Introduction

The band degeneracy is a fundamental phenomenon at the core of understanding topological properties in condensed matter physics1. The degenerate points appearing therein can be viewed as massless quasi-particles, establishing a bridge with cutting-edge researches in high-energy physics2,3. In 2D and 3D systems, Dirac points (DPs) and Weyl points (WPs) respectively exhibit linear crossing dispersions in all directions, making them ideal platforms to study Dirac and Weyl quasi-particles under an applied magnetic field4,5,6,7. Unlike the DPs which can split into discrete flat Landau levels (LLs) under the magnetic field8,9 (Fig. 1a, b), the paired WPs split into chiral LLs (Fig. 1c, d), with the zeroth LL exhibiting linear dispersion along the magnetic field direction10,11. The opposite slopes, i.e., the different chiralities, are caused by the opposite topological charges of WPs.

Fig. 1: Landau levels in different systems.
figure 1

a, b Schematics of a Dirac system (a), with the gray dot symbolizing the Dirac point (DP) and the flat achiral Landau levels (LLs) induced by the external magnetic field (b). c, d Schematics of a Weyl system (c), where the dots represent a pair of Weyl points (WPs) with opposite topological charges, and the chiral LLs induced by the external magnetic field (d). The red (blue) curves indicate the zeroth chiral LLs under opposite directions of the magnetic field.

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Starting by some pioneering theoretical works12,13, classical wave systems have gained prominent attentions for showcasing phenomena analogous to those in condensed matter physics. These systems allow for precise structural design and advanced measurement techniques, enabling the exploration of topological effects with feasible control14,15,16,17,18,19,20,21,22,23,24,25,26,27,28. To generate LLs in 2D classical wave systems, the method of constructing out-of-plane pseudo magnetic fields (PMFs) has been applied to Dirac systems, advancing the potential for controlled topological effects in these platforms29,30,31,32,33,34,35. Similar approaches have also enabled the realization of chiral LLs in Weyl systems36,37. However, these studies rely on complex 3D crystal structures that pose challenges for practical applications. Note that a recent study has introduced in-plane PMFs in sonic crystals to induce chiral LLs, highlighting an alternative pathway for realizing these phenomena in more accessible 2D setups38. The effectiveness of out-of-plane PMFs in 2D acoustic Weyl systems still remains a compelling and unresolved topic, warranting further investigation. Moreover, the precise role of chirality in Weyl systems remains unexplored through direct experimental observation, particularly for the chiral LLs.

In this work, we construct a 2D sonic crystal consisting of cylindrical scatterers arranged in a triangular lattice, treating it as a 3D crystal by introducing a synthetic dimension based on the rotational angles of the units. Thus, synthetic WPs can be obtained in the parameter space. Next, we generate an out-of-plane PMF by applying a gradient deformation, which shifts the WPs linearly in the Brillouin zone (BZ) without breaking time-reversal symmetry, distinguishing it from a real magnetic field39. Under this PMF, we experimentally observe the zeroth chiral LLs with opposite linear dispersions, corresponding to WPs with opposite topological charges, as well as higher LLs displaying parabolic dispersions. Additionally, the distinctive sub-lattice polarization of the zeroth chiral LLs is experimentally detected as well, further confirming the distinct topological properties of the system.

Results and discussion

Synthetic WPs

We begin with a 2D sonic crystal embedded in air, arranged in a triangular lattice with a lattice constant of a = 21.7 mm. Each primitive cell contains three rigid cylindrical scatterers positioned in an equilateral triangle (Fig. 2a) with a diameter of 2r = 6.0 mm and a center-to-center distance of d0 = 6.9 mm. When these three scatterers are treated as a single unit, their symmetry can be modulated between C3v and C3 by adjusting the orientation angle θ. In Fig. 2b, we plot the frequency shifts of the first two valley states at the K point in the 1st BZ as θ varies. From the perspective of valley band physics, the effective Hamiltonian can be derived using the k ⋅ p method as15

$$\delta {H}_{{{\mbox{K/K}}}^{{\prime} }}(\theta )={v}_{{{{\rm{D}}}}}\delta {k}_{x}{\sigma }_{x}\pm {v}_{{{{\rm{D}}}}}\delta {k}_{y}{\sigma }_{y}+m(\theta ){\sigma }_{z},$$
(1)

where vD is the Dirac velocity, σx,y,z are Pauli matrices, and δkx and δky represent the x and y components of the Bloch wave vector measuring the momentum deviation from \({{{\rm{K}}}}/{{{{\rm{K}}}}}^{{\prime} }\) points. For simplicity, we denote δkx(y) as kx(y) in the following discussion. The term m(θ), pertaining to the gap size with Δω = 2∣m∣, indicates an effective mass introduced by the local parity inversion symmetry disruption. The variation of the effective mass term Δm with the rotation angle θ is further described in Fig. 2b. The system possesses a periodicity of θ = 2π/3, and band degeneracies occur at θ = π/6 + nπ/3 with n for the integer. This periodicity allows us to treat θ as an extra synthetic dimension, kz, transforming the 2D BZ into a synthesized 3D BZ as shown in Fig. 2c. The band structure calculated along the path in the synthesized 3D BZ (Fig. 2d) shows linear dispersions around the K/H points in all three directions. This characteristic confirms that the degenerate points at the zone corners can be viewed as WPs in this synthetic system, and the Hamiltonian can be reformulated as an anisotropic Weyl equation:

$${H}_{{{\mbox{K/K}}}^{{\prime} }}^{{\mbox{W}}\,}={v}_{x}{k}_{x}{\sigma }_{x}+{v}_{y}{k}_{y}{\sigma }_{y}+{v}_{z}{k}_{z}{\sigma }_{z},$$
(2)

where vz = m(θ)/Δθ and Δθ = kz represents the deviation of θ from  ± π/6. The WPs located at the K and \({{{{\rm{K}}}}}^{{\prime} }\) point host different topological charges identified by sgn(vxvyvz). Their positions in the first BZ are illustrated in Fig. 2c, with topological charges of  +1 (red) and  −1 (blue) indicated by spheres.

Fig. 2: Synthetic acoustic Weyl semimetal.
figure 2

a Illustration of the primitive cell of a 2D triangular sonic crystal composed of cylindrical scatterers (blue) in an air matrix (gray). b Phase diagram for the effective mass term Δm and the band gap width while scanning the synthetic degree of freedom θ. c The synthetic 3D Brillouin zone (BZ), in which the red and blue dots denote the synthetic Weyl points (WPs) with topological charges of  +1 and  −1, respectively. d Calculated band diagram of the synthetic acoustic weyl semimetal. Blue shadowed regions represent the scanning paths traversing the synthetic axis kz(θ).

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Deformation-induced deviation of WPs

In order to observe the chiral LLs, an analog or pseudo- magnetic field must be introduced in the 2D acoustic system. We achieve this by applying a gradient deformation to the scatterers in the finite sonic crystal. For instance, with θ = π/2 (Fig. 3a), the distance between two pillars can be modulated by a deformation parameter, η, while keeping the third pillar fixed. Due to spatial constraints, the focus is on cases where η ≥ 0. The bottom panel of Fig. 3b shows the projected band diagram for the first band with η = 0, where the synthetic WPs emerge exactly at BZ corners (\({{{\rm{K}}}}/{{{{\rm{K}}}}}^{{\prime} }\) points). Once η deviates from 0, the WPs shift away from these high-symmetric points, as shown in the top panel of Fig. 3b for η = 0.4. The relationship between the introduced deformation η and the displacement of WPs is carefully studied in Fig. 3c, which demonstrates a linear slope fitted by the blue line.

Fig. 3: Deformation-induced acoustic pseudo-magnetic field (PMF).
figure 3

a Structural deformation is introduced through modulating the interval distance between two cylinders as characterized by η. b The projected band diagrams of the first band with η = 0 (bottom) and η = 0.4 (top). Right panel shows the zoom-in view of the region near the Brillouin zone (BZ) corner. Blue and red dots represent the positions of the Weyl points (WPs). c Variations of the WPs' positions along kx while tuning the deformation η. Dashed curves indicate the first two bands with the red dots for the WPs. The shift Δk exhibits good linearity with respect to η while the blue line being the fitted line. d Dispersion relations evolve with θ at point W when η = 0.4. e WPs shifted in the synthesized 3D BZ. f Band diagram calculated along the path labeled by the green frame in (e). Dashed lines indicate the positions of the WPs which are shifted from the high-symmetric points.

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We then examine the WPs along the synthetic θ axis. With η = 0.4, positioning the WP at kx = 1.4π/a, we plot the band-edge frequencies along the θ axis (Fig. 3d). Two notes should be emphasized. (i) A pair of WPs exists, maintaining linear dispersion even without the C3 symmetry. (ii) The period of synthetic θ extends to 2π rather than 2π/3, with WPs located at  ±π/2. Scanning along the path in the 3D BZ shown in Fig. 3e, we obtain the band structure in Fig. 3f, which clearly demonstrates the WPs shifting from the K/H to the W1/W2 points.

Acoustic PMFs

To build a PMF in the proposed synthetic acoustic system, we introduce the gradient distributions of WPs according to certain rules, which could be regarded as a pseudo-vector potential, denoted as A. This potential functions as an effective canonical momentum, modifying the wave vector to k + A. In practice, we vary the deformation parameter η from 0.4 to 0 along the y axis, as shown in Fig. 4a. This gradient in η induces a corresponding shift in WPs along the kx direction, establishing the PMF. By linearly decreasing η along the y direction (Fig. 4b), the WPs shift linearly from their original positions, as illustrated in Fig. 4c. In this case, the pseudo-vector potential is expressed as A = (By, 0, 0). The resulting PMF, given by B = ∇ × A, is orthogonal to both kx and ky directions. Without loss of generality, we assume this PMF to be parallel to the synthetic out-of-plane kz axis (i.e., θ). Although our discussion primarily focuses on the PMF induced by WPs near the K point, the same analysis can apply to other WPs. Due to the time-reversal symmetry, the PMFs at the \({{{{\rm{K}}}}}^{{\prime} }\)/\({{{{\rm{H}}}}}^{{\prime} }\) point are oriented in opposite directions. Note that the PMF we create through introducing structural deformations of scatterers still exist even without the introduction of the synthetic degree of freedom, leading to the flat zeroth LLs.

Fig. 4: Experimental observation of the chiral Landau levels (LLs).
figure 4

a Photo of a fabricated 2D sonic crystal composed of 20a × 20a units, where the pseudo-magnetic field (PMF) is engineered by the gradient deformations along the y direction. Right-top inset shows the structure designed for small angle adjustment of θ, which consists of a nonagonal base and a octagonal disk with the structural parameters provided in Supplementary Note 3. b Distributions of the deformations η with respect to the y direction. Solid line shows the theoretical result and the brown triangles represent the values selected to design the structure. c The corresponding positions' shift Δk of the Weyl points (WPs) at different places. d, e Experimentally measured projected dispersions of the chiral LLs near the WPs W1 (d) and W2 (e). The curves denote the simulated results. f, g Frequency spectra for the total sound intensity measured at all sub-lattices p (blue) and q (red) while excited by a point source placed at the position q (f) and p (g) near the sample center.

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We demonstrate that applying the synthetic gauge field along kz could lead to the quantization of energy levels as

$${\epsilon }_{n}=\left\{\begin{array}{c}{{{\rm{sgn}}}}(n)\sqrt{{v}_{z}^{2}{k}_{z}^{2}+2| n| B{v}_{D}^{2}},\quad n\,\ne\, 0\\ \chi | {v}_{z}| {{{\rm{sgn}}}}(B){k}_{z}.\quad n=0\end{array}\right.$$
(3)

Here \(\chi ={{{\rm{sgn}}}}({v}_{x}{v}_{y}{v}_{z})\) is the topological charge of the WPs, and B = κμ, where μ = Δη/Δy represents the decaying rate of the deformation magnitude η along the y axis. Detailed derivations can be found in Supplementary Note 1. It can be deduced from the above expression that, under the raised PMF in the synthetic 3D Weyl system, the WPs with different topological charges will split into zeroth chiral LLs with opposite group velocities, as depicted in the schematic of Fig. 1d. Meanwhile, other orders of LLs exhibit hyperbolic dispersions. We demonstrate that the chiral dispersion curves of the zeroth LLs can be observed and fully understood along the synthetic kz direction for a fixed kx under the proposed out-of-plane PMF. This chiral dispersion relation is one of the key phenomena for the chiral LLs we are committed to observing in our following experiments.

Observation of synthetic chiral Landau levels

To experimentally observe the zeroth chiral LLs within synthetic dimensions, we construct a finite sonic crystal consisting of 20 × 20 units. This crystal has a deformation gradient η oriented along y-axis, following the configuration in Fig. 4b, with an initially fixed rotational angle for all scatterers. By modulating the deformation gradient, the PMF strength can be adjusted as desired, which is carefully discussed in Supplementary Note 2. These scatterers are sandwiched between two plates without any intervals, which can be thus regarded as a 2D system. Along the x-axis, we position an array of microphones to capture the amplitude and phase of the sound field, excited by a point source–implemented via a microspeaker placed inside the crystal. By applying a Fourier transform, we extract the dispersion relation between frequency and kx at this specific rotational angle. In order to track the topological phenomenon along the synthetic dimension with high resolution, we need to vary the rotational angle θ incrementally. Thus, achieving precise control over these small rotations is essential, and to this end, we design a multilayer inlay structure for each unit (see the inset of Fig. 4a). The precise control on the 5° rotation on the unit cell can be implemented through the difference of the rotation angles between the octagon disk and the nonagonal base (see Supplementary Note 3 for details). Subsequently, by gradually adjusting θ across all scatterers and repeating measurements, we obtain an experimental visualization of the dispersion relation along the kz (θ) direction, as shown in the color maps of Fig. 4d, e. These maps illustrate the chiral LLs around the W1 and W2 points, respectively, at kx = 1.36π/a. The close alignment between our experimental data and simulations confirms the distinct chiralities of the zeroth LLs associated with WPs of opposite topological charge. Additional results for varying kx are provided in Supplementary Note 4, and the discussions on the factors affecting the measurement accuracy can be found in Supplementary Note 5.

Besides, we are committed to observing another significant characteristic of the zeroth chiral LLs, which is the sub-lattice polarization. We demonstrate that under the current PMF, the relation of sgn(vxvyB) = 1 always maintains for the WPs, leading to the eigenvector of the zeroth LLs being polarized at one of the two sub-lattices, q40, as indicated by the red dot in the inset of Fig. 4f. Note that reversing the PMF–by applying the deformation gradient η in the opposite direction–would shift this polarization to the other sub-lattice, p. To experimentally identify the property of sub-lattice polarization, we position a point source at site q in the center of the sample (θ = −π/2) and measure the sound intensity at each sub-lattice p and q. The summarized total intensity ∑i=p, qIi calculated from the experimental data is shown in Fig. 4f. We can observe three distinct peaks at 8.4 kHz, 9.0 kHz, and 9.7 kHz from the measurements at sub-lattice q (red curve), which correspond to LLs of n = −1, 0 and 1, respectively. The zeroth LL, however, is not localized at sub-lattice p (blue curve), indicating strong sub-lattice polarization of the zeroth LL. For further validation, we reposition the point source at site p in the unit cell center and repeat the measurements. As shown in Fig. 4g, the zeroth LL remains absent, reinforcing that the chiral LL in this structure is exclusively polarized at sub-lattice q. We emphasize that the observed sub-lattice phenomenon is another important chirality-related effect of chiral LLs, besides the opposite group velocities with different chiralities.

Conclusions

In conclusion, we have successfully constructed a synthetic 3D Weyl acoustic system by introducing the orientation angle θ as an additional dimension within a triangular lattice composed of scatterer units, each featuring three pillars. By applying an out-of-plane PMF through gradual deformation along the y-axis, we observe the emergence of LLs, including the zeroth chiral LLs and their distinctive sub-lattice polarization. The experimental results align well with both simulation and theoretical derivations. This design methodology, leveraging the out-of-plane PMFs and synthetic dimensions, offers versatility and precision, making it well-suited for investigating further unexplored phenomena in Weyl physics. Potential applications include studies of topological phase transitions in 5D Weyl systems41,42,43,44, defect-induced Weyl phenomena45, and the 3D quantum Hall effect46,47. Moreover, the exotic characteristics of zeroth chiral LLs, such as sub-lattice polarization, open up possibilities for manipulating acoustic waves.

Methods

Numerical simulations

The numerical simulations in this study were conducted using the commercial finite-element software COMSOL Multiphysics. In the simulations, the density for the background medium air was set to be 1.15 kg/m3, and the sound speed in air was set to be 351.87 m/s. Due to the significant acoustic impedance mismatch between air and the 3D-printed epoxy resin cylinders, the cylinder boundaries were modeled with hard-wall boundary conditions. For band structure calculations, Floquet periodic boundary conditions were applied at the edges of the periodic unit cells or strips. In pressure-field calculations, the perfect matching boundary conditions were implemented along all the edges to eliminate interference from reflected waves.

Sample design and experimental measurements

The acoustic lattice was fabricated using epoxy resin via 3D printing. Each unite cell comprised a three-pillar scatterer with a nonagon base, which can be inserted into an octagon disk with a nonagon groove. Thus, a 5° adjustment could be realized by combining a 40° rotation of the nonagon base with a 45° rotation of the octagonal disk (see Supplementary Note 3 for details). The complete sample consisted of 20 × 20 unit cells, enclosed between two plexiglass plates. The lower plexiglass plate is machined with regular-octagonal grooves to secure scattering columns. And the upper plexiglass plate incorporates precisely designed apertures for mounting microphones or speakers, accompanied by corresponding sealing plugs to seal unused apertures and prevent acoustic leakage. For acoustic excitation, a microspeaker mounted on a large plug served as a point sound source. The acoustic signals were generated by an arbitrary waveform generator (NI PXI-5421), then amplified using an audio power amplifier. Local pressure fields were measured by inserting a 1/4-inch condenser microphone (GRAS 40PH) into the top plate at designated positions. The outputs of the microphones were acquired by a digitizer (NI PXI-4499) and processed using LabVIEW program. A photograph of the experimental setup is provided in Supplementary Note 6. To determine the dispersion, we measured the sound pressure field along x-axis using an array of 16 microphones, scanning frequencies from 7.9 kHz to 10.4 kHz, and performed spatial Fourier analysis. For sub-lattice polarization measurements, the microspeaker was placed at the p and q positions sequentially, and the responses at each p and q positions were recorded by moving the microphone array along the y-axis.