Universal fluctuations in spectra of the lattice Dirac operator

Abstract

Recently, Kalkreuter obtained the complete Dirac spectrum for an $SU(2)$ lattice gauge theory with dynamical staggered fermions on a $12^4$ lattice for $\beta =1.8$ and $\beta =2.8$. 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 ${\Sigma }_2(n)\sim \frac{1}{2{\pi }^2}(\log n+const.)$ 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 the continuuum theory. For $N_c=3$ we predict that the eigenvalue correlations are given by the Gaussian Unitary Ensemble.

Comments: 8 pages + 3 figures (will be faxed on request)