Uncovering Meaning for the Z₄ Classification of Inversion-Symmetric Nonmagnetic 3D Insulators through Domain-Wall Symmetry Fractionalization

It was unexpectedly recently discovered that inversion-symmetric 3D insulators have a stable Z₄ classification, as opposed to the Z₂ predicted by Fu and Kane. Confusingly, the Z₄ = 1, 3 states are both standard 3D topological insulators governed by θ = π axion electrodynamics, whereas the Z₄ = 2 topological crystalline insulator (TCI) states, despite having abundant and accessible material candidates, have unknown quantized response signatures. We here theoretically and numerically demonstrate that the full Z₄ group can be understood via an anomalous sequence of quantized symmetry fractionalization exhibited by domain-wall fluxes that is stable to symmetric interactions. We further uncover a link to other Z₄ invariants in 1D and 3D, generalizing the well-established relationship in insulators with U(1) charge-conservation symmetry between the Z₂ Berry phase in 1D and θ angle in 3D. To aid our analysis, we introduce a generalization of spin-resolved topology that characterizes U(1)-broken insulators with appropriately chosen interactions. Our results concretely link the long-wavelength properties of interacting symmetry-protected topological phases and real-material free-fermion TCIs.