Toward Exact Bulk-boundary Correspondence for Zero-energy Corner States

We refine some of weakly established bulk-boundary correspondence regarding zero-energy corner states (ZCS) in two-dimensional (2D) crystalline insulators, and reveal that bulk polarization fails to be a relevant indicator of ZCSs. Our analysis using multilayer stacking construction of C^3-symmetric crystalline insulators demonstrates that ZCSs in breathing Kagome lattices are attributed to Z_2 Zak phase of their edge band and that Z_3 bulk polarization is not correlated well with the existence of ZCSs. Also, we show that the ZCSs in C^4-symmetric crystalline insulators are a result of chiral symmetry and topological half charge at half-filling, and that the bulk polarization at quarter-filling fails to explain the emergence of a ZCS. Lastly, we provide a 2D Hamiltonian model supporting a ZCS despite completely trivial topology both in its bulk and edge. Our work, therefore, elucidates that overlapping phase diagrams of a bulk invariant and a boundary signature don't necessarily guarantee a solid bulk-boundary correspondence.