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https://doi.org/10.1038/s41467-021-23574-2 OPEN Roton-like acoustical relations in 3D ✉ Yi Chen 1, Muamer Kadic2,3 & Martin Wegener 1,2

Roton dispersion relations have been restricted to correlated quantum systems at low temperatures, such as liquid Helium-4, thin films of Helium-3, and Bose–Einstein con- densates. This unusual kind of dispersion relation provides broadband acoustical backward fl “ fl ” 1234567890():,; , connected to energy ow vortices due to a return ow , in the words of Feynman, and three different coexisting acoustical modes with the same polarization at one . By building mechanisms into the unit cells of artificial materials, metamaterials allow for molding the flow of waves. So far, researchers have exploited mechanisms based on various types of local resonances, Bragg resonances, spatial and temporal symmetry breaking, topology, and nonlinearities. Here, we introduce beyond-nearest-neighbor interactions as a mechanism in elastic and airborne acoustical metamaterials. For a third-nearest-neighbor interaction that is sufficiently strong compared to the nearest-neighbor interaction, this mechanism allows us to engineer roton-like acoustical dispersion relations under ambient conditions.

1 Institute of Applied Physics, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany. 2 Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany. 3 Institut FEMTO-ST, UMR 6174, CNRS, Université de Bourgogne Franche-Comté, Besançon, France. ✉ email: [email protected]

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or usual acoustical waves or in gases, liquids, and It is well known that the Euler equation for classical gases or solids, energy and are proportional to each liquids22 and the Navier equation for elastic solids23 lead to F 1 ωð Þ¼ other . Rotons can be seen as highly unusual acoustical dispersion relations of the type k vphk for longitudinal or waves with a parabolic minimum of the energy versus momen- transverse acoustical waves in the bulk. Thus, neither of them tum at finite momentum and finite energy2. Based on a prediction captures roton-like behavior. In sharp contrast, it has recently by Landau2 and following a suggestion by Feynman3,4, the roton been shown that chiral Eringen micropolar continuum elasticity dispersion relation for longitudinal acoustical waves was observed theory24 can lead to roton-like dispersion relations for transverse in liquid Helium-4 (a Bose-liquid) at low temperatures by means acoustical elastic waves25. In their work25, chirality has been a – of inelastic neutron scattering5 7. A bulk of later theoretical work necessary mechanism, whereas mechanisms based on periodicity, interpreted rotons in terms of strong spatial correlations in this such as ordinary or extraordinary Bragg reflections, are not – quantum system8 10. More recently, rotons have been found accounted for in micropolar elasticity theory24. More broadly experimentally in two-dimensional (2D) liquid Helium-311 (a speaking, mechanisms such as ordinary Bragg reflection26,27, local Fermi-liquid) and in Bose–Einstein condensates of erbium atoms resonances28–31, near-ideal joints32–34 introducing soft modes, subject to weak magnetic dipole–dipole interactions12. The roton spatial or temporal symmetry breaking35–38, topology39,40, dispersion relation has also been predicted theoretically for other duality41,42, as well as geometrical nonlinearities34,43 have inde- ultracold dipolar13 and quadrupolar14 gases confined in one- pendently given rise to a wealth of other unusual dispersion – dimensional (1D) and 2D geometries15 17, as well as for Rydberg- relations and quasi-static behaviors of elastic and acoustical dressed atoms18. metamaterials. The roton dispersion relation is illustrated in Fig. 1,wherewe Here, we introduce and analyze a class of three-dimensional represent energy as _ω,withthewave’s ω,and (3D) microstructured elastic metamaterials supporting roton-like (quasi-)momentum by _k, with the k. Apart from its transverse as well as longitudinal dispersion relations. We engi- basic physics, the dispersion relation in Fig. 1 is interesting for neer these metamaterials by tailoring beyond-nearest-neighbor applications because it potentially allows to manipulate acoustical elastic interactions among the 3D crystal unit cells waves in unusual ways. First, it comprises a region of negative slope in addition to the usual nearest-neighbor interactions. Further- ωð Þ ’ ¼ ω= of k versus k,inwhichthewaves vph k>0 more, we apply the same concept to airborne acoustical waves in ¼ ω= and vgr d dk<0haveoppositesign.Such macroscopic three-dimensional channel-based metamaterials to behavior gives rise to backward waves and can lead to negative illustrate the general nature of the approach. at interfaces19–21. We will see below that this character- istic can occur without absorption/damping over a broad spectral Results regime. Second, the extrema of the dispersion relation in Fig. 1 One-dimensional toy model. Let us start our discussion by a correspond to zero group velocity and hence to peaks in the simple 1D mathematical toy model illustrated in Fig. 2a: Identical fi ω . Third, at a given xed angular frequency and for masses m separated by distance a are connected to their k>0, the single dispersion curve supports three (one backward, two immediate neighbors by linear Hooke’s springs with spring forward) wave modes with three different , phase constant K1. In addition, each mass shall be coupled to the velocities, and . Unfortunately, no natural or rationally masses separated by distance Na to the left and right (with integer designed artificial materials showing roton-like dispersion behavior ≥ ’ N 2) by springs with spring constant KN . Newton s equation of under ambient conditions have been reported so far. € motion for the acceleration un of the mass displacement un at

Fig. 1 Roton-like acoustical wave dispersion relation. a The acoustical wave’s angular frequency ωðÞk is depicted versus wavenumber k. The dispersion relation starts off with the usual linear increase of ω versus k.Atafinite characteristic wavenumber, the dispersion relation exhibits a parabolic minimum. In a certain frequency range (highlighted by the light-yellow background), a single frequency ω leads to three modes at different k (exemplified by the λ ¼ π=k v ¼ ω= k dashed line and the black dots), hence different wavelengths 2 . In the hatched wavenumber interval, the group velocity gr d d is negative, v ¼ ω=k a fi π=a whereas the phase velocity ph is positive. In a crystal with period , the edges of the rst Brillouin zone at wavenumbers ± are important. b Corresponding group velocity versus angular frequency ω, normalized by the phase velocity in the long- limit, k ! 0. The different colors serve to connect the different parts of the dispersion relation between a and b. Panels a qandffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib can be taken as schemes. They actually correspond to N ¼ K =K ¼ ω ¼ ðK þ K N2Þ=m solutions of the 1D toy model (cf. Fig. 2b) with parameters 3, N 1 3, and 0 1 N .

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flow (see Methods) 1 ÀÁ pkðÞ ¼ eu2ω K sinðÞþka NK sinðÞNka ð3Þ 2 1 N / has two contributions. The contribution K1 is always positive for 0 1 N. This aspect is analogous to what Feynman4 referred to as the “return flow” in the context of the roton. Therefore, the sign of the group velocity and that of the energy flow are identical and independent on the choice of the Brillouin zone. The sign and magnitude of the phase velocity ω=k, however, depend on the Brillouin zone. If and only fi ≠ if the nearest-neighbor interaction is nite, i.e., for K1 0, the spatial period is a, hence jjk ≤ π=a is the proper first Brillouin zone. This leads to a positive phase velocity in the interval k 2½0; π=aŠ, whereas the group velocity and the mean energy flow are negative for part of this interval (approximately for 2½π=ð Þ; π=ð ފ = = k 3a 2 3a ) for KN K1 > 1 N. This behavior corre- Fig. 2 One-dimensional toy model. a Masses m (yellow dots) are sponds to a backward wave. The two wavenumbers for which the fl connected to their nearest neighbors separated by distance a by Hooke’s total energy ow and hence the group velocity are zero (cf. K Fig. 2b) are merely special cases. On the basis of this discussion, springs with spring constants 1 (blue straight lines). In addition, all masses are connected to their Nth-nearest neighbors at distance Na by springs with the roton-like behavior can be seen as an unusual hybridization between two ordinary acoustical dispersion relations, one Hooke’s spring constants KN (red curved lines). As an example, we choose N ¼ 3. For K ≠ 0, the spatial period of this arrangement is a. Hence, the for a mass-and-spring model with period a and the other for a 1 ¼ first Brillouin zone is given by wavenumber jjk  π=a. b Dispersion relation three-fold degenerate mass-and-spring model with period Na ωðÞ¼k ωðÀkÞ. The differently colored curves (see legend) represent 3a (cf. Fig. 2b). different ratios of the spring constants K =K , increasing from top to In other words, the roton-like minimum in the dispersion 3 1 ωð Þ bottom. For clarity,q weffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifix the phase velocity in the long-wavelength limit relation k of the 1D toy model with spatial period a results 2 from extraordinary Bragg reflections with reciprocal lattice vector k ! 0, i.e., v ¼ a ðK þ KNN Þ=m ¼ const ¼ aω . ph 1 0 2π=l, corresponding to the length l ¼ Na of the beyond-nearest- ¼À1; ¼ ; 1 neighbor interaction. A roton-like minimum occurs if and only if lattice site n (with integer n ) is given by fi ≥ ÀÁÀÁthe interaction has suf ciently long range (i.e., N 3) and the fi mu€ ¼ K u þ À 2u þ u À þ K u þ À 2u þ u À : strength of the beyond-nearest-neighbor interaction is suf ciently n 1 n 1 n n 1 N n N n n N = = ≥ ð Þ large (i.e., KN K1>1 N). For N 4, even several minima can 1 occur within the first Brillouin zone. To test our interpretation of the roton-like dispersion relation, we prescribe the displacement of a single mass in The solution of this equation of motion are Bloch waves, the middle of the 1D toy model chain (cf. Fig. 2a), u ðÞ¼t eu expðiðkna À ωtÞÞ with amplitude eu and the imaginary ÀÁ n ðÞ¼e ðω Þ ðÀ =τ 2Þ unit i, following the dispersion relation u0 t ucos t exp t , by a temporal pulse with Gaus- sian and carrier frequency ω ¼ 0:5ω in the spectral sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 region of the roton-like dispersion relation for which kðωÞ has ωðÞ¼ωðÀ Þ¼ K1 2 ka þ KN 2 Nka : ð Þ =  k k 2 sin sin 2 three solutions. For the parameter range from K3 K1 2to m 2 m 2 =  K3 K1 5, this excitation launches two clearly visible triplets of ≠ ≠ Gaussian wave packets (see Supplementary Figs. 1 and 2). This We emphasize that, for K1 0 and K3 0, the underlying = fi dependence on the ratio K3 K1 indicates that the effects of the spatial period is strictly a. Therefore, the borders of the rst discussed hybridization are most pronounced if the two Brillouin zone lie at jjk ¼ ±π=a. For 0 ≤ k  π=a, we recover a ωðÞ¼ ingredient phonon dispersion relations effectively have about usual acoustic-wave dispersionpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi relation with k vphk and equal weight. The right-going (left-going) triplet has positive ¼ ð þ 2Þ= ≥ fl phase velocity vph a K1 KN N m . For N 3 and (negative) mean energy ow and group velocity. Each triplet = = ωð Þ fi KN K1 > 1 N, a minimum of k occurs inside of the rst contains two forward waves and one backward wave. For the Brillouin zone at wavenumber k  2π=ðNaÞ < π=a. latter, group and phase velocity have opposite sign (see insets in Example dispersion relations for N ¼ 3 and different ratios of Supplementary Fig. 1). These findings are consistent with the = ¼ fi K3 K1 (see different colors) are depicted in Fig. 2b. For K3 0 expectation from Fig. 1b and con rm our reasoning. Further- ≠ and K1 0, we obtain the usual acoustical phonon dispersion more, Supplementary Fig. 3 shows that each of the three right- relation in the first Brillouin zone corresponding to spatial period propagating modes can be excited selectively by tailoring of the ≤ ≤ π= ¼ ≠ a, i.e., 0 k a. For the opposite limit of K1 0 and K3 0, excitation conditions. the mass-and-spring model shown in Fig. 2a falls apart into N ¼ 3 disconnected staggered one-dimensional chains, each with Three-dimensional microstructured elastic metamaterial. Next, spatial period Na ¼ 3a. We thus again obtain a usual acoustical we translate the behavior of the 1D mathematical toy model into phonon dispersion relation, however, the first Brillouin zone is a practical metamaterial structure. From Fig. 2a it is clear that the given by jjk ≤ π=ð3aÞ. For k 2½π=ð3aÞ; 2π=ð3aފ, the group (red) beyond-nearest-neighbor springs unavoidably overlap in velocity is negative, regardless of which Brillouin zone we use two dimensions, making it necessary to go to three dimensions. for the representation of the dispersion relation. The mean energy The 3D architecture depicted in Fig. 3 (also see Supplementary

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Fig. 3 Designed three-dimensional elastic metamaterial structure. a The architecture incorporates nearest-neighbor as well as beyond-nearest-neighbor interactions (cf. Fig. 2a) and is composed of a single ordinary linearly elastic constituent material. The colors are for illustration only. Elements mediating the elastic interaction between one layer and its third-nearest-neighbor along the z-direction are highlighted in red. The blue and red cylindrical rods have a r =a ¼ : r =a ¼ : fl radius of 1 z 0 08 and 2 z 0 12, respectively. The structure has no center of inversion but two mirror planes and a rotation-re ection symmetry, making it achiral and leading to a degeneracy of the lowest two transverse acoustical bands (cf. Fig. 4). The period of the structure along the z-direction is az, the corresponding first Brillouin zone edges lye at wavenumbers kz ¼ ± π=az (cf. Fig. 4). The period or lattice constant along the x- and y-directions is a ¼ a t =a ¼ : t =a ¼ : a ¼ μ ´ ´ xy 2 z. The other geometrical parameters are 1 z 0 40, 2 z 0 60, and z 100 m. b 3 3 5 unit cells out of a corresponding bulk metamaterial, with the front corner cut out to allow for a view inside. The part highlighted in red illustrates the beyond-nearest-neighbor interaction. Two red rods connect a first cube to the red frame (made partly transparent at the corner). Two further red rods connect this frame to a second cube, which has a distance 3az with respect to the first cube. An animated view of the structure is given in Supplementary Movie 1.

Movie 1) is composed of a single ordinary linear elastic con- displacement vector field ~u (see Methods). The used parameters stituent material, e.g., a polymer. The masses in the 1D toy model for the ordinary elastic constituent material refer to a typical ’ ¼ : ’ are replaced by the small cubes with side length t1. The effective polymer with Young s modulus E 4 2 GPa, Poisson s ratio spring constants of the nearest-neighbor (beyond-nearest-neigh- ν ¼ 0:4, and mass density ρ = 1140 kg m−3. These parameters bor) interaction between these cubes are tailored by the radius of merely serve as an example. The frequency axis in Fig. 4 can easily ð Þ ρ fi ν the thin (thick) cylindrical rods r1 r2 . The frame with height t2 be scaled to other values of E, , and a at xed . In Fig. 4a, we serves as an auxiliary structure to mediate the beyond-nearest- find roton-like dispersion relations for the two degenerate neighbor interaction for N ¼ 3. Starting from any one cube, the transverse acoustical bands as well as for the longitudinal oblique rods connect to the auxiliary frame, from which another acoustical phonon mode—as expected from our discussion of set of oblique rods connects to the third-nearest-neighbor of the the 1D toy model. The higher bands plotted in gray are of lesser ¼ fi ω starting cube. Clearly, the two types of rods and the auxiliary importance here. These bands emerging from kz 0 with nite frames introduce substantial additional mass, which needs to be do not occur in the toy model. They are partly due to local considered. The resulting metamaterial structure in Fig. 3 is resonances of the long connecting cylindrical beams. highly anisotropic, has no center of inversion, but two mirror In Fig. 4b, we illustrate the energy flux along the z-direction in planes and a rotation-reflection symmetry, namely a 90-degree one unit cell for three different Bloch modes of the longitudinal rotation around the z-axis combined with a reflection of a plane band at the same frequency of 0:65 MHz. For the two eigenmodes parallel to the xy-plane. The structure is therefore not chiral and A and C with positive group velocity, the mean energy flow in the the lowest two transverse acoustical bands are degenerate by vertical rods, acting as nearest-neighbor springs, and in the symmetry for propagation along the z-axis with wave vector oblique rods, mediating the beyond-nearest-neighbor coupling, is ~ ¼ð ; ; Þ B k 0 0 kz . When designing the elements mediating the positive. In contrast, for mode with negative group velocity, the beyond-nearest-neighbor interaction, it is important that the oblique rods support a backward propagating partial wave, while (local) resonance of these elements are pushed to the partial wave in the vertical rods is a forward wave. The sum of much higher frequencies than the frequencies of the lowest- the two energy flows is negative, consistent with negative group frequency acoustical bands. Otherwise, one obtains a complex velocity. This behavior is the same as for the above 1D toy model. band diagram comprising band crossings and avoided crossings In both cases, the partial forward and partial backward energy such that the roton-like dispersion relation is obscured. Clearly, flow lead to a vortex-like behavior of the energy flow. In his work fulfilling this condition becomes increasingly difficult with on rotons4, Feynman referred to the backward energy flow increasing range of the beyond-nearest-neighbor interaction (i.e., contribution as a “return flow”. with increasing integer N in the toy model) because increasing The complete phonon band structure for all high-symmetry length of beams clearly leads to decreasing beam resonance fre- directions in three dimensions is shown in Fig. 5. Roton behavior quency at otherwise fixed parameters. only occurs along the ΓZ-direction (cf. Fig. 4a). In Fig. 4a, we depict the calculated phonon band structure of The roton-like behavior discussed thus far refers to the bulk, the microstructure shown in Fig. 3. Here, we have numerically i.e., to a metamaterial crystal which is infinitely extended along solved the eigenvalue problem for the Navier equation23 for the all three spatial directions. It is interesting to ask whether the

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ÀÁ Fig. 4 Elastic metamaterial phonon band structure along z-direction. a The dispersion relation ω kz ¼ ωðÀkzÞ of the architecture in Fig. 3 is shown for z k a fi propagation of elastic waves along the -direction with wavenumber z. The spatial period is z, corresponding to the rst Brillouin zone given by the condition kz  π=az (cf. Fig. 1). The two transverse acoustical bands, which are degenerate by symmetry, are plotted in red, the single longitudinal acoustical band in blue. Higher dispersion branches are of lesser importance here and are depicted in gray. They partly result from local resonances within the unit cell, leading to finite values of ω at zero wavenumber kz ¼ 0. b Mean energy flux Iz along the z-direction (on a false-color scale) corresponding to three eigenmodes marked as A, B, and C of the longitudinal band for the same frequency 0:65 MHz. The left column shows an oblique view of the unit cell, the right column a cut through the xy-plane. The mean energy flux in the thin vertical rods in the middle is positive for all three modes. The same holds true for the mean energy flow through the thicker oblique rods for modes A and C. The oblique rods mediate the beyond-nearest-neighbor interactions. In contrast, the energy flux through the oblique rods for mode B is negative, indicating a backward-wave behavior. Integration of the energy flux over the xy-plane for this mode also leads to a negative total energy flow, consistent with a negative group velocity. Parameters are az ¼ 100 μm (cf. Fig. 3), aspect ratios as given in Fig. 3, Young’s modulus E ¼ 4:2 GPa, Poisson’s ratio ν ¼ 0:4, and mass density ρ = 1140 kg m−3 for the constituent material.

Fig. 5 Elastic metamaterial phonon band structure in 3D. As Fig. 4 (for the metamaterial structure shown in Fig. 3), but for many high-symmetry directions rather than only the ΓZ or z-direction as in Fig. 4. a Illustration of the first Brillouin zone of the tetragonal-symmetry real-space lattice and selected high-symmetric directions in reciprocal space (marked in blue). b Calculated three-dimensional phonon band structure with the characteristic directions as indicated in a. Clearly, due to the used tetragonal symmetry, roton-like acoustical dispersion relations only occur for the ΓZ direction. The corresponding colored bands (blue and red) are the same as the ones shown in Fig. 4. behavior is robust and could also be observed in a metamaterial Furthermore, we have emphasized that the structure shown in beam with finite cross section. Therefore, in Supplementary Fig. 3a, which contains the crucial beyond-nearest-neighbor Fig. 4, we show results for a beam with a cross section of merely interactions, is not chiral. Therefore, chirality is clearly not a 2 ´ 2 unit cells (cf. Fig. 3a). A roton-like dispersion relation necessary condition for roton-like behavior. Due to the absence for the transverse bands is maintained. Roton-like behavior of chirality, the two lowest transverse bands in Fig. 4a are is also found for the twist band in Supplementary Fig. 4, degenerate and the associated eigenmodes do not contain any sort which additionally appears due to the finite cross section of of rotation. However, we can introduce chirality into this the beam. metamaterial structure by “twisting the rods” mediating the

NATURE COMMUNICATIONS | (2021) 12:3278 | https://doi.org/10.1038/s41467-021-23574-2 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23574-2 beyond-nearest-neighbor interactions. We additionally double neglected friction between air and the walls (see Methods). The the number of these rods to obtain four-fold rotational symmetry latter assumption has been used many times in the literature46,47 around the z-axis. The resulting structure is illustrated in and is justified if the diameter of the channels is sufficiently large. Supplementary Fig. 5, which can be compared to its achiral Therefore, we consider rather macroscopic parameters in Fig. 6 ¼ counterpart in Fig. 3. The corresponding phonon band structure (az 10 cm). As air exclusively supports longitudinal pressure depicted in Supplementary Fig. 6 again shows a roton-like waves (and no transverse modes), the resulting overall band dispersion relation. In addition, as a result of chirality, structure in Fig. 6a is simpler than the one for elastic waves in the degeneracy of the two lowest transverse bands is lifted Fig. 4. Finally, we depict examples of the energy flow in Fig. 6b. (compare Fig. 4a and Supplementary Fig. 6) and the eigenmodes As expected from the mentioned analogy between forces and air become chiral (see Supplementary Fig. 6), which means that they currents, the behavior is closely similar to that shown in Fig. 4b are directly associated to rotations within the unit cells. This for the 3D elastic metamaterial. aspect makes the connection to the original interpretation of rotons according to Landau2 and Feynman3 in terms of rotations of groups of Helium-4 atoms even closer. Discussion The famous roton dispersion relation for acoustical waves has first been predicted for and later observed in the Bose-liquid Three-dimensional tube-based metamaterial for airborne Helium-4 at low temperatures2–6. More recently, it has also been sound. We now translate the findings of the 1D toy model to discussed for the Fermi-liquid Helium-3 at low temperatures11 airborne acoustical waves (or sound) rather than elastic waves in and for interacting atoms in Bose–Einstein condensates12–17. the previous section. Recall that classical forces (in the sense of Here, we have realized roton-like dispersion relations by designed Newton’s second law), which are mediated by the Hooke’s springs periodic metamaterials for both, elastic waves or phonons in in Fig. 2a or by the cylindrical solid elastic beams in Fig. 3, can be solids and pressure waves in gases. All of these are classical sys- interpreted as momentum currents44,45 in the language of con- tems operating under ambient conditions. The underlying tinuum mechanics. For airborne acoustical waves, the momen- mechanism is based on designed third-nearest-neighbor interac- tum current is directly related to the instantaneous air current. tions in addition to the usual nearest-neighbor interactions. The The air current along the tube axis in a cylindrical tube with rigid third-nearest-neighbor interaction gives rise to a hybridization of walls can obviously be controlled by the inner cross section of phonon branches with different spatial periods and hence to the tube. extraordinary Bragg reflections with reciprocal lattice vectors This analogy allows us to propose a 3D metamaterial smaller than the wave vector at the edge of the first Brillouin architecture for airborne sound. The structure is the complement zone. For both, the proposed (achiral and chiral) 3D microscopic of the one shown in Fig. 4a. This means that the constituent microstructures for elastic waves and the proposed 3D macro- material is replaced by voids and vice versa. Air propagates in the scopic channel-based structures for airborne sound waves, the 3D resulting channels inside a rigid material. By tailoring the inner additive manufacturing technology required to make the meta- diameter of the channels, we engineer the effective strength of material unit cells is readily available. However, large numbers of the nearest-neighbor and beyond-nearest-neighbor interaction, unit cells are needed to avoid edge effects. This aspect together respectively. The calculated acoustical wave dispersion relation with directly measuring the roton-like dispersion relations shown in Fig. 6a again exhibits roton-like behavior. Here we have represents a challenge.

Fig. 6 Roton-like behavior for airborne sound. We consider a structure which is the complementÀÁ of the one shown in Fig. 3. This leads to a network of fl ω k ¼ ωðÀk Þ channels inside a rigid material in which air can ow. a Acoustical dispersion relation z z for air pressure waves propagating along the z-direction, exhibiting a roton-like minimum within the first Brillouin zone kz  π=az. Higher bands at (much) higher frequencies are not shown. b Mean energy flux Iz on a false-color scale along the z-direction for the three eigenmodes A, B, and C marked in a at the same frequency of 150 Hz. As in Fig. 4b, backward-wave behavior is observed for mode B, for which the group velocity is negative. The lattice constant is chosen as az ¼ 10 cm. All other parameter v ¼ = ratios are the same as in Fig. 3. We assume ambient conditions, corresponding to an airborne of air 340 m s and a mass density of ρ ¼ : À3 air 1 29 kgm .

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The approach can be generalized to more than just two types of over one temporal oscillation period has been evaluated by interactions, i.e., to the combined action of nearest-neighbor, () ~ ~ 1 i e ~e* ð Þ¼ ~ ðÞ~ ∇ ~ ðÞ~ ; ð Þ second-nearest-neighbor, third-nearest-neighbor, etc. interac- Ii k Re ~ Pk;i r Pk;i r 10 2 ω ðkÞρ tions. This generalization would allow for tailoring almost any i air ρ ¼ : À3 wanted dispersion relation of acoustical waves or phonons in the with the imaginary unit i and the air mass density air 1 29 kgm . In Fig. 6b, spirit of a Fourier expansion. However, feasible corresponding the z-component of this vector, Iz, is plotted. For the numerical calculations, the Pressure Acoustics Module of Comsol Multiphysics has been used. The quantity Iz three-dimensional microstructures would need to be designed. has been obtained directly by this module. The idea of beyond-nearest-neighbor interactions can also be combined with a variety of established other mechanisms in Data availability metamaterial design to obtain further unusual and useful effective The data that support the plots within this paper and other findings of this study are material behaviors. available from the corresponding author upon reasonable request.

Methods Code availability Energy flow in the 1D toy model. Equation (3) for the energy flow in the 1D toy Numerical simulations in this work for the 1D toy model are all performed using the model has been derived as follows. For the Bloch waves with wavenumber k and commercial software MATLAB. Numerical simulations in this work for the elastic and ω angular frequency , the displacement of the mass at lattice site n is given by acoustic metamaterials are all performed using the commercial software COMSOL ðÞ¼e ð ð À ω ÞÞ ’ un t u exp i kna t . The energy, which is transmitted through the Hooke s Multiphysics. All related codes can be built with the instructions provided in the main springs that couple neighboring masses (cf. Fig. 2a), averaged over an oscillation text and in the Methods section. period, is given by  ÀÁ* 1 dun 1 2 Received: 6 January 2021; Accepted: 30 April 2021; p ðkÞ ¼ Re K u À À u ¼ eu ωK sinðÞka : ð4Þ 1 2 1 n 1 n dt 2 1 ÀÁ À Herein, the term K1 unÀ1 un is the force acting onto the mass at lattice site n by the Hooke’s spring to its left and the symbol * stands for the complex conjugate. Similarly, the mean energy flow through the Hooke’s springs that mediate the beyond-nearest-neighbor coupling is References  1. Kittel, C. Introduction to Solid State Physics (Wiley, 2005). ÀÁ* 2. Landau, L. Theory of the Superfluidity of Helium II. Phys. Rev. 60, 356 (1941). 1 dun 1 2 ð Þ p ðkÞ ¼ Re K u À À u ¼ eu ωK sinðÞNka : 5 fl N 2 N n N n dt 2 N 3. Feynman, R. P. Atomic theory of the two- uid model of liquid helium. Phys. Rev. 94, 262 (1954). The total mean energy flow through the 1D toy model is the sum of these two 4. Feynman, R. P. & Cohen, M. 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25. Kishine, J., Ovchinnikov, A. S. & Tereshchenko, A. A. Chirality-induced National Natural Science Foundation of China (contract 11802017). This research has phonon dispersion in a noncentrosymmetric micropolar crystal. Phys. Rev. additionally been funded by the Deutsche Forschungsgemeinschaft (DFG, German Lett. 125, 245302 (2020). Research Foundation) under Germany’s Excellence Strategy via the Excellence Cluster 26. Joannopoulos, J. D., Johnson, S. G., Winn, J. N. & Meade, R. D. Photonic “3D Matter Made to Order” (EXC-2082/1-390761711), which has also been supported by Crystal: Molding The Flow of Light (Princeton University Press, 2008). the Carl-Zeiss Foundation through the “Carl-Zeiss-Foundation-Focus@HEiKA”, by the 27. Deymier, P. A. Acoustic Metamaterials and Phononic Crystals (Springer, State of Baden-Württemberg, and by the Karlsruhe Institute of Technology (KIT). We 2013). further acknowledge support by the Helmholtz program “Materials Systems Engineer- 28. Liu, Z. Y. et al. Locally resonant sonic materials. Science 289,1734–1736 (2000). ing” (MSE). M.K. is grateful for support by the EIPHI Graduate School (contract ANR- 29. Kaina, N., Lemoult, F., Fink, M. & Lerosey, G. Negative refractive index and 17-EURE-0002). We acknowledge support by the KIT-Publication Fund of the Karlsruhe acoustic superlens from multiple scattering in single negative metamaterials. Institute of Technology. Nature 525,77–81 (2015). 30. Ma, G. & Sheng, P. Acoustic metamaterials: from local resonances to broad Author contributions horizons. Sci. Adv. 2, 1501595 (2016). 31. Cummer, S. A., Christensen, J. & Alù, A. Controlling sound with acoustic M.W. and M.K. had the idea to incorporate beyond-nearest-neighbor interactions, Y.C. metamaterials. Nat. Rev. Mater. 1, 16001 (2016). and M.K. designed the metamaterials. Y.C. performed the numerical simulations. M.W. 32. Kadic, M., Bückmann, T., Stenger, N., Thiel, M. & Wegener, M. 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Phys. 12, 621 (2016). 41. Fruchart, M., Zhou, Y. & Vitelli, V. Dualities and non-Abelian mechanics. Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in 577 – Nature , 636 640 (2020). published maps and institutional affiliations. 42. Fruchart, M. & Vitelli, V. Symmetries and dualities in the theory of elasticity. Phys. Rev. Lett. 124, 248001 (2020). 43. Jin, L. et al. Guided transition waves in multistable mechanical metamaterials. Open Access Proc. Natl Acad. Sci. USA 117, 2319–2325 (2020). This article is licensed under a Creative Commons 44. Planck, M. Bemerkungen zum Prinzip der Aktion und Reaktion in der Attribution 4.0 International License, which permits use, sharing, allgemeinen Dynamik. Phys. Z 98, 30 (1908). adaptation, distribution and reproduction in any medium or format, as long as you give 45. Disessa, A. A. Momentum flow as an alternative perspective in elementary appropriate credit to the original author(s) and the source, provide a link to the Creative mechanics. Am. J. Phys. 48, 365–369 (1980). Commons license, and indicate if changes were made. The images or other third party ’ 46. Liang, Z. X. & Li, J. Extreme acoustic metamaterial by coiling up space. Phys. material in this article are included in the article s Creative Commons license, unless Rev. Lett. 108, 114301 (2012). indicated otherwise in a credit line to the material. If material is not included in the 47. Peri, V. et al. Experimental characterization of fragile topology in an acoustic article’s Creative Commons license and your intended use is not permitted by statutory metamaterial. Science 367, 797–800 (2020). 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/. Acknowledgements We thank Tobias Frenzel (KIT) and Jörg Schmalian (KIT) for stimulating discussions. © The Author(s) 2021 Y.C. acknowledges support by the Alexander von Humboldt Foundation and by the

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