Topological Lifshitz transition in twisted hyperbolic acoustic metasurfaces

Recent works in condensed matter physics have shown that the twist between two-dimensional (2D) bilayer materials is responsible for inducing exciting properties. Its fine tuning can result in magic-angle flatband superconductivity,¹ moiré atomic photonic crystals,^2,3 the creation of moiré excitons,^4–9 interlayer magnetism,¹⁰ quasi-crystalline phases with multiple Dirac cones,¹¹ or helical topological states mosaics,^8,12 for instance, giving birth to the thriving twistronics domain.¹³ These exciting properties arise from the hybridization of the top and bottom band structures related to the creation of moiré superlattices whose parameters are directly related to the twist angle. Some of these concepts have been recently transposed in the wave domain using photonic and phononic composite materials,^14,15 namely, flat bands in bilayer graphene analogs,^16–21 wave localization,^22–24 and steering²⁵ driven by moiré interference patterns, topological synthetic dimensions,²⁶ control of valley topological protection,²⁷ and artificial gauge fields.^28,29

Recent studies³⁰ have extended the concept of twistronics for photons leveraging optical 2D metamaterials featuring extreme anisotropy, namely, hyperbolic metasurfaces (HMTs).^31–35 These structures support in-plane wave propagation with directional features, offering large broadband enhancement of the local density of states, sub-wavelength imaging, negative refraction, and canalization. In Ref. 30, it was shown that two metasurfaces operated in the hyperbolic regime and rotated with respect to one another can enable a topological transition,³⁶ going from hyperbolic to elliptical propagation of light as a function of the twist angle. The latter is the analog of a Lifshitz transition in the electronic band structure,³⁷ which is known for playing a crucial role in the physics of Weyl and Dirac semimetals.³⁸ This transition from open to closed isofrequency contours (IFCs) occurs at specific magic-angles which are tightly linked to the number of anti-crossings between the dispersion of the individual metasurfaces.³⁹ This phenomenon has been experimentally demonstrated in polaritonic systems,^39–42 showing great potential for enhanced nanophotonic devices.⁴³ 

In this paper, we transpose these concepts to airborne sound harnessing the properties of acoustic metamaterials.^44–47 In particular, acoustic hyperbolic metamaterials,⁴⁸ whose strong anisotropic features are exploited in the context of not only guiding⁴⁹ but also sub-diffraction imaging,⁵⁰ have been demonstrated in media whose effective parameters show opposite signs along orthogonal directions of space. Recently, these extreme anisotropic properties have been transposed to the 2D version of metamaterials, namely, metasurfaces,^51,52 which are less prone to dissipation due to their lower dimensionality and whose sound confinement at the surface enables easier access to acoustic field enhancement. In practice, this hyperbolic configuration is obtained using spatially directional resonant modes, whose resonance frequency differs depending on the spatial direction. Knowing that above (below) resonance the medium presents negative (positive) effective parameters in the frequency range between the two orthogonal resonances in the plane, the metasurface effective response flips its sign depending on the spatial direction. If such directional resonances are straightforward to obtain for vector waves, such as in electromagnetism⁵³ or elastodynamics,²⁵ it is not trivial for longitudinal acoustic waves due to their associated scalar symmetry constraints.⁵¹ Nevertheless, this limitation can be circumvented by adding non-local properties in the medium through resonant couplings⁵¹ or structural band folding.⁵² The precise design of the latter permits to introduce dipole-like mode symmetries whose inherent spatial asymmetry and resonant nature provide the requirements for hyperbolic sound propagation.

Hence, inspired by these non-local hyperbolic metasurface designs,^51,52 we achieve hyperbolic sound propagation confined to the close vicinity of the openings of an array of suitably tailored resonators. We then show that a proper out-of-plane coupling between two of such metasurfaces preserves the global hyperbolic features. Then, inducing a twist between the two layers, we demonstrate a topological phase transition within the system. Such a transition arises at a magic-angle, associated with a topological invariant corresponding to the integer number of anti-crossing points $NACP$ between the IFCs of each metasurface when isolated. Finally, we investigate the directional excitation of these guided surface acoustic waves through complex multiple source patterns. This work evidences the great potential of twisted hyperbolic metasurfaces for enhanced airborne sound manipulation.

As a starting point, we exploit the benefits of hyperbolic sound propagation over an acoustic hyperbolic metasurface, as shown in Fig. 1(a). Inspired by Ref. 51, our structure consists of a sub-wavelength scaled square lattice of tubes opened on both sides (gray) and tightly binded to each other with coupling tubes in the $x$ (green) and $y$ (blue) directions. The step of the lattice is $a=4 cm$⁠, the side length of the opened tubes is $a/2$⁠, and their height is $h=15 cm$⁠. As mentioned, one solution to circumvent the symmetry constraints of scalar acoustic waves for the realization of hyperbolic metasurfaces is to exploit non-locality. Here, the necessary non-local features are obtained through coupling tubes with an elbow, responsible for their resonant behavior. A simple tuning of its height allows to genuinely control the corresponding resonant frequency. Hence, detuning these resonances along the $x$ and $y$ axes leads to a negative (positive) coupling along the $x$ (⁠$y$⁠) direction for the frequency range between the two resonant modes. This results in the band shown in Fig. 1(b), computed with COMSOL Multiphysics, which presents a positive (negative) slope along the $y$ (⁠$x$⁠) direction consistent with the coupling signs, yielding hyperbolic IFCs. They lay outside the sound cone (centered circles), which evidences the deeply sub-wavelength nature of the corresponding modes in the medium.

Thus, when an acoustic point source with a frequency of $1040 Hz$ is placed near the top of the metasurface [yellow star in Fig. 1(c)], the excited waves remain confined in the close vicinity of the tube openings, as they propagate along the bottom and top interfaces [Fig. 1(c)]. The pressure field map taken $5 mm$ away from the bottom interface shows propagation of a guided surface wave with negative curvature wavefronts, proving its hyperbolicity [Fig. 1(e)]. This effect is confirmed by the hyperbolicity of the IFC obtained by the bidimensional spatial Fourier transform (FT) of the corresponding field map shown in Fig. 1(d) [corresponding to the black line in Fig. 1(b)]. The IFC is almost flat close to the free space IFC (white circle), denoting a canalization⁵⁴ regime at this frequency, in good agreement with the guided acoustic wave shown in Fig. 1(e).

We have designed a hyperbolic metasurface able to guide acoustic surface waves on both of its sides. This allows us to envision the superposition of multiple of these devices along the $z$ axis. As depicted in Fig. 2(a), we now study the interaction of two aligned hyperbolic metasurfaces on top of each other, separated by a distance $hs$ in air. Figure 2(b) shows the band structure of this binary system as a function of $hs/a$⁠. For the sake of clarity, the latter is represented as projections along the main directions of the first Brillouin zone (⁠$ΓM, ΓX, ΓY,MX, and MY$⁠).

First, when $hs≫a$⁠, the dispersion of the system is simply a superposition of two identical band structures: the metasurfaces are completely uncoupled. Then, when $hs=0.5*a$⁠, a degeneracy lift occurs. Notably by decreasing $hs$ even further to $0.05*a$⁠, one of the two dispersions is strongly shifted toward the lower frequencies, while the other one is slightly shifted toward the high frequencies. The corresponding field profiles clearly evidence a symmetric mode along the $z$ direction (⁠$S$⁠, black circle) for the low frequency band, while the high frequency one is related to an antisymmetric profile (⁠$AS$⁠, black triangle). This asymmetric degeneracy lift is explained by the combination of near field coupling and resonant multiple scattering mediated by the wave propagating in the interspacing. Hence, finely tuning $hs$⁠, it is possible to obtain a bilayer system, which behaves similarly to a single hyperbolic metasurface. Indeed, the band structure remains out of the free-space cone [gray zone in Fig. 2(b)], and the corresponding waves are still confined at the surface of the two sides of the coupled metasurface system, as evidenced by the symmetric and antisymmetric (⁠$x,z$⁠) pressure field maps.

Now, we are ready to investigate the effect of rotation within this bilayer. As displayed in Fig. 3(a), we rotate the Top metasurface (green), while the Bottom one stands still (red), exciting the system with a source near the upper interface of the top metasurface (yellow star). In reciprocal space, this amounts to hybridize the IFC contours of the bottom (static, red line) and top (rotated, green line) metasurfaces [Fig. 3(b)], leading to two IFCs anti-crossing each other at specific locations of the first Brillouin zone. According to the twist angle $Θ$⁠, the number of anti-crossing points (⁠$NACP$⁠) is either $2$ or $4$⁠. In the first case, the hybridization of the two hyperbolic IFCs leads to an IFC, which is still open. In the second case however, the IFC closes. These two scenarios are related to two topologically different phases: the elliptic phase, where the waves are allowed to propagate in every direction of the plane, and the hyperbolic phase, where some directions are forbidden, leading to collimated surface wave propagation. By tuning the twist angle $Θ$⁠, we can drive a transition between these two phases, whose transition point occurs at the magic angle $Θ*,$³⁹ where the hybridized IFC is flat in the first Brillouin zone. This phenomenon occurs if the IFCs are geometrically able to cross each other four times, which adds constraints to the hyperbolicity of the metasurface. Namely, the corresponding IFCs should cross the first Brillouin zone orthogonal borders. This is the case at $f=1040 Hz$ but not at $f=1200 Hz,$ for instance [Fig. 1(b)]. We note here that in twisted bilayer graphene, the magic angle is associated with the emergence of flat energy bands whose combination with the fermionic nature of electrons leads to dissipation-less (superconductivity) electronic transport.¹ In the acoustic analog presented here, the magic-angle is associated with flat contours associated with diffraction-free (canalized) propagation of sound at the surface, which is also intrinsically associated with an enhanced wave–matter interaction.⁵⁵ If in both cases, the effect is due to the modification of the interlayer coupling with the twist, superconductive electronic flat bands are linked to moiré (structural) features of the twisted bilayer stacking, while the physics of acoustic twisted hyperbolic metasurfaces is ruled by the contour's hybridization of homogenized layers.

We carried out finite-size simulations of the bilayer in COMSOL Multiphysics for a spacing distance $hs=0.05*a$⁠. We set a point source near the top of the medium, at a frequency of $1040 Hz$⁠, and obtained the (⁠$x,y$⁠) pressure field map $5 mm$ at the bottom interface. We then computed its bidimensional spatial Fourier transform (FT) to retrieve the corresponding IFC. We carried out this procedure as we rotate the top layer (green) by an angle $Θ$ between $0°$ and $90°$⁠. The corresponding field maps are displayed on Fig. 4(a) above their FT on Fig. 4(b) for each angle. For the sake of clarity, Fig. 4(c) shows the rotated top (green), static bottom (red), and hybridized IFCs of the metasurface, in order to track the topological phase transition as a function of the twist angle $Θ$⁠. The dotted line is the free-space IFC.

For $Θ=0°$⁠, the field map and FT are clearly hyperbolic and identical to those of a single metasurface, as shown in Figs. 1(e) and 1(f). The center white circle is the free-space IFC. As we rotate the Top layer by $Θ=15°$⁠, a similar guided wave occurs but rotated by the same angle. The FT map shows two separate IFCs with anti-crossing at the location of the green and red IFCs superposition point [Fig. 4(c)]. For $Θ=30°≃Θ*,$ not only the guided wave direction rotates by $Θ$⁠, but also the wavefronts have changed. Indeed, they are now flat, evidencing the emergence of canalization, confirmed by the FT where the IFCs are flatter.

When $Θ=60°>Θ*$⁠, the corresponding wavefronts curvature undergoes a progressive inversion, and it is now positive, which is directly related to the progressive closing of the IFC. This is confirmed by the FT map, which shows that the IFC closest to the center of the first Brillouin zone is almost closed and elliptical. For angles larger than $Θ=60°$⁠, the twisted bilayer is in another topological phase, as proven by the elliptical (⁠$Θ=75°)$ or circular (⁠$Θ=90°)$ IFCs shown by the FT maps. The corresponding surface waved is now totally delocalized. Interestingly, the pressure field amplitude is enhanced for twist angles below $Θ=60°$⁠, coinciding with flatter contours. This is in good agreement with the canalized regime observed in the vicinity of the topological phase transition.⁵⁵ 

These results undoubtedly demonstrate the sonic equivalent of a Lifshitz transition from a hyperbolic phase to an elliptic one driven by the twist between two hybridized acoustic hyperbolic metasurfaces. Interestingly, its topological nature accounts for its robustness: as long as the twist angle is smaller than the magic-angle, the bilayer system is able to guide and steer surface waves. Moreover, unlike moiré phenomena studied in condensed matter, the physics at play here does not depend on superlattice effects, which undeniably loosens the constraints on the specific value of the magic-angle. The less precise and larger twists allow us to envision designs whose functionality is significantly less sensitive to errors and sample fabrication. Furthermore, the fact that some of these contours are in the vicinity of the free-space circle [dotted line in Fig. 4(c)] allows for an efficient coupling with outside radiation, with promising opportunities for directional far-field radiation of a local point source mediated by the twisted bilayer device.

The presence of dissipation, such as thermo-viscous losses, has to be taken into account when moving toward experimental implementation. Nevertheless, an accurate design of the operating frequency and dimensions of the device can mitigate such damping phenomena, preserving the topological transition (see the supplementary material, Fig. S1).

Finally, we investigate the interplay between the twist and the chirality of the propagating waves. Here, we take advantage of the different chiral features of the hyperbolic branches.⁵⁶ To serve this purpose, we carry out similar simulations using two sources, as depicted in Fig. 5(a) (red and blue stars). Tailoring the phase shift between them, we can preferentially excite the bottom or top branch of the corresponding IFC, which feature opposite chirality of their eigen-modes. Figure 5(b) demonstrates this effect in the case of a bilayer system (⁠$hs=0.1*a$⁠) without twist, at $f=1105 Hz$⁠, with an almost flat IFC, as shown by the corresponding 2D spatial FT [Fig. 5(c)], supporting directional canalization of sound at the surface of the device. Rotating the bottom layer by $Θ=90°$ adds two propagation directions along $x$⁠, as displayed in Fig. 5(d), which is consistent with the square shape of the modified IFC [Fig. 5(e)]. Furthermore, by increasing the interlayer coupling by getting the two twisted metasurfaces closer, we modify the shape of the IFC, orienting the flat parts along diagonal directions in reciprocal space. This is shown in Figs. 5(f) and 5(g) for $hs=0.01*a$ and $Θ=90°$⁠, where sound is now guided toward the bottom corners of the metasurface. Hence, we clearly evidence directional wave guiding and steering at the surface of the device, as shown on the angular transmission profile in Fig. 5(h), controlled by both the twist and spacing between the metasurfaces.

To conclude, in this paper, we demonstrated the sonic analog of a Lifshitz transition with twisted bilayer hyperbolic acoustic metasurfaces. Like its optics counterpart, this transition is described by the variation of an integer quantity, namely, the number of anti-crossing points between the IFCs of the bottom and top layer. The topological nature of the transition points, known as magic-angles, accounts for a great robustness, allowing to envision applicable designs whose functionality is significantly less sensitive to errors and sample fabrication. We also showed the straightforward tunability of the system by implementing tunable canalization and directional excitation of acoustic surface waves over this platform. This work transposes fundamental concepts from condensed matter physics to classical waves, and it provides great opportunities for applications related to acoustic surface waves manipulation, namely, twistronics for sound. For instance, enhanced surface acoustic wave control is particularly relevant in the domains of microelectronics, filters, haptics, and microfluidics, where the precise manipulation of enhanced pressure fields at the surface may improve particle or cell trapping, sorting, and tweezing. Twistronics for sound also opens research avenues for active wave control through dynamical twisting, for the design of these devices' far-field features in the context of reconfigurable sonic leaky-wave antennas and for their transposition to on-chip scales.

See the supplementary material for the results in Fig. 4 taking into account realistic dissipation in the form of thermo-viscous losses in the involved materials.

This work was supported by the Air Force Office of Scientific Research and the Simons Foundation.