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

• Published in: The journal of high energy physics Vol. 2011; no. 12
• Main Authors: Kanazawa, Takuya, Wettig, Tilo, Yamamoto, Naoki
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-12-2011; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Condensates; Density; Eigenvalues; Elementary Particles; High energy physics; Linear algebra; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Quarks; Relativity Theory; Spectral correlation; String Theory; Sum rules; Chiral Lagrangians; Spontaneous Symmetry Breaking; Lattice Gauge Field Theories
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP12(2011)007

A
bstract
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 Diracoperators. Our results can in principle be tested in lattice simulations.