The theory of electric polarization in crystals defines the dipole moment of an insulator in terms of a Berry phase (geometric phase) associated with its electronic ground state. This concept not only solves the long-standing puzzle of how to calculate dipole moments in crystals, but also explains topological band structures in insulators and superconductors, including the quantum anomalous Hall insulator and the quantum spin Hall insulator, as well as quantized adiabatic pumping processes. A recent theoretical study has extended the Berry phase framework to also account for higher electric multipole moments, revealing the existence of higher-order topological phases that have not previously been observed. Here we demonstrate experimentally a member of this predicted class of materials—a quantized quadrupole topological insulator—produced using a gigahertz-frequency reconfigurable microwave circuit. We confirm the non-trivial topological phase using spectroscopic measurements and by identifying corner states that result from the bulk topology. In addition, we test the critical prediction that these corner states are protected by the topology of the bulk, and are not due to surface artefacts, by deforming the edges of the crystal lattice from the topological to the trivial regime. Our results provide conclusive evidence of a unique form of robustness against disorder and deformation, which is characteristic of higher-order topological insulators.

中文翻译：

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具有拓扑保护角态的量子化微波四极子绝缘体

晶体中的电极化理论根据与其电子基态相关的 Berry 相（几何相）来定义绝缘体的偶极矩。这一概念不仅解决了长期以来如何计算晶体偶极矩的难题，而且解释了绝缘体和超导体中的拓扑能带结构，包括量子反常霍尔绝缘体和量子自旋霍尔绝缘体，以及量子化绝热泵浦过程. 最近的一项理论研究扩展了 Berry 相框架，以解释更高的电多极矩，揭示了以前未观察到的高阶拓扑相的存在。在这里，我们通过实验证明了这一类预测材料的成员——量子化四极拓扑绝缘体——使用千兆赫频率可重构微波电路生产。我们使用光谱测量并通过识别由体拓扑产生的角状态来确认非平凡的拓扑相。此外，我们通过将晶格的边缘从拓扑结构变形到微不足道的状态来测试关键预测，即这些角状态受体积拓扑保护，而不是由于表面人工制品。我们的结果提供了对无序和变形的独特形式的鲁棒性的确凿证据，这是高阶拓扑绝缘体的特征。我们使用光谱测量并通过识别由体拓扑产生的角状态来确认非平凡的拓扑相。此外，我们通过将晶格的边缘从拓扑结构变形到微不足道的状态来测试关键预测，即这些角状态受体积拓扑保护，而不是由于表面人工制品。我们的结果提供了对无序和变形的独特形式的鲁棒性的确凿证据，这是高阶拓扑绝缘体的特征。我们使用光谱测量并通过识别由体拓扑产生的角状态来确认非平凡的拓扑相。此外，我们通过将晶格的边缘从拓扑结构变形到微不足道的状态来测试关键预测，即这些角状态受体积拓扑保护，而不是由于表面人工制品。我们的结果提供了对无序和变形的独特形式的鲁棒性的确凿证据，这是高阶拓扑绝缘体的特征。通过将晶格的边缘从拓扑变形到平凡状态。我们的结果提供了对无序和变形的独特形式的鲁棒性的确凿证据，这是高阶拓扑绝缘体的特征。通过将晶格的边缘从拓扑变形到平凡状态。我们的结果提供了对无序和变形的独特形式的鲁棒性的确凿证据，这是高阶拓扑绝缘体的特征。