Characterization of zero-energy corner states in higher-order topological systems with chiral symmetry

Corner-localized states represent intriguing aspects of higher-order topological systems. Despite their importance, the topological invariant that distinguishes between zero-energy and non-zero-energy corner states has received limited attention in the literature. Therefore, we introduce ``modified multipole chiral numbers,'' utilizing the 2-norm of the pseudo wave vector to characterize zero-energy corner states and to categorize the topological phase in chiral-symmetric two-dimensional and one-dimensional systems. The quantified topological invariant, protected by chiral symmetry, is associated with the quantity and spatial distribution of the zero-energy corner states. Theoretical analyses were conducted using multiple models, and both simulations and experimental data gathered from honeycomb lattices in acoustic systems corroborate these insights. These results offer valuable guidelines for designing systems that feature topologically protected corner states across various platforms and open avenues for the exploration of boundary-obstructed topological phases.