Singular values of the Dirac operator at nonzero density

TL;DR

This paper investigates the spectral properties of the Dirac operator at nonzero density, deriving exact relations and sum rules for its singular values, which reveal insights into condensates in QCD-like theories.

Contribution

The authors construct low-energy effective theories across different density regimes and derive new exact results for the Dirac singular values, including index theorems and sum rules.

Findings

Derived Banks-Casher-type relations for diquark and pionic condensates.

Established Smilga-Stern-type relations for the slope of the singular-value density.

Presented a rigorous index theorem for non-Hermitian Dirac operators.

Abstract

At nonzero density the eigenvalues of the Dirac operator move into the complex plane, while its singular values remain real and nonnegative. In QCD-like theories, the singular-value spectrum carries information on the diquark (or pionic) condensate. We have constructed low-energy effective theories in different density regimes and derived a number of exact results for the Dirac singular values, including Banks-Casher-type relations for the diquark (or pionic) 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 also present a rigorous index theorem for non-Hermitian Dirac operators.