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arXiv:0803.2177 · Physical Review E

Geometric properties of the three-dimensional Ising and XY models

Frank Winter, Wolfhard Janke E, Adriaan M. J. Schakel

Published Jun 9, 2008• 15 citations
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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 models.

DOI
10.1103/physreve.77.061108

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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 models.
The fractal structure of the Ising model at the critical point Tc is studied in the present paper.The fractal dimensionality D of the total magnetization at Tc was estimated numerically as D = 1.86 ±O.Ol for the two-dimensional square lattice and D=2.46±O.Ol for the three-dimensional simple cubic lattice by Monte Carlo simulations.These values agree very well with the value D=1.875 obtained from the exact critical exponents and D=2.48 obtained from the known critical exponents, re-~pectively, through the relation D=d-fJ/v=(d+r/v)/2.This fractalness yields the hyperscaling relation dv=2fJ+r.It was also observed how the fractal nature of the relevant system disappears as the system deviates from the critical point.The dimensionality d of the relevant lattice is observed at temperatures lower than Tc and the random percolation value d/2 at higher temperatures.
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.
Geometric representations provide a useful perspective on critical phenomena in the Ising model. In a recent study [Phys. Rev. E 112, 034118 (2025)], we found that the two-dimensional critical Ising model exhibits two consecutive percolation transitions for geometric spin clusters as the bond-occupation probability ppp between parallel spins increases. Here, through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations. Further theoretical analysis of the Ising model on the complete graph also yields the same scenario. In addition, we study percolation on a two-dimensional layer embedded in the three-dimensional critical Ising model. For this layer system, we estimate the red-bond exponent yp=0.426(6)y_p = 0.426(6)yp​=0.426(6) and the fractal dimensions of the largest cluster, hull, and shortest path as df=1.8926(20)d_f = 1.8926(20)df​=1.8926(20), dhull=1.663(4)d_{\rm hull} = 1.663(4)dhull​=1.663(4), and dmin=1.080(10)d_{\rm min} = 1.080(10)dmin​=1.080(10), respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations.
The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds, with probability p, between any pair of parallel spins within an extended range beyond nearest neighbors. At the Ising criticality, we observe two percolation transitions as p increases: starting from a disordered phase with only small clusters, the percolation system enters into a stable critical phase that persists over a wide range p_{c_{1}}<p<p_{c_{2}}, and then develops a long-ranged percolation order with giant clusters for both up and down spins. At p_{c1} and for the stable critical phase, the critical behaviors agree well with those for the Fortuin-Kasteleyn random clusters and the spin domains of the Ising model, respectively. At p_{c2}, the fractal dimension of clusters and the scaling exponent along p direction are estimated as y_{h2}=1.9580(6) and y_{p2}=0.552(9), of which the exact values remain unknown. These findings reveal interesting geometric properties of the two-dimensional Ising model that has been studied for more than 100 years.