Localization and chiral properties near the ordering transition of an Anderson-like toy model for QCD

Abstract: The Dirac operator in finite temperature QCD is equivalent to the Hamiltonian of an unconventional Anderson model, with on-site noise provided by the fluctuations of the Polyakov lines. The main features of its spectrum and eigenvectors, concerning the density of low modes and their localization properties, are qualitatively reproduced by a toy-model random Hamiltonian, based on an Ising-type spin model mimicking the dynamics of the Polyakov lines. Here we study the low modes of this toy model in the vicinity of the ordering transition of the spin model, and show that at the critical point the spectral density at the origin has a singularity, and the localization properties of the lowest modes change. This provides further evidence of the close relation between deconfinement, chiral transition and localization of the low modes.