Topological quantum chemistry I: Global band topology and topological phases

For the past century, chemists and physicists have advocated fundamentally complementary perspectives on materials: while chemists have adopted a ``local'' viewpoint, through the theory of chemical bonding and hybridization, physicists have thought about materials predominantly through band-structures in a nonlocal, momentum-space picture. The contrast between these two descriptions has been highlighted by the advent of topological insulators, the understanding of which overwhelmingly used the momentum-space picture. In this talk, I will present our method for unifying these two descriptions. By exploiting the constraints of symmetry on the relation between Bloch and Wannier functions, I will show how simple chemical input can be used to constrain the global band topology for materials in all 230 space groups. From this I will derive a predictive classification of topological insulators and semimetals.