Statistics of geometric clusters in the Ising model on a Bethe lattice: statistical mechanics approach

TL;DR

This paper develops a statistical mechanics approach to analyze the size distribution of like-sign spin clusters in the Ising model on a Bethe lattice, revealing their relation to classical percolation theory.

Contribution

It introduces an analytical method for calculating the generating function of spin cluster sizes in the Ising model on a Bethe lattice, extending previous models.

Findings

Derived the phase diagram for spin clusters and percolation transition.

Showed that cluster size distribution relates to classical bond percolation.

Provided analytical expressions for cluster size distribution depending on temperature.

Abstract

The statistical mechanics method is developed for determination of generating function of like-sign spin clusters' size distribution in Ising model as modification of Ising-Potts model by K. K. Murata (1979). It is applied to the ferromagnetic Ising model on Bethe lattice. The analytical results for the field-temperature percolation phase diagram of + spin clusters and their size distribution are obtained. The last appears to be proportional to that of the classical non-correlated bond percolation with the bond probability, which depends on temperature and Ising model parameters.