Spin-Resolved Topology, Partial Axion Angles, and Surface Anomalies in Time-Reversal-Invariant Insulators: Numerical Analysis Techniques

3D higher-order topological crystalline insulators (HOTIs) exhibit 1D hinge states whose configuration depends on the details of the sample termination. In this work, we identify sample-independent bulk experimental signatures of time-reversal- (T-) invariant (helical) HOTIs. We develop numerical techniques to characterize the bulk topological properties of 2D and 3D insulators: (nested) spin-resolved Wilson loops, position-space Chern numbers (which we relate to spin-resolved layer constructions), and spin-resolved entanglement spectra. We introduce spin-resolved topological phases along with these numerical techniques. We find that helical HOTIs realize one of three spin-resolved phases: (i) 3D quantum spin Hall insulators, (ii) spin-Weyl semimetals with gapless spin spectra, and (iii) T-doubled axion insulators with nontrivial partial axion angles indicative of a 3D spin-magnetoelectric bulk response and whose surfaces realize anomalous halves of a 2D topological insulator. We provide experimental signatures of each of these spin-resolved phases. We also use ab-initio calculations to demonstrate that the candidate HOTI β-MoTe₂ realizes a spin-Weyl semimetal state. Lastly, we consider the relationship between symmetry and U(1) magnetic flux insertion in helical HOTIs.