奇异平带晶格中的体-孔对应与内部鲁棒边界模
Bulk-hole correspondence and inner robust boundary modes in singular flatband lattices
AI总结：
提出奇异平带晶格的体-孔对应原理，揭示内部鲁棒边界模与孔数及 Betti 数的关系，并在激光直写 Kagome 晶格中实验验证。
摘要：
基于体-边界对应的拓扑实体无处不在，从常规拓扑绝缘体到高阶拓扑绝缘体皆是如此，其中受保护状态通常局域在外边界（边缘或角）。一种较少被探索的情形涉及局域在内边界、并与体态共享相同能量的受保护状态。在这里，我们提出并演示了我们所称的体-孔对应——即内部鲁棒边界模（RBM）与奇异平带晶格中多个“孔”的存在之间的关系，这种关系由体 Bloch 波函数不可移动的不连续性所介导。我们发现，独立平带态的数量总是等于独立紧致局域态数量与非平凡内部 RBM 数量之和，这一点由 Betti 数刻画；该数也可从拓扑数据分析中计数孔的数量。这种对应对于奇异平带晶格具有普适性，与晶格形状和孔的形状无关。利用激光直写 Kagome 晶格作为平台，我们在实验上观测到了这种内部 RBM，证明了它们的实空间拓扑性质和鲁棒性。我们的结果或许可推广到光子学之外的其他奇异平带系统，包括非欧几里得晶格，从而为理解含孔晶格系统中的非平凡平带态和拓扑提供一种新途径。
英文摘要：
Topological entities based on bulk-boundary correspondence are ubiquitous, from conventional to higher-order topological insulators, where the protected states are typically localized at the outer boundaries (edges or corners). A less explored scenario involves protected states that are localized at the inner boundaries, sharing the same energy as the bulk states. Here, we propose and demonstrate what we refer to as the bulk-hole correspondence - a relation between the inner robust boundary modes (RBMs) and the existence of multiple "holes" in singular flatband lattices, mediated by the immovable discontinuity of the bulk Bloch wavefunctions. We find that the number of independent flatband states always equals the sum of the number of independent compact localized states and the number of nontrivial inner RBMs, as captured by the Betti number that also counts the hole number from topological data analysis. This correspondence is universal for singular flatband lattices, regardless of the lattice shape and the hole shape. Using laser-written Kagome lattices as a platform, we experimentally observe such inner RBMs, demonstrating their real-space topological nature and robustness. Our results may extend to other singular flatband systems beyond photonics, including non-Euclidean lattices, providing a new approach for understanding nontrivial flatband states and topology in hole-bearing lattice systems.
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