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Recent studies on the interplay between band topology and layer degree offreedom provide an effective way to realize exotic topological phases. Here wesystematically study the $C_6$- and $C_3$-symmetric higher-order topologicalphases in bilayer spinless tight-binding lattice models. For concreteness, weconsider bilayer phononic crystals as the realizations of these models. We findthat for mirror-symmetric-stacking bilayer lattices, the interlayer couplingscontrol the emergence and disappearance of the topological bound states in thecontinuum where we consider the corner states as possible bound states in thebulk continuum. For the bilayer phononic crystals formed by two differentlattices with identical symmetry, the band topology is determined by both theband topology of each layer as well as their mutual couplings. The bilayerphononic crystals experience a phase transition from nontrivial to trivial bandtopology when the interlayer couplings are gradually increased. Our workunveils the rich physics and topological phases emerging in bilayer latticesystems that can be used to engineer interesting phenomena and induce emergenttopological phases.

Higher-order topological phases in bilayer phononic crystals and topological bound states in the continuum