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