Acoustic higher-order topology derived from first-order with built-in Zeeman-like fields

Higher-order topological insulators (HOTIs), with topological corner or hinge states, have emerged as a thriving topic in the field of topological physics. However, few connections have been found for HOTIs with well-explored first-order topological insulators. Recently a proposal asserted that a significant bridge can be established between the HOTIs and Z₂ topological insulators. When subjected to an in-plane Zeeman field, corner states, the signature of the HOTIs, can be induced in a Z₂ topological insulator. Such Zeeman fields can be produced, for example, by the ferromagnetic proximity effect or magnetic atom doping, which drastically increases the experimental complexity. Here, we show that a phononic crystal, designed as a bilayer of coupled acoustic cavities, exactly hosts the Kane-Mele model with built-in in-plane Zeeman fields. The helical edge states along the zigzag edges are gapped, and the corner states, localized spatially at the corners of the samples, appear in the gap. This verifies the Zeeman field induced higher-order topology. We further demonstrate the intriguing contrast properties of the corner states at the outer and inner corners in a hexagonal ring-shaped sample.