The puckered lattice geometry, along with $p$ orbitals, is often overlooked in the study of topological physics. Here, we investigate the higher-order topology of the $p_{x,y}$-orbital bands in acoustic metamaterials using a simplified two-dimensional phosphorene lattice which possesses a puckered structure. Notably, unlike the $s$-orbital bands in planar lattices, the unique higher-order topology observed here is specific to $p$ orbitals and the puckered geometry due to the unusual hopping patterns induced by them. Using acoustic pump-probe measurements in metamaterials, we confirm the emergence of the edge and corner states due to the unconventional higher-order topology. We reveal the uniqueness of the higher-order topological physics here via complimentary tight-binding calculations, finite-element simulations, and acoustic experiments. We analyze the underlying physics of the special properties of the edge and corner states in the puckered lattice acoustic metamaterials from the picture of Wannier orbitals. Our work sheds light on the intriguing physics of $p$-orbital topological physics in puckered lattices and acoustic metamaterials which lead to unconventional topological boundary states.

中文翻译：

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褶皱晶格声学超材料中 ap 轨道高阶拓扑绝缘体相的观察

褶皱的晶格几何形状，以及$p$轨道，在拓扑物理的研究中经常被忽视。在这里，我们研究了高阶拓扑$p_{X,y}$-使用具有褶皱结构的简化二维磷烯晶格的声学超材料中的轨道带。值得注意的是，与$s$-平面晶格中的轨道带，这里观察到的独特的高阶拓扑特定于$p$轨道和褶皱几何形状是由于它们引起的不寻常的跳跃模式而产生的。使用超材料中的声泵浦探针测量，我们确认了由于非常规的高阶拓扑而出现的边缘和角态。我们通过免费的紧束缚计算、有限元模拟和声学实验揭示了高阶拓扑物理的独特性。我们从万尼尔轨道图片中分析了褶皱晶格声学超材料中边缘和角态特殊性质的基本物理原理。我们的工作揭示了有趣的物理学$p$-褶皱晶格和声学超材料中的轨道拓扑物理，导致非常规的拓扑边界态。