Equations of motion approach to the spin-1/2 Ising model on the Bethe lattice

TL;DR

This paper presents an exact solution for the spin-1/2 Ising model on the Bethe lattice using equations of motion and Green's functions, providing detailed thermodynamic and correlation insights.

Contribution

It introduces an exact method to solve the Ising model on the Bethe lattice, including non-local correlations and thermodynamics, for any coordination number.

Findings

Exact eigenoperators and eigenvalues derived

Green's and correlation functions expressed in self-consistent parameters

Thermodynamic quantities and non-local correlations computed

Abstract

We exactly solve the ferromagnetic spin-1/2 Ising model on the Bethe lattice in the presence of an external magnetic field by means of the equations of motion method within the Green's function formalism. In particular, such an approach is applied to an isomorphic model of localized Fermi particles interacting via an intersite Coulomb interaction. A complete set of eigenoperators is found together with the corresponding eigenvalues. The Green's functions and the correlation functions are written in terms of a finite set of parameters to be self-consistently determined. A procedure is developed, that allows us to exactly fix the unknown parameters in the case of a Bethe lattice with any coordination number z. Non-local correlation functions up to four points are also provided together with a study of the relevant thermodynamic quantities.