Exact solution of the two-dimensional (2D) Ising model at an external magnetic field

TL;DR

This paper presents an exact analytical solution for the 2D Ising model in an external magnetic field using a modified Clifford algebra approach, revealing topological effects and phase transition behaviors.

Contribution

It introduces a novel method combining Clifford algebra and topological transformations to solve the 2D Ising model with magnetic field, extending techniques from 3D models.

Findings

Magnetization increases with magnetic field, shifting the critical temperature.

First-order magnetization jump occurs at a critical field.

Partition function and magnetization elucidate physical properties of 2D magnetic materials.

Abstract

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.