Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Here g(x) = 1 on the boundary circle S

= @D,and

thus can be viewed as a function S

!G.The

three-dimensional unit ball B has S

as a boundary

and g is extended in an arbitrary way from the

boundary to the ball B. The extension is possible

since 

(G) = 0 for any finite-dimensional Lie

group. The value of C(g) depends on the extension

only modulo an integer and therefore e

2iC(g)

well defined.

The central extension is then defined as

LG ¼ðMapðD; GÞS

Þ=N

One can show easily that the Lie algebra of

is indeed given through the cocycle [17].

When G = SU(n) in the defining representation,

this central extension is the basic extension:

The cohomology class is the generator of

(LG, Z). In general, to obtain the basic exten-

sion one has to correct [23] and [24] by a

normalization factor.

This construction generalizes to the higher loop

groups Map(S, G) for compact odd-dimensional

manifolds S. For example, in the case of a

3-manifold, one starts from an extension of

Map(D, G), where D is a 4-manifold with bound-

ary S. The extension is defined by a 2-cocycle ,

but now for given g, g

the cocycle  is a real-

valued function of a point g

2 Map(S, G), which is

a certain differential polynomial in the Maurer–

Cartan 1-forms g

1

, g

1

dg, g

1

dg. The normal

subgroup N is defined in a similar way; now C(g)is

the integral of the 5-form !

over a 5-manifold

B with boundary @B identified as D=,the

equivalence shrinking the boundary of D to one

point. This gives the extension only over the

connected component of identity in Map(S, G), but

it can be generalized to the whole group. For

example, when S = S

and G is simple, the con-

nected components are labeled by elements of the

third homotopy group 

G = Z.

In some cases, the de Rham cohomology class of

the extension vanishes but the extension still

contains interesting torsion information. In quan-

tum field theory this comes from Hamiltonian

formulation of global anomalies. A typical example

of this phenomenon is the Witten SU(2) anomaly in

four spacetime dimensions. In the Hamiltonian

formulation, we take S

as the physical space, the

gauge group G = SU(2). In this case, the second

cohomology of Map(S

, G) becomes pure torsion,

related to the fact that the 5-form !

on SU(2)

vanishes for dimensional reasons. Here the homo-

topy group 

(G) = Z

leads the nontrivial funda-

mental group Z

in each connected component of

Map(S

, G). Using this fact, one can show that

there is a nontrivial Z

extension of the group

Map(S

, G).

See also: Anomalies; Bosons and Fermions in External

Fields; Characteristic Classes; Dirac Operator and Dirac

Field; Index Theorems; K-Theory.

Further Reading

Araki H (1987) Bogoliubov automorphisms and Fockrepresentations

of canonical anticommutation relations. Contemporary Mathe-

matics, vol. 62. Providence: American Mathematical Society.

Berline N, Getzler E, and Vergne M (1992) Heat Kernels and

Dirac Operators, Die Grundlehren der mathematischen

Wissenschaften, vol. 298. Berlin: Springer.

Brylinski J-L Loop Spaces, Characteristic Classes, and Geometric

Quantization, Progress in Mathematics, vol. 107. Boston:

Birkha¨user.

Carey AL, Mickelsson J, and Murray M (2000) Bundle gerbes

applied to quantum field theory. Reviews in Mathematical

Physics 12: 65–90.

Gawedzki K and Reis N (2002) WZW branes and gerbes.

Reviews in Mathematical Physics 14: 1281–1334.

Giraud J (1971) Cohomologie Non Abelienne, (French) Die

Grundlehren der Mathematischen Wissenschaften, vol. 179.

Berlin: Springer.

Hitchin N (2001) Lectures on special Lagrangian submanifolds.

In: Winter School on Mirror Symmetry, Vector Bundles and

Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP

Stud. Adv. Math., vol. 23, pp. 151–182. Providence, RI:

American Mathematical Society.

Mickelsson J (1989) Current Algebras and Groups. London:

Plenum.

Treiman SB, Jackiw R, Zumino B, and Witten E (1985) Current

Algebra and Anomalies, Princeton Series in Physics. Princeton,

NJ: Princeton University Press.

546 Gerbes in Quantum Field Theory

Ginzburg–Landau Equation

Y Morita, Ryukoku University, Otsu, Japan

Introduction

In the Ginzburg–Landau theory of superconduc-

tivity, a complex order parameter  characterizes

a macroscopic/mesoscopic superconducting state

in a bulk superconductor. The square of the

magnitude jj

expresses the density of super-

conducting electrons and  is regarded as a

macroscopic wav e function. With a magne tic

vector potential A and the order parameter ,

the Helmholtz free energy density in a super-

conducting material near the critical temperature

is given by

F ¼F

þ jj



jj

ihr







jHj

8

where F

denotes the energy density of the normal

state, c is the light speed, H = curl A,andm

and e

are mass and charge of a superconducting electron,

respectively. The parameters  and  depend on

temperature an d are determined by the material.

Moreover, below the criti cal temperature T

 = (T) and  = (T) take negative and positive

values, respectively. In the presence of an applied

magnetic field H

, we have to consider the Gibbs

free energy density, G = F  H  H

=4.

Introduce the following physical parameters:



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

=

; H

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4

=

 ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

m

=4e

;¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

h

=2m



 ¼ =

½1

The value 

implies the equilibrium density and H

is the thermodynamic critical field, which is

obtained by equating G = F

jH

=8 (for the

normal state =0, H = H

) with G = F

 

=2

(for the perfect superconductivity jj

=

, A = 0).

The parameters  and  stand for penetration depth

and coherence length, respectively. The ratio  of

these characteristic lengths is called the Ginzburg–

Landau parameter, which determines the type of

superconducting material: type I for <1=

ﬃﬃﬃ

and

type II for >1=

ﬃﬃﬃ

We use the nondimensional variables x

, 

, A

, and

x ¼x

;  ¼ 



A ¼

ﬃﬃﬃ

A

ðH

¼ curl

Þ;

ﬃﬃﬃ

=

F ¼F

þð

G=

 1=2

þ 2H

 H

=

jH

=

ÞH

=4

½2

Dropping the primes after the change of variables

and integrating

G over a domain   R

(n = 2, 3),

which is occupied by a superconducting sample,

yields a functional of  and A , called the Ginzburg–

Landau energy in a nondimensional form,

Eð; AÞ¼





riAðÞ



ð1 jj

þjcurl A  H



dx ½3

The Ginzburg–Landau equations are the Euler–

Lagrange equations of this energy, which are given

riAðÞ

 ¼ 

ðjj

 1Þ in  ½4

curl

A ¼ J þ curl H

in  ½5

where

J :¼

ð



r  r



Þjj

A ½6





stands for the complex conjugate of . In a two-

dimensional domain , the differential operator

‘‘curl’’ acts on A = (A

, A

):R

! R

such that

curl A ¼@

 @

curl H ¼ð@

H; @

HÞ

H :¼curl A

and H

is replaced by a scalar-valued function.

Note that J represents a supercurrent in the

material. Every critical point of the energy is

obtained by solving the Ginzburg– Landau equations

with appropriate boundary conditions and, thus, a

physical state in the superconducting sample is

realized by a solution of the equations. A minimizer

of [3] is a solution of [4]–[5] that minimizes the

energy [3] in an appropriate function space, whereas

a local minimizer is a solution minimizing the energy

locally in the space. A solution is called a stable

solution if it is a local minimizer of the energy.

A physically stable phenomenon could be realized

by a minimizer or at least a local minimizer.

Ginzburg–Landau Equation 547

The Ginzburg–Landau energy and the equations

are gauge invariant under the transformation

ð; AÞ7!ðe

i

; A þrÞ½7

for a smooth scalar function (x). Therefore, we can

identify two solutions which have the correspon-

dence through the transformation [7]. The following

London (Coulomb) gauge is often chosen:

div A ¼ 0in ½8

(with a boundary condition if necessary).

Let (, A) be a smooth solution of [4]–[5].Inaregion

for j(x)j > 0, the expression =w(x)exp(i(x))

(w = j(x)j)leadsto

w ¼jr  Aj

w þ 

ðw

 1Þw ½9

divðw

ðr  AÞÞ ¼ 0 ½10

curl

A ¼ J ¼ w

ðr  AÞ½11

where the gauge [8] is fixed and curl H

= 0is

assumed. Let S be a surface in  bounded by a

closed curve @S. Suppose w (x) > 0on@S. Then

from [11],

:¼

ðJ=w

þ AÞds

J  ds þ

curl A  dS

r  ds ¼ 2d ½12

where d is an integer; in fact, d = deg(, @S) is the

winding number of (@S) in the complex plane.

Thus, the identity [12] relates the magnetic field to a

topological degree of the order parameter. The

quantity , multiplied by an appropriate constant,

is called the fluxoid. A connected component of

vanishing points of  generally has codimension 2 in

the domain, and it is called a vortex.

From the expression [9], the asymptotic behavior

w ! 1as !1 is expected under a suitable

condition. Then, by [11], H = curl A enjoys the

property curl

H þ curl H = 0, which is known as the

London equation. However, this is valid for jj > 0.

Otherwise, a singularity appears around zeros of .

There are several characteristic phenomena

observed in a bulk superconductor. Typical phenom-

ena are: perfect conductivity (persistent current),

perfect diamagnetism (Meissner effect), nucleation

of superconductivity, and vortices (quantization of a

penetrating magnetic field). These phenomena can be

expressed by solutions of the Ginzburg–Landau

equations in various settings.

Ginzburg–Landau Equations in R

A standard model of the Ginzburg–La ndau energy is

considered in the whole space R

. Let A = (A

, A

)

and assume H

= 0in[3]. Consider then the energy

functional

Eð; AÞ¼

j

þjcurl Aj



ð1 jj

dx ½13

where D

:= riA. Then the Ginzburg–Landau

equations are

 ¼ 

ðjj

 1Þ in R

½14

curl

A ¼ Imð



Þ in R

½15

In the gauge theory, this model can be regarded as a

two-dimensional abelian (U(1)) Higgs model. In that

context,  is a scalar (Higgs) field, A is a connection

on the U(1) bundle R

 U(1), and D

is the

covariant derivative.

Equations [14]–[15] are useful in observing quan-

tization of the magnetic field, although it is an ideal

model for superconductivity. By the natural condition

that the right-hand side of [13] is finite, we may

assume that jD

j, jcurl Aj!0andjj!1as

jxj!1.From[12], the flux quantization follows:

curl A dx ¼ 2d ½16

If  has a finite number of zeros {a

}

j=1

, [16] implies

curl A dx ¼ 2

j¼1

degð;@Bða

;ÞÞ

for a small positive number , where B(a

, ) stands

for the disk with the center a

and the radius .

A zero of  represents a vortex, at which the

magnetic field is quantized, and a supercurrent

moves around the field.

To characterize the configuration analytically, we

find a solution (, A) expressed by the polar

coordinate in the form

 ¼ f ðrÞexpðidÞ; AðrÞ¼ðrÞðsin ; cos Þ

Substituting these into [14]–[15], one obtains

ðrf



 



f ¼ 

ðf

 1Þf

ðrÞ



¼ f

 



(

= d=dr) with the boundary conditions

f ð0Þ¼0; f ð1Þ ¼ 1;ð1Þ ¼ 0

548 Ginzburg–Landau Equation

This system of the equations has a solution for >0.

In addition to these types of solutions, when

 = 1=

ﬃﬃﬃ

, a special transformation reduces the

system of [14]–[15] to a scalar nonlinear equation

with a singular term. Then, it is proved that for an

arbitrary d 2 Z, under the constraint of [16] there

exists a minimizer of [13] with zeros of prescribed

points {a

}

jdj

j¼1

(Jaffe and Taubes 1980).

Solutions for Persistent Current

A current flowing in a superconducting ring with no

decay even in the absence of an applied magnetic

field is called a persistent current. Assume that a

superconducting sample  in R

is surrounded by

vacuum and adopt the energy functional as

Eð; AÞ¼



j



ð1 jj

jcurl Aj

½17

Although the functional [17] is minimized by a

trivial solution (, A) = (exp (ic), 0)(c 2 R), which is

the case for perfect diamagnetism, this is not the

solution describing a persistent current since J = 0

everywhere. We have to look for a nontrivial

solution that locally minimizes the energy, that is,

a local minimizer of [17]. To characterize a solution

representing the persistent current, we define a

mapping from  to S

 C by x 2  ! (x)=j(x)j

for a solution (, A) of the corresponding

Ginzburg–Landau equations to [17]. Consider a domain

having infinitely many homotopy classes in the

space of continuous functions C

(



, S

)(e.g.,a

solid torus). If (, A) is a local minimizer and

=jj is not homotopic to a constant map of

(



, S

), then it is a solution describing a

persistent current. The existence of such a solution

has been established mathematically for large 

(Jimbo and Morita 1996, Rubinstein and Sternberg

1996).

Configuration of Solutions under an

Applied Magnetic Field

In the presence of an applied magnetic field,

according to the magnitude of the field, a sample

exhibits the transition from the superconducting

state to the normal state and vice versa. This

transition can be considered mathematically as a

bifurcation of solutions to the Ginzburg– Landau

equations with a parameter measuring the magni-

tude of the applied magnetic field. In fact, let H

an applied magnetic field perpendicular to the

horizontal plane and assume that it is constant

along the vertical axis, that is, H

= (0, 0, H

Then a rich bifurcation structure is suggested by

numerical and analytical studies in the parameter

space of (H

, ). Mathematical developments for

variational methods and nonlinear analysis reveal

the configuration of the solutions and provide

rigorous estimates for critical fields in a parameter

regime for a two-dimensional model, pr edicted by

physicists.

Throughout this section, we consider the Ginzburg–

Landau model in an infinite cylinder =D  R

(D  R

) with a constant applied magnetic field

= H

= (0, 0, H

), H

> 0. Assuming the uni-

formity along the vertical axis, we may write

A = (A

, A

) and H = curl A = @

 @

as in

the previous section. Then the Ginzburg–Landau

energy on D is

Eð; AÞ¼



j



ð1 jj

þjcurl A  H



dx ½18

With the London gauge

div A ¼ 0inD; A  n ¼ 0on@D

the Ginzburg–Landau equations in the prese nt

setting are written as

 ¼ðjj

 1Þ in D ½19

r

A ¼ Imð



Þ in D ½ 20

n r ¼ 0on@D ½21

curl A ¼ H

on @D ½ 22

where n denotes the outer unit normal.

Meissner Solutions

As seen in the case of no applied magnetic field, the

trivial solution (, A) = (exp(ic), 0) is a minimizer of

[18]. This solution expresses no magnetic field in the

sample. In a superconducting sample, the diamag-

netism holds even in the presence of an applied

magnetic field if the field is weak. Namely, the

sample is shielded so that penetration of the field is

only allowed near the surface of the sample. This

phenomenon is called the Meissner effect. A solution

expressing Meissner effect is called a Meissner

solution. Mathematically, it is understood that as

increases, such a Meissner solution continues

from the trivial solution. Then the solution preserves

the configuration 0 < j(x)j < 1. A study of the

asymptotic behavior of the Meissner solution as 

tends to 1 shows that the Meissner solution is a

Ginzburg–Landau Equation 549

minimizer up to H

= O( log ) for sufficiently large

 (Serfaty 1999).

Nucleation of Superconductivity

In an experiment, the Meissner state breaks down by

a stronger applied magnetic field. Then the sample

turns to be the nor mal state (in a type I conductor)

or it allows a mixed state of superconductivity and

normal state (in a type II cond uctor). In the former

case, the critical magnitude of the field is denoted by

, which corresponds to the one of [1], while it is

denoted by H

in the latter case. Moreover, the

mixed state eventually breaks down to be normal

state by further increasing the applied field up to

another critical field H

. To characterize these

two types mathematically, we consider a transition

from the normal state to the superconducting state

by reducing the magnitude of the field.

Let A

satisfy curl A

= H

(x 2 D)andA

 n =

0(x 2 @D). Then eqns [19]–[22] have a trivial

solution (, A) = (0, A

), which stands for the

normal state. Consider the second variation of the

energy functional [18] at this trivial solution

Eðs ; A

þ sBÞ



s¼0

jðr  iA

Þ j

 

j j

þjcurl Bj

If the minimum of this second variation for nonzero

( , B) is positive (or negative), then the trivial

solution is stable (or unstable). The minimum gives

the least eigenvalue of the linearized problem of

[19]–[20] around the trivial solution. Seeking such a

least eigenvalue  is reduced to studying an

eigenvalue problem of the Schro¨ dinger operator

L[ ]:= (riA

)

If the domain D is the whole space R

,itis

proved that  = H

. Back to the original variable of

[2], we can define a critical field H

ﬃﬃﬃ

;

 = 1=

ﬃﬃﬃ

separates a class of superconductors into

type I by <1=

ﬃﬃﬃ

< H

) and type II by

>1=

ﬃﬃﬃ

> H

In the bounded domain D, however, the critical

field at which superconductivity nucleates in the

interior of a sample is larger than H

(it is denoted

by H

), since the eigenvalue problem of L is

considered in the domain with the Neumann

boundary condition. A study of the least eigenvalue

 shows that the critical field has the asymptotics as

ﬃﬃﬃ





þ Oð1Þ;!1

where 0 <<1. If the applied field is very close to

and  is sufficiently large, the amplitude of the

eigenfunction associated with the leas t eigenvalue of

L (with the Neumann boundary condition) is very

small except for a 1= neighborhood of the

boundary. This implies that the nucleation of super-

conductivity takes place at the boundary. This

phenomenon is called surface nucleation (Del Pino

et al. 2000, Lu and Pan 1999).

Solutions of Vortices

In a type II superconductor, it is well known that

there exists a mixed state of superconductivity and

normal state in a parameter regime H

< H

In the mixed state, the magnetic field penetrating in

the sample is quantized such that it delivers a finite

number of lines or curves in the sample. This

configuration (called vortex) is characterized by

zero sets of the order parameter of the Ginzburg–

Landau equations. In a two-dimensional domain,

isolating vanishing points of the order parameter are

called vortices. Thus, it is quite an interesting

problem how such a vortex configuration can be

described mathematically by a minimizer of the

energy funct ional. In the sect ion ‘‘Ginz burg–Lan dau

equati ons in R

,’’ a specific configurat ion for vorte x

solutions is stated under very special conditions,

 = 1=

ﬃﬃﬃ

, on the whole space and no applied

magnetic field. However, this result is not general-

ized in the present setting.

A standard approach to a solution with the vortex

configuration is using a bifurcation analysis near the

critical field H

(or H

) by expanding a solution and

the difference H

 H

in a small parameter. Then the

leading term is given by an eigenfunction of the least

eigenvalue of the Schro¨ dinger operator coming from the

linearization. Under the doubly periodic conditions in

the whole space R

, the spatial pattern of vortices, called

Abrikosov’s vortex lattice, is studied by a local bifurca-

tion theory.

However, this kind of bifurcation analysis only

works near the critical field and the trivial solution

(, A) = (0, A

), which implies that only a small-

amplitude solution can be found. To realize a sharp

configuration of vortices, we need to consider a

parameter regime far from the bifurcation point. As

a matter of fact, mat hematical and numerical studies

for sufficiently large  exhibit nice configurations of

vortex solutions. In this case, in a neighborhood of

each vortex, with radius O(1=), a sharp layer

arises, and there exists a solution with multivortices

in an appropriate parameter region for H

.In

addition, as H

increases (up to H

), the number

of vortices also increases. This implies that the

minimizer of the energy functional [18] admits a

larger number of zeros for a higher magnitude of

applied magnetic field. However, it is a puzzle since

550 Ginzburg–Landau Equation

a solution with a smaller number of vortices seems

to have less energy. Thus, there is some balance

mechanism between contributions of the vortices

and the applied magnetic field to the total energy.

Mathematically, it is possible to estimate

E(, A) for the vortex solution to [19]–[22] as

follows: consider a family of square tiles K

with

side-length  which are periodically arranged over

the whole space. Assume each square in the

domain D has a single vortex. For an appropriate

test function, the energy over K

is estimate d as

O(log()). Since the number of vortices in the

domain is O(jDj=

)(jDj: the measure of D), we

obtain an upper bound O((jDj=

)log()). This

bound is less than E(0, A

) = jDj

=2forH

=

o(1) and  = 1=

ﬃﬃﬃﬃﬃﬃ

. Although in a general case it is

difficult to estimate the energy of the minimizer from

below, the leading order can be precisely determined in

some range of the interval (H

, H

)if is sufficiently

large (Sandier and Serfaty 2000).

A Simplified Model

Since the Ginzburg–Landau equations [4]–[5] are

coupled equations for  and A, we often encounter

mathematical difficulty in realizing a solution with

the configuration shown by a numerical experiment.

To look at a specific configuration, we may use a

simpler model equation. A typical simplification is

to neglect the magnetic field, which leads to the

equation for the order parameter :

þ 

ð1 j j

Þ ¼ 0in ½23

This equation is also called the Ginzburg–Landau

equation and it is the Euler–Lagrange equation of

the energy

Gð Þ¼



jr j



ð1 j j

dx ½24

in an appropriate function space. Under no constraint,

a constant solution with j j= 1 is a minimizer. If a

domain is topologically nontrivial, eqn [23] also

allows local minimizers of [24] for large  as seen in

the section ‘‘Solutions for Persistent Current.’’

On the other hand, [23] in a simply connected

domain D  R

with a boundary condition

= g(x)(x 2 @D) is used for a study of a vortex

solution for large .Let = 1=. Under the constraint

deg(g, @D) = d, a minimizer



must have at least jdj

zeros. The leading order of the energy around each

vortex is estimated as 2 log (1=). The result of Bethuel

et al. (1994) describes the energy for a minimizer

Gð



Þ¼2jdj logð1=Þþ þ Wða



; ...; a



jdj

Þþoð1Þ

where {a



} are zeros of



and  is a universal

constant. The function W is explicitly given as

Wða

; ...; a

jdj

Þ¼2

1j;kjdj;j6¼k

log ja

 a

jþR

where R is derived from a Green function satisfying

some boundary condition depending on g. More-

over, as  ! 0, the zeros converge to a minimizer of

W, which implies that the asymptotic position of

every zero (vortex) is determined by the explicit

function W. The first term of W shows that vortices

with the same sign of the degree are repulsive to one

another and the optimal arrangement of vortices

never allows the superposition of multivortices.

Although the boundary condition is rather artificial,

their mathematical formulation promoted the devel-

opment of variational methods applied to the

Ginzburg–Landau equatio n.

Time-Dependent Ginzburg–Landau

Equations

The Ginzburg–Landau equations in the preceding

sections are static models. We consider time evolu-

tion models called the time-dependent Ginzburg–

Landau equations. The evolut ion equati ons serve

various numerical simulations exhibiting dynamical

properties of solutions. They also provide mathe-

matical pro blems on global time behaviors of

solutions, stability of stationary solutions, dynami-

cal laws of vortices, etc. The Ginzburg–Landau

energy is denoted by E(u), u = (, A). The simplest

model for the time-dependent problem is the

gradient flow for E(u)

u ¼

E

u

where E=u is the first variation of the energy.

A more standard evolution equation in a nondimen-

sional form is given by

ð@

þ iÞ  D

 ¼ 

ð1 jj

Þ ½25

ð@

A þrÞþcurl

A ¼ Imð



Þþcurl H

½26

where (x, t) is the electric (scalar) potential and  is

a positive parameter with a physical quantity. In

fact, this equation was derived by Gor’kov and

Eliashberg from the Bardeen, Cooper, and Schrieffer

(BCS) theory.

The system of the equations [25]–[26] is invariant

under the following time-dependent gauge

transformation:

ð;;AÞ7!ð expðiÞ; @

; A þrÞ

Ginzburg–Landau Equation 551

The equations in the bounded domain D  R

are

considered subject to boundary and initial

conditions

  n ¼ 0on@ ð0; TÞ

curl A ¼ H

on @ ð0; TÞ

ðx; 0Þ¼

in 

Aðx; 0Þ¼A

ðxÞ in 

½27

Then, besides the Coulomb gauge [8], we can

choose the Lorentz gauge as follows:

div A þ  ¼ 0inD;

 dx ¼ 0

A  n ¼ 0on@D

For a smooth solution u(x, t)to[25]–[26] with [27],

EðuÞ¼2



jð@

þ iÞj

þ j@

A þrj

dx  0

holds if H

is time independent. This is also true in

the case of the whole space R

with a condition for

the asymptotic behavior as jxj!1.

Suppose that a domain   R

is occupied by a

superconducting sample and it is surrounded by a

medium (or vacuum). Then the electromagnetic

behavior in the outside domain, caused by the

induced magnetic field of a supercurrent in  and

an applied magnetic field, should be expressed

by the Maxwell equations. With the electric field

E = (@

A þr), we obtain

@

E  E þ curl

A ¼ curl H

in R





where , , and  are physical parameters (e.g.,  = 0

in the vacuum). To match the inside and the outside

of , appropriate boundary conditions are required.

From a point of the gauge theory as in the section

‘‘Ginz burg–Lan dau equation s in R

,’’ the following

time-dependent equations in the whole space are

also considered:

ð@

þ iÞ

  D

 ¼ 

ð1 jj

Þ

@

E þ curl

A ¼ Imð



Þ

r  E ¼ Imð



ð@

þ iÞÞ

Further Reading

Bethuel F, Brezis H, and He´lein F (1994) Ginzburg–Landau

Vortices. Boston: Birkha¨ user.

Chapman SJ, Howison SD, and Ockendon JR (1992) Macro-

scopic models for superconductivity. SIAM Review 34:

529–560.

Del Pino M, Felmer PL, and Sternberg P (2000) Boundary

concentration for the eigenvalue problems related to the onset

of superconductivity. Communications in Mathematical Phy-

sics 210: 413–446.

Du Q, Gunzberger MD, and Peterson JS (1992) Analysis and

approximation of the Ginzburg–Landau model of super-

conductivity. SIAM Review 34: 54–81.

Helffer B and Morame A (2001) Magnetic bottles in connection

with superconductivity. Journal of Functional Analysis 185:

604–680.

Hoffmann K-H and Tang Q (2001) Ginzburg–Landau Phase

Transition Theory and Superconductivity. Basel: Birkha¨user.

Jaffe A and Taubes C (1980) Vortices and Monopoles. Boston:

Birkha¨user.

Jimbo S and Morita Y (1996) Ginzburg–Landau equations and

stable solutions in a rotational domain. SIAM Journal of

Mathematical Analysis 27: 1360–1385.

Lin FH and Du Q (1997) Ginzburg–Landau vortices: dynamics,

pinning and hysteresis. SIAM Journal of Mathematical

Analysis 28: 1265–1293.

Lu K and Pan X-B (1999) Estimates of the upper critical field for

the Ginzburg–Landau equations of superconductivity. Physica D

127: 73–104.

Rubinstein J and Sternberg P (1996) Homotopy classification of

minimizers of the Ginzburg–Landau energy and the existence

of permanent currents. Communications in Mathematical

Physics 179: 257–263.

Sandier E and Serfaty S (2000) On the energy of type-II

superconductors in the mixed phase. Reviews in Mathematical

Physics 12: 1219–1257.

Serfaty S (1999) Stable configurations in superconductivity:

uniqueness, multiplicity, and vortex-nucleation. Archives for

Rational and Mechanical Analysis 149: 329–365.

Tinkham M (1996) Introduction to Superconductivity, 2nd edn.

New York: McGraw-Hill.

552 Ginzburg–Landau Equation

Glassy Disordered Systems: Dynamical Evolution

S Franz, The Abdus Salam ICTP, Trieste, Italy

Introduction

Many macroscopic systems if left to evolve in

isolation or in contact with a bath, are able to

relax, after a finite time, to history-independent

equilibrium states characterized by time-independent

values of the state variables and time-translation

invariance correlations. In glassy systems, the relaxa-

tion time becomes so large that equilibrium behavior

is never observed. On short timescales, the micro-

scopic degrees of freedom appear to be frozen in

far-from-equilibrium disordered states. On longer

timescales slow, history-dependent, off-equilibrium

relaxation phenomena become detectable.

The list of physical systems falling in disordered

glassy states at low temperature is long, just to mention

a few examples one can cite the canonical case of simple

and complex liquid systems undergoing a glass transi-

tion, polymeric glasses, dipolar glasses, spin glasses,

charge density wave systems, vortex systems in type II

superconductors, and many other systems.

Experimental and theoretical research has pointed

out the existence of dynamical scaling laws char-

acterizing the off-equilibrium evolution of glassy

systems. These laws, in turn, reflect the statisti cal

properties of the regions of configuration space

explored during relaxation.

The goal of a theory of glassy systems is the

comprehension of the mechanisms that lead to the

growth of relaxation time and the nature of

the scaling laws in off-equilibrium relaxation.

A well-developed description of glassy phenomena

is provided by mean-field theory based on spin glass

models, which gives a coherent framework that is

able to describe the dynamics of glassy systems and

provides a statistical interpretation of glassy relaxa-

tion. Despite important limitations of the mean-field

description for finite-dimensional systems, it allows

precise discussions of general concepts such as

effective temperatures and configurational entropies

that have been successfully applied to the descrip-

tion of glassy systems.

In the following, examples of two different ways of

freezing will be discussed: spin glasses, where

disorder is built in the random nature of the coupling

between the dynamical variables, and structural

glasses, where the disordered nature of the frozen

state has a self-induced character. These systems are

examples of two different ways of freezing.

A Glimpse of Freezing Phenomenology

Spin Glasses

The archetypical example of systems undergoing the

complex dynamical phenomena described in this

article is the case of spin glasses (Fischer and Hertz

1991, Young 1997). Spin glass materials are

magnetic systems where the magnetic atoms occupy

random position in lattices formed by nonmagnetic

matrices fixed at the moment of the preparation of

the material. The exchange interaction between the

spin of the magnetic impurities in these materials is

an oscillating function, taking positive and negative

values according to the distance between the atoms.

Spin glass models (see Spin Glasses, Mean Field

Spin Glasses and Neural Networks, and Short-

Range Spin Glasses: The Metastate Approach) are

defined by giving the form of the exchange

Hamiltonian, describing the interaction between

the spins S

of the magnetic atom s. In the presence

of an external magnetic field h, the exchange

Hamiltonian can be written as

H ¼

i; j2

 S

 h

i2

½1

The spin variable can have classical or quantum

nature. This article will be limited to the physics of

classical systems. The most common choice in

models is to use Ising variables S

= 1. The

couplings J

, which in real material depend on the

distance, are most commonly chosen to be indepen-

dent random variables with a distribu tion with

support on both positive and negative values. Most

commonly, one considers either a symmetric bimo-

dal distribution on {1, 1} or a symmetric Gaussian.

The sums are restricted to lattices  of various types.

The most common choices are =Z

for the

Edwards–Anderson model, the complete graph

={(i, j)ji < j; i, j = 1, ..., N} for the Sherrington–

Kirkpatrick (SK) model, and the Erdos–Renyi ran-

dom graph for the Viana–Bray (VB) model.

The presence of interactions of both signs induces

frustration in the system: the impossibility of

minimizing all the terms of the Hamiltonian at the

same time. One then has a complex energy land-

scape, where relaxation to equilibrium is hampered

by barriers of energetic and entropic nature.

Spin glass materials, which have a paramagnetic

behavior at high temperature, show glassy behavior

at low temperature, where magnetic degrees of

freedom appear to be frozen for long times in

apparently random dire ctions. There is quite a

Glassy Disordered Systems: Dynamical Evolution 553

general consensus, based on the analysis of the

experimental data and the numerical simulations,

that in three dimensions and in the absence of a

magnetic field, the two regimes are separated by a

thermodynamic phase transition at a temperature T

where the magnetic response  exhibits a cusp (see

Figure 1). By linear response,  is related to the

equilibrium spin correlation function

 ¼



hS



having denoted by hi the Boltzmann–Gibbs aver-

age. A cusp in  indicates a second-order transition

where the so-called Edwards–Anderson parameter

q = (1=N)

becomes different from zero,

indicating freezing of the spins in ran dom directions.

In the presence of a magnetic field, although the

low-temperature phenomenology is similar to

the one at zero field, the thermodynamic nature of

the freezing transition is more controversial. Theo-

retically, mean-field theory, based on the SK model,

predicts a phase transition with a cusp in the

susceptibility both in the absence and in the

presence of a magnetic field. Unfortunately, no

firm theoretical result is available on the existence

and the nature of phase transitions in finite-

dimensional spin glass models which is a completely

open problem.

Structural Glasses

Analogous freezing of dynamical variables is

observed in a variety of systems. Some of them

share with the spin glasses the presence of quenched

disorder; in many others, this feature is absent. This

is the case of structural glasses (Debenedetti 1996).

Many liquids under fast enough cooling, instead

of crystallizing, as dictated by equilibrium thermo-

dynamics, form glasses. Simple liquids can be

modeled as classical systems of particles with

pairwise interactions. In the simplest example of a

monoatomic liquid, the potential energy of a

configuration is then written as

Vðr

; ...; r

Þ¼

i<j

ðr

 r

Þ½2

In the case of atomic mixtures, the potential 

acquires a dependence on the species of the inter-

acting atoms.

Liquids can be characterized as good or bad glass

formers depending on the facility by which they

form glasses. In good glass formers, in order to

avoid crystallization, it is in general sufficient to

cross the region around the liquid–crystal transition

point fast enough, so that the systems can set in a

supercooled liquid metastable equilibrium. On low-

ering the temperature, the sup ercooled liquid

becomes denser and more viscous while the relaxa-

tion time of the system, related to the viscosity

through the Maxwell relation  = =G (G is the

instantaneous shear modulus of the liquid), under-

goes a rapid growth. One defines a con ventional

glass transition temperature T

as the point where 

takes the solid-like value  = 10

Poise, correspond-

ing to a relaxation time   100 s. After that point,

the system falls out of equilibrium; under usual

experimental conditions, it does not have the time to

adjust to external solicitations and behaves mechani-

cally like a solid. The glass transition temperature is

then characterized as the point where the liquid goes

out of equilibrium, the relaxation time becomes

larger than the external timescale and the positions

of the atoms appear as frozen on that scale.

A great effort has been devoted to understand the

behavior of the temperature dependence of the

relaxation time and the nature of the dynamical

processes in supercooled liquids. In deeply super-

cooled liquids, the empirical behavior of the relaxa-

tion time ranges from the Arrhenius form for

‘‘strong glasses’’   exp(=T) to the Vogel–Fulcher

form (T)  exp(D=(T  T

)) for ‘‘fragile glasses.’’

The Vogel–Fulcher law predicts a finite-temperature

divergence of the relaxation time at the temperature

. Unfortunately, in typical cases, the T

results are

χ (10

–5

emu/g)

1.0

2.0

3.0

4.0

5.0

12345

T(K)

678910

Cu – 0.1% Mn

Ag – 0.5% Mn

Au – 0.5% Mn

Ag – 1.0% Mn

Au – 0.2% Cr

(×10)

Figure 1 Magnetic susceptibility as a function of temperature

in spin glass materials. Reproduced from Fischer KH and

Hertz JA (1991) Spin Glasses. Cambridge, UK: Cambridge

University Press.

554 Glassy Disordered Systems: Dynamical Evolution

estimated to be 10–15% lower then T

so that it is

not possible to verify the law close enough to T

support the divergent behavior.

As a consequence of freezing, one observes

important qualitative changes in the behavior of

thermodynamic quantities similar to those encoun-

tered in equilibrium phase transitions. In a narrow

interval around T

, specific heat and compressibility

undergo jumps from liquid-like values to much

lower solid-like values.

Aging and Slow Dynamics

While the crudest picture of the glass transition

describes freezing as complete structural arrest, both

for the cases where the glass transition is a gen-

uine off-equilibrium phenomenon, as in structural

glasses, and in the case where it has a thermo-

dynamical character as in spin glasses, the study of

dynamical quantities reveals the existence of persist-

ing, history-dependent, slow relax ation processes in

the frozen phase (Norblad and Svendlidh 1997).

This is the phenomenon of aging, which is a

constitutive feature of the glassy state. Its theoreti cal

analysis occupies a central theoretical role in the

comprehension of the way glassy systems explore

configuration space. A first characterization of

relaxation is given by the behavior of ‘‘one-time

quantities’’ like internal energy, density, etc., which

slowly evolve in the course of time towards values

corresponding to states of lower free energy. More

interesting is the behavior of ‘‘two-time quantities,’’

time-dependent correlation functions and responses,

which reveal the deep off-equilibrium nature of

glassy relaxation. In experimental, numerical, and

theoretical studies, a special position is occupied by

the linear response function. Using the language of

magnetic systems , apt to the spin glasses, one

considers the response of the magnetization to an

applied magnetic field. To deal with other systems,

different conjugated couples of variables are con-

sidered and simple changes of language are needed.

Linear perturbations allow to reveal the dynamics of

the systems without affecting its evolution. Denoting

by M(t) the magnetization at a time t and by h(t

)

the magnetic field at time t

, the instantaneous linear

response function is defined as

Rðt; t

Þ¼

MðtÞ

hð t



h¼0

½3

Measures of the time integral of R(t, t

) are com-

monly performed to reveal the presence of aging in

glassy systems. Aging is usually studied observing the

dynamics that follows a rapid quench from high

temperature, at an instant that marks the origin of

time. One can reconstruct the response function

measuring the zero-field-cooled (ZFC) magnetization

as the response to a magnetic field acting from a

waiting time t

to the measuring time t,



ZFC

ðt; t

Þ¼

Rðt; t

Þ½4

or its complement, the thermoremanent magnetization

(TRM) corresponding to the response to a magnetic

field acting from the time of the quench up to t



TRM

ðt; t

Þ¼

Rðt; t

Þ½5

In Figure 2, the behavior of the susceptibility 

ZFC

shown as a function of t  t

in a typical example

of aging experiment at low temperature. Out-of-

equilibrium behavior is manifest in the dependence

of the curves on the waiting time t

. The relaxation

appears slower and slower for larger waiting times,

and the t

dependence does not disappear even for

very large times. Two nontrivial dynamical regimes

can be identified: a first regime for small t  t

, that

is, t  t

<< t

where the relaxation is independent

of t

and a second regime roughly valid for t  t

 t

where time-translation invariance is manifestly vio-

lated. The analysis of experimental and simulation data

shows a scale-invariant behavior according to which

curves corresponding to different waiting times can be

superimposed rescaling the time difference t  t

with a

suitable t

-dependent relaxation time (t

). This is a

growing function of t

whichseemstodivergeforlarge

. Up to the waiting times where it has been possible to

test the relation, (t

) behaves as a power (t

)  t



where in different materials and models,  = 0.8–0.9.

= 30 s

100 s

300 s

1000 s

3000 s

10000 s

10% M

T = 0.85 T

–1

M(t)/h

t [s]

Figure 2 ZFC magnetization in an aging experiment. The

curves, from bottom to top, correspond to increasing waiting

times. Reproduced from Norblad P and Svendlidh P (1997)

Experiments in spin glasses. In: Young AP (ed.) Spin Glasses

and Random Fields. Singapore: World Scientific, with permission

from World Scientific Publishing Co. Pte Ltd.

Glassy Disordered Systems: Dynamical Evolution 555