The landscape of symmetry enhancement in tight-binding models

Band structures are ubiquitous in condensed matter physics and their symmetries constrain possible degeneracies, topology and response functions across a broad range of different systems. Here we address the question: given a parent crystal, what is the symmetry of hopping models on that lattice at a given shell number? We find that the parent structure does not, in general, determine the symmetry of the tight-binding model. Instead, the symmetry is dependent on the hopping range. The key to symmetry breakdown on the lattice is the existence of different {\it bond equivalence classes} whose number is related to group-subgroup indices for a broad classes of cases. We find all bond equivalence classes for $s$-wave hopping out to 20th neighbor across the different space groups and Wyckoff positions and the symmetries of the associated tight-binding models. These observations naturally lead to the definition of a {\it bond complex} $-$ the possible classes of networks of bonds to which symmetries may be enhanced from a given parent structure.