Chiral symmetry (CS) in energy bands appears as perfectly symmetric antibonding and bonding pairs of energy levels. It has only been observed in a few classes of models when bipartite hopping occurs among their pairwise bases. We find that nonbipartite kagome and pyrochlore lattices can host CS when strong spin-orbit coupling is introduced. Unfortunately, the model exhibits an intricate frustrated hopping accompanied by spin rotation with no apparent sign of realizing CS. However, by making use of a local gauge transformation that properly rotates the spin quantization axis site-dependently, complex hopping among up- and down-spin electrons is untangled to bipartite-hopping ones. The clue to find such a transformation and to construct a chiral operator is given by the gauge invariant Wilson loop operator on a triangular unit. The framework allows us to disclose the symmetry and structures of spin-textured CS wave functions and energy bands.

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• Spin-orbit coupling
• Topological materials
• Kagome lattice
• Pyrochlores
• Discrete symmetries in condensed matter
• Tight-binding model