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. 20
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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


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],

(c11 ’ c66 )qx qy
2 2
φ0 c11 qx + c66 qy
¦q = , (12.168)
(c11 ’ c66 )qx qy 2 2
c66 qx + c11 qy
B
488 12. Classical statistical dynamics


where q belongs to the ¬rst Brillouin zone, and B is the magnitude of the external
magnetic ¬eld, and φ0 /B is therefore equal to the area, a2 , of the unit cell of the
vortex lattice. In the continuum limit we obtain a longitudinal branch, el (q) · q = 1,
ˆ
with corresponding eigenvalues Kl (q) = c11 a q , and a transverse branch, et (q) · q =
22 ˆ
22
0, with corresponding eigenvalues Kt (q) = c66 a q .

12.6.1 High-velocity limit
√ 2
At high velocities, v ν0 /(·rp ), where lowest-order perturbation theory in the
disorder is valid, we can neglect the self-energies in the self-consistent expression for
the pinning force, Eq. (12.133), i.e. we can insert the free retarded Green™s function
for the lattice and, assuming v T /(·rp ), neglect the ¬‚uctuation exponent, and
obtain for the pinning force

·k · v (k · eb (k))2
dk
Fp = ’ k ν(k) . (12.169)
(·k · v)2 + (Kb (k))2
(2π)2
b=l,t


The maximum values, attained at the boundaries of the Brillouin zones, of the trans-
verse and longitudinal eigenvalues are speci¬ed by the compression and shear moduli,
Kt ∼ c66 and Kl ∼ c11 . The compression modulus is much greater than the shear
modulus, c11 c66 , in thin ¬lms and high-temperature superconductors (see for ex-
ample reference [137]). The order of magnitude of the ¬rst term in the denominator
’2
of Eq. (12.169) is ·v 2 rp , since the range of the impurity correlator is rp , and at
intermediate velocities, c66 rp /· v c11 rp /·, only the transverse mode therefore
contributes to the pinning force, and we obtain

(k · et (k))2
dk
=’ k ν(k) . (12.170)
Fp
·k · v
(2π)2

The eigenvalues et (k) are periodic in the reciprocal lattice and, assuming short-range
disorder, rp a, the rest of the integrand is slowly varying, and we obtain for the
pinning force


dk ν(k)k 2
1 ν0
Fp = ’ =’ v. (12.171)
k
(2π)2 ·k · v 8πrp ·v 2
4
2

At very high velocities, v c11 rp /·, the eigenvalues of the dynamic matrix in
Eq. (12.169) can be neglected compared with the velocity-dependent term in the
denominator, and the longitudinal and transverse parts of the free retarded Green™s
function give equal contributions to the pinning force, and we obtain

ν0
Fp = ’ v. (12.172)
4πrp ·v 2
4
12.6. Vortex lattice 489


This result is identical to the expression for the pinning force on a single vortex,
√ 2
Eq. (12.142), in the high velocity regime, v ν0 /(·rp ), since the in¬‚uence of the
elastic interaction is negligible.

12.6.2 Numerical results
In this section we consider the pinning force on the vortex lattice obtained from
the self-consistent theory. For any given average velocity of the lattice, the coupled
equations of Green™s functions and self-energies, Eq. (12.124) and Eq. (12.125), may
be solved numerically by iteration. In order to simplify the numerical calculation,
the self-consistent equations are brought on dimensionless form by introducing the
1/2 1/2
following units for length, time, and mass, a, ·a3 /ν0 , and · 2 a4 /ν0 . Starting
by neglecting the self-energies, we obtain numerically the response and correlation
functions. From Eq. (12.133) we can then determine the pinning force as a function
of the velocity. We have calculated the velocity dependence of the pinning force for
vortex lattices of sizes 4 — 4, 8 — 8, and 16 — 16 using the self-consistent theory, and
the results are shown in Figure 12.12.




5.5
5
4.5
Fp
4
3.5
3
2.5
10 20 30 40 50
v

Figure 12.12 Pinning force (in units of ν0 a’2 ) as a function of velocity (in units of
1/2

· ’1 ν0 a’2 ) obtained from the self-consistent theory for three di¬erent lattice sizes.
1/2

The stars correspond to a 4 — 4 lattice, and the two curves correspond to 8 — 8 and
16 — 16 lattices, respectively. The mass and temperature are chosen to be zero, and
1/2 1/2
the elastic constants are given by c66 a3 = 100ν0 and c11 a3 = 104 ν0 , and the
range of the disorder correlator is chosen to be rp = 0.1a.
490 12. Classical statistical dynamics


The di¬erence between the results obtained for the 8 — 8 and the 16 — 16 lattice
is small, and we conclude that the pinning force is fairly insensitive to the size of the
lattice.
In Figure 12.13 we compare the pinning force as a function of the velocity for
lattices of di¬erent sti¬nesses, and we ¬nd that the pinning force decreases with
increasing sti¬ness of the lattice.




6

5

4
Fp
3

2

1
0 10 20 30 40 50
v

Figure 12.13 Pinning force (in units of ν0 a’2 ) on a vortex lattice of size 16 — 16
1/2

as a function of velocity (in units of · ’1 ν0 a’2 ) obtained from the self-consistent
1/2

1/2
theory for the compression modulus given by c11 a3 = 104 ν0 and three di¬erent
1/2 1/2
shear moduli: c66 a3 = 50ν0 (upper dashed line), c66 a3 = 100ν0 (solid line) and
1/2
c66 a3 = 200ν0 (lower dashed line). The mass and temperature are both chosen to
be zero, and rp = 0.1a.




Generally, the interaction between the vortices lowers the pinning force, since the
neighboring vortices in a moving lattice drag a vortex over the potential barriers.
This can be inferred from the self-consistent theory by comparing the pinning forces
depicted in Figures 12.3 and 12.12, and in perturbation theory by noting the extra
term originating from the elastic interaction in the denominator of the expression for
12.6. Vortex lattice 491


the pinning force, Eq. (12.169).
When the temperature is increased, the pinning force decreases, except at very
high velocity, as apparent from Figure 12.14. This feature is common to the single
vortex case, and simply re¬‚ects that thermal noise helps a vortex over the potential
barriers.




3.7
3.6
3.5
3.4
Fp
3.3
3.2
3.1
3
2.9
10 20 30 40 50
v

Figure 12.14 Pinning force (in units of ν0 a’2 ) on a vortex lattice of size 16—16 as
1/2

a function of velocity (in units of · ’1 ν0 a’2 ) obtained from the self-consistent theory
1/2

1/2
for two di¬erent temperatures. The elastic constants are given by c66 a3 = 100ν0
’1/2
and c11 a3 = 104 ν0 , and rp = 0.1a and m = 1.0 · 10’4 · 2 a3 ν0
1/2
. The dashed line
1/2 ’1
corresponds to T = 0, and the solid line to T = 0.5ν0 a .




The convergence of the iterative procedure is monitored by checking that energy
conservation is ful¬lled. The energy conservation relation for a vortex lattice is
obtained as in Section 12.5.5, and since the term originating from the harmonic
interaction between the vortices disappears owing to the symmetry property of the
dynamic matrix, Eq. (12.76), we obtain for a vortex lattice the energy conservation
relation
· ‚t tr ’i‚t GK (R, t; R, t ) + 2 T GR (R, t; R, t ) |t =t = ’v · Fp . (12.173)
492 12. Classical statistical dynamics


The convergence of the iteration procedure, employed when solving the self-consistent
equations, has been checked by numerically calculating the terms in Eq. (12.173). We
¬nd that the right- and left-hand sides of the energy conservation relation di¬er by
no more than a few percent after twenty iterations.

12.6.3 Hall force
We now consider the in¬‚uence of a Hall force on the dynamics of a vortex lattice. The
motion of the vortex lattice, with its associated magnetic ¬eld, induces an average
electric ¬eld. The relationship between the average vortex velocity and the induced
electric ¬eld, E = v — B, and the expression for the Lorentz force, yields for the
resistivity tensor of a superconducting ¬lm
φ0 B ·e¬ ±
ρ= , (12.174)
’±
2 ·e¬
·e¬ + ±2

where the e¬ective friction coe¬cient, ·e¬ , was introduced in Eq. (12.79).13 Accord-
ing to Eq. (12.174), the following relationship between the transverse, ρxy , and the
longitudinal resistivities, ρxx , is obtained

±2
±
ρxy = ρ2 1+ . (12.175)
xx 2
Bφ0 ·e¬

If the Hall force is small, ± ·e¬ , the scaling relation
±
ρxy = ρ2 (12.176)
xx
Bφ0
is seen to be obeyed. This scaling law is valid for all velocities of the vortex, provided
the Hall force is small compared with the friction force, ± ·. We note that the
scaling law is also valid at small vortex velocities for arbitrary values of the Hall
force, if the e¬ective friction coe¬cient diverges at small velocities. This occurs if
the pinning force decreases slower than linearly in the vortex velocity. This is indeed
the case, according to the self-consistent theory, at temperatures lower than the

average barrier height, T ν0 /rp , as indicated by the low velocity behavior of the
pinning force in Figure 12.15. This behavior of the pinning force is also obtained for
non-interacting vortices as apparent from Figure 12.8.
In Figure 12.15 is shown the pinning force obtained from the self-consistent theory
as a function of velocity for the case of zero temperature. As expected there is
no in¬‚uence of the Hall force on the pinning force at low velocities, but we ¬nd
a suppression at intermediate velocities, and at very high velocities, v c11 a/·,
we recover the high velocity limit of the single vortex result, i.e. Eq. (12.165). By
comparison of Figures 12.8 and 12.15, we ¬nd that the Hall force has a much weaker
in¬‚uence at intermediate velocities on the pinning of an interacting vortex lattice
than on a system of non-interacting vortices. Furthermore, the in¬‚uence of the Hall
force on the pinning force is more pronounced for a sti¬ lattice than a soft lattice, as
seen from the inset in Figure 12.15.
13 The e¬ective friction coe¬cient was determined to lowest order in the disorder in reference [138].
12.7. Dynamic melting 493




3.6
3.5
3.4
3.3
3.2
Fp 1.6
1.5
3.1 Fp
1.4
3 1.3
2.9 1.2
0 50
2.8
v
2.7
0 10 20 30 40 50
v

Figure 12.15 Pinning force (in units of ν0 a’2 ) on a vortex lattice of size 16 — 16
1/2

as a function of velocity (in units of · ’1 ν0 a’2 ) obtained from the self-consistent
1/2

theory. The solid and dashed lines correspond to ± = 0 and ± = ·, respectively.
The temperature and mass are both chosen to be zero, and rp = 0.1a. The elastic
1/2 1/2
constants are given by c11 a3 = 104 ν0 and c66 a3 = 100ν0 . Inset: pinning force as
1/2
a function of velocity for ± = 0 and ± = ·, respectively. Here c66 a3 = 300ν0 and
the other parameters are unchanged.



In Figure 12.16 the dependence of the Hall angle on the velocity is presented for
various sti¬nesses of the vortex lattice; the sti¬est lattice has the greatest Hall angle.
Since the pinning force is reduced by the interaction between the vortices, the Hall
angle for a lattice is larger than for an independent vortex, except at high velocities
where they saturate at the same value. A similar behavior of the Hall angle at zero
velocity, as observed for a single vortex in Section 12.5.5, pertains to a vortex lattice.


12.7 Dynamic melting
In this section we consider the in¬‚uence of quenched disorder on the dynamic melting
of a vortex lattice. This non-equilibrium phase transition has been studied experi-
mentally [139, 140, 141, 142, 143, 144], as well as through numerical simulation and
a phenomenological theory and perturbation theory [145, 146, 147]. The notion of
dynamic melting refers to the melting of a moving vortex lattice where, in addition to
the thermal ¬‚uctuations, ¬‚uctuations in vortex positions are induced by the disorder.
A temperature-dependent critical velocity distinguishes a transition between a phase
494 12. Classical statistical dynamics



0.8

0.7

0.6
θ
0.5

0.4

0.3
0 10 20 30 40 50
v

Figure 12.16 Hall angle obtained from the self-consistent theory for a vortex lattice
of size 16 — 16 as a function of velocity (in units of · ’1 ν0 a’1 ) for a moderately
1/2

1/2
strong Hall force, ± = ·. The compression modulus is given by c11 a3 = 104 ν0 ,
and the three curves correspond to decreasing values of the shear modulus c66 a3 =
1/2 1/2 1/2
200ν0 , 100ν0 , 50ν0 . The mass and temperature are both chosen to be zero, and
rp = 0.1a.



where the vortices form a moving lattice, the solid phase, and a vortex liquid phase.
Before solving the self-consistent equations by numerical iteration in order to
obtain the phase diagram, we consider the heuristic argument for determining the
phase diagram for dynamic melting of a vortex lattice presented in reference [145].
There, the disorder induced ¬‚uctuations were estimated by considering the correlation
function
(p)
(p)
κ±± (x, t) = f± (x, t) f± (0, 0) (12.177)

of the pinning force density

f (p) (x, t) = ’ δ(x ’ R ’ uRt ) ∇V (x ’ vt) . (12.178)
R

Neglecting the interdependence of the ¬‚uctuations of the vortex positions and the
¬‚uctuations in the disorder potential, the pinning force correlation function factorizes

δ(x ’ R ’ uRt ) δ(R ’ uR 0 ) ∇±∇± V (x ’ vt)V (0) .
κ±± (x, t)
RR
(12.179)

Introducing the Fourier transform (A is the area of the ¬lm)

CRR (q, t) = A’1 e’iq·(R+uRt ’R ’uR 0 ) (12.180)
12.7. Dynamic melting 495


of the vortex density-density correlation function

δ(x ’ R ’ uRt ) δ(R ’ uR 0 )
CRR (x, t) = (12.181)

and employing the translational invariance yields

’nV δ(x ’ R ’ uRt ’ R ’ uR 0 ) ∇± ∇± ν(x ’ vt) ,
κ±± (x, t) =
RR
(12.182)

where nV is the density of vortices. In the ¬‚uidlike phase the motion of di¬erent
vortices is incoherent and the o¬-diagonal terms, R = R , can be neglected yielding

κ±± (x, t) = ’nV δ(x) ∇± ∇± ν(vt) . (12.183)

In analogy with the noise correlator, the e¬ect of disorder-induced ¬‚uctuations is
then represented by a shaking temperature

1 1 ν0 1 ν0
dx dt κ±± (x, t) √ =√
Tsh = , (12.184)
3 3
4·nV 4 2π ·vrp 4 2π F rp
’∞
±

where in the last equality it is assumed that the pinning force is small compared
with the friction force, i.e. ·v F . An e¬ective temperature is then obtained by
adding the shaking temperature to the temperature, Te¬ = T + Tsh , and according to
Eq. (12.184) the e¬ective temperature decreases with increasing external force, i.e.
with increasing average velocity of the vortices. As the external force is increased the
¬‚uid thus freezes into a lattice. The value of the external force for which the moving
lattice melts, the transition force Ft , is in this shaking theory de¬ned as the value for
which the e¬ective temperature equals the melting temperature, Tm , in the absence
of disorder
Te¬ (F=Ft ) = Tm (12.185)
and has therefore in the shaking theory the temperature dependence
ν0

Ft (T ) = (12.186)
4 2πrp (Tm ’ T )
3


for temperatures below the melting temperature of the ideal lattice. We note that
the transition force for strong enough disorder exceeds the critical force for which
1/2 2
the lattice is pinned Ft > Fc ∼ ν0 /rp .
We now describe the calculation within the self-consistent theory of the physical
quantities of interest for dynamic melting. The conventional way of determining a
melting transition is to use the Lindemann criterion, which states that the lattice
melts when the displacement ¬‚uctuations reach a critical value u2 = c2 a2 , where cL
L
is the Lindemann parameter, which is typically in the interval ranging from 0.1 to 0.2,
and a2 is the area of the unit cell of the vortex lattice. In two dimensions the position
¬‚uctuations of a vortex diverge even for a clean system, and the Lindemann crite-
rion implies that a two-dimensional vortex lattice is always unstable against thermal
¬‚uctuations. However, a quasi-long-range translational order persists up to a certain
496 12. Classical statistical dynamics


melting temperature [146]. As a criterion for the loss of long-range translational
order a modi¬ed Lindemann criterion involving the relative vortex ¬‚uctuations

(u(R + a0 , t) ’ u(R, t))2 = 2c2 a2 , (12.187)
L

where a0 is a primitive lattice vector, has successfully been employed [146], and
its validity veri¬ed within a variational treatment [148]. The relative displacement
¬‚uctuations of the vortices are speci¬ed in terms of the correlation function according
to

(u(R+a0 , t) ’ u(R, t))2 = 2itr GK (0, 0) ’ GK (a0 , 0) , (12.188)

where the translation invariance of the Green™s functions has been exploited. The
correlation function is determined by the Dyson equation, Eq. (12.147), where the in-
¬‚uence of the quenched disorder appears explicitly through ΣK and implicitly through
ΣR and ΣA in the retarded and advanced response functions. Furthermore, the self-
energies depend self-consistently on the response and correlation functions. We have
calculated numerically the Green™s functions and self-energies and thereby the vortex
¬‚uctuations for a vortex lattice of size 8 — 8, and evaluated the pinning force from
Eq. (12.133).
We determine the phase diagram for dynamic melting of the vortex lattice by cal-
culating the relative displacement ¬‚uctuations for a set of velocities, and interpolate
to ¬nd the transition velocity, vt , i.e. the value of the velocity at which the ¬‚uctua-
tions ful¬ll the modi¬ed Lindemann criterion (the determination of the Lindemann
parameter is discussed shortly). An example of such a set of velocities is presented
in the lower inset in Figure 12.17, where the relative displacement ¬‚uctuations as a
function of velocity are shown. The magnitude of the transition force is determined
by the averaged equation of motion

Ft = ·vt + Fp (vt ) (12.189)

and is then obtained by using the numerically calculated pinning force. Repeating the
calculation of the transition force for various temperatures determines the melting
curve, i.e. the temperature dependence of the transition force, Ft (T ), separating two
phases in the F T -plane: a high-velocity phase where the vortices form a moving solid
when the external force exceeds the transition force, F > Ft (T ), and a liquid phase
for forces less than the transition force.
In order to be able to compare the results of the self-consistent theory with the
simulation results, we use the same parameters as input to the self-consistent theory
as used in the literature [145]. There, the melting temperature in the absence of
disorder is given by Tm = 0.007 (the unit of energy per unit length is taken to be
2(φ0 /4π»)2 ) as obtained by simulations of clean systems [149], and assumed equal
to the Kosterlitz“Thouless temperature [150, 151]

c66 a2
TKT = . (12.190)

The shear modulus is therefore determined to have the value c66 = 0.088 (as a is taken
as the unit of length). The range of the vortex interaction, », was approximately
12.7. Dynamic melting 497


equal to the lattice spacing, a0 , giving for the compression modulus [130]
16π»2 c66
c11 = 50 c66 4.4 . (12.191)
a2
0

The range and strength of the disorder correlator in the simulations are in the chosen
units, rp = 0.2 and ν0 = 1.42 · 10’5 , and since the simulations are done for an over-
damped system, the vortex mass in the self-consistent theory should be set to zero.
As described above, our numerical results for the relative displacement ¬‚uctu-
ations can be used to obtain the dynamic phase diagram once the Lindemann pa-
rameter is determined. In order to do so we calculate melting curves by using the
self-consistent theory for a set of di¬erent values of the Lindemann parameter. We
¬nd that these curves have the same shape, close to the melting temperature, as the
melting curve obtained from the shaking theory, Eq. (12.186),
C2
T = C1 ’ . (12.192)
Ft
The curve which intersects at the melting temperature Tm = 0.007, the one depicted
in the upper inset in Figure 12.17, i.e. the one for which C1 is closest to the value
0.007, is then chosen, determining the Lindemann parameter to be given by the value
cL = 0.124.
Having determined the Lindemann parameter, we can determine the melting
curve, and the corresponding phase diagram obtained from the self-consistent theory
is shown in Figure 12.17. The simulation results of reference [145] are also presented,
as well as the melting curve obtained from the shaking theory. We note the agree-
ment of the simulation with the self-consistent theory, as well as with the shaking
theory, although the simulation data are not in the large-velocity regime and the
shaking argument is therefore not a priori valid.
In view of the good agreement between the self-consistent theory, the shaking
theory and the simulation, and the fact that we have only one ¬tting parameter at
our disposal, the melting temperature in the absence of disorder, it is of interest
to recall that while the melting curve obtained from the shaking theory was based
on an argument only valid in the liquid phase, i.e. freezing of the vortex liquid was
considered, the melting curve we obtained from the self-consistent theory is calculated
in the solid phase, i.e. we consider melting of the moving lattice. Furthermore, the
melting of the vortex lattice was indicated in the simulation by an abrupt increase
in the structural disorder [145], yet another melting criterion, and the agreement of
the self-consistent theory with the simulation data are therefore further validating
the use of the modi¬ed Lindemann criterion.
As is apparent from the upper inset in Figure 12.17, the critical exponent obtained
from the self-consistent theory, 1.0, is in excellent agreement with the prediction of
the shaking theory, where the critical exponent equals one. Furthermore, we ¬nd
that the self-consistent theory yields the value 1.65 · 10’4 for the magnitude of the

slope C2 , which is in good agreement with the value, ν0 /(4 2πrp ) = 1.77 · 10’4 ,
3

predicted by the shaking theory, represented by the lower dashed line. That the
values are so close testi¬es to the appropriateness of characterizing the disorder-
induced ¬‚uctuations e¬ectively by a temperature.
498 12. Classical statistical dynamics




Figure 12.17 Phase diagram for the dynamic melting transition. The melting curve
separates the two phases “ for values of the external force larger than the transition
force the moving vortices form a solid, and for smaller values they form a liquid.
The dots in the boxes represent points on the melting curve obtained from the self-
consistent theory using a vortex lattice of size 8 — 8, while the three stars represent
the simulation results of reference 6. The crosses represent the lowest-order pertur-
bation theory results. The dashed line is the curve Ft (T ) = 1.77 · 10’4 /(0.007 ’ T ),
the melting curve predicted by the shaking theory. Upper inset: relationship between
temperature and the inverse transition force obtained from the self-consistent theory,
close to the melting temperature, for the particular value of the Lindemann parame-
ter cL = 0.124, for which the curve intersects the vertical axis at Tm = 0.00701. The
set of points calculated from the self-consistent theory (plus signs) coincides with
a straight line in excellent agreement with the prediction for the critical exponent
by the shaking theory being 1. Lower inset: relative displacement ¬‚uctuations as a
function of velocity. The dots to the left are calculated by using the self-consistent
theory and the dots to the right are calculated by using lowest-order perturbation
theory (for the temperature T = 0.0065).
12.7. Dynamic melting 499


It is of interest to compare the melting curves obtained from the self-consistent
theory and perturbation theory. Expanding the kinetic component of the Dyson
equation, Eq. (12.124), to lowest order in the disorder we obtain

Dqω ΣK(1) ’ 2i·T Dqω
GK(1) R A
=
qω qω



’ 2i·kB T Dqω ΣR(1) Dqω + Dqω ΣA(1) Dqω ,
R R A A
(12.193)
qω qω


where ΣR(1) , ΣA(1) and ΣK(1) are the lowest-order approximations of the self-energies,
i.e. calculated to ¬rst order in ν0 . The relative vortex displacement ¬‚uctuations,
Eq. (12.188), can then be obtained in perturbation theory from Eq. (12.193). In Fig-
ure 12.17 is shown the melting curve predicted by perturbation theory, i.e. where for
the transition velocity interpolation we use the relative vortex ¬‚uctuations obtained
from perturbation theory, an example of which is shown in the lower inset. As is
to be expected, the perturbation theory result is in good agreement with the self-
consistent theory, and the shaking theory, at high velocities. However, we observe
from Figure 12.17 that the melting curve obtained from lowest-order perturbation
theory deviates markedly at intermediate velocities from the prediction of the non-
perturbative self-consistent theory, and thereby from the shaking theory, which is
known to account well for the measured melting curve [142].
The shaking theory is seen to be in remarkable good agreement with the self-
consistent theory for the parameter values considered above. We have investigated
whether this feature persists for stronger disorder. As apparent from Figure 12.18,
there is a more pronounced di¬erence between the shaking theory and the self-
consistent theory at stronger disorder. Whereas the deviation between the self-
consistent and shaking theory for the previous parameter values typically is 5%,
in the case of a ¬ve-fold stronger disorder, ν0 = 7.1 · 10’5 , it is more than 15%.
We have studied the in¬‚uence of pinning on vortex dynamics in the ¬‚ux ¬‚ow
regime. A self-consistent theory for the vortex correlation and response functions
was constructed, allowing a non-perturbative treatment of the disorder using the
powerful functional methods of quantum ¬eld theory presented in Chapter 10. The
validity of the self-consistent theory was established by comparison with numerical
simulations of the Langevin equation.
The self-consistent theory was ¬rst applied to a single vortex, appropriate for
low magnetic ¬elds where the vortices are so widely separated that the interaction
between them can be neglected. The result for the pinning force was compared with
lowest-order perturbation theory and good agreement was found at high velocities,
whereas perturbation theory failed to capture the non-monotonic behavior at low
velocities, a feature captured by the self-consistent theory. The in¬‚uence of the Hall
force on the pinning force on a single vortex was then considered using the self-
consistent theory. The Hall force was observed to suppress the pinning force, an
e¬ect also con¬rmed by our simulations. The suppression of the pinning force was
shown at high velocities to be in agreement with the analytical result obtained from
lowest-order perturbation theory. The suppression of the pinning force was caused
by the Hall force through its reduction of the response function, while the e¬ect of
¬‚uctuations through the ¬‚uctuation exponent at not too high temperatures could be
500 12. Classical statistical dynamics



6
5
4
Ft 3
2
1
0
0.006 0.0065 0.007
T

Figure 12.18 Phase diagram for the dynamic melting transition for the disorder
strength ν0 = 7.1 · 10’5 . The plus signs represent points on the melting curve
obtained from the self-consistent theory for a vortex lattice of size 8 — 8, while the
dashed curve is the curve Ft (T ) = 8.85 ·10’4 /(0.007’T ), the melting curve predicted
by the shaking theory.



neglected. The situation at high temperatures was the opposite, since in that case
the thermal ¬‚uctuations were of importance, and the Hall force then increased the
pinning force because it suppressed the ¬‚uctuation exponent.
We also studied a vortex lattice treating the interaction between the vortices in
the harmonic approximation. The pinning force on the vortex lattice was found to
be reduced by the interaction. The pinning force as a function of velocity displayed a
plateau at intermediate velocities, before eventually approaching at very high veloci-
ties the pinning force on a single vortex. Analytical results for the pinning force were
obtained in di¬erent velocity regimes depending on the magnitude of the compression
modulus of the vortex lattice. Furthermore, we included the Hall force and showed
that its in¬‚uence on the pinning force was much weaker on a vortex lattice than on
a single vortex.
We developed a self-consistent theory of the dynamic melting transition of a vortex
lattice, enabling us to determine numerically the melting curve directly from the
dynamics of the vortices. The presented self-consistent theory corroborated the phase
diagram obtained from the phenomenological shaking theory far better than lowest-
order perturbation theory. The melting curve obtained from the self-consistent theory
was found to be in good quantitative agreement with simulations and experimental
data.


12.8 Summary
In this chapter we have considered the theory of classical statistical dynamics treating
systems coupled to a heat bath and classical stochastic forces. In particular we
12.8. Summary 501


studied Langevin dynamics and quenched disorder, and applied the method to study
the dynamics of the Abrikosov ¬‚ux line lattice. As to be expected, the formalism of
classical statistical dynamics is the classical limit of the general formalism of non-
equilibrium states, Schwinger™s closed time path formulation of quantum statistical
mechanics, the general technique to treat non-equilibrium states we have developed
and applied in this book. The language of quantum ¬eld theory is thus the tool to
study ¬‚uctuations whatever their nature might be.
Appendix A

Path integrals

Quantum dynamics was stated in Chapter 1 in terms of operator calculus, viz.
through the Schr¨dinger equation or equivalently via the Hamiltonian as in the evo-
o
lution operator. Alternatively, quantum dynamics can be expressed in terms of path
integrals which directly exposes the basic principle of quantum mechanics, the su-
perposition principle1 . To acquaint ourselves with path integrals we show here for
the case of a single particle the equivalence of the two formulations by deriving the
path integral formulation from the operator expression for Dirac™s transformation
ˆ
function of Eq. (1.16), x, t|x , t = x|U (t, t )|x = G(x, t; x , t ) ≡ K(x, t; x , t ).
Propagating in small steps by inserting complete sets at intermediate times we have
for the propagator

dx1 dx2 . . . dxN x, t|xN , tN xN , tN |xN ’1 , tN ’1
x, t|x , t =

— xN ’1 , tN ’1 |xN ’2 , tN ’2 · · · x1 , t1 |x , t . (A.1)

We are consequently interested in the transformation function for in¬nitesimal times,
and from Eq. (1.16) we obtain

xn |e’
ˆ
i
xn , tn |xn’1 , tn’1 |xn’1
”tH(tn )
=
”t ˆ
δ(xn ’ xn’1 ) + xn |H(tn )|xn’1 + O(”t2 ) ,
= (A.2)
i
where ”t = tn ’ tn’1 = (t ’ t )/(N + 1)), as we have inserted N intermediate
resolutions of the identity.
In the following we shall consider a particle of mass m in a potential V for which
we have the Hamiltonian
p2
ˆ
ˆ
H(t) = + V (ˆ , t) , (A.3)
x
2m
ˆ pˆ
i.e. H = H(ˆ , x, t), where H by correspondence is Hamilton™s function.
1 For a detailed exposition of how the superposition principle for alternative paths leads to the
Schr¨dinger equation, we refer the reader to chapter 1 of reference [1].
o


503
504 Appendix A. Path integrals


Inserting a complete set of momentum states, we get

xn |H(ˆ , p, tn )|xn’1 xn |H(xn , p, tn )|xn’1
xˆ ˆ
=

dpn
e pn ·(xn ’xn ’1 ) H(xn , pn , tn ) ,
i
= (A.4)
d
(2π )

where we encounter Hamilton™s function on phase space

p2
n
H(xn , pn , tn ) = + V (xn , tn ) . (A.5)
2m
Inserting into Eq. (A.2), we get

dpn ”t
e pn ·(xn ’xn ’1 ) 1 +
i
xn , tn |xn’1 , tn’1 H(xn , pn , tn ) + O(”t2 )
= d
(2π ) i

dpn
e [pn ·(xn ’xn ’1 )’”tH(xn ,pn ,tn )] + O(”t2 ) .
i
= (A.6)
d
(2π )

Inserting additional internal times, we approach the limit ”t ’ 0, or equivalently
N ’ ∞, obtaining for the transformation function
N N +1
dpn
e [pn ·(xn ’xn ’1 )’”tH(xn ,pn ,tn )]
i
x, t|x , t = lim dxn
(2π )d
N ’∞
n=1 n=1


Dxt Dpt i
¯ ¯ t
dt [pt ·xt ’H(xt ,pt ,t)]
¯ ¯ ™¯ ¯ ¯¯
≡ e , (A.7)
t
d
(2π )

where x0 ≡ x , and xN +1 ≡ x. In the last equation we have just written the limit of
the sum as a path integral, and the integration measure has been identi¬ed by the
explicit limiting procedure.
The Hamilton function is quadratic in the momentum variable, and we have
Gaussian integrals which can be performed
∞ xn ’xn ’1 2
dpn p2 m d/2
xn ’xn ’1 i
e ”t(pn · ”t ’ 2m ) =
m ”t
i n
e2 (A.8)
”t
d
’∞ (2π ) 2πi ”t

and we thus get for the propagator
N +1 m (xn ’xn ’1 )2
N
’V (xn ,tn )
i
1 dxn ”t 2”t
K(x, t; x , t ) = lim e n =1
’d/2 ’d/2
N ’∞ m m
n=1 2πi ”t
2πi ”t

xt =x
t
dt L(xt ,xt ,t)
i ¯ ¯ ™¯ ¯
≡ Dxt e , (A.9)
t
¯

xt =x
Appendix A. Path integrals 505


where L in the continuum limit is seen to be Lagrange™s function
1
mx2 ’ V (xt , t) = xt · pt ’ H(xt , pt , t)
™ ™t ™
L(xt , xt , t) = (A.10)
2
related to Hamilton™s function through a Legendre transformation. The integration
measure has here been obtained for the case where we take the piecewise linear
approximation for a path.2
Instead of formulating quantum dynamics in terms of operator calculus we have
thus exhibited it in a way revealing the underlying superposition principle, viz. ac-
cording to Feynman™s principle: each possible alternative path contributes a pure
phase factor to the propagator, exp{iS/ }, where
t
¯ ¯ ™¯ ¯
S[xt ] = dt L(xt , xt , t) (A.11)
t

is the classical action expression for the path, xt , in question.3
The classical path is determined by stationarity of the action

δS
=0 (A.12)
δxt xt =xc l
t


the principle of least action,4 or explicitly through the Euler-Lagrange equations

d ‚L ‚L
’ =0 (A.13)

dt ‚x ‚x

the classical equation of motion.
Formulating quantum mechanics of a single particle as the zero-dimensional limit
of quantum ¬eld theory amounts to focussing on the correlation functions of, for
example, the position operator in the Heisenberg picture, say the time-ordered cor-
relation function
GH (t, t ) ≡ T (ˆ H (t) xH (t ))
ˆ (A.14)
x
where the bracket refers to averaging with respect to some state of the particle, pure

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