Clusters and Critical Behavior in the Ising Model

Monte-Carlo simulations of two and three dimensional Ising systems and a cluster search have been performed and lead to a scaling analysis of the cluster distribution. Scaling exponents of the distribution have been carefully measured and successfully related to the thermodynamic exponents. The clusters are observed to have fractal surface properties, and they are consistently larger than the magnetic fluctuations, as their linear scale diverges faster than the correlation length. This is observed for d = 2, for which an infinite cluster appears at the Ising transition, and for d = 3, for which the percolation threshold seems to appear below the Ising critical point. A scaling analysis is possible only below this percolation threshold; the data cannot resolve the ambiguity of a scaling with respect to the Ising transition or the percolation threshold; some possible interpretations are discussed. The same analysis has been done also on the random field Ising model for d = 3; the clusters are still fractal, but pinned in position by the quenched random field configuration. The magnetic fluctuation is mostly contained near the surface. The magnetization has an apparent discontinuity, and suggests a first order transition with an effective dimensional reduction to d = 2. A percolation of the lattice also occurs below the magnetic transition. A more detailed structure analysis is done for d = 2, and the clusters, or domains, are seen to be stable at low temperatures, suggesting that the lower critical dimensionality is 2. Monte-Carlo and local mean field simulations of the same samples have been done and compared; the magnetization as a function of the temperature has the structure of a random staircase for strong field and finite lattice. Monte-Carlo and mean field simulations give the same qualitative results and suggest an hierarchy of cluster nucleations, according to their field configurations.