ABSTRACT

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 similar robust, quantized, interface transport. This talk considers systems modeled by elliptic partial differential operators. We review a general topological classification by means of confining domain walls, which leads to a bulk-difference invariant encoding a phase difference between insulating bulks even when absolute bulk phases are ill-defined. We next introduce a physical observable quantifying the asymmetry of the edge transport. A bulk-edge correspondence (BEC), a pillar of topological phases of matter, states that the two agree. We prove this mathematically for elliptic systems and present other systems for which it fails. We also briefly address the origin of such macroscopic models. Starting from tight-binding Hamiltonians, we derive continuous approximations that are valid over long times. Somewhat surprisingly, higher-order approximations that improve the description of transport may lead to improper topological characterizations. The framework is finally illustrated on two families of applications. In

electronics, we compute the edge invariant of gated rhombohedral multilayer graphene explicitly and exhibit a non-trivial dependence on the number of layers and on the interlayer coupling strength, to be compared with a recently measured quantum anomalous Hall effect with large Chern number. In photonics, we consider

magnetically biased cold plasmas, whose invariants require a regularization, as well as interfaces of finite width, for which the BEC holds when the magnetic bias varies continuously while discontinuities in that bias generate additional localized modes and an anomalous BEC.