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Journal of Physics A Mathematical and General

Exact solution of the Ising model on a pentagonal lattice

Viktor Urumov

Published Aug 13, 2002• 23 citations
Publisher

Abstract

It is shown that the Ising model on a two-dimensional lattice with pentagonal tiling and nearest-neighbour interactions is equivalent to the Ising model on the Union-Jack lattice with nearest- and second nearest-neighbour non-crossing interactions. Besides the critical temperature, exact expressions are obtained for the spontaneous magnetization for both types of lattice sites.

DOI
10.1088/0305-4470/35/34/306

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Journal of the Physical Society of Japan•Dec 15, 1993•13 citations
It is shown that the Ising model on a two-dimensional lattice with pentagonal tiling and nearest-neighbour interactions is equivalent to the Ising model on the Union-Jack lattice with nearest- and second nearest-neighbour non-crossing interactions. Besides the critical temperature, exact expressions are obtained for the spontaneous magnetization for both types of lattice sites.
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.
We show that the square two-dimensional (2D) Ising lattice with nearest- (J) and weak next-nearest-neighbour interactions (Jd ) can be mapped on a square 2D Ising lattice that has only nearest neighbour interactions (J*). For J d /J << 1 the transformation equation has the simple form . This result can be used to derive expressions for several thermodynamic functions of the square 2D Ising lattice with weak next-nearest-neighbour interactions. As an example we consider the spontaneous magnetization and compare it with low-temperature series expansion results.
The diagrammatical approach to the exact solution of two-dimensional Ising model, developed for the system with spin 1/2 and 1, is extended to be applicable to the Ising model of spin S greater than 1/2. The exact solution for the system on the square lattice and other lattices with non-crossing interactions is obtained only when the spin is half an odd integer, and the interaction between two spins s j and s k is expressed as - J j k φ j ( s j )φ k ( s k ) in terms of one paramater J j k and odd functions φ j ( s ) and φ k ( s ) of s satisfying |φ j ( s )|=|φ k ( s )|=1 for s =- S , - S +1,· ··, and S . The exact solution on the honeycomb lattice is obtained for all S , for the cases when the interactions divided by the temperature are expressed in terms of S +1/2 or S parameters.