Three Dimensional Ising Model, Percolation Theory and Conformal Invariance

TL;DR

This paper investigates the fractal and scaling properties of a 2D slice of the 3D Ising model, revealing critical behavior, conformal invariance, and universality classes through Monte Carlo simulations and theoretical analysis.

Contribution

It demonstrates that geometric spin clusters at the critical point can be described by SLE with specific ppa values, linking 3D Ising features to 2D conformal invariance and universality classes.

Findings

Percolation transition of geometric spin clusters occurs at the Curie point.

Fractal dimensions and winding angle statistics suggest conformal invariance and SLE description.

FK clusters undergo a percolation transition in the same class as 2D critical percolation.

Abstract

The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the critical behavior of the 3d model. The fractal dimension and the winding angle statistics of the perimeter and external perimeter of the geometric spin clusters at the critical point suggest that, if conformally invariant in the scaling limit, they can be described by the theory of Schramm-Loewner evolution (SLE_\kappa) with diffusivity of \kappa=5 and 16/5, respectively, putting them in the same universality class as the interfaces in 2d tricritical Ising model. It is also found that the Fortuin-Kasteleyn (FK) clusters associated with the cross sections undergo a nontrivial percolation transition, in the same universality class as the ordinary 2d critical percolation.