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

TL;DR

This paper introduces a class of Monte Carlo algorithms for nonlinear sigma-models based on Wolff-type embeddings, demonstrating that optimal dynamic critical behavior occurs under specific geometric conditions of the target manifold, supported by numerical simulations.

Contribution

It proposes a new class of embedding algorithms for sigma-models and identifies geometric conditions for their efficiency, supported by heuristic analysis and numerical results.

Findings

Optimal algorithms require isometries with fixed-point manifolds of codimension 1.

Numerical simulations for the $O(4)$ model show a dynamic critical exponent around 1.5.

Theoretical analysis links the efficiency of algorithms to the geometry of the target manifold.

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 $z_{int,{\cal M}^2} = 1.5 \pm 0.5$ (subjective 68\% confidence interval), in agreement with our heuristic argument.