Universal fluctuations in spectra of the lattice Dirac operator

Recently, Kalkreuter obtained the complete Dirac spectrum for an $SU(2)$ lattice gauge theory with dynamical staggered fermions on a $12^4$ lattice for $β=1.8$ and $β=2.8$. We performed a statistical analysis of his data and found that the eigenvalue correlations can be described by the Gaussian Symplectic Ensemble. In particular, long range fluctuations are strongly suppressed: the variance of a sequence of levels containing $n$ eigenvalues on average is given by $Σ_2(n) \sim\frac 1{2π^2}(\log n + {\rm const.})$ instead of $Σ_2(n) = n$ 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 the continuuum theory. For $N_c = 3$ we predict that the eigenvalue correlations are given by the Gaussian Unitary Ensemble.

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