Relevant spontaneous magnetization relations for the triangular and the cubic lattice Ising model

TL;DR

This paper introduces a new mathematical method to derive spontaneous magnetization relations for 2D triangular and 3D cubic lattice Ising models, showing good agreement with known results and providing new insights for 3D cases.

Contribution

A novel, easily calculable mathematical approach for deriving spontaneous magnetization in Ising models, applicable to both 2D and 3D lattices, with validation against existing results.

Findings

The method accurately reproduces known 2D triangular lattice results.

It provides a new expression for 3D cubic lattice spontaneous magnetization.

Results are consistent with existing numerical data for the 3D lattice.

Abstract

The spontaneous magnetization relations for the 2D triangular and the 3D cubic lattices of the Ising model are derived by a new tractable easily calculable mathematical method. The result obtained for the triangular lattice is compared with the already available result to test and investigate the relevance the new mathematical method. From this comparison, it is seen that the agreement of our result is almost the same or almost equivalent to the previously obtained exact result. The new approach is, then, applied to the long-standing 3D cubic lattice, and the corresponding expression for the spontaneous magnetism is derived. The relation obtained is compared with the already existing numerical results for the 3D lattice. The essence of the method going to used in this paper is based on exploiting the main characteristic of the order parameter of a second order phase transition which provides a more direct physical insight into the calculation of the spontaneous magnetization of the Ising model.