Topological triviality of flat Hamiltonians

Landau levels have played a pivotal role in advancing our understanding of topological phases of matter. Motivated by Landau levels, which are flat and topological, the authors answer here the following question: can a tight-binding Hamiltonian that exclusively has flat bands be topologically nontrivial? The authors show that, under broad conditions, energy levels in such flat-band Hamiltonians have vanishing Chern numbers. They also identify the exotic condition that is necessary for topologically nontrivial flat-band Hamiltonians. Importantly, all the proofs extend even to Hamiltonians that lack translational invariance.