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 on 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 $(3\le q\le 12)$ 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 five-figure accuracy. Finally, we propose several efficient schemes aimed at estimating the microcanonical temperature.

20 pages; to appear in Phys. Rev. E

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• Universality aspects of the 2d random-bond Ising and 3d Blume-Capel models
• Optimized multicanonical simulations: a new proposal based on classical fluctuation theory
• Multicanonical Sampling of the Space of States of H(2,n)-Vector Models
• Re-orientation Transition in Molecular Thin Films: Potts Model with Dipolar Interaction
• Extended Defects in the Potts-Percolation Model of a Solid: Renormalization Group and Monte Carlo Analysis
• On the nature of the phase transitions in two-dimensional type II superconductors