Abstract
Precise control of solidstate elastic waves’ mode content and coherence is of great use nowadays in reinforcing mechanical energy harvesting/storage, nondestructive material testing, wavematter interaction, high sensitivity sensing, and information processing, etc. Its efficacy is highly dependent on having elastic transmission channels with lower loss and higher degree of freedom. Here, we demonstrate experimentally an elastic analog of the quantum spin Hall effects in a monolithically scalable configuration, which opens up a route in manipulating elastic waves represented by elastic pseudospins with spinmomentum locking. Their unique features including robustness and negligible propagation loss may enhance elastic planarintegrated circuitlevel and systemlevel performance. Our approach promotes topological materials that can interact with solidstate phonons in both static and timedependent regimes. It thus can be immediately applied to multifarious chipscale topological phononic devices, such as patharbitrary elastic waveguiding, elastic splitters and elastic resonators with highquality factors.
Introduction
Phonons play a fundamental role in modern industrial civilization, for they are elaborately used to carry and process information with great accuracy in time, frequency and phase domains in a wide variety of fields. It is especially the case in the solidstate elastic world, since elastic phonons simultaneously possess several crucial advantages compared with fluid/airborne ones, such as: scalability towards integrated devices, antijamming capability, energy capacity, and extremely low losses. Elastic phonons found in solids are nowadays key to high signaltonoise ratio information processing^{1}, highsensitive and remote sensing, nondestructive testing of the internal structure of matters as well as intense wavematter interaction for future quantum acoustics^{2}. Fully unleashing the potential of these phonons, however, requires a precise manipulation of their temporal as well as spatial dynamics (i.e., waveguiding), which is often not so trivial compared with other particles (electrons, photons, or fluid phonons). Acoustic impedances of solidstate materials are relatively high and close, so it is basically ineffective in realizing conventional index guiding with large impedance contrast for elastic phonons in any continuous medium. Worse still, defects in solids are much more ubiquitous, further aggravating elastic dissipation induced by scattering of the phonons. Hence, it will be highly welcome to have a material that provides practically viable means of manipulating elastic transport dynamics, thereby offering a high degree of freedom in elastic waveguiding along with low transmission losses.
Topological states have been discovered in various electronic systems in past decades. Recently, the core principle behind these states has been adapted to bosonic systems, leading to the discovery and demonstration of topological photonics^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17} and mechanics^{18} (in both platforms built from discrete/weaklyinteracting manmade atoms^{19,20,21,22} and fluid acoustic systems^{23,24,25,26,27}). An essential impetus of these explorations is about gapless edge states, for they can be transported in a backscatteringimmune fashion with unparalleled tolerance towards any “nonmagnetic” defects, arbitrary bends, and fabrication imperfections. This performance superiority thus has promoted an era of functional largescale topological photonic circuits^{28,29,30}.
Topological states (in particular, the quantum Hall family) for elastic waves (specifically, linear stress waves in continuous solids) have been predicted in theoretical works^{31,32,33} using, e.g., gyroscopic inertial effects to break the timereversal (TR) symmetry^{31} and subwavelength metastructures to create effective spins^{32}. None of them have been demonstrated experimentally, mainly due to herculean engineering difficulties of these material phases. To date, almost all the preceding acoustic topological phases are realized in the only scalar (longitudinal wave) system of fluid airborne sound, whose practical role is largely limited. It is thus not only a remaining intellectual challenge but also of great practical value to realize the elastic topological states, especially in a generally concise configuration that can be scaled accordingly for future chipscale applications.
Here, we demonstrate an elastic quantum spin Hall effect (QSHE) in probably the simplest solidstate structure, i.e., a plain plate consisting of identical perforated holes in wavelength scales. The structural material possesses an elastic accidentally double Dirac cone with fourfold degeneracy. These four degenerate modes can be hybridized to form two elastic pseudospin½ and then mimicking QSHE via geometric tuning. The topologically protected elastic transport against structural imperfections and disorders has allowed us to demonstrate several unprecedented functional components for elastic phonons. Examples include arbitrary elastic pathway, elastic beam splitter, and highquality elastic resonator in arbitrary geometries. These demonstrated highperformance components may serve as building blocks for the coming era for largescale phononic circuits and networks (e.g., elastic interferometers), paving the way for advance acoustic signal processing, sensing and even analogue computing. Our configuration can be fully scaled down in size to operate in RF region, boosting the performance of and/or adding new functionality into the existing integrated surface/bulk acoustic wave devices.
Results
Elastic pseudospin½ via an accidental fourfold degeneracy
In bosonic systems, the most common methods to break TR symmetry is using a gauge field, e.g., a magnetic field for photons^{34} or an effective gauge field generated with a dynamic modulation for phonons^{35,36}/photons^{37}. However, in order to obtain TRinvariant bosonic QSHE, a totally different approach is needed. TR operator of bosons (T_{b}^{2} = + 1) is essentially distinct from that of fermions (T_{f}^{2} = −1), meaning that bosonic systems cannot naturally guarantee Kramers degeneracies, a prerequisite for QSHE. Consequently, the quest of realizing bosonic QSHE solely relies on the construction of artificial bosonic spin½ states (herein referred as pseudospins) under a pseudo (fermilike) TR symmetry (T_{p}^{2} = −1). Generally, these pseudospins can be emulated through polarization or modal hybridization. For instance, two degenerate modes M_{1} and M_{2} can be hybridized to construct these pseudospins: spin+/− ≡ M_{1} + iM_{2}/M_{1 }– iM_{2}, if only \({\it{M}}_{\mathrm{1}}\mathop{\longrightarrow}\limits^{{{\it{T}}_{\it{p}}}}{\it{M}}_{\mathrm{2}}\) while \({\it{M}}_{\mathrm{2}}\mathop{\longrightarrow}\limits^{{{\it{T}}_{\it{p}}}}  {\it{M}}_{\mathrm{1}}\). In our elastic plate, an elastic fourfold degeneracy can be accidentally formed [see details in Supplementary Note 1], further leading to the pseudospins for elastic waves under the pseudo TR symmetry, as exemplified in Fig. 1.
In a single plate (Fig. 1a), two different 2D elastic insulators (plate phononic crystals with elastic bandgap) are fabricated next to each other, forming an interface. These insulators have the same point group (C_{6v}) and lattice constant a. The only difference between them is the holecenter distance b (measured from the center of the six perforated holes to the center of the unitcell). The elastic insulator on the right side of the interface has a relatively small b (equals to a_{0}, where 3a_{0} = a), and it supports two pairs of twofold degeneracy at the Brillouin zone center (k = 0), forming an elastic bandgap. The degeneracy at the band edges corresponds to p_{x}/p_{y} (like) bulk modes at higher frequencies and d_{x2−y2}/d_{xy} (like) bulk modes at lower frequencies, respectively, which are similar to p and d orbitals of electrons. Here p_{x} obeys symmetry σ_{x}/σ_{y} = −1/+1; p_{y} obeys + 1/−1; d_{x2−y2} obeys + 1/+1; and d_{xy} obeys −1/−1, where σ_{x(y)} = +1, −1 represents the even or odd symmetry along the x or y axis, respectively. The other elastic insulator on the left has a relatively large b (equal to 1.12a_{0}) resulting in an inversion of the p and d modes at the band edges. In between these two cases, there exists a configuration (a_{0} < b≈1.0873a_{0} < 1.12a_{0}), where the elastic bandgap completely vanishes, leading to a double Dirac point with the fourfold degeneracy. i.e., a topological transition point between an ordinary insulator (OI) and a topological insulator (TI, see Supplementary Note 2) with an overlapped bulk bandgap.
Projected band structure of the OI–TI interface is calculated and shown in Fig. 1b. Ignoring the dispersion of shearhorizontal modes (SH modes, see more in Supplementary Note 3) denoted by grey lines, a clean single Dirac cone, formed by two gapless dispersions, appears at the Brillouin zone center (k_{‖} = 0) inside the overlapped bulk bandgap. These two gapless dispersions represent two helical edge states, originated from the degenerate p and d bulk modes of the two elastic insulators. As demonstrated in Fig. 1c, bulk p and d modes hybridize to form a pair of normal modes, i.e., one symmetric mode \(S = ( {p_x + d_{x2  y2}} ){\mathrm{/}}\sqrt 2\) and one antisymmetric mode \(A = ( {p_y + d_{xy}} ){\mathrm{/}}\sqrt 2\). Then, these two normal modes are used as the basis to construct the required two pseudospins, S + iA and S − iA, which are protected by the pseudo TR symmetry (T_{p}^{2} = −1), as validated by placing the T_{p} operator on the S/A basis, i.e., \(+ {\it{S}}\mathop{\longrightarrow}\limits^{{{\it{T}}_{\it{p}}}} + {\it{A}}\mathop{\longrightarrow}\limits^{{{\it{T}}_{\it{p}}}}  {\it{S}}\mathop{\longrightarrow}\limits^{{{\it{T}}_{\it{p}}}} \cdot \cdot \cdot\). Specific to our system, this T_{p} equals to \(\left( {{\it{C}}_{\mathrm{6}}  \sigma _{\mathrm{v}} \cdot \sigma _{\mathrm{d}} \cdot {\it{C}}_{\mathrm{3}}} \right){\mathrm{/}}\sqrt 3\) (see more detailed visualized operation of T_{p} in Supplementary Note 1).
Visualized elastic pseudospin transport with spinmomentumlocking
These elastic pseudospins and the corresponding spinmomentum locked propagation at the TI–OI interface can be directly imaged by a laser vibrometer (see “Methods”). The propagating states at the interface are excited by an ultrasonic transducer placed on the surface of the plate. The amplitude of the outofplane displacement of the elastic wave [actually surface acoustic waves (SAWs)] is mapped out in the vicinity of the TI–OI interface. When excited from the bottom, as illustrated in Fig. 1d, the upwardtraveling elastic wave exhibits a characteristic temporal evolution pattern, i.e., \(+ {\it{S}} \to + {\it{A}} \to  {\it{S}} \to \cdot \cdot \cdot\). By contrast, when excited from the top, the downwardtraveling elastic wave, as illustrated in Fig. 1e, exhibits also a characteristic but different pattern, i.e., \(+ {\it{S}} \to  {\it{A}} \to  {\it{S}} \to \cdot \cdot \cdot\). In addition, regardless of the excitation conditions, the elastic energy is found to be highly located near the interface. All these dynamic visual results are conclusive evidences demonstrating 1) the existence of timedependent anticlockwise elastic pseudospin+ (i.e., S + iA) and the clockwise elastic pseudospin− (i.e., S − iA), as illustrated in the S/A normal mode basis (bottom insets of Figs. 1d, e and 2) edge state transport with spinmomentum locking at the TI–OI interface, i.e., pseudospin+ only in the upwardtraveling direction while pseudospin− only in the downwardtraveling direction, characterizing intuitively an elastic counterpart of QSHE.
Meanwhile, this spinmomentum locked elastic transport enables elastic pseudospin separation with arbitrary excitation states, which can be faithfully confirmed in an elastic pseudospinselective waveguide coupler (i.e., an elastic beam splitter, see Supplementary Note 5).
Integratable elastic phononic circuits with backscattering immunity
An ideal elastic waveguide can be constructed using our topologically protected TI–OI interface. Due to the spinmomentumlocking characteristic of the helical edge states, the backscattering of the elastic edge propagating modes (Mode_{+k}\(\mathop{\longleftrightarrow}\limits^{{{\mathrm{scattering}}}}\)Mode_{−k}) is intrinsically suppressed, making this elastic waveguide robust against various defects and distortions and mostly importantly allowing for a flexible geometry configuration of its pathways.
Experimentally, different defects are intentionally introduced into our TI–OI interfaces (elastic waveguides) to study the elastic wave propagation. The lower panels of Fig. 2a–d show schematics of the samples, in which the geometries of the topological TI–OI interfaces are denoted by the yellow dashed lines. Two different defects, including a cavity with randomly filled holes (Fig. 2b) and a randomly disordered lattice of perforated holes (Fig. 2c) are placed in the original interface (Fig. 2a). In addition, a Zshaped interface with two 120° sharp bends (Fig. 2d) is also fabrication for test. Note that in all these configurations, spinmixing mechanisms (effective magnetic impurities) are absent. Elastic energy distributions obtained in simulations and measurements are shown in the upper panels of Fig. 2a–d with the measured transmission spectra shown in Fig. 2e, confirming the robustness and flexibility of this elastic waveguiding scheme, which is characterized by the elastic waves detouring around all these arbitrary defects and bends while maintaining nearly a lossfree elastic transmission. In comparison, for traditional elastic waveguides formed by OIOI interface, the same defects cause distinct elastic resonances and the bends severely inhibit the elastic forward transmission. As a result, a decreased transmission or even a total reflection can be observed due to strong elastic backscattering (see Supplementary Note 4). It should be noticed that the robustness in our system against bends is valid for any bending angle besides the one, 120°, shown in our experiments.
Remarkably, since its structural simplicity and (electromechanical) transducing accessibility, our elastic waveguide is thoroughly scalable and CMOS/MEMS compatible (see discussion in Supplementary Note 4). It can be exploited as a standard component/platform with great versatility for future largescale planner integration of phononic devices in, e.g., piezo(opto)mechanical systems^{38,39,40} operating over GHz and even beyond.
Monolithic elastic topological whispergallery resonator
The above novel waveguiding scheme for the elastic waves can be leveraged to create counterintuitive devices that have immediate applications. An example of significance is a monolithically integrated elastic topological whispergallery (WG) resonator. WG resonators have found key applications and have become an integral part in many fields such as coherent energy storage and source generation. However, the circulating WG modes in conventional resonators lose their energy gradually, owning to scattering losses from surface imperfection and geometric variation along the circulating path. Many efforts have been devoted to reduce scattering losses by developing advanced material processing and fabrication techniques^{41,42,43}. Here the topological design offers a completely novel and probably the most advantageous alternative for combating the scattering losses. For illustration, two TI–OI interfaces are constructed as shown in Fig. 3a: one straight waveguide of twoports on the left, and one closed hexagonshaped waveguide on the right, i.e., the WG resonator with six abrupt corners. Our WG resonator has a much reduced rotation symmetry compared with traditional ring or disk resonators, thus completely against common wisdom for highperformance resonator design. It is, however, very efficient and practically viable thanks to the topological protection. Specifically, the spinmomentum locking restricts the pseudospin+, excited by the bottom transducer, to only two spinpreserving scattering channels as no spinflipping scatters presented in our case. These two scattering channels are the upward channel in the straight waveguide and the counterclockwise circulating channel in the hexagonal resonator, respectively.
The spectrum for the elastic energy density inside the hexagonal resonator is measured and shown in Fig. 3c. Two strong peaks exist, suggesting the two scattering channels can efficiently couple with each other via multiple scattering processes in the vicinity between the straight waveguide and the WG resonator. Their resonance frequencies are 108.55 kHz and 110.09 kHz, respectively. By examining the calculated dispersions of our elastic helical edge states (shown in Fig. 3b, symmetric to the center of the Brillouin zone), it is found that these two frequencies are almost symmetric about the Dirac frequency (109.34 kHz). Thanks to the helical edge states’ linear dispersion near the Dirac frequency, the Bloch momentums corresponding to these two frequencies have the same magnitude but opposite direction. Since the selfconsistence condition imposed by the WG resonance involves only the magnitude of the Bloch momentum, the resonance always emerges as a pair around the Dirac frequency, which is very unique to topological systems.
These two resonances are further examined in real space by imaging the field of this device. The obtained normalized elastic energy distribution at these two resonance frequencies, shown in Fig. 3d, e indicates that the elastic energy is mostly localized inside the hexagonal resonator. The loaded Qfactor for the resonance at 110.09 kHz is measured to be more than 1200 (with a 3 dB bandwidth of 0.09 kHz and a mode volume of ~500 cm^{3}), a remarkable value for acoustic devices at this large scale when compared with that (usually about 500–1000) of a highquality tuning fork or a commercial miniaturized SAW resonator^{44}. Considering the fact that our prototype device has not gone through any rigorous optimization, even higher Qfactor is very likely by further exploiting other substrates or geometrical configurations. Our study here clearly exemplifies the advantage that is readily obtained in practice from the topological robustness of the edge states. It thus could be applied straightforwardly into a broad range of relevant fields, where coherence enhancement is in great need, e.g., highsensitive shear force detection, elastic energy harvesting^{45}, optomechanical sensing^{46}, and coherent phononic/photonic laser sources^{47}, etc.
Elastic pure pseudospin current at TI boundary
As the signature of the QSHE, pure spin current with zero charge current (j_{c} = 0, j_{s} ≠ 0) can be imaged and studied using a configuration shown in Fig. 4a. It contains a straight topologically protected waveguide with two identical transducers placed face to face on the waveguide, each of which will excite both an upward traveling pseudospin+ and a downward traveling pseudospin− at the same frequency simultaneously. In the top region (marked by red), there only exists an upwardtraveling pseudospin+ , superimposed from two individual pseudospin + , each originated from one of the two transducers. This fact is precisely observed and confirmed in experiment (see timeharmonic Supplementary Movie 1). Similar results can be obtained in the bottom region (marked by blue), where only downward traveling pseudospin− exists (see Supplementary Movie 2).
Remarkably in the region (marked by purple) between the two transducers, there exist simultaneously an upward traveling pseudospin+ excited by the bottom transducer and a downward traveling pseudospin− excited by the top transducer. Pure pseudospin current (without energy flow) in this region is thus determined by the coherent superposition of these two pseudospins, as illustrated in Fig. 4b. When the pseudospins exited by two transducers have the same amplitude, they (in red and blue arrows) are effectively annihilated anywhere in this region, due to their same rotating phase and group velocity but opposite direction. Consequently, this process leaves behind only a nonspinning or an equivalent linear vector field component in the S/A normal mode basis. Since the phase difference between the two pseudospins varies linearly with location (e.g., from position#1 to position#7), the resulted vector direction of the linear component rotates correspondingly in a linear fashion to form a welldefined spatial chirality, as illustrated by the DNAlike double helix marked by the violet and green traces. Note that in this case the helix is lefthanded at the right boundary of TI, and a righthanded chirality can also be obtained but at the left boundary of TI.
In experiment, the outofplane displacement distribution in this region is obtained at a deep subwavelength resolution and shown in Fig. 4c (see Supplementary Note 6 for complete field information, and Supplementary Movie 3 for its temporal evolution). At any location in this region, e.g., the top position #7, the field pattern in a unit cell remains the same, but its intensity blinks with time, as illustrated in the right panel. This indicates the formation of a standing wave pattern. Along the waveguide in this region, the symmetry of the field in the unit cell undergoes periodic changes, e.g., symmetric—antisymmetric—symmetric or equivalently \(\pm {\it{S}}\cdot\cdot\cdot \mp {\it{A}}\cdot\cdot\cdot \mp {\it{S}}\). Consequently, the standingwave pattern is seen to spiral in oneway fashion, representing the signature of pure pseudospin current, i.e., no energy flow but a pseudo angular momentum transport in real space.
Up to date, a satisfactory visualization of the edge states of the QSHE, especially the corresponding spin dynamics in spatial and temporal space, remains a challenge in electronic systems, because of the limitations set by the extreme scale of electron waves and difficulties lying in material growth and device fabrication. Our observation in elastic systems provides a comprehensive scenario to the fundamental role of the spin/pseudospin dynamics in topological states, opening up a possibility towards potential bosonic pseudospintronics with great flexibility, stability and signal fidelity.
Discussion
We realized and confirmed with abundant visual data an elastic topological material in a continuous solidstate medium. Using this new material, we have captured convincing real space evidences for elastic helical edge states, pseudospin dynamics and other related novel phenomena with a neverbefore deep subwavelength resolution. All our findings obtained here may be directly leveraged to investigate and ultimately pave the way for future phononbased high performance (in terms of energy efficiency, information capacity, and signal integrity) information processing devices on a variety of commonly used material systems, e.g., Si^{48}, AlN^{49}, LiNbO_{3}^{50}, and on many emerging molecular or twodimensional piezoelectric materials (like MoSe_{2}, WTe_{2} in the transition metal chalcogenide families^{51}). This material may also reinforce the recent developed study of quantum acoustics, since a precise controlled elastic phonon may interact with various quantum systems, e.g., superconducting qubits coupled via surface^{52} and bulk^{53} acoustic waves. To further demonstrate the capability of this new material, some unprecedented and highperformance elastic devices such as an elastic topologically protected whispergallery resonator are constructed and studied thoroughly in realspace, potentially impacting a broad range of relevant fields. Last, in the view point of fundamental sciences, our findings may even underpin the foundations for understanding the topological properties and spin related transport behaviors in other bosonic and fermionic materials under the theory of great unity of waves.
Methods
Sample preparation
Our samples are prepared exclusively on polished stainlesssteel plates (Type 201, mass density 7903 kg m^{−3}) with a fixed plate thickness 7.82 mm. Their elastic parameters are determined by ultrasonic scattering echo method, i.e., 201.075 GPa for Young’s modulus and 0.3254 for Poison ratio. The plates are perforated on a precision CNC milling machine to create phononic crystals with identical hole radius r = 3.52 mm and lattice constant a_{0} = 11.29 mm. An impedance matched sound proof adhesive (epoxy resin and tungsten powder) is coated on the circumference of the sample to prevent unwanted back reflection.
Numerical calculation
Elastic band structure calculations, eigen mode distributions and surface acoustic energy distributions shown in this work are conducted by a threedimensional finite element method using acoustic module of commercial software COMSOL MULTIPHYSICS.
Experimental apparatus
Broadband piezoelectric ultrasonic transducers (center frequency at 110 kHz) are attached to the sample surface as an excitation source. They are driven by a function generator followed by a power amplifier. A fiber laser based twowave mixing interferometer with a lockin amplifier is used to interrogate the amplitude and the phase information of the back reflected optical measurement beam from the sample surface. The field profile of the elastic wave (SAW) is then mapped out in a point by point fashion based on the information of the reflected optical measurement beam. The radiating pressure exerted by our measurement optical beam on the plate surface is estimated to be negligible to the shearforce pressure caused by the SAWs. Thus, the SAW propagation is unaffected by our optical beam. In addition, the imaging resolution is determined primarily by the size of our optical beam, which can be focused down to less than 1/50 of the SAW wavelength. All these advantages combined enables us to present the first nondisturbing deep subwavelength imaging of topological edge states and helical pseudospin transport.
Data availability
The datasets within the article and supplementary information in the current study are available from the authors upon request.
Change history
31 January 2019
The original version of this Article contained errors in the second sentence in the legend of Fig. 1, which incorrectly read ‘These two elastic insulators are identical in lattice constant a (3a_{0}), plate thickness (0.4a_{0}), and radius of perforated holes r (0.18a_{0}) but different holecenter distance characterized by b.’ The correct version states ‘plate thickness (√3 × 0.4a_{0})’ in place of ‘plate thickness (0.4a_{0})’ and ‘radius of perforated holes r (√3 × 0.18a_{0})’ rather than ‘radius of perforated holes r (0.18a_{0})’.The first sentence of the ‘Sample preparation’ section of the Methods originally incorrectly read ‘Our samples are prepared exclusively on polished stainlesssteel plates (Type 201, mass density 7803 kg m^{−3}) with a fixed plate thickness 7.82 mm.’ In the corrected version, ‘mass density 7903 kg m^{−3}’ replaces ‘mass density 7803 kg m^{−3}’.The second sentence in the legend of Supplementary Fig. 3, originally incorrectly read ‘The symmetry of the phononic crystal remains unchanged as C_{6ν}, and thickness of the substrates H (equals to 0.4a_{0}), lattice constant a (equals to 3a_{0}) and radius of perforated holes r (equals to 0.18 a_{0}) maintain constant.’ The correct version states ‘√3 × 0.4a_{0}’ in place of ‘0.4a_{0}’ and ‘√3 × 0.18a_{0}’ rather than ‘0.18a_{0}’.This has been corrected in both the PDF and HTML versions of the Article.
References
 1.
Hashimoto, K. Y. Surface Acoustic Wave Devices in Telecommunications. (Springer, Berlin, 2010).
 2.
Aref, T. et al. Quantum Acoustics with Surface Acoustic Waves. (Springer International Publishing, Cham, 2016) 217–244.
 3.
Lu, L., Joannopoulos, J. D. & Soljacic, M. Topological photonics. Nat. Photon 8, 821–829 (2014).
 4.
Khanikaev, A. B. & Shvets, G. Twodimensional topological photonics. Nat. Photon 11, 763 (2017).
 5.
Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional backscatteringimmune topological electromagnetic states. Nature 461, 772–775 (2009).
 6.
Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon 7, 1001–1005 (2013).
 7.
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
 8.
Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).
 9.
Chen, W. J. et al. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nat. Commun. 5, 5782 (2014).
 10.
Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015).
 11.
Bliokh, K. Y., Smirnova, D. & Nori, F. Quantum spin Hall effect of light. Science 348, 1448–1451 (2015).
 12.
Wu, L.H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015).
 13.
Ma, T., Khanikaev, A. B., Mousavi, S. H. & Shvets, G. Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides. Phys. Rev. Lett. 114, 127401 (2015).
 14.
Gao, F. et al. Probing topological protection using a designer surface plasmon structure. Nat. Commun. 7, 11619 (2016).
 15.
Cheng, X. et al. Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nat. Maters 15, 542–548 (2016).
 16.
Slobozhanyuk, A. et al. Threedimensional alldielectric photonic topological insulator. Nat. Photon 11, 130–136 (2017).
 17.
Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346–350 (2018).
 18.
Huber, S. D. Topological mechanics. Nat. Phys. 12, 621–623 (2016).
 19.
Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).
 20.
Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials. PNAS 112, 14495–14500 (2015).
 21.
Mitchell, N. P., Nash, L. M., Hexner, D., Turner, A. M. & Irvine, W. T. Amorphous topological insulators constructed from random point sets. Nat. Phys. 14, 380–385 (2018).
 22.
SerraGarcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).
 23.
Xiao, M. et al. Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11, 240–244 (2015).
 24.
He, C. et al. Acoustic topological insulator and robust oneway sound transport. Nat. Phys. 12, 1124–1129 (2016).
 25.
Peng, Y. G. et al. Experimental demonstration of anomalous Floquet topological insulator for sound. Nat. Commun. 7, 13368 (2016).
 26.
Yves, S. et al. Crystalline metamaterials for topological properties at subwavelength scales. Nat. Commun. 8, 16023 (2017).
 27.
Li, F., Huang, X., Lu, J., Ma, J. & Liu, Z. Weyl points and Fermi arcs in a chiral phononic crystal. Nat. Phys. 14, 30 (2018).
 28.
Bahari, B. et al. Nonreciprocal lasing in topological cavities of arbitrary geometries. Science 358, 636–640 (2017).
 29.
Harari, G. et al. Topological insulator laser: theory. Science 359, 1230 (2018).
 30.
Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, 1231 (2018).
 31.
Wang, P., Lu, L. & Bertoldi, K. Topological phononic crystals with oneway elastic edge waves. Phys. Rev. Lett. 115, 104302 (2015).
 32.
Mousavi, S. H., Khanikaev, A. B. & Wang, Z. Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 6, 9682 (2015).
 33.
Brendel, C., Peano, V., Painter, O. & Marquardt, F. Snowflake phononic topological insulator at the nanoscale. Phys. Rev. B. 97, 020102 (2018).
 34.
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).
 35.
Fleury, R., Sounas, D. L., Sieck, C. F., Haberman, M. R. & Alù, A. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator. Science 343, 516–519 (2014).
 36.
Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).
 37.
Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photons 6, 782 (2012).
 38.
Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391 (2014).
 39.
SafaviNaeini, A. H. et al. Twodimensional phononicphotonic band gap optomechanical crystal cavity. Phys. Rev. Lett. 112, 153603 (2014).
 40.
Balram, K. C., Davanço, M. I., Song, J. D. & Srinivasan, K. Coherent coupling between radiofrequency, optical and acoustic waves in piezooptomechanical circuits. Nat. Photons 10, 346–352 (2016).
 41.
Vahala, K. Optical Microcavities. (World Scientific, Singapore, 2004).
 42.
Chiasera, A. et al. Spherical whispering‐gallery‐mode microresonators. Laser & Photonics Rev. 4, 457–482 (2010).
 43.
Kippenberg, T. J., Holzwarth, R. & Diddams, S. A. Microresonatorbased optical frequency combs. Science 332, 555–559 (2011).
 44.
Campanella, H. Acoustic Wave and Electromechanical Resonators: Concept to Key Applications. (Artech House, Boston/London, 2010).
 45.
Erturk, A., & Inman, D.J. Piezoelectric Energy Harvesting. (John Wiley & Sons, New York, 2011).
 46.
Matsko, A. B., Savchenkov, A. A., Ilchenko, V. S., Seidel, D. & Maleki, L. Optomechanics with surfaceacousticwave whisperinggallery modes. Phys. Rev. Lett. 103, 257403 (2009).
 47.
Jing, H. et al. PTsymmetric phonon laser. Phys. Rev. Lett. 113, 053604 (2014).
 48.
Graczykowski, B. et al. Phonon dispersion in hypersonic twodimensional phononic crystal membranes. Phys. Rev. B 91, 075414 (2015).
 49.
Lin, C. M., Chen, Y. Y., Felmetsger, V. V., Senesky, D. G. & Pisano, A. P. AlN/3C–SiC composite plate enabling high‐frequency and high‐Q micromechanical resonators. Adv. Mater. 24, 2722–2727 (2012).
 50.
Yudistira, D. et al. Monolithic phononic crystals with a surface acoustic band gap from surface phononpolariton coupling. Phys. Rev. Lett. 113, 215503 (2014).
 51.
Duerloo, K.A. N., Ong, M. T. & Reed, E. J. Intrinsic piezoelectricity in twodimensional materials. J. Phys. Chem. Lett. 3, 2871–2876 (2012).
 52.
Gustafsson, M. V. et al. Propagating phonons coupled to an artificial atom. Science 346, 207–211 (2014).
 53.
Chu, Y. et al. Quantum acoustics with superconducting qubits. Science 358, 199–202 (2017).
Acknowledgements
The work was jointly supported by the National Key R&D Program of China (Grant No. 2017YFA0305100, No. 2017YFA0303702 and No. 2016YFA0301700), the National Basic Research Program of China (Grant No. 2015CB659400), and the National Natural Science Foundation of China (Grant No.51721001, No.51472114, No.51732006, No.11625418, No.11474158 and No.51702152). We also acknowledge the support of the Natural Science Foundation of Jiangsu Province (Grant No. BK20140019 and No. BK20150057), the Guangdong Innovative and Entrepreneurial Research Team Program (Grant No. 2016ZT06D348), Fundamental Research Funds for Central Universities (Grant No. 021314380093 and No. 021314380100) and the Academic Program Development of Jiangsu Higher Education (PAPD).
Author information
Affiliations
Contributions
All authors contributed extensively to the work presented in this paper. M.H.L., X.P.L., and Y.F.C. supervised all aspects of this work. S.Y.Y. managed this project. S.Y.Y. and C.H. carried out the theoretical analysis and experiments with the assist of Z.W., F.K.L., X.C.S. Z.L. and H.Z.L.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Yu, SY., He, C., Wang, Z. et al. Elastic pseudospin transport for integratable topological phononic circuits. Nat Commun 9, 3072 (2018). https://doi.org/10.1038/s41467018054615
Received:
Accepted:
Published:
Further reading

Computation and data driven discovery of topological phononic materials
Nature Communications (2021)

Phononic topological insulators based on sixpetal holey silicon structures
Scientific Reports (2019)

Experimental demonstration of coherent superpositions in an ultrasonic pseudospin
Scientific Reports (2019)

Topological analog signal processing
Nature Communications (2019)

Rainbow guiding of the lowestorder antisymmetric Lamb mode in phononic crystal plate
Science China Technological Sciences (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.