Non-Hermitian and topological quantum physics

The mathematics underpinning topological matter is of a large and growing interest, especially since the Nobel Prize in Physics was awarded in 2016 to the British theoretical physicists Thouless, Haldane and Kosterlitz "for theoretical discoveries of topological phase transitions and topological phases of matter". At its heart, topological physics allows one to explain the behaviour of complicated systems very simply, usually through some important number: a so-called topological invariant. A famous example is in the much-celebrated quantum Hall effect, where the resistance in the material surprisingly plateaus at certain integer values. This intriguing phenomenon can be understood elegantly through topological numbers known as Chern numbers. We have so far only discussed conventional materials, which are primarily governed by the behaviour of their constituent electrons. However, much less is known when it is the particle of light (the photon) which instead plays an important role in the system. In the field of quantum nanophotonics, we are concerned with nanoscale materials where the light-matter interaction is decisive. In particular, in the strong light-matter coupling regime, electrons and photons may hybridize to form a new particle (a polariton), carrying desirable traits of both of its component parts. This sets the stage for the proposed project, which aims to make fundamental mathematical physics advances in topological quantum matter. Namely, inspired by the success of topological theories in describing electronic systems, we will develop theories to describe and exploit topological light-matter excitations for use in the transportation of both energy and information at the nanoscale. We will also develop non-Hermitian theories to account for the inevitable losses in such systems.