Unification of the complex Langevin method and the Lefschetz-thimble method

TL;DR

This paper proposes a unified framework combining the complex Langevin and Lefschetz thimble methods to address the sign problem in complex-weight statistical systems, demonstrating interpolation between the two approaches.

Contribution

The paper introduces a unified formulation that integrates Langevin dynamics and holomorphic gradient flow, clarifying the relationship between two key methods for the sign problem.

Findings

The unified formulation effectively addresses the sign problem.

Interpolation between the two methods is achieved by varying flow time.

Application to a simple model demonstrates the approach's validity.

Abstract

Recently there has been remarkable progress in solving the sign problem, which occurs in investigating statistical systems with a complex weight. The two promising methods, the complex Langevin method and the Lefschetz thimble method, share the idea of complexifying the dynamical variables, but their relationship has not been clear. Here we propose a unified formulation, in which the sign problem is taken care of by both the Langevin dynamics and the holomorphic gradient flow. We apply our formulation to a simple model in three different ways and show that one of them interpolates the two methods by changing the flow time.