Topological rainbow trapping of sound waves in synthesized three-dimensional space for square lattice

The three-dimensional (3D) synthetic space offers a platform for exploring the valley Hall insulator, which is usually constructed in graphene lattice. Here, based on the valley transports in the square lattice, we construct a synthetic space by treating the rotation angle as the third dimension and observe the emergent Weyl points in this synthetic space. Since the construction of synthetic Weyl points results in the formation of edge states between the phononic crystal (PC) and the hard boundary, and the factors of rotation angle and the distance from the PC to the hard boundary affect the frequency of these edge states, we can construct topological rainbow concentrator based on the above principles. We conduct a comprehensive numerical and experimental study to explore the characteristics of topological rainbow concentrators. This work may play a driving role in the development of topological rainbow devices.