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arXiv:1101.5015

Numerical Simulations of the Ising Model on the Union Jack Lattice

Vincent A. Mellor, Katrina Hibberd

Published Jan 26, 2011• 1 citation
arXivHTMLPDF

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

DOI
10.48550/arxiv.1101.5015

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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
This paper reviews the work of Wu and Lin on the Union Jack lattice Ising model. This model is of interest as it one of the few to display re-entrant phase transitions. Specifically we re-examine their result for the general anisotropic sublattice magnetisations, comparing these with the works of Vaks, Larkin and Ovchinnikov, and our own numerical simulations. We discuss the disagreements found in both sublattice predictions including non-zero antiferromagnetic results and a rotational variance. We will then suggest additional conditions and modified formulae that will allow valid results to be produced.
The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.
Though the idea to use numerical techniques, in order to solve complex three-dimensional problems, has become quite old, computational techniques have gained immense importance in past few decades because of the advent of new generation fast and efficient computers and development of algorithms as parallel computing.Many mathematical problems have no exact solutions.Depending on the complexity of the equations, one needs to use approximate methods.But there are problems, which are beyond our limits, and need support of computers.Ernst Ising published his PhD dissertation in the form of a scientific report in 1925.He used a string of magnetic moments; spin up (+1/2) and spin down (À1/2), and applied periodic boundary conditions to prove that magnetic phase transition does not exist in one dimensions.Lars Onsager, latter, exactly solved the phase transition problem in two dimensions in 1944.It is going to be a century-old problem now.A variety of potential applications of Ising model are possible now a days; classified as Ising universality class models.It has now become possible to solve phase transition problems in complex three-dimensional geometries.Though the area of spinotronics still needs more engagements of computational techniques, its limited use have provided good insights at molecular scale in recent past.This chapter gives a brief introduction to Ising model and its applications, highlighting the developments in the field of magnetism relevant to the area of solid state physics.
Abstract In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.