Underwater acoustic metamaterial based on double Dirac cone characteristics in rectangular phononic crystals

We theoretically construct a rectangular phononic crystal (PC) structure surrounded by water with C_2v symmetry, and then place a steel rectangular scatterer at each quarter position inside each cell. The final complex crystal has two forms:the vertical type, in which the distance s between the center of the scatterer and its right-angle point is greater than 0.5a, and the transverse type, in which s is smaller than 0.5a (where a is the crystal constant in the x direction). Each rectangular scatterer has three variables:length L, width D, and rotation angle θ around its centroid. We find that, when L and D change and θ is kept at zero, there is always a linear quadruply degenerate state at the corner of the irreducible Brillouin zone. Then, we vary θ and find that the quadruply degenerate point splits into two doubly-degenerate states with odd and even parities. At the same time, the band structure reverses and undergoes a phase change from topologically non-trivial to topologically trivial. Then we construct an acoustic system consisting of a trivial and a non-trivial PC with equal numbers of layers, and calculate the projected band structure. A helical one-way transmission edge state is found in the frequency range of the body band gap. Then, we use the finite-element software Comsol to simulate the unidirectional transmission of this edge state and the backscattering suppression of right-angle, disorder, and cavity defects. This acoustic wave system with rectangular phononic crystal form broadens the scope of acoustic wave topology and provides a platform for easy acoustic operation.