theory, we have performed simulations of the velocity ¬‚uctuations. In Figure 12.7,

the velocity ¬‚uctuations obtained from the self-consistent theory are compared with

simulations.

1.05

1.04

1.03

1.02

1.01

1

0 0.5 1 1.5

v

Figure 12.7 Longitudinal and transverse velocity ¬‚uctuations (in units of · ’2 rp ν0 )

’4

as a function of the average velocity (in units of · ’1 rp ν0 ). The solid and dashed

’21/2

lines represent the results for the longitudinal (parallel to the external force), (xt ’

™

2 2

v) , and transverse, yt , velocity ¬‚uctuations obtained from the self-consistent

™

theory, respectively. The plus signs and crosses represent the simulation results for

1/2 ’1

the longitudinal and transverse velocity ¬‚uctuations, respectively (T = 0.1ν0 rp

’1/2

and m = 0.1· 2 rp ν0

3

). At low average velocities the ¬‚uctuations approach their

thermal value, T /m, which for the parameters and units in question equals 1. At

intermediate average velocities the longitudinal velocity ¬‚uctuations are larger than

the transverse, owing to the jerky motion of the particle along the preferred direction

of the external force, before reaching the same value at high average velocities where

the e¬ect of the disorder simply acts as an additional contribution to the temperature.

The agreement between the self-consistent theory and the numerical simulations

is seen to be good, indicating that ¬‚uctuations calculated from the self-consistent

theory are quantitatively correct. At low average velocities the velocity ¬‚uctuations

approach their thermal value T /m. The relative velocity ¬‚uctuations diverge at zero

velocity even at zero temperature. This can be inferred from the energy conservation

relation, Eq. (12.158), and the sub-linear velocity dependence of the pinning force at

low velocities, as for example is apparent from Figure 12.3. At intermediate average

482 12. Classical statistical dynamics

velocities, the velocity ¬‚uctuations in the direction parallel to the average velocity

(chosen along the x-axis), the longitudinal velocity ¬‚uctuations, (xt ’v)2 , are found

ˆ ™

to be larger than the ¬‚uctuations perpendicular to the average velocity, the transverse

™2

velocity ¬‚uctuations, yt . The reason behind this is that at not too high velocities,

where the force due to the disorder is strong compared with the friction force, the

motion of the particle is jerky since the particle slowly makes it to the disorder

potential tops, and subsequently is accelerated by the disorder potential. Since the

average motion of the particle is caused by the external driving force, the jerky

motion and the velocity ¬‚uctuations are largest in that preferred direction. At high

average velocity, the longitudinal and transverse velocity ¬‚uctuations saturate and

are seen to become equal, owing to the strong friction force causing a steadier motion.

In this connection we should also mention that we have noticed from our numerical

calculations that the second term on the left-hand side of Eq. (12.158) is independent

of the average velocity (and disorder), as is also apparent by comparing Figures

12.6 and 12.7. This thermal ¬‚uctuation contribution to the velocity ¬‚uctuations is

therefore given by its zero velocity value, and according to Eq. (12.158) is speci¬ed by

the equilibrium velocity ¬‚uctuations and therefore determined by equipartition. The

saturation value of the velocity ¬‚uctuations can therefore be determined from the

energy conservation relation, Eq. (12.158). For example, in the case of a small vortex

3√

· 2 rp / ν0 , we can use the high velocity expression for the pinning force,

mass, m

Eq. (12.142), and obtain the result that the saturation value equals T /m+ν0 /8πrp · 2 ,

4

a result in good agreement with Figure 12.7. At high average velocity, the velocity

¬‚uctuations saturate, and the e¬ect of the disorder simply acts as an additional

contribution to the temperature.

12.5.5 Hall force

In this section the e¬ect of a Hall force is considered, and the previous analysis of

the dynamics of a single vortex is extended to include the Hall force

m¨ t + · xt = ±xt — z ’ ∇V (xt ) + Ft + ξ t .

™ ™ ˆ (12.160)

x

We shall use the self-consistent theory to calculate the pinning force, the velocity

¬‚uctuations, and the Hall angle

FH ±

θ = arctan = arctan , (12.161)

v·F

ˆ ·e¬

which can be expressed in terms of the e¬ective friction coe¬cient.

Analytical results

The inverse of the free retarded Green™s function acquires, according to Eq. (12.160),

o¬-diagonal elements

’i±ω

mω 2 + i·ω

’1

DR (ω) = (12.162)

mω 2 + i·ω

i±ω

12.5. Single vortex 483

and the free retarded Green™s function is given by

1 mω + i· i±

R

Dω = . (12.163)

’i±

(ω +i0) ((mω +i·)2 ’±2 ) mω + i·

√ 2

In the high-velocity regime, v ν0 /(·rp ), where lowest-order perturbation theory

in the disorder is valid, we can neglect the self-energies in the self-consistent expres-

sion for the pinning force, Eq. (12.152), i.e. we can insert the free retarded Green™s

function and neglect the ¬‚uctuation exponent. Since the free retarded Green™s func-

tion is antisymmetric in the Cartesian indices, only the diagonal elements make a

contribution to the pinning force. The diagonal elements of the free retarded Green™s

Ryy

Rxx

function are identical, Dt0 = Dt0 , and given by

’· ± ±t ±t

e’·t/m

’ cos

Rxx

Dt0 = θ(t) 1+ sin (12.164)

·2 + ±2 · m m

3√

· 2 rp / ν0 ,

and we obtain for the pinning force, for vanishing mass, m

·ν0

Fp = ’ v. (12.165)

4π(· 2 + ±2 )rp v 2

4

We observe that the pinning force is suppressed by the Hall force in the high-velocity

√

ν0 (· 2 + ±2 )’1/2 rp , and the high-velocity regime therefore sets in at a

’2

limit, v

lower value in the presence of the Hall force.

√ √

ν0 /rp , and moderate velocities, v < · ν0 /((· 2 +

At high temperatures, T

±2 )rp ), the Hall force has the opposite e¬ect, i.e. it increases the pinning force, as a

2

calculation similar to the one leading to Eq. (12.154) shows that the pinning force is

3√

· 2 rp / ν0 ):

(m

ν0 (· 2 + ±2 )

Fp = ’ v. (12.166)

8π·T 2 rp

2

We have found by solving the self-consistent equations numerically at high temper-

√

ature, T = 10 ν0 /rp , that the pinning force is linear at low velocities and increases

with increasing Hall force. The deviation from the linear behavior in the presence of

the Hall force starts at a lower velocity value in accordance with the high-velocity

regime starting at a lower value in the presence of the Hall force.

Numerical results

For any given average velocity of the vortex, the pinning force can be calculated

from the self-consistent theory. We have numerically calculated the pinning force

for various strengths of the Hall force. In Figure 12.8, the resulting pinning force

as a function of the velocity is shown for di¬erent strengths of the Hall force for a

√

temperature lower than the average barrier height, T < ν0 /rp . The Hall force is

seen to reduce the pinning force in this temperature regime except, of course, at low

velocities.

484 12. Classical statistical dynamics

0.14

0.12

0.1

Fp

0.08

0.06

0.04

0.02

0 0.5 1 1.5

v

Figure 12.8 Pinning force (in units of ν0 rp ’2 ) on a single vortex as a func-

1/2

tion of velocity (in units of · ’1 rp ’2 ν0 ) obtained from the self-consistent the-

1/2

ory for various strengths of the Hall force. The di¬erent curves correspond to

±/· = 0, 0.2, 0.4, 0.6, 0.8, 1, where the uppermost curve corresponds to ± = 0

’1/2 1/2 ’1

(m = 0.1· 2 rp 3 ν0 and T = 0.1ν0 rp ).

0.15

0.1

Fp

0.05

0

0 0.5 1 1.5

v

Figure 12.9 Pinning force (in units of 10’4 ν0 rp ’2 ) on a single vortex as a function

1/2

of velocity. Comparison of the simulation results and the results of the self-consistent

and lowest order perturbation theory, Eq. (12.165), for the case of no Hall force,

’1/2

± = 0, and a moderately strong Hall force, ± = · (m = 0.1· 2 rp 3 ν0 and T =

1/2 ’1

0.1ν0 rp ). The solid line represents the self-consistent result and the crosses the

simulation result, while the upper dash-dotted line represents the perturbation theory

result, all for the case ± = 0. The dashed line and the plus symbols represent the

self-consistent and simulation results, while the lower dash-dotted line represents the

perturbation theory result, all for the case ± = ·.

12.5. Single vortex 485

In Figure 12.9 we compare the pinning force obtained from the self-consistent

theory with the result of perturbation theory valid at high velocities, Eq. (12.165),

and simulations. According to Figure 12.9, the reduction of the pinning force due to

the Hall force predicted by the self-consistent and the perturbation theory is in good

agreement at high velocities. The pinning force obtained from the self-consistent

theory and the simulations are also in good agreement in the presence of a Hall force,

even at lower velocities. In fact in much better agreement than in the absence of

the Hall force, in accordance with the fact that the Hall force suppresses the velocity

¬‚uctuations, as we demonstrate shortly.

The Hall angle calculated from the self-consistent theory approaches from below

the disorder-independent value arctan(±/·) at high velocities, as shown in Figure

12.10.

0.8

0.7

0.6

0.5

θ

0.4

0.3

0.2

0.1

0 0.5 1 1.5

v

Figure 12.10 Hall angle as a function of velocity for a single vortex. The curves

represent the self-consistent results for the three temperatures T = 0, 0.1, 1 (in units

1/2 ’1

of ν0 rp ), where the uppermost curve corresponds to the highest temperature. The

’1 1/2

plus symbols represent the simulation results for the temperature T = 0.1ν0 rp .

The parameter ±/· is one and m = 0.1· 2 rp ν0 1/2 .

3

In Figure 12.10, the Hall angle obtained from the self-consistent theory is also

compared with simulations, and the agreement is seen to be good. As apparent from

Figure 12.10, an increase in the temperature increases the Hall angle at low velocities,

because the e¬ective friction coe¬cient decreases with increasing temperature, and

this feature vanishes at high velocities. From Figure 12.10 we can also infer the

following behavior of the Hall angle at zero velocity: at low temperatures it is zero,

since the dependence of the pinning force at low velocities is sub-linear. At a certain

temperature, the Hall angle at zero velocity jumps to a ¬nite value, since the pinning

force then depends linearly on the velocity, and saturates at high temperatures at

the disorder independent value.

486 12. Classical statistical dynamics

We have also determined the in¬‚uence of the Hall force on the velocity ¬‚uctuations

as shown in Figure 12.11.

1.04

1.035

1.03

1.025

(x ’ v)2

1.02

™

1.015

1.01

1.005

1

0 0.5 1 1.5

v

Figure 12.11 Dependence of the single vortex velocity ¬‚uctuations (in units of

· ’2 rp ν0 ) on the average velocity (in units of · ’1 rp ν0 ) for ± = · and ± = 0

’4 ’2 1/2

’1/2

1/2

(T = 0.1ν0 /rp and m = 0.1· 2 rp ν03

). The solid and dashed lines represent the

longitudinal and transverse velocity ¬‚uctuations, respectively, calculated by using the

self-consistent theory for the case ± = ·, and the plus symbols and crosses represent

the corresponding simulation results. The two dash-dotted lines represent the lon-

gitudinal and transverse velocity ¬‚uctuations, respectively, calculated by using the

self-consistent theory in the absence of the Hall force, ± = 0, which were compared

with simulations in Figure 12.7.

We observe that the Hall force at low velocities slightly increases the transverse

velocity ¬‚uctuations, and decreases the longitudinal ¬‚uctuations, whereas the longi-

tudinal and transverse velocity ¬‚uctuations are strongly suppressed by the Hall force

at higher velocities, in particular the longitudinal ¬‚uctuations. The suppression of

the velocity ¬‚uctuations is caused by the blurring by the Hall force of the preferred

direction of motion due to the external force, resulting in a less jerky motion. At

high average velocity, the longitudinal and transverse velocity ¬‚uctuations saturate

and become equal because of the strong friction. As previously discussed in the

12.6. Vortex lattice 487

absence of the Hall force, the saturation value can be determined from the energy

conservation relation (which take the same form, Eq. (12.159), as in the absence of

the Hall force, since the Hall force does not perform any work) and the high-velocity

expression for the pinning force, Eq. (12.165), since our numerical results show that

the second term on the left-hand side of Eq. (12.158) is independent of the Hall force

and velocity (and disorder). This observation tells us that the suppression of the

velocity ¬‚uctuations caused by the Hall force, according to the energy conservation

relation, Eq. (12.158), is in correspondence with the suppression of the pinning force.

We note from Figure 12.11 that the high-velocity regime sets in at lower velocities

than in the absence of the Hall force. In Figure 12.11, the velocity ¬‚uctuations cal-

culated by using the self-consistent theory are also compared with simulations, and

the agreement is seen to be good.

We have ascertained the convergence of the numerical iteration process by testing

that the obtained solutions satisfy the energy conservation relation. We ¬nd that the

energy conservation relation is ful¬lled within an accuracy of 2%, except at the lowest

velocities.

12.6 Vortex lattice

After having gained con¬dence in the Hartree approximation studying the case of a

single vortex, we consider in this section the in¬‚uence of pinning on a vortex lattice

in the ¬‚ux ¬‚ow regime, where the lattice moves with a constant average velocity,

™

uRt = v, since the external force is assumed independent of time. We consider a

triangular Abrikosov vortex lattice, and treat the interaction between the vortices in

the harmonic approximation. The free retarded Green™s function of the vortex lattice

eb (q) eb (q)

R

Dqω = (12.167)

+ i·ω ’ Kb (q)

mω 2

b

is obtained by diagonalizing the dynamic matrix, and inverting the inverse free re-

tarded Green™s function speci¬ed by Eq. (12.84) (for the moment we neglect the Hall

force). The sum in Eq. (12.167) is over the two modes, b = 1, 2, corresponding to

eigenvectors eb (q) and eigenvalues Kb (q), respectively. The eigenvalues and eigen-

vectors of the dynamic matrix are periodic with respect to translations by reciprocal

lattice vectors.

Since the lattice distortions of interest are of small wave length compared to

the lattice constant, the dynamic matrix of the vortex lattice is speci¬ed by the

continuum theory of elastic media, i.e. through the compression modulus c11 and

the shear modulus c66 , and in accordance with reference [136],