By Todd Neve, The University of Melbourne
The Ising Model is a mathematical model of ferromagnetism in statistical mechanics. It was first studied by Ernst Ising in the 1920s to analyse critical points of phase transition in ferromagnets. It gives insight into how short-range interactions between atoms can give rise to long-range, correlative behaviour.
Background
Ferromagnetism is a mechanism by which certain materials, such as iron, can form permanent magnets. In a lattice of iron atoms, each atom is effectively a tiny magnet. The magnetic moments of each iron atom spontaneously align and combine to create a strong magnetic field. However, as the temperature increases we find that beyond a certain point, the lattice loses its magnetisation.
The Project
In this project I considered a 2D square lattice, where each lattice site is occupied by an atom with a magnetic moment that points either up or down.
At low temperatures we expect the magnetic moment of each atom to point in the same direction. As the temperature increases however, we expect half will point up and half down.
Our aim is to understand how this change occurs, and to predict the system’s potential for this change to occur.
In several seminal papers released by Yang and Lee in the 1940s, they proposed a mathematical definition of a phase transition.
To understand it, we first consider the thermodynamic functions of a system, such as our 2D lattice. Each function gives information about the system which depends on certain parameters. An example is the total energy, which depends on multiple parameters, including the temperature.
If we consider physical values for these parameters, we find that the function will vary smoothly, as in the figure above. However, if we generalise these parameters to ‘complex’ values, we find that in certain cases, the function has points of discontinuity.
Yang and Lee suggested that as the system gets larger and larger, these points of discontinuity, which occur in the complex plane, pinch the real line. That is, the points of discontinuity approach the physical values for the parameters. Phase transitions arise from this.
So by studying the lattice’s behaviour using complex parameters, we can learn about its behaviour for physical parameters. In this way we hope to gain insight into how phase transitions occur.
Todd Neve was one of the recipients of a 2015/16 AMSI Vacation Research Scholarship.