Abstract
Threedimensional (3D) gapless topological phases can be classified by the dimensionality of the band degeneracies, including zerodimensional (0D) nodal points, onedimensional (1D) nodal lines, and twodimensional (2D) nodal surfaces. Both nodal points and nodal lines have been realized recently in photonics and acoustics. However, a nodal surface has never been observed in any classicalwave system. Here, we report on the experimental observation of a twofold symmetryenforced nodal surface in a 3D chiral acoustic crystal. In particular, the demonstrated nodal surface carries a topological charge of 2, constituting the first realization of a higherdimensional topologicallycharged band degeneracy. Using direct acoustic field measurements, we observe the projected nodal surface and its Fermiarclike surface states and demonstrate topologicallyinduced robustness of the surface states against disorders. This discovery of a higherdimensional topologicallycharged band degeneracy paves the way toward further explorations of the physics and applications of new topological semimetal phases.
Introduction
Specially designed structures including metamaterials and artificial crystals in classicalwave systems can possess topological properties, which offer fundamentally new degrees of freedom to manipulate the classical waves in photonics, acoustics, and mechanics^{1,2,3,4,5}. The most representative examples are the photonic/acoustic analogs of topological insulators (generally referred to as “photonic/acoustic topological insulators”) that behave as “insulating” materials in bulk but support chiral or helical states on their edges^{2,6,7,8,9}. An important property of such edge states is that they are robust against backscattering caused by disorders and defects, which is guaranteed by the nontrivial band topology. These photonic/acoustic topological insulators have found a lot of applications, including backscatteringimmune waveguides^{9,10}, robust delay lines^{11}, and topological waveguide splitters^{12,13}. The discovery of the photonic/acoustic topological insulators has soon stimulated broad research interest, giving birth to the emerging fields of topological photonics/acoustics^{1,2,4,5}.
Unlike topological insulators, which have band gaps, topological semimetals are gapless, and their topological properties are characterized by band degeneracies^{14,15}. Such gapless topological phases have also found their classical analogs in photonics and acoustics^{4,5}. Weyl photonic/acoustic crystals^{16,17,18,19,20,21,22}, for example, host twofold zerodimensional (0D) point degeneracies known as Weyl points which are monopoles of quantized Berry flux in threedimensional (3D) momentum space and carry topological charge ±1; when projected to a surface, two oppositely charged Weyl points are connected in momentum space by open surfacestate arcs. As it turns out, Weyl points are only one member of a broader family of topologically nontrivial band degeneracies. Researchers have demonstrated photonic/acoustic crystals with topological nodal lines—band degeneracies lying along onedimensional (1D) curves in momentum space—which are associated with Berry phase winding of π and drumhead surface states^{23,24,25,26}. Recent theoretical studies have also revealed the possibility of higherdimensional topological band degeneracies, such as twofold twodimensional (2D) nodal surfaces^{27,28,29,30,31,32,33,34}. Along the normal of nodal surfaces, the Bloch states possess a pseudospin degree of freedom associated with the band index and have an effective description in terms of massless 1D Dirac particles^{30}. Interestingly, in the acoustic/photonic systems^{29,33}, there are two linear Weyl points pared with the topological nodal surface and the total topological charges of all bands are zero, being consistent with the Nielsen–Ninomiya theorem^{35}. In comparison to the condensed matter systems where the nodal surfaces usually are fragile to spin–orbit interaction effects^{30} or coexist with the other trivial bands, the classicalwave systems are advantageous in realizing the 2D nodal surfaces, as the structure of each unit cell can be precisely tailored and the resulting bandstructure is easily probed at various frequencies, and also because classical waves experience no intrinsic spin–orbit coupling. Despite the great interest in topological nodal surfaces, there has been no experimental observation in any classicalwave system to date^{29,33}.
This work reports on the experimental demonstration of a twofold topological nodal surface in a chiral 3D phononic crystal. The nodal surface is stabilized by a nonsymmorphic lattice symmetry and timereversal symmetry and takes the form of a nodal plane spanning the Brillouin zone (i.e., a torus in momentum space). As described below, the nodal surface carries a topological charge of 2, associated with inversion symmetry breaking in the underlying lattice. To the best of our knowledge, this is the first realization of a nodal surface in classicalwave systems, and also the first realization of a higherdimensional topologically charged band degeneracy beyond the previously discovered topologically charged 0D points including the Weyl points and their variants in all physical systems^{15,29}. The nodal surface exhibits a topological bulkedge correspondence in the form of a pair of open arcs that emanate from the projected nodal surface, connecting to two Weyl cones elsewhere in the projected Brillouin zone. Using acoustic measurements, we directly image the projected nodal surface and its Fermiarclike surface states and demonstrate the robustness of surface states against disorder.
Results
Design of a 3D acoustic crystal with the topological nodal surface
The 3D artificial acoustic crystal, fabricated by 3D printing, is shown in Fig. 1a, b. The 3D hexagonal unit cell, which has three length parameters h, r, and t, is shown in Fig. 1c. Here, the background is solid resin, with three types of hollow air tubes indicated by three different colors. The blue tubes run along the zdirection; the red tubes connect nextnearestneighbor sites on the z = h/2 + n × h planes (where n is an integer), and the green tubes connect nextnearestneighbor sites on the z = n × h planes. Note that the green and red tubes do intersect. Acoustic (sound) waves propagate in the network of air tubes without penetrating the solid resin background. The crystal has the nonsymmorphic space group symmetry P6_{3} (No. 173), which lacks inversion, mirror, or other rotoinversion symmetries; it is a socalled topological chiral crystal^{32,36,37,38,39}. The resulting acoustic crystal has twofold screw rotations along the zdirection S_{2z}: (x, y, z)→(−x, −y, z + 0.5 h), as well as bosonic timereversal symmetry (T^{2} = 1). The combination TS_{2z} satisfies \((TS_{2z})^2 = {\rm{e}}^{  {\rm{i}}k_zh}\), where k_{z} is the wavevector in the zdirection. Along the k_{z} = π/h plane, \((TS_{2z})^2 =  1\). Therefore, twofold Kramers degeneracy applies at every point on this plane, forming a topological nodal surface^{28,29,30}.
The 3D Brillouin zone and the corresponding numerically calculated acoustic band structures are shown in Fig. 1d, e. There is a nodal surface along the k_{z} = π/h plane, shown in the right inset of Fig. 1f. Away from the topological nodal surface, the dispersion is highly linear, as shown by the R–Σ and K–H highsymmetry lines in Fig. 1e. The nodal surface is stabilized by the twofold screw rotation and timereversal symmetries, and its nonzero topological charge is specifically due to the lack of inversion symmetry^{29,32,33}.
To find the topological charge, we compute the Chern numbers of the acoustic Bloch wavefunctions on a surface enclosing the topological nodal surface using the Wilson loop method^{32}. This firstprinciples calculation reveals that the topological charge of the nodal surface is +2. Accompanying the topological nodal surface, there exist a pair of Weyl points at K/K′ (marked as red dots in Fig. 1d, e, and shown in the left inset of Fig. 1f), which both have topological charge −1, in agreement with the NielsenNinomiya theorem which states that the topological charges must sum to zero^{35}. For details of these calculations, refer to Supplementary Note 1. Over the frequency range 6.15–7.9 kHz, indicated by the horizontal red lines in Fig. 1e, Fermiarclike surface states can be observed, as described below. Besides, we should note that in comparison to Xiao et al.’s design^{29}, our design has two advantages that facilitate experiment. Firstly, our design has a very large frequency range where the topological surface states exist (relative bandwidth ~20%), while the design in ref. ^{29} is small (relative bandwidth <2%). Secondly, there are zoriented tubes running through the sample. The acoustic probes then can be inserted into the sample through these tubes, allowing the direct characterization of bandstructure.
Experimental observation of the topological nodal surface and its surfacestate arcs
We perform three experiments to characterize the topological properties of the topological nodal surface (see experimental details in Methods). First, we measure the projected bulk bandstructure by probing the sound pressure along a plane passing through the middle of the sample, as shown in Fig. 2a. Here, both the probe and the source are inserted into the sample via the vertical air tubes indicated in blue in Fig. 1c. The point source is located 4 cm away from the measurement plane, and therefore the sound waves measured along the plane carry both inplane and outofplane momenta. We apply spatial Fourier transforms to the measured complex acoustic field distributions, and hence obtain the projected bulk bandstructure shown in Fig. 2c. For comparison, the numerically calculated projected band diagram is shown in Fig. 2d. As can be seen, there is a good quantitative agreement between experimental and numerical results, and the projections of the nodal surface and Weyl points are clearly visible in both sets of results.
Next, we probe for surface waves at a boundary between the acoustic crystal and acrylic board. The experimental configuration is shown in Fig. 3a. In the results shown in Fig. 3b, and the corresponding numerical results of Fig. 3c, we observe a family of surface states in the 6.0 to 8.0 kHz frequency range (see Fig. 1e). Note that the slight difference between the measured (Fig. 3b) and simulated (Fig. 3c) results is due to the experimental error caused by a limited resolution in momentum space. The numerical isofrequency plots in Fig. 3d, f indicate that there are two surface states that emanate from the projected nodal surface, each connecting to one of the two projected Weyl cones. This is consistent with the aforementioned fact that the nodal surface possesses a nontrivial topological charge of +2. Note that due to the timereversal symmetry preserved in our system, we have employed the inversion symmetry operator to make the measured dispersion more symmetric. The experimental results, shown in Fig. 3e, g, are in good agreement. Besides, due to the limited accuracy of the fabrication and measurement, there are slight differences between the simulated and measured results. As the frequency decreases and the corresponding wavelength increases, the acoustic dispersion becomes more insensitive to the details. Therefore, the agreement between simulation and measured data look better at lower frequencies.
Finally, we probe the robustness of the Fermiarclike surface states. As indicated in Fig. 4a, iron rods of radius 1.5 mm are inserted into the vertical air tubes in a region of the lattice adjacent to the boundary with the acrylic board. The iron rods block the flow of sound within the affected tubes, which are the principal route over which energy can pass between different layers of the structure (see Fig. 1c); hence, they should have a dramatic effect on the acoustic modes. Surface waves are then launched from one side of the disorder region, and the sound pressure is measured along the y = 0 mm plane. The results in Fig. 4b clearly show the surface waves passing through the disorder region. A spatial Fourier transform is applied to these results, producing the momentum space plot shown in Fig. 4c. This reveals that the acoustic waves are predominantly concentrated in the forward direction, with negligible backreflection from the disorder region. It can be well explained as follows: in our experiment, the excitation is optimized and almost only the openarc surface states with kz located in [0.1π/pz, 0.75π/pz] are excited at 6.45 kHz. In this range, one can view each kx, ky plane for a fixed kz (preserved as a result of translational symmetry along the z axis) as a 2D Chern insulator^{15} (see Fig. 3d, e). Therefore, the backreflection is strongly suppressed.
Discussion
We have experimentally observed a twofold topological nodal surface stabilized by the symmetries of a 3D nonsymmorphic acoustic crystal. Our results clearly demonstrate that the nodal surface carries a topological charge of 2, and therefore constitutes the first realization of topologically charged band degeneracy beyond the previously realized Weyl points and their variants. We observe topological bulkedge correspondence in the form of paired Fermiarclike surface states, which connect the topological nodal surface to two different Weyl points. The present acoustic crystal provides an excellent platform for studying the topological nodal surface phenomenon since the surfacestate arcs exist over a relatively broad (~20%) frequency range. In future work, similar acoustic crystals may be used to explore other types of band degeneracies. It would be interesting to perform a detailed study of the topological nodal surface states, due to the possibility of realizing exotic oneway chiral zero modes that cannot be achieved with standard Weyl points^{40,41}.
Methods
Numerical methods
All simulations are performed in the acoustic module of commercial finite element method software COMSOL Multiphysics. Due to the large impedance difference between air and the photosensitive resin the surfaces of the resin region are simulated as hard boundaries. Besides, due to the limited sound pressure in our experiment, the nonlinear effects are negligible, as manifested in previous works^{9,19,21}. The air density is 1.18 kg m^{−3} and the speed of sound is 343 m s^{−1}. To calculate the bulk dispersion of the unit cell, periodic boundary conditions are applied in all directions. To calculate the surface wave modes, periodic boundary conditions are applied in the x and z directions, with a hard boundary in the ydirection. The simulations use only 10 unit cells in the ydirection, which is sufficient since the surface waves are extremely well confined.
Sample fabrication
The sample is manufactured via an additive manufacturing technique (stereolithography). The material is photosensitive resin with modulus 2880 MPa and density 1.10 g cm^{−3}.
Experimental setup
In the surfacestate measurement, a square acrylic plate (length of 500 mm), whose surfaces act as hard boundaries, covers the right side of the sample. A balanced armature speaker (radius of about 1 mm) driven by a power amplifier, working as a broadband sound source, is placed at the center of the interface between the acrylic plate and the sample. For the bulk state measurement, the broadband sound signal is launched from a narrow tube (radius of about 1.5 mm and length of about 200 mm), which penetrates into the sample from sample’s bottom. The distance between the tube and the right surface is about 100 mm.
To measure acoustic pressure fields, two microphones (radius of about 3.2 mm, Brüel&Kjær Type 4961) are separately placed in a sealed sleeve with a tube (radius of 1 mm and length of 350 mm) that penetrates deep into the sample During each measurement, microphone 1 is scanned pointbypoint through one of the vertical air holes of the structure, to detect the input acoustic signal. Microphone 2 is fixed towards the sound source to detect the reference acoustic signal. The scanning steps in the x and z directions are 20 mm and 15 mm, respectively. The distances between the measured plane and the right surface of the sample are about 5 mm and 140 mm for the surface and bulk states, respectively. Both the amplitude and phase of the acoustic pressure field are recorded by Brüel&Kjær 3160A022 module. The bulk and surfacestate dispersions are obtained by applying Fourier transform to the measured complex acoustic pressure fields.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.
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Acknowledgements
This work was sponsored by the Singapore Ministry of Education under Grants No. MOE2018T2–1–022 (S), MOE2015T2–2–008, MOE2017T22108, MOE2016T3–1–006, and Tier 1 RG174/16 (S). H.S. acknowledges the support of the National Natural Science Foundation of China (11774137). S.Y. acknowledges the support of the National Natural Science Foundation of China (51779107).
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All authors contributed extensively to this work. Y.Y. created the design and performed the simulations. Y.Y. and H.S. fabricated the sample and designed the experiments. H.S., J.X., Y.G., and D.J. performed measurements. Y.Y., S.Y.Y., S.Q.Y., Y.C., and B.Z. analyzed data and wrote the paper. Y.C. and B.Z. supervised the project.
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Yang, Y., Xia, Jp., Sun, Hx. et al. Observation of a topological nodal surface and its surfacestate arcs in an artificial acoustic crystal. Nat Commun 10, 5185 (2019). https://doi.org/10.1038/s41467019132583
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