Multi-Grid Monte Carlo via $XY$ Embedding. 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 $\xi$ up to $\sim 4\times 10^5$, using a multi-grid 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 $\approx 12$ at $\xi \sim 25$, decreases to $\approx 2$ at $\xi \sim 4\times 10^5$. We also analyze the dynamic critical behavior of the MGMC algorithm using lattices up to $256\times 256$, finding the dynamic critical exponent $z_{int,{\mathcal{M}}^2}\approx 0.45\pm 0.02$ (subjective 68% confidence interval). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated.

Comments: self-unpacking archive including .tex, .sty and .ps files; 126 pages including all figures