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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 321

as ﬂuctuations of energy minima, i.e. θ = θ

as assumed above, and is the

energy landscape characterized by a unique energy scale, (ii) is the use of

the Arrhenius law v ∼ exp(−U

/T ) for a single barrier fully justiﬁed or

does the distribution of barriers (eventually broad) have an eﬀect, (iii) after

the optimal barrier has been passed, what is the subsequent motion, is it

deterministic and looks like avalanches near T = 0 depinning? The closely

related question of the thermal rounding of the depinning transition is also

not fully understood.

139

The one-loop FRG in the presence of temperature

amounts to adding the term

∆

(u) in Eq. (63). In the presence of this term,

the correlator

∆(u) remains a smooth function in a very narrow boundary

layer u ∼

T ∼ T R

−θ

. The dynamical exponent t ∼ R

is controlled by the

curvature, z = 2 −

∆

(0) ∼ 1/

T , and blows up leading to diverging barriers.

The FRG thus leads to a derivation of the creep law with θ = θ

It also

allows to study the eﬀect of barrier distributions

140

and to predict that the

behavior at intermediate scale in the creep process is indeed governed by

T = 0 depinning.

Novel algorithms,

142

exploiting the so-called no passing

rule for interfaces,

141

have conﬁrmed that the roughness exponent is ζ = ζ

dep

in the critical region above the scale R

opt

(and below a much larger scale set

by the velocity).

Since for r

 a the Bragg glass is characterized by two distinct scales

and R

, the creep barrier crosses over from the random manifold (RM)

regime, U

(f) ∼ U

/f)

for f

 f  f

to U

(f) ∼ U

/f)

BrG

for f  f

with U

= U

)

and f

= (R

)

2−ζ

The asymptotic exponent for the divergence of barriers in the Bragg

glass is thus predicted to be µ

BrG

= 1/2, corresponding to ζ = 0 and

θ = d − 2.

47,50

Although in other pinned systems, such as domain walls in magnetic

ﬁlms,

143

or in ferroelectric ﬁlms,

144

the creep law has been nicely conﬁrmed

over many orders of magnitude, precise and unambiguous transport mea-

surements for vortices are a challenge. This is due to the many length scales

and complicated eﬀects in the spatial distribution of the current due to

surface barriers.

145

Indeed in BSCCO crystals near melting T

, the bulk

pinning is weak (an ideal situation to measure creep) and surface barriers

dominate. In a remarkable experiment this eﬀect was circumvented by po-

sitioning the contacts far from the edge.

146

The results are represented in

Fig. 15. It demonstrates a pure Arrhenius law for activated motion just be-

low the melting transition in the vortex solid. The measured barrier U

(j)

(inset) shows a dependence in current consistent with the exponent µ = 1/2

expected for the Bragg glass in its asymptotic regime.

October 7, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

322 P. Le Doussal

83 77 71 67 63 5659

-6

-5

-4

-3

1.2 1.3 1.4 1.5 1.6 1.7 1.8

R(T) / R(95K)

100 / T [100 / K]

T [K]

Figure 5, Fuchs et al.

30 mA

103.3

3 mA

strip

100 Oe

square

1 10 100

U [K]

I [mA]

Fig. 15. Arrhenius plot of the normalized resistance at 100 G and various currents

between 1 and 30 mA for a BSCCO strip. The dashed curve is the normalized

resistance of the strip at 3 mA for comparison. Inset: Activation energy as a

function of current in the vortex solid phase at applied ﬁelds of 100 G (dots), 300 G

(crosses) and 500 G (squares). Solid line: U = CI

1/2

. From Ref. 146.

5. Other Glasses

5.1. Vortex glass and gauge glass

It was proposed early on,

44,45

by analogy with spin glasses, that in presence

of disorder the Abrikosov lattice was replaced by a phase, called the vortex

glass, characterized by random but frozen phase coherence, with correlation

deﬁned as:

V G

(x) =

∗

(x)

ψ(0)i|

(74)

with non-zero limit C

V G

(x → ∞), where

ψ(x) = ψ(x)e

−i

2π

A·d`

to main-

tain gauge invariance. Besides some mean ﬁeld calculations,

152

additional

material was put on this proposal by considering the gauge glass model,

i.e. an XY model with random gauge. In this approach, more suitable to

strong disorder, the vortex lattice is fully ignored and the superconductor is

represented by grains with superconducting phase φ

at positions x

on the

sites of a d-dimensional lattice

F = −

hiji

cos



− φ

− A

−

a(x)dx





(∇ × a)

(75)

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

Novel Phases of Vortices in Superconductors 323

where A

is a random variable uniformly distributed in [0, 2π] and a de-

notes the ﬂuctuations of the vector potential which are limited by the bare

screening length λ

. This model was found numerically to exhibit a vortex

glass phase in d = 3 for λ

= ∞, with a continuous transition to a high

temperature phase.

This d = 3 vortex glass phase was, however, found

to be unstable at non-zero temperature in the more realistic case of ﬁnite

153

A continuous vortex liquid to glass transition can be described phe-

nomenologically assuming a single diverging length scale ξ ∼ |T − T

−ν

and time scale τ ∼ ξ

, as:

E ∼ ξ

−z−1

(jξ

d−1

) , σ(ω) ∼ ξ

2+z−d

(ωξ

) (76)

with e

(x) ∼ x and e

−

(x) ∼ exp(−a/x

) at small x, tailored to match lin-

ear response and creep, respectively. At the transition, consistency requires

E ∼ j

z+1

d−1

and σ(ω) ∼ (−iω)

(d−z−2)/z

, and above the transition, the resistiv-

ity vanishes as ρ ∼ (T −T

)

ν(z+2−d)

. Note that this phenomenological theory

does not say much about the detailed nature of the glass phase, and should

hold as long as the transition is continuous and scaling holds (look for a sim-

ple but instructive toy example Ref. 154). Many experiments were analyzed

using (76) leading to values ν = 1.1 ± 0.4 and z = 5.2 ± 0.6, however there

has been much debate on the interpretation in terms of a transition to a vor-

tex glass. Pinning by dislocation lines in ﬁlms, or by twin boundaries in bulk

samples, lead to a diﬀerent Bose glass phase (see below). Some of the more

clear-cut experiments have been done on the isotropic BaKBiO supercon-

ductor.

155

In BSCCO similar ﬁts fail to produce reasonable (i.e. universal)

values for the exponents.

156

5.2. Bose glass and splay glass

Correlated disorder occurs either naturally as dislocation lines or twin

boundary planes, or can be induced through irradiations by heavy ions.

The latter produce lines of damage in the material which act as columnar

defects very eﬃcient to pin the ﬂux lines, hence of technological importance.

Figure 16 shows the phase diagram of BSCCO and the irreversibility line

before and after irradiation producing columnar defects of density d

−2

and

matching ﬁeld B

= Φ

. The irreversible regime where the vortex solid

is pinned is considerably enhanced in all compounds,

and predicted to

form a distinct phase, the Bose glass.

The simplest description uses the

mapping from ﬂux lines to quantum mechanics. The partition function of

a single ﬂux line at temperature T obeys, as a function of its endpoint r,

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

324 P. Le Doussal

depinning in bulk

depinning in surface

first-order melting

Fig. 16. Left: Upward shift in irreversibility line in BSCCO after irradiations by

5.8 GeV Pb ions at various ﬂuences. A saturation of the maximum temperature

with d

−2

signals diﬀusion of two-dimensional pancake vortices between vortex lines

and lost linelike character of vortices. From Ref. 159 and courtesy C. J. van der

Beek. Right: A 3D thermally ﬂuctuating ﬂux line around a columnar defect map

onto a quantum particle in a 2D well.

the Schrodinger equation with imaginary time replaced by z and ~ by T .

Its binding energy to a columnar defect is obtained from solving the quan-

tum problem in dimension d − 1 for a particle in a well (see Fig. 16), which

yields U(T ) = U

f(T /T

∗

) with T

∗

= b

√

˜

and U

∼ 

/2ξ)

where

= max(c

√

2ξ) is the eﬀective width of the well and c

the defect radius.

Here ˜

is the local tilt modulus, see Sec. 3.3. It shows that a single vortex

line in a forest of random defects has bounded wandering δr(z) ∼ `

⊥

(T )

and obtains estimates of its localization length `

⊥

(T ) and (eﬀective) critical

current j

(T ) ≈ cU(T )/Φ

⊥

(T ). The many vortex line 3D problem maps

to repulsive bosons in a 2D random potential, known from numerics to ex-

hibit a Bose glass phase with ﬁnite localization length.

157

This mapping

together with numerics and phenomenological arguments predicts

(i) con-

tinuous Bose glass to vortex liquid transition with a diverging localization

length `

⊥

(T ) ∼ |T − T

−ν

⊥

with ν

⊥

≈ 1, `

∼ `

⊥

and anisotropic vor-

tex glass-like scaling at the transition,

(ii) divergent barriers in the glass

phase with two regimes of creep as E ∼ exp



−

/j)



for intermediate

= cU

/Φ

 j  j

where the nucleation barrier arises from a half-

loop kinks away from a single column, and E ∼ exp



−

/j)

1/3



at very

small j, when optimization of the nucleation barrier over many defects can

occur, by an extension of Mott’s variable range hopping argument for bosons,

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

Novel Phases of Vortices in Superconductors 325

(iii) transverse Meissner eﬀect: in the BoG phase the eﬀective tilt modu-

lus is inﬁnite and it requires a ﬁnite transverse threshold ﬁeld H

⊥

(T ) ∼

(T )/Φ

, with E

(T ) =

˜

U(T )d, to tilt the vortex lines away from

the columns and produce a non-zero bulk B

⊥

. Many of the predicted prop-

erties of the BoG have been checked in transport and magnetization ex-

periments.

92,160

Remarkable double-sided decoration experiments

158

show

localization and Coulomb gap eﬀects very much as in semiconductors. Sur-

prisingly though, detecting the transverse Meissner eﬀect has been a chal-

lenge.

161

Finally, in very anisotropic compounds, localization of pancakes

by columns can be studied from some discrete extension of the mapping

to quantum mechanics,

162

and phenomena such as recoupling by columnar

defects can occur.

One can also describe the Bose glass within an elastic theory, starting

from the vortex lattice and adding correlated disorder along z. The GVM

and the FRG can both be used,

163

very much as in Sec. 4.4 above, and also

lead to predictions of localization and transverse Meissner eﬀect. However in

the case of heavy ion irradiations, the columnar disorder is very strong and

immediately creates dislocations: the elastic description works then only in

some high temperature region just below the transition where the eﬀective

disorder is weakened and L

disloc

is large (e.g. R

∼ 30a

was estimated in

Ref. 90). In fact, if R

is large enough, the BoG-liquid transition should

become ﬁrst order, as is indeed observed for very small ion ﬂuence.

One

notes that the ground state with columnar disorder is identical, up to trans-

lations along z, to the ground state in d − 1 dimension with point disorder.

Hence, even weak columnar disorder in d = 3 produces dislocations at large

scales, and exponential decay of C

(r). A Bragg–Bose glass phase should

however exist in the d = 3 + 1 problem, which has no realization in real su-

perconductors, but could be realized in a genuine 3D quantum crystal with

substrate quenched point disorder.

It was proposed

that a moderate dispersion in the directions of the irra-

diation tracks would keep the vortices localized along the splayed columns,

leading to yet another glass phase, the splay glass, with further improved

transport properties: (i) the vortices being now entangled, the energy cost

of ﬂux cutting increases the barrier for activated motion, (ii) the vari-

able range vortex creep should be suppressed since the dominant excita-

tions occur within the families of similarly inclined tracks, leading e.g. to

E ∼ exp



−

/j)

3/5



. This was checked in experiments, which did show

cases of reduced relaxation, see Fig. 17, the eﬀect being more pronounced in

the less anisotropic compounds, such as YBCCO.

96,164

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

326 P. Le Doussal

VOLUME 76, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 APRIL 1996

FIG. 1. Cross-sectional TEM image of YBCO irradiated with

1.08 GeV Au, taken from a region at !9 mm from the entry

surface of a "19 mm thick crystal. This crystal was rocked

around an axis ! c (upper left sketch) by an angle Q ! 610.

The viewing direction is nearly along the axis of 610

rotation

(i.e., " along [010]).

half maximum (FWHM) of P

#Q$ was estimated from

the TRIM Monte Carlo calculations [8,10]. The ion energy

was "0.6 GeV at the entrance to crystals, setting the

projected range at "21 mm [10]. The energy deposition

rate was above the threshold for columnar track formation

(1.8 keV/Å) throughout the crystal thickness [8,10,11]. In

all cases, the total pin density corresponded to a dose-

equivalent matching ﬁeld B

! 3 T.

Figure 2 shows the temperature variation of persistent

currents J(T) for different conﬁgurations of columnar pins

in a ﬁeld of 1T applied along the c axis. J(H,T) was

obtained from the irreversible magnetization M(H,T) in the

critical state [12], measured with a SQUID magnetometer

[7,8]. The variation in J in crystals before irradiation was

less than 20% [2,8]. Inspection of a crystal with P

#65

and the same (virgin) crystal before irradiation shows that

after irradiation J #5 K$%2 3 10

A&cm

—an increase

by over an order of magnitude. The enhancement at

77 K is by about 4 orders of magnitude. A maximum in

, shown in the lower inset of Fig. 2, indicates that

the 65

planar splay supports the largest current, well

above the parallel #k$ conﬁguration. At the same time

the strong pinning range is nearly intact. Both the k and

pins shift the irreversibility line, H

irr

, towards high

temperatures in a predictable way [9], as shown in the

upper inset of Fig. 2. A slight downward displacement

FIG. 2. Persistent current density J(T) for YBCO crystals with

three splay conﬁgurations of columnar pins, each with B

3 T: planar splay (solid dots) with Q ! 65

, parallel tracks

(open circles), and a Gaussian distribution (solid triangles) with

FWHM of 64.47

. The best result is for the 65

planar

splay (also lower inset). J(T) for the same crystal before

irradiation (encircled stars) is also shown. H

irr

is shifted

to higher temperatures after irradiation (upper inset). J(T) for

the Gaussian splay is smaller than for the k pins above "16 K

(the arrow).

of H

irr

for the 65

splay, relative to P

, is expected

from the decrease of glass melting temperature due to the

average tilt [5].

Intuitively, one might expect a greater degree of en-

tanglement for a Gaussian splay [5], and thus the largest

. This is not observed. J(T) for P

#Q$ with 64.47

FWHM (Fig. 2) falls below parallel conﬁguration above

T ' 16 K. The detrimental effect of large-angle tails of

is suggested by the different ﬁeld dependencies of

J(H,T) for the Gaussian and planar (or P

) arrangements.

It is particularly striking at high temperatures. At 70 K,

J(H) for the parallel-pin and planar-splay arrangements for

sufﬁciently small #,45

$ angles, all give a distinct “dip” in

J(H) around zero ﬁeld [Fig. 3(a)], which articulates above

40 K and disappears near T

. J(H) for the Gaussian has

a spike-like feature near H % 0. Such shape difference

was seen in M(H) for H k defects and H misaligned with

defects by a large (60

) angle in the experiment of Civale

et al. [2] (Fig. 3 inset).

The effect on vortex dynamics is shown in

Fig. 3(b), which contrasts the thermal relaxation rates

S ! 2dlnJ&dlnt for the Gaussian, the 65

, and the k

pin conﬁgurations. S(H,T) was obtained from the time

evolution of M(H,T,t) from the critical state for times

60 , t , 7200 sec [13]. Below H " 1.5 T and outside

2564

VOLUME 76, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 APRIL 1996

FIG. 1. Cross-sectional TEM image of YBCO irradiated with

1.08 GeV Au, taken from a region at !9 mm from the entry

surface of a "19 mm thick crystal. This crystal was rocked

around an axis ! c (upper left sketch) by an angle Q ! 610.

The viewing direction is nearly along the axis of 610

rotation

(i.e., " along [010]).

half maximum (FWHM) of P

#Q$ was estimated from

the TRIM Monte Carlo calculations [8,10]. The ion energy

was "0.6 GeV at the entrance to crystals, setting the

projected range at "21 mm [10]. The energy deposition

rate was above the threshold for columnar track formation

(1.8 keV/Å) throughout the crystal thickness [8,10,11]. In

all cases, the total pin density corresponded to a dose-

equivalent matching ﬁeld B

! 3 T.

Figure 2 shows the temperature variation of persistent

currents J(T) for different conﬁgurations of columnar pins

in a ﬁeld of 1T applied along the c axis. J(H,T) was

obtained from the irreversible magnetization M(H,T) in the

critical state [12], measured with a SQUID magnetometer

[7,8]. The variation in J in crystals before irradiation was

less than 20% [2,8]. Inspection of a crystal with P

#65

and the same (virgin) crystal before irradiation shows that

after irradiation J#5 K$%2 3 10

A&cm

—an increase

by over an order of magnitude. The enhancement at

77 K is by about 4 orders of magnitude. A maximum in

, shown in the lower inset of Fig. 2, indicates that

the 65

planar splay supports the largest current, well

above the parallel #k$ conﬁguration. At the same time

the strong pinning range is nearly intact. Both the k and

pins shift the irreversibility line, H

irr

, towards high

temperatures in a predictable way [9], as shown in the

upper inset of Fig. 2. A slight downward displacement

FIG. 2. Persistent current density J(T) for YBCO crystals with

three splay conﬁgurations of columnar pins, each with B

3 T: planar splay (solid dots) with Q ! 65

, parallel tracks

(open circles), and a Gaussian distribution (solid triangles) with

FWHM of 64.47

. The best result is for the 65

planar

splay (also lower inset). J(T) for the same crystal before

irradiation (encircled stars) is also shown. H

irr

is shifted

to higher temperatures after irradiation (upper inset). J(T) for

the Gaussian splay is smaller than for the k pins above "16 K

(the arrow).

of H

irr

for the 65

splay, relative to P

, is expected

from the decrease of glass melting temperature due to the

average tilt [5].

Intuitively, one might expect a greater degree of en-

tanglement for a Gaussian splay [5], and thus the largest

. This is not observed. J(T) for P

#Q$ with 64.47

FWHM (Fig. 2) falls below parallel conﬁguration above

T ' 16 K. The detrimental effect of large-angle tails of

is suggested by the different ﬁeld dependencies of

J(H,T) for the Gaussian and planar (or P

) arrangements.

It is particularly striking at high temperatures. At 70 K,

J(H) for the parallel-pin and planar-splay arrangements for

sufﬁciently small #,45

$ angles, all give a distinct “dip” in

J(H) around zero ﬁeld [Fig. 3(a)], which articulates above

40 K and disappears near T

. J(H) for the Gaussian has

a spike-like feature near H % 0. Such shape difference

was seen in M(H) for H k defects and H misaligned with

defects by a large (60

) angle in the experiment of Civale

et al. [2] (Fig. 3 inset).

The effect on vortex dynamics is shown in

Fig. 3(b), which contrasts the thermal relaxation rates

S ! 2dlnJ&dlnt for the Gaussian, the 65

, and the k

pin conﬁgurations. S(H,T) was obtained from the time

evolution of M(H,T,t) from the critical state for times

60 , t , 7200 sec [13]. Below H " 1.5 T and outside

2564

Fig. 17. Left: TEM image of YBCO irradiated with 1.08 GeV Au ions in two

splayed family. Right: Persistent current density J(T ) for splayed conﬁguration

±5

(solid circles), Gaussian distribution (triangles) compared to parallel columns

(open circles). From Ref. 96.

5.3. Moving glasses

Finally, we now brieﬂy discuss moving phases of vortices. Consider the vor-

tex lattice undergoing overdamped dynamics under the action of the Lorentz

force produced by an external current, which, for simplicity we take as uni-

form in space. If j  j

, i.e. f  f

the system is in the ﬂux ﬂow regime

v ∼ f /η at large velocity. There, it is natural to argue that disorder is

weak, being averaged out by fast motion, hence the elastic description is a

reasonable starting point:

η(∂

+ v ·∇)u

(x, t) =

αβ

(x −x

, t) + f

(u, x, t) + f

−ηv

(77)

where vt + u(x, t) is the deformation expressed in laboratory frame coordi-

nates, and the pinning force is:

(u, x, t) = −

δF

dis

[u]

δu

(x, t)

= U(x)ρ

iK·(x−vt−u(x,t))

(78)

At large velocity (78) oscillates very fast and one can perform a weak disor-

der/large velocity expansion

165

and compute the deformations u in powers

of 1/v. This was done in Ref. 166 and it was concluded that above some

velocity the VL becomes a perfect crystal again at an eﬀective temperature

= T + T

, the eﬀect of disorder being described by an eﬀective shaking

temperature T

∼ 1/v

. This approach would suggest bounded displace-

ments in d > 2 and the absence of glassy properties in the moving solid.

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

Novel Phases of Vortices in Superconductors 327

The moving vortex lattice turns out to be a much more subtle and inter-

esting object. An examination of (78) immediately shows what is missing

167

a part of the pinning force along the direction (denoted by y) perpendicular

to motion, is static! This part comes from the contribution of the vectors

K = (0, K

), such that K · v = 0, and is not averaged by motion. Hence

(x, t) sees a resulting static random potential and as a result exhibits glassy

properties. This is also apparent in the weak disorder perturbation theory

which leads to infrared divergences for corrections to u

. These can again be

treated using the dynamical FRG on the equation retaining only transverse

deformations, the so-called moving glass equation

167

η∂

(x, t)+ηv·∇u

(x, t) = c∇

(x, t)+U(x)ρ

(y−u

(x,t))

(79)

Apart from the convective term, which retains the memory that the system

is moving and dissipates, it is very similar to the standard pinning equation

for a periodic elastic object. Analysis

167

shows that u

(x, t) is pinned, hence

(i) motion occurs along static channels directed along the average velocity,

in which the vortices ﬂow like beads on a string (see Fig. 18 right), (ii) these

channels are rough, hence translational order decays for d ≤ 3 (analogous to

Larkin’s d ≤ 4 for static, with here anisotropic scaling L

∼ vL

/c). In d = 3

smectic type quasi-long range translational order persists in the y direction,

(iii) there is transverse pinning, i.e. the system resists if an additional force is

applied transverse to motion, there are transverse barriers. Two dynamical

phases with these properties are predicted

167,168

(i) a moving Bragg glass

(MBrG) where topological order is retained in all directions (transverse and

longitudinal), (ii) a ﬂowing smectic glass (MSG), where dislocation exists

in between the channels (decoupled channels) but transverse (i.e. smectic)

topological order survives. All these predictions have been veriﬁed in numer-

ics.

169

Decoration in motion have shown ﬂow along channels

170,171

and both

predicted phases can be seen in the experiment in Fig. 18. At weak disorder

the BrG should depin elastically and transform continuously into the MBrG:

a creeping moving glass was seen in STM experiments.

172

Another signa-

ture of a topologically ordered MBrG, narrow band noise at the washboard

frequency ω = a/v, has been reported.

173

The detection of the transverse

critical force f

in vortex lattices has been a challenge, although it was re-

ported in superconducting metal glasses Fe

1−x

174

It has been neatly

observed in Wigner crystals.

175

CDW in presence of a transverse current is

also described by Eq. (79) and also exhibits the moving glass physics.

176

The fact that moving phases may be glassy and exhibit barriers came as

a surprise. In fact, one shows that with point disorder the MBrG and MSG

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

328 P. Le Doussal

Nature © Macmillan Publishers Ltd 1998

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(a)

(b)

(c)

FIG. 9. (a) The motion in the moving glass occurs through rough static channels. Relative deformations grow with distance

and become of order a at distances R

∼ R

. Only the channels t hemselves are elastically coupled along the y direction.

Depending on the dimension, velocity and disorder strength two main cases can occur: (b) an elastic ﬂow where the particles

positions are elastically coupled between channels (in d = 3 and weak disorder or large velocities). In this regime the lattice is

topologically ordered (no dislocations) and the rows of the lattice follow the channels. This is a Moving Bragg Glass. (c) In

d = 2 or at stronger disorder in d = 3 the positions of particles in diﬀerent channels may decouple. Dislocations with Burgers

vectors along x (indicated by the square) are then injected between some channels beyond the length R

. This situation

describes a Moving transverse glass (with a smectic or a line crystal type of topological order).

(a) (b)

FIG. 10. Diﬀerent types of topological order for the manifold of channels in d = 3 (a) for line lattices in motion the channels

are “sheets” an d thus form an anisotropic type of smectic layering (b) for 3d lattices of pointlike objects or for triply periodic

structures (triple Q CDW) they have instead the topology of a line crystal

B. Dynamical Larkin length and transverse critical force

Another important property of the moving glass intimately related to the ex istence of stable channels is the existence

of “transverse barriers”. Inde e d it is natural physically that once the pattern of channels is established the system

does not respond in the transverse direction along which it is pinned. Thus we have predicted in Ref. 68 that the

respo nse to an additional small exter nal force F

in the direction transverse to motion vanishes at T = 0. A true

Fig. 18. Left: Decoration in motion images in NbSe

at 4 K from Ref. 170. Right

column: Real space image (Fourier ﬁltered) which shows the static channels along

direction of motion. Left column: Its Fourier transform, which shows transverse

order (two peaks) or full triangular lattice order (six peaks). Top: Moving smec-

tic (low-ﬁeld, higher-velocity). Bottom: Moving Bragg glass (high-ﬁeld, lower-

velocity). Right: (top) Moving Bragg glass on static channels, (bottom) moving

smectic, channels are decoupled by dislocations (black square).

do not possess diverging barriers,

167,168

hence they are weaker glasses than

their static parent. However, in the presence of columnar disorder, a truly

glassy moving phase with diverging barriers was predicted,

177

and seen in

numerics,

178

the moving Bose Glass. Its detection in experiments remains a

challenge.

Let us close by mentioning that dynamical phases and eﬀects in mov-

ing vortices are much richer, there are also strong disorder or nonlinear

physics situations such as plastic depinning, ﬁlamentary ﬂow and chaos,

179

incommensurate easy ﬂow channels,

180

avalanche motion,

181

dendritic and

ﬁngering phenomena,

182

helical instabilities,

5,183

turbulent ﬂow,

185

vortex

twisters,

184

which are often more diﬃcult to model.

6. Conclusion

We close this chapter by (very few) examples of topics of current research.

First, some eﬀects require a description of the vortex lattice beyond the

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

Novel Phases of Vortices in Superconductors 329

FIG. 1: (Color online). The VL diﬀraction patterns taken at 2 K in applied ﬁelds, B of (a) 1.5 T after FC, (b) 1.5 T after

OFC, (c) 4 T after FC, (d) 6.5 T after FC and (e) 10.8 T after FC. Overlaid patterns indicate the diﬀerent VL structures that

make up the overall diﬀraction patterns, and the angles between certain Bragg spots. In (b) the angle φ is bisected by t he b

∗

direction, in (c) ρ is bisected by a

∗

whilst in (d) ν is bisected by b

∗

. In all cases, white arrows indicate the {110} directions.

The diﬀraction patterns were constructed by summing the detector measurements taken for a series of sample angles about

the horizontal and vertical axes. Statistical noise around the main beam at the center of the image has been masked, and the

pixelated data has been smoothed with a Gaussian envelope smaller than the instrument resolution. The real space V L can be

visualized by rotating the reciprocal space image by 90

◦

about the ﬁeld axis and add ing an additional spot at the center.

measurements were taken at T >T

, and subtrac ted from

2 K foregr ounds, to leave just the VL signal.

Figs. 1 (a) to (e) show VL diﬀraction patterns col-

lected a t 2 K and at various ﬁelds. The results at 1.5 T

(Fig. 1(a) and (b)), reveal the marked diﬀerence in VL

structure we obse rve depending on whether the VL at low

ﬁelds is prepared by FC or OFC. In Fig. 1(a ), clos e in-

spec tion of the VL structure reveals it to be composed of

two distorted hexagonal domains, each with two of their

six spots corresponding to {110} planes. These struc-

tures are indicated in Fig. 1(a). Similar VL structures

arise in one of the crystal domains of a twinned crys-

tal [3], with the VL orientation c ontrolled by twin planes.

In the present case, there was no detectable signal from

VLs with a distortion corresponding to the other crys-

tal domain. Nevertheless, we cannot rule out pinning

by twin planes associated with a very small fraction of

minority crystal domains in our sample. However, the

pinning potential may instead be due to minor crystal

disorder along {110} planes remaining after the removal

of twins. After OFC at 1.5 T, Fig. 1(b) shows that the

VL structure is instead compose d of a single distor ted

hexagonal domain. The structure change is also ac c om-

panied by a > 50% mean re duc tion in the rocking curve

width o f the Br agg spots, and > 20% mean reduction

in their azimuthal width. These correspond to substan-

tial improvements in the longitudinal cor relation of the

vortices in the ﬁeld direction, and in the orientational or-

der about the ﬁeld axis re spectively. This improvement

in VL perfection sugg e sts that the vortex motion caused

by OFC allows the VL to explore the true minimum F

of its free energy F , at temp e ratures below the FC irre-

versibility line. Since anisotropic London theory predicts

all orientations to be degenerate [9], we expect F

to be

shallow at low ﬁelds, and OFC required to reveal it.

At ﬁelds from 2.5 to 6 T, with stronger vortex-vortex

interaction, the eﬀect of OFC is observed to be less im-

portant; whether the VL is prepared by FC or OFC, the

VL structure is composed of a single distorted hexag-

onal domain like that in Fig . 1(c). Note that this

intermediate-ﬁeld structur e is rotated by 90

◦

relative to

that in Fig. 1(b). However, in the ﬁeld region between

1.5 and 2.5 T, the structures illustrated in Figs. 1(b) and

so the free energy F must vary little with VL orienta-

tion. Under these circumstances, pinning may again be

expec ted to play a role. Indeed, on FC at 2 T, we observe

that the diﬀrac ted intensity forms an almost continuous

ellipse, indicating VLs pinned in a range of orientations

of similar F . Even with OFC at 2 T, due to the absence

of a dominant intrinsic F

, pinning remains important,

as we observe the re-emergence of the {110 }-pinned VL

structure of Fig. 1(a). However, now this structure co-

exists up to 2.25 T with that illustrated in Fig. 1(c). In no

case do we observe a smooth transition between the two

VL orientations repre sented by Figs. 1(b) and (c), which

indicates that the intrinsic transition between them is

likely to be ﬁrst-order. Further evide nc e for this is given

by the ﬁeld-dependence of the hexagonal structure dis-

tortion. For each VL structur e , this is characterized by

the axial ratio , η, of the ellipse that overlays the Bragg

spots in each domain. The value of η may be calculated

from the measured angle s between VL reciprocal lattice

vectors (all angles would be 60

◦

for the isotropic case of

η = 1). Angles (approximately) bisected by symmetry

axes (i.e. ω in Fig. 1(a), φ in Fig. 1(b) and ρ in Fig. 1(c))

provide the most sensitive me asures of η. The resulting

ﬁeld-dependence of η is presented in Fig. 2. We note that

for the OFC data, Fig. 2 reveals a clear discontinuity in

η at ≈ 2 T, which is further evidence for the ﬁrst-or de r

nature of the re- orientation transitio n.

Further increase of the ﬁeld yields another ﬁrst-or de r

VL structure transition. Fig. 1(d) shows that by 6.5 T

the single distorted hexagonal structure like that in

Fig. 19. The VL diﬀraction patterns for untwinned YBCO at T = 2K and

between hexagonal and rhombic structures with distortions of opposite sign, pro-

viding evidence for a d + s-wave admixture for the order parameter, from Ref. 189.

s-wave Ginzburg–Landau model. It has been predicted that the d-wave

symmetry of the order parameter should favor a square vortex lattice over a

triangular one at high ﬁeld, leading to a ﬁrst order structural transition as

the ﬁeld is increased.

186,187

Small-angle neutron-scattering experiments on

YBCO have indeed revealed a structural transition.

188

More recent experi-

ments on untwinned samples

189

are shown in Fig. 19: there, the coexistence

region suggests a ﬁrst order transition around 6T . A distinct transition at

lower ﬁelds around 2T is also seen

189

between a low-ﬁeld hexagonal to an

intermediate-ﬁeld distorted hexagonal phase and is claimed to be driven by

Fermi surface eﬀects. Recent speciﬁc heat experiments, performed at higher

temperature, do ﬁnd signature of ﬁrst order transition at about this ﬁeld

which suggest a FOT structural transition line in the phase diagram. Note

that these transitions occur very deep within the vortex solid, i.e. within the

Bragg glass.

Next, we mention that the FRG and concepts such as quasi translational

order, topological order and Bragg glass phases have been useful in other

problems with quenched disorder

150

: e.g. liquid crystals

147,149

conﬁned in a

random matrix, the spontaneous vortex lattice in ferromagnetic supercon-

ductors (such as ErNi

C),

148

and random ﬁeld magnets.

151

A direction

of present research, mentioned in Sec. 4.3.3, is to study avalanches phe-

nomena in other pinned systems, as well as relaxation from a prescribed

initial state,

191

with close contact between theory

126

and experiments.

128

Remarkable recent developments

192

allow to image and manipulate individ-

ual vortices using magnetic force microscopy. It allows to directly measure

the interaction of a moving vortex with the local disorder potential. Being

near the realization of the quadratic well setting described in Sec. 4.3.3 it

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

330 P. Le Doussal

could be a useful probe to perform more precise comparison with theory,

e.g. for processes such as depinning, vortex creep and vortex avalanches

181

and to study relaxation from a chosen at will initial conﬁguration.

Acknowledgments

I am particularly indebted to T. Giamarchi and K. Wiese for multiple and

fruitful collaborations. I am grateful to B. Horovitz, D. S. Fisher, T. Nat-

termann, D. R. Nelson and V. Vinokur, as well as L. Balents, P. Chauve,

D. Carpentier, L. Cugliandolo, A. Fedorenko, A. Koshelev, C. Marchetti,

A. Middleton, E. Olive, A. Rosso, G. Schehr, J. C. Soret, T. Hwa for col-

laborations, and numerous other colleagues for useful discussions. I have

learned a lot about experiments from C. J. van der Beek, M. Charalambous,

T. Klein, C. Simon, A. Kapitulnik, E. Zeldov and many others. I am very

grateful to C. J. van der Beek, D. Feldman, B. Horovitz, T. Giamarchi and

E. Zeldov for careful reading of the manuscript and useful suggestions.

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