The modern theory of charge polarization in solids is based on a generalization of Berry’s phase. The possibility of the quantization of this phase arising from parallel transport in momentum space is essential to our understanding of systems with topological band structures. Although based on the concept of charge polarization, this same theory can also be used to characterize the Bloch bands of neutral bosonic systems such as photonic or phononic crystals. The theory of this quantized polarization has recently been extended from the dipole moment to higher multipole moments. In particular, a two-dimensional quantized quadrupole insulator is predicted to have gapped yet topological one-dimensional edge modes, which stabilize zero-dimensional in-gap corner states. However, such a state of matter has not previously been observed experimentally. Here we report measurements of a phononic quadrupole topological insulator. We experimentally characterize the bulk, edge and corner physics of a mechanical metamaterial (a material with tailored mechanical properties) and find the predicted gapped edge and in-gap corner states. We corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases that are predicted by the quadrupole theory. These topological corner states are an important stepping stone to the experimental realization of topologically protected wave guides in higher dimensions, and thereby open up a new path for the design of metamaterials.

中文翻译：

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声子四极拓扑绝缘体的观察

固体中电荷极化的现代理论基于 Berry 相的概括。由动量空间中的平行输运引起的这一阶段的量化的可能性对于我们理解具有拓扑能带结构的系统至关重要。虽然基于电荷极化的概念，但同样的理论也可用于表征中性玻色子系统（如光子或声子晶体）的布洛赫带。这种量化极化的理论最近已从偶极矩扩展到更高的多极矩。特别是，预测二维量化四极子绝缘体具有带隙但拓扑的一维边缘模式，可稳定零维带隙角状态。然而，这种物质状态以前没有在实验中观察到。在这里，我们报告了声子四极拓扑绝缘体的测量结果。我们通过实验表征机械超材料（一种具有定制机械性能的材料）的体积、边缘和角落物理特性，并找到预测的有间隙边缘和有间隙角状态。我们通过将拓扑非平凡系统的机械性能与四极理论预测的其他相中的样品进行比较来证实我们的发现。这些拓扑角态是实验实现更高维度拓扑保护波导的重要垫脚石，从而为超材料设计开辟了一条新途径。机械超材料（一种具有定制机械特性的材料）的边缘和角落物理，并找到预测的有间隙边缘和有间隙角状态。我们通过将拓扑非平凡系统的机械性能与四极理论预测的其他相中的样品进行比较来证实我们的发现。这些拓扑角态是实验实现更高维度拓扑保护波导的重要垫脚石，从而为超材料设计开辟了一条新途径。机械超材料（一种具有定制机械特性的材料）的边缘和角落物理，并找到预测的有间隙边缘和有间隙角状态。我们通过将拓扑非平凡系统的机械性能与四极理论预测的其他相中的样品进行比较来证实我们的发现。这些拓扑角态是实验实现更高维度拓扑保护波导的重要垫脚石，从而为超材料设计开辟了一条新途径。