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We compute the chiral condensate in 2+1-flavor QCD through the spectrum oflow-lying eigenmodes of Dirac operator. The number of eigenvalues of the Diracoperator is evaluated using a stochastic method with an eigenvalue filteringtechnique on the background gauge configurations generated by lattice QCDsimulations including the effects of dynamical up, down and strange quarksdescribed by the Mobius domain-wall fermion formulation. The low-lying spectrumis related to the chiral condensate, which is one of the leading orderlow-energy constants in chiral effective theory, as dictated by theBanks-Casher relation. The spectrum shape and its dependence on the sea quarkmasses calculated in numerical simulations are consistent with the expectationfrom one-loop chiral perturbation theory. After taking the chiral limit as wellas the continuum limit using the data at three lattice spacings ranging0.080-0.045 fm, we obtain $\Sigma^{1/3}$(2 GeV) = 270.0(4.9) MeV, with theerror combining those from statistical and from various sources of systematicerrors. Finite volume effect is confirmed to be under control by a directcomparison of the results from two different volumes at the lightest availablesea quarks corresponding to 230 MeV pions.

Stochastic calculation of the Dirac spectrum on the lattice and a determination of chiral condensate in 2+1-flavor QCD