Symmetry and filling enforced topological insulators in two and three dimensions

Due to the difficulty of either analytical or numerical calculations, in more than one spatial dimensions, it proves hard to reliably predict topological states in strongly correlated systems. Here we prove a set of theorems of the Lieb-Schultz-Matthis type, which dictates the nature of a non-fractionalized insulating ground state in certain systems. More concretely, with certain crystalline symmetries, at a certain filling fraction, a non-fractionalized insulating ground state must be a topological insulator. These results shed new light on how to guide the search for topological states in strongly correlated systems.