The well-known Landau theory of phase transitions classifies phases of matter according to broken symmetries and local order parameters, such as solids that break translational symmetry, or magnets that break magnetic rotation symmetry. It has been long known that there are phases of matter that defy this classification — the quantum Hall state being the most obvious (but by no means only) example. With the discovery of topological insulators about 10 years ago, interest in this field has exploded, and we now know of many distinct phases of matter with no local order parameter, but instead characterised by a topological invariant. This short lecture course will focus mostly on non-interacting band theory, and introduce topological invariants, boundary states, and the bulk-boundary correspondence necessary to understand the modern topic of topological insulators. Other manifestations of topology in modern condensed matter physics will also be exposed, although not discussed in detail.
Sam works on the theory of strongly correlated systems, specialising in low-dimensional systems both in and out of equilibrium. He has worked in groups in the US, Italy and Germany, and since 2013 has been a lecturer at the University of Kent in Canterbury.