Nature of project: **theory**, software

Available to full-time physicists or joint students.

In many ferro-magnetic systems there is a temperature above which the system ceases to exhibit spontaneous magnetisation. This temperature is known as the Curie temperature. In 1920, Lenz and Ising introduced a model for the ferro-magnetic to para-magnetic transition. This model (known as the Ising model) is based on a lattice of 'spins' at fixed positions. Each spin can attain two orientations (e.g. 'up' or 'down'). Each spin interacts with its nearest neighbours on the lattice. Each spin also interacts with an external magnetic field. This model has been shown to exhibit a phase transition in two and three dimensions.

For a square lattice in two dimensions, Lars Onsager derived an analytical solution for the free energy in the absence of an external field. This explicitly shows the phase transition and pinpoints the Curie temperature exactly. All attempts to generalise Onsager's approach to three dimensions have failed.

This project is about studying the Ising model using numerical simulation, in particular using various versions of the Monte Carlo algorithm. A Metropolis type algorithm with simple flips of individual spins is fairly straightforward to implement.

*A successful project will develop beyond the above in one/some of the following directions:*

The 2D results can be compared to Onsager's solution. To develop the project further, the three dimensional case can be studied. In the 3D case, there is no analytical solution, so there is a genuine need for simulating the Ising magnet.

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

*Additional scope or challenge if taken as a Year-4 project:* For a Y4 project, the more complicated Swendsen-Wang algorithm [3] could be considered.

*Initial literature for students:*

- K. Huang, Statistical Mechanics, Wiley, 2nd edition 1987.
- L. E. Reichl, A modern course in Statistical Physics,Wiley, 3rd edition, 2009.
- https://en.wikipedia.org/wiki/Swendsen%E2%80%93Wang_algorithm

Many algorithms have already been developed for simulating the Ising model. The student(s) can choose any of them and implement it in a programming language of their choice. (Fortran or C would probably be most appropriate.) The ferro-magnetic to para-magnetic transition is a classic example of a second order phase transition. The student(s) can study the thermodynamic behaviour of the system near the critical point (i.e the Curie temperature).

milestone | to be completed by |
---|---|

Familiarisation with thermodynamics of the Ising model | end of November |

Familiarisation with existing algorithms for simulating the Ising model | Christmas |

Implementation of algorithm for simulating the Ising model | mid-March |

Analysis of data | Easter |