Band touching from real-space topology in frustrated hopping models

We study frustrated hopping models, in which at least one energy band-at the maximum or minimum of the spectrum - is dispersionless. The states of the flat band (s) can be represented in a basis, which is fully localized, having support on a vanishing fraction of the system in the thermodynamic limit. In the majority of examples, a dispersive band touches the flat band (s) at a number of discrete points in momentum space. We demonstrate that this band touching is related to states which exhibit nontrivial topology in real-space. Specifically, these states have support on one-dimensional loops which wind around the entire system (with periodic boundary conditions). A counting argument is given that determines, in each case, whether there is band touching or none, in precise correspondence to the result of straightforward diagonalization. When they are present, the topological structure protects the band touchings in the sense that they can only be removed by perturbations, which also split the degeneracy of the flat band.