Papers in Physics, vol. 2, art. 020009 (2010)

Received: 25 November 2010, Accepted: 6 January 2011
Edited by: A. Vindigni
Licence: Creative Commons Attribution 3.0
DOI: 10.4279/PIP.020009

www.papersinphysics.org

ISSN 1852-4249

Commentary on “Anisotropic finite-size scaling of an elastic string at the
depinning threshold in a random-periodic medium”

Andrei A. Fedorenko1∗

In their paper [1] S. Bustinagorry and
A.B. Kolton study the finite-size scaling properties
of an elastic string driven in a disordered medium
at the depinning transition. The zero-temperature
dynamics of the model is given by the equation of
motion for the displacement field u(z, t):

γ∂tu(z, t) = c∂
2
z u(z, t) + Fp(u, z) + F, (1)

where γ is the friction, c the elasticity, and F the
driving force. Fp is the Gaussian random force due
to disorder with zero mean and variance

Fp(u, z)Fp(u′, z′) = ∆(u − u′)δ(z − z′). (2)

The model of an elastic object in a disordered
medium can be used to describe diverse physical
systems. They can be cast into two classes: (i) peri-
odic systems, such as charge density waves (CDW)
in solids and vortex lattice in disordered super-
conductors; (ii) interfaces such as domain walls in
magnets. ∆(u) is a periodic function for the ran-
dom periodic systems and a short range function
for the interfaces. Due to the competition between
elasticity and disorder, these systems exhibit a rich
glassy behavior and a non-trivial response to exter-
nal perturbations. In particular, a driving force F
exceeding a certain threshold value Fc is required

∗E-mail: andrey.fedorenko@ens-lyon.fr

1 CNRS-Laboratoire de Physique de l’Ecole Normale
Supérieure de Lyon, 46, Allée d’Italie, 69007 Lyon,
France.

to set the elastic object into steady motion. To
some extent, the depinning transition shares many
features with critical phenomena where the veloc-
ity v plays the role of the order parameter. For
instance, the correlation length defined through
the velocity-velocity correlation function diverges
when approaching the transition from above as
ξ ∼ (F − Fc)−ν . However, in comparison with the
ordinary critical phenomena, the problem of disor-
dered elastic systems is notably difficult due to the
so-called dimensional reduction. The latter takes
place also in random field spin systems and states
that a d-dimensional disordered system at zero tem-
perature is equivalent to all orders in perturbation
theory to a pure system in d−2 dimensions at finite
temperature. It turns out that the metastability
renders the zero-temperature perturbation theory
useless: it breaks down on scales larger than the so-
called Larkin length. The peculiarity of the prob-
lem is that for d < 4 there is an infinite set of rele-
vant operators, which can be parameterized by the
function ∆(u) = −R′′(u). The corresponding func-
tional renormalization group (FRG) equation for
the renormalized disorder correlator has two fixed
points: the random field fixed point (RFFP) which
describes the depinning transition for the interfaces
and the random periodic fixed point (RPFP) which
describes the random periodic systems. The run-
ning (renormalized) disorder correlator becomes a
non-analytic function beyond the Larkin scale. The
appearance of a non-analyticity in the form of a
cusp at the origin is related to metastability, and

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Papers in Physics, vol. 2, art. 020009 (2010) / A. A. Fedorenko

nicely accounts for the generation of a threshold
force at the depinning transition: Fc ∼ −∆∗′(0+).
Note, that the latter is zero in the bare (analytic)
theory. Thus, FRG provides a way to analytically
compute the threshold force which turns out to be
nonuniversal (similar to the critical temperature
in critical phenomena) and the critical exponents
which depend only on the universality class. In
particular, one can compute the roughness expo-
nent which gives the scaling of the average width of
the interface with its size: w2 ∼ L2ζ . The observ-
ables computed using FRG in the thermodynamic
limit are already averaged over different disorder
realizations. The picture for finite systems is more
involved.

In numerical simulations, one usually considers
a finite system of linear size L moving in a box of
size M with periodic boundary conditions. Then,
depending on how one performs the finite-size scal-
ing analysis, one can approach different scaling
regimes. In the limit of L → ∞ at fixed M one
approaches the scaling behavior of a random peri-
odic system. In the opposite limit of M → ∞ at
fixed L, the system falls in one of the universality
classes of a particle moving in a random landscape
described by some particle fixed point (PFP) [2].
The random field universality class describing the
scaling properties of a pinned interface corresponds
to the limit of both L, M → ∞. However, one has
to specify the precise way of taking this limit in or-
der to separate the basin of attraction of the RFFP
from those of the RPFP and the PFP. Studying the
crossover between these three universality classes is
the main aim of the paper [1].

The key parameter is k = M/Lζdep where ζdep is
the roughness exponent of the interface (a string)
at the depinning transition. It has been found in [1]
that w2L−2ζdep is a universal function of k, G(k),
which has a minimum at k∗ < 1. For k � k∗,
one reveals that the periodicity is important and
the system is described by the RPFP. In this case,
there are two relevant contributions to the dis-
placement correlation function: the universal loga-
rithmic growth with zero roughness exponent and
the Larkin type growth due to instability of RPFP
which gives the roughness exponent ζL = (4−d)/2.

For k ≈ k∗, one expects that the system ap-
proaches the RFFP universality class in the limit
L → ∞. The function G(k) for 0 < k < k∗ de-
scribes the crossover between the RPRF and the

RFFP. The main result of the paper is that in or-
der to verify the predictions of FRG and numeri-
cally study the properties of the RFFP universality
class, one has to keep k = k∗. In the case of devia-
tions from k = k∗, the results are modified by the
crossover either to the RPFP or to the PFP.

It has been found that G(k) increases slower than
any power-law for k > k∗. In this regime, the sys-
tem is expected to show crossover to the PFP [2].
The properties of the critical configuration are de-
termined by the extreme value statistics. For very
large k, one can split the system in k almost un-
correlated regions with their own subcritical con-
figurations. The true critical configuration corre-
sponds to the subcritical configuration with the
maximal critical force. As a result, in the limit
k → ∞, the distribution of the threshold force ex-
hibits crossover from the Gaussian distribution to
the Gumbel distribution. The average critical force
increases as Fc ∼ log k since the critical force can be
regarded as the maximum among k different sub-
critical forces. The authors argue that the increase
in the critical force might be correlated with the
slow increase of roughness. They suggest that the
larger critical force can be achieved either by ac-
cidental extended defects or by rare uncorrelated
strong pinning forces. They argue that the sec-
ond scenario is more plausible since the presence
of accidental correlation should decrease roughness.
However, as it was shown in [3], the presence of cor-
related disorder and extended defects also enhances
roughness. Therefore, one can expect the simulta-
neous enhancement of the roughness and the criti-
cal force.

The similar crossover effects have been observed
by the authors in the behavior of the structure fac-
tor and the interface width distribution.

Summarizing, the authors numerically observe
that the average width and its probability distribu-
tion depend, in a universal way, on the parameter
k that is encoded in the universal function G(k).
The parameter k is decisive regarding the univer-
sality class which will be approached in the thermo-
dynamic limit or probed by the finite size scaling
analysis. Knowing the function G(k), and espe-
cially the position of its minima k∗, is required for
the correct analysis of the numerical simulations.
Analytical calculation of G(k) using, for example,
FRG remains, however, an open challenging prob-
lem.

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Papers in Physics, vol. 2, art. 020009 (2010) / A. A. Fedorenko

[1] S Bustingorry, A B Kolton, Anisotropic finite-
size scaling of an elastic string at the depin-
ning threshold in a random-periodic medium,
Pap. Phys. 2, 020008 (2010).

[2] P Le Doussal, K J Wiese, Driven parti-
cle in a random landscape: disorder correla-
tor, avalanche distribution and extreme value
statistics of records, Phys. Rev. E 79, 051105
(2009).

[3] A A Fedorenko, P Le Doussal, K J Wiese, Stat-
ics and dynamics of elastic manifolds in media
with long-range correlated disorder, Phys. Rev.
E 74, 061109 (2006).

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