A directed walk in probability space that locates mean field solutions to spin models

TL;DR

This paper introduces a novel functional optimization method to efficiently find mean field solutions for complex spin models, outperforming traditional Monte Carlo techniques in speed and accuracy.

Contribution

The paper presents a new approach using functional optimization to derive self-consistent mean field equations for classical continuous spins with non-local interactions.

Findings

Outperforms Monte Carlo in convergence speed

Achieves higher accuracy in mean field solutions

Extends to complex and continuum spin systems

Abstract

Despite their formal simplicity, most lattice spin models cannot be easily solved, even under the simplifying assumptions of mean field theory. In this manuscript, we present a method for generating mean field solutions to classical continuous spins. We focus our attention on systems with non-local interactions and non-periodic boundaries, which require careful handling with existing approaches, such as Monte Carlo sampling. Our approach utilizes functional optimization to derive a closed-form optimality condition and arrive at self-consistent mean field equations. We show that this approach significantly outperforms conventional Monte Carlo sampling in convergence speed and accuracy. To convey the general concept behind the approach, we first demonstrate its application to a simple system - a finite one-dimensional dipolar chain in an external electric field. We then describe how the approach naturally extends to more complicated spin systems and to continuum field theories. Furthermore, we numerically illustrate the efficacy of our approach by highlighting its utility on nonperiodic spin models of various dimensionality.