The low-lying eigenvalue spectrum of the QCD Dirac operator in the $ϵ$ regime is expected to match with that of chiral random matrix theory (ChRMT). We study this correspondence for the case including sea quarks by performing two-flavor QCD simulations on the lattice. Using the overlap fermion formulation, which preserves exact chiral symmetry at finite lattice spacings, we push the sea quark mass down to $\sim 3\mathrm{MeV}$ on a ${16}^3\times 32$ lattice at a lattice spacing $a\simeq 0.11\mathrm{fm}$. We compare the low-lying eigenvalue distributions and find a good agreement with the analytical predictions of ChRMT. By matching the lowest-lying eigenvalue we extract the chiral condensate, ${\Sigma }^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=(251\pm 7\pm 11\mathrm{MeV}{)}^3$, where errors represent statistical and higher order effects in the $ϵ$ expansion. We also calculate the eigenvalue distributions on the lattices with heavier sea quarks at two lattice spacings. Although the $ϵ$ expansion is not applied for those sea quarks, we find a reasonable agreement of the Dirac operator spectrum with ChRMT. The value of $\Sigma$, after extrapolating to the chiral limit, is consistent with the estimate in the $ϵ$ regime.

• Received 23 May 2007

DOI:https://doi.org/10.1103/PhysRevD.76.054503