321 research outputs found

### Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules

We consider the non-equilibrium dynamics of disordered systems as defined by
a master equation involving transition rates between configurations (detailed
balance is not assumed). To compute the important dynamical time scales in
finite-size systems without simulating the actual time evolution which can be
extremely slow, we propose to focus on first-passage times that satisfy
'backward master equations'. Upon the iterative elimination of configurations,
we obtain the exact renormalization rules that can be followed numerically. To
test this approach, we study the statistics of some first-passage times for two
disordered models : (i) for the random walk in a two-dimensional self-affine
random potential of Hurst exponent $H$, we focus on the first exit time from a
square of size $L \times L$ if one starts at the square center. (ii) for the
dynamics of the ferromagnetic Sherrington-Kirkpatrick model of $N$ spins, we
consider the first passage time $t_f$ to zero-magnetization when starting from
a fully magnetized configuration. Besides the expected linear growth of the
averaged barrier $\bar{\ln t_{f}} \sim N$, we find that the rescaled
distribution of the barrier $(\ln t_{f})$ decays as $e^{- u^{\eta}}$ for large
$u$ with a tail exponent of order $\eta \simeq 1.72$. This value can be simply
interpreted in terms of rare events if the sample-to-sample fluctuation
exponent for the barrier is $\psi_{width}=1/3$.Comment: 8 pages, 4 figure

### Duality symmetries and effective dynamics in disordered hopping models

We identify a duality transformation in one-dimensional hopping models that
relates propagators in general disordered potentials linked by an up-down
inversion of the energy landscape. This significantly generalises previous
results for a duality between trap and barrier models. We use the resulting
insights into the symmetries of these models to develop a real-space
renormalisation scheme that can be implemented computationally and allows
rather accurate prediction of propagation in these models. We also discuss the
relation of this renormalisation scheme to earlier analytical treatments.Comment: 29 pages, 7 figs. Final version, some extra context and references
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### Anomalous diffusion in disordered multi-channel systems

We study diffusion of a particle in a system composed of K parallel channels,
where the transition rates within the channels are quenched random variables
whereas the inter-channel transition rate v is homogeneous. A variant of the
strong disorder renormalization group method and Monte Carlo simulations are
used. Generally, we observe anomalous diffusion, where the average distance
travelled by the particle, []_{av}, has a power-law time-dependence
[]_{av} ~ t^{\mu_K(v)}, with a diffusion exponent 0 \le \mu_K(v) \le 1.
In the presence of left-right symmetry of the distribution of random rates, the
recurrent point of the multi-channel system is independent of K, and the
diffusion exponent is found to increase with K and decrease with v. In the
absence of this symmetry, the recurrent point may be shifted with K and the
current can be reversed by varying the lane change rate v.Comment: 16 pages, 7 figure

### Phases of random antiferromagnetic spin-1 chains

We formulate a real-space renormalization scheme that allows the study of the
effects of bond randomness in the Heisenberg antiferromagnetic spin-1 chain.
There are four types of bonds that appear during the renormalization flow. We
implement numerically the decimation procedure. We give a detailed study of the
probability distributions of all these bonds in the phases that occur when the
strength of the disorder is varied. Approximate flow equations are obtained in
the weak-disorder regime as well as in the strong disorder case where the
physics is that of the random singlet phase.Comment: 29 pages, 12 encapsulated Postscript figures, REVTeX 3.

### Localization Properties in One Dimensional Disordered Supersymmetric Quantum Mechanics

A model of localization based on the Witten Hamiltonian of supersymmetric
quantum mechanics is considered. The case where the superpotential $\phi(x)$ is
a random telegraph process is solved exactly. Both the localization length and
the density of states are obtained analytically. A detailed study of the low
energy behaviour is presented. Analytical and numerical results are presented
in the case where the intervals over which $\phi(x)$ is kept constant are
distributed according to a broad distribution. Various applications of this
model are considered.Comment: 43 pages, plain TEX, 8 figures not included, available upon request
from the Authors

### Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension $d=1,2,3$ via a Monte-Carlo procedure in the disorder

In order to probe with high precision the tails of the ground-state energy
distribution of disordered spin systems, K\"orner, Katzgraber and Hartmann
\cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-Carlo
Markov chain in the disorder. In this paper, we combine their Monte-Carlo
procedure in the disorder with exact transfer matrix calculations in each
sample to measure the negative tail of ground state energy distribution
$P_d(E_0)$ for the directed polymer in a random medium of dimension $d=1,2,3$.
In $d=1$, we check the validity of the algorithm by a direct comparison with
the exact result, namely the Tracy-Widom distribution. In dimensions $d=2$ and
$d=3$, we measure the negative tail up to ten standard deviations, which
correspond to probabilities of order $P_d(E_0) \sim 10^{-22}$. Our results are
in agreement with Zhang's argument, stating that the negative tail exponent
$\eta(d)$ of the asymptotic behavior $\ln P_d (E_0) \sim - | E_0 |^{\eta(d)}$
as $E_0 \to -\infty$ is directly related to the fluctuation exponent
$\theta(d)$ (which governs the fluctuations $\Delta E_0(L) \sim L^{\theta(d)}$
of the ground state energy $E_0$ for polymers of length $L$) via the simple
formula $\eta(d)=1/(1-\theta(d))$. Along the paper, we comment on the
similarities and differences with spin-glasses.Comment: 13 pages, 16 figure

### Random elastic networks : strong disorder renormalization approach

For arbitrary networks of random masses connected by random springs, we
define a general strong disorder real-space renormalization (RG) approach that
generalizes the procedures introduced previously by Hastings [Phys. Rev. Lett.
90, 148702 (2003)] and by Amir, Oreg and Imry [Phys. Rev. Lett. 105, 070601
(2010)] respectively. The principle is to eliminate iteratively the elementary
oscillating mode of highest frequency associated with either a mass or a spring
constant. To explain the accuracy of the strong disorder RG rules, we compare
with the Aoki RG rules that are exact at fixed frequency.Comment: 8 pages, v2=final versio

### Smoothening of Depinning Transitions for Directed Polymers with Quenched Disorder

We consider disordered models of pinning of directed polymers on a defect
line, including (1+1)-dimensional interface wetting models, disordered
Poland--Scheraga models of DNA denaturation and other (1+d)-dimensional
polymers in interaction with columnar defects. We consider also random
copolymers at a selective interface. These models are known to have a
(de)pinning transition at some critical line in the phase diagram. In this work
we prove that, as soon as disorder is present, the transition is at least of
second order: the free energy is differentiable at the critical line, and the
order parameter (contact fraction) vanishes continuously at the transition. On
the other hand, it is known that the corresponding non-disordered models can
have a first order (de)pinning transition, with a jump in the order parameter.
Our results confirm predictions based on the Harris criterion.Comment: 4 pages, 1 figure. Version 2: references added, minor changes made.
To appear on Phys. Rev. Let

### Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums

For Anderson tight-binding models in dimension $d$ with random on-site
energies $\epsilon_{\vec r}$ and critical long-ranged hoppings decaying
typically as $V^{typ}(r) \sim V/r^d$, we show that the strong multifractality
regime corresponding to small $V$ can be studied via the standard perturbation
theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios
$Y_q(L)$, which are the order parameters of Anderson transitions, can be
written in terms of weighted L\'evy sums of broadly distributed variables (as a
consequence of the presence of on-site random energies in the denominators of
the perturbation theory). We compute at leading order the typical and
disorder-averaged multifractal spectra $\tau_{typ}(q)$ and $\tau_{av}(q)$ as a
function of $q$. For $q<1/2$, we obtain the non-vanishing limiting spectrum
$\tau_{typ}(q)=\tau_{av}(q)=d(2q-1)$ as $V \to 0^+$. For $q>1/2$, this method
yields the same disorder-averaged spectrum $\tau_{av}(q)$ of order $O(V)$ as
obtained previously via the Levitov renormalization method by Mirlin and Evers
[Phys. Rev. B 62, 7920 (2000)]. In addition, it allows to compute explicitly
the typical spectrum, also of order $O(V)$, but with a different $q$-dependence
$\tau_{typ}(q) \ne \tau_{av}(q)$ for all $q>q_c=1/2$. As a consequence, we find
that the corresponding singularity spectra $f_{typ}(\alpha)$ and
$f_{av}(\alpha)$ differ even in the positive region $f>0$, and vanish at
different values $\alpha_+^{typ} > \alpha_+^{av}$, in contrast to the standard
picture. We also obtain that the saddle value $\alpha_{typ}(q)$ of the Legendre
transform reaches the termination point $\alpha_+^{typ}$ where
$f_{typ}(\alpha_+^{typ})=0$ only in the limit $q \to +\infty$.Comment: 13 pages, 2 figures, v2=final versio

### Permutation-Symmetric Multicritical Points in Random Antiferromagnetic Spin Chains

The low-energy properties of a system at a critical point may have additional
symmetries not present in the microscopic Hamiltonian. This letter presents the
theory of a class of multicritical points that provide an interesting example
of this in the phase diagrams of random antiferromagnetic spin chains. One case
provides an analytic theory of the quantum critical point in the random
spin-3/2 chain, studied in recent work by Refael, Kehrein and Fisher
(cond-mat/0111295).Comment: Revtex, 4 pages (2 column format), 2 eps figure

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