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

## TL;DR

This paper develops a multi-grid Monte Carlo method using XY embedding for 2D nonlinear sigma models, demonstrating reduced critical slowing-down and confirming theoretical predictions for correlation lengths.

## Contribution

It introduces a novel MGMC variant based on XY embedding and analyzes its effectiveness on 2D $O(N)$ models, providing both theoretical insights and empirical results.

## Key findings

- Critical exponents $z_{int,{\
- static correlation length matches four-loop predictions
- Critical slowing-down is significantly reduced but not eliminated

## 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 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 $z_{int,{\cal M}^2} \approx 0.70 \pm 0.08$, $0.60 \pm 0.07$, $0.52 \pm 0.10$ 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 $\xi$ agrees with the four-loop asymptotic-freedom prediction to within $\approx 1\%$ over the interval $12 < \xi < 650$.