## New tools from summer school

Nick Jones is a second-year postgraduate researcher in the School of Mathematics. His research is rooted in physics and the problem of understanding how a large number of interacting particles behave. Particularly interesting is the case where the emerging cooperative behaviour of the whole system is very different to that of the smaller pieces that make it up.

Having just returned to my desk after a summer of attending conferences and workshops I thought I’d say something about the merits of travelling – a common feature of ‘a year in the life of a PhD’. First, though, I should explain some context about my research – I am studying a physical phenomenon called the fractional quantum Hall effect. The Hall effect, discovered in 1879, is the bunching up of electrons on one side of a current-carrying wire once you place that wire in a magnetic field. You could think of the wire as a horizontal pipe containing fast flowing water which carries bubbles of air and, while there would bubbles all through the pipe, you might expect more towards the top as they drift upwards (of course the real motion of bubbles can get very complicated, but it’s a picture to have in mind). You can explain the Hall effect using the theory of electromagnetism from the 19th century, and characterise the bunching in terms of the conductance. The conductance can be measured in an experiment and is seen to go down as you increase the applied field.

In the 1980s Klaus von Klitzing looked at what happens when you cool a particular ‘wire’ (something called a MOSFET) down to very low temperatures and place this in a very strong magnetic field. These extreme conditions bring the quantum mechanics (the counterintuitive microscopic theory) of the electrons to the forefront and remarkably he saw that the conductance no longer moves up and down continuously with the field, rather it gets stuck on special values – integers. You can explain these special integer values using the quantum mechanics of electrons that we assume to not interact with each other except for the famous rule that any two electrons can’t be in the same state. The pattern the electrons make is very rigid and doesn’t change at all if you prod it a little bit (with the only tool available to you – the magnet). At some point though, as you change how strong the magnet is, the integer jumps up or down. The behaviour in these jumps is even more interesting and is called the fractional quantum Hall effect – this is because you see some further special values inside the jump where the conductance is some fraction (the easiest one to see is 1/3).

There’s still lots of research going into how to describe the system at these fractional conductances but one point that is particularly important to me is that very nice mathematical objects represent the states of the electrons – functions of complex numbers (these are numbers where you are able to take the square root of negative numbers). Another beautiful object appears if the electrons aren’t completely free to move (they’re bound somewhat to atoms) – the Hofstadter butterfly.

The Hofstadter butterfly shows you which energies the electrons are allowed to have as you change the strength of the magnetic field. We’re trying to see how these two sides of the same physical system fit together in terms of some ideas from another mathematical community – the probability theorists. They work on randomness in all its manifestations but in particular have looked at these complex functions on lattices – exactly the setting where the Hofstadter butterfly appears.

To see how these pieces fit together, I need to understand the two sides, and also the potential glue. In August I was lucky to get the chance to participate in a month-long school on topological condensed matter physics – a booming area of research which grew out of work on the quantum Hall effect, and in which a central theoretical concept is the Berry phase (discovered in Bristol!). The school brought together experimentalists and theorists from all over the world, just outside the alpine village of les Houches. The participants all lived together, went to lectures together and in between ate cheese on hikes together. This kind of proximity meant we had plenty of opportunities to talk to each other and I now have a much better understanding of the physics I’m studying. The conference closed with an excellent talk by von Klitzing on the history of the quantum Hall effect and some of the more recent work he has done – a nice way to finish discussions on the field that had since exploded.

Up next was probability, and luckily the probability group at Bristol were hosting a week long workshop as soon as I got back. The topic was using probability to get results about quantum systems – exactly what I’m thinking about. This was followed by another workshop in Oxford on recent work in the field and lots of inspiring ideas. I met another student there who had been at les Houches and had explained a lot about Hofstadter’s butterfly to me. He told me that he was trying to compare some electron states he had found to a mathematical object called a Jack polynomial. He couldn’t generate these things on his computer and so it was taking a long time, but I realised that Bristol had a resident expert on Jack polynomials and sure enough she knew a computer program that could do what he needed. Lucky coincidences like this made me appreciate even more the importance of travelling.  Now I’m back where I started in July, with the same problem to do, but a few more tools to tackle it and am looking forward to this term.