Thouless Energy and Correlations of QCD Dirac Eigenvalues

Abstract

Eigenvalues and eigenfunctions of the QCD Dirac operator are studied for an instanton liquid partition function. We find that for energy differences $\delta E$ below an energy scale $E_c$, identified as the Thouless energy, the eigenvalue correlations are given by random matrix theory. The value of $E_c$ shows a weak volume dependence for eigenvalues near zero and is consistent with a scaling of $E_c\sim 1/L^2$ in the bulk of the spectrum in agreement with estimates from chiral perturbation theory that $E_c/\Delta \approx F_{\pi }^2{L}^2/\pi$ (with average level spacing $\Delta$). For $\delta E>\mathrm{}{E}_c$ the number variance shows a linear dependence. For the wave functions we find a small nonzero multifractality index.