Topological insulators with unique edge states have revolutionized the understanding of solid-state materials. Recently, higher-order topological insulators (HOTIs), which host both gapped edge states and in-gap corner/hinge states, protected concurrently by band topology, were predicted and observed in experiments, unveiling a new horizon beyond the conventional bulk-edge correspondence. However, the control and manifestation of band topology in a hierarchy of dimensions, which is at the heart of HOTIs, have not yet been witnessed. Here, we propose theoretically and observe experimentally that tunable two-dimensional sonic crystals can be versatile systems to visualize and harness higher-order topology. In our systems, the two-dimensional acoustic bands mimic the quantum spin Hall effect, while the resultant one-dimensional helical edge states are gapped due to broken space-symmetry and carry quantized Zak phases, which then lead to zero-dimensional topological corner states. We demonstrate that topological transitions in the bulk and edges can be triggered independently by tuning the geometry of the sonic crystals. With complementary experiments and theories, our study reveals rich physics in HOTIs, opening a new route towards tunable topological metamaterials where novel applications, such as the topological transfer of acoustic energy among two-, one- and zero-dimensional modes, can be achieved. By tuning the geometry of a two-dimensional sonic crystal, its one-dimensional helical edge states become gapped and zero-dimensional topological corner states emerge. The band topology is thus manifested in a hierarchy of dimensions.

Second-order topology and multidimensional topological transitions in sonic crystals