Topological Effects and Critical Phenomena in the Three-Dimensional (3D) Ising Model

Young C. Kim, Mikhail A. Anisimov, Jan V. Sengers +1 more

E. H. Boubcheur, H. T. Diep

We study a three-dimensional (3D) Ising model with ferromagnetic interaction and elastic (cohesive) interaction between neighboring atoms by Monte Carlo simulation. When the elastic interaction strength is much larger than that of the magnetic one, the transition from ferromagnetic state to paramagnetic phase is of the 3D Ising universality class. However, the critical behavior of the 3D Ising model changes into a 3D XY universality when the elastic interaction is smaller than a critical value.

Zhang Zhi-dong

The Ising model and its applications,the procedure for deriving the exact solution of the two-dimensional Ising model,the difficulty for solving explicitly the three-dimensional(3D)Ising model have been introduced briefly.The progresses in studying physical properties and critical phenomena of the 3D Ising model by various approximation methods such as the mean field theory and its extensions,renormalization group theory and Monte Carlo simulations etc.have been reviewed.Finally,our recent advances in the conjectures and the putative exact solution of the 3D Ising model are introduced.

Subinay Dasgupta, Dietrich Stauffer, V. Dohm

Zhang Zhi-dong

The exact solution of the two-dimensional (2D) Ising model at an external magnetic field is derived by a modified Clifford algebraic approach. At first, the transfer matrices are analyzed in three representations, i.e., Clifford algebraic representation, transfer tensor representation and schematic representation, to inspect nonlocal effects in this many-body interacting system. It is ensured that nontrivial topological structures exist in this system, which is analogous to (but different with) those in the three-dimensional (3D) Ising model at zero magnetic field. Therefore, the approaches developed for the 3D Ising models are modified to be appropriable for solving analytically the solution of the 2D Ising model at a magnetic field. An additional rotation, serving as a topological Lorentz transformation, is applied for dealing with the topological problems in the present system. The rotation angle for the transformation is determined by Yang-Baxter relations and a subsequent average of rotation angles treating the linear change of the topological actions. Application of a magnetic field increases the magnetization, shifting the critical point to higher temperatures. At the temperature above the critical point, the magnetization keeps zero until a critical field at which it jumps rapidly as a first-order magnetization process. The partition function and the magnetization obtained are helpful for understanding the physical properties, in particular, the magnetization processes of the 2D magnetic materials.

Hans-Otto Heuer