New insights into the problem with a singular drift term in the complex Langevin method

TL;DR

This paper investigates the failure of the complex Langevin method caused by singular drift terms, identifying the underlying issues and demonstrating a parameter region where the method remains effective despite these challenges.

Contribution

It clarifies the cause of failures in the complex Langevin method due to singular drifts and shows conditions under which the method can still succeed.

Findings

Failure linked to breakdown of Fokker-Planck relation

Singular drift terms cause stochastic process issues

Existence of parameter regions where the method works

Abstract

The complex Langevin method aims at performing path integral with a complex action numerically based on complexification of the original real dynamical variables. One of the poorly understood issues concerns occasional failure in the presence of logarithmic singularities in the action, which appear, for instance, from the fermion determinant in finite density QCD. We point out that the failure should be attributed to the breakdown of the relation between the complex weight that satisfies the Fokker-Planck equation and the probability distribution associated with the stochastic process. In fact, this problem can occur in general when the stochastic process involves a singular drift term. We show, however, in a simple example that there exists a parameter region in which the method works although the standard reweighting method is hardly applicable.