Polyakov loops and spectral properties of the staggered Dirac operator

TL;DR

This paper investigates the spectral properties of the staggered Dirac operator in SU(2) gauge fields near the free limit, revealing a multi-scale cluster structure influenced by lattice geometry and Polyakov loops, with analysis based on random matrix theory.

Contribution

It introduces an analytical formula explaining the emergence of multiple spectral scales and links spectral behavior to Polyakov loops and lattice geometry in SU(2) gauge fields.

Findings

Identification of a three-scale cluster structure in the spectrum.

Derivation of an analytical formula for spectral scale emergence.

Spectral statistics align with random matrix theory predictions.

Abstract

We study the spectrum of the staggered Dirac operator in SU(2) gauge fields close to the free limit, for both the fundamental and the adjoint representation. Numerically we find a characteristic cluster structure with spacings of adjacent levels separating into three scales. We derive an analytical formula which explains the emergence of these different spectral scales. The behavior on the two coarser scales is determined by the lattice geometry and the Polyakov loops, respectively. Furthermore, we analyze the spectral statistics on all three scales, comparing to predictions from random matrix theory.