Abstract
Following the discovery of the quantum Hall effect^{1,2} and topological insulators^{3,4}, the topological properties of classical waves began to draw attention^{5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}. Topologically nontrivial bands characterized by nonzero Chern numbers are realized through either the breaking of timereversal symmetry using an external magnetic field^{5,6,7,15,16} or dynamic modulation^{8,17}. Owing to the absence of a Faradaylike effect, the breaking of timereversal symmetry in an acoustic system is commonly realized with moving background fluids^{20,22}, which drastically increases the engineering complexity. Here we show that we can realize effective inversion symmetry breaking and create an effective gauge flux in a reduced twodimensional system by engineering interlayer couplings, achieving an acoustic analogue of the topological Haldane model^{2,23}. We show that the synthetic gauge flux is closely related to Weyl points^{24,25,26} in the threedimensional band structure and the system supports chiral edge states for fixed values of k_{z}.
Main
Sound is probably the ‘simplest’ of all classical waves. It has no intrinsic spin and does not respond to magnetic fields; hence, the fundamental interactions underlying the quantum spin Hall effect (QSHE) and quantum Hall effect (QHE) do not apply to acoustic waves. However, by engineering the coupling in the z direction, we can create synthetic staggered flux and hence k_{z}preserved unidirectional edge modes in the x–y plane in acoustic metacrystals which have simple ‘static’ structures with no moving fluid and no dynamic modulation. The acoustic metacrystals have Weyl points^{24,25,26} in the threedimensional (3D) band structure, and the systems are acoustic analogues of the topological Haldane model^{2,23} for fixed values of k_{z}.
Tightbinding model
To illustrate how the idea works, we start with a simple nearestneighbour tightbinding model for an AAstacked honeycomb lattice (Fig. 1a). Its acoustic implementation will be discussed later. The Hamiltonian H of this system consists of the intralayer part H_{0} and the interlayer part H_{1}.
where a (b) and a^{†} (b^{†}) are the annihilation and creation operators on the sublattice sites, ɛ represents the onsite energy difference. Each lattice is specified by subscripts (i, k), where the first labels the position in each layer and the second labels the number of layers. The first term in equation (1) represents the sublattice onsite energy difference. The second term represents the intralayer hopping between nearest sublattices. The intralayer hopping t_{n} is real and constant. The first Brillouin zone (BZ) of this lattice is shown in Fig. 1c. As this system is periodic along the z direction, k_{z} is a good quantum number. For each fixed k_{z}, and if we consider the dispersion and transport in the x–y plane, the 3D system can be reduced to an effective twodimensional (2D) system with a unit cell, as shown in Fig. 1b. The corresponding first BZ is illustrated in Fig. 1d, which represents a plane cut with the specified k_{z} in the original first BZ in Fig. 1c.
We can now introduce different kinds of interlayer coupling, from which we choose two special examples (Fig. 1e, g, where hopping is nonzero only between connected sites). In Fig. 1e, the hopping amplitudes are different for the two different sublattices. In this case, the interlayer hopping part H_{1} is given by
where t_{a} and t_{b} (t_{a} ≠ t_{b} and both are real) represent the interlayer hopping terms for sublattices a and b. The corresponding Bloch Hamiltonian H(k) is given by
where , a is the distance between the two sublattices, d_{h} is the interlayer distance and k = (k_{x}, k_{y}, k_{z}) is the Bloch wavevector. The eigenvalue of H(k) is given by . The first term under the square vanishes along the KH line. If [ɛ + (t_{a} − t_{b})cos(k_{z}d_{h})] is nonzero, the inversion symmetry of the reduced 2D hexagonal lattice is broken, as illustrated in Fig. 1f. When t_{a} − t_{b} > ɛ, there exist special values k_{z} = ±arccos[ɛ/(t_{b} − t_{a})]/d_{h}, where H(x, y, k_{z}) = H(−x, −y, k_{z}). This implies that Dirac cones can form in the k_{x}–k_{y} plane for these values of k_{z}. These special points can give rise to Weyl points in the 3D band structure, as we will discuss later.
Another interesting example is shown in Fig. 1g, in which we consider a chiral kind of interlayer coupling. We will see that if we consider the propagation in the x–y plane for a fixed k_{z}, this is effectively a realization of the topological Haldane model^{2}. The interlayer coupling coefficients as indicated by the cyan bonds in Fig. 1g are denoted by t_{c}, which are taken to be identical and real. For a fixed k_{z}, the interlayer hopping becomes nextnearestneighbour hopping in the reduced 2D system, with a complex hopping coefficient t_{c}e^{iφ}, where φ = ±k_{z}d_{h} depends on whether the hopping proceeds in the clockwise or anticlockwise direction. In Fig. 1h, we use red arrows to indicate the direction along which φ = −k_{z}d_{h}. After a complete loop of hopping in the direction indicated, the total phase accumulated is −3k_{z}d_{h}. This means the gauge flux enclosed by this loop is −3k_{z}d_{h}. In Fig. 1h, we use dotted and crossed black circles to represent the sign of local flux. Although the total flux inside each unit cell is zero, we have nonzero local flux. The chiral interlayer coupling introduces Peierls phase^{27} (Supplementary Information II) for the hopping parameters in 2D for any nonzero k_{z}, thereby achieving a staggered synthetic gauge flux in a ‘static’ system. This is different from the standard paradigm of using dynamical perturbation to induce synthetic gauge flux^{17}. For a system with both the coupling shown in Fig. 1e and that in Fig. 1g, one can then tune the system across a topological transition point by changing the value of (t_{a} −t_{b})/t_{c} or k_{z}. The phase diagram is shown in Supplementary Information I.
Breaking of inversion symmetry
Let us now consider real acoustic systems. We start with a periodic array of acoustic cavities linked together by tubes, as shown in Fig. 2a (top view) and b (side view). The resonance cavities can be viewed as ‘metaatoms’ and the hopping strength between the metaatoms can be tuned by changing the radius of the connecting tubes. The light blue colour in Fig. 2b denotes the area where the hard boundary condition is applied and the system is filled with air. The intralayer couplings are set to be equal by giving all horizontal connecting tubes the same radius (w_{0}). The interlayer couplings (along the z direction) are set to different values by choosing different radii (w_{1} ≠ w_{2}) for different sublattice sites. Here we consider the mode (Fig. 2c) whose pressure is described by a sinusoidal function along the z direction and does not vary in the horizontal plane. We note here that different mode profiles give basically the same results, except that the working frequency will be different. In Fig. 2d, we show the band dispersions of this mode in the k_{x}–k_{y} plane with different values of k_{z}. At k_{z} = 0 (red lines in Fig. 2d), the dispersion in the k_{x}–k_{y} plane has a gap at the point owing to inversion symmetry breaking. As this reduced 2D system still has mirror symmetry with respect to the x–z plane, the Berry curvature is an odd function in the reciprocal space and the two red bands are topologically trivial, with zero Chern numbers. At k_{z} = 0.623, there is a Dirac point at , as shown by the black lines in Fig. 2d, consistent with the tightbinding model’s prediction that degeneracy for the reduced 2D system can be recovered at some finite k_{z} value. The value of k_{z}, where the system has a Dirac cone in the k_{x}–k_{y} plane, is slightly different from the tightbinding prediction. This is because the introduction of the connecting tubes changes the resonance frequency of the cavities, which is equivalent to ɛ ≠ 0 in equation (2). In Fig. 2e, we show the band dispersion (black curves) along the KH direction (z direction). The difference in frequency between the two bands along k_{z} reflects the strength of inversion symmetry breaking as a function of k_{z}. As shown in Fig. 2e, the band dispersion near the Dirac point in Fig. 2d is also linear along the z direction, indicating that the degeneracy point is a Weyl point^{24,25,26,28} in the 3D band structure.
Synthetic gauge flux
We now consider the realization of effective acoustic gauge flux. In fact, creating effective acoustic gauge flux for a fixed k_{z} with chiral interlayer coupling can go beyond the tightbinding description. Figure 3a, b shows the top and side views of a unit cell of an acoustic system that exhibits synthetic gauge flux. The chiral coupling is characterized by the relative rotation angle θ between the holes on the upper and lower boundaries of the planar (x–y plane) waveguides. When θ = 0, the coupling is nonchiral and the synthetic gauge flux vanishes. The strength of the gauge flux depends on the rotational angle as well as the radius of the connecting tube r_{1} (Supplementary Information IV).
We consider the lowestorder acoustic mode and, as before, the waveguide is filled with air. In Fig. 3c, red/black curves indicate the band dispersion in the reduced 2D BZ at k_{z} ≠ 0/k_{z} = 0, representing the system with/without the effective gauge flux. The degeneracy at is lifted by the gauge flux. The effect of this synthetic gauge flux can also be seen from the Chern number of each isolated band, which is found to be +1/ −1 for the lower/upper band when k_{z} is positive. The strength of the gauge flux can be seen from the width of the gap at the point. In Fig. 3d, we show the band dispersion along the KH direction. The two bands are required to be degenerate at the K and H points by a combination of timereversal and C_{6} rotational symmetry. Along the KH line, the two states repel and the width of the gap reaches its maximum near the middle of the KH line.
A nonzero Chern number for a nonzero k_{z} implies the existence of a topologically protected chiral edge mode in the boundary between this system and a topologically trivial system inside the common gap region. We construct a hexagonal ribbon with finite width along the y direction and periodic along the x direction (see Supplementary Information V). We use hard boundaries to confine the sound wave in the y direction to within the acoustic system. The hard boundary condition can be regarded as a trivial band gap with zero decay length. Figure 3e shows the projection band (grey) along the x direction and the dispersions of two surface states with k_{z} = 0.5π/(h_{c} + l). The red and blue colours denote surface states localized at the lower and upper boundaries in the y direction, respectively (see Supplementary Information V). The two degeneracy points in Fig. 3d correspond to Weyl points^{24,25,26,28} in the 3D band structure. If we consider a surface BZ spanned by k_{x} and k_{z}, the allowed boundary modes for a given excitation frequency will trace out trajectories connecting two Weyl points that are analogous to ‘Fermi arcs’ in electronic Weyl semimetals^{25,28,29} (see Supplementary Information VI). If the width of the ribbon is large enough (larger than the decay length of the edge state), the edge state localized at one boundary cannot be scattered backwards as long as k_{z} is still preserved. In Fig. 3f, we show the property of the edge state. The edge state is excited on the left boundary (marked by the purple star) inside the gap region, and propagates clockwise around the corners and the defect without being backscattered. Black and purple arrows are drawn to show the direction of propagation of the sound wave. The direction, either clockwise or anticlockwise, depends on the sign of k_{z}. We supplemented our full wave simulation for an infinite system (with periodic boundary conditions) with simulations using a 3D finitesized tightbinding model (Supplementary Information III). Different from the oneway edge states of 2D acoustic systems published recently^{20}, the timereversal symmetry in our 3D system is preserved. We note that the timereversal partner of a clockwise state for a particular k_{z} is an anticlockwise state at −k_{z}.
Weyl points
This synthetic gauge flux is related to the Weyl points. In 3D, Weyl point dispersion is governed by the Weyl Hamiltonian H(k) = ∑ k_{i}v_{ij}σ_{j}, i, j ∈ {x, y, z} (refs 24, 25, 26, 28, 30, 31, 32), where v_{ij} are the group velocities and σ_{j} are the Pauli matrices. Weyl points have associated topological charges (or chirality, c = sgn[det(v_{ij})] = ±1), which can be regarded as monopoles of Berry flux^{26}. In Fig. 4a, b, we show the Weyl points of the two acoustic systems considered in Figs 2 and 3, respectively. The systems considered in Fig. 2 have mirror symmetry with respect to the x–z plane, which ensures that the two mirrorsymmetric Weyl points in the reciprocal space have charges with opposite signs, and hence the net charge in the horizontal light blue plane in Fig. 4a is zero. For any 2D band with an arbitrary (but fixed) value of k_{z}, the net Berry flux vanishes and its Chern number is zero. In contrast, the system with chiral coupling (for example, Fig. 3) does not possess mirror symmetry and the remaining C_{6} symmetry ensures all the Weyl points on the same k_{z} plane have charges with the same sign. Meanwhile, the net charge of Weyl points inside the first BZ must vanish^{29}, which means there must be at least two planes with different k_{z} possessing Weyl points of different charges. For example, the k_{z}d_{h}/π = 0 and k_{z}d_{h}/π = ±1 planes in Fig. 4b carry net topological charges of +2 and −2 respectively. Thus, for an arbitrary fixed k_{z} lying between these two planes, the net Berry flux through the reduced 2D BZ is 2π and the Chern number is ±1, with the sign determined by the sign of k_{z} (refs 24, 25, 29). The chiral coupling guarantees the nonzero Chern number, which corroborates the existence of a synthetic gauge flux.
Our idea of manipulating acoustic waves can also be extended to electromagnetic wave systems. The vector property of electromagnetic waves as well as the possibility of breaking timereversal symmetry using magnetic fields in electromagnetic systems can offer more flexibility in introducing interesting phenomena such as analogues of chiral anomaly^{33}. The results shown in Figs 2 and 3 demonstrate separately and respectively the consequences of onsite coupling difference and chiral coupling. If both symmetrybreaking mechanisms are simultaneously incorporated in the design of the acoustic metacrystal, we can in principle enter all the regimes in the phase diagram of the Haldane model as we vary k_{z}. The effective gauge flux induced by chiral coupling may stimulate new ideas for manipulating sound wave propagation, and may have implications in fields such as sound signal processing, sound energy harvesting, and noise protection.
Methods
All simulations were performed using the commercial solver package COMSOL Multiphysics. The 3D geometry was implemented by imposing periodic boundary conditions on some specified boundaries. The systems were filled with air (density ρ = 1.3 kg m^{−3} and speed of sound ν = 343 m s^{−1}). Eigenmode calculations were carried out to find the band structures as well as eigenmodes in Figs 2c–e and 3c–e. Frequency domain calculations were carried out to find the k_{z} conserved backscattering immune transport behaviour of the edge mode shown in Fig. 3f.
Figure 2a and b, respectively, show the top view and side view of the unit cell studied in Fig. 2. The parameters used were r_{c} = 3 cm, a = 9 cm, w_{0} = w_{2} = 0.6 cm, w_{1} = 1 cm, h_{c} = 8 cm and l = 3 cm. The light blue colour in Fig. 2b denotes the area where the hard boundary condition was applied, and the white colour indicates the area where the periodic boundary condition was applied with a given Bloch wavevector. Figure 2c shows the real part of the pressure field of a cavity mode found at 2,144 Hz, where the parameters of the cavity were the same as before—that is, r_{c} = 3 cm and h_{c} = 8 cm. The hard boundary condition was applied over all the cavity boundaries and the eigenmode calculation was then performed to find the eigenpressure distribution, where the red/blue colour indicates positive/negative local pressure.
Figure 3a and b, respectively, show the top and side views of the unit cell studied in Fig. 3. The parameters of the unit cell were r_{0} = 3 cm, r_{t} = 5.5 cm, a = 8 cm, r_{1} = 1.2 cm, θ = 90°, l = 5 cm, h_{c} = 8 cm. The dashed circles and solid circles in Fig. 3a represent the holes opened at the upper and lower sides of the sound waveguide in the x–y plane, respectively. θ represents the rotation angle of the connecting tubes. In Fig. 3b, the light blue colour marks the area where the hard boundary condition was applied, and solid white circles represent areas where Floquet periodic boundary conditions were applied with a given Bloch wavevector. Floquet periodic boundary conditions were also applied to the side walls (they are transparent to expose the interior structure of the unit cell). To calculate the projection band shown in Fig. 3e, we constructed a ribbon with finite length (16 unit cells) along the y direction. Floquet periodic boundary conditions were then applied to the side walls in the x and z directions. The remaining boundaries were all set as hard boundaries. The Bloch wavevector along the z direction was fixed at k_{z} = 0.5π/(l + h_{c}) in this calculation. In Fig. 3f, the purple star marks the position of our source. To couple waves inside our system, we adopted the plane wave radiation boundary condition for one of the side boundaries of a unit cell with a plane wave excited at this port and a working frequency of 1,360 Hz. All the side boundaries in the negative y direction were also set as plane wave radiation boundaries to couple waves outside of our system, and we used purple arrows to indicate the direction of wave propagation through these ports. Here the ‘plane wave radiation boundary’ plays two roles in our simulation. The one at the side boundary serves as the source and those in the negative y direction at the bottom are used to couple the wave from our system to the outside so that the surface wave will not go around all the side boundaries and back to the source port boundary. The plane wave radiation boundary, which serves as source, can also be replaced by other kinds of source, such as a line source. All the boundaries on the upper and lower sides in the z direction are set as Floquet periodic boundary conditions with a Bloch wavevector along the z direction given by k_{z} = −0.5π/(l + h_{c}), where the ‘+’ or ‘−’ sign of k_{z} determines the direction (clockwise or anticlockwise) of surface wave propagation. All the remaining boundaries are set as hard boundaries.
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Acknowledgements
The authors would like to thank Z. Q. Zhang and K. T. Law for discussions. This work was supported by the Hong Kong Research Grants Council (grant no. AoE/P02/12).
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C.T.C. initiated the programme. M.X. and W.J.C. contributed equally to this work. All authors contributed to the analysis and discussion of the results.
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Xiao, M., Chen, WJ., He, WY. et al. Synthetic gauge flux and Weyl points in acoustic systems. Nature Phys 11, 920–924 (2015). https://doi.org/10.1038/nphys3458
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