Three Dimensional Ising Model, Percolation Theory and Conformal Invariance

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_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.

5 pages, 5 figures, accepted for publication in EuroPhysics Letters (EPL)

References in corpus (27)

• SLE for theoretical physicists
• 2D growth processes: SLE and Loewner chains
• Conformal invariance in two-dimensional turbulence
• A Measure of data-collapse for scaling
• Conformally Invariant Fractals and Potential Theory
• Low-temperature behavior of two-dimensional Gaussian Ising spin glasses
• Inverse turbulent cascades and conformally invariant curves
• The Cluster Processor: New Results
• Conformal Invariance and SLE in Two-Dimensional Ising Spin Glasses
• Are Domain Walls in Spin Glasses Described by Stochastic Loewner Evolutions?
• Geometrical vs. Fortuin-Kasteleyn Clusters in the Two-Dimensional $q$-State Potts Model
• Winding angle variance of Fortuin-Kasteleyn contours
• Conformal Curves on $W{O}_3$ Surface
• Fractal Structure of Spin Clusters and Domain Walls in 2D Ising Model
• Nodal Domain Statistics for Quantum Maps, Percolation and SLE
• Conformal Invariance of Iso-height Lines in two-dimensional KPZ Surface
• Critical and Topological Properties of Cluster Boundaries in the $3d$ Ising Model
• Classification of (2+1)-Dimensional Growing Surfaces Using Schramm-Loewner Evolution
• Thermal Behavior of Spin Clusters and Interfaces in two-dimensional Ising Model on Square Lattice
• Critical interfaces in the random-bond Potts model
• Direct Evidence for Conformal Invariance of Avalanche Frontier in Sandpile Models
• Universal amplitude-exponent relation for the Ising model on sphere-like lattices
• Self Avoiding Surfaces in the 3D Ising Model
• Scaling of Clusters and Winding Angle Statistics of Iso-height Lines in two-dimensional KPZ Surface
• Universal amplitudes in the FSS of three-dimensional spin models
• Geometrical Phase Transition on WO${}_3$ Surface
• Geometric and Stochastic Clusters of Gravitating Potts Models

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