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Physical review. B, Condensed matter

Distribution of fractal clusters and scaling in the Ising model

J.‐L. Cambier, Michael Nauenberg

Published Dec 1, 1986• 48 citations
Publisher

Abstract

The distribution of clusters of aligned spins in the Ising model in two and three dimensions is evaluated by Monte Carlo simulations below the critical temperature. These clusters are found to be fractals and the distribution for large-size clusters satisfies scaling properties. The scaling exponents for the distribution function, the surface exponent, and the fractal dimension of the clusters are evaluated and related to the critical exponents of the Ising model.

DOI
10.1103/physrevb.34.8071

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Progress of Theoretical Physics•Jun 1, 1987•24 citations
The distribution of clusters of aligned spins in the Ising model in two and three dimensions is evaluated by Monte Carlo simulations below the critical temperature. These clusters are found to be fractals and the distribution for large-size clusters satisfies scaling properties. The scaling exponents for the distribution function, the surface exponent, and the fractal dimension of the clusters are evaluated and related to the critical exponents of the Ising model.
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.
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.
The dynamics of Ising spins on a two-dimensional fractal lattice has been studied by Monte Carlo simulations. Both the limits $L>>\ensuremath{\xi}$ and $L<<\ensuremath{\xi}$ were studied where LLL is the size of the sample and $\ensuremath{\xi}$ the correlation length. Exact transfer matrix calculations were also carried out to check that the correct equilibrium values were reached. In both limits the predicted breakdown of conventional dynamic scaling was obtained. Using finite-size scaling $\ensuremath{\tau}\ensuremath{\sim}{L}^{z}$ for $L<<\ensuremath{\xi}$, an effective temperature-dependent critical exponent zzz was obtained consistent with the recent prediction $z\ensuremath{\propto}\frac{1}{T}$ for small TTT.
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.