Wolff-Type Embedding Algorithms for General Nonlinear $\sigma$-Models

Abstract

We study a class of Monte Carlo algorithms for the nonlinear $\sigma$-model, based on a Wolff-type embedding of Ising spins into the target manifold $M$. We argue heuristically that, at least for an asymptotically free model, such an algorithm can have dynamic critical exponent $z\ll 2$ only if the embedding is based on an (involutive) isometry of $M$ whose fixed-point manifold has codimension 1. Such an isometry exists only if the manifold is a discrete quotient of a product of spheres. Numerical simulations of the idealized codimension-2 algorithm for the two-dimensional $O(4)$-symmetric $\sigma$-model yield $z_{int,{\mathcal{M}}^2}=1.5\pm 0.5$ (subjective 68\% confidence interval), in agreement with our heuristic argument.

70 pages, 7 postscript figures

References in corpus (2)

• Multi-Grid Monte Carlo III. Two-Dimensional O(4)-Symmetric Nonlinear $\sigma$-Model
• Dynamic Critical Behaviour of Wolff's Algorithm for $R{P}^N$ $\sigma$-Models

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