Applications of a novel interband topological index in condensed matter: optical selection rules and valley physics

We introduce a novel gauge-invariant, quantized interband index for condensed matter systems. We show how this index may be used to potentially: A) strengthen exciton optical selection rule theory into gauge invariant forms; and B) improve the topological description of so-called valleys in 2D materials. We provide numerical calculations of the interband index in model Hamiltonian systems as evidence for both applications A) and B).