∞

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)

4π

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