Fig. 1. (a) Schematic of two selected supercells (red dashed lines) based on one common graphene-like lattice, the corners and the edges of the supercells represent the pseudo-atoms and the corresponding hoppings, respectively. The corresponding energy band structures labeled with parities are presented in (b) and (c). (d) and (e) The pressure field distributions of the first two double-degenerate bands from bottom to top at M point of the two supercells, respectively. The opposite parities indicate the parity inversion.

Fig. 1. Pseudo-spin states at (a) Γ and (b) M. In (a), d_{mod 1,2} represents the double-degenerate modes below the upper gap at Γ, and p_{mod 1,2} represents the modes upon the upper gap. In (b), the labels of the four dotted boxes represent the count of the double-degenerate bands from bottom to top. The (±) represents the parity of each mode, and circulation arrow of energy flow demonstrates the certain pseudo-spin state. The relative band structure is shown in Fig. 3(b).

Fig. 1. Comparison of the simulated acoustic field distribution at 6.84 kHz for topological interface and ordinary defects, corresponding to (a) a cavity and (b) a disturbance in the waveguide. The red and black arrows represent propagation of the topological states and ordinary sound, respectively. (c) Simulated transmission spectra for perfect topological interface (red curve), topological interface with a cavity (green curve), topological interface with a disturbance (black curve) and ordinary defects (blue dashed curve). The white region represents the predicted range of edge states.

Fig. 2. Band structures of the ribbons composed of (a) 30 trivial supercells and (b) 15 trivial and 15 nontrivial supercells, being finite along y direction and periodic along x direction.

Fig. 2. Schematic diagrams of the two conditions: (a) the upper and the lower materials are the same and (b) the upper and the lower materials are different. The structure in (a) appears to be insulating, and the structure in (b) appears to be conductive.

Fig. 3. (a) Schematic of the structure divided into independent layers. Inset: Details of the configuration near the interface. (b) β_A-dependent band-edge frequencies for the supercells with k = 0 within a translation period. g₁ and g₂ represent the ranges between the two topological bands. Energy band structures with distinct edge states for (c) β_A = 0.125, (d) 0.25, and (e) 0.375, respectively.

Fig. 4. (a) Energy band structure of the ribbon with two independent interfaces, the two pairs of the edge states are independently emerging on the specific interface. (b) Pressure field distributions and intensity flows of the pseudo-spin-dependent edge states for k = ± 0.05 × 2 π / a_x on the interfaces of the ribbon.

Fig. 5. (a) Experimental setup of the two-dimensional acoustic topological filter. (b) Measured bulk (green) and port 3 (red) transmission spectra for β_P1 = β_P3 = 0.25. (c) Measured bulk (green) and port 3 (red) transmission spectra for β_P1 = β_P3 = 0.5. The gray regions in both (b) and (c) represent the theoretic range of the edge states.

Fig. 6. (a) Measured transmission spectra for port 2 (red), port 3 (green), and port 4 (blue) when β_Pi = 0. (b) Measured transmission spectra for port 2 (red), port 3 (green), port 4 (blue) and the sum of the three ports for β_P1 = β_P3 = 0.5 and β_P2 = β_P4 = 0.25. The green dashed curve representing the total transmission loss exhibits lossless transmission in the two theoretical frequency ranges marked with grey. (c) and (d) The simulated acoustic energy transmission distributions for 6400 Hz and 7125 Hz, respectively.