$PT$ Symmetry's Real Topology

Symmetry-protected topological phases have been a central theme in condensed matter physics and beyond over the past two decades. Most efforts have focused on topological classifications of physical systems under given symmetries, while the intrinsic topology of the symmetries themselves has received much less attention. Here, we show that, in generic non-interacting spinless crystals, the spacetime inversion symmetry $PT$ naturally carries a real vector-bundle structure whose topology is characterized by Stiefel--Whitney (SW) classes. In contrast to previous work, where SW classes were used to describe the topology of real valence bundles protected by $PT$, we identify SW classes associated to the $PT$ symmetry itself. These symmetry SW classes can endow the \emph{total} real bundle of a $PT$-symmetric band structure with nontrivial topology, overturning the common assumption that the total bundle is always trivial. As a consequence, valence and conduction bands can exhibit asymmetric SW classes, in sharp contrast to the usual symmetric scenario. We further demonstrate that the symmetry SW classes provide a refined distinction between atomic insulator phases. Our results underscore the importance of treating crystal symmetries as topological objects in their own right, rather than focusing solely on the topology of energy bands.