Robust asymmetric transport at the interface between two-dimensional insulating bulks has been observed in many areas of (geo)physical and materials sciences. The main practical appeal of this asymmetry is its immunity to large classes of perturbations. This stability is explained by topological considerations.
A physical observable, a one-dimensional conductivity, is assigned to the asymmetric transport. Interface Hamiltonians modeling the transition between the bulk phases are next introduced and classified by a topological charge, the index of an appropriate Fredholm operator. A general principle, the bulk-edge correspondence, then states that the conductivity is quantized and equal to the topological charge, which may be interpreted as a difference of bulk topologies.
While ubiquitous in the physical and engineering literatures, the bulk-edge correspondence remains difficult to establish mathematically or in fact even heuristically. This talk presents recent results on the derivation of the correspondence for reasonably large algebras of (pseudo-)differential operators that appear generically as low-energy large-wavelength models in the applications. We use the correspondence to compute the asymmetry in several settings where a direct estimation seems hopeless, with applications, e.g., in graphene-based Floquet topological insulators and topological properties of twisted bilayer graphene.
Time permitting, we will contrast the above spectral properties with the practically more relevant temporal picture and, for instance, the propagation of semi-classical wavepackets along curved interfaces.