Stochastic calculation of the Dirac spectrum on the lattice and a   determination of chiral condensate in 2+1-flavor QCD
  We compute the chiral condensate in 2+1-flavor QCD through the spectrum of low-lying eigenmodes of Dirac operator. The number of eigenvalues of the Dirac operator is evaluated using a stochastic method with an eigenvalue filtering technique on the background gauge configurations generated by lattice QCD simulations including the effects of dynamical up, down and strange quarks described by the Mobius domain-wall fermion formulation. The low-lying spectrum is related to the chiral condensate, which is one of the leading order low-energy constants in chiral effective theory, as dictated by the Banks-Casher relation. The spectrum shape and its dependence on the sea quark masses calculated in numerical simulations are consistent with the expectation from one-loop chiral perturbation theory. After taking the chiral limit as well as the continuum limit using the data at three lattice spacings ranging 0.080-0.045 fm, we obtain $\Sigma^{1/3}$(2 GeV) = 270.0(4.9) MeV, with the error combining those from statistical and from various sources of systematic errors. Finite volume effect is confirmed to be under control by a direct comparison of the results from two different volumes at the lightest available sea quarks corresponding to 230 MeV pions. 
Guido Cossu, Hidenori Fukaya, Shoji Hashimoto, Takashi Kaneko, and   Jun-Ichi Noaki
hep-lat