Polynomial hybrid Monte Carlo algorithm for lattice QCD with an odd number of flavors

We present a polynomial hybrid Monte Carlo (PHMC) algorithm for lattice QCD with odd numbers of flavors of $O\left(a\right)\mathrm{}$-improved Wilson quark action. The algorithm makes use of the non-Hermitian Chebyshev polynomial to approximate the inverse square root of the fermion matrix required for an odd number of flavors. The systematic error from the polynomial approximation is removed by a noisy Metropolis test for which a new method is developed. Investigating the property of our PHMC algorithm in the $N_f=2\mathrm{}$ QCD case, we find that it is as efficient as the conventional HMC algorithm for a moderately large lattice size ${(16\mathrm{}}^3\times 48)$ with intermediate quark masses ${(m}_{\mathrm{PS}}{/m}_V\sim 0.7–0.8).$ We test our odd-flavor algorithm through extensive simulations of two-flavor QCD treated as an $N_f=1+1\mathrm{}$ system, and comparing the results with those of the established algorithms for $N_f=2\mathrm{}$ QCD. These tests establish that our PHMC algorithm works on a moderately large lattice size with intermediate quark masses ${(16\mathrm{}}^3\times {48,m}_{\mathrm{PS}}{/m}_V\sim 0.7–0.8).$ Finally we experiment with the $\mathrm{}(2+1)\mathrm{}$-flavor QCD simulation on small lattices ${(4\mathrm{}}^3\times 8$ and $8^3\times 16),$ and confirm the agreement of our results with those obtained with the R algorithm and extrapolated to a zero molecular dynamics step size.