Analytical Expressions for Ising Models on High Dimensional Lattices

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 the results of computer simulations. For d=5,6,7, our theoretical estimates match the numerical results both qualitatively and quantitatively. For d=3,4, our method is sufficiently accurate for the 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 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.

Characteristics of an Ising-like Model with Ferromagnetic and Antiferromagnetic Interactions

“…The discussion and our conclusions are given in Section 6. To conclude, we would like to mention that in our paper [25], for the first time, we used a similar technique.…”

Generalized Solution of Inverse Problem for Ising Connection Matrix on d-Dimensional Hypercubic Lattice

“…Some time ago, we developed an m-vicinity method for calculation of the partition function [ 5 ], which showed itself rather effective. The necessity to calculate exact expressions for the eigenvalues of the Ising connection matrix arose from this research.…”

Ising Model: Recent Developments and Exotic Applications

“…In yet another paper of the Special Issue, Kryzhanovsky et al developed an analytical method to examine the Ising model on lattices of large dimensions d [ 14 ]. Their approach enables them to calculate the critical temperature, free energy, or heat capacity and gives some insight into the behaviour of the density of state.…”