DMS Colloquium: Prof. Guillaume Bal

Speaker: Prof. Guillaume Bal (University of Chicago)

Title: Asymmetric transport and topological invariants

Abstract: The field of topological (acoustic, electromagnetic, electronic, mechanical) insulators analyzes asymmetric transport phenomena observed along interfaces that separate insulating bulks. It finds applications in many areas of physical and materials sciences. The topological nature of the asymmetric transport ensures that it persists in the presence of perturbations of the underlying model, which forms its main practical appeal.

A physical observable taking the form of an edge conductivity models the asymmetric transport of a system's Hamiltonian. To simplify the computation of this observable, we associate a natural Fredholm operator to the (partial differential) Hamiltonian and compute its topological charge by means of a Fedosov-H\"ormander formula, which implements in Euclidean space an Atiyah-Singer index theorem. We then relate the index of the Fredholm operator to a bulk-difference invariant, a Chern number, that is readily computable for a large class of problems. The main remaining difficulty is to equate the physical observable to the aforementioned index. The talk will present such a bulk-edge correspondence for a large class of Hamiltonians and sketch its derivation.

We then use the correspondence to compute the asymmetry in several settings, with applications in graphene-based regular and Floquet topological insulators as well as topological properties of twisted bilayer graphene. Time permitting, we will contrast the above spectral properties with the practically relevant temporal picture, in particular the propagation of semi-classical wavepackets along curved interfaces.

Faculty host: Junshan Lin