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 ΓA. 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.
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机译：3D Dirac Sonic晶体的照片。 B，用于B 3D Dirac Sonic Crystal的单元电池的C几何形状，以及具有额外的手性跳跃管的Weyl Sonic晶体。 B和C的插入显示了3D Dirac和Weyl Sonic Crystals的顶视图。一个单元电池用红色六边形概述，每个声波晶体。 D 3D Dirac Sonic Crystal的Brillouin区（BZ）。 e CANCAC（DP）位于γA的3D DIRAC SONON晶体的带结构。 F紧密绑定模型3D Dirac Sonic Crystal，包括最近邻居跳跃和下一个最近的邻居跳跃。 G威尔Sonic晶体的BZ，其中每个3D Dirac点分裂成一对Weyl点，WP1和WP2，分别为+1和-1的电荷。 H威尔·莫尔康星的带结构。具有阳性角度角动量（OAM）和负OAM的状态分别由红色和蓝线表示。我辅助手性交换器耦合在威尔Sonic晶体中。箭头代表正阶段跳跃的方向。
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