Exact higher-order bulk-boundary correspondence of corner-localized states

We demonstrate that the presence of a localized state at the corner of an insulating domain is not always a predictor nor a direct consequence of a certain nontrivial higher-order topological invariant, even though they appear to coexist in the same Hamiltonian parameter space. Our analysis of Cn-symmetric crystalline insulators and their multilayer stacks reveals that topological corner states are not necessarily correlated with other well-established higher-order boundary observables, such as fractional corner charge or filling anomaly. In a C3-symmetric breathing Kagome lattice, for example, we show that the bulk polarization, which successfully predicts the fractional corner anomaly, fails to be the relevant topological invariant for zero-energy corner states; instead, these corner states are explained by the decoration of topological edges. Also, while the corner states at the interface between C4-symmetric topological crystalline insulators and their trivial counterpart have long been reported to be the result of the bulk polarization of the lowest band, we reveal that such embedded corner states are trivial defect states. By refining several bulk-corner correspondences in two-dimensional topological crystalline insulators, our work motivates further development of rigorous theoretical grounds for associating the existence of corner states with higher-order topology of host materials.

Automated summary

Our study demonstrates that the presence of a localized state at the corner of an insulating domain is not always related to a certain nontrivial higher-order topological invariant, despite appearing in the same Hamiltonian parameter space. Topological corner states may not be correlated with other well-established higher-order boundary observables, motivating further development for associating corner state existence with the higher-order topology of host materials.