Назад
September 2, 2010 15:31 World Scientific Review Volume - 9.75in x 6.5in ch13
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:
˜
f
sol
d
=
a
2
T
2β
A
+ 2.848|a
T
|
1/2
+
2.4
a
T
+ ·· (33)
It is found that they intersect at a
T
9.5, leading to a first order VL
melting transition.
104
It compares reasonably with numerics
37
at high fields
(the same method in 2D gives a
T
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 =
1
2
K
2
a
2
0
c
2
L
. From the scaled
magnetization m = f
0
(a
T
) 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
40
which is also
observed to be accompanied by a jump.
64
The prediction c = 0.0075((2
2b + t)/t)
2
0.20Gi
1/3
(b 1 t)(b/t
2
)
2/3
is about twice larger than the one
observed in Ref. 64. The LLL approximation should hold for 4Bξ
2
(0)/Φ
0
>
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
a
2
ik
× u(k)
k
2
(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
Φ
0
R
x
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
4
with discrete 2D layers indexed by z = n. The layers
are coupled via (i) the Josephson coupling energy E
J
P
n
R
d
2
r cos
˜
φ(r, n+
1, n)
with
˜
φ(r, m, n) =
˜
φ(r, m)
˜
φ(r, n) and E
J
=
2
2
0
d (also noted J
2
).
(ii) the magnetic energy (the term B
2
). Glazman and Koshelev
19
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
h
˜
ψ(x)
˜
ψ
(x
0
)i with
˜
ψ(x) = |ψ
0
|e
i
˜
φ(x)
does not always decay to zero along z.
Along z, they found that for B > B
cr
Φ
0
2
d the melting transition is
given by its 2D limit T
m
= T
2D
m
=
0.62
8π
3
d
0
and the phase fluctuations are
h
˜
φ(0, n + 1, n)
2
i T/T
0
(B) with T
0
(B) T
2D
m
(B
cr
/B)
1/2
. Hence they con-
cluded that for B > B
cr
there is a range of temperatures T
0
(B) < T < T
m
where phase fluctuations are large. This result, and the further Gaussian
self-consistent treatment
20
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,
21
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
77
) (ii) a true transition
requires topological defects. In the case of decoupling, these are interstitial
and vacancies in a single layer.
80
These defects have also been called quar-
tets of dislocations.
79
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.
81
In be-
tween these two extremes the topological transition merges with the above
September 2, 2010 15:31 World Scientific Review Volume - 9.75in x 6.5in ch13
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
c
materials, which are found to account for the pinning disorder.
4,109
The coupling energy to the disorder is modeled as:
F
dis
=
Z
d
3
x˜ρ(x)
˜
U(x) (35)
where x = (r, z) and ˜ρ(r, z) =
P
i
p(r R
i
u
i
(z)). Within the GL model,
one can consider uncorrelated δT
c
-disorder,
˜
U(x) = |ψ
0
|
2
α(x) and a form
factor equal to p(r) = 1|ψ
v
(r)|
2
from the solution ψ
v
(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):
F
dis
=
Z
d
3
(x)U(x) , ρ(r, z) =
X
i
δ
(2)
(r R
i
u
i
(z)) (36)
with, in Fourier U(q) = p(q
)
˜
U(q). The correlator of the random potential
is then
U(x)U(x
0
) = R
U
(r r
0
)δ(z z
0
) (37)
where here and below
... denote average over the random potential, i.e. sam-
ple averages. Here R
U
(q) = ˜γ p(q
)
2
, which leads to R
U
(r) = γξ
2
k(r)
with k(0) = 1 and k(y) y
2
ln y at large y. For simplicity, one often con-
siders R
U
(r) = dU
2
p
e
r
2
/r
2
f
where d is the layer separation, and U
p
typical
pinning energy per unit length along z. The correlation length of the disor-
der seen by a vortex is r
f
ξ since this is the smallest length that it can
resolve. For a single vortex line one obtains:
F
dis
=
Z
dz U(u(z), z) (38)
A phenomenological description in terms of a 3D density n
i
of independent
impurities with random individual 3D pinning forces of r.m.s f
p
can also
be used. If there are a few defects within the volume ξ
2
d, as typically for
September 2, 2010 15:31 World Scientific Review Volume - 9.75in x 6.5in ch13
304 P. Le Doussal
oxygen vacancies, the resulting random potential is nearly Gaussian with
dU
2
p
= γξ
2
= f
2
p
n
i
ξ
4
.
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
C
K
(x) =
he
iK·u(x)
e
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
13
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
el
[u]
Z
d
d
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:
hu
α
(q)i = Φ(q)
1
αβ
f
β
(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
0
) = W δ
d
(x x
0
),
one finds, upon disorder averaging:
hu
γ
(q)u
γ
(q)i =
T
Φ
γ
(q)
+
W
Φ
2
γ
(q)
(42)
with γ = L, T the longitudinal and transverse components. For isotropic
elasticity Φ(q) = cq
2
and at T = 0 this leads to:
˜
B(x) =
(u(x) u(0))
2
= 4W
Z
q,BZ
1
c
2
q
4
(1 cos(qx)) =
4W
c
2
b(x)
C
K
(x) = e
K
2
˜
B(x)/4
, b(x) |x|
2ζ
L
, ζ
L
=
4 d
2
(43)
September 2, 2010 15:31 World Scientific Review Volume - 9.75in x 6.5in ch13
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
0
,
(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
c
. Since the random potential is smooth and varies on scale
r
f
, one expects this approximation to break down when the deformations
reach u r
f
, i.e. at scale such that
˜
B(x = R
c
) r
2
f
which, from (43) gives:
R
c
= (c
2
r
2
f
/W )
1/(4d)
(44)
a scale which, for weak disorder W W
c
= c
2
r
2
f
/a
4d
can be much larger
than a. The case R
c
< a, called single vortex pinning, is mentioned below.
From Sec. 4.1 one estimates W =
1
2a
4
P
K
K
2
R
U
(K) for the 3D vortex lat-
tice, as W
1
2a
2
trR
U00
(0) dU
2
p
/a
2
r
2
f
when r
f
a, using a
2
P
K
R
k
in
that case.
With remarkable insight, Larkin and Ovchnnikov
14
(LO) understood that
the scale R
c
also determines the threshold depinning force. In their picture,
the system breaks into domains of size R
c
which are independently pinned.
This leads to the collective pinning theory. To unpin the system one must
apply a total external force F R
d
c
on a Larkin domain at least equal to the
typical pinning force acting on the same domain, which scales as
R
d
d
xf(x)
W
1/2
R
d/2
c
. This leads to the LO estimate for the depinning threshold force
per unit volume:
F
c
=
f
c
a
2
W
1/2
R
d/2
c
=
cr
f
R
2
c
(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
c
= cr
2
f
R
d2
c
W
1/2
r
f
R
d/2
c
F
c
r
f
R
d
c
. 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
j
c
= cf
c
/Φ
0
= cF
c
/B.
110
For the triangular VL in d = 3 the deformation splits into
˜
B(x) =
˜
B
T
(x)+
˜
B
L
(x) and there are two Larkin lengths, R
c
in ab plane, and L
c
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:
˜
B
T
(x)= 2W
Z
q,BZ
(1cos(qx))
(c
66
q
2
+c
44
(q)q
2
z
)
2
2W
c
3/2
66
c
1/2
44
b(ρ) , ρ=
r
2
+
c
66
c
44
z
2
1/2
for large enough R
c
(weak disorder) when one can neglect non-locality and
use c
44
= c
44
(0). In that case b(ρ)
1
4π
2
ρ and R
c
is given by (44) with
c
2
c
3/2
66
c
1/2
44
. The Larkin length along z is L
c
= R
c
p
c
44
/c
66
λ
a
R
c
.
For R
c
< λ/ one cannot neglect non-locality: since c
44
1/q
2
one finds
much slower logarithmic growth, i.e. b(ρ) λ
c
r
2
a
2
+
z
a
. The Larkin length
thus varies exponentially with the disorder for a < R
c
< λ/ and one has
L
c
= R
2
c
/a.
The energy scale U
c
r
2
f
c
66
L
c
r
f
F
c
L
c
R
2
c
determines the critical force
per unit volume F
c
and the critical current j
c
as:
F
c
=
j
c
c
B c
66
r
f
R
2
c
(46)
The above arguments also apply to a single flux line, setting d = 1
and c = e
l
and f (z) = U (0, z) in (38). This gives the single vortex
Larkin length (along the field direction z), L
sv
c
= (e
2
l
r
2
f
/W
sv
)
1/3
, with
W
sv
=
1
2
2
R
U
(0) γ, leading to f
c
=
0
/c)j
c
e
l
r
f
/(L
sv
c
)
2
(in case
of a dispersive line tension the value e
l
(q
z
1/L
sv
c
) should be used). The
interactions with the neighboring vortex lines can be neglected if L
sv
c
< a.
One checks from above and the previous sections that the condition L
sv
c
< a
is equivalent to R
c
< a, that the term c
66
q
2
c
44
(q)q
2
z
for q
K
BZ
and
that j
c
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
c
. 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
el
[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
i
(z) u
i
(z) + a
0
. This
turns out to be crucial: since there is no energy gain to shift the lattice
by a
0
, 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
i
(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
0
1
α
u
α
(x) +
X
K6=0
e
iKxiKu(x)
!
(48)
where ρ
0
is the average density, in terms of the smooth field u(x) =
R
q,BZ
u(q)e
iqx
such that u(R
i
, z) = u
i
(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
i
+ u
i
(z), z) = R
i
, 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:
F
dis
q 0
= ρ
0
Z
d
d
xU
0
(x) · u(x) (49)
where U
0
(x) =
R
q0
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
September 2, 2010 15:31 World Scientific Review Volume - 9.75in x 6.5in ch13
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
2d
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
ij
(r)u
ij
(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
U
qK
close in periodicity to the lattice, which couple as:
F
dis
q K
= ρ
0
Z
d
d
xU
K
(x)e
iKu(x)
(50)
where U
K
(x) =
R
qK
U(q)e
iqx
. Their effect is to shift locally the crystal.
As a result, an important new length scale appears, R
a
, defined as the scale
over which u varies by a. It can be estimated as follows. The phase u,
roughly constant on scale R
a
, adjusts to cancel the phase of sum of random
complex numbers
R
d
d
rU
K
(r) over a volume R
d
a
. A complex plane ran-
dom walk argument yields that the gain in energy density from disorder as
ρ
0
R
U
(K)
1/2
/R
d/2
a
. On the other hand, the cost in energy density of such
a deformation a on scale R
a
scales as c(a/R
a
)
2
. The optimum occurs at
a length scale:
R
a
(c
2
a
8
/R
U
(K))
1/(4d)
(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
f
a and R
U
(K) e
r
2
f
K
2
2
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
U
K
(x)U
K
0
(x
0
)
R
U
(K)δ
K,K
0
δ
d
(x x
0
), the model (36) can thus be replaced by:
F [u]=F
el
[u]+
Z
d
d
xV (u(x), x) ,
V (u, x)V (u
0
, x
0
)= R(uu
0
)δ
d
(xx
0
) (52)
where, for a periodic object the correlator of the disorder reads:
R(u) =
X
K6=0
ρ
2
0
R
U
(K) cos(Ku) (53)
hence it is a periodic function with the periodicity of the lattice.
September 2, 2010 15:31 World Scientific Review Volume - 9.75in x 6.5in ch13
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
2
/r
2
f
) 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
a
(r), a = 1, · · n and average their Gibbs mea-
sure over disorder, defining the replicated Hamiltonian through e
βF
rep
[u]
=
e
β
P
a
F [u
a
]
. One can then show that all disorder averaged correlation
functions can be expressed as suitable averages of u
a
, e.g.
hu(x)u(x
0
)i =
hu
a
(x)u
a
(x
0
)i
H
rep
in the limit n = 0. For model (52)
F
rep
[u] =
n
X
a=1
F
el
[u
a
]
1
2T
Z
d
d
x
n
X
a,b=1
R(u
a
(x) u
b
(x)) (54)
where the disorder term is highly nonlinear and couples the replica. Ex-
panding (54) to quadratic order at small u
a
u
b
< r
f
one nicely recovers
the Larkin model, in its replicated version: the Larkin regime x < R
c
thus
exists for all pinned systems.
The Bragg glass, however, is characterized by two scales R
c
and R
a
,
and that leads to an interesting scale dependence. When H H
c2
(and
temperature is not too high) the correlation length of the disorder r
f
a,
thus R
c
R
a
. In that case the function R(u) is the sum of many harmonics
and in fact, for r
f
u a it looks like a short range function, e.g. if
R
U
K
e
K
2
r
2
f
/2
then R(u) e
u
2
/2r
2
f
, obtained by replacing the sum in
(53) by an integral. Thus one may expect, as confirmed below, that in the
range of scales R
c
< x < R
a
, 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
a
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
c
. 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))
2
i x
2ζ
, F
2
c
F
2
F
2
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
0
which approximates (54):
F
0
=
1
2
Z
q
αβ
(q)δ
ab
+ σ
ab
δ
αβ
]u
a
α
(q)u
b
β
(q) (56)
By extremization of the variational free energy F
var
= F
0
+hF
rep
F
0
i
F
0
one
obtains the saddle point equation for n ×n matrix of variational parameters
σ
ab
. 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
ab
= σ
c
δ
ab
+ 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
c
, 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
2
):
P (u(q))
X
i
e
f
(i)
/T
e
c
2T
(q
2
+R
2
c
)|u(q)u
(i)
(q)|
2
(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
c
sets the internal size of the