Topological elastic interface states in hyperuniform pillared metabeams

Artificially structured materials such as phononic crystals1–4 and acoustic metamaterials,5–7 which consist of periodic or non-periodic arranged units, have attracted much interest owing to their unusual properties in terms of wave attenuation, energy harvesting, waveguiding, and sensing. In recent years, topological physics, as an emerging field, has provided anomalous wave properties and is being used to design novel artificially structured materials to manipulate acoustic and elastic waves robustly.8–11 The key feature of a topological system is the existence of topologically protected states at the interface between two systems with distinct topological invariants, whose existence is robust against defects and impurities and makes them immune to backscattering in two-dimensional systems. In a one-dimensional (1D) system, the topological invariant is expressed by the Zak phase, which is a special kind of Berry phase associated with each 1D bulk band.¹² Then, various topological effects, such as topological interface states,^13,14 topological exceptional points,¹⁵ and robust Fano resonances,^16,17 have been investigated.

One of the most widely used models to predict and investigate the topological states in one-dimensional systems is the Su–Schrieffer–Heeger (SSH) model.¹⁸ This model describes a discrete system with a dimerized unit cell, where the trivial or nontrivial character is defined by the fact that the ratio between the hopping integrals of intracell and intercell atoms is less or greater than one. Accordingly, the existence of a topological edge (or surface) mode localized at the free boundary of the system is demonstrated. Based on the original SSH model, many effective SSH-like models have been proposed in photonics as well as in acoustic and mechanical continuous systems.^12,19–22 The SSH-like models are constructed by associating two subsystems with different topological phases, usually accompanied by symmetry inversion of the modes at the band edges.^17,22 The topological edge states appear now at the interface between the two topological phases instead of at the open boundary of the subsystems. The bulk-boundary correspondence for the SSH-like models relates the difference between the topological phases of the two subsystems and the topological interface mode of the combined system. To predict the existence of the topological interface modes, the Zak phases of the 1D systems can be determined by the symmetry of the band-edge states of the subsystems.¹² If the Zak phases of the common bandgaps for the two subsystems are different, the bandgaps are nontrivial, and a topological interface mode is expected in the nontrivial bandgap. In practice, these interface modes can be studied in two ways: either by associating two semi-infinite (or sufficiently long finite) subsystems or by considering a periodic structure in which the unit cell is a large supercell made up of the association of two finite subsystems. In both cases, they are localized around the interface and decay when going far from the interface. Both approaches are used in the following study. Note that in the periodic case, the interface modes appear ideally as flat (dispersionless) branches among the dispersion curves; however, the degree of their flatness is related to the size of the supercell that avoids the interaction between the topological interfaces in two neighboring unit cells.

Recently, the topological states in non-Hermitian systems have been studied with the development of non-Hermitian physics.^23–25 Fan et al. presented an SSH-like elastic metamaterial and showed that topological edge states can be obtained when uneven absorptive dampings are applied to the unit cell, even if the hopping strengths are identical.²⁶ The interplay between non-Hermiticity and topological systems will lead to topological physics beyond the Bloch band theory and bulk-boundary correspondence, in which the non-Hermitian skin effect appears and has drawn lots of attention.^27–29 However, most of the topological states are studied in periodic systems.

Lately, the concept of hyperuniform disorder has been introduced to metamaterials, which were first used to estimate point patterns according to their local density fluctuations^30–33 and exhibit the properties of liquids (amorphous) and crystals (periodic) simultaneously.³⁴ Hyperuniform systems can suppress large-scale density fluctuations, but they are statistically isotropic and lack long-range order.^32,35 Hyperuniform systems represent the low-k limit of stealth systems, which can completely suppress the scattering of incident waves for a set of wave vectors; namely, the hyperuniform systems are transparent in the long-wavelength limit.^36–38 Consequently, periodic systems belong to the hyperuniform systems, which can suppress the scattering for all wavelengths except those associated with Bragg scattering. The hyperuniform metamaterials were found to possess large isotropic photonic/phononic bandgaps^39,40 and have been efficiently used for free-form waveguides,⁴¹ high-Q cavities,⁴² integrated waveguide polarizers,⁴³ vibration concentration,⁴⁴ and graded effective index materials.⁴⁵ Recently, the topological properties of hyperuniform systems have been studied. Mitchell et al. constructed amorphous Chern insulators from arbitrary underlying structures, including hyperuniform systems, where local decorations control the topology of the vibrational spectrum.⁴⁶ Kuznetsova et al. experimentally presented the topological interface modes between two SSH-like one-dimensional hyperuniform acoustic systems with different geometrical representations.³⁴ However, few explorations of hyperuniform systems for topological elastic waves have been reported, especially owing to the fact that elastic structures such as beams support multiple modes.

In this work, we present the topological interface modes between two SSH-like hyperuniform pillared metabeams with different Zak phases. We first state the model of the pillared metabeam and illustrate the hyperuniform distribution with the structure factors. Then we calculate the band structures of hyperuniform pillared metabeams by the supercell method and obtain the Zak phases of bandgaps via the symmetry of band-edge states. Furthermore, we construct the SSH-like model by combining the two hyperuniform metabeams, which have inversed left half and right half, and demonstrate the topological interface modes by band structures, eigenmodes of finite metabeams, and transmissions. We emphasize the increase in the number of bandgaps at low frequencies and associated interface modes when increasing the level of disorder in the hyperuniform supercell. Finally, the robustness of the topological interface modes is checked by introducing perturbations in the pillars’ positions. Our study connects hyperuniform systems and topological elastic waves and could open new pathways to applications, such as broadband wave attenuation, sensing, and robust energy localization. The outline of the paper is as follows: After the introduction in Sec. I, the model of the metabeam and hyperuniform distribution are presented in Sec. II; in Sec. III, the Zak phases of the hyperuniform metabeam are obtained, and the topological interface states of the SSH-like model are demonstrated and discussed, in particular, as a function of the disorder level. Section IV is devoted to the conclusions and remarks.

A hyperuniform pillared metabeam is constructed by arranging the pillars symmetrically on the upper and lower surfaces of the host beam, as shown in Fig. 1(a). The number N of pillars is 20, and the positions of pillars follow the hyperuniform distributions. The pillar distribution has mirror symmetry, with the mirror plane indicated by the orange dashed line in Fig. 1(a). The left and right halves, each with ten pillars, are, respectively, called S and S*. To construct an SSH-like model, we invert the left and right halves and get the counterpart S*S. Then we put the supercells SS* and S*S together and obtain the combined supercell SS*S*S, as shown in Fig. 1(b). The topological interface modes will occur at the interface of SS* and S*S, as marked by the vertical black dotted line in Fig. 1(b).

The host beam and pillars are made of aluminum, with a density of 2730 kg/m³, Young’s modulus of 77.6 GPa, and Poisson’s ratio of 0.35. The diameter d and height h₁ of the pillar are 1 and 1.35 cm, respectively. The width w and height h₀ of the host beam are 2 and 1 cm, respectively.

Note that the stealthiness parameter χ is bounded in the interval [0, 1]. The larger the stealthiness parameter χ is, the more ordered the structure is, namely, the closer it is to a periodic structure.

Therefore, the collective coordinate C(q) has the minimum value of −N/2. For a stealthiness parameter χ, the corresponding point positions x_j can be obtained by making the collective coordinate C(q) in the constrained region q ≤ q_c reach the minimum value. We utilize an optimization program to deal with this, which starts with a random distribution. In addition, only the points in the left half or the right half need to be optimized due to the mirror symmetry in our model. To avoid the overlap of adjacent pillars, we generate many samples and choose the distributions that meet the non-overlapping conditions. In addition, the target hyperuniform point patterns can also be designed by simultaneously minimizing the summation of the structure factor and the standard deviation function.⁴⁷ 

The generated point positions and corresponding structure factors for periodic and one realization of χ = 0.4, χ = 0.2, and random distributions are shown in Fig. 2. The specific positions of the hyperuniform distributions can be found in the supplementary material, Sec. I. For periodic distribution with the periodicity of a shown in Fig. 2(a), the structure factor map presents the Bragg peaks at q = 2πn/a. For the wavelength out of the Bragg peaks, the structure factor is 0. This means that a periodic distribution of scatterers can suppress scattering for all wavelengths (namely, waves can propagate almost without scattering) except those associated with Bragg scattering.³⁶ The Bragg bandgap will occur around the wavelength of 2πn/a due to the Bragg scattering. For the hyperuniform distribution with χ = 0.4 shown in Fig. 2(b), only the structure factor around q = 0.5 deviates from zero compared to the periodic case. Moreover, the structure factor shows peaks at wavelengths corresponding to the Bragg scattering, which means that the system presents some hints of periodicity. This implies that the hyperuniform system keeps part of the bandgap property of the periodic structure (while the transparency range decreases). For the hyperuniform distribution with χ = 0.2 shown in Fig. 2(c), the system becomes more disordered, and the structure factor deviates more from zero, representing the more and stronger scattering effects from the scatters. As soon as the distribution becomes fully random, as shown in Fig. 2(d), there is no wavelength with a zero structure factor, and random scattering is produced.

We start with the properties of periodic pillared metabeams. It is worth noting that the waves propagating in the metabeam can be characterized by their symmetries and divided into four types. Indeed, the considered metabeam has two symmetry planes, xOy and xOz. Consequently, the waves propagating in the beam can be classified into four independent categories according to their symmetric or antisymmetric character with respect to the two symmetry planes. They are usually named flexural, symmetric shear, antisymmetric shear, and longitudinal modes. In this work, we focus on antisymmetric shear waves to illustrate disorder-induced multiple gaps and several topological interface states. However, the results for all four modes are presented in the supplementary material, Sec. II.

Band structures of the antisymmetric shear modes of the periodic case are shown in Fig. 3, where the first bandgap is highlighted by dark gray. The pillars have bending resonance at 31.4 kHz under antisymmetric shear wave excitation, which induces this hybrid resonant bandgap with the avoiding crossing effect. In the following study, we focus on the first hybrid resonant bandgap and study the novel effects associated with the hyperuniform structure.

In this and the following sections, we regard the hyperuniform pillared metabeam SS* (or S*S) as a supercell and calculate the band structures of the corresponding periodic systems, which is a common approach in the study of hyperuniform structures.^40,44 First, we investigate the properties of a hyperuniform pillared metabeam with χ = 0.4. Pillar positions for the hyperuniform supercell SS* and its SSH-like counterpart S*S are shown in Figs. 4(a) and 4(b), respectively. It is worth noting that the SS* and S*S are both hyperuniform structures with the same structure factors due to the mirror symmetry. We first present the band structure of the metabeam with the supercell of 20 unit cells in the periodic case in Fig. 4(c). For the traditional discrete SSH model, the Hamiltonian or equivalent dynamical matrix can be easily written and investigated analytically by using the intracell and intercell hopping amplitudes. However, in the case of continuous systems such as our metabeams, the dynamical matrix cannot be expressed by a finite matrix Hamiltonian, and numerical methods become necessary to calculate the eigenfrequencies and eigenmodes. At best, in periodic structures with simple unit cells, the dispersion curves can be approximately reproduced by using fitting parameters for the hopping integrals, but in a disordered system, these interactions become dependent on the distances between the pillars and are most likely non-local. In Fig. 4(c), one can see that the bands fold compared to those in Fig. 3, and a hybrid bandgap appears as indicated by the dark gray region as expected, which we call the main bandgap. The band structures of the metabeams with the supercell SS* or S*S in the hyperuniform case are exactly the same, as shown in Fig. 4(d). It is noticed that new bandgaps appear for the hyperuniform metabeam compared to Fig. 4(c), which originate from the multiple scattering caused by the disordered hyperuniform distribution of pillars in the supercell. In addition, the lower boundary of the main bandgap shifts upward, and the bands near the main bandgap (namely, around the 20th band) become separated and flat. As a matter of illustration, we show in Figs. 4(e) and 4(f) the total displacement fields for the lower and upper cyan branches at k = 0, which appear as localized modes near the mirror plane in the considered supercell. Note that for better visualization, the displacement fields in Fig. 4 and the following are shown only in the upper half of the metabeam. Such localized modes near the main bandgap are a significant characteristic of hyperuniform systems.³⁸ 

To explore the topological properties of hyperuniform metabeams, we calculate the Zak phase of the bandgaps. Thanks to the mirror symmetry of the supercell, the symmetry of band-edge states can be used to calculate the values of the Zak phase.¹² The band-edge states have either even or odd symmetry with respect to the mirror planes indicated by the orange dashed lines. For instance, the modes shown in Figs. 4(e) and 4(f) are of even and odd symmetry, respectively. If the two band-edge states of a band have the same symmetry, the Zak phase of this band is zero. Otherwise, the Zak phase is π. The Zak phase of a bandgap can be determined by the summation of the Zak phases of bands below this bandgap.⁴⁹ The values of Zak phases of bandgaps for hyperuniform metabeam SS* and S*S are labeled on the left and right in Fig. 4(d), respectively. When we move to the association of the metabeams SS* and S*S, the common bandgap can be considered nontrivial if the Zak phases of both metabeams are different; otherwise, it is trivial. According to the bulk-boundary correspondence, topological interface states localized between the metabeams SS* and S*S can appear within the nontrivial bandgaps when we combine the two metabeams together. The nontrivial and trivial bandgaps are indicated by the light and dark gray regions, respectively. It is found that multiple nontrivial bandgaps exist, which can be further used to design multiple topological interface states.

Then we combine the hyperuniform pillared structures SS* and S*S and construct the SSH-like model SS*S*S. As the topological interface states are localized in the middle of the combined structures rather than the edges, they can exist independently of the outermost boundary conditions. We first regard the combined structure SS*S*S as a supercell and apply periodic boundary conditions to calculate its band structure, as shown in Fig. 5(a). One can see that there are two new red bands in each nontrivial bandgap [indicated by the light gray region in Fig. 4(d)], which are the topological interface bands. The reason for the two interface branches is that in the super-periodic structure, each supercell contains two interfaces, namely, the interfaces in the middle and both ends. In addition, the small dispersion (deviation from flatness) of the interface branches results from the finite size of the supercell, which does not sufficiently isolate the consecutive interfaces in the neighboring supercells, especially in narrow bandgaps where the interface modes are more extended. To isolate the consecutive interfaces, each subsystem in the supercell needs to have a larger extension. On the contrary, there is no new band appearing in the trivial bandgaps. In addition, in a finite structure with open boundary conditions, the topological interface modes still exist due to their localized character at the interface. To illustrate this, we further calculate the eigenfrequencies of one supercell SS*S*S with free boundary conditions at both ends, and the results are shown as the green and yellow dots in Fig. 5(a). It is found that there is an eigenmode corresponding to the red topological interface bands within each nontrivial bandgap marked by a yellow dot.

The distributions of the five topological interface eigenmodes M₁–M₅ indicated by yellow dots are shown in Fig. 5(b), where the color scale represents the amplitude of the total displacement. One can see that most of the energy is localized near the interfaces indicated by the black dashed line, which also demonstrates the topological interface modes in analogy to the SSH model. Moreover, the displacements of mode M₅ are strongly localized near the interface. These results also demonstrate that one supercell of the combined metabeam SS*S*S can produce topological interface modes as well.

Furthermore, we calculate the transmission spectra through one supercell of the combined structure SS*S*S, as shown in Fig. 6(a). The transmission model consists of 40 pairs of pillars, which are located according to the SS*S*S distribution with χ = 0.4, and port boundary conditions are applied at both ends to simulate the incoming and outgoing waves. The transmission is large in the passband frequency range (white region) and small in the bandgap frequency range (gray region), as expected. There is a transmission peak in each nontrivial bandgap, as indicated by the black dot, which represents the topological interface mode. The total displacement fields of transmission peaks M₁–M₅ under the left incoming waves are shown in Fig. 6(b). One can see that the displacements are localized near the interface indicated by the black dashed line, and the displacement distributions are very similar to those in Fig. 5(b), which further illustrates the topological interface modes.

Now we investigate the properties of hyperuniform pillared metabeams with χ = 0.2, which means increasing the level of disorder. Pillar positions for hyperuniform supercell SS* and its SSH-like counterpart S*S are shown in Figs. 7(a) and 7(b), respectively. The band structures of the metabeams with the supercell SS* or S*S are shown in Fig. 7(c). More bandgaps occur below the main bandgap as compared to the results for χ = 0.4, originating from the stronger scattering by the more disordered pillars’ distribution in the supercell, which is indicated by the decrease in the zero structure factor in Fig. 2(c). Therefore, a strongly disordered distribution is conducive to opening bandgaps at low frequencies. However, their widths remain small. Similar to the bands for χ = 0.4, the 20th band marked by cyan near the main bandgap is flat, and the distribution of the displacement mode at k = 0 is given in Fig. 7(d). One can see that in the considered structure, vibrations are mostly localized near the two edges, which is different from Figs. 4(e) and 4(f).

The Zak phases of bandgaps for hyperuniform metabeam with supercells SS* and S*S are labeled on the left and right in Fig. 7(c), respectively. The nontrivial bandgaps with different Zak phases and the trivial bandgaps with the same Zak phases are, respectively, indicated by the light and dark gray regions. In this case, more nontrivial bandgaps appear, and a few nontrivial bandgaps exist at lower frequencies. In addition, there are two wide trivial bandgaps in this model, which would be beneficial for broadband low-frequency wave attenuation.

The band structures of the metabeam for SSH-like combined supercell SS*S*S corresponding to χ = 0.2 are shown in Fig. 8(a), in which red bands represent the topological interface bands. It is noticed that the first four red bands are relatively dispersive rather than flat, which means the topological states are weakly localized and relatively penetrating into the bulk parts. The eigenfrequencies of one combined supercell SS*S*S with free boundary conditions at both ends are shown as green and yellow dots, where the seven yellow dots labeled P₁–P₇ represent the topological interface modes in the nontrivial bandgaps (indicated by the light gray regions). The transmission spectra of the metabeam with one combined supercell SS*S*S with χ = 0.2 are calculated and shown in Fig. 8(b), where the transmission peaks in light gray regions represent the seven topological interface modes P₁–P₇. It is worth noting that the quality factors of the first two topological interface modes, P₁ and P₂, are low, which is related to the narrow bandgaps, dispersive bands, and expanding states. Furthermore, the transmissions in the two wide trivial bandgaps are small, which can be utilized for broadband wave attenuation.

Finally, we examine the robustness of the topological modes by introducing randomness in the pillars’ positions in two ways: maximum position perturbation δ_x along the x-axis (while keeping the xOz symmetry plane) and δ_y along the y-axis (without the xOz symmetry plane). For the sake of simplicity, we assume that any pair of upper and lower pillars with central symmetry moves simultaneously. We chose the first topological mode M₁ for χ = 0.4 and the second topological mode P₂ for χ = 0.2 to show the influences of the pillar’s position perturbation on the transmission, and the results are given in Fig. 9. We show the effect of δ_x in Figs. 9(a) and 9(b), where one can notice that the transmission curves have almost no change for 1 and 2 mm perturbations, except for a small peak frequency shift of less than 0.3% for χ = 0.4 and 0.2, respectively. The transmission curves start to change significantly for a 3 mm perturbation, including a reduction and blueshift of the peak amplitude while remaining below 1.5%. For the influences of δ_y shown in Figs. 9(c) and 9(d), the transmission peaks gradually redshift with the increasing perturbation strength δ_y for both hyperuniform metabeams, where the shifts under 1 and 2 mm perturbations are small and become larger under 3 mm perturbations. In addition, the transmission peak amplitudes for 2 mm perturbations decrease slightly, while those for 3 mm perturbations decrease significantly. However, it is observed that the transmission shapes are always conserved for any of the considered perturbations, and, in particular, the topological states remain inside the bandgap range. Thus, the results can demonstrate that the topological modes have a certain robustness to external disturbances. In addition, immunity to perturbations is better satisfied when the symmetry of the structure is conserved (as for δ_x) than when it is broken (as for δ_y).

Furthermore, we investigate the influences of stealthiness parameters χ on the main bandgaps. The results show that the central frequency and width of the main bandgap decrease with decreasing χ values (see the supplementary material, Sec. III).

In summary, we presented the multiple topological interface modes in SSH-like hyperuniform metabeams. The metabeams consist of pillars symmetrically positioned on the upper and lower surfaces of the host beam, with hyperuniform distribution along the x direction and a mirror symmetry plane. We showed that the hyperuniform metabeam could open new bandgaps at low frequencies, including nontrivial and trivial bandgaps, and have localized modes near the main bandgap, which expands the research scope of traditional metamaterials and phononic crystals and provides more freedom to design various passbands and bandgaps. Furthermore, the SSH-like model is constructed by combining the two hyperuniform metabeams, which have an inverted left half and right half. We demonstrated that multiple topological interface modes occur in the nontrivial bandgaps based on several arguments, such as the Zak phases of the band structures, eigenmodes of finite metabeams, and transmissions. Moreover, we found that the hyperuniform metabeam with lower stealthy parameters χ (namely, more disorder) could open more bandgaps at low frequencies and support more topological interface modes. The bandgaps at low frequencies of hyperuniform structures also have great potential for wave attenuation. Finally, the robustness of the topologically modes is verified by introducing perturbations in the pillars’ positions along the x and y directions. The proposed multiple topologically localized states can make up for the shortcomings of the single operating frequency of the traditional single topological localized state and improve the efficiency of potential applications such as energy harvesting and sensing. Our work presents a versatile platform for studying disordered hyperuniform metamaterials and topological properties and may open promising avenues for broadband wave attenuation, energy localization, sensing, and robust transport.

See the supplementary material for the specific positions of the points of the hyperuniform configuration used in the study, the topological states for all types of waves in the hyperuniform metabeams, and the variation of the main bandgap with χ-values for antisymmetric shear waves.

This work was supported by the National Natural Science Foundation of China (Grant Nos.12272267 and 52278411), the Young Elite Scientists Sponsorship Program by CAST (Grant No. 2021QNRC001), the Shanghai Science and Technology Commission (Grant Nos. 22JC1404100 and 21JC1405600), the Special Funds of the Tongji University for “Sino-German Cooperation 2.0 Strategy,” and the Fundamental Research Funds for the Central Universities. The first author is grateful for the support of the China Scholarship Council (Grant No. 202206260205). This work was part of the project MAGNIFIC, which has received funding from the European Union’s Horizon Europe research and innovation program under Grant Agreement No. 101091968. The project is also supported by the French national research agency ANR under Grant Agreement No. ANR-19-CE24-0014.

The authors have no conflicts to disclose.

Runcheng Cai: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Yan Pennec: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Laurent Carpentier: Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Yabin Jin: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Timon Rabczuk: Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Xiaoying Zhuang: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Bahram Djafari-Rouhani: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.