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454 12. Classical statistical dynamics


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

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.



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