Localization of Dirac modes in finite-temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> gauge theory on the lattice

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