No-go theorem for topological insulators and sure-fire recipe for Chern insulators

For any symmorphic magnetic space group $G$, it is proven that topological band insulators with vanishing first Chern numbers cannot have a groundstate composed of a single, energetically-isolated band. This no-go statement implies that the minimal dimension of tight-binding Hamiltonians for such topological insulators is four if the groundstate is stable to addition of trivial bands, and three if the groundstate is unstable. A sure-fire recipe is provided to design models for Chern and unstable topological insulators by splitting elementary band representations; this recipe, combined with recently-constructed Bilbao tables on such representations, can be systematized for mass identification of topological materials. All results follow from our theorem which applies to any single, isolated energy band of a $G$-symmetric Schr\"odinger-type or tight-binding Hamiltonian: for such bands, being topologically trivial is equivalent to being a band representation of $G$.