Multi-Grid Monte Carlo via XY Embedding I. General Theory and Two-Dimensional O(N)-Symmetric Nonlinear σ-Models

We introduce a variant of the multi-grid Monte Carlo (MGMC) method, based on the embedding of an XY model into the target model, and we study its mathematical properties for a variety of nonlinear σ-models. We then apply the method to the twodimensional O(N)-symmetric nonlinear σ-models (also called N -vector models) with N = 3, 4, 8 and study its dynamic critical behavior. Using lattices up to 256 × 256, we find dynamic critical exponents zint,M2 ≈ 0.70 ± 0.08, 0.60 ± 0.07, 0.52 ± 0.10 for N = 3, 4, 8, respectively (subjective 68% confidence intervals). Thus, for these asymptotically free models, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated; and the dynamic critical exponent does apparently vary with N . We also analyze the static data for N = 8 using a finitesize-scaling extrapolation method. The correlation length ξ agrees with the four-loop asymptotic-freedom prediction to within ≈ 1% over the interval 12 ∼< ξ ∼< 650.