Topological phononic insulator with robust pseudospin-dependent transport

Topological phononic states, which facilitate unique acoustic transport around defects and disorders, have significantly revolutionized our scientific cognition of acoustic systems. Here, by introducing a zone folding mechanism, we realize the topological phase transition in a double Dirac cone of the rotatable triangular phononic crystal with $C_{3v}$ symmetry. We then investigate the distinct topological edge states on two types of interfaces of our phononic insulators. The first one is a zigzag interface which simultaneously possesses a symmetric mode and an antisymmetric mode. Hybridization of the two modes leads to a robust pseudospin-dependent one-way propagation. The second one is a linear interface with a symmetric mode or an antisymmetric mode. The type of mode is dependent on the topological phase transition of the phononic insulators. Based on the rotatability of triangular phononic crystals, we consider several complicated contours defined by the topological zigzag interfaces. Along these contours, the acoustic waves can unimpededly transmit without backscattering. Our research develops a route for the exploration of the topological phenomena in experiments and provides an excellent framework for freely steering the acoustic backscattering-immune propagation within topological phononic structures.

Figure 1

Schematic of phononic crystal and its band structures. (a) Left panel: A geometric arrangement of phononic crystal consisting of rotatable triangular prisms with a lattice constant of 10 mm. The length of the triangular prism is 6.8 mm. Right panel: BZs of the primitive and larger hexagonal unit cells. (b) Band structure with a Dirac cone at the $K$ point in the BZ of the primitive unit cell. (c) Band structure with a double Dirac cone at the Γ point in the BZ of the larger hexagonal unit cell. (d) Band structure with a complete band gap induced by rotating triangular prisms of ${30}^{\circ }$. Pressure field distributions at the Dirac and double Dirac cones are inserted in (b–d). Dashed lines are the symmetric axis. Numerical methods are presented in Method 1 of the Supplemental Material [65].

Figure 2

Topological phononic insulator and its edge band structures. (a) A schematic of our topological phononic insulator with different interfaces along the $x$ direction and the $y$ direction. (b) Left panel: Band structure of a supercell consisting of $18\times 2$ phononic crystals with a zigzag interface along the $x$ direction. Right panel: Pressure field distributions of the supercell at $k=0$. Edge modes of the blue and red lines are respectively symmetric and antisymmetric about the light blue node due to the point symmetry of the geometrical structure. (c) Top panel: Band structure of a supercell consisting of $12\times 2$ phononic crystals (the left part turns left and the right part turns right) with a linear interface along the $y$ direction. Bottom panel: The pressure field distribution of the supercell at $k=0$. The edge mode is mirror symmetric about the dashed line. (d) Top panel: Band structure of a supercell consisting of $12\times 2$ phononic crystals (the left part turns right and the right part turns left) with a linear interface along the $y$ direction. Bottom panel: The pressure field distribution of the supercell at $k=0$. The edge mode is antisymmetric about the dashed line. Gray lines are bulk bands. Pale green shaded regions are topological band gaps. Numerical methods are presented in Method 1 of the Supplemental Material [65].

Figure 3

Acoustic pseudospin-dependent edge mode at topological cross-waveguide splitter. (a) Photo of our topological cross-waveguide splitter. The anticlockwise (clockwise) edge circulating propagation is indicated by the purple (green) circular arrows. (b, c) Simulated acoustic pressure field for cases with zigzag interfaces at a frequency of 18.42 kHz. (d) Simulated acoustic pressure field for the case with linear interfaces at a frequency of 18.42 kHz. (e, f, g) Experimental transmission spectra for cases (b)–(d). $T_{ij}$ indicates the transmission spectra from port $i$ to port $j(i,j=1,2,3,4)$. Shadow regions indicate topological band gaps. Experimental methods and measurements are presented in Method 2 of Supplemental Material [65].

Figure 4

Robust one-way transport. (a, b) Simulated acoustic pressure field at a frequency of 18.42 kHz in the topological phononic insulator with cavities and disorders. (c, d) Experimental transmission spectra. T12, T13, and T14 indicate the transmission spectra from port 1 to ports 2, 3, and 4. Shadow regions indicate topological band gaps.

Figure 5

Reconfigurable guiding of topological edge modes. (a, c) Simulated acoustic pressure field along the pathway through nodes A3 and A4. (c) Simulated acoustic pressure field distribution along the pathway through nodes A4 and A3. (b, d) Experimental transmission spectra for two pathways. The blue and red curves respectively indicate the transmission spectra in topological phononic insulators and ordinary phononic crystals. Shadow regions are topological band gaps.