Clusters and Critical Behavior in the Ising Model

Ricardo Paredes V., Carlos Vásquez

The influence of correlated impurities on the critical behavior of the three-dimensional (3D) Ising model is studied using Monte Carlo simulations. Spins are confined into the pores of simulated aerogels (diffusion-limited cluster-cluster aggregation) in order to study the effect of quenched disorder on the critical behavior of this magnetic system. Finite-size scaling is used to estimate critical couplings and exponents. Long-range correlated disorder does not affect the critical behavior. Asymptotic exponents differ from those of the pure 3D Ising model, but it is impossible, with our precision, to distinguish them from the randomly diluted Ising model.

J. Hubbard

Tao Chen, Jinhong Zhu, Wei Zhong +2 more

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.

Hans Gerd Evertz, H. M. Erkinger, Wolfgang von der Linden

Yutaka Okabe, Kazuhisa Kaneda, Yusuke Tomita +2 more

Chengxiang Ding

We study a cluster Ising model with multispin interactions which can be exactly solved in the framework of free fermions. The model can realize topological phases with any integer winding numbers; we study the critical and multicritical behaviors of the phase transitions between these topological phases. For the ordinary critical point, we find that the critical exponent that governs the divergence of the correlation length is ν=1, and the critical exponent that describes the scaling behaviors of the order parameter is β=ΔN_{w}/8, with ΔN_{w} the difference of the winding numbers of the two phases at the two sides of the critical point. However, these results are not applicable for some multicritical points which have much more complicated behaviors, such as path dependent scaling behaviors and quasicritical behaviors and so forth.