Robust edge states of planar phononic crystals beyond high-symmetry points of Brillouin zones

Robust edge states of the phononic crystals (PnCs) with the Dirac points fixed at corners or centers of the Brillouin zones (BZs) have received wide attentions. Here, we show that the Dirac points can degenerate at the high symmetrical boundaries of the first irreducible BZs. The mechanism of such Dirac dispersion is systematically discussed in the square lattice, the rectangular lattice, the centered rectangular lattice and the triangular lattice. These degenerated points, characterized by the vortex structure in a momentum space, are attributed to the band crossing in the presence of the crystalline mirror reflection symmetries. Variations of the geometric parameters can shift the positions of these Dirac points, but not lift them. By breaking the mirror reflection symmetries, the corresponding Dirac points will be lifted, yielding the directional or complete bandgaps. The phononic bandgaps efficiently block the propagation of the acoustic waves inside the bulks of the phononic systems, while their edge states localized at the domain walls between the PnCs with the distinct topological phases ensure the robust propagation of the sound energy flows. These results are unambiguously verified and determined by the experimental tests on the PnCs with the crystalline planar group P3m1.