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 of a certain non-trivial higher-order topological invariant, even though they appear to co-exist in the same Hamiltonian parameter space. Our analysis of $C^n$-symmetric crystalline insulators and their multi-layer 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 $C^3$-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 can be exactly explained by the decoration of topological edges. Also, while the zero-energy corner states in $C^4$-symmetric topological crystalline insulators have long been conjectured to be the result of the bulk polarization at quarter-filling, we correct this misconception by introducing a proper bulk invariant at half-filling and establishing a precise bulk-corner correspondence. 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.