Multi-grid Monte Carlo via XY embedding. General theory and two-dimensional O( N)-symmetric non-linear σ-models

Abstract

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 non-linear σ-models. We then apply the method to the two-dimensional 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 Z int, M 2 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 finite-size scaling extrapolation method. The correlation length ξ agrees with the four-loop asymptotic-freedom prediction to within 1 % over the interval 12 ⪅ ξ ⪅ 650.