Understanding the problem with logarithmic singularities in the complex Langevin method

TL;DR

This paper investigates why the complex Langevin method fails in cases with logarithmic singularities, revealing that the failure stems from the breakdown of key relations, but also identifies parameter regions where the method remains effective.

Contribution

It clarifies the cause of failures in the complex Langevin method due to singular drift terms and demonstrates 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

In recent years, there has been remarkable progress in theoretical justification of the complex Langevin method, which is a promising method for evading the sign problem in the path integral with a complex weight. There still remains, however, an issue concerning occasional failure of this method in the case where the action involves logarithmic singularities such as the one appearing from the fermion determinant in finite density QCD. In this talk, we point out that this failure is due to the breakdown of the relation between the complex weight which satisfies the Fokker-Planck equation and the probability distribution generated by the stochastic process. In fact, this kind of failure can occur in general when the stochastic process involves a singular drift term. We show, however, in simple examples, that there exists a parameter region in which the method works although the standard reweighting method is hardly applicable.