Fast Flat-Histogram Method for Generalized Spin Models

TL;DR

This paper introduces a fast Monte Carlo method for calculating the density of states in generalized spin models, significantly improving efficiency and accuracy, especially for models with long-range interactions and large system sizes.

Contribution

The authors develop a versatile Monte Carlo approach combining collective updates and microcanonical temperature control, applicable to any density of states estimation scheme, enhancing performance for complex spin systems.

Findings

Reduces algorithm complexity for long-range interactions to that of short-range models.

Outperforms local-update algorithms for chains with over a few hundred spins.

Achieves transition temperature estimates with five-figure accuracy for large systems.

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 $(3\leq q \leq 12)$ 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.