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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.




0.25

0.2

0.15
Fp
0.1

0.05

0
0 0.5 1 1.5
v


’2 1/2
Figure 12.3 Pinning force (in units of ν0 rp ) on a single vortex as a function
of velocity (in units of · ’1 rp ν0 ) obtained from the self-consistent theory. The
’2 1/2

curves correspond to the di¬erent temperatures T = 0.005, 0.05, 0.1, 0.2, 0.4, 0.5 (in
1/2
units of ν0 /rp ), where the uppermost curve corresponds to T = 0.005, and m =
3 ’1/2
0.1· 2 rp ν0 .




We ¬nd that the pinning force has a non-monotonic dependence as a function
of velocity, and that the peak in the pinning force decreases rapidly with increasing
temperature, and develops into a plateau once the thermal energy is of the order of the
average barrier height. At the highest temperature, the velocity dependence of the
478 12. Classical statistical dynamics


pinning force is seen in Figure 12.3 to approach the linear regime at low velocities
in accordance with the analytical result obtained in the high temperature limit,
Eq. (12.154). At high velocities, the pinning force is independent of the temperature
as apparent from Figure 12.3.
In fact, the pinning force is inversely proportional to the velocity at high veloci-
ties in agreement with the perturbation theory result, Eq. (12.142), as apparent from
Figure 12.4, where a comparison is made between the pinning force obtained from
lowest-order perturbation theory and the numerically evaluated self-consistent result.
The two results agree as expected in the large velocity regime, whereas the pertur-
bation theory result has an unphysical divergence at low velocities due to the neglect
of ¬‚uctuations, and a consequent absence of damping by the ¬‚uctuation exponent in
Eq. (12.152).




0.2

0.15
Fp
0.1

0.05

0
0 0.5 1 1.5
v

’21/2
Figure 12.4 Pinning force (in units of ν0 rp ) on a single vortex as a function of
velocity (in units of · ’1 rp ν0 ). The solid line represents the result obtained from
’2 1/2

the self-consistent theory, while the dashed line represents the result of lowest-order
3 ’1/2
1/2 ’1
perturbation theory in the disorder (T = 0.005ν0 rp and m = 0.1· 2 rp ν0 ).




In order to check the validity of the self-consistent theory beyond lowest-order
perturbation theory, we have performed numerical simulations. In Figure 12.5, a
comparison between the self-consistent theory and a numerical simulation of the
pinning force as a function of velocity is presented. The agreement between the
self-consistent theory and the simulation is good, except around the maximum value
of the pinning force, where the simulation is found to yield a higher pinning force
12.5. Single vortex 479


in comparison to the self-consistent theory. In this region the relative velocity ¬‚uc-
tuations are large, and in fact the self-consistent theory predicts that the relative
velocity ¬‚uctuations are diverging at zero velocity even at zero temperature, as we
discuss shortly. The self-consistent equations and their numerical solution, as well as
the simulations, can therefore be expected to be less accurate at low velocities.




0.16
0.14
0.12
Fp
0.1
0.08
0.06
0.04
0 0.5 1 1.5
v

’2 1/2
Figure 12.5 Comparison of the pinning force (in units of ν0 rp ) on a single vortex
as a function of velocity (in units of · ’1 rp ν0 ) obtained from the self-consistent
’2 1/2

’1 1/2
theory, solid line, and the numerical simulation, plus signs (T = 0.1ν0 rp and
’1/2
m = 0.1· 2 rp ν0
3
).




The convergence of the iterative procedure can be monitored by checking that
energy conservation is ful¬lled. The energy conservation relation is obtained by
multiplying the Langevin equation by the velocity of the vortex and averaging over
the thermal noise and the quenched disorder

m xt · xt = ’ xt · ∇V (xt ) + F·v + xt · ξ t .
+ · x2
™¨ ™t ™ ™ (12.157)

The ¬rst term is proportional to ‚t x2 , and vanishes since averaged quantities are
™t
independent of time, as the external force is assumed to be independent of time. The
¬rst term on the right-hand side, the term originating from the disorder, vanishes for
the same reason, since it can be rewritten as ’‚t V (xt ) . The energy conservation

relation therefore becomes, v = xt ,

· (xt ’ v)2 ’ xt · ξ t = ’v · Fp
™ ™ (12.158)
480 12. Classical statistical dynamics


or, in terms of the Green™s functions,

’i·‚t trGK = ’v · Fp .
2
+ 2·T ‚t trGR (12.159)
tt tt
t =t t =t

where tr denotes the trace with respect to the Cartesian indices. The energy conser-
vation relation simply states that, on average, the work performed by the external
and thermal noise forces is dissipated owing to friction.
In order to ascertain the convergence of the iteration process, employed when
solving the self-consistent equations, we test how accurately the iterated solution
satis¬es the energy conservation relation. In Figure 12.6 the velocity dependence
of the left- and right-hand sides of the energy conservation relation, Eq. (12.159),
is shown. After at the most twenty iterations, the energy conservation relation is
satis¬ed by the iterated solution to within an accuracy of 1%.




0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0 0.5 1 1.5
v

Figure 12.6 The values (in units of ν0 · ’1 rp ) of the expressions on the two sides of
’4

the energy conservation relation, Eq. (12.159), are shown as a function of the velocity
(in units of · ’1 rp ν0 ). The dashed line and the plus symbols correspond to the
’2 1/2
’1/2
’1
1/2
left- and right-hand side, respectively (T = 0.05ν0 rp and m = 0.1· 2 rp ν0
3
). The
energy conservation relation is ful¬lled to within an accuracy of 1%.




In Section 12.7 we shall consider dynamic melting of the vortex lattice, and it is
therefore of interest to check the validity of the ¬‚uctuations predicted by the self-
consistent theory against direct simulations of the Langevin equation. In order to
check the accuracy of the velocity ¬‚uctuations calculated within the self-consistent
12.5. Single vortex 481

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