On the condition for correct convergence in the complex Langevin method

TL;DR

This paper revisits the justification of the complex Langevin method, identifying a key criterion involving the decay of the drift term's probability distribution that determines when the method yields correct results.

Contribution

It clarifies a subtlety in the CLM's justification, proposing a new criterion based on the decay rate of the drift term's distribution to assess result reliability.

Findings

The probability distribution of the drift term should decay exponentially or faster.

The criterion successfully predicts the trustworthiness of CLM results in examples.

Application to chiral Random Matrix Theory demonstrates the criterion's effectiveness.

Abstract

The complex Langevin method (CLM) provides a promising way to perform the path integral with a complex action using a stochastic equation for complexified dynamical variables. It is known, however, that the method gives wrong results in some cases, while it works, for instance, in finite density QCD in the deconfinement phase or in the heavy dense limit. Here we revisit the argument for justification of the CLM and point out a subtlety in using the time-evolved observables, which play a crucial role in the argument. This subtlety requires that the probability distribution of the drift term should fall off exponentially or faster at large magnitude. We demonstrate our claim in some examples such as chiral Random Matrix Theory and show that our criterion is indeed useful in judging whether the results obtained by the CLM are trustable or not.