259 research outputs found
Statistics of the two-point transmission at Anderson localization transitions
At Anderson critical points, the statistics of the two-point transmission
for disordered samples of linear size is expected to be multifractal
with the following properties [Janssen {\it et al} PRB 59, 15836 (1999)] : (i)
the probability to have behaves as ,
where the multifractal spectrum terminates at as a
consequence of the physical bound ; (ii) the exponents that
govern the moments become frozen above some
threshold: , i.e. all moments of order are governed by the measure of the rare samples having a finite
transmission (). In the present paper, we test numerically these
predictions for the ensemble of power-law random banded matrices,
where the random hopping decays as a power-law . This
model is known to present an Anderson transition at between localized
() and extended () states, with critical properties that depend
continuously on the parameter . Our numerical results for the multifractal
spectra for various are in agreement with the relation
in terms of the
singularity spectrum of individual critical eigenfunctions, in
particular the typical exponents are related via the relation . We also discuss the statistics of the two-point
transmission in the delocalized phase and in the localized phase.Comment: v2=final version with two new appendices with respect to v1; 12
pages, 10 figure
A critical Dyson hierarchical model for the Anderson localization transition
A Dyson hierarchical model for Anderson localization, containing non-random
hierarchical hoppings and random on-site energies, has been studied in the
mathematical literature since its introduction by Bovier [J. Stat. Phys. 59,
745 (1990)], with the conclusion that this model is always in the localized
phase. Here we show that if one introduces alternating signs in the hoppings
along the hierarchy (instead of choosing all hoppings of the same sign), it is
possible to reach an Anderson localization critical point presenting
multifractal eigenfunctions and intermediate spectral statistics. The advantage
of this model is that one can write exact renormalization equations for some
observables. In particular, we obtain that the renormalized on-site energies
have the Cauchy distributions for exact fixed points. Another output of this
renormalization analysis is that the typical exponent of critical
eigenfunctions is always , independently of the disorder
strength. We present numerical results concerning the whole multifractal
spectrum and the compressibility of the level statistics,
both for the box and the Cauchy distributions of the random on-site energies.
We discuss the similarities and differences with the ensemble of ultrametric
random matrices introduced recently by Fyodorov, Ossipov and Rodriguez [J.
Stat. Mech. L12001 (2009)].Comment: 21 pages, 11 figures; v2=final versio
Zero-temperature spinglass-ferromagnetic transition : scaling analysis of the domain-wall energy
For the Ising model with Gaussian random coupling of average and unit
variance, the zero-temperature spinglass-ferromagnetic transition as a function
of the control parameter can be studied via the size- dependent
renormalized coupling defined as the domain-wall energy (i.e. the difference between the ground state
energies corresponding to AntiFerromagnetic and and Ferromagnetic boundary
conditions in one direction). We study numerically the critical exponents of
this zero-temperature transition within the Migdal-Kadanoff approximation as a
function of the dimension . We then compare with the mean-field
spherical model. Our main conclusion is that in low dimensions, the critical
stiffness exponent is clearly bigger than the spin-glass stiffness
exponent , but that they turn out to coincide in high enough
dimension and in the mean-field spherical model. We also discuss the
finite-size scaling properties of the averaged value and of the width of the
distribution of the renormalized couplings.Comment: v2=final version, 19 pages, 8 figure
Chaos properties of the one-dimensional long-range Ising spin-glass
For the long-range one-dimensional Ising spin-glass with random couplings
decaying as , the scaling of the effective coupling
defined as the difference between the free-energies corresponding to Periodic
and Antiperiodic boundary conditions defines the droplet exponent . Here we
study numerically the instability of the renormalization flow of the effective
coupling with respect to magnetic, disorder and temperature
perturbations respectively, in order to extract the corresponding chaos
exponents , and as a
function of . Our results for are interpreted in
terms of the entropy exponent which governs the
scaling of the entropy difference . We also study the instability of the ground state
configuration with respect to perturbations, as measured by the spin overlap
between the unperturbed and the perturbed ground states, in order to extract
the corresponding chaos exponents and
.Comment: 14 pages, 15 figure
Random Transverse Field Ising model in : analysis via Boundary Strong Disorder Renormalization
To avoid the complicated topology of surviving clusters induced by standard
Strong Disorder RG in dimension , we introduce a modified procedure called
'Boundary Strong Disorder RG' where the order of decimations is chosen a
priori. We apply numerically this modified procedure to the Random Transverse
Field Ising model in dimension . We find that the location of the critical
point, the activated exponent of the Infinite Disorder
scaling, and the finite-size correlation exponent are
compatible with the values obtained previously by standard Strong Disorder
RG.Our conclusion is thus that Strong Disorder RG is very robust with respect
to changes in the order of decimations. In addition, we analyze in more details
the RG flows within the two phases to show explicitly the presence of various
correlation length exponents : we measure the typical correlation exponent
in the disordered phase (this value is very close to
the correlation exponent of the {\it pure}
two-dimensional quantum Ising Model), and the typical exponent
within the ordered phase. These values satisfy the relations between critical
exponents imposed by the expected finite-size scaling properties at Infinite
Disorder critical points. Within the disordered phase, we also measure the
fluctuation exponent which is compatible with the Directed
Polymer exponent in dimensions.Comment: 10 pages, 10 figure
Random polymers and delocalization transitions
In these proceedings, we first summarize some general properties of phase
transitions in the presence of quenched disorder, with emphasis on the
following points: the need to distinguish typical and averaged correlations,
the possible existence of two correlation length exponents , the general
bound , the lack of self-averaging of thermodynamic
observables at criticality, the scaling properties of the distribution of
pseudo-critical temperatures over the ensemble of samples of size
. We then review our recent works on the critical properties of various
delocalization transitions involving random polymers, namely (i) the
bidimensional wetting (ii) the Poland-Scheraga model of DNA denaturation (iii)
the depinning transition of the selective interface model (iv) the freezing
transition of the directed polymer in a random medium.Comment: 20 pages, Conference Proceedings "Inhomogeneous Random Systems",
I.H.P., Paris, France, January 200
Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin-glass
The long-range one-dimensional Ising spin-glass with random couplings
decaying as presents a spin-glass phase
for (the limit corresponds to the
mean-field SK-model). We use the eigenvalue method introduced in our previous
work [C. Monthus and T. Garel, J. Stat. Mech. P12017 (2009)] to measure the
equilibrium time at temperature as a function of
the number of spins. We find the activated scaling with the same barrier exponent in the whole region
.Comment: v3=final version (12 pages
Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices
We consider the stochastic dynamics of the pure and random ferromagnetic
Ising model on the hierarchical diamond lattice of branching ratio with
fractal dimension . We adapt the Real Space
Renormalization procedure introduced in our previous work [C. Monthus and T.
Garel, J. Stat. Mech. P02037 (2013)] to study the equilibrium time
as a function of the system size near zero-temperature. For the pure Ising
model, we obtain the behavior
where is the interface dimension, and we compute the prefactor
exponent . For the random ferromagnetic Ising model, we derive the
renormalization rules for dynamical barriers near zero temperature. For the fractal dimension , we
obtain that the dynamical barrier scales as where
is a Gaussian random variable of non-zero-mean. While the non-random term
scaling as corresponds to the energy-cost of the creation of a system-size
domain-wall, the fluctuation part scaling as characterizes the
barriers for the motion of the system-size domain-wall after its creation. This
scaling corresponds to the dynamical exponent , in agreement with the
conjecture proposed in [C. Monthus and T. Garel, J. Phys. A 41,
115002 (2008)]. In particular, it is clearly different from the droplet
exponent involved in the statics of the random
ferromagnet on the same lattice.Comment: 24 pages, 7 figure
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