Topological Triviality of Strictly Local Projectors

The topological properties of a Bloch band are closely related to the localization properties of wavefunctions spanning it. It has been shown that if a set of compactly supported Wannier-type functions spans a band or a set of bands, then the band(s) are necessarily topologically trivial. This situation arises, for example, when a band is described by a strictly local projector. We show that if the occupied (gapped) subspace of a single-particle Hamiltonian is described by a projector that is strictly local, then the system is topologically trivial. We do not rely on the assumption of lattice translational invariance.