Complete Topological Classification with P and T Symmetries: Revealing a Topological Invariant Invisible to K-Theory

Abstract

The K-theoretic framework provides a complete topological classification of the tenfold symmetry classes and has been generalized to incorporate crystalline symmetries. Here, we show that this classification is incomplete even in the elementary case of spinless systems possessing both P and T symmetries. We obtain the complete classification through a first-principles analysis of the topological classes of P- and T-symmetric bands, namely, by classifying the corresponding symmetric clutching data. We identify a topological invariant that is invisible from the K-theoretic perspective when the occupied states at each inversion-invariant momentum have uniformly positive or uniformly negative parity. In special cases, such as when all inversion-invariant momenta have uniformly positive parity, this invariant can be interpreted as the second Stiefel--Whitney class or the Euler number defined over an inversion fundamental domain, namely, half of the Brillouin zone. Our work not only reveals a new topological invariant that cannot be determined from the parity spectra at inversion-invariant momenta under P and T symmetries, but also demonstrates the existence of crystalline topological phases that are absent from the K-theoretic classification.