Microscopic Spectrum of the Wilson Dirac Operator

We calculate the leading contribution to the spectral density of the Wilson Dirac operator using chiral perturbation theory where volume and lattice spacing corrections are given by universal scaling functions. We find analytical expressions for the spectral density on the scale of the average level spacing, and introduce a chiral random matrix theory that reproduces these results. Our work opens up a novel approach to the infinite-volume limit of lattice gauge theory at finite lattice spacing and new ways to extract coefficients of Wilson chiral perturbation theory.

Figure 1

The microscopic spectrum of $D_5$ for $\widehat{m}=3$, $\nu =0$, and $\widehat{a}=0$, 0.03, and 0.250. The $\nu =0$ spectrum is reflection symmetric about $\widehat{x}=0$.

Figure 2

The microscopic spectrum of $D_5$ for $\widehat{m}=3$, $\nu =2$ and $\widehat{a}=0.125$, $\widehat{a}=0.250$, and $\widehat{a}=0.500$, respectively.

Figure 3

The quenched density of the real eigenvalues of $D_W$.