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

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