Event Details

The role of topology in designing electromagnetic materials

Abstract:

Topology is an interesting area of mathematics, concerned with finding properties that stay the same after a continuous deformation. To the topologist there is famously no difference between a coffee cup and a doughnut, because one can be continuously transformed into the other. To think like a topologist you need to imagine that the only important number in the cup to doughnut transformation is the number of holes in the surface - the genus - which stays the same. The genus is therefore known as a topological invariant. But what has this got to do with physics? In the last few decades there has been a flurry of activity in physics, because it was realised that the mathematics of topology plays a very important role in condensed matter physics. Each band gap can be characterized by a number, which is called the Chern number, and is analogous to the genus of a surface. For a given band gap, this number stays constant if the material properties are continuously changed - for example via a continuous distortion of the material lattice - so long as the band gap doesn't close. Even more interestingly, the Chern number tells us about what happens at the edge of the solid if two different materials are butted up against one another. Within a common band gap of the two materials, conduction cannot take place within either solid. But if their Chern numbers are different, conduction can take place at their interface! This talk is concerned with the application of these ideas to the design of electromagnetic and acoustic materials. Can we use topological invariants to design materials for light or sound with particular wave transport properties? We will talk less about the electron wave in a lattice of ions, and more about the propagation of light and sound through a continuous material, or a lattice of scatterers. In the first part of the talk I shall show that the macroscopic Maxwell equations can be naturally cast into the form of a Dirac-like equation, with electromagnetic material parameters playing the role of `mass', `velocity', and `gauge field.' The zero energy solutions of the Dirac equation are already known to depend on the topology of a space and any associated gauge field, and I shall show that we can accordingly use topological invariants to predict the existence and dispersion of electromagnetic modes in inhomogeneous electromagnetic materials. In the second part of the talk I shall show that there is also a much simpler way to understand some of the interface modes that are predicted using topology. Extending the concept of the refractive index to include complex as well as real directions, we find that a generalization of the zero index condition can be used to design materials that support one-way propagation at their edges. We shall see some special cases of this, including the recently highlighted phenomenon of `spin-momentum' locking, which to date has been predicted using a rather unintuitive topological argument.