Topological insulators that can host special symmetry-protected boundary states and corner states have attracted increasing intention in acoustic engineering. Recently, the concept of the boundary-obstructed topological (BOT) phases has defined a class of topological phases without bulk energy band closing around zero energy, which greatly broadens the applications of the topological states. In this work, based on the two-dimensional Su-Schrieffer-Heeger network, we show that the band degeneracies around zero energy can be removed to open a complete bandgap by judiciously tuning the hopping terms to break C 4 v symmetry down to C 2 v symmetry but with the topological phase invariant, which can be directly characterized by the BOT phase. Furthermore, we experimentally propose a rigorous acoustic sample to visualize the hierarchy of the in-gap higher-order topological states exactly. Crucially, by designedly connecting the lattice with outside environment, we show that these spectrally isolated states still response to the specific frequencies robustly. Our results are expected to be helpful for manipulating wave propagation and sound energy harvesting. Published under an exclusive license by AIP Publishing.
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