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

TL;DR

This work establishes a rigorous, first-principles connection between chiral-symmetry restoration in the two-flavor chiral limit and the behavior of the Dirac spectrum in finite-temperature gauge theories with Ginsparg–Wilson fermions. By deriving an explicit generating-function representation of scalar and pseudoscalar susceptibilities in terms of Dirac eigenvalues and proving that restoration is equivalent to finiteness (and hence $m^2$-differentiability) of these susceptibilities, the paper translates a symmetry requirement into concrete spectral constraints. Under mild extensions (nonlocal restoration), spectral observables such as the Dirac spectral density inherit the same differentiability properties, effectively justifying $m^2$-analyticity assumptions used in prior work. The author then derives a set of first- and second-order spectral constraints, linking near-zero-mode correlations to topology, and shows that in the symmetric phase the topological cumulants reproduce ideal instanton-gas behavior to leading order if $U(1)_A$ remains effectively broken; conversely, a nonzero $U(1)_A$-breaking order parameter implies an instanton-gas-like topological structure in the chiral limit. These results lay a principled foundation for a detailed spectral analysis in the symmetric phase and pave the way for further examination of eigenvalue correlators in the subsequent paper in this series.

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.