Applied Math Seminar Series - Topological insulators and robust edge transport

Surprising asymmetric transport phenomena along interfaces separating insulating bulks have been observed in many areas of applied sciences, e.g., electronics, photonics, and geophysics. Such transport displays strong robustness to perturbations as an obstruction to Anderson localization. In fact, it affords a topological origin: systems in the same topological class display the same robust, quantized, edge/interface transport. Perturbations may then be interpreted as continuous deformations that preserve the topological class.

This talk considers such systems modeled by general elliptic partial differential operators on the Euclidean plane. We review a recent simple topological classification, which provides an explicit computation of a topological invariant, technically the index of a Fredholm operator obtained by means of confining domain walls. We next introduce a physical observable that allows us to quantify the asymmetry of the edge transport. The evaluation of such an observable is challenging in practice. We then present a bulk-edge correspondence, a pillar of topological phases of matter in the physical literature, stating that the interface current observable is in fact equal to the aforementioned simple topological invariant. The theoretical findings are illustrated with examples ranging from electronics applications to geophysical fluid flows.