Thouless energy and correlations of QCD Dirac eigenvalues

Abstract

Eigenvalues and eigenfunctions of the QCD Dirac operator are studied for gauge field configurations given by a liquid of instantons. We find that for energy differences δE below an energy scale E c the eigenvalue correlations are given by random matrix theories with the chiral symmetries of the QCD partition function. For eigenvalues near zero this energy scale shows a weak volume dependence that is not consistent with E c ∼ 1 L 2 which might be expected from the pion Compton wavelength and from the behavior of the Thouless energy in mesoscopic systems. However, the numerical value of E c for our largest volumes is in rough agreement with estimates from the pion Compton wavelength. A scaling behavior consistent with E c ∼ 1/ L 2 is found in the bulk of the spectrum. For δE > E c the number variance shows a linear dependence with a slope which is larger than the non-zero multifractality index of the wave functions. Finally, the average spectral density and the scalar susceptibilities are discussed in the context of quenched chiral perturbation theory. We argue that a non-zero value of the disconnected scalar susceptibility requires a linear dependence of the number variance on δE. © 1998 Elsevier Science B.V.