Thouless Energy and Correlations of QCD Dirac Eigenvalues

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 Ec 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 very weak volume dependence that is not consistent with Ec ∼ 1/L which might be expected from the pion Compton wave length and from the behavior of the Thouless energy in mesoscopic systems. However, the numerical value of Ec for our largest volumes is in rough agreement with estimates from the pion Compton wavelength. A scaling behaviour consistent with Ec ∼ 1/L is found in the bulk of the spectrum. For δE > Ec the number variance shows a linear dependence with a slope which is larger than the nonzero 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 nonzero value of the disconnected scalar susceptibility requires a linear dependence of number variance on δE.