Partition function and magnetization of two-dimensional Ising models in non-zero magnetic field: A semi-empirical approach

TL;DR

This paper introduces a semi-empirical, graph-theoretical approach to derive the partition function and magnetization of 2D Ising models in a magnetic field, aligning with Onsager's solutions at zero field.

Contribution

It presents a novel heuristic method to approximate the partition function and magnetization in non-zero magnetic fields, extending Onsager's exact zero-field solution.

Findings

Partition functions derived for low and high temperature regimes.

Magnetization matches Onsager's zero-field solution.

Series expansions of magnetization and susceptibility obtained.

Abstract

The partition functions of ferromagnetic Ising models of square lattices in a finite magnetic field is deduced using topological considerations within a heuristic graph-theoretical approach. These equations are derived separately for low and high temperature regimes while the exact solution of Onsager is obtained therefrom when the magnetic field is zero. The derived partition function equations here are almost similar to those given by Onsager, thus indicating a straight-forward protocol, even when the magnetic field is present. The spontaneous magnetization derived here using the Helmholtz free energy is identical with that arising from the exact solution. The partition functions lead to the known series expansions of the magnetization and zero-field susceptibility.