Higher-Order Flatband-Touching Points Mediate Unconventional Bulk-Boundary Correspondence
The bulk-boundary correspondence establishes a fundamental connection between bulk band topology and the emergence of boundary states, underpinning topological phenomena across electronic, acoustic, photonic, and many other wave systems. A broad class of singular flatband systems, however, exhibits an unconventional form of bulk-boundary correspondence, in which robust boundary modes (RBMs) originate from flatband-touching points (FBTPs) and the real-space topology associated with incomplete sets of compact localized states (CLSs), rather than conventional momentum-space bulk topological invariants. To date, experimental studies of such boundary modes have been limited to systems hosting lower-order FBTPs. Here, we experimentally demonstrate unconventional bulk-boundary correspondence in a photonic cross-bedded Kagome-like lattice featuring a higher-order FBTP formed by the touching of two degenerate flatbands and a dispersive band. Using continuous-wave laser writing, we observe shape-preserving propagation of both bulk CLSs and boundary RBMs. Remarkably, unlike previously reported flatband systems, the RBMs retain their boundary localization under input-field perturbations involving both singular and nonsingular CLSs, revealing an unconventional boundary response associated with the higher-order FBTP in this lattice. Our results establish the Kagome-like lattice as a new platform for exploring flatband physics and unconventional bulk-boundary correspondence, with potential applications in wave localization, flatband lasers, and other emerging topological photonic devices.