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 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 $256 \times 256$, we find dynamic critical exponents $z_{int,{\cal M}^2} \approx 0.70 \pm 0.08$, $0.60 \pm 0.07$, $0.52 \pm 0.10$ 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 $\approx 1\%$ over the interval $12 < ξ< 650$.

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