The Ising Model with Hybrid Monte Carlo

TL;DR

This paper introduces a novel formalism applying Hybrid Monte Carlo to the Ising model, enabling simulations with continuous variables and greater flexibility for complex lattice and coupling configurations.

Contribution

The authors develop a formalism that extends Hybrid Monte Carlo to the Ising model, allowing for simulations on arbitrary lattices and couplings, and facilitating advanced algorithm integration.

Findings

Enables HMC simulation of the Ising model with discrete spins

Allows for flexible modifications and generalizations of the model

Facilitates the use of advanced algorithms like shift preconditioners

Abstract

The Ising model is a simple statistical model for ferromagnetism. There are analytic solutions for low dimensions and very efficient Monte Carlo methods, such as cluster algorithms, for simulating this model in special cases. However most approaches do not generalise to arbitrary lattices and couplings. We present a formalism that allows one to apply Hybrid Monte Carlo (HMC) simulations to the Ising model, demonstrating how a system with discrete degrees of freedom can be simulated with continuous variables. Because of the flexibility of HMC, our formalism is easily generalizable to arbitrary modifications of the model, creating a route to leverage advanced algorithms such as shift preconditioners and multi-level methods, developed in conjunction with HMC.