Hydrodynamics of the Chiral Dirac Spectrum

TL;DR

This paper develops a hydrodynamical framework to describe the evolution of the chiral Dirac spectrum eigenvalues, revealing sound wave propagation and instanton-driven relaxation in QCD-like systems.

Contribution

It introduces a hydrodynamical model for the chiral Dirac spectrum eigenvalues, connecting spectral relaxation to Euler equations and sound wave dynamics.

Findings

Eigenvalues exhibit sound wave propagation in the hydrodynamical description.

Relaxation time from localized to delocalized eigenvalues is proportional to spectral density and inverse of N.

Hydrodynamical instantons describe stochastic relaxation processes in the spectrum.

Abstract

We derive a hydrodynamical description of the eigenvalues of the chiral Dirac spectrum in the vacuum and in the large $N$ (volume) limit. The linearized hydrodynamics supports sound waves. The stochastic relaxation of the eigenvalues is captured by a hydrodynamical instanton configuration which follows from a pertinent form of Euler equation. The relaxation from a phase of localized eigenvalues and unbroken chiral symmetry to a phase of de-localized eigenvalues and broken chiral symmetry occurs over a time set by the speed of sound. We show that the time is $\Delta \tau=\pi\rho(0)/2\beta N$ with $\rho(0)$ the spectral density at zero virtuality and $\beta=1,2,4$ for the three Dyson ensembles that characterize QCD with different quark representations in the ergodic regime.