3,834 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
Collapse transitions of a periodic hydrophilic hydrophobic chain
We study a single self avoiding hydrophilic hydrophobic polymer chain,
through Monte Carlo lattice simulations. The affinity of monomer for water
is characterized by a (scalar) charge , and the monomer-water
interaction is short-ranged. Assuming incompressibility yields an effective
short ranged interaction between monomer pairs , proportional to
. In this article, we take (resp.
()) for hydrophilic (resp. hydrophobic) monomers and consider a
chain with (i) an equal number of hydro-philic and -phobic monomers (ii) a
periodic distribution of the along the chain, with periodicity
. The simulations are done for various chain lengths , in (square
lattice) and (cubic lattice). There is a critical value of the
periodicity, which distinguishes between different low temperature structures.
For , the ground state corresponds to a macroscopic phase separation
between a dense hydrophobic core and hydrophilic loops. For (but not
too small), one gets a microscopic (finite scale) phase separation, and the
ground state corresponds to a chain or network of hydrophobic droplets, coated
by hydrophilic monomers. We restrict our study to two extreme cases, and to illustrate the physics of the various phase
transitions. A tentative variational approach is also presented.Comment: 21 pages, 17 eps figures, accepted for publication in Eur. Phys. J.
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
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
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