Detecting favors of vacuum from the Dirac operator spectrum

From simulations on 2+1 flavor domain wall fermion ensembles at three lattice spacings with two of them at physical quark masses, we compute the spectrum of the Dirac operator up to the eigenvalue $\lambda\sim$100 MeV using the overlap fermion. The spectrum is close to a constant below $\lambda\le$ 20 MeV as predicted by the 2-flavor chiral perturbative theory ($\chi$PT) up to the finite volume correction, and then increases linearly due to the non-vainishing strange quark mass. Furthermore, one can extract the light and strange quark masses with $\sim$20\% uncertainties from the spectrum data with sub-percentage statistical uncertainty, using the next to leading order $\chi$PT. Using the non-perturbative RI/MOM renormalization, we obtain the chiral condensates at $\overline{\textrm{MS}}$ 2GeV as $\Sigma=(260.3(0.7)(1.3)(0.7)(0.8)\ \textrm{MeV})^3$ in the $N_f=2$ (keeping the strange quark mass at the physical point) chiral limit and $\Sigma_0=(232.6(0.9)(1.2)(0.7)(0.8)\ \textrm{MeV})^3$ in the $N_f=3$ chiral limit, where the four uncertainties come from the statistical fluctuation, renormalization constant, continuum extrapolation and lattice spacing determination. Note that $\Sigma$ and $\Sigma_0$ are very different from each other due to the strange quark mass effect.