Edge States for Tight-Binding Operators with Soft Walls

We study one- and two-dimensional periodic tight-binding models in the presence of a potential that grows to infinity in one direction, hence preventing the particles to escape in this direction (the soft wall). We prove that a spectral flow appears in these edge models, as the wall is shifted with respect to the lattice. We identity this flow with the number of Bloch bands. This provides a lower bound for the number of edge states appearing in such models. For the two-dimensional case, we compute the spectral flow for edges that have any rational orientation with respect to the lattice. The results are illustrated by applying them to the one-dimensional SSH chain and the Wallace model for graphene.