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Journal of Hunan University

A Simple Deduction of the Partition Function of the Ising Model

Xie Zi-fang

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

External parameter is introduced to deduce the partition function of one-dimensional Ising model and two-dimensional Ising model,so the sum of spinor can be simply replaced by the partial derivative of the external parameter.

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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 NNN lattice sites. It is a sum of 2N2^N2N exponential functions and holds for DDD-dimensional (D=1,2,3,...D=1,2,3,...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.
The partition function of the 2D Ising model coupled to an external magnetic field is studied. We show that the sum over the spin variables can be reduced to an integration over a finite number of variables. This integration must be performed numerically. But in order to reduce the partition function we must introduce as many different coupling constants as spin variables. The total memory that we need in order to store these coupling constants imposed important restrictions on the number of spin variables.