Monopole Quantum Numbers and Projective Representations in Stable and Fragile Topological Crystalline Insulators

An abundance of noninteracting topological (crystalline) insulator (TI and TCI) phases have recently been theoretically predicted and identified in real materials.

The TCI states, comprising the majority of topological materials in nature, exhibit complicated classification groups and boundary states, and carry more ambiguous response signatures.

A powerful tool for understanding 3D symmetry-protected topological states is the theoretical insertion of a U(1) magnetic monopole, which can exhibit fractionalized quantum numbers or projective representations of the local many-body symmetry group, indicating the presence of quantized bulk responses that are stable to symmetric interactions.

We introduce numerical methods to model the insertion of magnetic monopoles into 3D TCIs and to extract their many-body quantum numbers using the reduced density matrix.

We surprisingly find that when crystal (point group) symmetries are accounted for, magnetic monopoles can appear to transform projectively even in unstable (fragile and Wannierizable) insulators, implying that magnetic monopoles have more limited applicability to 3D TCIs than previously realized.