Detecting the flavor content of the vacuum using the Dirac operator spectrum

We compute the overlap Dirac spectrum on three gauge ensembles generated using $2+1$-flavor domain wall fermions. The three ensembles have different lattice spacings and two of them have quark masses tuned to the physical point. The spectral density is determined up to $\lambda\sim$100 MeV with subpercentage statistical uncertainty. We find that the density is close to a constant below $\lambda\sim$ 20 MeV as predicted by chiral perturbative theory ($\chi$PT), and then increases linearly due to the strange quark mass. By fitting to the next-to-leading order $\chi$PT form and using the non-perturbative RI/MOM renormalization, the $\rm SU(2)$ (keeping the strange quark mass at the physical point) and $\rm SU(3)$ chiral condensates at $\overline{\textrm{MS}}$ 2 GeV are determined to be $\Sigma=(265.4(0.5)(4.2)\ \textrm{MeV})^3$ and $\Sigma_0=(234.3(0.5)(25.8)\ \textrm{MeV})^3$, respectively. The pion decay constants are also determined to be $F=84.1(1.9)(8.0)$ and $F_0=58.6(0.5)(10.0)$ MeV. The systematic errors are carefully estimated including the effects of fitting ranges and the uncertainty of low-energy constant $L_6$. We also show that one can resolve the sea flavor content of the sea quarks and constrain their masses with {$\sim10\%-20\%$} statistical uncertainties using the Dirac spectral density.