Individual Eigenvalue Distributions for the Wilson Dirac Operator

TL;DR

This paper derives the distributions of individual eigenvalues for the Wilson Dirac Operator and its Hermitian counterpart within the epsilon regime, providing explicit formulas and expansions applicable to various parameters and lattice spacings.

Contribution

It introduces a perturbative framework for eigenvalue distributions of the Wilson Dirac Operator applicable to any number of flavors and low energy constants, with explicit examples in the microscopic domain.

Findings

Explicit formulas for eigenvalue distributions derived.

Expansion truncates after nu terms for small lattice spacing.

Application demonstrated with quenched D5 and DW eigenvalues.

Abstract

We derive the distributions of individual eigenvalues for the Hermitian Wilson Dirac Operator D5 as well as for real eigenvalues of the Wilson Dirac Operator DW. The framework we provide is valid in the epsilon regime of chiral perturbation theory for any number of flavours Nf and for non-zero low energy constants W6, W7, W8. It is given as a perturbative expansion in terms of the k-point spectral density correlation functions and integrals thereof, which in some cases reduces to a Fredholm Pfaffian. For the real eigenvalues of DW at fixed chirality nu this expansion truncates after at most nu terms for small lattice spacing "a". Explicit examples for the distribution of the first and second eigenvalue are given in the microscopic domain as a truncated expansion of the Fredholm Pfaffian for quenched D5, where all k-point densities are explicitly known from random matrix theory. For the real eigenvalues of quenched DW at small "a" we illustrate our method by the finite expansion of the corresponding Fredholm determinant of size nu.