Two-Flavor Lattice-QCD Simulation in the $ϵ$ Regime with Exact Chiral Symmetry

We perform lattice simulations of two-flavor QCD using Neuberger’s overlap fermion, with which the exact chiral symmetry is realized at finite lattice spacings. The $ϵ$ regime is reached by decreasing the light quark mass down to 3 MeV on a ${16}^3\times 32$ lattice with a lattice spacing $\sim 0.11\mathrm{fm}$. We find a good agreement of the low-lying Dirac eigenvalue spectrum with the analytical predictions of the chiral random matrix theory, which reduces to the chiral perturbation theory in the $ϵ$ regime. The chiral condensate is extracted as ${\Sigma }^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=(251\pm 7\pm 11\mathrm{MeV}{)}^3$, where the errors are statistical and an estimate of the higher order effects in the $ϵ$ expansion.

Figure 1

Number of the Wilson-Dirac operator multiplication per trajectory (upper panel) and per an overlap inversion (lower panel). The curves are fit to data above $m=0.030$ with the form $\propto 1/m^{\alpha }$.

Figure 2

Lowest 5 eigenvalues ${\lambda }_k$ as a function of sea quark mass. Dashed line shows $\lambda =m$.

Figure 3

Ratio of the eigenvalues $⟨{\zeta }_k⟩/⟨{\zeta }_l⟩$ for combinations of $k$ and $l\in 1–4$ (denoted in the plot as $k/l$). In addition to the two-flavor QCD data (middle), quenched data at $Q=0$ (left) and 2 (right) are also shown. Lattice data (circles) are compared with the ChRMT predictions (bars).

Figure 4

Cumulative distribution of the low-lying eigenvalues. The horizontal error comes from the statistical error of $\Sigma$. The solid curves are the ChRMT results with the input from the average $⟨{\lambda }_1⟩$.