# An empirical partition function for the simple cubic Ising Model with a zero external magnetic field

## TL;DR

This paper proposes an empirical partition function for the 3D Ising model without an external magnetic field, aligning well with high-temperature expansion results and providing insights into its critical behavior.

## Contribution

It introduces a novel empirical partition function for the 3D Ising model based on lattice Green function connections, which approximates known results and explores critical phenomena.

## Key findings

- EPF matches high-temperature expansion data
- Specific heat diverges non-logarithmically at T_c
- Critical temperature estimate exceeds Monte Carlo results

## Abstract

There is no an accepted exact partition function (PF) for the three dimensional (3D) Ising model to our knowledge. Mainly based on the connection between the lattice Green function (LGF) for the simple cubic lattice and that for the honeycomb lattice, we infer an empirical partition function (EPF) for the simple cubic Ising model in the absence of an external magnetic field. This ${\rm EPF}_{_{\rm 3D}}=\frac{1}{{2{\pi ^3}}}\int_0^\pi \int_0^\pi \int_0^\pi \log [2(2{{\cosh }^3}2z + 3{{\sinh }^2}2z + 2)^{\frac{1}{2}} -2\alpha\sinh 2z \ (\cos\omega_1 + \cos \omega_2 + \cos\omega_3)] {\rm{d}}\omega_1{\rm{d}}\omega_2{\rm{d}}\omega_3, \alpha\in[\sqrt{2},\sqrt{3}]$ (where $z=\frac{\epsilon}{kT}$, $\epsilon$ the interaction energy, $T$ the temperature, and $k$ Boltzmann constant). When $\alpha=\sqrt{2}$, this EPF is consistent well numerically with the result from high temperature expansions by Guttmann and Enting (1993). The specific heat from this EPF approaches infinity non-logarithmically at the critical temperature $T_c$. $\frac{\epsilon}{kT_c}=\cosh^{-1}[\frac{1}{4} (17-3\sqrt{17})]/2\approx 0.277212$, which is greater than 0.221654 from the recent Monte Carlo study.