Geometrical properties of parafermionic spin models

TL;DR

This paper investigates the fractal dimensions of geometrical and FK clusters in Z_4 and Z_5 spin models, revealing differences from simpler models like Ising and Potts, and compares these with SLE interface dimensions.

Contribution

It provides the first measurements of fractal dimensions for geometrical clusters in Z_4 and Z_5 models and highlights their non-percolating nature at criticality.

Findings

Geometrical clusters have specific fractal dimensions at criticality.

FK clusters do not percolate at the critical point in these models.

Differences are observed compared to Ising and 3-state Potts models.

Abstract

We present measurements of the fractal dimensions associated to the geometrical clusters for Z_4 and Z_5 spin models. We also attempted to measure similar fractal dimensions for the generalised Fortuyin Kastelyn (FK) clusters in these models but we discovered that these clusters do not percolate at the critical point of the model under consideration. These results clearly mark a difference in the behaviour of these non local objects compared to the Ising model or the 3-state Potts model which corresponds to the simplest cases of Z_N spin models with N=2 and N=3 respectively. We compare these fractal dimensions with the ones obtained for SLE interfaces.