Fig. 1. Schematics of left- and right-handed unit cells: (a) Left-handed unit cell with four distinct air tubes, the top view of the unit cell shows a counterclockwise variation; (b) right-handed unit cell with the clockwise length variation of air tubes; (c) the bulk band structure with four bulk modes, where the eigenmodes for left-handed and right-handed unit cells are shown at the sides.左手性和右手性原胞示意图　(a)四个共振的空气柱子组成左手性原胞, 其俯视图表现出逆时针旋向; (b)相同空气柱排成的右手性原胞, 其俯视图呈现顺时针旋向; (c)原胞的体能带图有四种声学模式, 左右两侧为不同手性原胞在高对称点的模式分布

Fig. 2. Band structures of the interface states: (a) The mirror symmetric interface constructed by oppositely chiral unit cells; (b) the projected band structure of a supercell, the right part shows the supercell slab comprises 6 left-handed and 6 right-handed unit-cells with an even mode (f_a1) localized at the interface; (c) the robustness of the even mode against randomly introduced position and height disorders of air tubes. 超胞色散和局域界面态　(a)相反手性原胞构成镜像对称界面; (b)带状超胞能带色散, 其中五条蓝色曲线代表界面态, 灰色曲线代表体带, 超胞由六个左手性的原胞和六个右手性的原胞组成, 其单极子模式局域在界面处; (c)单极子界面模式对于位置微扰和高度微扰有较强鲁棒性

Fig. 3. Robust transport of sound in subwavelength scale: (a) The schematic of the mirror interface between two oppositely chiral sonic crystals; (b) the propagation of even interface modes along the waveguide channel without defects; (c) the propagation of even interface modes along the waveguide channel with position variations of air tubes; (d) the propagation of even interface modes along the waveguide channel with length variations of air tubes; (e) acoustic transmission for interface states and bulk states, the gray ribbon marks the band gap from 1407 Hz to 1487 Hz; (f) pressure amplitude distributions in the transverse direction.亚波长声波鲁棒传输　(a)镜像界面沿着x方向周期排布成直线波导通道; (b)单极子界面模式沿着波导通道局域传输; (c)在引入位置缺陷时, 声波沿直线局域的传输没有明显的背散射; (d)在引入高度微扰时, 声波沿直线局域的传输没有明显的背向散射; (e)几种不同情况下的传输效率曲线, 黑红蓝色曲线表示界面传输透射, 绿色曲线表示统一手性晶体中体态传输透射, 灰色区域表示禁带范围; (f)沿着横向路径的声压幅值分布

Fig. 4. The robust edge transport of sound: (a) The interface transport of odd modes along the interface with a dipole source; (b) the edge transport of odd modes along the soft boundary denoted by the green line; (c) the interface transport of even modes along the interface with a monopole source; (b) the edge transport of even modes along the right boundary denoted by the red line.声波边缘传输　(a)借助偶极源, 奇模式声压场沿界面局域传输; (b)当软边界条件施加在截断的晶体表面时, 声学奇模式沿着边缘传输, 绿线标记软边界条件; (c)借助单极源, 偶模式声压场沿界面局域传输; (d)当硬边界条件施加在截断的晶体表面时, 声学偶模式沿着边缘传输, 红线标记硬边界条件