An empirical partition function for two-dimensional Ising Model in an external magnetic field

TL;DR

This paper proposes an empirical partition function for the 2D Ising model with an external magnetic field, enabling analysis of thermodynamic properties and phase transition behavior where no exact solution exists.

Contribution

It introduces an empirical partition function for the 2D Ising model with a magnetic field and validates it against existing finite-size solutions, facilitating thermodynamic analysis.

Findings

Empirical PF aligns well with existing finite-size solutions.

The model exhibits spontaneous magnetization and phase transition behavior.

Magnetization decreases with temperature, showing qualitative agreement with known physics.

Abstract

There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices ($4\times 4$ and $4\times 6$) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.