Topological invariants of disconnected elementary band representations*

Elementary band representations transform according to a minimal set of symmetry-related orbitals. When there is a gap between two groups of bands that together transform as an elementary band representation, at least one of those groups of bands is topological, in the sense that it cannot be smoothly deformed to an atomic limit that preserves the crystal symmetry. These topological bands come in two varieties, depending on whether or not they have a stable topological index (for example, a Chern number). We show that in either case, we can construct a nontrivial topological invariant that is a physical observable.