Numerical Simulations of the Ising Model on the Union Jack Lattice

TL;DR

This paper investigates the anisotropic Ising model on the Union Jack lattice, combining theoretical analysis and Monte Carlo simulations to explore phase transitions and magnetic behaviors in this complex lattice structure.

Contribution

It introduces numerical simulations of the Ising model on the Union Jack lattice, extending previous analytical work with Monte Carlo methods to study phase transitions.

Findings

Identification of re-entrant phase transitions on the Union Jack lattice

Validation of Monte Carlo simulations against known models

Insights into ferromagnetic, antiferromagnetic, and metamagnetic behaviors

Abstract

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