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Macroscopic two-dimensional sonic crystals with inversion symmetry arestudied to reveal higher-order topological physics in classical wave systems.By tuning a single geometry parameter, the band topology of the bulk and theedges can be controlled simultaneously. The bulk band gap forms an acousticanalog of topological crystalline insulators with edge states which are gappeddue to symmetry reduction on the edges. In the presence of mirror symmetry, theband topology of the edge states can be characterized by the Zak phase,illustrating the band topology in a hierarchy of dimensions, which is at theheart of higher-order topology. Moreover, the edge band gap can be closedwithout closing the bulk band gap, revealing an independent topologicaltransition on the edges. The rich topological transitions in both bulk andedges can be well-described by the symmetry eigenvalues at the high-symmetrypoints in the bulk and surface Brillouin zones. We further analyze thehigher-order topology in the shrunken sonic crystals where slightly differentphysics but richer corner and edge phenomena are revealed. In these systems,the rich, multidimensional topological transitions can be exploited fortopological transfer among zero-, one- and two- dimensional acoustic modes bycontrolling the geometry.

Corner states and topological transitions in two-dimensional higher-order topological sonic crystals with inversion symmetry