Multi-Grid Monte Carlo via $XY$ Embedding I. General Theory and Two-Dimensional $O(N)$-Symmetric Nonlinear $\sigma$-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 $\sigma$-models. We then apply the method to the two-dimensional $O(N)$-symmetric nonlinear $\sigma$-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 $z_{int,{\mathcal{M}}^2}\approx 0.70\pm 0.08$, $0.60 \pm 0.07$,$0.52 \pm 0.10$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 $\xi$ agrees with the four-loop asymptotic-freedom prediction to within $\approx 1\mathrm{\%}$ over the interval $12 < \xi < 650$.