Poisson Statistics in the High Temperature QCD Dirac Spectrum

TL;DR

This paper demonstrates that at high temperatures in QCD, the low-end Dirac spectrum consists of localized eigenmodes with eigenvalues following a Poisson distribution, contrasting with the random matrix theory predictions at low temperatures.

Contribution

It provides the first detailed statistical description of the high-temperature Dirac spectrum, revealing localized modes and their Poissonian eigenvalue distribution using lattice simulations.

Findings

Localized eigenmodes appear at high temperature in the Dirac spectrum.

Eigenvalues of these modes follow a Poisson distribution.

Transition from Poisson to RMT distribution occurs higher in the spectrum.

Abstract

At low temperature in the epsilon regime of QCD the low-end of the Dirac spectrum is described by random matrix theory. In contrast, there has been no similarly well established staistical description in the high temperature, chirally symmetric phase. Using lattice simulations we show that at high temperature a band of extremely localized eigenmodes appear at the low-end of the Dirac spectrum. The corresponding eigenvalues are statistically independent and obey a generalized Poisson distribution. Higher up in the spectrum the Poisson distribution rapidly crosses over into the bulk distribution predicted by the random matrix ensemble with the corresponding symmetry. Our results are based on quenched lattice simulations with the overlap and the staggered Dirac operator done well above the critical temperature at several volumes and values of $N_t$. We also discuss the crucial role played by the fermionic boundary condition and the Polyakov-loop in this phenomenon.