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 12⁴. 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 Σ₂(n) ∼ 2(log n)/βπ² (β is equal to 4 and 1, respectively) instead of Σ₂(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.