Comparative studies of the deformation techniques for the singular-drift problem in the complex Langevin method

TL;DR

This paper compares three deformation techniques in the complex Langevin method applied to a matrix model, aiming to address the singular-drift problem caused by near-zero eigenvalues of the Dirac operator.

Contribution

It introduces and tests three different deformation methods in a simplified matrix model to evaluate their effectiveness in resolving the singular-drift issue.

Findings

Deformation methods successfully avoid near-zero eigenvalues.

Results show consistency among the different deformation techniques.

The approach provides a viable way to improve complex Langevin simulations in QCD-like models.

Abstract

In application of the complex Langevin method to QCD at high density and low temperature, the singular-drift problem occurs due to the appearance of near-zero eigenvalues of the Dirac operator. In order to avoid this problem, we proposed to deform the Dirac operator in such a way that the near-zero eigenvalues do not appear and to extrapolate the deformation parameter to zero from the available data points. Here we test three different types of deformation in a simple large-$N$ matrix model, which undergoes an SSB due to the phase of the fermion determinant, and compare them to see the consistency with one another.