Complex Langevin Simulation of a Random Matrix Model at Nonzero Chemical Potential

TL;DR

This paper evaluates the complex Langevin algorithm for simulating a random matrix model of QCD at nonzero chemical potential, identifying effective fixes for convergence issues and comparing different methods' performances.

Contribution

It systematically tests various fixes for complex Langevin convergence problems in a QCD-like matrix model, highlighting the effectiveness of reweighted complex Langevin.

Findings

Reweighted complex Langevin reproduces analytical results where other methods fail.

Gauge cooling solves convergence issues in a related model with milder sign problem.

Naive implementation yields phase quenched results, matching analytical predictions.

Abstract

In this paper we test the complex Langevin algorithm for numerical simulations of a random matrix model of QCD with a first order phase transition to a phase of finite baryon density. We observe that a naive implementation of the algorithm leads to phase quenched results, which were also derived analytically in this article. We test several fixes for the convergence issues of the algorithm, in particular the method of gauge cooling, the shifted representation, the deformation technique and reweighted complex Langevin, but only the latter method reproduces the correct analytical results in the region where the quark mass is inside the domain of the eigenvalues. In order to shed more light on the issues of the methods we also apply them to a similar random matrix model with a milder sign problem and no phase transition, and in that case gauge cooling cooling solves the convergence problems as was shown before in the literature.