Higher-Order Topology Embedded in First-Order Topological Bands

Topological insulators, including the first-order and higher-order topological insulators, drive transformative advances in condensed-matter physics and material sciences. Although nonzero first-order topological indices (e.g., Chern and spin-Chern numbers) fundamentally render the conventional higher-order topological indices ill-defined, first-order topological systems can still support higher-order topological corner states as Jackiw-Rebbi-type modes from edge band topologies. Here, by contrast, we experimentally implement a distinct relative-polarization mechanism and realize higher-order topological corner states embedded in first-order topological bands using an acoustic spin-Chern insulator. The higher-order topology in our system originates from the quantized differences in spin-resolved bulk polarizations at domain walls, notwithstanding the ill definition of the polarizations themselves in the crystal bulk due to the nonzero spin-Chern numbers. As a result, topological corner states emerge, coexisting with helical edge states within the same bulk band gap. Our Letter evidences a polarization-difference-induced higher-order topology embedded in first-order topological bands and may facilitate the development of topological devices in acoustics.