Abstract
Topologically protected surface modes of classical waves hold the promise to enable a variety of applications ranging from robust transport of energy to reliable information processing networks. However, both the route of implementing an analogue of the quantum Hall effect as well as the quantum spin Hall effect are obstructed for acoustics by the requirement of a magnetic field, or the presence of fermionic quantum statistics, respectively. Here, we construct a twodimensional topological acoustic crystal induced by the synthetic spinorbit coupling, a crucial ingredient of topological insulators, with spin nonconservation. Our setup allows us to free ourselves of symmetry constraints as we rely on the concept of a nonvanishing “spin” Chern number. We experimentally characterize the emerging boundary states which we show to be gapless and helical. More importantly, we observe the spin flipping transport in an Hshaped device, demonstrating evidently the spin nonconservation of the boundary states.
Introduction
The discovery of topological insulators (TIs), featuring a bulk gap and gapless boundary states, opened new avenues for condensedmatter physics^{1,2}. In two spatial dimensions, TIs come in two different classes, either described by a Z or Z_{2} topological index. The first^{3,4,5} breaks timereversal symmetry (TRS), and are commonly called Chern insulators (CIs). They host an (anomalous) integer quantum Hall effect and their surfaces are characterized by chiral, i.e., unidirectional, surface states. The latter Z_{2} insulators^{6,7,8}, such as the quantum spin Hall effect, are characterized by a pair of gapless helical boundary states. In the presence of spin conservation, the Z_{2} insulators can equivalently be described by spinChern numbers, where the spin sectors might carry an opposite but nonzero Chern number^{9}. In fact, spinChern numbers are well defined even in the absence of spin conservation or for TRSbroken systems^{9,10,11}. The spinChern numbers have been employed to identify TRSbroken spin1/2 electronic TIs and pseudospin TIs, giving rise to the concept of spinChern insulators (SCIs)^{10,11,12,13,14,15,16,17}. The SCIs feature helical boundary states, but whether gapless or not, depends on the system symmetry and microstructure of the sample boundary^{17}.
Recently, intense efforts have been devoted to realizing classical analogs of TIs for electromagnetic, mechanical, and acoustic waves^{18,19,20,21}. Photonic CIs have been realized in magnetooptic systems^{20,21,22,23}, mechanical CIs have been proposed in gyroscopic metamaterials^{24,25}, and acoustic CIs have been proposed in systems with circulating fluid^{26,27,28,29} and experimentally implemented recently^{30}. Hafezi et al. achieved a photonic SCI and observed the helical boundary states in a silicon photonic crystal. The pseudospin–orbit coupling was induced by the differential optical paths based on ring resonators^{31,32}. Mechanical SCIs were realized in bilayer structures, which rely on opposite interlayer and intralayer couplings^{33,34}.
Helical edge states have been observed in acoustic systems^{35,36,37,38,39,40,41,42,43,44}. Although these systems all host pseudospin, since the spin–orbit coupling, the essential ingredient for a SCI, is not available, they are not the acoustic SCIs. This can be reflected by fact that the helical edge states in the systems exist on the domain walls or interfaces, rather than on the boundaries or surfaces, as in SCIs. Returning to the photonic or mechanical SCIs aforementioned^{31,32,33,34}, it can be noted that the pseudospin in the SCIs is conserved, which means that these SCIs can actually be viewed as two independent copies of CIs and the topological properties can be described by Chern numbers. When the pseudospin conservation is broken, the description in terms of Chern numbers is no longer valid. One then needs to rely on the spinChern number to characterize the SCIs, in which the spin can vary or flip during transport. However, this general case has not yet been explored in all classical scenarios. A natural question arises: can we achieve acoustic SCIs (with spin–orbit coupling) without spin conservation?
In this work, to answer this question, we realize an acoustic spinChern insulator (ASCI) in a bilayer phononic crystal (PC). By introducing a layer pseudospin degree of freedom and the proper interlayer coupling, a synthetic spin–orbit interaction is successfully induced, which, particularly, also breaks the pseudospin conservation. We will first introduce the tightbinding model built on a bilayer Lieb lattice, which hosts all the physics of the ASCI. Then we will map this discrete model to a practical PC and demonstrate the topological properties of the ASCI, including the robust, transport of the helical or spinmomentum locking boundary states in the ASCI. In particular, we will present the observation of the spinflipping transport in a Hshaped device, evidencing the spin nonconservation of the boundary states.
Results
Tightbinding model for SCI
To illustrate how to realize an ASCI, we construct a tightbinding model on a bilayer Lieb lattice with a unit cell containing three sites in each layer, denoted A (red sphere), B (green sphere), and C (blue sphere) in Fig. 1a. The Hamiltonian is
where \(c_{i\alpha }^\dagger\) is the creation operator of layer pseudospin α on site i. The first term describes the nearestneighbor intralayer hopping with strength t_{0}. The second term denotes the onsite energy m on site A. The last term represents the chiral interlayer coupling with strength λ, where \(v_{ij,\alpha } = [ {\varepsilon _\alpha ( {\hat e_{kj} \times \hat e_{ik}} )_z + 1} ]/2\) with \(\varepsilon _{ \uparrow \downarrow } = \pm 1\), where i and j are two nextnearestneighbor sites with \(i \,\ne\, j\), k is their unique common nearestneighbor site, and the unit vector \(\hat e_{kj}\) points from j to k (see Supplementary Note 1 for details). The band dispersion of the model is presented in Fig. 1b. The interlayer coupling opens two bulk gaps and can induce topological phase transitions in the model.
The topological properties of this system can be captured by a spinChern number. One can introduce a pseudospin \(\tau _\alpha = \sigma _\alpha \otimes I_3\), where the Pauli matrices \(\sigma _\alpha\) act on the layer degree of freedom. While none of the components of τ is conserved, one can use the projection of, say \(\sigma _y\) into pairs of bands below the gap to split them. These split bands lead to welldefined fiber bundles that may carry a nonzero Chern number: the spinChern number. These spinChern numbers are a tool well tailored to classical systems, as they neither require any symmetry nor the presence of a fermionic timereversal operator. However, it is important to note that one relies not only on a spectral gap, but also on the spinprojection gap that allows for the splitting of the bands. Moreover, the details of the edge physics have to be inspected independently of the bulk, as there might be a spingap closing induced by the surface termination^{10,11}, see Supplementary Note 2 for details.
For illustrations, we focus on the tological properties of the lower gap. Figure 1c shows the spinChern number of the lower two bands \(C_s^l\) as a function of λ/t_{0} and m/t_{0}. Three topologically distinct phases exist. At the phase boundaries, indicated by the white lines, the energetic bulk gap closes. In the absence of spin–orbit interaction (λ = 0, the mass term opens a trivial gap: \(C_s^l = 0\)). The projected band dispersion for a ribbon with \(C_s^l = 2\) is plotted in Fig. 1d. The boundary states of the two lines oriented along the dotted black arrows localize at one boundary of the ribbon (the inset), while the others localize at the other boundary. The complete topological phases, the generalized bulkboundary correspondence, and associated helical boundary states are shown in Supplementary Note 2–4, respectively.
The SCI for acoustic waves
We now consider a PC implementation of the SCI for acoustic waves. As shown in Fig. 2a, the PC sample, fabricated by 3D printing, consists of a bilayer structure with interlayer connections realized by chiral tubes. Each layer of the unit cell contains three nonequivalent cavities connected by intralayer tubes (Fig. 2b). Mapping the PC to a tightbinding model, the cavities can be regarded as lattice sites, while the tubes provide hopping terms. The square unit cell has inplane length a = 20 mm and height h = 12.5 mm. The three square cavities composing a layer of the unit cell have the same height h_{c} = 5 mm and different inplane dimensions: L_{A} = 7 mm and L_{B} = L_{C} = L_{0} = 8 mm. The width and height of the intralayer tubes are L_{t} = 3.2 mm and h_{t} = 3 mm, respectively. The diameter of the interlayer tubes is d = 3.2 mm. Since the volume of cavity A is smaller than those of the other cavities, modes localized there are detuned to higher frequencies. This corresponds to a regime m/t_{0} < 0 for the tightbinding model of Eq. (1): a region of the phase diagram with \(C_s^l = 2\).
In Fig. 2c, we present the measured bulk band dispersion along high symmetry lines. Overlaid to the experimental data, we show the simulated bands. A bulk gap at M opens, thanks to the chiral interlayer couplers. To confirm the topologically nontrivial nature of the gap, we calculate the spindependent Berry curvature and spinChern number of the lowest two bands in the real PC. As the pressure field is mainly localized at the cavities, we construct the normalized wavefunctions φ(k) of the PC by using the pressure field sampled at the center of each cavity. Using these wavefunctions (Supplementary Note 2), we obtain the spindependent Berry curvatures \({\mathrm{\Omega }}_ \pm ^l\left( {\boldsymbol{k}} \right)\), shown in Fig. 2d. The two spinprojection sectors have opposite Berry curvature. By integrating separately \({\mathrm{\Omega }}_ \pm ^l\left( {\boldsymbol{k}} \right)\) in the whole Brillouin zone, we determine the spinChern number of the PC. The result, \(C_s^l = 2\), confirms that the PC has a gap with the same nontrivial topology predicted by the tightbinding model. The other phases of the ASCI, corresponding to the cases of λ/t_{0} > 0, are studied in Supplementary Note 5.
Helical boundary states in an ASCI
The nonzero \(C_s^l\) can induce a pair of helical boundary modes, even in the absence of crystalline symmetries (Supplementary Note 6). The projected band dispersions along the k_{x} direction are plotted in Fig. 3a and b for the wholecell and halfcell boundaries, respectively. The color maps represent the experimental dispersions, while the overlaid lines are the result of fullwave simulations. A pair of counterpropagating gapless boundary states (solid white lines) exists in the gap for both boundaries. The spin polarization along the y direction is defined as \(\langle \sigma _y\rangle = \langle \psi _k{\mathrm{}}\sigma _y{\mathrm{}}\psi _k\rangle\), where \(\psi _k\) is the eigenmode of the projected dispersion of the PC ribbon sampled at the center of each cavity. The spin nonconservation of the boundary states results in \( {\langle \sigma _y\rangle }  < 1\). On the other hand, although the spin is nonconserved, the boundary states still host spinmomentum locking property on each boundary. This is because that the boundary states satisfy \(\psi _{  k} = \psi _k^ \ast\) in the presence of TRS, and thus possess opposite spin polarization as \(\langle \psi _{  k} {\sigma _y} \psi _{  k}\rangle =  \langle \psi _k {\sigma _y} \psi _k\rangle\). This can be understood as a generalization of the spinmomentum locking for the helical boundary states. These results are demonstrated both by the simulations (the lines) and experiments (circles) in Fig. 3c and d. We note that the \(\sigma _y\) for the boundary states with k_{x} < 0 and with k_{x} > 0 are opposite, both satisfying \( {\langle \sigma _y\rangle }  < 1\).
Although the spin is nonconserved, the boundary waves are still robust against backscattering induced by the defect with TRS, because the spinmomentum locking property leads to destructive interference of the backscattering wave^{2}. The acoustic waves are nonreciprocal or nontimereversal in a flowing fluid^{26,27,28,29,30}, so a “magnetic defect” for an acoustic system can be created by circulating the fluid component somewhere locally in the structure. For our system, a “magnetic defect” on the boundary may be created by circulating the air inside a certain tube close to the boundary connecting the upper and lower layers. Practically, creating circulating flow brings much complexity in design and in experiment. Therefore, an equivalent “magnetic defect” for acoustic waves is rare, which means that the helical edge states observed in our acoustic system are topologically protected. Figure 4a shows the transport of acoustic boundary wave at 7.44 kHz in a sample possessing a rectangular defect. The boundary waves propagate smoothly around the defect. The experimental (upper panel) result is consistent with the simulation one (lower panel). However, the measured transmission is lower than that by simulation, as shown in Fig. 4b, because the boundary waves attenuate during propagation due to the loss in air. But the loss would not change the topology of the systems. The influence of air loss on the transmission and topology is discussed in Supplementary Note 7. Figure 4b also gives the comparison of the measured transmissions through a path along the rectangular defect and through a straight one of the same length. The transmission of the two samples agree in the bulk gap, indicating the robustness of the surface modes against backscattering induced by the defect. This result renders the realization of a spinfiltered oneway waveguide.
Spin flipping in an Hshaped ASCI
Finally, we show the spinflipping transport in an Hshaped device to evidently demonstrate the spin nonconservation of the helical boundary states. Actually, the spin cannot flip along a ribbon structure, because the spin polarization \(\langle \sigma _y\rangle\) is independent of the spatial position along a uniform periodic structure without bending. To show the spinflipping effect, we should consider \(\langle \sigma _y\rangle\) in, for example, a Hshaped structure, without uniform periodicity or with bending. In a Hshaped sample, \(\langle \sigma _y\rangle\) in the middle ribbon with small width is different from that in the left/right one. Figure 5a shows a schematic of the Hshaped structure of the ASCI, where the width of the left and right ribbons is W = 20a, and that of the middle ribbon is W_{m} = 1.5a. The boundary waves with spindown polarization excited from channel 1 not only can propagate to terminals 2 and 3 kept with the same polarization, but also may transport to terminal 4 with spin flipping, because of spin nonconservation. In Fig. 5b, we calculate the transmissions from channel 1 to terminals 2, 3, and 4 as a function of L_{m}, in which the operated frequency 7.6 kHz is tuned to the passing band of the middle ribbon. One can see that S_{41} is periodically oscillated, and the boundary waves can flip the spin polarization and reach terminal 4 at the right L_{m}. In Fig. 5c, for a fixed L_{m} = 2a, the measured and simulated field distributions clearly show that the boundary wave excited at channel 1 can propagate to channel 4, demonstrating that the spin polarization flips from spin down to spin up. The Hshaped structure can be designed to act as a spin flipper, and may also serve as a splitter with a switch effect, as discussed in Supplementary Note 8 and 9.
Discussion
In summary, we have realized an ASCI with a pair of helical boundary states with spin conservation breaking, which is of fundamental interest and opens up an avenue for applications of topological acoustics. This work implies two basic aspects different from earlier works of acoustic topological systems^{35,36,37,38,39,40,41,42,43,44}: first, the ASCI is induced by spin–orbit coupling, which is independent of any crystalline symmetries, and exhibits gapless edge states on the boundaries. Second, the ASCI breaks the spin conservation, which greatly expands the field of current topological physics, limited to spin conservation. The helical boundary states may have potential applications in innovative acoustic devices, such as topological splitters/switches with high tolerance. It should be noted that a similar acoustic structure has been employed to realize a fragile TI very recently^{45}, which however is essentially different from the presented ASCI (Supplementary Note 10).
Methods
Numerical simulations
All numerical simulations were performed by the commercial COMSOL Multiphysics solver package. The systems were filled with air with a mass density 1.3 kg m^{−3} and sound velocity 343 m s^{−1} at room temperature. Because of the huge acoustic impedance mismatch compared with air, the 3D printing plastic material was considered as hard boundary.
Experimental measurement
A subwavelength headphone with a diameter of 6 mm was used to generate acoustic waves. A microphone probe with a diameter of 1.5 mm was used to measure the acoustic pressure field distributions. A network analyzer (Keysight 5061B) was used to send and record the acoustic signals. The dispersions of bulk and boundary states were obtained by Fourier transforming the scanned acoustic pressure field distributions inside and on the boundary of the samples.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
This work is supported by the National Key R&D Program of China (Grant No. 2018YFA0305800), the National Natural Science Foundation of China (Grant Nos. 11804101, 11890701, 11704128, 11774275, 11974120, and 11974005), Guangdong Innovative and Entrepreneurial Research Team Program (Grant No. 2016ZT06C594), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019B151502012), and the Fundamental Research Funds for the Central Universities (Grant Nos. 2018MS93, 2019JQ07, and 2019ZD49). V.P. and S.H. acknowledge support from the European Research Council under the Grant Agreement No. 771503.
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W.D., X.H., F.L., and Z.L. conceived the original idea. W.D., X.H., J.L., V.P., S.H., and Z.L. performed the theoretical part of this work. F.L. and X.H. carried out the experiments. Z.L. supervised the project. All authors contributed to the analyses and the preparation of the paper.
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Deng, W., Huang, X., Lu, J. et al. Acoustic spinChern insulator induced by synthetic spin–orbit coupling with spin conservation breaking. Nat Commun 11, 3227 (2020). https://doi.org/10.1038/s41467020170391
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