In the past few years, acoustic topological insulators (ATIs) have become a hot topic in the area of acoustics for its great potential applications, such as acoustic-noise reduction [1], acoustic beam splitter [2] and acoustic communications [3]. However, the intrinsic difference between electrons and acoustics poses a challenge to realize a spin-like degree of freedom for acoustics due to the longitudinal polarization characteristics of sound. To overcome it, four types of ATIs have been proposed successively. To mimic the quantum Hall effect and break time-reversal symmetry, topologically protected edge states were observed in acoustic waveguides by introducing circular flowing air [4–9]. Later, analogous Floquet ATIs were proposed based on coupled acoustic trimers or strong coupling ring resonator waveguides [10–13]. Moreover, to realize acoustic valley Hall (AVH) states, the nonzero Berry curvature located near the valley K/Kʹ was obtained by breaking mirror or inversion symmetry, and valley-projected topological edge states were observed at the interface between two sonic crystals (SCs) with distinct AVH phases [14–28]. Furthermore, to mimic quantum spin Hall states in acoustics, the SCs with C ₆ crystal symmetry were introduced to realize artificial pseudospins-1/2 states by hybridizing degenerate modes [29–39]. Generally, the observed double Dirac cones are accidental degeneracy in these pseudospin-dependent ATIs, and the sizes or shapes of rods in the SCs are different, which is still a challenge for practical applications.

Recently, by introducing the zone folding mechanism, a type of pseudospin-dependent ATI without depending on a fixed filling ratio was proposed [30, 31], in which the observed double Dirac cones were a deterministic degeneracy. These types of pseudospin-dependent ATIs were designed by fictitious soft metamaterial rods [30] and three circular rods with different radius [31], and the complex structures greatly limit their applications. Based on the SCs composed of same rods, the experimental realization of the pseudospin-dependent ATIs with a deterministic degeneracy is still a challenge.

In this work, we propose a pseudospin-dependent ATI constructed by two airborne SCs which consist of the same rods. Based on the zone folding mechanism, we obtain double Dirac cones with the deterministic degeneracy, and realize a band inversion by simply rotating the rods in the SCs. Moreover, we design an ATI composed of two SCs with different rotation angles of the rods, and observe a one-way sound propagation of topological edge states. Finally, we experimentally demonstrate the robustness of the topologically protected sound propagation against two types of defects of a disorder and a bend.

Figure 1A shows an airborne SC composed of regular hexagonal rods with the side length l = 14.4 mm. The proposed SC can be described by the unit A or B, in which the lattice constants of the units A and B are a ₀ = 30 mm and a = 30 3 mm, respectively. By simply rotating these rods, we obtain another type of SC composed of the unit B (shown in Figure 1B), in which the rotation angles of a central rod and six surrounding rods are θ ₀ and θ, respectively. Figure 1C shows the folding mechanism of the Brillouin zones (BZs) for the units A and B. In the BZ, the point K₀ with high symmetry of the unit A can be folded into the point Γ of the unit B. Here, we use the COMSOL Multiphysics software to numerically simulate propagation characteristics of sound. The regular hexagon rods are fabricated with polymethyl methacrylate (PMMA), which is used to meet the condition of sound hard boundary in numerical models. The material parameters are adopted as follows: the density ρ = 1180 kg/m³, the longitudinal wave velocity c _l = 2730 m/s and the transversal wave velocity c _t = 1430 m/s for PMMA; the density ρ ₀ = 1.18 kg/m³ and sound velocity c ₀ = 346 m/s for air.

Figures 2A, B show the dispersion relations of the units A and B in the SC-I (θ ₀ = 0^o and θ = 0^o), respectively. As shown in Figure 2A, there exists a Dirac cone with the two-fold degeneracy at the point K₀ for the unit A, which is attributed to its C _6v symmetry. However, for the unit B, we observe double Dirac cones with a four-fold deterministic degeneracy at the point Γ (Figure 2B). The dispersion characteristics of both units agree well with the folding of the BZ.

Figures 2C,D show the dispersion relations of the unit B in the SC-II (θ ₀ = 30^o and θ = 0^o) and SC-III (θ ₀ = 0^o and θ = 30^o), respectively, in which the four-fold degeneracy is broken, and is divided into a double two-fold degeneracy (p and d types) for both cases. We can see that the bands of the d type are above the p type in the SC-II (Figure 2C), while the bands of the p type are above the d type in the SC-III. The phenomenon of band inversion arises from the increasing coupling strength between the hexagonal rods by changing the values of θ ₀ and θ. Additionally, the pressure eigenfunctions of the dipolar (p _x and p _y ) and quadrupolar (d _xy and d x 2 − y 2 ) modes which correspond to the blue and red dots are displayed in eight insets of Figures 2C,D, further showing a typical characteristic of the band inversion. Additionally, we also calculated the Chern numbers of SC-II and SC-III based on the Wilson-loop method (shown in the supplementary material), which further verify that both SCs have different topological phases. Here, it is worth mentioning that the pseudospin-1/2 modes of sound are the basis for imitating quantum spin Hall effect in acoustics, which can be realized by hybridizing both modes p and d as p ± = ( p x ± i p y ) / 2 and d ± = ( d x 2 − y 2 ± i d x y ) / 2 .

Figure 3A shows the dispersion relations of the designed ATI composed of the SC-II and SC-III, in which the SC-III is placed on the upside of the SC-II. We observe a pair of edge states (red and blue solid lines) in the overlapped bulk band gap of both SCs, which is located at the domain wall between the SC-II and SC-III. Here, the group velocity of the edge states is determined by the slope of the bands (red and blue solid lines), and thus the edge states have the same acoustic velocity, but propagate in the opposite directions. Moreover, it is obvious that there exists a mini gap located between the bands of two edge states, which mainly arises from the reduction of the C _6ν point group symmetry at the domain wall between SC-II and SC-III. Figure 3B shows the eigen-modes for the real (M₁) and imaginary (M₂) parts of the edge state at the point A. The edge states at the points A and B can be obtained by hybridizing the modes M₁ and M₂ as the forms of M₁+iM₂ and M₁−iM₂, respectively. To further exhibit the pseudospin-dependent characteristics, we simulate the distributions of energy flow around the domain wall of a supercell at the points A and B (shown in Figure 3C). We find that there exist energy vortexes around the domain wall, and the chirality of the vortex is clockwise at the point A, but is anticlockwise at the point B, which corresponds to the pseudospin− (red circular arrow) and pseudospin+ (blue circular arrow), respectively.

Next, we simulate the pseudospin-dependent one-way propagation of sound in the ATI composed of the SC-II and SC-III, in which the intensity distributions excited by two types of chiral sources (green stars) with the opposite chirality at 16.0 kHz are shown in Figure 4. The chiral sources are realized by six point sound sources arranged in a circular array with different initial phase delays, which are placed at the center of the domain wall. As shown in Figure 4A, when the chirality of the source is anticlockwise, the edge state propagates toward the left side along the domain wall with almost negligible backscattering. In addition, it is found from the zoom of R1 that, the chirality of the energy vortex remains the same during its propagation along the domain wall, which further demonstrates the one-way propagation of the pseudospin + edge state. However, when the chirality of the source is clockwise, the pseudospin edge state is excited and propagates toward the right side. Therefore, we demonstrate the one-way sound propagation of the pseudospin-dependent edge states.

To demonstrate the robustness feature of the pseudospin-dependent edge states, we separately introduce two kinds of defects (a lattice disorder and a bend) around the domain wall of the ATI. Figure 5A shows the photograph of the sample without the defects. The description of experimental set-up is displayed in the supplementary material. Figures 5B–D show the simulated intensity distributions in the ATIs without defects and with the defects of a disorder and a bend, respectively, in which a point source is placed at the left side of the domain wall. Meanwhile, the measured intensity distributions in the scanning region (green open rectangle) are shown at the right side. We find that the smooth propagations of the pseudospin edge states still exist with both types of defects (Figures 5C,D), and the corresponding intensity distributions are almost the same as those in Figure 5B. The experimental results agree well with the simulations. Beyond that, in the zooms of R3-R5 (Figures 5B–D), the chirality of edge states is not affected by both defects, showing high robustness of the pseudospin-dependent edge states.

Finally, to demonstrate the topologically protected sound propagation, we also experimentally measure the transmission spectra for the three types of ATIs in Figures 5B–D, which is shown in Figure 6. Note that the transmittance spectra for the three cases are almost the same, especially in the overlapped bulk band gap (black shaded region). Therefore, we further experimentally demonstrate the robustness of the pseudospin-dependent edge states.

In conclusion, we have demonstrated a pseudospin-dependent ATI composed of two SCs with the same regular hexagonal rods. The results show that, by using the zone folding mechanism, the double Dirac cones with a four-fold deterministic degeneracy are observed in the SC, and the band inversion is realized by rotating the rods. Moreover, the ATI composed of two SCs with different rotation angles of the rods is designed, and the one-way sound propagation of the pseudospin-dependent edge states excited by the chiral sound sources in the ATI is observed. Furthermore, by measuring the transmission spectra in the ATIs with and without defects, the robustness of sound propagation against two types of defects (a lattice disorder and a bend) is demonstrated. The simulated and measured results agree well with each other. Our work opens a new venue to design tunable pseudospin-dependent ATIs with various functionalities and applications.