Testing the criterion for correct convergence in the complex Langevin method

TL;DR

This paper evaluates a criterion based on the drift term distribution to identify correct convergence regions of the complex Langevin method, improving reliability in sign problem scenarios.

Contribution

It demonstrates the effectiveness of a new convergence criterion in two complex theories, enhancing the CLM's reliability for practical applications.

Findings

The criterion accurately identifies correct convergence regions.

It successfully applied to two-dimensional Yang-Mills theory.

It validated the CLM results in finite density QCD simulations.

Abstract

Recently the complex Langevin method (CLM) has been attracting attention as a solution to the sign problem, which occurs in Monte Carlo calculations when the effective Boltzmann weight is not real positive. An undesirable feature of the method, however, was that it can happen in some parameter regions that the method yields wrong results even if the Langevin process reaches equilibrium without any problem. In our previous work, we proposed a practical criterion for correct convergence based on the probability distribution of the drift term that appears in the complex Langevin equation. Here we demonstrate the usefulness of this criterion in two solvable theories with many dynamical degrees of freedom, i.e., two-dimensional Yang-Mills theory with a complex coupling constant and the chiral Random Matrix Theory for finite density QCD, which were studied by the CLM before. Our criterion can indeed tell the parameter regions in which the CLM gives correct results.