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Physical Review E

Statistical investigation of the cross sections of wave clusters in the three-dimensional Bak-Tang-Wiesenfeld model

Hor Dashti-Naserabadi, M. N. Najafi

Published May 27, 2015• 16 citations
Publisher

Abstract

We consider the three-dimensional (3D) Bak-Tang-Wiesenfeld model in a cubic lattice. Along with analyzing the 3D problem, the geometrical structure of the two-dimensional (2D) cross section of waves is investigated. By analyzing the statistical observables defined in the cross sections, it is shown that the model in that plane (named as 2D-induced model) is in the critical state and fulfills the finite-size scaling hypothesis. The analysis of the critical loops that are interfaces of the 2D-induced model is of special importance in this paper. Most importantly, we see that their fractal dimension is D(f)=1.387±0.005, which is compatible with the fractal dimension of the external perimeter of geometrical spin clusters of 2D critical Ising model. Some hyperscaling relations between the exponents of the model are proposed and numerically confirmed. We then address the problem of conformal invariance of the mentioned domain walls using Schramm-Lowener evolution (SLE). We found that they are described by SLE with the diffusivity parameter κ=2.8±0.2, nearly consistent with observed fractal dimension.

DOI
10.1103/physreve.91.052145

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We consider the three-dimensional (3D) Bak-Tang-Wiesenfeld model in a cubic lattice. Along with analyzing the 3D problem, the geometrical structure of the two-dimensional (2D) cross section of waves is investigated. By analyzing the statistical observables defined in the cross sections, it is shown that the model in that plane (named as 2D-induced model) is in the critical state and fulfills the finite-size scaling hypothesis. The analysis of the critical loops that are interfaces of the 2D-induced model is of special importance in this paper. Most importantly, we see that their fractal dimension is D(f)=1.387±0.005, which is compatible with the fractal dimension of the external perimeter of geometrical spin clusters of 2D critical Ising model. Some hyperscaling relations between the exponents of the model are proposed and numerically confirmed. We then address the problem of conformal invariance of the mentioned domain walls using Schramm-Lowener evolution (SLE). We found that they are described by SLE with the diffusivity parameter κ=2.8±0.2, nearly consistent with observed fractal dimension.
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.
Abstract The scaling behaviour and conformal properties of a triangular Ising model with three-spin interactions at the critical point are studied using extensive Monte Carlo simulations. We show that in the continuum limit the external perimeter of spin clusters are conformally invariant and statistically equivalent to Schramm–Loewner evolution (SLE κ ) curves with diffusivity κ = 4, indicating that the loop ensembles associated with the critical Ising model with three-spin interactions, belonging to the universality class of domain walls in the four-states Potts model. The left-passage probabilities, distribution of the winding angle, and the driving function distribution for the interfaces are consistent with each other. We also study many geometrical properties of the critical interfaces to characterize the fractal structure and the other scaling properties of conformal loop ensembles. To this end, the fractal dimension of contours, the scaling properties of length and areas distributions and Zipf’s laws, are demonstrated numerically. Our results are compatible with the hyper-scaling relation for spin models.
The fractal dimensions and the percolation exponents of geometrical spin clusters of like sign at criticality are obtained numerically for an Ising model with temperature dependent annealed bond dilution, also known as the thermalized bond Ising model (TBIM), in two dimensions. For this purpose, a modified Wolff single-cluster Monte Carlo simulation is used to generate equilibrium spin configurations on square lattices in the critical region. A tie-breaking rule is employed to identify nonintersecting spin cluster boundaries along the edges of the dual lattice. The values obtained for the fractal dimensions of the spanning geometrical clusters D c , and their interfaces D I , are in perfect agreement with those reported for the standard two-dimensional ferromagnetic Ising model. Furthermore, the variance of the winding angles results in a diffusivity κ = 3 for the two-dimensional thermalized bond Ising model, thus placing it in the universality class of the regular Ising model. A finite-size scaling analysis of the largest geometrical clusters results in a reliable estimation of the critical percolation exponents for the geometrical clusters in the limit of an infinite lattice size. The percolation exponents thus obtained are also found to be consistent with those reported for the regular Ising model. These consistencies are explained in terms of the Fisher renormalization relations, which express the thermodynamic critical exponents of systems with annealed bond dilution in terms of those of the regular model system.
An extensive Monte Carlo study of the two-dimensional Ising model is made to investigate the statistical behavior of spin clusters and interfaces as a function of temperature, T . We use a tie-breaking rule to define interfaces of spin clusters on a square lattice with strip geometry and show that such definitions are consistent with conformally invariant properties of interfaces at the critical temperature, T c . The effective fractal dimensions of spin clusters and interfaces ( d c and d I , respectively) are obtained as a function of temperature. We find that the effective fractal dimension of the spin clusters varies almost linearly with temperature in three different regimes. It is also found that the effective fractal dimension of the interfaces undergoes a sharp crossover around T c , between the values 1 and 1.75 at low and high temperatures, respectively. We also check the finite-size scaling hypothesis for the percolation probability, and the average mass of the largest spin cluster is in a good agreement with the theoretical predictions.
The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ds=2,{d}_{s}=2,ds​=2, the scaling of the roughness of the domain walls, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}},$ is consistent with the theoretical prediction $\ensuremath{\zeta}=2/3.$ As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent $\ensuremath{\theta},$ which determines the violation of hyperscaling, the correlation length exponent $\ensuremath{\nu},$ and the magnetization exponent $\ensuremath{\beta}.$ The value $\ensuremath{\beta}=0.017\ifmmode\pm\else\textpm\fi{}0.005$ computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.