Multigrid Monte Carlo simulation viaXYembedding. II. Two-dimensional SU(3) principal chiral model

Abstract

We carry out a high-precision simulation of the two-dimensional SU(3) principal chiral model at correlation lengths \ensuremath{\xi} up to $\ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}{10}^{5}$, using a multigrid Monte Carlo (MGMC) algorithm and approximately one year of Cray C-90 CPU time. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and we discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is \ensuremath{\approx}12% at \ensuremath{\xi}\ensuremath{\sim}25, decreases to \ensuremath{\approx}2% at $\ensuremath{\xi}\ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}{10}^{5}$. We also analyze the dynamic critical behavior of the MGMC algorithm using lattices up to 256\ifmmode\times\else\texttimes\fi{}256, finding the dynamic critical exponent ${z}_{\mathrm{int},{\mathcal{M}}^{2}}\ensuremath{\approx}0.45\ifmmode\pm\else\textpm\fi{}0.02$ (subjective 68% confidence interval). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated.