Surface impedance and generalized chiral symmetry in acoustic higher-order topological insulators

The emergence of higher-order topological states is always considered to be determined by nontrivial topology of the bulk bands, and such the topological invariants strongly depend on specific symmetry of the bulk lattice. However, for real-physical systems, the intrinsic chiral-symmetry breaking can hinder the measurement of these states, and a spectrally isolated topological state always requires compensation to the on-site energy on the boundary sites. In this work, we reveal that in acoustic systems, the compensation is intensively determined by the surface impedance. Based on a regular Su-Schrieffer-Heeger (SSH) topological chain, we derive the analytical solutions of the topological states in acoustic systems and show that generalized chiral symmetry requires acoustic soft-wall boundary condition. Further, we propose a SSH-like chain and show that the surface impedance also crucially impacts the bulk topology. Finally, we extend our results to two-dimensional condition and demonstrate the consistency of the results. Our results provide new insights into the origin of the acoustic topological phenomena, which in turn offers a platform to design topological materials with desired topological properties.