Spectrum of the Wilson Dirac Operator at Finite Lattice Spacings

TL;DR

This paper investigates how discretization errors affect the microscopic spectrum of the Wilson Dirac operator at finite lattice spacings, using chiral Perturbation Theory and Random Matrix Theory to analyze spectral densities and eigenvalue distributions.

Contribution

It demonstrates that chiral Random Matrix Theory reproduces key features of Wilson chiral Perturbation Theory and constrains low-energy constants based on Hermiticity properties.

Findings

Random Matrix Theory matches Wilson chiral Perturbation Theory predictions

Spectral density and chirality distribution are characterized at fixed index

Low-energy constants are constrained by Hermiticity

Abstract

We consider the effect of discretization errors on the microscopic spectrum of the Wilson Dirac operator using both chiral Perturbation Theory and chiral Random Matrix Theory. A graded chiral Lagrangian is used to evaluate the microscopic spectral density of the Hermitian Wilson Dirac operator as well as the distribution of the chirality over the real eigenvalues of the Wilson Dirac operator. It is shown that a chiral Random Matrix Theory for the Wilson Dirac operator reproduces the leading zero-momentum terms of Wilson chiral Perturbation Theory. All results are obtained for fixed index of the Wilson Dirac operator. The low-energy constants of Wilson chiral Perturbation theory are shown to be constrained by the Hermiticity properties of the Wilson Dirac operator.