Particle-Hole Symmetry Groups and Topological Quantum Chemistry

With the recent completion of Topological Quantum Chemistry (TQC) for magnetic symmetry groups (SGs), efforts have now shifted towards generalizing TQC to other ordered states, including weakly spin-orbit-coupled magnets (via spin space groups) and weak-coupling superconductors (SCs). However, to construct on equal footing TQC- (Wannier-) based characterizations of topological states in all of the classes of the tenfold way with additional crystal symmetries, it is necessary to formulate notions of elementary band representations (EBRs) and topological invariants that are agnostic as to the strength of SC pairing and sublattice, orbital, and spin-orbit interactions. We here accomplish this for crystals in Classes AIII and A with additional combinations of crystal and unitary particle-hole (chiral) symmetry S by introducing a new notion of S-SGs. We compute the EBRs, real-space invariants, and bulk-boundary correspondences for all nontrivial bands in the 1D and 2D S-SGs. We relate our findings to quantum anomalies and recent discussions of fermion doubling theorems, highlighting cases where non-on-site S symmetry in nonsymmorphic S-SGs leads to frequently overlooked gauge-invariance issues in the conventional 1D winding number.