Most Probable Cluster-Size Distribution and the Property of an Ising Model with Four-spin Interactions

Yuming Yang, Baohua Teng, Hongchun Yang +1 more

M. Ghasemi Nezhadhaghighi

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.

C. Ruge, Friedrich Wagner

Youjin Deng, Henk W. J. Blöte

Using the Wolff and geometric cluster algorithms and finite-size scaling analysis, we investigate the critical Ising and the tricritical Blume-Capel models with nearest-neighbor interactions on the simple-cubic lattice. The sampling procedure involves the decomposition of the Ising configuration into geometric clusters, each of which consists of a set of nearest-neighboring spins of the same sign connected with bond probability p. These clusters include the well-known Kasteleyn-Fortuin clusters as a special case for p=1-exp(-2K) , where K is the Ising spin-spin coupling. Along the critical line K=Kc , the size distribution of geometric clusters is investigated as a function of p . We observe that, unlike in the case of two-dimensional tricriticality, the percolation threshold in both models lies at pc =1-exp(-2Kc) . Further, we determine the corresponding red-bond exponents as yr =0.757(2) and 0.501(5) for the critical Ising and the tricritical Blume-Capel models, respectively. On this basis, we conjecture yr =1/2 for the latter model.

J.‐L. Cambier, Michael Nauenberg

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.

C.L. Wang, Z.K. Qin, D. L. Lin