The concept of topological insulator (TI) has drawn extensive attention since it was proposed. In condensed matter physics, TI has been proved to be an excellent platform for new types of lossless information transmission because of its defects-immune property based on gapless topological-protected edge states.^[1–5] In recent years, optical TIs^[6–16] and acoustic TIs^[17–26] have also been successfully demonstrated. More recently, the concept of higher-order topological insulator (HOTI) hosting gapped edge states and corner states has shown that the traditional bulk–boundary correspondence is no longer applicable to it.^[27–34] Different from the topological phases of the traditional TIs, the reported works have shown that the topologies of the HOTIs are characterized by the nonzero bulk polarization, which are strongly related to the point group symmetry (PGS) of the periodic structures.^[27,34–36] Moreover, some works have demonstrated that HOTIs can also be implemented in two-dimensional elastic waves,^[37,38] microwaves,^[39] toroidal circuits,^[40] optical^[41–44] and acoustic systems.^[45–49]

The previous researches on the HOTIs have always focus on the 0D corner states which are capable of confining the energy on the corners exactly, however, we note that the normally-gapped edge states of the HOTIs also provide unprecedented platforms for the control of wave propagation. In the classical wave systems, the behaviors of the topological states crucially depend on the structure parameters of the material, which implies the possibilities to adjust the bands of the gapped edge states.

In this work, we propose a graphene-like sonic HOTI and focus on the behaviors of the edge states. We show that the parity inversion of the proposed HOTI depends on the choices of the primitive cells, and the inverted parities of the eigenstates at the high symmetry points near the nontrivial gaps, which indicate nonzero bulk polarization, lead to the generation of the edge states on the interface. Meanwhile, the pseudo-spins carried by the states originating from the graphene-like lattice ensure the robust wave propagation. Importantly, we further demonstrate that the translation of the ‘layer’ leads to the changing of the coupling strength of the multipolar states between the layers, and then results in the expansion or contraction of the gapped edge states. It indicates that the edge states can be adjusted by simply controlling the translation rates. Based on these, a prototype of two-dimensional acoustic topological filter with adjustable edge states is experimentally proposed. By simply sliding the sections of the filter, the bandwidths of the edge states can be controlled, and the propagation of the sound waves at specific frequencies band to the selected ports can be freely manipulated.

We start with a regular 2D SC with a C_6v symmetry lattice consisting of solid hexagonal columns embedded in the air background as shown in Fig. 1(a). The side length of the column is r₀ = 6 mm, and the distance between the nearest-neighbor columns is R = 2.778r₀. The lattice constant is a=3R. In the perspective of HOTI, the nontrivial topology is determined by the bulk polarization, which in turn results in the emergence of the edge polarization (P_x,P_y). A nontrivial edge polarization indicates the existence of the in-gap edge states located on the boundaries of the nontrivial domains, and the robust 0D topological corner state can also emerge on the specific corner as the intersected boundaries are all polarized, which are different from the ordinary defects.^[50–52] Meanwhile, the topology of the material is strongly related to the point group symmetry, which means that the parity inversion indicating the topological phase transition may be induced simply by selecting different definition of the unit cell in one common lattice.^[53] In our work, the solid columns can be treated as hard boundries and the energy is actually propagating in the background air, which is different from the previous works.^[24,49] To simplify the model of our work, we only consider two nearest-neighbor hoppings of the aerial meta-atoms respectively along x and y directions, and the structure can then be redefined with two types of rectangular supercells [red dashed lines in Fig. 1(a)]. Figures 1(b) and 1(c) depict the energy band structures of the two supercells, respectively. Due to the same lattice constants a_x = a, ay=3a, and the same periods along both directions, the energy band structures of the two supercells shall also be the same. However, the parity of the modal distribution of each band labeled with “+” (even parity) and “–” (odd parity) at the high symmetry points represents distinct topology.

The topological non-triviality of the parity inversion can be characterized by the integration of the Berry connection over the 1st BZ^[27]

$$ \begin{eqnarray}\begin{array}{lll} & & {P}_{i}=\displaystyle \frac{1}{{(2\pi)}^{2}}\displaystyle \iint {{\rm{d}}}^{2}k{\rm{Tr}}[{A}_{i}],\\ & & {({A}_{i})}_{mn}(k)={\rm{i}}\langle {u}_{m}(k)|{\partial }_{{k}_{i}}|{u}_{n}(k)\rangle,\end{array}\end{eqnarray}$$ (1)

$$ \begin{eqnarray}\begin{array}{lll} & & {P}_{i}=\displaystyle \frac{1}{2}\left(\displaystyle \sum _{m}{q}_{i}^{m}\,\mathrm{mod}\,2\right),\end{array}\end{eqnarray}$$ (2)

$$ \begin{eqnarray}\begin{array}{lll} & & {(-1)}^{{q}_{i}^{m}}=\displaystyle \frac{{\eta }_{m}(M)}{{\eta }_{m}(\Gamma)},\end{array}\end{eqnarray}$$ (3)

Before introducing the topological states in this crystal, it is important to note that the nontrivial polarization of the domain does not require either the spin-like degrees of freedom or the time-reversal relation.^[53,54] However, the pseudo-spins can still be introduced by multiple scattering for the special C_6v lattice.^[24] We calculate the real-space distributions of the root-mean-square intensity of the energy flow I=1T∫Re(p)Re(v)dt to describe the pseudo-spins in the two supercells as illustrated in Figs. 1(d) and 1(e) (see details in Appendix A). Crucially, the chiral flows on each corresponding band of the two supercells imply the promotion of the wave propagation between the domains with distinct topology.

The bulk–edge–corner correspondence of the HOTIs predicts both the edge and corner states. In this work, we focus on the behaviors of the gapped edge states. Figures 2(a) and 2(b) present the energy band structures of two ribbons, which are periodic along y direction, composed of 30 trivial supercells or 15 trivial and 15 nontrivial supercells along x direction, respectively. Apparently, the predicted edge states vanish on the edge of the complete trivial domain [Fig. 2(a)], but emerge on the nontrivial interface and span the band gaps [Fig. 2(b)]. Meanwhile, due to the strong coupling of the pseudo-spins on the interface, the edge states appear to be gapless (see Fig. A2 in Appendix A).

We emphasize that the two primitive cells with distinct topology corresponding to the same lattice, which implies that the band inversion can be induced by simply sliding the parts of the structure. As shown in Fig. 3(a), the structure is divided into layers, which can be slid along x direction independently. We define the translation rate β_i = r_i / a_x (i = A, B, C, …), where r_i indicates the actual amount of the translation of the i-th layer. We first study one interface between A-layer and B-layer [inset in Fig. 3(a)], and only consider the slide of A-layer. Figure 3(b) illustrates the correlation between the gap of the edge states and β_A. As β_A increases, the symmetry of the interface reduces and the coupling of the pseudo-spins strengthens, which results in the expansion of the edge states. When β_A = 0.5, the anti-symmetry of the interface corresponds to the strongest coupling of the pseudo-spins, which meets the parity inversion exactly. Therefore, the edge states are steep and different from the flat cases without considering pseudo-spins.^[36] Figures 3(c)–3(e) present the corresponding band structures for β_A being 0.125, 0.25, and 0.375, respectively.

We further consider three independent layers with β_A = 0.5, β_C = 0.25, and β_B = 0. Figure 4(a) shows the energy band structure of the corresponding ribbon. Significantly, there are two pairs of edge states in each gap, which represents the independent topological propagating modes on the two interfaces, respectively. The real-space distributions and the flows of intensity of the pressure field for k = ± 0.05 × 2 π / a_x shown in Fig. 4(b) depict the predicted topological modes confined on the specific interface. These results provide the possibility to design a type of multi-channel topological acoustic waveguide to achieve filtering at specific frequencies.

Here, we construct a prototype of two-dimensional acoustic topological filter based on the crystal proposed above and experimentally measure the corresponding sound wave transmission. As shown in Fig. 5(a), the filter is composed of 3D printed solid hexagonal columns, which can be treated as the acoustic hard boundaries. The whole structure is placed between two hard-boundary waveguide plates to prevent the possible sound energy from leaking into the free space, and is surrounded by sound absorption boundaries to eliminate the interference of the environmental noise. The filter is divided into the four sections labeled with A, B, C, and D, and the four ports are labeled as input, port 2, port 3, and port 4 corresponding to the four inter-paths labeled with P₁, P₂, P₃, and P₄, respectively. A broadband sound source is placed at the input, and three transducers are placed at the other three ports to detect the sound energy transmission. We introduce the translation rates β_Pi (i = 1, 2, 3, 4) to indicate the translation ratio of the corresponding path. If we set β_P2 = β_P4 = 0 and β_P1 = β_P3 = 0.25, the symmetry of the structure at P₁ and P₃ will be broken and two transport paths will be generated, which means that the sound wave can only propagate in one dimension. Figure 5(b) exhibits the high transmission of the sound wave at 6.75–7.39 kHz to port 3 with very low leakage of the energy into the bulk, which corresponds to the energy band structure shown in Fig. 3(d). If β_P1 and β_P3 are increased to 0.5, the symmetry is the weakest and the sections on both sides are antisymmetric along the path. Hence, the two edge states in the band gaps connect to each other. Figure 5(c) shows the high-transmission of the sound wave at 5.74–7.39 kHz, which corresponds to the energy band structure shown in Fig. 2(b). These results experimentally demonstrate the adjustable edge states hosted by the filter.

We further study the two-dimensional transmission of the sound wave in the filter. Figure 6(a) shows the low transmission of the filter’s three ports for β_Pi = 0 (i = 1,2,3,4). If we set β_P1 = β_P3 = 0.5 and β_P2 = β_P4 = 0.25, the four transport paths will all be opened, but only P₁ and P₃ are antisymmetric. The corresponding transmissions illustrated in Fig. 6(b) indicate that the sound waves from 5.74 kHz to 6.75 kHz can only propagate along P₁ and P₃, but are hindered within P₂ and P₄. We show the numerical results of the energy fields for 6400 Hz and 7125 Hz in Figs. 6(c) and 6(d), respectively. Moreover, the total transmission efficiency of the three ports of the filter is almost 100% [dashed line in Fig. 6(b)] if the energy loss of the leakage or viscosity is ignored, which is consistent with the inherent property of TIs. Crucially, the independent translations in two directions indicate that the bandwidth of the edge states on the selected path can be adjusted freely. Therefore, an acoustic topological filter with multiple transmission channels can also be proposed by simply sliding the specific sections. In addition, it is noted that the filter is robust and topological protected, which is distinct from the ordinary defects.^[55,56] Simulated results are given in Appendix B. This significant property to freely adjust the bandwidth of the edge states provides excellent ways for the design of adjustable topological acoustic materials.

We have numerically and experimentally demonstrated the adjustable edge states of the two-dimensional graphene-like sonic HOTI. By tuning the translation rate of the layers of the material, we can change the coupling strength of the pseudo-spins on the interface, and then control the gapped edge states. Besides, we have proposed a prototype of multi-channel topological filter with independent adjustable edge states, and experimentally demonstrated its almost lossless transmission of the acoustic waves. These results provide new perspective to control the edge states. Based on this work, we expect to expand the design of the topological insulators, and provide the novel application prospects of acoustic topological materials such as filters, waveguides, and acoustic switches.