The landscape of symmetry enhancement in tight-binding models

TL;DR

This paper investigates how the symmetry of tight-binding models depends on hopping range and bond equivalence classes, revealing that the parent crystal structure alone does not determine the model's symmetry.

Contribution

It systematically characterizes bond equivalence classes and introduces the concept of bond complex to understand symmetry enhancement in tight-binding models.

Findings

Bond symmetry depends on hopping range, not just parent structure.

All bond equivalence classes for s-wave hopping up to 20th neighbor are identified.

The concept of bond complex explains symmetry enhancement possibilities.

Abstract

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.