Beyond hinge states: bulk spin-based signatures of non-axionic higher-order topology

Higher-order topological insulators (HOTIs) in 3D are characterized by a novel bulk-boundary correspondence, featuring protected modes on 1D boundary hinges in highly-symmetric model geometries. However, the ability to unambiguously detect these non-trivial hinge modes can be hampered by details of the surface and hinge termination, motivating the search for bulk observables characterizing HOTIs. For electronic materials, the existence of a spin degree of freedom can provide information on the non-trivial band topology. In this work, we introduce generalized Wilson loop numerical methods to derive quantized bulk indicators of non-axionic higher-order topology in the spin spectrum of 3D systems. In addition, we apply these techniques in position space to describe signatures of the gapped surfaces of HOTIs. We then relate these bulk and surface signatures to experimentally relevant observables beyond the axionic magnetoelectric effect. We also investigate the implications of non-trivial topology on the spin texture of the energy bands for strong TIs. We conclude with a discussion of potential applications for spintronic devices.