Probing Two Distinct Types of Topological Rainbow Concentrators Related to the Acoustic Valley Hall Insulator in Synthesized Three-Dimensional Space

Synthesized three-dimensional space provides us with a platform to explore the intriguing acoustic valley Hall insulator. In this paper, a rotational degree of freedom is introduced to construct a synthetic three-dimensional space. Based on simulations, our results show that Weyl points and Fermi arcs and topologically nontrivial edge modes appear in a synthetic space. Not only rotational operation but also boundary truncation can modulate the frequencies of nontrivial edge states, which result in two completely different edge states at the upper and lower boundaries. Using frequencies of edge states protected by Weyl points, we devise two different acoustic topological rainbow devices. The two devices are able to trap states with different frequency components at different spatial positions: one device is capable of capturing low-frequency components at close range, while the other has the ability to confine low-frequency components at long distances. This work may boost further the development of topological rainbow devices in a synthetic space related to the acoustic valley Hall insulator.