Topological Triviality of Strictly Local Flat Hamiltonians

The topological properties of an electronic 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. An interesting implication is that a flat band in a strictly local Hamiltonian is topologically trivial. We investigate this connection between flatness and topological triviality in the absence of lattice translational invariance. We argue that a 2d strictly local Hamiltonian without symmetries is trivial if all its bands are flat (or equivalently, if it has a finite number of distinct energies).