148)

mω 2 + i·ω ’ ΣR (ω) mω 2 + i·ω ’ ΣR (ω)

⊥

where the subscripts and ⊥ denote longitudinal and transverse components of the

retarded self-energy with respect to the direction of the velocity

ΣR (ω) = v± ΣR (ω) v±

ˆ ˆ (12.149)

±±

±,±

12.5. Single vortex 475

and

ΣR (ω) (δ±± ’ v± v± ) .

ΣR (ω) = ˆˆ (12.150)

⊥ ±±

±,±

The advanced Green™s function is obtained from the retarded by complex conjugation

and interchange of Cartesian indices

= [GR ± (ω)]— .

GA (ω) (12.151)

±± ±

The expression for the pinning force, Eq. (12.133), reduces for a single vortex to

∞

dk

k ν(k) (k GR k) eik·v(t’t ) ’ •k (t,t ) .

Fp = i dt (12.152)

tt

2

(2π)

’∞

The previous discussion of the high-velocity regime, where lowest-order pertur-

bation theory in the disorder is valid, can be generalized to nonzero temperature.

√ 2

At high velocities, v ν0 /·rp , the self-energies are, according to Eqs. (12.143)“

(12.145), inversely proportional to the velocity, and they can accordingly be neglected

in the calculation of the pinning force. We can therefore in this limit insert the free

retarded Green™s functions in the self-consistent expression for the pinning force,

Eq. (12.152), thereby obtaining an expression valid to lowest order in the disorder

strength, ν0 ,

∞

i dk

k k 2 ν0 e’rp k

22 2

=’ dt eik·vt’k T t/·

, (12.153)

Fp

(2π)2

· 0

3√

· 2 rp / ν0 . The

where again we only display the result for vanishing mass, m

integration over time can then be performed, and we obtain the result that the

pinning force for large velocities, v T /(rp ·), is given by the perturbation theory

expression, Eq. (12.142).

It is also possible to obtain an analytical expression for the pinning force at mod-

erate velocities, provided the temperature is high enough. At high temperatures,

√

T ν0 /rp , the kinetic component of the self-energy is inversely proportional to

the temperature, ΣK (ω = v/rp ) ∼ ν0 ·/(rp T ), and its contribution to the ¬‚uctua-

2

tion exponent is much smaller than the contribution from the thermal ¬‚uctuations.

√ 3

Similarly, at temperatures T ν0 /(·rp v), the retarded self-energy is of order

√

ΣR (ω = v/rp ) ∼ ν0 /(rp T ). At moderate velocities, v ¤ ν0 /(·rp ), the free retarded

4 2

Green™s function can therefore be inserted in the expression for the pinning force,

and we can expand the exponential exp{ik · vt}, and keep only the lowest-order term

in the velocity, since the inequality v T /(·rp ) is satis¬ed, and obtain the result

that the pinning force is proportional to the velocity and inversely proportional to

the square of the temperature

ν0 ·

Fp = ’ v. (12.154)

8πrp T 2

2

√

Thus, when the thermal energy exceeds the average disorder barrier height, ν0 /rp ,

the pinning force is very small compared with the friction force, and pinning just leads

476 12. Classical statistical dynamics

to a slight renormalization of the bare friction coe¬cient. In this high-temperature

limit, which can be realized in high-temperature superconductors, we observe that

the self-consistent theory, at not too high velocities, yields a pinning force that has

a linear velocity dependence, in contrast to the case of low temperatures where we

obtain from the self-consistent theory, as apparent from for example Figure 12.3, the

fact that the velocity dependence of the pinning force is sub-linear.

12.5.3 Simulations

In order to ascertain the validity of the self-consistent theory beyond the high-velocity

regime, where perturbation theory is valid, we perform numerical simulations of the

Langevin equation, Eq. (12.134). The pinning force is obtained from Eq. (12.77),

once the simulation result for the average velocity as a function of the external force

is determined. We simulate the two-dimensional motion of a vortex in a region of

linear size L = 20rp , and use periodic boundary conditions. The disorder is generated

on a grid consisting of 1024 — 1024 points.

The disorder correlator is diagonal in the wave vectors, since averaged quantities

are translationally invariant,

V (k)V (k ) = ν(k)L2 δk+k =0 (12.155)

and the real and imaginary parts of the disorder potential can be generated indepen-

dently according to

√ √

ν0 L ’rp k2 /2 ν0 L

m V (k) = √ e’rp k /2 δ ,

2 22

e V (k) = √ e σ, (12.156)

2 2

where σ and δ are normally distributed stochastic variables with zero mean and

unit standard deviation. The gradient of the disorder potential at the grid points

is obtained by employing the ¬nite di¬erence scheme. The potential gradient at the

vortex position is then obtained by interpolation of the values of the potential at the

four nearest grid points.

The simulations show that the vortex follows a fairly narrow channel through the

potential landscape. In the absence of the Hall force, the vortex will traverse only a

very limited region of the generated potential owing to the imposed periodic boundary

condition. To make better use of the generated potential, we therefore randomize the

vortex position at equidistant moments in time, and run the simulation for a short

time without measuring the velocity, in order for the velocity to relax, before again

starting to measure the velocity. In this way the number of generated potentials can

be kept at a minimum of twenty.

12.5.4 Numerical results

For any given average velocity of the lattice, the coupled equations of Green™s func-

tions and self-energies may be solved numerically by iteration. We start the itera-

tion procedure by ¬rst calculating the Green™s functions for vanishing self-energies,

corresponding to the absence of disorder, and the self-energies are then calculated

from Eqs. (12.143)“(12.145). The procedure is then iterated until convergence is

12.5. Single vortex 477

reached. The pinning force on a single vortex can then be evaluated numerically

from Eq. (12.152).

In the numerical calculations we shall always assume that the correlator of the

pinning potential is the Gaussian function, Eq. (12.135), with range rp and strength

ν0 . In order to simplify the numerical calculation, the self-consistent equations for the

self-energies and the Green™s functions, Eq. (12.147) and Eq. (12.148), are brought

on dimensionless form by introducing the following units for length, time and mass,

1/2 1/2

rp , ·rp /ν0 , · 2 rp /ν0 .

3 4

We have solved the set of self-consistent equations numerically by iteration. In

Figure 12.3, the pinning force as a function of velocity is shown for di¬erent values

of the temperature.