Multi-branch valley-chiral edge states of antisymmetric plate wave in phononic crystal plates with double-sided symmetric pillars

We build up phononic crystals (PCs) by attaching triangular pillars symmetrically to the double sides of a thin plate in the triangular lattice. The degeneracy of multiple Dirac cones is simultaneously lifted for the same PC plate, due to the mirror-inversion symmetry breaking caused by the rotation of triangular pillars; opposite Berry curves and valley Chern number are got at the lower and upper branches of each valley band. The multi-branch topological protected edge states are then created on the interface between PC plates with opposite topological phase. Interestingly enough, those edge states present various transverse symmetries and distinct robustness against sample imperfection such as cavity defect, rotation angle disorder, or sharp bending. Our results present the rich behaviors of multiple branches of topological edges.