New perspectives on the Ising model

TL;DR

This paper presents a unified solution to the Ising model with an external magnetic field across any dimension, using an isomorphism to fermionic particles, and derives exact expressions for Green's and correlation functions.

Contribution

It introduces a general, self-consistent framework for solving the Ising model via fermionic isomorphism, applicable in any dimension, and connects properties through a small set of parameters.

Findings

Exact solutions for Green's and correlation functions

Self-consistent parameters calculated for 1D case

Results agree with known exact solutions

Abstract

The Ising model, in presence of an external magnetic field, is isomorphic to a model of localized interacting particles satisfying the Fermi statistics. By using this isomorphism, we construct a general solution of the Ising model which holds for any dimensionality of the system. The Hamiltonian of the model is solved in terms of a complete finite set of eigenoperators and eigenvalues. The Green's function and the correlation functions of the fermionic model are exactly known and are expressed in terms of a finite small number of parameters that have to be self-consistently determined. By using the equation of the motion method, we derive a set of equations which connect different spin correlation functions. The scheme that emerges is that it is possible to describe the Ising model from a unified point of view where all the properties are connected to a small number of local parameters, and where the critical behavior is controlled by the energy scales fixed by the eigenvalues of the Hamiltonian. By using algebra and symmetry considerations, we calculate the self-consistent parameters for the one-dimensional case. All the properties of the system are calculated and obviously agree with the exact results reported in the literature.