Localization of Dirac modes in finite-temperature $\mathbb{Z}_2$ gauge theory on the lattice

TL;DR

This study investigates how the localization of Dirac eigenmodes in finite-temperature $ ext{Z}_2$ gauge theory correlates with the deconfinement transition, revealing sector-dependent localization behavior and novel high-spectrum localized modes.

Contribution

It demonstrates the connection between deconfinement and Dirac mode localization in a simple gauge theory, including new observations of localized modes at high energies across phases.

Findings

Localization of low Dirac modes occurs in the deconfined phase in the physical sector.

Localized modes correlate with Polyakov loop fluctuations and negative plaquettes.

Localized modes are present at high spectrum ends in all phases and sectors.

Abstract

We study the localization properties of the eigenmodes of the staggered Dirac operator in finite-temperature $\mathbb{Z}_2$ pure gauge theory on the lattice in 2+1 dimensions. We find that the low modes turn from delocalized to localized as the system crosses over from the confined to the deconfined phase in the "physical" sector (positive average Polyakov loop) selected by external fermionic probes, while they remain delocalized in the "unphysical" sector (negative average Polyakov loop). This confirms that the close connection between deconfinement and localization of the low Dirac modes in the physical sector, already observed in other models, holds also in the simplest gauge theory displaying a deconfinement transition. We also observe a clear correlation of localized modes with fluctuations of the Polyakov loop away from the ordered value, as expected according to the "sea/islands" picture of localization, and with clusters of negative plaquettes. A novel finding is the presence of localized modes at the high end of the Dirac spectrum in all phases/sectors of the theory.