$PT$ Symmetry's Real Topology

TL;DR

This paper reveals that the spacetime inversion symmetry $PT$ in non-interacting spinless crystals has an intrinsic real vector-bundle topology characterized by Stiefel--Whitney classes, leading to new insights into topological phases and symmetry classifications.

Contribution

It introduces the concept of symmetry Stiefel--Whitney classes associated with $PT$ symmetry itself, showing they can induce nontrivial topology in the total real bundle of band structures.

Findings

Symmetry SW classes can make the total bundle topology nontrivial.

Valence and conduction bands can have asymmetric SW classes.

Symmetry SW classes refine the classification of atomic insulator phases.

Abstract

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.