Lossy phononic metamaterials for valley nonreciprocity

Abstract

Non-Hermitian physics — characterized by complex band spectra — established a new paradigm in condensed matter systems and metamaterials. Combined to nonreciprocity, non-Hermitian gain can be arbitrarily injected in a system to manipulate its valley dynamics, leading to phenomena such as the amplified topological whispering gallery modes for acoustic lasing. Here, we reveal that the coupling loss can also be used to manipulate valley degrees of freedom in a phononic metamaterial. We theoretically and experimentally show three distinct effects induced by valley nonreciprocity, including unidirectional transport of valley bulk states that functions as a valley filter, valley-dependent skin effects, where bulk states from different valleys localize at opposite boundaries, and valley-projected edge states with boundary-dependent lifetimes that leads to an anomalous beam routing. Owing to the simple control over losses demonstrated here, our findings shed light on the interplay between non-Hermitian and valley physics, providing a route to applications of valley-based devices.

Introduction

In recent years, non-Hermitian (NH) physics, which explores the unique properties and novel phenomena of open systems described by NH Hamiltonians, has attracted extensive interest in condensed matter systems and metamaterials^1,2,3. Under NH interactions of gain, loss, or nonreciprocity, the eigenvalues of the NH Hamiltonian are generally complex numbers, with their real and imaginary parts corresponding to energy storage and input/output. Meanwhile, the NH eigenstates are nonorthogonal and can even coalesce with each other⁴. Regarding topological properties, in addition to the wave-function topology that predicts conventional boundary states^5,6, complex eigenvalues introduce a new type of topology, that is, spectral topology^7,8. The spectral topology is characterized by the winding number of the complex eigenvalues, which guarantees the NH skin effect, that is, allowing bulk states to manifest as skin modes collapsing to open boundaries^9,10,11. So far, various configurations of skin effects, typically enabled by nonreciprocity of couplings, have been experimentally visualized on different platforms, including one-dimensional (1D) skin effect^12,13,14,15, higher-order skin effect^16,17, geometry-dependent skin effect^18,19,20,21, and hybrid skin-topological effect^22,23,24,25. These not only greatly promote the development of the NH physics, but also facilitate advances in applications^26,27,28, including topological switch and sensor.

Discrete valleys, as a versatile degree of freedom, have been exploited as efficient carriers for processing and storing information and energy^29,30. A variety of valley-related phenomena, such as valley Hall effect^31,32,33,34, valley filters^35,36,37,38, and valley Zeeman effect^39,40,41,42, have been explored both theoretically and experimentally. Moreover, valley bulk transport with locked chirality^43,44,45,46 and topological valley edge transport with immunity to lattice imperfections^{47,48,49,50,51,52} have been extensively reported in optical, acoustic, and on-chip metamaterials. Recently, the introduction of the non-Hermiticity has opened alternative avenues for valley manipulations^53,54,55,56. For instance, a proposed NH Moiré valley filter can be driven by a valley-resolved NH skin effect with the valley-resolved nonreciprocal transport⁵⁴. In practice, with a gain medium tuned by the electro-thermoacoustic coupling, amplified topological whispering gallery modes can be engineered as acoustic lasers in phononic metamaterials⁵⁵. And NH Dirac cones with valley-dependent lifetimes enabled by nonreciprocal coupling with an operational amplifier have been realized in electric circuits⁵⁶. However, the gain or nonreciprocity interactions are typically implemented by active devices, which must be deliberately designed and complicate the structure. In contrast, pure loss naturally occurs in materials and is easy to implement, yet it is generally regarded as a detrimental factor. It is therefore desirable to explore the beneficial effects of loss and its potential applications in valley physics.

In this work, we reveal three interesting NH valley phenomena induced by valley nonreciprocity in a honeycomb lossy phononic metamaterial. The coupling loss is readily implemented by drilling holes on the coupling tubes and sealing them with the sound-absorbing sponges. Remarkably, the complex bulk dispersions exhibit valley nonreciprocity, that is, the imaginary energy reverses in a single valley and is opposite between the two valleys. This unique feature can give rise to unidirectional transport of valley bulk states, therefore functioning as an NH valley filter. Meanwhile, it leads to opposite spectral topologies for the two valleys and to a valley-dependent skin effect, in which the bulk states near different valleys are localized at opposite boundaries. And it is inherited by edge states, leading to NH valley-projected edge states with boundary-dependent lifetimes that exhibit anomalous beam routing beyond the Hermitian scenario. These unique NH valley phenomena are corroborated by airborne sound experiments that are in excellent agreement with theoretical predictions and numerical simulations.

Results

Tight-binding model

We start from a honeycomb lattice with coupling loss and staggered on-site energy, as depicted in Fig. 1a. The corresponding Hamiltonian in momentum space reads

$$H=({d}_{x}-i\gamma ){\sigma }_{x}+{d}_{y}{\sigma }_{y}+{d}_{z}{\sigma }_{z},$$

(1)

where ${d}_{x}={t}_{0}+2{t}_{0}\cos \frac{\sqrt{3}{k}_{y}}{2}\cos \frac{{k}_{x}}{2}$, ${d}_{y}=-2{t}_{0}\sin \frac{\sqrt{3}{k}_{y}}{2}\cos \frac{{k}_{x}}{2}$, ${d}_{z}=m$, ${\sigma }_{i}$ ($i=x,y,z$) are Pauli matrixes. Here, ${{{\bf{k}}}}=\left({k}_{x},{k}_{y}\right)$ is the Bloch wavevector, m is on-site energy, ${t}_{0}-i\gamma$ is the intracell NH hopping with γ marks the strength of loss, while t₀ is the intercell Hermitian hopping. The lattice constant is set to unity for simplicity. Under the coupling loss, the system possesses the spinless anomalous time-reversal symmetry (aTRS), expressed as ${U}_{T}{H}^{t}\left({{{\bf{k}}}}\right){U}_{T}^{-1}=H\left(-{{{\bf{k}}}}\right)$, where U_T is a unitary matrix and the superscript t denotes transpose operation. Note that, H^t(k) and H(k) have the same eigenvalues, giving rise to energy spectra $E\left({{{\mathbf{k}}}}\right)=E\left(-{{{\mathbf{k}}}}\right)$ (Supplementary Note 1A).

Fig. 1: Non-Hermitian valley phenomena in a lossy honeycomb lattice.

a Schematic of the Non-Hermitian (NH) lattice model. The yellow area denotes the unit cell, which consists of two nonequivalent sites 1 (red) and 2 (blue) with opposite on-site energies. b Complex bulk band dispersions around the K and K' valleys. The colors denote the imaginary part of the dispersions. The black lines mark the first BZ. Inset: Schematic of the valley nonreciprocal transport. c Spectral winding numbers along the k_y' direction varies with ${k}_{{x}^{{\prime} }}$. Inset: Schematic of the parameter evolution path. d Real part of the projected dispersions of a ribbon in the inset. The positive and negative values of the colors indicate the localization degree of the bulk states at the upper-left and lower-right boundaries of the ribbon, respectively. Inset: Field distributions of two eigenstates denoted by blue and red stars in the dispersions. e Phase diagram determined by the valley Chern number C_K in the $\gamma /{t}_{0}$ and $m/{t}_{0}$ plane. The gray regions represent bandgap closure. The cyan stars denote the phases with the specific parameters used in (f), that is, $m=-0.3$ for phase I and $m=0.3$ for phase II. f Complex projected dispersions of a ribbon with periodicity in the x direction, containing interfaces I-II and II-I. The colors denote the imaginary part of the dispersions. The other parameters are chosen as ${t}_{0}=1$ and $\gamma=0.2$ throughout; additionally, $m=-0.3$ in (b–d).

The complex bulk band dispersions near the K and K' valleys are shown in Fig. 1b, with band gaps opened. Interestingly, as the real and imaginary parts of the dispersions are shown, the complex bulk band dispersions reveal the valley nonreciprocity, arising jointly from aTRS and the imaginary energy reversal properties (Supplementary Note 1B). Such a distinctive feature leads to unidirectional transport of valley bulk states. Physically, the imaginary part of the eigenvalues determines the lifetimes of the states: the smaller the imaginary eigenvalues, the longer the lifetimes. The left-moving bulk states near the K valley possess longer lifetimes than the right-moving bulk states within the same valley, while the behavior is reversed at the K' valley as required by aTRS. That is, our model can only support the right-moving bulk states near the K' valley and the left-moving bulk states near the K valley in long-time limit, thus exhibiting unidirectional transport, as depicted by the inset in Fig. 1b.

Next, we analyze the topological properties of the system, including both spectral and wave-function topologies. The complex bulk band dispersions with valley nonreciprocity can induce valley-related spectral topology, which give rise to valley-dependent skin effects. The spectral winding numbers along the k_y' direction as a function of ${k}_{{x}^{{\prime} }}$ are presented in Fig. 1c. It shows that the spectral winding numbers are inverted near the K and K' valleys, indicating that the states near different valleys will be localized at the opposite boundaries along the x' direction, constituting the so-called valley-dependent skin effect. To verify this, we consider a ribbon with periodic boundary conditions in the x' direction and open boundary conditions in the y' direction, and calculate the projected dispersions. The corresponding results are shown in Fig. 1d, where the positive (negative) values of the colors indicate the localization degree of the bulk states at the upper-left (lower-right) boundary. Clearly, the states near the K and K' valleys are located at the opposite boundaries, exhibiting a valley-dependent feature. This can be further visualized by the eigenfield distributions of the states near the K and K' valleys shown in the inset of Fig. 1d, which are confined at the upper-left and lower-right boundaries, respectively, demonstrating the valley-dependent skin effect. See detailed calculations in Supplementary Note 1C.

The wave-function topology of the system can be characterized by the valley Chern number, which guarantees the appearance of the valley-projected edge states. The phase diagram described by the valley Chern number C_K as a function of $\gamma /{t}_{0}$ and $m/{t}_{0}$ (${C}_{{K}^{{\prime} }}=-{C}_{K}$, as required by aTRS) is shown in Fig. 1e, where exist three topologically distinct phases. In the orange and purple regions (referred to as phases I and II, respectively), the system has a bulk band gap with ${C}_{K}=\frac{1}{2}$ and $-\frac{1}{2}$, respectively; at the gray region, the system has no bulk band gap and the valley Chern number cannot be well defined. According to the bulk-boundary correspondence, the valley-projected edge states will emerge at the interface between two phases with opposite valley Chern number. Therefore, when phases I and II are combined to form an I-II-I heterostructure, the valley-projected edge states appear at both interfaces I-II and II-I, as shown by the complex projected dispersions in Fig. 1f. Notably, due to inheriting the valley nonreciprocity of the bulk, only the valley-projected edge states at the interface I-II persist in the long-time limit, whereas those at the interface II-I decay rapidly, exhibiting a boundary-dependent lifetime. More details can be seen in Supplementary Note 1D–F.

Unidirectional transport of valley bulk states

The tight-binding model can be directly implemented in a phononic metamaterial of cavity-tube structure filled with air, as illustrated in the enlarged view inserted in Fig. 2a. The schematic of the acoustic unit cell of the NH honeycomb lattice is depicted in Fig. 2b, corresponding to the yellow rhombus region in the inset in Fig. 2a. Structurally, the unit cell is composed of two nonequivalent acoustic cavities (yellow), labeled as 1 and 2, which are coupled through some identical rectangular waveguides (brown). The acoustic cavities have a side length of ${w}_{1}=6$ mm, and heights of ${h}_{1}=30.4$ mm and ${h}_{2}=29.6$ mm. The width of the waveguides is ${w}_{2}=5$ mm, and the lattice constant is $a=40$ mm. Green circles with diameters of $d=3.6$ mm on the waveguide represent three holes, which are used to introduce loss by sealing with the sound-absorbing sponges in the experiments. In simulations, the designed loss is realized by adding an imaginary part ($30i$ ${{{\rm{m}}}}/{{{\rm{s}}}}$) on the sound velocity of the corresponding waveguides (More details can be seen in Supplementary Notes 2 and 3). The simulated complex bulk band dispersions near the K valley for the acoustic unit cell and the corresponding complex projected dispersions are shown in Fig. 2c and d, respectively. One can see that the left-moving states near the K valley and the right-moving states near the K' valley possess longer lifetimes and thus exhibit a valley nonreciprocity, consistent with the theoretical results.

Fig. 2: Realization of the acoustic non-Hermitian valley filter.

a Photograph of the sample. Inset: Enlarged view of the domain B with coupling loss, where the yellow shadow marks a unit cell. The designed loss is achieved by the holes on the waveguides sealed with the sound-absorbing sponges. b Schematic of the acoustic unit cell. Green circles represent the holes on the waveguide. c Simulated complex bulk band dispersions near the K valley. d Simulated projected dispersions of a ribbon with periodic boundary conditions in the x direction. e Simulated pressure field distribution at 5.86 ${{{\rm{kHz}}}}$ for the ABA sandwich-shaped structure, which is excited by ten point-sources (blue stars) on the left  boundary. f Top panel: Measured spatial Fourier spectra of the acoustic field within the green rectangles at the input (left) and output (right) ports of ${{{\rm{ABA}}}}$, respectively. Bottom panel: The corresponding simulated results.

Based on the valley nonreciprocity, we realize unidirectional transport of valley bulk states and design an acoustic valley filter. The designed acoustic valley filter is shown in Fig. 2a, which consists of two types of domains A and B (separated by the blue dashed lines), corresponding to acoustic Hermitian and NH honeycomb lattices, respectively. The Hermitian domain is constructed similarly to the NH domain but without holes on their waveguides, and it hosts two valleys with real eigenfrequencies at the K and K' points. Such a sandwich-shaped structure (denoted as ABA) is a valley filter. To show this, ten point-sources are placed on the left boundary of the sample to excite the acoustic pressure field, and the sound pressure and phase were measured by drilling holes on the top of the entire sample, which are sealed when not in use. The simulated pressure field distribution of ABA at 5.86 kHz is shown in Fig. 2e. In experiment, we measure the sound signals within the green boxes at the input and output ports of ABA, and perform a 2D Fourier transform. The corresponding results are respectively shown in the upper panels of Fig. 2f. As we can see, the hybrid valley states containing both K and K' valleys are excited at the input port, while only the states near the K' valley arrive at the output port. The experimental data well capture the simulations in the lower panels of Fig. 2f, demonstrating the unidirectional transport of valley bulk states and valley filtering. It is worth noting that, compared to the previous proposals^38,54, the current scheme utilizes a NH strategy and has the advantages of a simple structure and no need to modify the system boundaries.

Valley-dependent skin effect

In the following, we numerically and experimentally validate the valley-dependent skin effect. The simulated real part of the projected dispersions for the ribbon, which is periodic in the x' direction and finite in the y' direction, is presented in Fig. 3a. The positive (negative) values of the colors represent that the states are localized at the upper-left (lower-right) boundary, and the eigenfield distributions of the states denoted by stars are displayed in the inset. It shows that the states near the K and K' valleys are confined at the opposite boundaries, exhibiting the valley-dependent property and in good agreement with the theoretical results.

Fig. 3: Observation of valley-dependent skin effect.

a Simulated real part of the projected dispersions of a ribbon, which is periodic in the x' direction and finite in the y' direction. The positive and negative values of the colors indicate the localization degree of the bulk states at the upper-left and lower-right boundaries, respectively. Inset: Field distributions of two eigenstates denoted by blue and red stars in the dispersions. b Measured pressure field distribution at 5.86 kHz for a finite-size sample with radiative boundaries in the x' direction. It is mainly localized on the upper-left and lower-right boundaries, visualizing the skin modes. c Measured (bright color) projected dispersions by Fourier transforming the pressure field at the lower-right boundary. White dots represent the simulated results. d The same as (c), but for the upper-left boundary.

Such valley-dependent skin effect can be clearly revealed by our airborne sound experiments. We construct a finite-size phononic metamaterial with open boundaries in the y' direction and radiative boundaries in the x' direction, and measure the pressure field distribution for fixed frequencies. The source excites at each cavity, and a microphone probes the pressure response at the cavity at the next period along the x' direction. The measured pressure field distribution at 5.86 kHz is shown in Fig. 3b. It is strongly confined at the upper-left and lower-right boundaries, confirming the skin states in the finite-size sample. We further verify that these skin states are valley-dependent. We position a point source at the center close to the lower-right boundary to excite skin states there (Supplementary Note 4). The measured projected band dispersions are displayed visually with the bright colors in Fig. 3c, the experimental data agrees well with the simulation denoted by white dots, and shows that the skin states confined at the lower-right boundary are mostly located near the K' valley. Similarly, the point source positioned close to the upper-left boundary can excite the skin states there well, and the measured projected dispersions show that the skin states localized at the upper-left boundary are mostly located near the K valley, as shown in Fig. 3d. The simulations together with experiments demonstrate the valley-dependent feature of the skin effect.

Valley-projected edge states with boundary-dependent lifetimes

We then study the valley-projected edge states with boundary-dependent lifetimes. The phase diagram, revealed by the real part of the band-edge frequencies at K point versus the height difference between the two nonequivalent cavities of the acoustic unit cell, is presented Fig. 4a. Evidently, the band gap remains closed when ∆h is small (here $\left|\Delta h\right| < 0.4$ mm), while opens and forms valleys as |∆h| gradually increases ($\left|\Delta h\right| > 0.4$ mm). We have checked that the valleys for $\Delta h < -0.4$ mm and $\Delta h > 0.4$ mm possess opposite valley Chern numbers, that is, ${C}_{K}=\frac{1}{2}$ for the former and ${C}_{K}=-\frac{1}{2}$ for the latter, referred to as phases I and II, respectively. The bulk band dispersions (with band gap of 5.51–6.00 kHz) of phase I with $\Delta h=-3$ mm are displayed in Fig. 4b. The mirror counterpart (phase II) with $\Delta h=3$ mm has exactly the same bulk band dispersions but the inversed valley topology. Similarly, we consider a ribbon of heterostructure I-II-I composed of phases I and II and calculate its complex projected dispersions, as shown in Fig. 4c. Evidently, owing to inheriting the valley nonreciprocity from the bulk states, the valley-projected edge states (blue) along the interface I-II possess longer lifetimes than those (red) along the interface II-I. This indicates the boundary dependence of their lifetimes, which is consistent with the tight-binding results.

Fig. 4: Valley-projected edge states with boundary-dependent lifetimes and their anomalous beam routing.

a Phase diagram revealed by the real part of the band-edge frequencies at K point as the function of $\Delta h={h}_{2}-{h}_{1}$, where $({h}_{1}+{h}_{2})/2=30$ mm. The gray shadow ($\left|\Delta h\right| < 0.4$ mm) corresponds to bulk band gap closure. The light orange ($\Delta h < -0.4$ mm) and light purple ($\Delta h > 0.4$ mm) areas with opposite valley Chern numbers are denoted as phase I and II, respectively. b Simulated bulk band dispersions of phase I with $\Delta h=-3$ mm (gray dashed line in a). c Simulated complex projected dispersions for heterostructure I-II-I. The colors denote the imaginary part of the dispersions. d Simulated pressure field distributions at $5.75$ kHz for interface structures I-II (top) and II-I (bottom). e, f Measured (bright color) dispersions by Fourier transforming the fields at the interfaces for structures I-II and II-I, respectively. White dots represent the simulated results. g Simulated pressure field distribution at $5.75$ ${{{\rm{kHz}}}}$ in the sample of an arrow-shaped interface intersection. h Measured transmitted pressure at port L, D, and R. The yellow region indicates the range of the band gap. The blue stars and lines in (d and g) represent the sound sources and the interfaces, respectively.

The valley-projected edge states with boundary-dependent lifetimes can be demonstrated by the simulations and experiments. As shown in Fig. 4d, we fabricate two samples of interfaces I-II and II-I, and place a point source at the center of the interfaces to excite the valley-projected edge states. As exemplified by the simulated pressure field distributions at 5.75 kHz, the valley-projected edge states on interface I-II propagate along $\pm x$ directions with negligible attenuation, while those on interface II-I scarcely propagate, even though neither interface involves lossy couplings. In experiments, we measure the projected dispersions for the two cases by Fourier transforming the measured pressure field at the interfaces, respectively. As displayed in Fig. 4e and f, the experimental data (color map) precisely captures the simulation results (white dots) and shows the good excitation of both the right- and left-moving valley-projected edge states on interface I-II, but no excitation of that on interface II-I. (The valley-momentum locking properties of the surviving edge states can be seen in Supplementary Note 5.) All these simulations and experiments explicitly demonstrate the boundary dependence of the lifetimes for the valley-projected edge states.

Such a property can lead to an anomalous beam routing at the intersection. As presented in Fig. 4g, we precisely fabricate an arrow-shaped interface intersection, which contains four ports marked by U, L, D, and R. A point source is positioned at port U to excite the valley-projected edge states, and the corresponding simulated pressure field distribution is presented by color map in Fig. 4g. It shows that the K valley-projected edge states can propagate predominantly along the interface I-II and reach the output port L, while being negligible at ports D and R. This behavior is noticeably distinct from the Hermitian case, in which the K valley-projected edge states can reach both ports L and R (Supplementary Note 6). To emphasize this fundamental difference, we term the non-Hermitian behavior as anomalous beam routing. Experimentally, we measure the transmitted pressure at ports L, D and R. The results are presented in Fig. 4h, which shows that port L allows pressure transmission, while ports D and R almost forbid it, in agreement with the simulations.

Discussion

In summary, we have observed three unique NH valley-related phenomena enabled by valley nonreciprocity in a lossy phononic metamaterial. Benefiting from such unique properties, we have designed a NH valley filter and demonstrated unidirectional transport of valley bulk states. Unlike previous proposals that require complicated structural designs or boundary modifications, our design provides a simple and universal approach. Based on the spectral topology, our work has built an experimental bridge between valley and skin effects and demonstrated the valley-dependent skin effects. At the boundary, the NH valley-projected edge states, protected by the wave-function topology, exhibit boundary-dependent lifetimes and can be exploited to achieve anomalous beam routing. Our work reveals a fascinating interplay between non-Hermiticity and valley physics, and paves the way for applications of NH valley-related devices.

Methods

Numerical simulation

All the simulations are performed by the commercial COMSOL Multiphysics solver package. The systems are filled with air (with a sound velocity of 343 ${{{\rm{m}}}}/{{{\rm{s}}}}$ and air density of 1.29 ${{{\rm{kg}}}}/{{{{\rm{m}}}}}^{2}$ at room temperature). We focus on the dipole mode of the cavity along the z direction. The resin used for 3D printing is modeled as an acoustic rigid wall, considering its huge impedance compared to the air. The simulated band dispersions and pressure field distributions of the ribbon in the main text are calculated by applying Bloch boundary conditions in the periodic directions in the corresponding unit cell or supercells. For brevity, trivial edge states localized at the edges rather than the interfaces are eliminated from all projected dispersions in both tight-binding calculations and acoustic simulations. All the simulated pressure field distributions are normalized by the corresponding maximum values. The spatial Fourier spectra in the lower panel of Fig. 2f are obtained by Fourier transforming the corresponding simulated acoustic pressure fields. In Fig. 3a, the specific value of the color is calculated by $D={\sum }_{x\in {L}_{1}}{\left|\psi \left(x\right)\right|}^{2}-{\sum }_{x\in {L}_{2}}{\left|\psi \left(x\right)\right|}^{2}$, where $\psi \left(x\right)$ is normalized eigenstate, L₁ and L₂ are the defined boundary lengths for upper-left and lower-right boundary, respectively. Specifically, the ribbon used for the calculation contains 48 cavities, and each of L₁ and L₂ contains the 8 cavities closest to the corresponding boundary.

Experimental measurement

The experimental sample is fabricated using 3D printing techniques with resin material. The geometric tolerance is ~0.1 mm. The thicknesses of the cavities and tubes are set to 2 mm. Circular holes with diameters of 3.7 mm are perforated on each cavity for the insertion of loudspeakers (with a diameter of 1.4 mm, can be viewed as point sound sources) or microphones (B&K Type 4183) to excite or probe the pressure field distributions of the crystal. The amplitude and phase of the acoustic field are recorded and frequency-resolved by a multi-analyzer system (B&K Type 3560B). The spatial Fourier spectra in the upper panel of Fig. 2f, and the projected dispersions in Figs. 3c, 3d, 4e and 4f are obtained by Fourier transforming the corresponding measured acoustic pressure fields. In Fig. 3b, the pressure field distribution is normalized by the maximum value.