# Universal correlations in spectra of the lattice QCD Dirac operator

## TL;DR

This paper analyzes the spectral correlations of the lattice QCD Dirac operator, revealing that they follow specific random matrix ensembles depending on fermion type, with suppressed long-range fluctuations consistent with underlying symmetries.

## Contribution

It provides a statistical analysis linking lattice QCD Dirac spectra to random matrix theory ensembles, highlighting symmetry-dependent spectral correlations.

## Key findings

- Eigenvalue correlations follow Gaussian Symplectic Ensemble for staggered fermions.
- Eigenvalue correlations follow Gaussian Orthogonal Ensemble for Wilson fermions.
- Long-range spectral fluctuations are strongly suppressed, with variance scaling as (log n)/β.

## Abstract

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) \sim 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.