Dynamic Critical Behavior of Multi-Grid Monte Carlo for Two-Dimensional Nonlinear $\sigma$-Models

TL;DR

This paper presents a new multi-grid Monte Carlo method for nonlinear sigma-models, embedding an XY model to analyze dynamic critical behavior across different models, revealing systematic variation in critical exponents.

Contribution

It introduces an embedding-based MGMC algorithm for nonlinear sigma-models and studies its dynamic critical behavior in various models, a novel approach in this context.

Findings

Dynamic critical exponent z varies between models.

z approximately 0.70 for O(3), 0.60 for O(4), 0.50 for O(8), 0.45 for SU(3).

No theoretical explanation provided for the observed behavior.

Abstract

We introduce a new and very convenient approach to multi-grid Monte Carlo (MGMC) algorithms for general nonlinear $\sigma$-models: it is based on embedding an $XY$ model into the given $\sigma$-model, and then updating the induced $XY$ model using a standard $XY$-model MGMC code. We study the dynamic critical behavior of this algorithm for the two-dimensional $O(N)$ $\sigma$-models with $N = 3,4,8$ and for the $SU(3)$ principal chiral model. We find that the dynamic critical exponent $z$ varies systematically between these different asymptotically free models: it is approximately 0.70 for $O(3)$, 0.60 for $O(4)$, 0.50 for $O(8)$, and 0.45 for $SU(3)$. It goes without saying that we have no theoretical explanation of this behavior.