Mathematical structure of three - dimensional (3D) Ising model

TL;DR

This paper explores the mathematical structure of the 3D Ising model through topological, algebraic, and geometric perspectives, revealing connections to relativistic quantum mechanics, topology, and integrability conditions.

Contribution

It introduces a novel algebraic and topological framework for understanding the 3D Ising model, including quaternion bases, transfer matrix smoothing, and tetrahedron relations, advancing the theoretical understanding.

Findings

Quaternion basis relates to 4D space-time rotations.

Unitary transformations smooth transfer matrix crossings.

Tetrahedron relation ensures integrability and commutativity.

Abstract

An overview of the mathematical structure of the three-dimensional (3D) Ising model is given, from the viewpoints of topologic, algebraic and geometric aspects. By analyzing the relations among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model: 1) The complexified quaternion basis constructed for the 3D Ising model represents naturally the rotation in a (3 + 1) - dimensional space-time, as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function by taking the time average. 2) A unitary transformation with a matrix being a spin representation in 2^(nlo)-space corresponds to a rotation in 2nlo-space, which serves to smooth all the crossings in the transfer matrices and contributes as the non-trivial topologic part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed also by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases {\Phi}x, {\Phi}y, and {\Phi}z. The relation with quantum field and gauge theories, physical significance of weight factors are discussed in details. The conjectured exact solution is compared with numerical results, and singularities at/near infinite temperature are inspected. The analyticity in {\beta} = 1/(kB T) of both the hard-core and Ising models has been proved for {\beta} > 0, not for {\beta} = 0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.