Singular values of the Dirac operator in dense QCD-like theories

TL;DR

This paper investigates the singular values of the Dirac operator in dense QCD-like theories, revealing their spectral properties, deriving exact formulas, and constructing random matrix models, with potential for lattice simulation validation.

Contribution

It introduces a comprehensive analysis of Dirac singular values across different density regimes, including new formulas, effective theories, and a rigorous index theorem for non-Hermitian operators.

Findings

Singular values are real and nonnegative at any nonzero density.

Spectral scale set by diquark condensate, differing from eigenvalues.

Derived exact formulas and constructed random matrix models.

Abstract

We study the singular values of the Dirac operator in dense QCD-like theories at zero temperature. The Dirac singular values are real and nonnegative at any nonzero quark density. The scale of their spectrum is set by the diquark condensate, in contrast to the complex Dirac eigenvalues whose scale is set by the chiral condensate at low density and by the BCS gap at high density. We identify three different low-energy effective theories with diquark sources applicable at low, intermediate, and high density, together with their overlapping domains of validity. We derive a number of exact formulas for the Dirac singular values, including Banks-Casher-type relations for the diquark condensate, Smilga-Stern-type relations for the slope of the singular value density, and Leutwyler-Smilga-type sum rules for the inverse singular values. We construct random matrix theories and determine the form of the microscopic spectral correlation functions of the singular values for all nonzero quark densities. We also derive a rigorous index theorem for non-Hermitian Dirac operators. Our results can in principle be tested in lattice simulations.