Robust zero-energy states in two-dimensional Su-Schrieffer-Heeger topological insulators

The Su-Schrieffer-Heeger (SSH) model on a two-dimensional square lattice has been considered as a significant platform for studying topological multipole insulators. However, due to the highly-degenerate bulk energy bands protected by $ C_{4v} $ and chiral symmetry, the discussion of the zero-energy topological corner states and the corresponding physical realization have been rarely presented. In this work, by tuning the hopping terms to break $ C_{4v} $ symmetry down to $ C_{2v} $ symmetry but with the topological phase invariant, we show that the degeneracies can be removed and a complete band gap can be opened, which provides robust protection for the spectrally isolated zero-energy corner states. Meanwhile, we propose a rigorous acoustic crystalline insulator and therefore these states can be observed directly. Our work reveals the topological properties of the robust zero-energy states, and provides a new way to explore novel topological phenomena.