Gauge cooling for the singular-drift problem in the complex Langevin method --a test in Random Matrix Theory for finite density QCD

TL;DR

This paper introduces a gauge cooling technique to address the singular-drift problem in the complex Langevin method, demonstrating its effectiveness in Random Matrix Theory models of finite density QCD, especially with light quarks.

Contribution

The study proposes a new gauge cooling criterion to overcome singular drift issues in the complex Langevin method for light quark regimes, validated in Random Matrix Theory.

Findings

Gauge cooling alters eigenvalue distributions of the Dirac operator.

Eigenvalue distribution exhibits universal divergence at the origin in the chiral limit.

Exact results are reproduced at zero temperature with light quarks.

Abstract

Recently, the complex Langevin method has been applied successfully to finite density QCD either in the deconfinement phase or in the heavy dense limit with the aid of a new technique called the gauge cooling. In the confinement phase with light quarks, however, convergence to wrong limits occurs due to the singularity in the drift term caused by small eigenvalues of the Dirac operator including the mass term. We propose that this singular-drift problem should also be overcome by the gauge cooling with different criteria for choosing the complexified gauge transformation. The idea is tested in chiral Random Matrix Theory for finite density QCD, where exact results are reproduced at zero temperature with light quarks. It is shown that the gauge cooling indeed changes drastically the eigenvalue distribution of the Dirac operator measured during the Langevin process. Despite its non-holomorphic nature, this eigenvalue distribution has a universal diverging behavior at the origin in the chiral limit due to a generalized Banks-Casher relation as we confirm explicitly.