Fast flat-histogram method for generalized spin models

Abstract

We present a Monte Carlo method that efficiently computes the density of states for spin models having any number of interaction per spin. By combining a random-walk in the energy space with collective updates controlled by the microcanonical temperature, our method yields dynamic exponents close to their ideal random-walk values, reduced equilibrium times, and very low statistical error in the density of states. The method can host any density of states estimation scheme, including the Wang-Landau algorithm and the transition matrix method. Our approach proves remarkably powerful in the numerical study of models governed by long-range interactions, where it is shown to reduce the algorithm complexity to that of a short-range model with the same number of spins. We apply the method to the q-state Potts chains (3 ≤ q ≤ 12) with power-law decaying interactions in their first-order regime; we find that conventional local-update algorithms are outperformed already for sizes above a few hundred spins. By considering chains containing up to 2 16 spins, which we simulated in fairly reasonable time, we obtain estimates of transition temperatures correct to fivefigure accuracy. Finally, we propose several efficient schemes aimed at estimating the microcanonical temperature.

Nature of phase transitions in two-dimensional type-II superconductors

“…so that the results no longer depend on the normalization of the wave function. The periodicity of the expression (52) shows that both m x and m y take integer values from 0 to N − 1. Thus…”

“…There are various refinements one can make to this process 45,[52][53][54][55][56] . In general, it has been shown that more complex patterns of choosing f can lead to faster convergence.…”

“…(C5) by a factor is not very effective at convergence. There are a number of alternative approaches where we increase as well as decrease these updates in an suitable pattern 45,[52][53][54][55][56]59 , many of which we attempted.…”

Fast Simulation for the Performance Evaluation of LDPC Codes using Fast Flat Histogram Method

“…In statistical physics, the Fast Flat Histogram method [9] employing Wang Landau algorithm [10,11] was introduced to estimate the density of states g(E), i.e., the number of all possible states for an energy level E of the system. The algorithm is based on the observation that if a random walk in energy space is performed by flipping spins randomly for a spin system and the probability to visit a given energy level E is proportional to the reciprocal of the density of states 1/g(E), then a flat histogram is generated for the energy distribution.…”

Optimized multicanonical simulations: A proposal based on classical fluctuation theory

“…Several proposals have been presented to generalize cluster update approaches in multicanonical ensemble simulations, either using spin-bond representations of the partition function, [24,25,26,27], or cluster building algorithms based on alternative ways of computing the microcanonical temperature, β(E) [28,29].…”