Fig. 1. (a) Cross section of the phononic crystal lattice in xy-plane and the vector representation of the minimum cell (red arrows) and compound cell (yellow arrows); (b) three-dimensional view of the minimum cell; (c) band structure of the first Brillouin zone of the minimum cell. The inset shows the first Brillouin zone. (a)晶格在xy平面截面图及最小元胞(红色箭头)和复合胞(黄色箭头)的矢量表示; (b) 最小元胞的三维图; (c) 最小元胞第一布里渊区的能带图, 插图为晶格的第一布里渊区

Fig. 2. (a) Three-dimensional view of the compound cell; (b) the folding mechanism from Bz to Bz_s. Red and yellow hexagon region represents the first Brillouin region of minimum cells and compound cells, respectively represented by Bz and Bz_s; (c) band structure of the compound cell. (a) 复合胞的三维图; (b) 从Bz到Bz_s的折叠机制, 红色和黄色六边形区域代表了最小元胞和复合胞的第一布里渊区, 分别用Bz和Bz_s表示; (c) 复合胞第一布里渊区的能带图

Fig. 3. Band structure of the compound cell for the case of (a) topologically trivial R₁ = 0.32b and (c) topologically nontrivial R₁ = 0.345b. The insets show the pressure field below the band gap around Г_s in corresponding lattice when the energy flow rotating clockwise. The pressure filed of the double degenerated state at Г_s point in the band structure of topologically (b) trivial and (d) nontrivial compound cell. The black arrows indicate the direction of energy flow. The energy flow rotating clockwise (anticlockwise) corresponds to the pseudospin-down state (pseudospin-up) represented by red (blue) arrow. 拓扑平庸　(a) R₁ = 0.32b和非平庸(c) R₁ = 0.345b复合胞的能带图; 插图给出了带隙频率下方能流顺时针流动时, 相应晶格在Г_s点附近的声压场分布. 拓扑平庸(b)和非平庸(d) 复合胞能带图中Г_s点两个双重简并态p态和d态的声压场分布图. 黑色箭头表示能流的运动方向. 能流顺时针转动, 对应向下赝自旋态, 用红色箭头表示; 能流逆时针转动, 对应向上赝自旋态, 用蓝色箭头表示

Fig. 4. (a) Schematic diagram and band structure of the ribbon-shaped supercell (in xy-plane). The ribbon-shaped supercell is composed of 10 topologically nontrivial compound cells sandwiched by 5 topologically trivial compound cells on both sides. The gray areas in the band diagram represent the bulk modes and the blue and red lines indicate the edge modes; (b) pressure fields of points A and B in (a). The four diamond-shaped color graphs in the middle are the edge modes of point A and B, respectively. The black arrows indicate the direction of energy flow. (a) 条状超胞示意图(xy平面)和能带图; 条状超胞是由中间的10个拓扑非平庸复合胞和上下各5个拓扑平庸复合胞构成的三明治结构, 能带图中灰色区域为体态, 蓝线和红线表示边界态; (b) 图(a)中A点与B点的声场分布图. 中间的四张菱形彩色图分别为A点和B点的两个边界态. 黑色箭头表示表面声波能流的运动方向

Fig. 5. Absolute pressure field of (a) the topologically trivial phonon crystals (Structure Ⅰ), (b) linear type topologically protected surface wave waveguide (Structure Ⅱ), (c) bending type topologically protected surface wave waveguide (Structure Ⅲ) at f = 7630 Hz. The insets in red diamonds show the absolute pressure field of the three structures at z = L in xy-plane; (d) transmission coefficient of Structure Ⅰ (black dashed line), Structure Ⅱ (red solid line) and Structure Ⅲ(blue dotted line). The shaded areas represent the gap frequency range of the compound cell. 频率为7630 Hz时　(a)拓扑平庸声子晶体(结构Ⅰ), (b) 受拓扑保护的直线表面波波导(结构Ⅱ), (c) 弯曲型拓扑保护表面波波导(结构Ⅲ)的声压绝对值分布; 红色菱形框内为三种结构在z = L处的xy平面上绝对值声压分布图; (d)结构Ⅰ(黑色虚线), 结构Ⅱ(红色实线)和结构Ⅲ(蓝色点划线)中的声波传输系数, 阴影部分表示复合胞超元胞带隙频率范围