Spectral properties of the Wilson Dirac operator in the $\epsilon$-regime

TL;DR

This paper studies the spectral behavior of the Wilson Dirac operator in quenched QCD within the epsilon-regime, comparing eigenvalue distributions across topological sectors to random matrix theory predictions.

Contribution

It demonstrates the agreement between spectral data and theoretical predictions, enabling extraction of low-energy constants for Wilson fermions from eigenvalue spectra.

Findings

Good agreement with random matrix theory for small volumes and coarse lattices

Eigenvalue distributions can distinguish topological sectors

Low-energy constants can be extracted from spectral properties

Abstract

We investigate the spectral properties of the Wilson Dirac operator in quenched QCD in the microscopic regime. We distinguish the topological sectors using the index as determined by the Wilson flow method. Consequently, the distributions of the low-lying eigenvalues of the Wilson Dirac operator can be compared in each of the topological sectors to predictions from random matrix theory applied to the $\epsilon$-regime of chiral perturbation theory. We find rather good agreement for volumes as small as $(1.5 \, \text{fm})^4$ and lattice spacings as coarse as $0.1\, \text{fm}$, and demonstrate that it is indeed possible to extract low-energy constants for Wilson fermions from the spectral properties of the Wilson Dirac operator.