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

Macroscopic two-dimensional (2D) sonic crystals with inversion symmetry are studied to reveal higher-order topological physics in classical wave systems. By tuning a single geometry parameter, the band topology of the bulk and the edges can be controlled simultaneously. The bulk band gap forms an acoustic analog of topological crystalline insulators with edge states which are gapped due to symmetry reduction on the edges. In the presence of mirror symmetry, the band topology of the edge states can be characterized by the Zak phase, illustrating the band topology in a hierarchy of dimensions, which is the key feature of higher-order topological phenomena. Moreover, the edge bands undergo topological transitions without closing the bulk band gap, revealing rich topological physics in both the bulk and edges. We characterize the higher-order topological bands using symmetry indicators in the Brillouin zone and link the symmetry indicators with the corner charge---a corner topological index. This approach allows us to identify higher-order topological boundary states through symmetry properties of the Bloch bands. We then generalize the higher-order topological phenomena protected by the inversion symmetry to arch-shaped sonic crystals with nonsymmorphic symmetries where similar but slightly different topological phenomena are revealed. In these systems, the rich, multidimensional topological transitions can be exploited for topological transfer among 0D (corner), 1D (edge), and 2D (bulk) acoustic modes by tuning the geometry of acoustic metamaterials.