Unifying Optical Selection Rules for Excitons in Two Dimensions: Band Topology and Winding Numbers.

We show that band topology can dramatically change the photophysics of two-dimensional semiconductors. For systems in which states near the band extrema are of multicomponent character, the spinors describing these components (pseudospins) can pick up nonzero winding numbers around the extremal k point. In these systems, we find that the strength and required light polarization of an excitonic optical transition are dictated by the optical matrix element winding number, a unique and heretofore unrecognized topological characteristic. We illustrate these findings in three gapped graphene systems-monolayer graphene with inequivalent sublattices and biased bi- and trilayer graphene, where the pseudospin textures manifest into nontrivial optical matrix element winding numbers associated with different valley and photon circular polarization. This winding-number physics leads to novel exciton series and optical selection rules, with each valley hosting multiple bright excitons coupled to light of different circular polarization. This valley-exciton selective circular dichroism can be unambiguously detected using optical spectroscopy.