a, A 3D Dirac sonic crystal and its transition to a Weyl sonic crystal. a Photograph of the 3D Dirac sonic crystal. b,c Geometry of the unit cells for b the 3D Dirac sonic crystal and c the Weyl sonic crystal with additional chiral hopping tubes. The insets of b and c show top-view images of the 3D Dirac and Weyl sonic crystals. One unit cell is outlined with a red hexagon for each sonic crystal. d The Brillouin zone (BZ) of the 3D Dirac sonic crystal. e The band structure of the 3D Dirac sonic crystal where a Dirac point (DP) is located on . f Tight-binding model of the 3D Dirac sonic crystal, including nearest-neighbour hopping and next-nearest-neighbour hopping. g The BZ of the Weyl sonic crystal, where each 3D Dirac point splits into a pair of Weyl points, WP1 and WP2, with charges of +1 and -1, respectively. h The band structure of the Weyl sonic crystal. States with positive orbital angular momentum (OAM) and negative OAM are represented by red and blue lines, respectively. i Additional chiral interlayer coupling in the Weyl sonic crystal. The arrows represent the direction of positive phase hopping.
by Boyang Xie, Hui Liu, Hua Cheng, Zhengyou Liu, Jianguo Tian, and Shuqi Chen