Detecting SPT Signatures and Projective Representations in Free-Fermion Topological Crystalline Insulators

The many-body quantum numbers of 0D collective excitations bound to crystal and electromagnetic defects (such as magnetic fluxes and monopoles) provide powerful indicators of bulk topology, in that they can indicate the presence of quantized responses in the bulk that are governed by long-wavelength topological field theories that are stable to symmetric interactions. In interacting symmetry-protected topological phases (SPTs), defect quantum numbers frequently indicate nontrivial bulk topology if they transform in projective representations of the local symmetry group. We here introduce numerical methods for computing defect quantum numbers in stable and fragile free-fermion topological crystalline insulators (TCIs) via the reduced density matrix, revealing a deep connection between defect quantum numbers and the entanglement spectrum. We surprisingly find that when crystal symmetries are included in the local symmetry group, defects can appear to transform projectively even in Wannierizable (fragile) insulators. Our framework systematically characterizes stable and fragile TCIs beyond filling anomalies and higher-order topology, allowing direct connections between free-fermion TCIs and interacting SPTs.