Recent studies on the interplay between band topology and the layer degree of freedom provide an effective way to realize exotic topological phases. Here we systematically study the $C_6$- and $C_3$-symmetric higher-order topological phases in bilayer spinless tight-binding lattice models. For concreteness, we consider bilayer phononic crystals as the realizations of these models. We find that for mirror-symmetric-stacking bilayer lattices the interlayer couplings control the emergence and disappearance of the topological bound states in the continuum where we consider the corner states as possible bound states in the bulk continuum. For the bilayer phononic crystals formed by two different lattices with identical symmetry, the band topology is determined by both the band topology of each layer as well as their mutual couplings. The bilayer phononic crystals experience various topological phase transitions when the interlayer couplings are gradually increased. Our paper reveals the rich physics and topological phases emerging in bilayer lattice systems that can be used to engineer interesting phenomena and topological effects.

中文翻译：

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双层声子晶体中的高阶拓扑相和连续体中的拓扑束缚态

最近对能带拓扑和层自由度之间相互作用的研究提供了实现奇异拓扑相的有效方法。下面我们系统地研究一下$C_6$- 和$C_3$-双层无自旋紧束缚晶格模型中的对称高阶拓扑相。为了具体起见，我们将双层声子晶体视为这些模型的实现。我们发现，对于镜像对称堆叠双层晶格，层间耦合控制连续体中拓扑束缚态的出现和消失，其中我们将角态视为体连续体中可能的束缚态。对于由两个具有相同对称性的不同晶格形成的双层声子晶体，其能带拓扑由各层的能带拓扑及其相互耦合决定。当层间耦合逐渐增加时，双层声子晶体会经历各种拓扑相变。我们的论文揭示了双层晶格系统中出现的丰富的物理和拓扑相，可用于设计有趣的现象和拓扑效应。