Formulation of the partition functions and magnetization for two-dimensional nearest neighbour Ising models for finite and infinite lattice sites

TL;DR

This paper introduces a combinatorial approach to derive partition functions and magnetization for 2D Ising models on finite and infinite lattices, offering a new hierarchical enumeration method without computational tools.

Contribution

It presents a novel hierarchical enumeration method for 2D Ising models, enabling analytical derivation of partition functions and magnetization without algorithms.

Findings

Derived partition functions for 16-site lattice

Proposed a semi-empirical magnetization equation for non-zero fields

Methodology applicable to infinite lattice magnetization analysis

Abstract

Using a combinatorial method, the partition functions for two-dimensional nearest neighbour Ising models have been derived for a square lattice of 16 sites in the presence of the magnetic field. A novel hierarchical method of enumeration of all the configurations for any arrangement of sites has been proposed. This enumeration has been executed by a systematic analysis of the appropriate diagrams without employing any algorithmic approach or computational tools. The resulting algebraic eqn in terms of the magnetic field and nearest neighbour interaction energies may then provide a methodology for deducing the magnetization in the thermodynamic limit of infinite sites. A semi-empirical eqn for magnetization is proposed for non-zero magnetic fields.