The nontrivial surface states excited by isolated Weyl points (IWPs) have been scarcely studied to date, primarily due to their circumvention from the Nielsen-Ninomiya no-go theorem. In a groundbreaking study on this topic [Adv. Sci., 10, 2207508 (2023)], we discovered that IWPs can generate a novel nontrivial surface state, namely the multi-fold fan-shaped surface state, which we named. Here, we report another type of fan-shaped surface state generated by an IWP surrounded by a closed Weyl nodal wall (WNW). Unlike previous findings, the fan-shaped surface state discovered here exhibits inhomogeneous in spatial distribution, with significantly varying sizes of the fan blades. Moreover, this surface state can be generated by a charge-four IWP protected by the rotation symmetries {C31+|000}, {C2z|12012}, {C2x|01212} and the time-reversal symmetry in the space group (SG) No. 198. Importantly, our simulation results of the acoustic crystals in this SG revel that the inhomogeneous fan-shaped surface state can provide multiple channels for acoustic wave transmission without energy dissipation, demonstrating that this kind of nontrivial surface state offers an effective mechanism for designing multi-frequency acoustic wave filters and selectors.

中文翻译：

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由声晶体中孤立的 Weyl 点和相关的多频声波滤波器诱导的不均匀扇形表面状态


迄今为止，由孤立的 Weyl 点 （IWP） 激发的非平凡表面态几乎没有被研究，主要是因为它们规避了 Nielsen-Ninomiya 不通定理。在关于该主题的开创性研究 [Adv. Sci.， 10， 2207508 （2023）] 中，我们发现 IWP 可以产生一种新的非平凡表面状态，即我们命名的多折扇形表面状态。在这里，我们报告了另一种类型的扇形表面状态，该表面状态是由 IWP 被封闭的外尔节点壁 （WNW） 包围的。与以前的发现不同，这里发现的扇形表面状态在空间分布上表现出不均匀，风扇叶片的大小显着变化。此外，这种表面状态可以由四电荷 IWP 产生，该 IWP 受旋转对称性 {C31+|000}、{C2z|12012}、{C2x|01212} 和空间群 （SG） 编号 198 中的时间反转对称性保护。重要的是，我们对该 SG 中声学晶体的仿真结果表明，不均匀的扇形表面状态可以为声波传输提供多个通道，而不会耗散能量，这表明这种非平凡的表面状态为设计多频声波滤波器和选择器提供了一种有效的机制。