Description

Statistical mechanics is one of the pillars of modern physics: it allows us to predict the relations between observable average properties of a macroscopic system starting from the properties of its microscopic constituents and the forces among them. It applies to the hot tea in your mug, to the magnet that keeps the door of your fridge closed, and even to the Sun and the cosmic microwave background that permeates the universe.

The link from the microscopic to the macroscopic relies on well understood statistical ideas, but in practice it is hard to carry out the necessary calculations explicitly unless the microscopic constituents interact very weakly. There is however a class of idealized interacting systems for which exact calculations are possible: they involve degrees of freedom localized at the sites of a lattice, a regular arrangement of points. The simplest example, which will be the first focus of the project, is the Ising model, which describes spins pointing up or down and interacting with their nearest neighbouring spins on the lattice and with an external magnetic field. For a lattice of dimension one or two, the model can be solved exactly.

In the first part of the project we will study the exact solution of the two-dimensional Ising model, which uncovers a rich mathematical and physical structure. You will encounter phase transitions (which lead to drastic changes in the macroscopic properties of the system), critical behaviour and universality (in which the system develops long range correlations and loses memory of its microscopic details), and the surprising Kramers-Wannier duality, which relates the high temperature and low temperature behaviours of the system in a highly non-trivial way. To get a feel of the main properties of the two-dimensional Ising model, play with this numerical simulation, in which white and black squares represent spins pointing up or down in a square lattice.

In the second part of the project you could explore further topics such as:

• the mathematics and physics of other lattice models, from generalizations of the Ising model such as the O(n) model, to the dimer model and the Miura-ori origami;

• the theory of phase transitions, from first order phase transitions like the one underlying boiling water, to second order phase transitions which lead to the idea of universality, and up to the phase transitions of infinite order discovered by Berezinskii, Kosterlitz and Thouless in the XY model, a generalization of the Ising model;

• numerical simulations of the Ising model or other lattice models in various dimensions.

Prerequisites

2H Mathematical Physics
3H Statistical Mechanics is highly recommended, but we will go through the basics at the beginning of the project.

Resources

• R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, 1989.
  The first chapter of this book gives a nice introduction to the subject.
• D. Tong, Statistical Physics (section 5) and Statistical Field Theory (sections 1 and 4).
• M. Kardar, Statistical Physics of Fields, Cambridge University Press, 2007 (sections 6 and 7).
• K. Huang, Statistical Mechanics, Wiley, 1987 (sections 14, 15, 16).
• R. Savit, Duality in field theory and statistical systems, Rev. Mod. Phys. 52, 453, 1980.