Numerical analysis of the spectrum of the Dirac operator in four-dimensional SU(2) gauge fields

TL;DR

This paper introduces two numerical algorithms for computing eigenvalues of Dirac operators in lattice gauge theories and applies them to analyze the spectral properties in SU(2) gauge fields, providing detailed spectra and spectral densities.

Contribution

It presents an accelerated conjugate gradient and a Lanczos method for eigenvalue computation, with application to SU(2) gauge fields in lattice QCD.

Findings

Complete spectra for lattices up to 8^3 x 12 are obtained.

Numerical values for fermionic determinants are derived.

Spectral densities are analyzed and presented.

Abstract

Two numerical algorithms for the computation of eigenvalues of Dirac operators in lattice gauge theories are described: one is an accelerated conjugate gradient method, the other one a standard Lanczos method. Results obtained by Cullum's and Willoughby's variant of the Lanczos method (whose convergence behaviour is closely linked with the local spectral density) are presented for euclidean Wilson fermions in quenched and unquenched SU(2) gauge fields. Complete spectra are determined on lattices up to $8^3 \cdot 12$, and we derive numerical values for fermionic determinants and results for spectral densities.