Geometric Properties of the Three-Dimensional Ising and XY Models

TL;DR

This paper investigates the fractal structure of high-temperature graphs in 3D Ising and XY models, revealing their percolation behavior and fractal dimensions at criticality through lattice simulations and percolation analysis.

Contribution

It provides the first detailed analysis of the fractal properties and percolation thresholds of high-temperature graphs in 3D Ising and XY models.

Findings

Ising graphs percolate at the Curie critical point

The correlation length diverges near the percolation threshold

Fractal dimensions are estimated as approximately 1.735 for Ising and 1.763 for XY models

Abstract

The fractal structure of high-temperature graphs of the three-dimensional Ising and XY models is investigated by simulating these graphs directly on a cubic lattice and analyzing them with the help of percolation observables. The Ising graphs are shown to percolate right at the Curie critical point. The diverging length scale relevant to the graphs in the vicinity of the percolation threshold is shown to be provided by the spin correlation length. The fractal dimension of the high-temperature graphs at criticality is estimated to be $D = 1.7349(65)$ for the Ising and $D = 1.7626(66)$ for the XY model.