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International Symposium on Lattice Field Theory (LATTICE)

Geometrical representation of the 2D Antiferromagnetic Ising Model with topological term theta=pi

Gennaro Cortese, V. Azcoiti, E. Follana, Matteo Giordano

Published Dec 20, 2012• 0 citations
Publisher

Abstract

We study the two-dimensional Antiferromagnetic Ising Model with an imaginary magnetic field iθ at θ = π.We use a new geometric algorithm which does not present a sign problem.This allows us to perform efficient numerical simulations of this system.
DOI
10.22323/1.164.0248

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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.
The partition function and the spontaneous magnetization of a two-dimensional (2D) Ising model with the next nearest interactions at zero magnetic field are derived analytically. At first, the transfer matrices are analyzed in three representations, i.e., Clifford algebraic representation, transfer tensor representation, and schematic representation, to inspect nontrivial topological structures in this system. The system is equivalent to a triangular Ising model plus an interaction along the z axis, so that the approaches developed for the 3D Ising models are modified to be appropriable for analytically solving the solution of the 2D Ising model with the next nearest interactions. The comparison with the exact solutions of other Ising lattices reveals that either the increase of the number of interactions in a unit cell or the presence/increase of topological contributions enhances the critical point of the Ising lattices. The results obtained in this work are helpful for understanding the physical properties of the 2D magnetic materials.