Introduction

Following the rise of graphene, layer-stacked graphene came up to the stage as a platform for exploring a plethora of intriguing physics phenomena1,2,3,4, such as unconventional quantum Hall effects5,6 and domain-wall valley transports7,8,9. These phenomena are closely tied to the quadratic Dirac cone. It is demonstrated that, in the presence of trigonal warping effect, the quadratic Dirac cone is unstable and evolves into a central linear Dirac cone accompanied by three additional linear Dirac cones surrounding it, collectively referred to as satellite Dirac cones1,6,10,11,12,13. Despite being proposed nearly two decades ago, satellite Dirac cones are experimentally challenging to identify, and their direct spectral observation remains elusive14.

Meanwhile, the Dirac cones inherently carry topological characteristics, as exemplified by the presence of edge states in graphene via the principle of bulk-edge correspondence15,16. Additionally, introducing gaps in the Dirac cones leads to topological insulating phases, featuring by topologically protected edge or interfacial states17. Dirac cones and the associated topological transport are not confined to electronic systems and can be generalized to classical wave systems18,19,20, where the flexibility in structural design allows the relaxation of locality constraints in interactions. For example, roton-like dispersions can be driven by dominant long-range couplings in metamaterials21,22,23,24,25, which enables the emergence of larger topological invariants26,27,28,29,30. From this perspective, an intriguing question arises: could symmetry-permitted longer-ranged couplings generate additional Dirac cones in the Brillouin zone (BZ), forming satellite Dirac cones? If so, what would be the associated bulk-edge correspondence, and what novel phenomena might emerge if these satellite Dirac cones are gapped into satellite valleys?

In this work, we achieve the realization of satellite Dirac cones by introducing strong third-nearest-neighbor (TNN) couplings. These couplings enable the symmetry-protected Dirac cones in a monolayer honeycomb structure to generate satellite Dirac cones, accompanied by a band reshaping. The resulting satellite Dirac cones carry Berry phases in an opposite sign to that of the central cone. Consequently, satellite Dirac cones induce a doubling of edge states compared to the original one. Our model is implemented in phononic crystals (PCs), which have long served as versatile platforms for exploring topological phenomena19,20. PCs offer flexibility for performing broadband measurements and facilitate counterintuitive couplings, including the strong TNN couplings required in this study. Moreover, by introducing staggered on-site potentials, the Dirac cones are gapped, forming satellite valleys that support multiple valley interfacial modes. The satellite Dirac cones and their associated transport phenomena are validated through both simulations and experiments. These findings highlight the potential of longer-range couplings in band engineering, enabling the emergence of topologically nontrivial phases.

Results

Lattice model with satellite Dirac cones

We incorporate TNN hoppings into a monolayer graphene lattice to achieve satellite Dirac cones, which significantly simplifies the scenario in Bernal-stacked bilayer graphene (see Supplementary Note 1). As illustrated in Fig. 1a, our model yields the following Bloch Hamiltonian in the (A, B) sublattice basis:

$$H\left({{{\bf{k}}}}\right)=\left(\begin{array}{cc}0 & {t}_{1}{\gamma }_{1}\left({{{\bf{k}}}}\right)+{t}_{3}{\gamma }_{3}\left({{{\bf{k}}}}\right)\\ {t}_{1}{\gamma }_{1}^{*}\left({{{\bf{k}}}}\right)+{t}_{3}{\gamma }_{3}^{*}\left({{{\bf{k}}}}\right) & 0\end{array}\right),$$
(1)

where \({t}_{1}\) and \({t}_{3}\) represent the nearest-neighbor and TNN hopping strengths, respectively, both of which are real-valued. The explicit forms of \({\gamma }_{1}\left({{{\bf{k}}}}\right)\) and \({\gamma }_{3}\left({{{\bf{k}}}}\right)\) are given as \({\gamma }_{1}=1+{e}^{i{{{\bf{k}}}}\cdot {{{{\bf{a}}}}}_{2}}+{e}^{i{{{\bf{k}}}}\cdot ({{{{\bf{a}}}}}_{2}-{{{{\bf{a}}}}}_{1})}\) and \({\gamma }_{3}={e}^{-i{{{\bf{k}}}}\cdot {{{{\bf{a}}}}}_{1}}+{e}^{i{{{\bf{k}}}}\cdot {{{{\bf{a}}}}}_{1}}+{e}^{i{{{\bf{k}}}}\cdot (2{{{{\bf{a}}}}}_{2}-{{{{\bf{a}}}}}_{1})}\), with primitive lattice vectors defined as \({{{{\bf{a}}}}}_{1}=({{\mathrm{1,0}}})\) and \({{{{\bf{a}}}}}_{2}=\left(1/2,\sqrt{3}/2\right).\) The eigenvalues of \(H({{{\bf{k}}}})\) are given by \({E}_{\pm }=\pm |{t}_{1}{\gamma }_{1}\left({{{\bf{k}}}}\right)+{t}_{3}{\gamma }_{3}\left({{{\bf{k}}}}\right)|\), and the band-touching points are determined by solving \({E}_{\pm }=0\). The existence of these solutions is guaranteed by the combined symmetries of time-reversal and spatial inversion. In real graphene, the TNN hopping strength (\({t}_{3}\)) is approximately 40 times weaker than \({t}_{1}\) 1\(,\) and thus \({t}_{3}\) is often neglected when analyzing electronic properties of graphene. This approximation gives rise to the well-known Dirac cones at the corners of the BZ.

Fig. 1: Lattice model and bulk-edge correspondence for satellite Dirac cones.
Fig. 1: Lattice model and bulk-edge correspondence for satellite Dirac cones.
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a Schematic of the graphene lattice with nearest-neighbor couplings \({t}_{1}\) and TNN couplings \({t}_{3}\). b Evolution of the band structure as \({t}_{3}\) increases. The red dashed box highlights the appearance of satellite Dirac cones with \({t}_{3}=3{t}_{1}\). c Distribution of Dirac nodes (black dots) in reciprocal space, with the colormap denoting \(\theta\). The dashed hexagon indicates the first BZ. d, e Projected band spectrum and the corresponding winding numbers along \({k}_{x}\), respectively. f Evolution of single-band pseudospins \(\hat{{{{\mathbf{n}}}}}\) over a periodic cycle in \({k}_{y}\) (toroidal direction). The left panel corresponds to \({k}_{x}=0.25\) with the winding number \(W=2\), while the right panel corresponds to \({k}_{x}=0.36\) with \(W=1\). g Eigen spectra for \({k}_{x}=0.25\) and \(0.36\), obtained using a strip geometry with 40 unit cells. (h) Distributions of the corresponding zero-energy edge states.

To explore the impact of stronger beyond-neighbor couplings (\({t}_{3} > {t}_{1}\)), we show the evolution of the Dirac points as \({t}_{3}\) increases, as depicted in Fig. 1b. Notably, this evolution induced by TNN hoppings resembles a Lifshitz transition in non-interacting systems31, and, in twist-bilayer graphene near magic angle, similar Dirac cone evolution and Berry curvature distributions can be observed32. Starting from \({t}_{3}=0\), where a regular Dirac cone is located at \(K\), increasing \({t}_{3}\) gradually reduces the gap at \(M\) until it closes at \({t}_{3}={t}_{1}/3\). This gap closure corresponds to the formation of a semi-Dirac cone characterized by an anisotropic Fermi velocity. As \({t}_{3}\) continues to increase, the semi-Dirac cone splits into two Dirac cones along \(K\)-\(K^{\prime}\). Subsequently, when \({t}_{3}={t}_{1}/2\), these Dirac cones merge with the central cone at \(K\), resulting in a quadratic node, which accounts for the low-energy band structure in Bernal-stacked bilayer graphene without trigonal warping. Further increasing \({t}_{3}\), the satellite Dirac cones reappear from the quadratic band crossing along the \(\varGamma K\) directions, resembling the trigonal warping effect. A summary of these Dirac cones with their low-energy Hamiltonians is provided in Supplementary Note 2 with Table S1. Hereafter, we concentrate on the regime of stronger \({t}_{3}\), using \({t}_{3}=3{t}_{1}\) as a representative case.

We investigate the topological property and bulk-edge correspondence associated with the satellite Dirac cones. The eigenstates of Hamiltonian in Eq. (1) take a spinor form, \(\left|{u}_{\pm }({{{\bf{k}}}})\right\rangle={1/\sqrt{2}\left(\pm 1,{e}^{i\theta \left({{{\bf{k}}}}\right)}\right)}^{T}\), where \(\theta\) represents the phase difference between the two sublattices. This phase accumulates over a closed loop around the Dirac cone and yields its Berry phase. Figure 1c shows the distribution of \(\theta ({{{\bf{k}}}})\) in reciprocal space. At each corner of the BZ, there exists a central Dirac cone, acting as a phase singularity for \(\theta ({{{\bf{k}}}})\). Due to the presence of strong TNN couplings, three satellite Dirac cones appear and are trigonally wrapped around the central one with reversed phase vorticity, resulting in a total of eight Dirac cones in the first BZ.

Figure 1d shows the spectra in projected 1D BZ for a ribbon with bearded edges, with spectral color denotes state localization. Localized zero-energy edge states are found threading the projections of Dirac cones in blue-shaded regions. The appearance of these edge states corresponds to the nontrivial values of the winding number \(W({k}_{x})={\left(2\pi i\right)}^{-1}\oint d{k}_{y}{\partial }_{{k}_{y}}[\log ({t}_{1}{\gamma }_{1}+{t}_{3}{\gamma }_{3})]\), as shown in Fig. 1e. Beyond \(W=1\) regions, higher \(W=2\) regions are clearly observed, which originates from a twice change in phase induced by additional satellite Dirac cones. The winding number can also be intuitively viewed by counting the winding number of pseudospin vector as \({k}_{y}\) traverses the BZ. Under a proper gauge choice, one can define a pseudospin vector \(\hat{{{{\bf{n}}}}}({{{\bf{k}}}})\,{{{\equiv }}}\,(\langle {\sigma }_{y}\rangle,-\langle {\sigma }_{x}\rangle )\) and track its evolution, where \(\langle {\sigma }_{s}\rangle\) represents the expectation value of the Pauli matrices for the eigenstate \(\left|{u}_{+}({{{\bf{k}}}})\right\rangle\). For example, as shown in Fig. 1f, at \({k}_{x}=0.25\), the pseudospin vector completes two full twists before returning to its starting point, indicating \(W=2\). In contrast, at \({k}_{x}=0.36\), the pseudospin vector exhibits a single twist, corresponding to \(W=1\). For \({k}_{x}\) with trivial \(W\) values, the pseudospin vector shows no winding at all. Consequently, a doubled number of edge states occur at \({k}_{x}=0.25\) in Fig. 1g, compared to a single pair at \({k}_{x}=0.36\). We further visualize the distributions of these edge states in Fig. 1h, where edge states with larger winding numbers exhibit shorter decay lengths. By extending the coupling range, additional Dirac points and higher winding numbers become attainable, as summarized in Table S2 of Supplementary Note 8, highlighting that long-range couplings provide a simple and systematic approach for engineering topological band degeneracies and manipulating topological invariants33,34.

Acoustic Realizations

To realize satellite Dirac points experimentally, we design an acoustic analog of graphene with enhanced TNN couplings. Figure 2a presents a photograph of our 3D-printed sample, where cylindrical cavities are connected by pipe to form a honeycomb structure with \(14\times 11\) unit cells in the \(x\)-\(y\) plane, with a lattice constant of \(a=25\) mm. Figure 2b shows the schematic of the sample design for realizing stronger acoustic TNN couplings (see Supplementary Note 3 with Fig. S3 for geometric details). All cavities in the structure are identical in height and diameter, ensuring that the unit cell belongs to the point group \({D}_{3}\), which guarantees the band crossings at BZ corners of PC. In the experiment, the system is driven near the frequency of the first longitudinal cavity mode, which features a wave node at the midplane of the cavity (lower-right inset in Fig. 2a). This mode strengthens the coupling for tubes connected at the cavity ends, while reducing the coupling for tubes connected near the midplane. This design facilitates the required TNN couplings for achieving the acoustic satellite Dirac points.

Fig. 2: Probing acoustic satellite Dirac cones.
Fig. 2: Probing acoustic satellite Dirac cones.
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a Photograph of the PC. The upper inset shows a zoomed-in view of the structure, while the lower inset illustrates the first longitudinal cavity mode. b Schematic of the PC design to achieve strong TNN couplings. The labeled parameters are \({h}_{0}=35\,{{\mathrm{mm}}}\), \({r}_{0}=6\,{{\mathrm{mm}}}\), \({r}_{1}={r}_{3}=1.55\,{{\mathrm{mm}}}\) and \(d=3.46\,{{\mathrm{mm}}}\). The dashed hexagon labels a unit cell. c Acoustic dispersions obtained from experiments (colormaps), simulations (orange curves) and the tight-binding model (white dashed lines), respectively. The inset denotes the schematic for BZ. d, e Measured acoustic dispersions (colormaps) along \(K{\varGamma }_{1}\) and \(K{\varGamma }_{2}\), respectively.

The measured bulk dispersion is depicted in Fig. 2c, which matches well with the simulated and analytical results represented by orange solid and white dashed lines. These results confirm the appearance of extra Dirac cone dispersions along the \(\varGamma\)-\(K\) line, arising from the interplay between strong TNN couplings and trigonal crystalline symmetry. Figure 2d, e present the measured acoustic dispersions along \(K\)-\({\varGamma }_{1}\) and \(K\)-\({\varGamma }_{2}\), further confirming the presence of satellite structure in band crossings around frequency \({f}_{{{{\rm{D}}}}}=4.94\) kHz. The measured frequency contours near \(K\) and \(K^{\prime}\) are present in Supplementary Fig. S4. Note that the absence of genuine chiral symmetry in the real system leads to slightly difference for the dispersions above and below \({f}_{{{{\rm{D}}}}}\). Nevertheless, our PC captures the key features and profiles of the satellite Dirac cones with high fidelity. One can further improve the resolution of measured dispersions by using samples with more unit cells.

We further experimentally validate the bulk-edge correspondence in the gapless PC with satellite Dirac cones. Excited by a point-source placed in an edge cavity of the PC, the scanned acoustic pressure field is displayed in Fig. 3a. The measured field decays more rapidly toward the bulk than along the edge, indicating the existence of edge states. The response spectra of the sample in edge and bulk area are given in Fig. S5. Figure 3b shows the projected dispersion as a function of \({k}_{x}\), where the colormaps represent experimental results and the dots correspond to simulated dispersions for a strip with bearded edges. Around the Dirac frequency \({f}_{{{{\rm{D}}}}}\), the bright colormaps align with the simulated edge states along \({k}_{x}\), covering the wavevectors between satellite Dirac point projections, as illustrated in Fig. 3c. Specifically, at \({k}_{x}=0.25\), four edge modes are present, with the simulated acoustic pressure fields shown in Fig. 3d (only the upper boundary is displayed for clarity). Similarly, at \({k}_{x}=0.36\), two edge modes are observed with the corresponding fields shown in Fig. 3e. Simulated results for acoustic edge states with zigzag boundaries are provided in Supplementary Fig. S6. These simulated and experimental results show good agreement, providing support for the satellite Dirac points in our acoustic system.

Fig. 3: Experimental validation of edge states and bulk-edge correspondence for acoustic satellite Dirac cones.
Fig. 3: Experimental validation of edge states and bulk-edge correspondence for acoustic satellite Dirac cones.
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a Measured acoustic pressure field of the PC with a source placed at the middle of the upper boundary. b Measured (colormaps) and simulated (gray dots) acoustic edge state dispersions. c Schematic of the first BZ. The vertical dashed lines in (b) mark the projections of Dirac points, as illustrated in (c), while the vertical solid lines in orange, red, and blue highlight \({k}_{x}=0\), \(0.25\), and \(0.36\), respectively. d, e Simulated acoustic eigenfields at the upper boundary for \({k}_{x}=0.25\) and \(0.36\), respectively. f–h Acoustic bulk dispersions along the colored vertical lines in (c). i–k Measured (dots) and simulated (lines) pseudospin trajectories of the first bands in (f-h).

In experiments, the response signal near the sample boundary generally results from a collective contribution of all edge modes. This complicates a clear experimental identification of the \({k}_{x}\)-dependent bulk-edge correspondence solely based on edge responses, as shown in Fig. 3b. To address this subtlety, we measure the variation of pseudospin \((\langle {\sigma }_{x}\rangle,\langle {\sigma }_{y}\rangle )\) across a \({k}_{y}\) path at three specific \({k}_{x}\), marked by colored solid lines in Fig. 3c, which provides an intuitive representation of the winding numbers that corresponds to the number of edge states, as discussed in the lattice model. Figure 3f–h shows the gapped bulk dispersions along \({k}_{y}\) at three representative \({k}_{x}\) values, which can alternatively be interpreted as the dispersions of reduced 1D subsystems parameterized by different \({k}_{x}\). For each subsystem, the spinor used to construct \(\left\langle {\sigma }_{i}\right\rangle\) is approximated by Fourier transforming the response of a single band from the two sublattices separately (see Methods and Supplementary Note 4 with Fig. S7). The corresponding pseudospin trajectories for the first band over a \({k}_{y}\)-cycle are depicted in Fig. 3i–k. In Fig. 3i, the pseudospin winds around the origin once, corresponding to the presence of the two edge states at \({k}_{x}=0.36\). At \({k}_{x}=0.25\), where four edge states are present, a double winding of the pseudospin spiral is clearly observed in Fig. 3j. Finally, at \({k}_{x}=0\), the pseudospin trajectory oscillates back and forth without forming a winding, as illustrated in Fig. 3k, which accounts for the absence of edge state responses. To show the extensibility of our approach, we further experimentally identified a winding number of 3 in a similar honeycomb PC with strong eighth-nearest-neighbor couplings (see Supplementary Note 8), further confirming that longer-range couplings enable larger winding numbers.

The satellite-valley interface states

We further explore the acoustic valley interface states related to the satellite Dirac cones. A typical sample with a zigzag-type interface is shown in Fig. 4a. The staggered sublattice potential is introduced by varying the cavity heights, \({h}_{B}-{h}_{A}=\pm \delta\) with \(\delta=0.05{h}_{0}\) for the two distinct phases I and II. As such, the satellite Dirac cones split into satellite valleys located around \(K(K^{\prime} )\) with opposite Berry curvatures, resulting in multi-valley insulating phases, as shown in Fig. 4b. Consequently, as shown in Fig. 4c, the valley interface channel holds four guiding modes, each exhibiting two valley polarizations propagating in the same direction (see Supplementary Note 9 for the lattice model results). Distinct from approaches rely on high-order modes35, these interface states arise from satellite valleys induced by long-range couplings, enabling a wider frequency window. The robustness of these valley guiding modes against sharp path bends is demonstrated in Supplementary Note 9.

Fig. 4: Experimental demonstration of satellite-valley interface states.
Fig. 4: Experimental demonstration of satellite-valley interface states.
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a Photograph of the sample, consisting of two phases with different on-site biases \(\delta\). The red dashed line denotes the interface. b Simulated Berry curvature distribution around \(K\) for the two phases. c Measured (colormaps) and simulated (dots) dispersions of guiding modes along the interface shown in (a). d, e Simulated and experimentally measured acoustic pressure field splitting. The red dashed lines denote the interfaces, and the numbers indicate the three output channels. The green dotted lines in (e) outline the region of the interface states, and the red and blue points further highlight the cavities with different \(\delta\). f–i Measured Fourier spectra of the acoustic field for the four channels, with the red dots indicating the positions of valleys.

As is well known, the preservation of valley pseudospin enables valley-locked one-way topological transport, which plays a key role in governing wave behavior in a prototypical four-channel beam splitter. These splitters serve as fundamental building block for functional topological valley-based networks. Note that in the multi-valley cases, the Berry curvature becomes more diffuse, facilitating intervalley transitions. We exploit these transitions to enable transmission through channels otherwise forbidden by valley locking. As illustrated in Fig. 4d, we construct such a beam splitter by combining two distinct phases (see Supplementary Note 5 with Figs. S8 and S9 for the configuration details of the splitter and eigenfields in the channels), with the input channel as shown in Fig. 4a. We then stimulate the acoustic interface state carrying valley index \(q\) at the input channel and examine how it bifurcates at the intersection. Remarkably, within a frequency window of approximately 4.73–4.78 kHz, the acoustic interface state can transport forward into channel 4, as demonstrated by both simulations and experiments shown in Fig. 4d, e. Note that such forward transport is previously heavily suppressed due to the mismatched valley polarization, while here the satellite valleys enable this anomalous forward transport. Outside this frequency window, the system exhibits normal wave-splitting behavior (see Supplementary Note 6 for the channel transmissions at different frequencies). The measured Fourier spectra at 4.75 kHz for the channels are presented in Fig. 4f–i. For a \(q\)-valley polarized input acoustic wave, the output waves from channels 2 and 3 retain the valley polarization as expected (Fig. 4g, h). However, the acoustic wave in channel 4 is converted to be \(q^{\prime}\)-valley polarized (Fig. 4i), breaking the valley conservation. This is probably caused by intervalley scattering, which occurs due to reduced momentum-space separation between the valleys36. (see Supplementary Note 7 with Fig. S13 for simulated results of the splitter with different interface configurations). This phenomenon may add fresh ingredients for valleytronics and valley-based transportation.

Discussion

In conclusion, we have demonstrated the satellite Dirac cones induced by strong TNN couplings. The acoustic satellite Dirac cones are directly observed in PCs through bulk dispersion measurements, and crucially, their wavevector-dependent nontrivial topology is experimentally determined for the first time, providing a valuable supplement to methods of measuring topological invariants37,38. According to bulk-edge correspondence, the satellite Dirac cones bring about a doubling of zero edge states. Moreover, TNN coupling enhanced multi-valley transport is observed by introducing staggered on-site potentials, enabling the formation of an acoustic splitter with anomalous forward propagation39,40,41,42,43,44,45,46, which offers a promising approach for realizing valley inverters47,48,49. Our findings introduce an easy-to-implement mechanism for engineering topological properties, adding fresh avenues for exploring novel topological phases and inspiring further research on beyond-nearest-neighbor couplings in metamaterials50,51. This design could be further scaled down to the on-chip level22,42, with significant potential for robust information telecommunications and particle manipulations44,52,53.

Methods

Experiments

The PC samples were fabricated by 3D printing, using photosensitive resin. The wall thickness of all cavities and connecting tubes was set at 2 mm. Due to the significant impedance mismatch, the walls act as rigid boundaries for sound waves. Small holes (3 mm in diameter) were drilled on top of each cavity to facilitate the excitation and probing of sound signals; these holes were sealed with stoppers when not in use. In experiments, a network analyzer (B&K Type 3560B) was employed to generate and analyze the signals in the frequency domain, with a frequency step of 4 Hz. Sound generation was achieved using loudspeakers (about 2 mm in diameter) inserted into designed cavities, while sound detection was carried out using two acoustic probes. One probe (B&K Type 4187 \(1/4\)-inch microphone) was fixed near the sound source to collect the reference signals, while the other (B&K Type 4182) was manually swept in the sampling area of desired cavities to measure the field distributions (Fig. 4e). Spatial Fourier transformations of the acoustic fields were performed to extract the corresponding dispersions for bulk states (Fig. 2c–e), edge states (Fig. 3b), and interface states (Fig. 4c and Fig. 4f–i).

The experimental method for extracting pseudospin windings in Fig. 3i–k is detailed as follows. A point source was placed in a bulk cavity of the sample shown in Fig. 2a to generate sound waves (see Supplementary Fig. S5 for the measured response spectrum). The acoustic fields in sublattice cavities A and B were separately recorded and subjected to Fourier transformations. From the transformed data, the one-dimensional bulk band versus \({k}_{y}\) was extracted at \({k}_{x}={k}_{{xi}}\) as shown in Fig. 3f–h. The response intensity of a single band at wavevector \({k}_{{yi}}\) and frequency \({f}_{i}\) was approximated by a peak, which was subsequently Fourier transformed to yield extended waves in real space \(y\). Using the transformed data, the field amplitudes at the same \(y\) from the two sublattices were adopted to construct the spinors, thereby obtaining the pseudospin vector at \({k}_{{yi}}\). This procedure is described in Supplementary Note 4 with Fig. S7. This process was repeated for each \({k}_{{yi}}\) and yielded the pseudospin windings shown in Fig. 3i–k.

Simulations

All simulations of PCs were implemented with COMSOL Multiphysics, a finite element solver package. The speed of sound in air was set to \(344\,{{{\rm{m}}}}/{{{\rm{s}}}}\) at room temperature and the air density was \(1.29\,{{\mathrm{kg}}}/{{{{\rm{m}}}}}^{3}\). The solid walls of the sample were treated as rigid boundaries. To obtain the bulk dispersions (orange line in Fig. 2c), periodic boundary conditions were applied to the unit cell in both directions. For the projected bands corresponding to edge states (dotted lines in Fig. 3b), a strip-like supercell consisting of 11 unit cells along the \(y\) direction was used, with solid boundary conditions on the edges and periodic boundary conditions along the \(x\) direction. The calculation of valley interface states in Fig. 4c was carried out in a similar way, with states localized at the external edges filtered out for clarity. To simulate the frequency-domain field distributions shown in Fig. 4d, two identical point sources were placed in two interface cavities at \(x=-10\). Sound damping in simulation was incorporated by assigning an imaginary part of the sound velocity with a value of \(0.5i{{{\rm{m}}}}/{{{\rm{s}}}}\). The height of the point sources was set to \(0.9({h}_{A}-\delta )\) for exciting the acoustic field.