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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

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