An exact solution to the partition function of the finite-size Ising Model

TL;DR

This paper presents an exact, comprehensive solution for the partition function of finite-size Ising models in any dimension, enabling precise calculations of thermodynamic properties and offering new insights into classic models.

Contribution

It provides the first exact partition function for finite-size Ising models in any dimension, including 3D, using a rigorous elementary method.

Findings

Partition function expressed as sum of 2^N exponential functions

Calculated specific heat and magnetization match experimental and simulation data

Solution applicable to other models like percolation and Potts

Abstract

There is no an exact solution to three-dimensional (3D) finite-size Ising model (referred to as the Ising model hereafter for simplicity) and even two-dimensional (2D) Ising model with non-zero external field to our knowledge. Here by using an elementary but rigorous method, we obtain an exact solution to the partition function of the Ising model with $N$ lattice sites. It is a sum of $2^N$ exponential functions and holds for $D$-dimensional ($D=1,2,3,...$) Ising model with or without the external field. This solution provides a new insight into the problem of the Ising model and the related difficulties, and new understanding of the classic exact solutions for one-dimensional (1D) (Kramers and Wannier, 1941) or 2D Ising model (Onsager, 1944). With this solution, the specific heat and magnetisation of a simple 3D Ising model are calculated, which are consistent with the results from experiments and/or numerical simulations. Furthermore, the solution here and the related approaches, can also be available to other models like the percolation and/or the Potts model.