Bulk and Surface Theories for Helical Higher-Order Topological Insulators

3D higher-order topological crystalline insulators (HOTIs) exhibit intrinsic 1D hinge modes in highly symmetric model geometries. Away from the unrealistic limit of perfect global crystal symmetry, topological phases can still be described by continuum field and response theories. Magnetic HOTIs with chiral hinge modes have recently been recognized to carry bulk nontrivial axion angles θ=π, clarifying their response. But for HOTIs with helical hinge modes, the analogous bulk and surface theories are not yet known. This significantly constrains currently available experimental signatures, despite the wealth of accessible material candidates including bismuth, MoTe₂, WTe₂, and BiBr. In this talk, we first use the recently-developed concept of spin-resolved topology to analyze helical HOTIs with (weakly) broken S^z symmetry, finding that they carry quantized “partial” axion angles, which lead to anomalous surface half quantum spin Hall states and a bulk spin-magnetoelectric response. We then use dimensional reduction and the insertion of magnetic flux and monopoles to theoretically characterize helical HOTIs with arbitrarily strong spin-orbit coupling.