Topological valley states in sonic crystals with Willis coupling

Acoustic media with Willis coupling possess intrinsic directionality owing to the vector nature of coupling coefficients in constitutive relations. Here, we report a type of sonic topological insulator that exhibits the valley Hall effect by using an acoustic fluid of the Willis type. We find that the valley Hall phase transition can be triggered by tuning the coupling vector. In addition, the Dirac cones or valley position are displaced away from high symmetry points in the Brillouin zone. The tunability of the valley offset offered by Willis coupling helps to realize equally robust one-way transport for both zigzag and armchair domain walls and for more tortuous wave channels.