Test for a universal behavior of Dirac eigenvalues in the complex Langevin method

TL;DR

This paper investigates the eigenvalue spacing distribution of Dirac operators in a complex Langevin simulation of chiral random matrix theory, revealing universal behaviors and their dependence on convergence correctness.

Contribution

It demonstrates that the universal Wigner distribution is preserved in the CL method for correct convergence, linking eigenvalue statistics to physical phase behavior.

Findings

Wigner distribution observed for correct convergence at small quark mass

Ginibre ensemble observed for wrong convergence cases

Universal eigenvalue behavior may be preserved in CL simulations

Abstract

We apply the complex Langevin (CL) method to a chiral random matrix theory (ChRMT) at non-zero chemical potential and study the nearest neighbor spacing (NNS) distribution of the Dirac eigenvalues. The NNS distribution is extracted using an unfolding procedure for the Dirac eigenvalues obtained in the CL method. For large quark mass, we find that the NNS distribution obeys the Ginibre ensemble as expected. For small quark mass, the NNS distribution follows the Wigner surmise for correct convergence case, while it follows the Ginibre ensemble for wrong convergence case. The Wigner surmise is physically reasonable from the chemical potential independence of the ChRMT. The Ginibre ensemble is known to be favored in a phase quenched QCD at finite chemical potential. Our result suggests a possibility that the originally universal behavior of the NNS distribution is preserved even in the CL method for correct convergence case.