Low-lying Eigenvalues of the QCD Dirac Operator at Finite Temperature

TL;DR

This study analyzes the low-lying eigenvalues of the QCD Dirac operator at finite temperature, revealing spectral behaviors and tails that inform the chiral phase transition in both quenched and dynamical QCD.

Contribution

It provides the first detailed comparison of low-lying Dirac eigenvalues across quenched and dynamical QCD near the phase transition, highlighting spectral features and their implications.

Findings

High-temperature phase exhibits a square root spectral density behavior.

Quenched simulations show a volume-independent tail extending to zero eigenvalues.

Smallest eigenvalue distribution accurately indicates the chiral phase transition point.

Abstract

We compute the low-lying spectrum of the staggered Dirac operator above and below the finite temperature phase transition in both quenched QCD and in dynamical four flavor QCD. In both cases we find, in the high temperature phase, a density with close to square root behavior, $\rho(\lambda) \sim (\lambda-\lambda_0)^{1/2}$. In the quenched simulations we find, in addition, a volume independent tail of small eigenvalues extending down to zero. In the dynamical simulations we also find a tail, decreasing with decreasing mass, at the small end of the spectrum. However, the tail falls off quite quickly and does not seem to extend to zero at these couplings. We find that the distribution of the smallest Dirac operator eigenvalues provides an efficient observable for an accurate determination of the location of the chiral phase transition, as first suggested by Jackson and Verbaarschot.