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

TL;DR

This paper investigates the relationship between Dirac mode localization and deconfinement in a 2+1D $ ext{Z}_2$ gauge theory, demonstrating that low modes localize at the deconfined phase and correlate with disorder and negative plaquettes.

Contribution

It provides the first evidence in a simple lattice gauge model that Dirac mode localization is associated with the deconfined phase, supporting the conjecture of a universal connection.

Findings

Low Dirac modes are delocalized in the confined phase.

Low Dirac modes become localized in the deconfined phase.

Localized modes correlate with disorder in Polyakov loop and negative plaquettes.

Abstract

The low-lying Dirac modes become localised at the finite-temperature transition in QCD and in other gauge theories, suggesting a general connection between their localisation and deconfinement. The simplest model where this connection can be tested is $\mathbb{Z}_2$ gauge theory in 2+1 dimensions. We show that in this model the low modes in the staggered Dirac spectrum are delocalised in the confined phase and become localised in the deconfined phase. We also show that localised modes correlate with disorder in the Polyakov loop configuration, in agreement with the "sea/island" picture of localisation, and with negative plaquettes. These results further support the conjecture that localisation and deconfinement are closely related.