Abstract
Topological valley states at the domain wall between two artificial crystals with opposite valley Chern numbers offer a feasible way to realize robust wave transport since only broken spatial symmetry is required. In addition to the valley, spin and crystal dimension are two other important degrees of freedom, particularly in realizing spinrelated topological phenomena. Here we experimentally demonstrate that it is possible to construct twodimensional acoustic topological pseudospinvalley coupled saddle surface states, designed from glide symmetry in a threedimensional system. By taking advantage of such twodimensional surface states, a full set of acoustic pseudospins can be realized, exhibiting pseudospinvalley dependent transport. Furthermore, due to the hyperbolic character of the dispersion of saddle surface states, multidirectional anisotropic controllable robust sound transport with little backscattering is observed. Our findings may open research frontiers for acoustic pseudospins and provide a satisfactory platform for exploring unique acoustic topological properties in threedimensional structures.
Introduction
The discovery of topological phases of matter has renewed our understanding of condensed matter physics over the past few decades^{1,2} and has inspired studies of classical bosonic systems such as photonics^{3,4,5,6,7,8,9,10,11,12} and phononics^{13,14,15,16,17,18,19,20}. Without considering the difference in spin between fermions (halfinteger spin) and bosons (integer spin), their wavefunctions share a similar form associated with a similar topology. This condition gives rise to the search for photonic/phononic analogues of quantum Hall effect with broken timereversal (TR) symmetry^{3,4,14}, topological valley states with a broken mirror or inversion symmetry^{10,11,12,18,19,20}, Floquet topological states due to temporal (or spatial) modulation^{21,22,23} and Weyl semimetals with chiral structures^{24,25}. However, regarding spinrelated topological phenomena, degenerate polarizations or Bloch states must be introduced to construct pseudospins (with pseudoTR squares to −1)^{26}. Therefore, counterparts of the twodimensional (2D) quantum spin Hall effect^{8,15} and of threedimensional (3D) topological states^{27,28} for photons/phonons can be designed in principle as electrons in electronic systems by using pseudoTR instead of natural TR.
For airborne sound such as a spinless wave, an additional degree of freedom (DOF) such as crystal symmetry needs to be considered to construct acoustic pseudospins^{29}. Here, we resort to 3D artificial acoustic systems, which provide more flexible platforms to search for this kind of pseudospin among all 230 types of space groups and provide a predesignable artificial unit structure. Typically, the topological nature is manifested in its character in lower dimension, e.g. 2D systems can possess topological protected onedimensional (1D) edge or zerodimensional (0D) corner states^{30}, therefore, the 3D models can provide rich topologically protected 2D surface states beyond 1D and 0D lowering from 2D systems. In addition, a typical artificial structure has no more than 10^{4} artificial atoms as a whole, which enable us to accurately manipulate every atom and to deliberately introduce defects, take measurements without limitation of the Fermi level^{31} and to create arbitrary interfaces^{32}.
In particular, due to the lack of strong spin–orbit coupling and efficiently TRbreaking method for airborne sound, the valley DOF can provide a convenient way to realize acoustic topological states since only broken mirror (or inversion) spatial symmetry is required. The degenerate point of band structures, such as Dirac degeneracy in 2D case, can be lifted to form the K (K′) valley in the momentum space associated with nontrivial Berry curvature. Although the summation of Berry curvatures over the whole Brillouin zone (BZ) is trivial in TR symmetric cases, robust edge transports still exist along some particular directions.
In this article, we focus on 3D valley acoustic crystals with pseudospinrelated topological phenomena. By elaborately designing the symmetries of 3D lattices, the fourfold degenerate point of bulk band structures is lifted to twofold degenerate valleys, hosting a pair of acoustic pseudospins. The 2D acoustic topological pseudospinvalley coupled saddle surface states and the corresponding anisotropic robust sound transport with little backscattering can be observed in our experiments.
Results
Crystal structure and the bulk band structure
Our 3D periodical crystal structure is composed of stacked doublelayer honeycomb lattices along the z axis, containing two kinds of acoustic atoms in each layer (Fig. 1a). Two adjacent layers show glide symmetry that is the combination of reflection symmetry (xz plane) and a translation by half a lattice constant h (along the z axis). This crystal structure belongs to the nonsymmorphic space group No. 194 (P6_{3}/mmc). The acoustic atom is a triangular prism cavity with five tubes connecting the nearest neighbours (Fig. 1b). For simplicity, different sidelengths (l_{g} and l_{r}) of the triangular prisms represent different kinds of acoustic atoms with the other parameters fixed (lattice constant a = 8.7 mm, height h = 0.54a, height of the prism h_{p} = a/3, radius of the tube r = 0.13a). Onehalf of the first BZ is shown in Fig. 1c.
To show the evolution of the band structure, we start from what is likely the simplest 3D structure: a stacked monolayer graphene structure (graphite). The primary unit cell (inset) with all its identical atoms and band structures is shown in Fig. 1d. The side length of the prism is l_{g} = l_{r} = 0.7a. The bulk band projected onto the k_{xy} and k_{yz} momentum planes at the frequency near the centre of the K–H direction is shown in the lower panel. Due to the D_{6h} symmetry, the k_{xy} projected bulk band is a Dirac cone, where the Dirac point is a twofold nodal line along the K–H direction. However, just twofold degenerate states are not enough to construct acoustic pseudospins. The key point is to increase the DOF to form fourfold degeneracy^{26}. An efficient method is the BZ folding approach. As shown in Fig. 1e, we choose a doublesized unit cell (associated with onehalf of the BZ). Then, the first two bands are folded into four bands with fourfold degenerate states at the H point. The k_{xy} projected bulk bands become doubly Dirac cones. Due to the unbroken TR symmetry, the states at the H′ point are also fourfold degenerate. Then, we introduce a sublattice structure by choosing two different acoustic atoms (l_{g} = 0.6a and l_{r} = 0.8a) with glide symmetry (Fig. 1f). In this case, a complete band gap is created. More importantly, the fourfold degeneracy is split into two twofold degeneracies, which can be used to form acoustic pseudospins. The glide symmetry here can be described as G:(x, y, z) → (x, −y, z+h). Then, G^{2}:(x, y, z) → (x, y, z+2h). Due to the Bloch theorem, the Bloch phase function under lattice translation can be described as \(e^{  i{\boldsymbol{k}} \cdot {\boldsymbol{r}}}\). At the H point (k_{z} = π/2 h), G = \(e^{  i\pi /2}\) (G^{2} = \(e^{  i\pi }\)). The pseudoTR written as GK, squares to −1 (the complex conjugation K represents the TR of sound)^{26,27}, ensuring that the completely twofold degenerate Bloch states on the k_{z} = π/2h plane form acoustic pseudospins (lower panel of Fig. 1f).
The pseudospinvalley Chern numbers
Our model is based on a fourband model, which can be treated as a kind of acoustic topological pseudospinvalley states. As shown in Fig. 1e, the fourfold degenerate states at the H (H′) point are doubly Dirac cones projecting onto the k_{xy} plane. After introducing the sublattice structure (glide), the mirror symmetry respective to the xzplane is broken, with the fourfold degeneracy splitting into two twofold degenerate acoustic pseudospin± associated with nonzero Berry curvatures. The valley Chern numbers at the H point for the lower two bands (acoustic pseudospin±) (Fig. 1f) are halfinteger with opposite signs \({\mathrm{C}}_{\mathrm{H}}^ \pm =\pm 1/2\), while \({\mathrm{C}}_{{\mathrm{H}}^\prime }^ \pm = \mp 1/2\) at the H′ point. Thus, the acoustic pseudospinvalley Chern numbers^{33} can be described as \({\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}} = ({\mathrm{C}}_{\mathrm{H}}^ +  {\mathrm{C}}_{{\mathrm{H}}^\prime }^ +  {\mathrm{C}}_{\mathrm{H}}^  + {\mathrm{C}}_{{\mathrm{H}}^\prime }^  )/2\) = +1 (see Supplementary Note 1). It should be noticed that unlike recently realized topological spinvalleylocked states in 2D photonic systems^{10} [where the spins are two intrinsic electromagnetic polarizations with the same signs for the valley Chern numbers at the H(H′) point (associated with \({\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}} = 0\))], our acoustic pseudospins are artificially constructed via crystal symmetry with nontrivial \({\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}}\).
Acoustic pseudospins for topological saddle surface states
To study the 2D topological pseudospinvalley surface states, we introduce a domain wall (zigzag) between two opposite domains on the xzplane. Figure 2a, b shows onehalf of the surface BZ projected onto the k_{xz} plane and a schematic for the interface. Notably, there are two pairs of topological surface states with surface nodal lines, constructed by two opposite saddle surfaces accidentally touching at their saddle points (Fig. 2c, d). Here, we choose four points (marked in Fig. 2c) near the centre of the BZ to investigate the acoustic pseudospins. The acoustic fields are constructed by a symmetric (S) real part and antisymmetric (A) imaginary part of the acoustic fields, forming S ± iA states (Fig. 2e). Then, acoustic pseudospin± can be defined as \(\nabla \times\) (S ± iA). More importantly, the A state has two independent components: antisymmetric respective to the yzplane (noted as A_{x}) and antisymmetric respective to the xzplane (noted as A_{z}). This result can be attributed to the existence of two mirror symmetric planes perpendicular to the domain wall. Consequently, a full set of acoustic pseudospins on the Bloch sphere (linear, circular or elliptical) can be constructed as shown in Fig. 2f, while in 2D systems, the pseudospins are limited in a 1D space^{7,15}. It should be noticed that such a zigzag domain wall has a pseudospinvalley Chern number difference: \(\Delta {\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}} = 2\) corresponding to two pairs of topological surface states (Supplementary Figure 3). We can also design a \(\Delta {\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}} = 1\) domain wall with only a pair of acoustic pseudospins (see Supplementary Figure 6). It is also worth noting that these acoustic surface states are gapped at the k_{xz} (armchair) or k_{xy} interfaces, because the H and H′ points are projected onto the same point of the surface BZ; thus, the valley Chern numbers with opposite signs will be cancelled out (see Supplementary Figure 4).
Anisotropic robust sound transport
In our experiment, we choose two orthogonal directions to show multidirectional robust acoustic pseudospinvalley transport (Supplementary Figure 1). Figure 3a shows the measured transmission spectra (10 periods) for surface states along the \({\tilde{\mathrm \Gamma }}\)–\({\tilde{\mathrm X}}\) direction with both straight (red line) and zshape (blue line) waveguides. In contrast, the measured transmission spectra for bulk states are shown by the black line, with a relative band gap width of over 15%. Following the incidence of the pseudospin+ acoustic wave, the transmissions maintain a very high transmission value in the bulk band gap frequency region for both straight and zshape waveguides, indicating a strongly suppressed backscattering property (Supplementary Figure 2). The overall transmission of the saddle surface states is 20 dB larger than that of the bulk within the bulk band gap, representing the surface’s gapless behaviour, except for a transmission dip near 19.678 kHz according to the quadratic saddle point. Near the saddle point, the extremely flat dispersion results in the rapidly enhanced state intensity of sound; thus, the transmission is intensely attenuated even at a very low loss. The simulation results for the pressure field at frequencies of 19 kHz (in bulk band gap) and 19.678 kHz (near the saddle point) are shown in Fig. 3b. In simulations, the loss is introduced by adding an imaginary part (10^{−3}) for the sound speed. On the other hand, even without the nonlinear effect of loss, the transmission near the saddle point is still sharply decreased because of the diminishing group velocity down to zero. A similar robust transmission along the \({\tilde{\mathrm \Gamma }}\)–\({\tilde{\mathrm Z}}\) direction is experimentally observed (Fig. 3c), matching well with the simulation results (Fig. 3d). The lossless condition at the saddle point is also provided for comparison (right panel).
To further verify such saddle surface states, we increase the height (h) between two layers from 0.54a to 0.67a (Fig. 4a). The interaction between the two layers is weaker. Thus, the whole band structures show a slight redshift. Interestingly, two opposite saddles are separated to form an eyeshape (Fig. 4b, c). The experimental transmission spectra along the \({\tilde{\mathrm \Gamma }}\)–\({\tilde{\mathrm X}}\) and \({\tilde{\mathrm \Gamma }}\)–\({\tilde{\mathrm Z}}\) directions are shown in Fig. 4d, e. There is no obviously decreased transmission along the \({\tilde{\mathrm \Gamma }}\)–\({\tilde{\mathrm X}}\) direction. However, due to the eye opening, a wider transmission dip along the \({\tilde{\mathrm \Gamma }}\)–\({\tilde{\mathrm Z}}\) direction can be found than that in Fig. 3c. The dashed box represents the frequency region of the eye. Based on such strong hyperbolic behaviour for the topological saddle surface states, multidirectional anisotropic controllable robust sound transport with little backscattering can be obtained (see Supplementary Figures 7–10).
Discussion
In summary, we experimentally demonstrate 2D acoustic pseudospinvalley coupled saddle surface states in 3D topological acoustic crystals generated due to glidesymmetry design^{29}. Compared to the electronic topological crystalline insulator with saddle dispersion in condensed matter physics^{34,35}, this acoustic model exhibits surface nodal lines for the surface states, which can hardly shrink to a Dirac cone because of the lack of intrinsic acoustic spins (Supplementary Figure 5). Our acoustic pseudospins (satisfying pseudoTR) are constructed by crystal symmetry which cannot be kept intact on the domain wall. Unlike the intrinsic spins of electrons, these acoustic pseudospins are gradually changed along the surface, e.g. becoming linear pseudospins at the saddle point. However, this 3D acoustic topological model still shows strongly suppressed backscattering behaviour on the whole 2D surface, which resembles the 2D topological cases with a tiny gap in the middle of the 1D edge states^{7,15,36} due to spatial symmetry breaking on the boundary. The results we revealed here may pave the way towards acoustic pseudospins and valleytronics in 3D structures^{37}. The saddle surface states could be applied to realize hyperbolic pseudospin filters^{37}. The robust pseudospinvalley propagation within a large topological band gap and the extremely flat dispersion near the saddle point may give rise to an ultraslow sound, ultrahighQ acoustic resonator.
Methods
Experiments
Our samples are fabricated by 3D printing with commercial lowviscosity liquid photopolymer materials (Somos Imagine 8000). The tolerance of the fabrication is ±0.1 mm, which is less than 5% compared to the smallest feature size of 2.2 mm in our model. Due to the fabrication tolerance of different samples, the measured transmission spectra for the saddle points in Fig. 3a, c show a slight blue or redshift. A B&K49392670 microphone acts as a detector, which is placed 1 cm from the boundary with its response acquired and analysed in B&K3560C. The frequencies are swept from 14 to 24 kHz with an increment of 0.02 kHz. The experimental transmission spectra plotted in this article are normalized to the acoustic wave transmission through the same distance in air. The slight deviation recorded in experiments is due primarily to the frequency dependent coupling into and out efficiency.
Simulations
Numerical investigations used to calculate band structures (Figs. 1d–f, 2c, d and 4b) and field distributions (Figs. 2e and 3b, d) are conducted by using an acoustic model in commercial FEM software (COMSOL MULTIPHYSICS). Due to the large acoustic impedance mismatch between air and photopolymer material (modulus 2765 MPa, density 1.3 g cm^{−3}), the models in the numerical calculation are constructed using only acoustic cavities with hard boundaries, without considering the polymer background. The density and velocity of sound are chosen to be 1.25 kg m^{−3} and 343 m s^{−1}, respectively. The numerical results agree well with experimental results.
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.
References
 1.
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
 2.
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
 3.
Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken timereversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).
 4.
Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljacic, M. Observation of unidirectional backscatteringimmune topological electromagnetic states. Nature 461, 772–775 (2009).
 5.
Lu, L., Joannopoulos, J. D. & Soljacic, M. Topological photonics. Nat. Photonics 8, 821–829 (2014).
 6.
Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).
 7.
Wu, L.H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015).
 8.
Cheng, X. et al. Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nat. Mater. 15, 542–548 (2016).
 9.
Bliokh, K. Y., Smirnova, D. & Nori, F. Quantum spin Hall effect of light. Science 348, 1448–1451 (2015).
 10.
Gao, F. et al. Topologically protected refraction of robust kink states in valley photonic crystals. Nat. Phys. 14, 140–144 (2018).
 11.
Dong, J.W., Chen, X.D., Zhu, H., Wang, Y. & Zhang, X. Valley photonic crystals for control of spin and topology. Nat. Mater. 16, 298–302 (2017).
 12.
Noh, J., Huang, S., Chen, K. P. & Rechtsman, M. C. Observation of photonic topological valley Hall edge states. Phys. Rev. Lett. 120, 063902 (2018).
 13.
Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).
 14.
Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).
 15.
He, C. et al. Acoustic topological insulator and robust oneway sound transport. Nat. Phys. 12, 1124–1129 (2016).
 16.
Mousavi, S. H., Khanikaev, A. B. & Wang, Z. Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 6, 8682 (2015).
 17.
Xiao, M., Chen, W.J., He, W.Y. & Chan, C. T. Synthetic gauge flux and Weyl points in acoustic systems. Nat. Phys. 11, 920–924 (2015).
 18.
Lu, J. et al. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 369–374 (2017).
 19.
Yang, Y., Yang, Z. & Zhang, B. Acoustic valley edge states in a graphenelike resonator system. J. Appl. Phys. 123, 091713 (2018).
 20.
Lu, J. et al. Valley topological phases in bilayer sonic crystals. Phys. Rev. Lett. 120, 116802 (2018).
 21.
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
 22.
Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photonics 7, 1001–1005 (2013).
 23.
Peng, Y.G. et al. Experimental demonstration of anomalous Floquet topological insulator for sound. Nat. Commun. 7, 13368 (2016).
 24.
Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015).
 25.
Li, F., Huang, X., Lu, J., Ma, J. & Liu, Z. Weyl points and Fermi arcs in a chiral phononic crystal. Nat. Phys. 14, 30–34 (2018).
 26.
He, C. et al. Photonic topological insulator with broken timereversal symmetry. Proc. Natl. Acad. Sci. USA 113, 4924–4928 (2016).
 27.
Lu, L. et al. Symmetryprotected topological photonic crystal in three dimensions. Nat. Phys. 12, 337–340 (2016).
 28.
Slobozhanyuk, A. et al. Threedimensional alldielectric photonic topological insulator. Nat. Photonics 11, 130 (2016).
 29.
Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).
 30.
SerraGarcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342 (2018).
 31.
Xiong, J. et al. Evidence for the chiral anomaly in the Dirac semimetal Na_{3}Bi. Science 350, 413–416 (2015).
 32.
Liu, C.C., Zhou, J.J., Yao, Y. & Zhang, F. Weak topological insulators and composite Weyl semimetals: bBi_{4} X _{4} (X=Br, I). Phys. Rev. Lett. 116, 066801 (2016).
 33.
Ezawa, M. Topological Kirchhoff law and bulkedge correspondence for valley Chern and spinvalley Chern numbers. Phys. Rev. B 88, 161406 (2013).
 34.
Okada, Y. et al. Observation of Dirac node formation and mass acquisition in a topological crystalline insulator. Science 341, 1496–1499 (2013).
 35.
Liu, J., Duan, W. & Fu, L. Two types of surface states in topological crystalline insulators. Phys. Rev. B 88, 241303 (2013).
 36.
Barik, S. et al. A topological quantum optics interface. Science 359, 666–668 (2018).
 37.
Schaibley, J. R. et al. Valleytronics in 2D materials. Nat. Rev. Mater. 1, 16055 (2016).
Acknowledgements
The work was jointly supported by the National Key R&D Program of China (Grant Nos. 2017YFA0303702 and 2017YFA0305100) and the National Natural Science Foundation of China (Grant Nos. 11874196, 11625418, 11474158, 51732006, 51721001, 51702152 and 51472114). We also acknowledge the support of the Fundamental Research Funds for the Central Universities (Grant No. 14380097) and the Natural Science Foundation of Jiangsu Province (BK20140019).
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C.H. conceived the idea and performed the numerical simulations. C.H., S.Y.Y., H.G. and Y.T. carried out the experimental measurements. H.W. and C.H. carried out the theoretical analysis. H.Z., X.C.S., Y.B.C. and J.Z. contributed to the discussion of the results. C.H., M.H.L. and Y.F.C. supervised all the aspects of this work and managed this project.
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He, C., Yu, SY., Ge, H. et al. Threedimensional topological acoustic crystals with pseudospinvalley coupled saddle surface states. Nat Commun 9, 4555 (2018). https://doi.org/10.1038/s41467018070302
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