Application of the path optimization method to a discrete spin system

TL;DR

This paper applies the path optimization method, originally designed for quantum field theories, to a discrete spin system, demonstrating its effectiveness in mitigating the sign problem and accurately reproducing known analytic results.

Contribution

It extends the path optimization method to discrete spin models using Hubbard-Stratonovich transformation, showing its potential to address the sign problem in such systems.

Findings

Enhanced average phase factor indicating weakened sign problem

Reproduction of analytic values with controlled errors

Demonstration of method's applicability to discrete spin systems

Abstract

The path optimization method, which is proposed to control the sign problem in quantum field theories with continuous degrees of freedom by machine learning, is applied to a spin model with discrete degrees of freedom. The path optimization method is applied by replacing the spins with dynamical variables via the Hubbard-Stratonovich transformation, and the sum with the integral. The one-dimensional (Lenz-)Ising model with a complex coupling constant is used as a laboratory for the sign problem in the spin model. The average phase factor is enhanced by the path optimization method, indicating that the method can weaken the sign problem. Our result reproduces the analytic values with controlled statistical errors.