Cluster Percolation and Critical Behaviour in Spin Models and SU(N) Gauge Theories

TL;DR

This paper introduces a simple, universal way to define clusters in spin models and SU(N) gauge theories, linking percolation phenomena with critical behavior and phase transitions.

Contribution

It proposes a minimal bond probability-based cluster definition that accurately captures critical points across various models, including gauge theories.

Findings

The cluster definition works for many classical spin models in 2D and 3D.

It accurately describes the confinement-deconfinement transition in SU(N) gauge theories.

The percolation exponents match the thermal critical exponents.

Abstract

The critical behaviour of several spin models can be simply described as percolation of some suitably defined clusters, or droplets: the onset of the geometrical transition coincides with the critical point and the percolation exponents are equal to the thermal exponents. It is still unknown whether, given a model, one can define at all the droplets. In the cases where this is possible, the droplet definition depends in general on the specific model at study and can be quite involved. We propose here a simple general definition for the droplets: they are clusters obtained by joining nearest-neighbour spins of the same sign with some bond probability p_B, which is the minimal probability that still allows the existence of a percolating cluster at the critical temperature T_c. By means of lattice Monte Carlo simulations we find that this definition indeed satisfies the conditions required for the droplets, for many classical spin models, discrete and continuous, both in two and in three dimensions. In particular, our prescription allows to describe exactly the confinement-deconfinement transition of SU(N) gauge theories as Polyakov loop percolation.