Cluster Percolation in O(n) Spin Models

TL;DR

This paper demonstrates that in 3D O(2), O(3), and O(4) spin models, Wolff clusters percolate at the physical phase transition, with percolation exponents matching the models' universality classes, confirmed analytically and numerically.

Contribution

It establishes the physical significance of Wolff clusters in O(n) models and confirms their percolation at criticality with correct universality class behavior.

Findings

Clusters percolate at the phase transition in O(2), O(3), and O(4) models.

Percolation exponents match the universality class of each model.

Analytical proof for O(2) and O(3), numerical for O(4).

Abstract

The spontaneous symmetry breaking in the Ising model can be equivalently described in terms of percolation of Wolff clusters. In O(n) spin models similar clusters can be built in a general way, and they are currently used to update these systems in Monte Carlo simulations. We show that for 3-dimensional O(2), O(3) and O(4) such clusters are indeed the physical `islands' of the systems, i.e., they percolate at the physical threshold and the percolation exponents are in the universality class of the corresponding model. For O(2) and O(3) the result is proven analytically, for O(4) we derived it by numerical simulations.