MATHEMATICAL STRUCTURE AND THE CONJECTURED EXACT SOLUTION OF THREEDIMENSIONAL (3D) ISING MODEL

In this article, the history of study on Ising model was first reviewed briefly, including a brief introduction of Ising model, the advances in the study of two-dimensional (2D) and three-dimensional (3D) Ising models, with a special interest in the exact solution of the 2D Ising model. Then two conjectures and putative exact solution of the 3D Ising model were introduced, and the mathematical structure of the 3D Ising model was investigated from the aspects of topology, algebra and geometry. The transfer matrices of the 3D Ising model, the knot theory in the topology, the relations between the Yang-Baxter equations and the tetrahedron equations were analysized. The non-local effect in the 3D Ising model, the relation between quantum field theory and gauge theory, the physical significance of weight factors, the singularity and the topological phase transition at/near infinite temperature in the 3D Ising model were also discussed. Finally, it was pointed out that some approximation techniques (for examples, low-temperature expansions, high-temperature expansions, renormalization group and Monte Carlo simulations) have disadvantages for studying the 3D Ising model.