Boundaries of boundaries: a systematic approach to lattice models with solvable boundary states of arbitrary codimension

We present a generic and systematic approach for constructing D-dimensional lattice models with exactly solvable d-dimensional boundary states localized to corners, edges, hinges and surfaces. These solvable models represent a class of "sweet spots" in the space of possible tight-binding models---the exact solutions remain valid for any tight-binding parameters as long as they obey simple locality conditions that are manifest in the underlying lattice structure. Consequently, our models capture the physics of both (higher-order) topological and non-topological phases as well as the transitions between them in a particularly illuminating and transparent manner.