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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

Ft (T ) = (12.186)
4 2πrp (Tm ’ T )

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
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)

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

(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)

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),
T = C1 ’ . (12.192)
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 ,

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 ,
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

Ft 3
0.006 0.0065 0.007

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

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-
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
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’
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)
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
H(t) = + V (ˆ , t) , (A.3)
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].

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

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

where we encounter Hamilton™s function on phase space

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

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

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

Inserting additional internal times, we approach the limit ”t ’ 0, or equivalently
N ’ ∞, obtaining for the transformation function
N N +1
e [pn ·(xn ’xn ’1 )’”tH(xn ,pn ,tn )]
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)
(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)

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