Three-Dimensional Valley Hall Phases in Phononic Crystals.

Valley Hall phases (VHPs) in electronic and classical wave systems have been extensively explored and brought fruitful wave manipulations and associated applications. However, the concept of VHPs is so far restricted to two-dimensional (2D) systems. VHPs in 3D platforms remain an open question, which is not a straightforward extension of theirs 2D counterparts. In this work, we generalize the 2D VHPs into the 3D case in phononic crystals, which are derived from a nodal-line semimetal. The variation of geometric parameters gives rise to band inversion and generates two distinct VHPs possessing a pair of valleys with opposite Berry curvature distributions in the Brillouin zone. Topological surface states protected by the 3-tuple valley Chern numbers emerge at the interface between two different VHPs. Thanks to the configuration of the equal-frequency contour, some intriguing surface wave propagations have been observed experimentally. At the corner surrounded by two VHPs, interestingly, circular propagation of the surface wave exhibits negative refraction to form a closed loop around the corner. Furthermore, 3D partition is demonstrated in a four-channel structure. The 3D VHPs will stimulate other exotic transports, such as 3D collimation and wave guiding, and provide new means for designing 3D acoustic devices.