This page provides details of 4th year Master's projects I have supervised.
Majorana Modes, Superconducting Vortices, and String Theory
A mapping exists between p-wave superconductors confined to a two dimensional region of radius R and s-wave superconductors excluded from a region 1/R by a flux vortex. This mapping sounds reminiscent of the so-called “T-duality” in string theory, which takes (for example) a type-IIA string compactified in a curled-up dimension of size R to a type-IIB string with a momentum in that dimension.
The superconductor mapping is between Hamiltonians, but the string theory literature tends to work with states. To make the equivalence more precise we propose to find the action of the superconductor mapping on the states - specifically, the Majorana edge states.
The project involves solving the Bogoliubov deGennes equation numerically, then solving analytically for the special case of zero energy (Majorana) modes, and applying the superconductor mapping. If the mapping does turn out to be a form of T-duality we may be able to add another entry to the string theory/condensed matter dictionary.
The Origin of Symmetry in Snowflakes
This project was carried out by my student John Watts under joint supervision with Prof. John Hannay. The aim was to address a deceptively simple question: why are snowflakes symmetrical? The hexagonal crystal structure of ice Ih ensures sixfold symmetry locally, but does not explain why one leg of the snowflake ‘knows about’ the others.
The standard explanation is as follows: the growth of a snow crystal depends very sensitively on temperature and humidity. These quantities vary little on the scale of the crystal, so all six legs experience identical growing conditions - and thus turn out the same. Additionally the odds of two crystals following the same path through the cloud (and hence growth conditions) are vanishingly small, meaning no two snowflakes look alike.
This argument is not entirely convincing, though. If the crystal is sensitive enough to the growth conditions even tiny differences between legs would cause asymmetrical formations. Our student John applied a combination of analytical and numerical methods (cellular automata) to model snowflake growth, and to test how robust the symmetry is to disorder in the crystal seed and growth conditions.
Click here to see John's report.
After the project we discovered that even the idea of snowflakes having six legs is not accepted by everyone, in particular certain less reputable Christmas decoration manufacturers. Please see the Fun Extras section for the Snowflake Wall of Shame.