Quadratic Dirac point and valley topological phases in acoustic analogues of AB-stacked bilayer graphene

The concepts of Dirac point and valley have been extended to phononic crystals, inspired by the development of valleytronic materials. Usually, the valley topological phases are induced by opening the gap of the linear Dirac points. In this work, we investigate the valley topological phases by breaking the quadratic Dirac point in two-dimensional bilayer phononic crystals, as analogues of the AB-stacked bilayer graphene. The acoustic edge states exist at the zigzag boundary of a single valley phase, attributed to the integer valley Chern number. In the domain wall between two distinct valley phases, the numbers of the acoustic edge states double in both the zigzag and armchair boundaries. Our work provides a macroscopic platform to explore the intriguing properties of the bilayer graphene.