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Novel Phases of Vortices in Superconductors 301

and g(x) is the function (called f (x) there) computed in Ref. 36. For the

solid a two-loop calculation perturbing around the perfect AL gives:

sol

= −

2β

+ 2.848|a

1/2

2.4

+ ·· (33)

It is found that they intersect at a

≈ −9.5, leading to a ﬁrst order VL

melting transition.

104

It compares reasonably with numerics

at high ﬁelds

(the same method in 2D gives a

≈ −13.2 in very good agreement with

numerics). The Debye–Waller factor is found e

−2W

≈ 0.50 to one loop which

predicts the Lindemann number through 2W =

. From the scaled

magnetization m = −f

) one gets a jump ∆M/M = 0.018 consistent with

experiments

105

if one takes into account additional weak pinning disorder.

104

This jump is responsible for the spike in the speciﬁc heat

which is also

observed to be accompanied by a jump.

The prediction ∆c = 0.0075((2 −

2b + t)/t)

−0.20Gi

1/3

(b −1 −t)(b/t

)

2/3

is about twice larger than the one

observed in Ref. 64. The LLL approximation should hold for 4Bξ

(0)/Φ

Gi or 1 − t − b < 2b. Extension of the theory including weak disorder has

been studied,

40,104

however it does not, at present day, provide a detailed

description of the Bragg glass (see below).

3.6. Superconducting phase coherence and decoupling

Thermal ﬂuctuations of the vortex positions lead to strong phase

ﬂuctuations:

δφ(k) ≈

2π

⊥

× u(k)

(34)

Since δφ(k) ∼ u(k)/k, it leads to suggestion of breakdown of superconducting

phase order for d ≤ 4.

106

There is still some debate on the validity and

consequences of this argument (see Refs. 4 and 6 for review). In particular,

the consideration of the gauge invariant phase

φ(x) = φ(x) −

2π

A · d`

led to the opposite conclusion that there is phase long range order in d = 3,

with cancellation of the divergences.

40,107

The question of the phase ﬂuctuations was much discussed for highly

anisotropic superconductors, described by the Laurence–Doniach model, a

generalization of GL

with discrete 2D layers indexed by z = n. The layers

are coupled via (i) the Josephson coupling energy −E

r cos



φ(r, n+

1, n)



with

φ(r, m, n) =

φ(r, m)−

φ(r, n) and E

= 



/πd (also noted J/ξ

(ii) the magnetic energy (the term B

). Glazman and Koshelev

computed,

within the London theory, the quadratic ﬂuctuations of the gauge invari-

ant phase. Their calculation suggests that the gauge invariant correlation

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

302 P. Le Doussal

ψ(x)

∗

)i with

ψ(x) = |ψ

φ(x)

does not always decay to zero along z.

Along z, they found that for B > B

∼ Φ

/γ

d the melting transition is

given by its 2D limit T

= T

0.62

8π

√

d

and the phase ﬂuctuations are

φ(0, n + 1, n)

i ∼ T/T

(B) with T

(B) ≈ T

/B)

1/2

. Hence they con-

cluded that for B > B

there is a range of temperatures T

(B) < T < T

where phase ﬂuctuations are large. This result, and the further Gaussian

self-consistent treatment

suggests the vanishing of critical current along z

at a decoupling transition T

dec

. In this transition, the eﬀective Josephson

coupling between layers vanishes while the lattice can be maintained by the

electromagnetic (EM) coupling between layers.

To predict more deﬁnitely a decoupling transition beyond the Gaus-

sian (self-consistent) approximation, one needs to (i) use the renormaliza-

tion group which predicts that the renormalized Josephson coupling ﬂows

to zero,

i.e. a decay of long range order in the phase, i.e. a decay of

hcos(

φ(r, n, n + 1)) cos(

φ(r, 1, 2))i at large n, while a ﬁnite Josephson cou-

pling is maintained locally between adjacent layers (and, in fact accounts

for the experimentally observed plasma resonance

) (ii) a true transition

requires topological defects. In the case of decoupling, these are interstitial

and vacancies in a single layer.

These defects have also been called quar-

tets of dislocations.

Indeed, in a layered superconductor one can think of

the vortices as pancake vortices in each layer, joined by a Josephson string.

The component of this string parallel to the layer is the Josephson vortex

or ﬂuxon. Such a ﬂuxon excitation can also exist as a loop in between two

layers, within which the phase diﬀerence between the layers varies by 2π. A

vacancy-interstitial VI pair in a single layer can then also be seen as a closed

Josephson loop connecting them. In the presence of a Josephson coupling J

these pairs are bound. At the decoupling transition these defects proliferate,

with loops of arbitrary sizes.

There are two limits in which the decoupling transition problem is sim-

pler. (i) When J → 0 and the layers are coupled only magnetically: then

there is an unbinding transition of the VI pairs, from a perfect VL where

phase coherence is maintained between the layers and a defected VL with

vacancies and interstitials where phase coherence is lost. In the presence

of pinning disorder this limit was studied in Ref. 83, (ii) the more isotropic

superconductor: one can keep the description in terms of vortex lines: a VI

pair then corresponds to a ﬂux line which wanders of order a in a height

z = d, or two ﬂux lines which exchange positions. It is thus an entan-

gled solid, the analogous of a super-solid in quantum problems.

In be-

tween these two extremes the topological transition merges with the above

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Novel Phases of Vortices in Superconductors 303

thermal decoupling once the Josephson coupling is ﬁnite, being two

anisotropic limits of the same transition,

108

at which superconducting or-

der is destroyed while the ﬂux line positional correlations are maintained.

4. From the Abrikosov Lattice to the Bragg Glass

4.1. Pinning disorder

Here we consider point disorder due to point defects perturbing the super-

conductor on a scale smaller than ξ. This is the case of oxygen defects in

high-T

materials, which are found to account for the pinning disorder.

4,109

The coupling energy to the disorder is modeled as:

dis

x˜ρ(x)

U(x) (35)

where x = (r, z) and ˜ρ(r, z) =

p(r − R

− u

(z)). Within the GL model,

one can consider uncorrelated δT

-disorder,

U(x) = |ψ

α(x) and a form

factor equal to p(r) = 1−|ψ

(r)|

from the solution ψ

(r) of the GL equation

for a vortex normalized to one at inﬁnity. One may also consider mean

free path disorder, i.e. random mass. Equation (35) can be rewritten as a

coupling to the vortex density ρ(x):

dis

xρ(x)U(x) , ρ(r, z) =

(2)

(r − R

− u

(z)) (36)

with, in Fourier U(q) = p(q

⊥

)

U(q). The correlator of the random potential

is then

U(x)U(x

) = R

(r − r

)δ(z − z

) (37)

where here and below

... denote average over the random potential, i.e. sam-

ple averages. Here R

(q) = ˜γ p(q

⊥

)

, which leads to R

(r) = γξ

k(r/ξ)

with k(0) = 1 and k(y) ∼ y

−2

ln y at large y. For simplicity, one often con-

siders R

(r) = dU

−r

where d is the layer separation, and U

typical

pinning energy per unit length along z. The correlation length of the disor-

der seen by a vortex is r

≈ ξ since this is the smallest length that it can

resolve. For a single vortex line one obtains:

dis

dz U(u(z), z) (38)

A phenomenological description in terms of a 3D density n

of independent

impurities with random individual 3D pinning forces of r.m.s f

can also

be used. If there are a few defects within the volume ξ

d, as typically for

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304 P. Le Doussal

oxygen vacancies, the resulting random potential is nearly Gaussian with

= γξ

= f

Pinning disorder deforms the vortex lattice and one ﬁrst describes these

deformations using elastic theory. The elastic description assumes small

nearest neighbor displacements, |u(x) − u(x + a)|  a and is a priori guar-

anteed to work at weak disorder and short scale (i.e. small system size).

Even for weak disorder, there is no guarantee that it holds at large scales,

and indeed we discuss later the possibility of more violent plastic deforma-

tions such as dislocations. An important correlation function, which probes

translational order (also called positional order) is deﬁned as

(x) =

iK·u(x)

−iK·u(0)

i (39)

measured in practice as a translational average over a large given sample.

4.2. Larkin Ovchinnikov theory

The ﬁrst assault on the Abrikosov lattice came from Larkin who carried

perturbation theory in the disorder

and found that the deformations of

the vortex lattice must grow with scale in space dimension d ≤ 4. His more

involved original calculation can be summarized in terms of the beautifully

simple, random force Larkin model:

F = F

[u] −

xf(x)u(x) (40)

which assumes a linear coupling of the displacement ﬁeld to the disorder.

Since this is a quadratic energy it can be solved for any realization of the

disorder:

(q)i = Φ(q)

−1

αβ

(q) (41)

where Φ(q) is the elastic matrix (12 and 14). For a short range isotropic

correlated random force f(x) of variance W , with

f(x)f(x

) = W δ

(x −x

one ﬁnds, upon disorder averaging:

(−q)u

(q)i =

(q)

(42)

with γ = L, T the longitudinal and transverse components. For isotropic

elasticity Φ(q) = cq

and at T = 0 this leads to:

B(x) =

(u(x) − u(0))

= 4W

q,BZ

(1 − cos(qx)) =

b(x)

(x) = e

−K

B(x)/4

, b(x) ∼ |x|

2ζ

, ζ

4 − d

(43)

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Novel Phases of Vortices in Superconductors 305

hence relative distortions grow as u ∼ |x|

1/2

in 3D and translational order

decays exponentially. Note that thermal ﬂuctuations are subdominant.

Although solvable and instructive, this linear model has seriously un-

wanted features (i) it is not invariant by a lattice translation u(r) →u(r)+a

(ii) if an external force is applied, the vortex lattice ﬂows freely, i.e. there

is no barrier to motion. One may wonder why it is relevant at all to the

problem. It was understood by Larkin that linearizing the random potential

seen by the vortices is a valid approximation for scales smaller than a charac-

teristic length R

. Since the random potential is smooth and varies on scale

, one expects this approximation to break down when the deformations

reach u ∼ r

, i.e. at scale such that

B(x = R

) ∼ r

which, from (43) gives:

= (c

/W )

1/(4−d)

(44)

a scale which, for weak disorder W  W

= c

4−d

can be much larger

than a. The case R

< a, called single vortex pinning, is mentioned below.

From Sec. 4.1 one estimates W =

(K) for the 3D vortex lat-

tice, as W ≈

trR

U00

(0) ∼ dU

when r

 a, using a

−2

≡

that case.

With remarkable insight, Larkin and Ovchnnikov

(LO) understood that

the scale R

also determines the threshold depinning force. In their picture,

the system breaks into domains of size R

which are independently pinned.

This leads to the collective pinning theory. To unpin the system one must

apply a total external force F R

on a Larkin domain at least equal to the

typical pinning force acting on the same domain, which scales as

xf(x) ∼

1/2

d/2

. This leads to the LO estimate for the depinning threshold force

per unit volume:

≈

1/2

d/2

(45)

the last equality means that the elastic, disorder and external force energies

within a Larkin domain are of the same order, deﬁned as U

= cr

d−2

∼

1/2

d/2

∼ F

. Although sometimes termed “non-universal physics”

since it occurs at short scale, the validity of (45) is by now proved by

more sophisticated methods (see below) and has been incredibly useful in

numerous experimental realizations of pinned systems. In superconduc-

tors, it determines quite well the magnitude of the critical current density

= cf

/Φ

= cF

/B.

110

For the triangular VL in d = 3 the deformation splits into

B(x) =

(x)+

(x) and there are two Larkin lengths, R

in ab plane, and L

along z. The

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

306 P. Le Doussal

dominant contribution arises from transverse modes:

(x)= 2W

q,BZ

(1−cos(qx))

⊥

(q)q

)

≈

3/2

1/2

b(ρ) , ρ=





1/2

for large enough R

(weak disorder) when one can neglect non-locality and

use c

= c

(0). In that case b(ρ) ∼

4π

ρ and R

is given by (44) with

→ c

3/2

1/2

. The Larkin length along z is L

= R

∼

For R

< λ/ one cannot neglect non-locality: since c

∼ 1/q

⊥

one ﬁnds

much slower logarithmic growth, i.e. b(ρ) → λ



a



. The Larkin length

thus varies exponentially with the disorder for a < R

< λ/ and one has

= R

/a.

The energy scale U

∼ r

determines the critical force

per unit volume F

and the critical current j

as:

B ≈ c

(46)

The above arguments also apply to a single ﬂux line, setting d = 1

and c = e

and f (z) = ∇U (0, z) in (38). This gives the single vortex

Larkin length (along the ﬁeld direction z), L

= (e

)

1/3

, with

∇

(0) ≈ γ, leading to f

= (Φ

/c)j

≈ e

/(L

)

(in case

of a dispersive line tension the value e

∼ 1/L

) should be used). The

interactions with the neighboring vortex lines can be neglected if L

< a.

One checks from above and the previous sections that the condition L

< a

is equivalent to R

< a, that the term c

⊥

 c

(q)q

for q

⊥

∼ K

and

that j

is then determined by single vortex pinning.

To summarize, the LO theory shows that disorder makes the Abrikosov

vortex lattice unstable. It does not predict however what is the new phase

of vortex matter at scales larger than the Larkin length R

. Beyond that

scale one expects multiple metastable states with barriers between them,

and to describe the system one must use modern tools which have emerged

in nonlinear physics of glasses and glassy systems.

4.3. The Bragg glass

To describe the vortex system beyond the LO theory one must study the

full model:

F [u] = F

[u] + F

dis

[u] (47)

As the Larkin model, it leads to a competition between the elastic energy

(12) — which tends to reduce deformations — and the disorder energy (36)

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Novel Phases of Vortices in Superconductors 307

— which tends to favor them — but where now the full nonlinear dependence

in u(x) is kept. The third ingredient is the periodicity of the lattice, i.e. the

exact invariance of F [u] by a lattice translation u

(z) → u

(z) + a

. This

turns out to be crucial: since there is no energy gain to shift the lattice

by a

, large deformations will not be favored, by contrast to Larkin model

which, it turns out, overestimates them. As discussed below, this simple fact

leaves room for the existence of a Bragg glass.

Since periodicity is so important, we now examine the coupling of a pe-

riodic object to a random potential. To simplify the model (47) we need to

trade the vortex position variables u

(z) for a smooth continuum deforma-

tion ﬁeld u(r, z). Since the density however is not a smooth function, this

raises some subtle issues.

4.3.1. The model: coupling a lattice to disorder

It turns out that the density of a periodic object can be written:

ρ(x) ≈ ρ

1 − ∂

(x) +

K6=0

iKx−iKu(x)

(48)

where ρ

is the average density, in terms of the smooth ﬁeld u(x) =

q,BZ

u(q)e

iqx

such that u(R

, z) = u

(z). In the absence of topological de-

fects an exact version of this decomposition can be given,

47,111

introducing

a labeling phase ﬁeld φ(R

+ u

(z), z) = R

, from which corrections to (48),

mainly higher gradients, can be computed. The gradient term is the change

in local density due to compression. For a slowly varying u(x) it contains

Fourier components only near q ≈ 0, while the other terms oscillate fast,

with the periodicity of the lattice, i.e. q

⊥

≈ K. They correspond to a trans-

lation of the crystal by u, with little change in the averaged density over a

period, and can be viewed as a local shift in the phase of the periodic density

wave, as in charge density wave theory, where usually a single K is retained.

The coupling to the random potential (36) thus splits into two very diﬀerent

contributions

47,50,112

(i) Long wavelength disorder: The modes of the disorder U (q) with small q

give rise to a coupling, in general dimension d:

dis

q ∼0

= −ρ

(x)∇ · u(x) (49)

where U

(x) =

q≈0

U(q)e

iqx

, which mostly compresses or dilates the lattice

locally. Although reminiscent of Larkin’s model it is down by a gradient and

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308 P. Le Doussal

is thus much less eﬃcient in destroying the lattice. For point impurities and

standard elasticity, it results (by itself) in displacements u ∼ L

2−d

which are

unbounded only for d ≤ 2 (and as ∼ ln L in d = 2). Furthermore, it does

not lead to pinning, since it is invariant by an arbitrary global translation of

the lattice. It is a particular case of the random stress disorder with linear

coupling energy F ∼ σ

(r)u

(r) to the strain matrix, a type of disorder also

generated by internal disorder, e.g. a ﬁxed connectivity crystal with atoms

of diﬀerent sizes.

103,113

In d = 2, it generates topological defects only above

a critical strength, if alone,

114

or for any strength if combined with pinning

disorder

115,116

(see below). It can however be neglected in d = 3.

(ii) Pinning (or commensurate) disorder : More dangerous are the modes

q≈K

close in periodicity to the lattice, which couple as:

dis

q ∼K

= ρ

−K

(x)e

−iKu(x)

(50)

where U

(x) =

q≈K

U(q)e

iqx

. Their eﬀect is to shift locally the crystal.

As a result, an important new length scale appears, R

, deﬁned as the scale

over which u varies by ∼ a. It can be estimated as follows. The phase u,

roughly constant on scale R

, adjusts to cancel the phase of sum of random

complex numbers −

(r) over a volume R

. A complex plane ran-

dom walk argument yields that the gain in energy density from disorder as

∼ρ

(K)

1/2

d/2

. On the other hand, the cost in energy density of such

a deformation ∼a on scale R

scales as ∼c(a/R

)

. The optimum occurs at

a length scale:

∼ (c

(K))

1/(4−d)

(51)

also called the Fukuyama–Lee length and was studied in the context of charge

density waves,

112

a particular case of a more general argument for disordered

system.

117

Note that in systems where r

 a and R

(K) ∼ e

−r

pinning

disorder can be much suppressed. This occurs in some 2D systems (such as

electrons on the surface of helium).

Keeping only pinning disorder (50), and using

(x)U

−K

) ≈

(K)δ

K,K

(x − x

), the model (36) can thus be replaced by:

F [u]=F

[u]+

xV (u(x), x) ,

V (u, x)V (u

, x

)= R(u−u

)δ

(x−x

) (52)

where, for a periodic object the correlator of the disorder reads:

R(u) =

K6=0

(K) cos(Ku) (53)

hence it is a periodic function with the periodicity of the lattice.

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Novel Phases of Vortices in Superconductors 309

It turns out that (52) has emerged as the universal model for pinning,

i.e. depending on the choice of space dimension d, number of components

N of u(x), and function R(u) it can model a variety of disordered elastic

systems — without topological defects. For d = D − 1, N = 1, it models

a magnetic interface (without large scale overhangs) in a D-dimensional

magnet with either random bond (RB) type disorder (a short range function

R(u) ∼ e

−u

) or random ﬁeld (RF) (a long range function R(u) ∼ |u|),

contact line of ﬂuids d = 1, N = 1 and RF disorder, as well as various

periodic systems R(u) periodic such as charge density waves N = 1, magnetic

bubbles (d = 2, N = 2) and of course vortex lattices in d = 3, N = 2 or in

d = 2, N = 2 (ﬁeld perpendicular to plane) N = 1 ﬁeld along the plane.

To handle disorder, it is customary to use replica, i.e. introduce n copies

of the displacement ﬁeld u

(r), a = 1, · · n and average their Gibbs mea-

sure over disorder, deﬁning the replicated Hamiltonian through e

−βF

rep

[u]

−β

F [u

]

. One can then show that all disorder averaged correlation

functions can be expressed as suitable averages of u

, e.g.

hu(x)u(x

)i =

(x)u

rep

in the limit n = 0. For model (52)

rep

[u] =

a=1

] −

a,b=1

R(u

(x) − u

(x)) (54)

where the disorder term is highly nonlinear and couples the replica. Ex-

panding (54) to quadratic order at small u

− u

< r

one nicely recovers

the Larkin model, in its replicated version: the Larkin regime x < R

thus

exists for all pinned systems.

The Bragg glass, however, is characterized by two scales R

and R

and that leads to an interesting scale dependence. When H  H

(and

temperature is not too high) the correlation length of the disorder r

 a,

thus R

 R

. In that case the function R(u) is the sum of many harmonics

and in fact, for r

 u  a it looks like a short range function, e.g. if

∼ e

−K

then R(u) ∼ e

−u

/2r

, obtained by replacing the sum in

(53) by an integral. Thus one may expect, as conﬁrmed below, that in the

range of scales R

< x < R

, the so-called “random manifold” regime, the

problem looks the same as the N = 2, d = 3 random bond (non-periodic)

elastic manifold: relative displacements between vortices being < a, each

vortex sees in eﬀect an independent disorder from its neighbor. Only at

scales x & R

does it feel the periodicity of the system and one expects

another behavior.

There are at present two main analytical methods to treat the pin-

ning model (52) and its replicated version (54), and to describe the vortex

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

310 P. Le Doussal

lattice beyond R

. We ﬁrst describe each in some generality, giving only the

main idea. Both predict that the pinning problem is characterized by two

exponents:

B(x) =

h(u(x) − u(0))

i ∼ x

2ζ

, F

≡ F

− F

∼ R

2θ

(55)

where ζ describes the roughness of the pinned elastic object, and θ = d −

2 + 2ζ the sample to sample ﬂuctuations of its free energy F , with only a

few universality classes depending on d, N, the type of disorder, i.e. of R(u),

and the type of elasticity (local, non-local, . . .). Finally, we present the main

results speciﬁc to the Bragg glass.

4.3.2. Variational method: a mean ﬁeld theory

The idea,

47,51,118,119

of the Gaussian variational method (GVM) is to look

for the best trial Gaussian Hamiltonian F

which approximates (54):

[Φ

αβ

(q)δ

+ σ

αβ

(q)u

(−q) (56)

By extremization of the variational free energy F

var

= F

+hF

rep

−F

one

obtains the saddle point equation for n ×n matrix of variational parameters

. This method is known to give a reasonable approximation in the case

of the pure one component Sine Gordon model, predicting a ﬂat (massive)

phase in d = 3, and the roughening transition in d = 2. Here, two types of

solutions exist. The simplest one of the form σ

= σ

+ σ, mimics the

distribution (thermal and over disorder) of each displacement mode u(q) by

a simple Gaussian, and leads to Larkin model behavior. The second type,

which is a better variational solution above the Larkin scale, i.e. for system

size R > R

, exhibits spontaneous replica permutation symmetry breaking

(RSB). This feature corresponds to the phase space breaking into separate

ergodic components and was found for the ﬁrst time in the mean ﬁeld de-

scription of spin glasses. In a nutshell, the RSB solution approximates the

distribution of displacements by a (hierarchical) superposition of Gaussians

centered at diﬀerent randomly located points in space

118

: all modes u(q) are

in eﬀect decoupled and in a given sample their Gibbs distribution takes the

form (for isotropic elasticity Φ(q) = cq

P (u(q)) ∝

−f

(i)

−

−2

)|u(q)−u

(i)

(q)|

(57)

Each mode thus ﬂuctuates around preferred conﬁgurations u

(i)

(q),

i.e. metastable states with an associated free energy f

(i)

. One recovers qual-

itatively the LO picture: the Larkin length R

sets the internal size of the