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International Journal of Modern Physics C

ISING CLUSTER FRAGMENTATION AT THE CRITICAL POINT

Muyoung Heo, Mookyung Cheon, Iksoo Chang, Dietrich Stauffer

Published Sep 1, 1999• 2 citations
Publisher

Abstract

The scaling law of Edwards et al., for cluster fragmentation of critical percolation clusters is not confirmed by analogous Monte Carlo simulations at the Curie point of the two-dimensional Ising model.
DOI
10.1142/s0129183199000851

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The scaling behavior for a binary fragmentation of critical percolation clusters is investigated by a large-cell Monte Carlo real-space renormalization group method in two and three dimensions. We obtain accurate values of critical exponents λ and ϕ describing the scaling of fragmentation rate and the distribution of fragments' masses produced by a binary fragmentation. Our results for λ and ϕ show that the fragmentation rate is proportional to the size of mother cluster, and the scaling relation σ = 1 + λ − ϕ conjectured by Edwards et al. to be valid for all dimensions is satisfied in two and three dimensions, where σ is the crossover exponent of the average cluster number in percolation theory, which excludes the other scaling relations.
In three dimensions, or more generally, below the upper critical dimension, scaling laws for critical phenomena seem well understood, for both infinite and for finite systems. Above the upper critical dimension of four, finite-size scaling is more difficult. Chen and Dohm predicted deviation in the universality of the Binder cumulants for three dimensions and more for the Ising model. This deviation occurs if the critical point T = Tc is approached along lines of constant A = L*L*(T-Tc)/Tc, then different exponents a function of system size L are found depending on whether this constant A is taken as positive, zero, or negative. This effect was confirmed by Monte Carlo simulations with Glauber and Creutz kinetics. Because of the importance of this effect and the unclear situation in the analogous percolation problem, we here reexamine the five-dimensional Glauber kinetics.