Wolff-Type Embedding Algorithms for General Nonlinear $σ$-Models

Details Wolff-Type Embedding Algorithms for General Nonlinear $σ$-Models Wolff-Type Embedding Algorithms for General Nonlinear $σ$-Models

1992-05-04

arxiv.org/abs/hep-lat/9205005v1

Nucl.Phys. B403 (1993) 475-541

Physics

High Energy Physics

High Energy Physics - Lattice

70 pages, 7 postscript figures

Scientific paper

10.1016/0550-3213(93)90044-P

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.

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