Bulk-Hole Correspondence and Inner Robust Boundary Modes in Singular Flatband Lattices

作     者：Limin Song Shenyi Gao Shiqi Xia Yongsheng Liang Liqin Tang Daohong Song Daniel Leykam Zhigang Chen 

作者机构：The MOE Key Laboratory of Weak-Light Nonlinear Photonics TEDA Applied Physics Institute and School of Physics Nankai University Tianjin 300457 China Collaborative Innovation Center of Extreme Optics Shanxi University Taiyuan Shanxi 030006 China Science Mathematics and Technology Cluster Singapore University of Technology and Design 8 Somapah Road 487372 Singapore Singapore 

出 版 物：《Physical Review Letters》 (Phys Rev Lett)

年 卷 期：2025年第134卷第6期

页      面：063803-063803页

核心收录：

学科分类：07[理学] 0702[理学-物理学] 

基　　金：Ministry of Education - Singapore, MOE, (SKI 20210501) Ministry of Education - Singapore, MOE National Natural Science Foundation of China, NSFC, (12134006, 12274242, 12374309) National Natural Science Foundation of China, NSFC Natural Science Foundation of Tianjin for Distinguished Young Scientists, (21JCJQJC00050) National Key Research and Development Program of China, NKRDPC, (2022YFA1404800) National Key Research and Development Program of China, NKRDPC Higher Education Discipline Innovation Project, (B23045) Higher Education Discipline Innovation Project Natural Science Foundation of Tianjin Municipality, (21JCYBJC00060) Natural Science Foundation of Tianjin Municipality 

主　　题：Topology 

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 wave functions. 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.