Analytical solutions for Ising models on high dimensional lattices

TL;DR

This paper develops an m-vicinity analytical method to study Ising models on high-dimensional hypercube lattices, accurately predicting critical temperatures and revealing insights into phase transition singularities across different dimensions.

Contribution

The paper introduces a new m-vicinity method applicable to high-dimensional Ising models, providing analytical expressions for critical temperatures and analyzing phase transition singularities.

Findings

Accurate critical temperature predictions for d=5,6,7

Method aligns well with simulations for high dimensions

Predicts finite heat capacity jump at critical point for d=3

Abstract

We use an m-vicinity method to examine Ising models on hypercube lattices of high dimensions d>=3. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice d. We obtain an expression for the critical temperature in terms of the interaction constants that is in a good agreement with results of computer simulations. For d=5, 6, 7, our theoretical estimates match the experiments both qualitatively and quantitatively. For d=3, 4, our method is sufficiently accurate for calculation of the critical temperatures, however, it predicts a finite jump of the heat capacity at the critical point. In the case of the three-dimensional lattice (d=3), this contradicts to the commonly accepted ideas of the type of the singularity at the critical point. For the four-dimensional lattice (d = 4) the character of the singularity is under current discussion. For the dimensions d=1, 2 the m-vicinity method is not applicable.