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Acta Physica Sinica

FREE-FERMION APPROXIMATE SOLUSION OF THE ISING MODEL ON UNION JACK LATTICE

WANG FU-GAO, HU JIA-ZHEN, 上海交通大学应用物理系凝聚态物理研究所,上海200030

Published Jan 1, 1993• 0 citations
Publisher

Abstract

In this paper,the Ising model with two-body anisotropic interactions and three-body interactions on union Jack lattice is solved by free-fermion approximation. We have obtained its free energy, spontaneous magnetisation and the equation of critical points. When the interaction coefficients are simplified as in a square lattice Ising model, the same solusion as Onsager's is obtained.

DOI
10.7498/aps.42.853

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In this paper,the Ising model with two-body anisotropic interactions and three-body interactions on union Jack lattice is solved by free-fermion approximation. We have obtained its free energy, spontaneous magnetisation and the equation of critical points. When the interaction coefficients are simplified as in a square lattice Ising model, the same solusion as Onsager's is obtained.
The Ising model is famous model for magnetic substances in Statistical Physics, and has been greatly studied in many forms. It was solved in one-dimension by Ernst Ising in 1925 and in two-dimensions without an external magnetic field by Lars Onsager in 1944. In this thesis we look at the anisotropic Ising model on the Union Jack lattice. This lattice is one of the few exactly solvable models which exhibits a re-entrant phase transition and so is of great interest. Initially we cover the history of the Ising model and some possible applications outside the traditional magnetic substances. Background theory will be presented before briefly discussing the calculations for the one-dimensional and two-dimensional models. After this we will focus on the Union Jack lattice and specifically the work of Wu and Lin in their 1987 paper "Ising model on the Union Jack lattice as a free fermion model." [WL87]. Next we will develop a mean field prediction for the Union Jack lattice after first discussing mean field theory for other lattices. Finally we will present the results of numerical simulations. These simulations will be performed using a Monte Carlo method, specifically the Metropolis-Hastings algorithm, to simulate a Markov chain. Initially we calibrate our simulation program using the triangular lattice, before going on to run simulations for Ferromagnetic, Antiferromagnetic and Metamagnetic systems on the Union Jack lattice