Critical statistics at the mobility edge of QCD Dirac spectra

TL;DR

This paper investigates the critical spectral statistics at the mobility edge of QCD Dirac spectra above the critical temperature, using deformed random matrix models to describe the transition and identify the mobility edge.

Contribution

It introduces a novel application of deformed random matrix ensembles to model the critical statistics of QCD Dirac eigenvalues at the mobility edge, linking it to Anderson localization theory.

Findings

Eigenvalues near the mobility edge follow critical statistics.

The best-fit deformation parameter aligns with the Anderson unitary class.

A method is proposed to locate the mobility edge at the origin of the spectrum.

Abstract

We examine statistical fluctuation of eigenvalues from the near-edge bulk of QCD Dirac spectra above the critical temperature. For completeness we start by reviewing on the spectral property of Anderson tight-binding Hamiltonians as described by nonlinear sigma models and random matrices, and on the scale-invariant intermediate spectral statistics at the mobility edge. By fitting the level spacing distributions, deformed random matrix ensembles which model multifractality of the wave functions typical of the Anderson localization transition, are shown to provide an excellent effective description for such a critical statistics. Next we carry over the above strategy for the Anderson Hamiltonians to the Dirac spectra. For the staggered Dirac operators of QCD with 2+1 flavors of dynamical quarks at the physical point and of SU(2) quenched gauge theory, we identify the precise location of the mobility edge as the scale-invariant fixed point of the level spacing distribution. The eigenvalues around the mobility edge are shown to obey critical statistics described by the aforementioned deformed random matrix ensembles of unitary and symplectic classes. The best-fitting deformation parameter for QCD at the physical point turns out to be consistent with the Anderson Hamiltonian in the unitary class. Finally, we propose a method of locating the mobility edge at the origin of QCD Dirac spectrum around the critical temperature, by the use of individual eigenvalue distributions of deformed chiral random matrices.