Spin-resolved topology and partial axion angles in three-dimensional insulators.

Symmetry-protected topological crystalline insulators (TCIs) have primarily been characterized by their gapless boundary states. However, in time-reversal- (T -) invariant (helical) 3D TCIs—termed higher-order TCIs (HOTIs)—the boundary signatures can manifest as a sample-dependent network of 1D hinge states. We here introduce nested spin-resolved Wilson loops and layer constructions as tools to characterize the intrinsic bulk topological properties of spinful 3D insulators. We discover that helical HOTIs realize one of three spin-resolved phases with distinct responses that are quantitatively robust to large deformations of the bulk spin-orbital texture: 3D quantum spin Hall insulators (QSHIs), "spin-Weyl" semimetals, and T -doubled axion insulator (T-DAXI) states with nontrivial partial axion angles indicative of a 3D spin-magnetoelectric bulk response and half-quantized 2D TI surface states originating from a partial parity anomaly. Using ab-initio calculations, we demonstrate that β-MoTe₂ realizes a spin-Weyl state and that α-BiBr hosts both 3D QSHI and T-DAXI regimes. 3D higher-order topological insulators (HOTIs) exhibit 1D hinge states depending on extrinsic sample details, while intrinsic features of HOTIs remain unknown. Here, K.S. Lin et al. introduce the framework of spin-resolved topology to show that helical HOTIs can realize a doubled axion insulator phase with nontrivial partial axion angles. [ABSTRACT FROM AUTHOR]