Topological insulating (TI) phases were originally highlighted for their disorder-robust bulk responses, such as the quantized Hall conductivity of 2D Chern insulators. With the discovery of time-reversal- (T-) invariant 2D TIs, and the recognition that their spin Hall conductivity is generically non-quantized, focus has since shifted to boundary states as signatures of 2D and 3D TIs and symmetry-enforced topological crystalline insulators (TCIs). However, in T-invariant (helical) 3D TCIs such as bismuth, BiBr, and MoTe2 - termed higher-order TCIs (HOTIs) - the boundary signatures manifest as 1D hinge states, whose configurations are dependent on sample details. It is hence desirable to elucidate bulk and surface signatures of helical TCIs, and their relationship to sample-independent experimental observables.
Using a range of theoretical and numerical probes, including a newly introduced principle of spin-resolved topology, we fully characterize the bulk topological properties of inversion- and T-protected helical HOTIs. 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, "spin-Weyl" semimetal states with gapless spin spectra, and T-doubled axion insulator (T-DAXI) states with nontrivial “partial” axion angles indicative of a 3D spin-magnetoelectric bulk response. We provide experimental signatures of each spin-stable regime of helical HOTIs, including surface Fermi arcs in spin-Weyl semimetals under strong Zeeman fields, and half-quantized 2D TI states on the gapped surfaces of T-DAXIs originating from a partial parity anomaly.
[1] K.-S.Lin, Palumbo, Z.Guo, Blackburn, Shoemaker, Mahmood, Z.-J.Wang, Fiete, Wieder, Bradlyn, arXiv:2207.10099 (2022).
[2] Schindler, Tsirkin, Neupert, Bernevig, Wieder, Nature Communications (2022).
