Dynamic Critical Behaviour of Wolff's Algorithm for $RP^N$ $\sigma$-Models

TL;DR

This paper investigates the efficiency of Wolff-type algorithms for $RP^N$ sigma-models, revealing that the choice of Ising spin update method significantly affects critical slowing-down, with perfect updates greatly reducing it.

Contribution

It demonstrates that the embedding method's performance depends on the Ising update algorithm, linking collective mode encoding to algorithm efficiency in frustrated systems.

Findings

Swendsen-Wang update exhibits critical slowing-down with $z_\chi \approx 1

Perfect independent updates reduce $z_\chi$ to approximately 0

Embedding encodes collective modes, affecting algorithm performance

Abstract

We study the performance of a Wolff-type embedding algorithm for $RP^N$ $\sigma$-models. We find that the algorithm in which we update the embedded Ising model \`a la Swendsen-Wang has critical slowing-down as $z_\chi \approx 1$. If instead we update the Ising spins with a perfect algorithm which at every iteration produces a new independent configuration, we obtain $z_\chi \approx 0$. This shows that the Ising embedding encodes well the collective modes of the system, and that the behaviour of the first algorithm is connected to the poor performance of the Swendsen-Wang algorithm in dealing with a frustrated Ising model.