Analysis of Random Number Generators Using Monte Carlo Simulation

Revisions are almost entirely in the introduction and conclusion. Results are unchanged, however the comments and recommendations on different generators were changed, and more references were added.Comment: Email: [email protected] 16 pages, Latex with 1 postscript figure. NPAC technical report SCCS-52

Particle-hole symmetric localization in two dimensions

We revisit two-dimensional particle-hole symmetric sublattice localization problem, focusing on the origin of the observed singularities in the density of states $\rho(E)$ at the band center E=0. The most general such system [R. Gade, Nucl. Phys. B {\bf 398}, 499 (1993)] exhibits critical behavior and has $\rho(E)$ that diverges stronger than any integrable power-law, while the special {\it random vector potential model} of Ludwiget al [Phys. Rev. B {\bf 50}, 7526 (1994)] has instead a power-law density of states with a continuously varying dynamical exponent. We show that the latter model undergoes a dynamical transition with increasing disorder--this transition is a counterpart of the static transition known to occur in this system; in the strong-disorder regime, we identify the low-energy states of this model with the local extrema of the defining two-dimensional Gaussian random surface. Furthermore, combining this ``surface fluctuation'' mechanism with a renormalization group treatment of a related vortex glass problem leads us to argue that the asymptotic low $E$ behavior of the density of states in the {\it general} case is $\rho(E) \sim E^{-1} e^{-|\ln E|^{2/3}}$, different from earlier prediction of Gade. We also study the localized phases of such particle-hole symmetric systems and identify a Griffiths ``string'' mechanism that generates singular power-law contributions to the low-energy density of states in this case.Comment: 18 pages (two-column PRB format), 10 eps figures include

Real-space renormalization group for the random-field Ising model

We present real--space renormalization group (RG) calculations of the critical properties of the random--field Ising model on a cubic lattice in three dimensions. We calculate the RG flows in a two--parameter truncation of the Hamiltonian space. As predicted, the transition at finite randomness is controlled by a zero temperature, disordered critical fixed point, and we exhibit the universal crossover trajectory from the pure Ising critical point. We extract scaling fields and critical exponents, and study the distribution of barrier heights between states as a function of length scale.Comment: 12 pages, CU-MSC-757

Universality aspects of the d=3 random-bond Blume-Capel model

The effects of bond randomness on the universality aspects of the simple cubic lattice ferromagnetic Blume-Capel model are discussed. The system is studied numerically in both its first- and second-order phase transition regimes by a comprehensive finite-size scaling analysis. We find that our data for the second-order phase transition, emerging under random bonds from the second-order regime of the pure model, are compatible with the universality class of the 3d random Ising model. Furthermore, we find evidence that, the second-order transition emerging under bond randomness from the first-order regime of the pure model, belongs to a new and distinctive universality class. The first finding reinforces the scenario of a single universality class for the 3d Ising model with the three well-known types of quenched uncorrelated disorder (bond randomness, site- and bond-dilution). The second, amounts to a strong violation of universality principle of critical phenomena. For this case of the ex-first-order 3d Blume-Capel model, we find sharp differences from the critical behaviors, emerging under randomness, in the cases of the ex-first-order transitions of the corresponding weak and strong first-order transitions in the 3d three-state and four-state Potts models.Comment: 12 pages, 12 figure

Singular Density of States of Disordered Dirac Fermions in the Chiral Models

The Dirac fermion in the random chiral models is studied which includes the random gauge field model and the random hopping model. We focus on a connection between continuum and lattice models to give a clear perspective for the random chiral models. Two distinct structures of density of states (DoS) around zero energy, one is a power-law dependence on energy in the intermediate energy range and the other is a diverging one at zero energy, are revealed by an extensive numerical study for large systems up to $250\times 250$. For the random hopping model, our finding of the diverging DoS within very narrow energy range reconciles previous inconsistencies between the lattice and the continuum models.Comment: 4 pages, 4 figure

Strong-disorder renormalization for interacting non-Abelian anyon systems in two dimensions

We consider the effect of quenched spatial disorder on systems of interacting, pinned non-Abelian anyons as might arise in disordered Hall samples at filling fractions \nu=5/2 or \nu=12/5. In one spatial dimension, such disordered anyon models have previously been shown to exhibit a hierarchy of infinite randomness phases. Here, we address systems in two spatial dimensions and report on the behavior of Ising and Fibonacci anyons under the numerical strong-disorder renormalization group (SDRG). In order to manage the topology-dependent interactions generated during the flow, we introduce a planar approximation to the SDRG treatment. We characterize this planar approximation by studying the flow of disordered hard-core bosons and the transverse field Ising model, where it successfully reproduces the known infinite randomness critical point with exponent \psi ~ 0.43. Our main conclusion for disordered anyon models in two spatial dimensions is that systems of Ising anyons as well as systems of Fibonacci anyons do not realize infinite randomness phases, but flow back to weaker disorder under the numerical SDRG treatment.Comment: 12 pages, 12 figures, 1 tabl