Localisation of Dirac modes in gauge theories and Goldstone's theorem at finite temperature

TL;DR

This paper explores how localized near-zero Dirac modes at finite temperature can affect Goldstone's theorem in gauge theories, potentially removing massless excitations despite a non-zero chiral condensate.

Contribution

It provides a generalized proof of Goldstone's theorem at finite temperature and analyzes how localized Dirac modes influence the existence of massless excitations.

Findings

Localized near-zero modes can cause divergence in pseudoscalar correlators.

Massless quasi-particles may disappear despite non-zero chiral condensate.

The ratio of the mobility edge to fermion mass determines different physical scenarios.

Abstract

I discuss the possible effects of a finite density of localised near-zero Dirac modes in the chiral limit of gauge theories with $N_f$ degenerate fermions. I focus in particular on the fate of the massless quasi-particle excitations predicted by the finite-temperature version of Goldstone's theorem, for which I provide an alternative and generalised proof based on a Euclidean ${\rm SU}(N_f)_A$ Ward-Takahashi identity. I show that localised near-zero modes can lead to a divergent pseudoscalar-pseudoscalar correlator that modifies this identity in the chiral limit. As a consequence, massless quasi-particle excitations can disappear from the spectrum of the theory in spite of a non-zero chiral condensate. Three different scenarios are possible, depending on the detailed behaviour in the chiral limit of the ratio of the mobility edge and the fermion mass, which I prove to be a renormalisation-group invariant quantity.