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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],

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