Dirac spectrum in the chirally symmetric phase of a gauge theory. I

TL;DR

This paper investigates how chiral symmetry restoration affects the Dirac spectrum in finite-temperature gauge theories, deriving conditions for symmetry restoration and implications for the U(1)_A symmetry and topological properties.

Contribution

It provides a rigorous proof linking the finiteness of susceptibilities to chiral symmetry restoration and explores the spectral constraints and topological implications in the chiral limit.

Findings

Chiral symmetry is restored iff susceptibilities do not diverge as m→0.

Finiteness of susceptibilities constrains the Dirac spectrum in the symmetric phase.

The topological properties resemble an instanton gas if U(1)_A remains broken.

Abstract

I study the consequences of chiral symmetry restoration for the Dirac spectrum in finite-temperature gauge theories in the two-flavor chiral limit, using Ginsparg--Wilson fermions on the lattice. I prove that chiral symmetry is restored at the level of the susceptibilities of scalar and pseudoscalar bilinears if and only if all these susceptibilities do not diverge in the chiral limit $m\to 0$, with $m$ the common mass of the light fermions. This implies in turn that they are infinitely differentiable functions of $m^2$ at $m=0$, or $m$ times such a function, depending on whether they contain an even or odd number of isosinglet bilinears. Expressing scalar and pseudoscalar susceptibilities in terms of the Dirac spectrum, I use their finiteness in the symmetric phase to derive constraints on the spectrum, and discuss the resulting implications for the fate of the anomalous $\mathrm{U}(1)_A$ symmetry in the chiral limit. I also discuss the differentiability properties of spectral quantities with respect to $m^2$, and show from first principles that the topological properties of the theory in the chiral limit are characterized by an instanton gas-like behavior if $\mathrm{U}(1)_A$ remains effectively broken.