Comparison of complex Langevin and mean field methods applied to effective Polyakov line models

TL;DR

This paper compares complex Langevin and mean field methods for solving effective Polyakov line models with a sign problem, highlighting where they agree or diverge due to complex Langevin's branch cut issues.

Contribution

It systematically applies and compares complex Langevin and mean field methods to SU(3) gauge-matter models at finite chemical potential, identifying the causes of divergence.

Findings

Methods agree almost perfectly when results match.

Divergence occurs due to branch cut crossing in complex Langevin.

Complex Langevin faces issues with logarithmic branch cuts.

Abstract

Effective Polyakov line models, derived from SU(3) gauge-matter systems at finite chemical potential, have a sign problem. In this article I solve two such models, derived from SU(3) gauge-Higgs and heavy quark theories by the relative weights method, over a range of chemical potentials where the sign problem is severe. Two values of the gauge-Higgs coupling are considered, corresponding to a heavier and a lighter scalar particle. Each model is solved via the complex Langevin method, following the approach of Aarts and James, and also by a mean field technique. It is shown that where the results of mean field and complex Langevin agree, they agree almost perfectly. Where the results of the two methods diverge, it is found that the complex Langevin evolution has a branch cut crossing problem, associated with a logarithm in the action, that was pointed out by Mollgaard and Splittorff.