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Résumé |
Over the past 15 years, a dizzying array of noninteracting topological insulator (TI) and topological crystalline
insulator (TCI) phases have been theoretically predicted and identified in real materials. While the TI states
are well understood, the TCI states – which comprise the majority of topological materials in nature – exhibit
more complicated classification groups and boundary states and carry more ambiguous response signatures.
For earlier variants of interacting symmetry-protected topological states (SPTs), both the classification and
response were clarified through the many-body quantum numbers of the 0D collective excitations bound to
crystal and electromagnetic defects, such as magnetic fluxes and monopoles. In particular, when 0D defects
exhibit fractionalized quantum numbers, or more generally projective representations of the local many-body
symmetry group, this 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 this talk, I will
introduce numerical methods for computing defect quantum numbers in stable and fragile TCI states via the
reduced density matrix, revealing a deep connection between defect quantum numbers and the entanglement
spectrum. Surprisingly, we find that when crystal symmetries are included in the local symmetry group,
defects can appear to transform projectively even in Wannierizable (fragile) insulators, casting doubt on the
suitability of magnetic monopoles for characterizing the TCI states present in real 3D materials. Our results
represent a crucial step towards describing TCIs beyond tight-binding models and frameworks like “higher-
order topology,” and facilitate more direct connections between free-fermion TCIs and interacting SPTs. |
