标题： 无量纲分层拓扑声子态 显示英文标题

标题： Dimensionless Hierarchical Topological Phononic States

摘要： 拓扑绝缘体表现出独特的边界态，这些边界态由体带的拓扑性质保护，这一现象现已扩展到声子学和力学等经典系统。 通常，体中具有非平凡拓扑的 $n$-维系统会导致出现 $(n-1)$-维的拓扑保护边界态。 然而，这些态常常可以通过破坏保护它们的对称性而打开能隙，从而可能产生新的能隙内的高阶拓扑模式。 一个显著的例子是高阶拓扑绝缘体（HOTI），其中表面态的打开能隙导致形成较低维度的拓扑模式，例如铰链态或角态。 这一过程将受保护模式的空间维度从 $(n-1)$降低到 $(n-2)$或更低。 在这项工作中，我们提出了一种利用一维 Su-Schrieffer-Heeger 模型实现高阶拓扑模式的替代方法。 我们通过操控畴壁的位置来打开原本拓扑保护的畴壁态，从而诱导出新的高阶拓扑态。 这些新的高阶拓扑态可以通过广义缠绕数计算进行表征。 这种方法允许在一维晶格中实现多种甚至无穷多的拓扑序，同时保持体-边界对应原则。 我们的研究揭示了一种新的机制，丰富了拓扑层次结构，超越了传统的分类。 这种机制还可以扩展到更高维度，可能创造出复杂的拓扑态网络，并推进我们对波现象的控制。 显示英文摘要

摘要： Topological insulators exhibit unique boundary states that are protected by the topology of the bulk bands, a phenomenon that has now been extended to classical systems such as phononics and mechanics. Typically, nontrivial topology in an $n$-dimensional bulk leads to the emergence of $(n-1)$-dimensional topologically protected boundary states. However, these states can often be gapped out by breaking the symmetry that protects them, resulting in the possible creation of new in-gap higher-order topological modes. A notable example of this is the higher-order topological insulator (HOTI), where gapping out surface states leads to the formation of lower-dimensional topological modes, such as hinge or corner states. This process reduces the spatial dimensionality of the protected modes from $(n-1)$ to $(n-2)$ or even lower. In this work, we propose an alternative method to achieve higher-order topological modes using a one-dimensional Su-Schrieffer-Heeger model. Instead of relying on dimensional reduction, we manipulate the positions of domain walls to gap out the originally topologically protected domain-wall states, thereby inducing new higher-order topological states. These new higher-order topological states can be characterized using a generalized winding number calculation. This approach allows for the realization of multiple (and even infinite) topological orders within simple 1D lattices while maintaining the principle of bulk-boundary correspondence. Our study reveals a new mechanism that enriches topological hierarchies beyond conventional classifications. Such a mechanism could also be extended to higher dimensions, potentially creating intricate networks of topological states and advancing our control over wave phenomena.