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We study the problems of chiral symmetry breaking and eigenmode localisationin finite-temperature QCD by looking at the lattice Dirac operator as a randomHamiltonian. We recast the staggered Dirac operator into an unconventionalthree-dimensional Anderson Hamiltonian ("Dirac-Anderson Hamiltonian") carryinginternal degrees of freedom, with disorder provided by the fluctuations of thegauge links. In this framework, we identify the features relevant to chiralsymmetry restoration and localisation of the low-lying Dirac eigenmodes in theordering of the local Polyakov lines, and in the related correlation betweenspatial links across time slices, thus tying the two phenomena to thedeconfinement transition. We then build a toy model based on QCD and on theDirac-Anderson approach, replacing the Polyakov lines with spin variables andsimplifying the dynamics of the spatial gauge links, but preserving theabove-mentioned relevant dynamical features. Our toy model successfullyreproduces the main features of the QCD spectrum and of the Dirac eigenmodesconcerning chiral symmetry breaking and localisation, both in the ordered(deconfined) and disordered (confined) phases. Moreover, it allows us to studyseparately the roles played in the two phenomena by the diagonal and theoff-diagonal terms of the Dirac-Anderson Hamiltonian. Our results support ourexpectation that chiral symmetry restoration and localisation of the low modesare closely related, and that both are triggered by the deconfinementtransition.

An Anderson-like model of the QCD chiral transition