the structure in the dynamical index space

⎛ ⎞

u± u± t

0 ˜Rt R

Gij = ’i ⎝ ⎠

u± u± t δu± δu± t

Rt ˜R Rt R

GA (R, t; R , t )

0 ±±

= , (12.108)

GR (R, t; R , t ) GK (R, t; R , t )

±± ±±

where we observe that the connected and unconnected retarded (or advanced) Green™s

functions are equal. Similarly, in the absence of sources the expectation value of the

auxiliary ¬eld vanishes, and the average ¬eld is therefore given by

¯ ˜

φRt = ( uRt , uRt ) = (0, vt) , (12.109)

where v is the average velocity of the vortex lattice.

The retarded Green™s function GR yields the linear response to the force F± ,

±±

i.e. to linear order in the external force we have

∞

dt GR (R, t; R , t ) F± (R , t ) ,

u± (R, t) = (12.110)

±±

R ’∞

and GK is the correlation function, both matrices in Cartesian indices as indicated.

±±

The matrix Green™s function in dynamical index space, Eq. (12.108), has only two

independent components, since the advanced Green™s function is given by

GA (R, t; R , t ) = GR ± (R , t ; R, t) . (12.111)

±± ±

Pursuing an equation for the pinning force, we introduce the e¬ective action, “,

the generator of two-particle irreducible vertex functions, i.e. the double Legendre

transform of the generator of connected Green™s functions, W (recall Section 10.5.1),

1 i

“[φ, G] = W [f, K] ’ f φ ’ φKφ ’ TrGK , (12.112)

2 2

where Tr denotes the trace over all variables and indices, i.e. TrGK denotes the

expression

∞∞

dt dt G±± (R, t; R , t ) Ki i ± (R , t ; R, t) .

±

TrGK = (12.113)

ii

’∞ ’∞

R,R

±,± =x,y

i,i =1,2

The e¬ective action satis¬es the equations

δ“

= ’f ’ Kφ (12.114)

δφ

12.4. Self-consistent theory of vortex dynamics 469

and

δ“ i

= ’ K. (12.115)

δG 2

The e¬ective action was shown in Section 10.5.1 to have the form

i i i

’1

= S[φ] + TrDS G ’ Tr ln D’1 G ’ Tr1

¯ ¯

“[φ, G]

2 2 2

¯

’ i ln eiSint [φ,ψ] 2PI

, (12.116)

G

’1

where the quantity DS is the second derivative of the action at the average ¬eld

¯

δ 2 S[φ]

’1

DS [φ](t, t ) = (12.117)

δφt δφt

¯

and Sint [φ, ψ] is the part of the action S[φ + ψ] that is higher than second order in

ψ in an expansion around the average ¬eld. The superscript “2PI” on the last term

indicates that only the two-particle irreducible vacuum diagrams should be included

in the interaction part of the e¬ective action, the last term in Eq. (12.116), and the

subscript that propagator lines represent G, i.e. the brackets with subscript G denote

the average

’1

= (det iG)’1/2

¯ ¯

i

Dψ e 2 ψG

eiSint [φ,ψ] ψ

eiSint [φ,ψ] . (12.118)

G

The ¬rst dynamical index component of Eq. (12.114) together with the equa-

tion for the average motion Eq. (12.77) provide an expression for the pinning force,

Eq. (12.78), in term of the dynamical matrix propagator of the theory. The general

expression is still intractable, and in the next section we shall introduce the main

approximation.

12.4 Self-consistent theory of vortex dynamics

Because of the disorder, the equation of motion describing the vortex dynamics has no

simple analytical solution. The employed ¬eld theoretical formulation of the pinning

problem will therefore be used in combination with a self-consistent approximation

for the e¬ective action for studying vortex motion in type-II superconductors. Since

we have constructed the two-particle irreducible e¬ective action, we expect that its

lowest-order approximation contains the main in¬‚uence of the quenched disorder

on the vortex dynamics. The validity of the self-consistent theory is ascertained by

comparing with numerical simulations of the Langevin equation. The e¬ective action

method will be used to study the dynamics of single vortices and vortex lattices,

and yields results for the pinning force, ¬‚uctuations in position and velocity, etc.

The dependence of the pinning force on vortex velocity, temperature and disorder

strength is calculated for independent vortices as well as for a vortex lattice, and both

analytical and numerical results for the pinning of vortices in the ¬‚ux ¬‚ow regime

are obtained. Finally, the in¬‚uence of pinning on the dynamic melting of a vortex

lattice is studied in Section 12.7.

470 12. Classical statistical dynamics

12.4.1 Hartree approximation

In order to obtain a closed expression for the self-energy in terms of the two-point

Green™s function, we expand the exponential and keep only the lowest-order term

¯ ¯ ¯

’i ln eiSint [φ,ψ] ’i ln 1 + iSint [φ, ψ]

2PI 2PI

Sint [φ, ψ] , (12.119)

G

G G

i.e. we consider the Hartree approximation, which in diagrammatic terms corresponds

to neglecting diagrams where di¬erent impurity correlators are connected by Green™s

functions.

Figure 12.2 Typical vacuum diagram not included in the Hartree approximation

for the e¬ective action. The solid line represents the correlation function or kinetic

component, GK , of the matrix Green™s function. The retarded Green™s function, GR ,

is depicted as a wiggly line ending up in a straight line, and vice versa for the advanced

Green™s function GA . The curly line ending up on the dot represents the ¬rst kinetic

component of the average ¬eld. A dashed line attached to circles represents the

impurity correlator and the additional dependence on the second component of the

average ¬eld as explicitly speci¬ed in Eq. (12.120).

A typical vacuum diagram not included in the Hartree approximation for the

e¬ective action is shown in Figure 12.2, and represents the expression

2 2

i 1 dk1 dk2 ¯

k2 · φ1 (R2 , t2 )

(2π)2 (2π)2

2 4!

—(k2 GR (R2 , t2 ; R1 , t1 )k1 )(k1 GR (R1 , t1 ; R1 , t1 )k1 )

—(k1 GR (R1 , t1 ; R2 , t2 )k2 )(k2 GK (R2 , t2 ; R2 , t2 )k2 )

—ν(k1 )eik1 ·(R1 ’R1 +v(t1 ’t1 )) ν(k2 )eik2 ·(R2 ’R2 +v(t2 ’t2 )) , (12.120)

where integrations over time and summations over vortex positions are implied, and

we have introduced the notation

kGR (R, t; R , t )k = k± GR (R, t; R , t ) k± (12.121)

±±

±±

12.4. Self-consistent theory of vortex dynamics 471

for Cartesian scalars.

In the Hartree approximation, Eq. (12.119), we drop the superscript “2PI” since

¯

the action Sint [φ, ψ] only generates two-particle-irreducible vacuum diagrams, due to

the appearance of only one impurity correlator. The Hartree approximation can be

expressed as a Gaussian ¬‚uctuation corrected saddle-point approximation [135].

The e¬ective action can in the Hartree approximation be rewritten on the form

i i i

“[φ, G] = S0 [φ] ’ Tr ln D’1 G + TrD’1 G ’ Tr1 + SV [φ + ψ]

¯ ¯ ¯ (12.122)

G

2 2 2

since

∞ ∞ ¯

δ 2 SV [φ]

i

¯ ¯ ¯

G ’ SV [φ] ’

Sint [φ, ψ] = SV [φ + ψ] Tr dt dt ¯ ¯ Gt t , (12.123)

G

2 ’∞ ’∞ δ φt δ φt

where the trace in the time variable has been written explicitly for clarity.

In the physical situation of interest the two-particle source, K, vanishes, and since

“ is two-particle-irreducible, Eq. (12.115) therefore becomes the Dyson equation

G’1 = D’1 ’ Σ , (12.124)

where the self-energy in the Hartree approximation is the matrix in dynamical index

space

¯

ΣK ΣR δ SV [φ + ψ] G

Σij = = 2i . (12.125)

ΣA 0 δGij K=0, J=0

The Dyson equation, Eq. (12.124), and the self-energy expression, Eq. (12.125), and

the equation relating the e¬ective action to the external force, Eq. (12.114), constitute

a set of self-consistent equations for the Green™s functions, the self-energies, and the

average ¬eld, in this non-equilibrium theory the latter speci¬es the velocity of the

vortex lattice.

The matrix self-energy in dynamical index space has only two independent com-

ponents since

ΣA (R, t; R , t ) = ΣR ± (R , t ; R, t) , (12.126)

±± ±

a simple consequence of Eq. (12.111) and the Dyson equation. From Eq. (12.125) we

obtain for a vortex lattice having a unit cell of area a2 and consisting of N vortices,

the self-energy components (each a matrix in Cartesian space)

i

ν(k) kk e’•(R,t;R ,t ;k;v)

ΣK (R, t; R , t ) = ’ ˜

(12.127)

N a2

k

and

∞

˜˜

ΣR (R, t; R , t ) = σ R (R, t; R , t ) ’ δRR δ(t ’ t ) ˜

dt σ R (R, t; R, t) ,

˜

R ’∞

(12.128)

472 12. Classical statistical dynamics

where

1

ν(k) kk (kGR (R, t; R , t )k) = e’•(R,t;R ,t ;k;v) .

σ R (R, t; R , t ) = ˜

N a2

k

(12.129)

We use dyadic notation, i.e. kk denotes the matrix with the Cartesian components

k± k± . The in¬‚uence of thermal and disorder-induced ¬‚uctuations are described by

the ¬‚uctuation or damping exponent

•k (R, t; R , t ) = ik GK(R, t; R, t) ’ GK(R, t; R , t ) k (12.130)

contained in

•(R, t; R , t ; k; v) = ’ik · (R ’ R + v(t ’ t )) + •k (R, t; R , t ) . (12.131)

˜

The pinning force on a vortex, Eq. (12.78), is determined by the averaged equation of

motion, Eq. (12.77), and the ¬rst dynamical index component of Eq. (12.114), which

in the Hartree approximation yields

∞

δ SV [φ + ψ]

’1±± G

’ ±

dt DR (R, t; R , t ) v± t = FR + (12.132)

±

δφ1 (R, t)

’∞

R ± φRt =(0,vt)

resulting in the expression for the pinning force

∞

dk

k ν(k)(kGR t k)e’•(R,t;R ,t ;k;v) .

˜

Fp = i dt (12.133)

RtR

2

(2π)

R ’∞

The self-consistent theory in the Hartree approximation is still intractable to

analytical treatment, except in the limiting cases considered in the following, but it

is manageable numerically.12 In the following we shall study numerically the vortex

dynamics in the Hartree approximation. The results obtained from the self-consistent

theory will then be compared with analytical results obtained in perturbation theory,

and with simulations of the vortex dynamics.

12.5 Single vortex

In order to study the essential features of the model and the self-consistent method,

we ¬rst consider the case of a single vortex, since this example will allow the important

test of comparing the results of the self-consistent theory with simulations. The case

of non-interacting vortices is appropriate for low magnetic ¬elds, where the vortices

are so widely separated that the interaction between them can be neglected. The

dynamics of a single vortex is described by the Langevin equation

m¨ t + · xt = ’∇V (xt ) + Ft + ξ t ,

™ (12.134)

x

12 In the rest of this chapter we follow reference [134].

12.5. Single vortex 473

where xt is the vortex position at time t. We defer the discussion of the Hall force

to Section 12.5.5.

When presenting analytical and numerical results obtained from the self-consistent

theory, we shall always choose the vortex mass (per unit length) to be small, in fact

3√

· 2 rp / ν0 , that the case of zero mass only deviates slightly from the

so small, m

presented results, i.e. at most a few percent.

In the analytical and numerical calculations, the correlator of the pinning poten-

tial shall be taken as the Gaussian function with range rp and strength ν0

ν0 ’(x’x )2 /2rp

ν(k) = ν0 e’rp k .

2 22

ν(x ’ x ) = e , (12.135)

2

2πrp

12.5.1 Perturbation theory

At high velocities, the pinning force can be obtained from lowest-order perturbation

theory in the disorder, since the pinning force then is small compared with the friction

force, and makes, according to Eq. (12.77), only a small contribution to the total force

on the vortex. We ¬rst consider the case of zero temperature, where we obtain the

following set of equations by collecting terms of equal powers in the pinning potential

∞

’1 (0)

’ dt DR (t, t ) xt = Ft (12.136)

’∞

and

∞

’1 (1) (0)

’ = ’∇V (xt )

dt DR (t, t ) xt (12.137)

’∞

and

∞

’1 (2) (1) (0)

’ = ’∇ xt · ∇V (xt ) .

dt DR (t, t ) xt (12.138)

’∞

Assuming that the external force is independent of time, the average vortex ve-

locity will be constant in time, and in the absence of disorder the average vortex

position is

Ft

(0)

= vt = , (12.139)

xt

·

i.e. the friction force balances the external force, ·v = F. The ¬rst-order contribution

to the vortex position vanishes upon averaging with respect to the pinning potential,

and the second-order contribution to the average vortex velocity becomes, according

to Eqs. (12.136) to (12.138),

∞

i dk

k k 2 ν0 e’k rp +ik·v(t’t )

22

(2)

’ R

™ = dt Dtt

xt

(2π)2

· ’∞

∞ 2 2

ν0 vt vt ’ vt

’

R

= dt Dt0 e . (12.140)

2r p

5·

4πrp 0 rp 2rp

474 12. Classical statistical dynamics

The second-order contribution is immediately calculated, and for example for the

3√

· 2 rp / ν0 , we obtain

case of a vanishing mass, m

ν0

(2)

=’

™ v. (12.141)

xt

4πrp · 2 v 2

4

The pinning force is then, according to Eq. (12.77), to lowest order in the disorder