The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way

TL;DR

This paper demonstrates that the near-zero spectrum of the QCD Dirac operator can be modeled by three classes of chiral random matrix ensembles, matching different gauge group symmetries and reproducing known sum rules.

Contribution

It establishes a correspondence between QCD Dirac spectra and three chiral random matrix classes based on gauge group symmetries, extending the threefold way to QCD.

Findings

Identifies three chiral ensembles corresponding to different gauge groups.

Shows the joint probability density reproduces Leutwyler-Smilga sum rules.

Connects spectral properties of QCD Dirac operator with random matrix theory.

Abstract

We argue that the spectrum of the QCD Dirac operator near zero virtuality can be described by random matrix theory. As in the case of classical random matrix ensembles of Dyson we have three distinct classes: the chiral orthogonal ensemble (chGOE), the chiral unitary ensemble (chGUE) and the chiral symplectic ensemble (chGSE). They correspond to gauge groups $SU(2)$ in the fundamental representation, $SU(N_c), N_c \ge 3$ in the fundamental representation, and gauge groups for all $N_c$ in the adjoint representation, respectively. The joint probability density reproduces Leutwyler-Smilga sum rules.