Investigation of the Ising-type models within cluster approach

TL;DR

This paper develops a cluster approximation method for calculating correlation functions and susceptibilities in Ising-type models with short-range interactions, providing exact solutions in 1D and applying to various models.

Contribution

It introduces a cluster approach for Ising models with arbitrary spin, deriving explicit correlation functions and susceptibilities, including exact 1D results and applications to specific models.

Findings

Exact expressions for 1D correlation functions are obtained.

The method provides explicit formulas for pair and ternary correlations.

Application to ferroelectric material KD2PO4 demonstrates practical utility.

Abstract

The method for calculation of the correlation functions of the Ising-type systems with short-range interaction and with arbitrary value of spin is developed within cluster approximation. For the Ising model (spin $S^z=\pm1$) the expressions for pair and ternary correlation functions within two-particle approximation in $\set{q}$-space are obtained for the hypercubic Bravais lattices. In the 1D case the exact expressions for them in the site space is obtained as well. On the basis of the Glauber equation within two-particle cluster approximation the longitudinal dynamical susceptibility $\chi(\set{q},E)$ is found. In the 1D case and in the absence of external field the expression for $\chi(\set{q},E)$ is exact. For the Emery-Blume-Griffiths model ($S^z=-2,0,2$) within two-particle approximation the pair correlation functions are calculated. The four-particle cluster approximation is used for calculation of static susceptibility $\chi(\set{q})$ of $KD_2PO_4$ ferroelectrics.