Compactly supported Wannier functions and strictly local projectors

Wannier functions that are maximally localized help in understanding many properties of crystalline materials. In the absence topological obstructions, they at least exponentially localized. some cases such as flat-band Hamiltonians, it is possible to construct even more localized, so compactly supported thus having zero support outside their corresponding locations. Under what general conditions functions? We answer this question paper. Specifically, we show 1d non-interacting tight-binding models, strict locality projection operator a necessary and sufficient condition for subspace be spanned by orthogonal basis, independent lattice translation symmetry. For any strictly local projector, provide procedure obtaining basis. higher dimensional systems, discuss additional under which an occupied projectors topologically trivial cases. also projector arbitrary dimensions if only chosen axis, its image hybrid along axis.