Universal correlations in spectra of the lattice QCD Dirac operator

Recently, Kalkreuter obtained complete Dirac spectra for SU(2) lattice gauge theory both for staggered fermions and for Wilson fermions. The lattice size was as large as 12(4). We performed a statistical analysis of these data and found that the eigenvalue correlations can be described by the Gaussian Symplectic Ensemble for staggered fermions and by the Gaussian Orthogonal Ensemble for Wilson fermions. In both cases long range spectral fluctuations are strongly suppressed: the variance of a sequence of levels containing n eigenvalues on average is given by Sigma(2)(n) similar to 2(log n)/beta pi(2) (beta is equal to 4 and 1, respectively) instead of Sigma(2)(n) = n for a random sequence of levels. Our findings are in agreement with the anti-unitary symmetry of the lattice Dirac operator for N-c = 2 with staggered fermions which differs from Wilson fermions (with the continuum anti-unitary symmetry). For N-c = 3, we predict that the eigenvalue correlations are given by the Gaussian Unitary Ensemble.