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

Novel Phases of Vortices in Superconductors 311

elastically correlated domains. In the simplest type of RSB solution, there is

only one level to the hierarchy: the f

are distributed according to an expo-

nential distribution P (f ) ∼ exp(u

f/T ), where u

= T /T

, and the u

(i)

(q)

according to:

P (u

(i)

(q)) ∼

−

(i)

(q)|

(58)

where T

is a temperature scale corresponding to thermal depinning. This

so-called one-step RSB holds for a periodic object in d = 2: there, the elastic

model (with topological defects forbidden) presents a genuine glass transition

at T

, the glassy analog of the roughening transition (in a real 2D crystal

however, this transition is usually preempted by melting). For the vortex

lattice in d = 3 however, no such transition exists and one needs the inﬁnite-

step RSB solution. There the states u

(i)

(q) are organized in a hierarchical

manner (clusters of states inside larger clusters and so on) generalizing the

above construction. The GVM gives the correlation function as

118

hu(−q)u(q)i =

Φ(q)



1 +

[σ](v)

Φ(q) + [σ](v)



∼

q→0

d+2ζ

(59)

where [σ](v) ∼ (v/T )

2/θ

at small v, with θ = [d−2+2ζ] and Z ﬁnite as T → 0.

For the “random manifold” problem, i.e. a short range correlator R(u), and

Φ(q) = cq

, the GVM predicts ζ = ζ

4−d

4+N

. This is the same result as

the “Flory approximation” which implies the balance u

d−2

∼ u

−N/2

d/2

of elastic energy and disorder energy in (52) estimated from a dimensional

estimate R(u) ∼ δ

(N)

(u). It predicts ζ = 3/5 in d = 1, N = 1 a case where

ζ = 2/3 is known to be exact. Being a mean ﬁeld method, the GVM cannot

be expected to be exact, but it does provide a quite consistent qualitative

picture and reasonable approximations. For a periodic object it correctly

predicts ζ = 0 and an amplitude Z not so far oﬀ from the correct value (see

below). The extension of the GVM to the dynamics provides an interesting

approach to aging phenomena

121

but is not at present the most convenient

or accurate method to describe creep and depinning.

122

We now turn to the more powerful FRG method, which is exact in a

d = 4 − ε expansion and treats more accurately the coupling between the

modes.

4.3.3. The functional renormalization group (FRG)

The method, pioneered in Refs. 47 and 120, is presented here it in its modern

form.

123–125

There, the central object, the renormalized disorder correlator,

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

312 P. Le Doussal

is not an abstract but a perfectly measurable quantity. To deﬁne it, imagine

a gedanken experiment where one puts the elastic system of size R in an

harmonic well:

V,w

[u] = F

[u] +

x V (u(x), x) +

(u(x) − w)

(60)

and denote u(x, w) the conﬁguration which minimizes F

V,w

and u(w) =

−d

x u(x, w) its barycenter. The role of the harmonic well is to ﬁx

the global translation mode of the system at u(x, w) = w in absence of

disorder, and, on average, at

u(x, w) = w in presence of disorder. It adds to

the elastic energy and provides an additional length scale R

such that the

system feels the well only on scales x ≥ R

: in the simplest case of local

isotropic elasticity cq

→ cq

+ m

and R

√

c/m.

A remarkable feature is that the minimum energy conﬁguration u(x, w)

experiences discontinuous jumps as w is varied. These jumps, also called

shocks, occur each time a distinct local energy minimum becomes more

favorable. The total force density exerted by the harmonic spring f =

(w − u(w)) is zero on average but its ﬂuctuations are measured by the

second cumulant

123

(u(w) − w)(u(w

) − w

)

= R

−d

∆(w − w

) (61)

which deﬁnes the function ∆(w), called renormalized force correlator. For

notational simplicity we focus here on N = 1 (interfaces), but for an N

component displacement ﬁeld, ∆(w) → ∆

αβ

( ~w). The factor R

−d

is natural

there from the central limit theorem, as the eﬀect of the well is to split

the system into ∼ (R/R

)

regions which may be viewed as independent.

Accordingly, roughness u ∼ x

is expected to saturate for scales x > R

This leads to deﬁne a rescaled force correlator through:

∆(w) = A

2ζ−(4−d)

∆(w/R

) (62)

where A

is a known constant. The main result of the FRG is that ∆(w)

satisﬁes a nonlinear diﬀerential equation, which can be put in the form:

−m∂

∆(w)=(4−d−2ζ)

∆(w)+ζw

∆

(w)−



∆(w)

−

∆(0)

∆(w)



n≥2

[

∆]

(63)

for N = 1, and similar generalizations for N > 1, where the nonlinear

term can be computed to successive orders β

∼ O(

∆

n+1

) in a so-called

loop expansion. Hence the original complicated, many degree of freedom

pinning problem has been reduced to solving a (relatively) simple diﬀerential

equation.

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

Novel Phases of Vortices in Superconductors 313

Analyzing Eq. (63) reveals striking features. First, one ﬁnds that in the

limit of interest R

large (m → 0)

∆(w) ﬂows to some ﬁxed point (FP)

∆

∗

(w) which is uniformly O(ε) = 4 − d) near d = 4, hence justifying the

small

∆ expansion in (63), and justifying the form (62) in the small m limit.

One ﬁnds several possible FP,

47,120

each associated to a distinct value of

the roughness exponent ζ and corresponding to the diﬀerent universality

classes of the pinning model (52). The most striking feature is that these

FP are non-analytic and exhibit a linear cusp

∆

∗

(w) −

∆

∗

(0) ∼ |w| at small

w. In fact, this feature becomes pronounced at scale R

≥ R

, the Larkin

length, and directly follows from the fact that u(w) exhibits jumps S

u(w

) −u(w

−

) at some discrete set of w

. More precisely one shows

126

from

(61) that the right derivative is:

|∆

)| = m

i/2hSi (64)

i.e. it contains information on the moments of the distribution P (S) of jump

sizes.

Until now, we focused on the minimal energy state, but the method can

be extended to Gibbs equilibrium at non-zero temperature T > 0, simply

replacing everywhere u(x, w) → hu(x)i

V,w

i.e. thermal averages at ﬁxed V

and w. The resulting FRG equation is discussed in Sec. 4.4(c).

It can also be extended to the dynamics. There w is changed with time,

e.g. w = vt. The limit v = 0

at T = 0 allows to study the depinning tran-

sition. There, as w increases adiabatically, the conﬁguration u(r, w) follows

a sequence of metastable states: at a given w the system is stuck in a local

energy minimum, distinct in general from the global minimum. When w is

increased, nothing happens until the barrier seen by the system vanishes at

some w

. At w = w

the system jumps to the next metastable state, and

so on. The attentive reader will guess that the average force with which the

harmonic spring drives the system is now non-zero and behaves as:

w − u(w) = F

(m) ≈ F

− C m

2−ζ

dep

(65)

at small m, where F

is the depinning threshold force density in the absence

of the harmonic well. Equation (61) still deﬁnes the renormalized force

correlator, and (64) also holds for these dynamical jumps, called avalanches.

Thanks to the FRG one can again compute ∆(w). Curiously it is found to

obey, to lowest (one-loop) order, the same Eq. (63) as for the statics, the

diﬀerence only starts in the two-loop term

125

[

∆] =



(

∆

(w)

− λ

∆

)

)(

∆(w) −

∆(0))



(66)

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

314 P. Le Doussal

with λ = 1 for the statics and λ = −1 for depinning. Solving for the FP one

ﬁnds ζ

= 0.20829804ε + 0.006858ε

+ ·· and ζ

dep

(1 + 0.14331ε + ··) for

the roughness of non-periodic objects with short range disorder (e.g. an Ising

interface with random bond disorder) in equilibrium and at the depinning

transition, respectively.

Hence thanks to the FRG, the function ∆(w) deﬁned in (61) has been

computed order by order in the ε = 4 − d expansion. It has also been

measured in numerical simulations for interfaces, both for the statics and for

the depinning transition.

127

The agreement was found remarkable. Recently,

∆(w) was measured in wetting experiments on contact line depinning,

128

and should be accessible in cracks and magnets. Whether it can be directly

measured in experiments on vortex systems remains open.

Knowing ∆(w) the FRG method allows to compute other quantities in

an expansion in

∆

∗

, i.e. in a ε = 4 − d expansion, such as the correlation

function

u(−q)u(q) = q

−(d+2ζ)

f(qL

), the avalanche size distribution P (S),

or the dynamical exponent z. It also allows to obtain the v–f characteristics

above and near the depinning transition, as v ∼ (f − f

)

where β = (z −

ζ)/(2 −ζ), with generally β < 1 for the depinning of an elastic object. This

diﬀers from the conventional picture, (3) and Fig. 2, and leads to the T = 0

characteristics of Fig. 14.

Let us close by indicating that while the setting using a harmonic well

makes the FRG well deﬁned, and allows e.g. reliable higher loop calculations,

if one only wants one-loop approximation the Wilson shell method, where

the small scale degrees of freedom Λ

`+d`

< q < Λ

= Λe

−`

are iteratively

eliminated, can also be used.

120

One recovers (63) with −m∂

→ ∂

4.4. Correlations and dynamics in the Bragg glass

We now present the main properties of the vortex lattice in presence of

pinning disorder which can be obtained using the above methods and the

resulting physical picture for the Bragg glass phase.

(a) Quasi-long range order:

Let us ﬁrst discuss the static, i.e. equilibrium correlations:

αβ

(x) =

h(u

(x) − u

(0))(u

(x) − u

(0))i (67)

The Gaussian variational method (GVM), complemented by the FRG, leads

to three main regimes in the case r

 a and allows to describe the

crossover between them.

For simplicity, we start with isotropic elasticity

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

Novel Phases of Vortices in Superconductors 315

Φ(q) = cq

, where

αβ

(x) =

αβ

B(x) and describe below the more realistic

case.

(i) Larkin regime

B(x)  r

(T ): The results of Sec. 4.2 are recovered. In

addition, although thermal ﬂuctuations are subdominant in (48), they also

average out the disorder locally, as captured by the variational method which

in eﬀects replaces r

→ r

(T ) = r

+hu

i, i.e. the thermal width of the vortex

line, leading to an eﬀectively temperature dependent Larkin length R

(T ), a

useful concept to study thermal depinning phenomena (discussed in details

e.g. in Ref. 4).

(ii) Random manifold regime r

(T )



B(x)  a

: There

B(x) ≈

(x/R

)

2ζ

, i.e. displacements grow as a power law with ζ = ζ

≈ 0.2

(ζ

) for 3D vortex lattices (N = 2). The decay of the translational or-

der in that regime is a stretched exponential C

(x) ∼ e

−Cx

2ζ

. This regime

was anticipated in Refs. 48 and 51.

(iii) Asymptotic regime

B(x) ≥ a

, i.e. x > R

. The periodicity of the

lattice becomes important and one ﬁnds that displacements should grow

only logarithmically at large scale.

50,129

More precisely

47,119

B(x) ≈

2NA

ln |x| (68)

with η = A = 4 −d in the GVM and K

the lowest reciprocal lattice vector.

Within the GVM the decay of translational order is found to be a power

law:

(x) ∼

|x|

(69)

as conﬁrmed by the one-loop FRG, which gives η = A =

ε ≈ 1.096

and for the N = 2 triangular lattice found within 5% of that value.

more accurate one loop estimate for the triangular lattice, A = η = 1.143,

was later obtained.

130

Higher loop corrections to (68), computed only for

N = 1:

125

A = ε(1 +

+ ··), are likely to increase again this result slightly.

The conclusion is thus that the translational order decays as a power law,

quite reminiscent of the quasi-long range order exhibited by 2D solids. The

value of the exponent η ≈ 1 means that the structure factor:

(q) =

iqx

(x) ≈

iqx

−

αβ

(x)

∼

3−η

(70)

exhibits divergent Bragg peaks

as q → 0 (it does so as long as η < 3).

Note that for r

 a, the scale R

is obtained by matching

B(x) between

regimes (ii) and (iii), which predicts R

∼ R

(a/r

)

1/ζ

. The variational

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

316 P. Le Doussal

method thus reproduces the Flory approximation result R

∼ R

(a/r

)

which is also what (51) predicts for K = K

. In principle, the FRG allows

to predict more accurately the crossover function with the correct value for

ζ but this has not yet been attempted. When r

(T ) ∼ a (high ﬁelds or

temperatures) the crossover occurs directly from the Larkin regime to the

asymptotic one, with R

∼ R

These results are not much changed by considering realistic elasticity ma-

trix Φ

αβ

(q) for the vortex lattice. The variational method allows to compute

the correlation

αβ

(x) = P

αβ

(r)

(x)+P

αβ

(r)

(x), which takes the form,

in the limit c

 c

L,T

(r, z) =

L,T





, (71)

with L

, and R

∼ a

3/2

1/2

/ρ

, where ρ

= a

−2

is the

average vortex density. It can also be written in a form similar to the Larkin

length, as R

= a

3/2

1/2

BrG

where the disorder parameter W

BrG

much weaker than W deﬁned in Sec. 4.2, as W

BrG

= W (r

/a)

, leading to

the Flory approximation R

= (a/r

)

. If we use local elastic constants

∼1/a

it leads to R

∼ 1/a

while R

∼ a

, hence as the ﬁeld is increased

the Larkin length increases, while R

decreases leading eventually to the

instability of the Bragg glass at some “disorder induced” melting ﬁeld. The

qualitative statement that “increasing the ﬁeld increases the disorder” is

thus justiﬁed in the Bragg glass regime (for R & R

): scaling the ﬁeld as

u = a˜u, then results in a theory with a-independent elastic constants and

a appearing only as ρ

in the disorder strength in front of the cosine term

in (53).

The in-plane anisotropy of the deformations is measured by the ratio

σ =

(r, 0)/

(r, 0) in decoration experiments in Fig. 14 (left). It is

predicted to cross over, as r increases, from the universal value

σ = 2ζ

1 ≈ 1.4, in the RM regime, to σ = 1, i.e. isotropic displacements, at large

distance.

Within the one loop FRG the exponents ζ

and η are found

to depend slightly on the ratio c

, however the eﬀect is very weak

130

(smaller than the two-loop correction): η varies between 1.143 and 1.159.

Finally, consideration of non-local elastic moduli yields further regimes and

crossovers at intermediate scales. If anything, the behavior c

∼ 1/q

⊥

for

⊥

> λ

−1

makes the Bragg glass more stable, i.e. the deformations grow more

slowly as

B(x) ∼ ln |x| in the Larkin and RM regimes, and as

B(x) ∼ ln ln |x|

for r > R

. Of course, the asymptotic regime (iii) remains unchanged at the

largest scales r > λ

and is dominated by local elasticity with c

≡ c

(0).

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

Novel Phases of Vortices in Superconductors 317

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0.4

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–1.5

–1 –0.5 0 0.5 1 1.5

0.10 T

0.61 T

0.73 T

1.00 T

1.40 T

ω = θ – θ

(deg.)

Neutron beam

Magnetic field

Sample

Vortices

I/F

(a.u.)

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C',&()"0"# &C,5) #E=P B=

Fig. 12. Top: (left) Neutron diﬀraction peaks in the vortex solid phase of a

(K,Ba)BiO3 superconductor at T = 2K (right) predicted angular dependence of

the intensity, taking into account the (ﬁxed) experimental resolution (called here

ξ, distinct from the coherence length). For R

> ξ, as R

decreases (H increases),

the peak width at half-maximum should remain constant ∼ 1/ξ. Bottom (right):

The diﬀracted intensity (rocking curves) indeed collapses without any broadening

above 0.7 T an evidence for the Bragg glass (left): L

≡ R

as a function of H

obtained from the maximum of the peak: resolution limited ξ ∼ 50a

at small

ﬁeld, it decreases near H

which coincides with the position of the second peak in

magnetization curves. From Ref. 74.

The prediction of divergent Bragg peaks was compared with experi-

ments,

131

and most directly veriﬁed

on (K, Ba)BiO

, an isotropic material

where no complications due to anisotropy or dimensional crossover can be

invoked. Taking into account the ﬁnite experimental resolution ξ, the di-

vergent Bragg peak (70) are predicted to take the shape shown in Fig. 12

(top right) if R

> ξ: (i) the peak width is determined only by ξ (ii) the

peak height allows to measure the positional correlation length R

. Thus

if disorder (i.e. here the magnetic ﬁeld) is increased, R

decreases and the

observed peaks should collapse without broadening. This is what they do in

Fig. 12 (bottom right), a direct evidence of the Bragg glass phase and its

algebraic positional order.

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

318 P. Le Doussal

(b) Topological order

The striking result that quasi-long range order survives was derived within

an elastic theory, assuming the absence of dislocations. However, it was

argued early on

45,129

that dislocation loops of size L could gain an energy

d/2

from disorder, larger than their core energy cost L

d−2

in d ≤ 4, hence

that they would always be generated even at weak disorder. The argument is

similar to the Imry Ma argument showing the destruction of a ferromagnetic

phase by random ﬁelds.

117

Although this estimate was implicitly based on

the Larkin model, it remained unchallenged, leading to the belief that the

elastic theory would hold only below some ﬁnite scale L

disloc

, beyond which

the lattice was destroyed, presumably leading to a vortex glass. Indeed,

such dislocations were always seen in early decoration pictures,

although

it was questioned whether these were a true equilibrium feature or simply

the result of the quench process (ﬁeld cooling).

131

These also suggested

some hexatic glass regime, where translational order decays exponentially

beyond L

disloc

, but disclinations being absent, a bond orientational order

could survive.

132

Ultimately however, and in absence of a locking eﬀect to a

lattice direction,

133

Larkin and Bragg glass type arguments would also apply

and perfect hexatic order would also be unstable and lead to quasi-order,

also potentially unstable to disclinations at large scale.

The energy argument was then re-examined in light of the more detailed

theory for lattice deformations beyond the Larkin scale.

One compares

the core energy cost ∼ L ln L of a dislocation loop in d = 3 on scale L, with

the cost of an elastic deformation, of the order of L

d−2+2ζ

, which can be

typically relaxed by allowing for a dislocation loop, provided u ∼ L

. So

if translational order is destroyed ζ > 0, i.e. if the Larkin or the random

manifold regime were true up to inﬁnite scales, it would indeed be favorable

to create dislocations. However, if quasi long range order persists ζ = 0 and

for weak disorder, dislocations may not be generated by disorder and the

elastic solution may thus be self-consistently stable. It was thus proposed

that a distinct, topologically ordered, phase with no large unbound disloca-

tion, the Bragg glass, survives at weak disorder, and that the vortex glass,

if it exists, would only occur at stronger disorder, i.e. higher ﬁelds in the

phase diagram. The original energy argument was strengthened in subse-

quent works

134,135

and received support from numerical simulations.

136

Note

that by contrast, d = 2 lattices (e.g. a single layer with H perpendicular to

the plane) are ultimately unstable to dislocations at large scales.

115,116

One

shows that pinning disorder, if present, renormalizes upward the long wave-

length disorder, which grows unboundedly and generates dislocations at a

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

Novel Phases of Vortices in Superconductors 319

Fig. 13. Decoration picture from Ref. 138 on BSCCO: the Delaunay triangulation

shows that each vortex has six neighbors and reveals the absence of dislocations.

scale L

disloc

∼ R

√

ln(R

/a)

∼ ae

∗

)

2/3

which however can be large for

large bare core energy E

, or weak disorder. A topologically ordered glass

however exists in d = 2 for lines directed along the plane, with however a

faster decay C

(r) ∼ e

−B(ln r)

of translational order than in the 3D Bragg

glass, and was studied in Ref. 137.

The most vivid illustration of what the Bragg glass looks like, i.e. a dis-

torted but topologically ordered lattice, is probably the decoration picture

shown in Fig. 13. In this later experiment non-equilibrium dislocations were

eliminated by tilting back and forth the external ﬁeld, and equilibrated re-

gions containing ∼10

vortices without dislocations were obtained in ﬁelds

of 70–120 G. The mean square distortions

L,T

(r, z = 0) were measured and

Fig. 14 shows a remarkable agreement with the predictions of the random

manifold regime (ii). The data also shows saturation of growth where it is

expected from the asymptotic regime (iii) but it was not possible in that

experiment to equilibrate the system at such large scales.

Although the Bragg glass retains the topological order of the Abrikosov lat-

tice, it diﬀers from it by being a glass, with many metastable states and

barriers to motion. At zero temperature, T = 0, this implies a non-zero

critical current j

(i.e. depinning threshold force f

) with a depinning tran-

sition, and the v − f or E − j curve represented in Fig. 14. At T > 0 ﬂux

creep occurs by thermal activation over barriers ∼ U

(j), even at very small

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

320 P. Le Doussal

longitudinal

(r )

and transverse

(r )

parts: B (r)

(r )" B

(r )

/2, where

# u„0…

•

. #1$

It is worth noting that in the random manifold regime, the

ratio of B

(r ) and B

(r ), is predicted to be 2

" 1,

and thus

an independent estimate of the roughening exponent. The

average value of this ratio measured from our data !inset to

Fig. 3#a$" is 1.40, which is consistent with the value of 2

obtained from B#r$. As shown in Fig. 3#a$, both B

and

(r ) are described well with the power-law behavior up to

r&L

. Beyond this range, however, the transverse displace-

ment B

(r ) ﬁrst deviates from power law causing deviations

in B#r$. Thus, we infer that shear motion of the FL lattice

should be responsible for the abnormal behavior of B#r$.

Since the shear modulus of FL lattice is much smaller in

magnitude than the compressional modulus,

(r ) is al-

ways larger than B

(r), and the shear motion dominates the

relaxation of the FL lattice during the ﬁeld cooling process.

As temperature decreases, the long-wavelength component

of shear motion is frozen out. We believe that the domainlike

structures seen in Fig. 1 are a snap shot of these frozen

long-wavelength shear motions. Note that the characteristic

length scale of these domainlike structures in Fig. 1 is again

, which explains the deviations in B#r$ for r $ L

Therefore, L

is the equilibrium length scale within which

FL’s can relax to the local equilibrium during our experi-

mental time scale.

This issue can also be addressed through Fourier space

analysis. Figure 3#b$ displays a blowup of one Bragg peak.

Several small satellite peaks appear around the relatively

sharp main peak; these satellite peaks indicate a large-scale

modulation of the FL lattice. If the FL’s were in equilibrium,

only one main peak should be expected. The corresponding

real-space distance between the main and satellite peaks is,

again, &L

. Hence, these satellite peaks provide another

evidence of the frozen-in dynamics beyond the equilibrium

length scale L

. In addition, we have prepared FL lattices in

different ways to address the nonequilibrium structures. For

example, we cooled the samples in the absence of a ﬁeld to

65 K, applied a ﬁeld 70 G, and then cooled slowly # 0.1

K/min$ to 4 K. Signiﬁcantly, we ﬁnd a similar density of

dislocations and FL structure compared to the rapid #10 K/s$

ﬁeld-cooled samples. Since 65 K is far below the melting

temperature,

this observation suggests that the nonequilib-

rium structures originate from the frozen-in dynamics far be-

low the melting temperature. Although we can probe FL’s up

to a length scale of &300a

, there is a much smaller length

scale L

that prohibits direct application of the theory de-

rived for equilibrium FL’s. Further studies should address

this important issue.

Finally, we consider the origin of dislocations observed in

our experiments, since the nonequilibrium vs equilibrium na-

ture of dislocations is critical to the existence of the Bragg

glass phase. We believe that our data, which exhibit a small

number of dislocations, in fact, favor the nonequilibrium na-

ture of dislocations in the FL lattice for the following rea-

sons. First, it is found that most dislocations are pinned in

between domain boundaries !see Fig. 1#b$ for example". If

there were a dislocation within the domainlike structures

where FL’s are locally in equilibrium, the dislocation should

be an equilibrium feature.

13,17

Second, L

% L

! n

# 1/2

&250a

, where L

and n

are the average distance between

FIG. 2. #a$ Mean-square relative displacement correlator B#r$

#open circles$ as a function of distance r calculated from the image

shown in Fig. 1#a$. The solid line is a power-law ﬁt: B (r )&r

0.44

#b$ Translational correlation function C

(r ) #dots$ calculated from

the same image. The open circles are a comparison with the Gauss-

ian approximation: C

(r )'e

# G

B(r)/2

FIG. 3. #a$ Transverse #open circle$ and longitudinal #solid

circle$ displacement correlators, i.e., B

(r ) and B

(r ), as a function

of distance calculated from Fig. 1#a$. The inset shows the ratio of

the two quantities. #b$ The Fourier space image showing all six

ﬁrst-order Bragg peaks. #c$ Detail of one of the Fourier peaks cal-

culated from the same image #inverted gray scale$. The dark arrows

highlight two satellite peaks.

RAPID COMMUNICATIONS

PRB 60 R12 591STRUCTURE OF FLUX LINE LATTICES WITH WEAK . . .

Fig. 14. Left: Mean squared deformation

L,T

(r, z = 0) as a function of r in the

decoration picture of Fig. 13. The exponent ζ = 0.22 of the random manifold regime

is the best ﬁt. Inset: Anisotropy ratio as a function of scale, in agreement with the

predicted value 1 + 2ζ. From Ref. 138. Right: v–f curve for an elastic object.

Elastic depinning v ∼ (f −f

)

at T = 0 (solid line) with β < 1, i.e. a vertical slope

at f

, and creep v ∼ e

−A/(T f

)

at small f and T > 0 (dashed line) with zero linear

resistivity. Both can be derived using the FRG theory.

j  j

(f  f

). The salient feature of a glass is the divergence of the

barrier U

(j) ∼ j

−µ

at small j. The classical argument

48,49

assumes that for

small j  j

, most of the time is spent near equilibrium and that the motion

proceeds via thermal nucleation of an excitation involving displacements of

order u(R) ∼ r

(R/R

)

 r

over an internal scale R, which costs a typi-

cal energy F

dis

∼ U

(R/R

)

. Taking into account the energy gain from the

external force, one sees that in order to move, the system must overcome

the barrier:

(f) ∼ max

dis

− fu(R)R

] , U

(f) ∼ U





µ =

d + ζ − θ

(72)

giv

en by the optimal nucleus of size R

opt

∼ f

1/(d+ζ−θ)

. This leads to the

creep law:

v ∼ v

−U

(f)/T

∼ v

−





µ =

d − 2 + 2ζ

2 − ζ

(73)

where

in the latter we have assumed that θ = θ

= d − 2 + 2ζ

, i.e. equi-

librium values for the exponents.

Although this argument has been quite successful, fascinating questions

remain, still mostly under investigation (i) do relevant barriers indeed scale