An improvement in complex Langevin dynamics from a view point of Lefschetz thimbles

TL;DR

This paper proposes a modified complex Langevin method inspired by Lefschetz thimbles, enabling accurate computation of observables in models with multiple thimbles by reducing to a single thimble scenario.

Contribution

It introduces a novel modification technique for complex Langevin dynamics based on Lefschetz thimble decomposition, improving accuracy in models with severe sign problems.

Findings

Correct results achieved with the modified method

Effective reduction to a single Lefschetz thimble

Overcomes failure of naive complex Langevin in toy model

Abstract

We develop a way of improving complex Langevin dynamics motivated by the Lefschetz-thimble decomposition of integrals. In our method, arbitrary observables of an original model with multiple Lefschetz thimbles are computed by a modified model with a single thimble. We apply our modification method to a one dimensional integral in which the naive implementation of the complex Langevin dynamics fails to reproduce the exact results due to the severe sign problem. We show that the toy model can be modified so that the new model consists of a single Lefschetz thimble. We find that correct results can be obtained by the improved complex Langevin dynamics.