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 nontopological phases as well as the transitions between them in a particularly illuminating and transparent manner.