Dynamic critical behavior of multi-grid Monte Carlo for two-dimensional nonlinear σ-models

We introduce a new and very convenient approach to multi-grid Monte Carlo (MGMC) algorithms for general nonlinear σ-models: it is based on embedding an XY model into the given σ-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) σ-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.