Deconfinement transition and localization of Dirac modes in finite-temperature $\mathbb{Z}_3$ gauge theory on the lattice

We study the localization properties of the eigenmodes of the staggered Dirac operator across the deconfinement transition in finite-temperature $\mathbb{Z}_3$ pure gauge theory on the lattice in 2+1 dimensions. This allows for nontrivial tests of the sea-islands picture of localization, according to which low modes should localize on favorable Polyakov-loop fluctuations in the deconfined phase of a gauge theory. We observe localized low modes in the deconfined phase of the theory, both in the real Polyakov-loop sector, where they are expected, and in the complex Polyakov-loop sectors, where they are not. Our findings expose the limitations of the standard sea-islands picture, and call for its refinement. An improved picture, where spatial hopping terms play a more prominent role, is proposed and found to be in excellent agreement with numerical results.