Three-dimensional Ising model, percolation theory and conformal invariance

¹ School of Physics, Institute for Research in Fundamental Sciences (IPM) - P.O. Box 19395-5531, Tehran, Iran
² Department of Physics, Sharif University of Technology - P.O. Box 11155-9161, Tehran, Iran

^a a_saberi@ipm.ir, ab.saberi@gmail.com

Received: 18 August 2010
Accepted: 7 December 2010

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-Löwner evolution (SLE_κ) with diffusivity of κ = 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.