Multi-Grid Monte Carlo III. Two-Dimensional O(4)-Symmetric Nonlinear $\sigma$-Model

Abstract

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation applied to the two-dimensional O(4)-symmetric nonlinear $\sigma$-model [= SU(2) principal chiral model], on lattices up to $256\times 256$. We find a dynamic critical exponent $z_{int,{\mathcal{M}}^2}=0.60\pm 0.07$ for the W-cycle and $z_{int,{\mathcal{M}}^2}=1.13\pm 0.11$ for the V-cycle, compared to $z_{int,{\mathcal{M}}^2}=2.0\pm 0.15$ for the single-site heat-bath algorithm (subjective 68% confidence intervals). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated. For a $256\times 256$ lattice, W-cycle MGMC is about 35 times as efficient as a single-site heat-bath algorithm.