Universal fluctuations in spectra of the lattice Dirac operator

TL;DR

This paper analyzes the spectral fluctuations of the lattice Dirac operator in SU(2) gauge theory, showing they follow the Gaussian Symplectic Ensemble, with implications for understanding eigenvalue correlations and symmetries.

Contribution

The study provides the first statistical analysis of lattice Dirac spectra, linking eigenvalue correlations to random matrix theory ensembles based on gauge group and fermion representation.

Findings

Eigenvalue correlations match Gaussian Symplectic Ensemble for SU(2).

Long-range fluctuations are suppressed compared to random levels.

Predictions for SU(3) eigenvalue correlations follow Gaussian Unitary Ensemble.

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 $\Sigma_2(n) \sim\frac 1{2\pi^2}(\log n + {\rm const.})$ 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.