Emergent chiral symmetry in non-bipartite kagome and pyrochlore lattices with spin-orbit coupling

Chiral symmetry in energy bands appears as perfectly symmetric anti-bonding and bonding pairs of energy levels. It has only been observed in a few classes of models with a bipartite lattice structure or Bogoliubov-de-Gennes systems having the pairwise basis. We show that the non-bipartite kagome and pyrochlore lattices can host chiral symmetric bands when the strong spin-orbit coupling is introduced. There, the electrons hop to their neighbors by always converting the spin orientation up-side-down, which allows the up and down spin bases to form fictitious bipartite connections. The gauge invariant Wilson loop operator defined on a triangular unit serves as a marker to detect the presence of chiral symmetry, and using this property, the chiral operator is constructed. This allows us to access their topological symmetry classes that can easily change with small perturbations.