Valley-protected topological interface states in metastructures with internal geometric rotation

Topological mechanical metamaterials have garnered significant attention for their ability to exhibit robust, defect-immune wave propagation and mechanical behaviors, inspired by topological protection mechanisms in condensed matter physics. In this study, we present a novel mechanical metamaterial design that introduces rotational geometric parameters within the unit cell to explicitly break spatial inversion symmetry. By precisely tuning the rotation angle of structural elements in the unit cell, we induce asymmetric valley states with opposite Berry curvatures, thereby realizing the valley Hall effect in a mechanical framework. This purely geometric approach avoids reliance on material composition gradients or external fields, offering intrinsic control over valley polarization through structural design alone. Numerical simulations and mechanical analyses demonstrate that the proposed metamaterial supports topologically protected interface states at the boundary between regions of distinct valley topologies. These interface states exhibit unidirectional propagation, confirming their topological protection. This work provides a universal geometric strategy to engineer topological phenomena in structural systems. The realized topologically protected interface states hold promise for applications in high-precision sensors, energy harvesting devices, and vibration isolation systems.

1 Introduction

In recent decades, metamaterials have garnered extensive research attention in the fields of mechanics and physics (Li et al., 2020; Jiao et al., 2023; Ni et al., 2023; Chen C. et al., 2025; Du et al., 2025; Chen et al., 2024c; Dong et al., 2024; Akbari-Farahani and Ebrahimi-Nejad, 2024; Ni and Shi, 2023; Chen et al., 2023). This is primarily attributed to their unique capability of achieving novel mechanical properties through rational design of structural architectures, rather than relying solely on the inherent composition of base materials.

Natural materials are inherently constrained in their acoustic and elastic wave propagation characteristics, as such capabilities are fundamentally limited by their intrinsic lattice structures and continuous physical properties. In sharp contrast, artificially engineered metamaterials—crafted with high-precision design and fabrication—can markedly enhance the flexibility of manipulating and controlling acoustic and elastic waves, breaking through the inherent restrictions of natural materials.

Researchers have now developed numerous types of metamaterials exhibiting extraordinary physical properties, such as negative effective density, negative elastic modulus, and negative refractive index. These anomalous characteristics are fundamentally rooted in theoretical frameworks including local resonance (Santoro et al., 2023; Wang et al., 2025), Bragg scattering (Jia et al., 2025; Rao et al., 2024), transformation acoustics (Cao et al., 2025; Pendry et al., 2006), topological bands (Chu et al., 2024), and non-Hermitian physics (Cao et al., 2022), enabling the realization of remarkable physical phenomena including but not limited to: (1) diffraction-limited acoustic focusing, (2) broadband noise dissipation, and (3) resonant vibration isolation.

Traditional metamaterials achieve equivalent parameters such as negative refractive index and negative modulus through specific structural designs (e.g., resonant units). These properties are highly sensitive to defects and disorder—any structural damage or manufacturing deviations can cause these characteristics to significantly weaken or disappear. In other words, traditional metamaterials macroscopically mimic material parameters not found in nature, and their functionality relies on structural perfection. In contrast, the extraordinary properties of topological metamaterials originate not from the emulation of specific parameters, but rather from the global topological characteristics inherent in their structural band structures. As a result, wave propagation in topological metamaterials (such as waves at the boundaries) is globally protected. Even in the presence of internal defects, manufacturing imperfections, or sharp bends in the path, the waves can propagate along the boundary—they are not only dissipationless and unidirectional but also unaffected by these disturbances or local perturbations. Inspired by topological insulators in quantum physics (Haldane, 1988; Klitzing et al., 1980; Den, 1982), researchers have developed three major categories of topological metamaterials within classical physics domains such as acoustics and mechanics.

The first approach draws on topological insulators based on the Quantum Hall Effect (QHE) (Susstrunk and Huber, 2015). By breaking time-reversal symmetry through globally rotating the coupled pendulum array, a topologically non-trivial phase is induced. The results demonstrate that vibrational energy propagates unidirectionally along the lattice boundary with remarkable robustness—even in the presence of defects or disordered structures, the energy bypasses obstacles without backscattering and continues to propagate. A hexagonal lattice structure designed by Nash et al. incorporates gyroscopes at its nodal points (Nash et al., 2015), utilizing their rotational effects to break time-reversal symmetry, thereby enabling the development of topological metamaterials based on the Quantum Hall Effect. The second way, inspired by Bernevig and Zhang (2006), does not need to break time-reversal symmetry. The fundamental principle of this class of metamaterials is similar to the spin-orbit coupling mechanism observed in QSHE. Chen et al. (2024a) designed a Kekule lattice-based elastic topological resonator with tunable coupling strip widths using the quantum spin Hall effect, achieved robust edge state transmission through topological phase transitions, and realized frequency-space separation by integrating the topological rainbow effect—providing a novel mechanism for efficient elastic wave manipulation and the development of programmable sensors. Zhang et al. developed a groundbreaking methodology by employing a flow-free metamaterial lattice with deliberately broken structural symmetry (Zhang et al., 2017), successfully constructing acoustic pseudo-spin multipole states. Through systematic investigation, the research team demonstrated that implementing symmetry-breaking operations in a honeycomb lattice and regulating the interaction strength between resonant units within the lattice can achieve band inversion between pseudo-spin dipole and quadrupole states. This mechanism serves as a pivotal factor in driving topological phase transitions. Chen et al. (2025c) for the first time introduces topological Wannier cycles into full-polarization micromechanical metamaterials, realizes broadband robust elastic dislocation states via helical dislocation structures, enhances energy harvesting and frequency identification capabilities using mode conversion and coupling systems, and provides a new paradigm for full-polarization elastic wave manipulation. Chen et al. (2018) developed an innovative approach to realize a mechanical quantum spin Hall insulator and successfully modeled this quantum phenomenon using a simplified mass-spring Kagome lattice structure. By employing the Brillouin zone folding technique, the model constructed a lattice system featuring double Dirac cones, opening new research dimensions and methodological pathways for exploring the mechanical quantum spin Hall Effect. Chen et al. (2025b) establishes an electro-acoustic analogy model to achieve second-order to quadrupole topological transition in acoustic metamaterials, investigates multi-dimensional localized states in dislocation structures, and provides a robust framework for acoustic topological manipulation with promising applications in sensing, signal processing, and energy harvesting. Mousavi et al. (2015) proposed the design of a dual-scale phononic crystal plate featuring a triangular lattice of air holes, which achieved degenerate Dirac cones for both symmetric and antisymmetric Lamb wave modes. Subsequently, by breaking spatial mirror symmetry, strong spin-orbit coupling was introduced, leading to a topological phase transition and the formation of a complete phononic bandgap. Chen et al. (2024b) innovatively combines topological insulators with the rainbow trapping effect, designs surface wave photonic crystals based on concrete-filled steel tubes (CFST), achieves robust transmission, frequency-spatial separation, and energy concentration of low-frequency surface waves, and provides a new scheme with both theoretical value and application potential for surface wave manipulation in fields such as reconfigurable seismic metamaterials, sensing, and energy harvesting.

The third method for achieving topologically protected edge states, based on the quantum valley Hall effect (QVHE) (Dong et al., 2024; Chen et al., 2023; Liu et al., 2019), is relatively more straightforward. This approach relies solely on breaking spatial inversion symmetry. QVHE-based topological metamaterials preserve time-reversal symmetry, simplifying geometric complexity. Lu et al. (2016) developed a novel mechanism for breaking mirror symmetry, successfully constructing an acoustic analogue of the topological semimetal-insulator phase transition. By precisely adjusting the rotation angles of anisotropic scatterers within the phononic crystal, it is able to control the shape of the frequency bandgap and phase domain walls. Li et al. (2019) put forward a creative design framework for elastic metamaterials, termed “valley anisotropy,” which can be achieved via an asymmetric structural scheme integrating bio-inspired hard helical scatterers and a compliant material matrix. The research team verified that Berry curvature—a crucial topological indicator—can be accurately regulated by modifying the geometric parameters. This regulation mechanism enables the topological manipulation of transverse elastic waves, thereby allowing for the control of energy propagation behaviors, including the redirection and termination of energy flow. Zhang et al. (2022) present a valley phononic crystal, which consists of a hexagonal aluminum plate equipped with six support rods arranged in a chiral configuration. By tuning the lengths of these support rods within the chiral unit cell, a novel complete bandgap is induced at the Dirac point, and this bandgap formation is accompanied by the appearance of topological edge modes. The authors analyzed the dispersion characteristics of the chiral system and applied them to interface waveguides, thereby enabling the implementation of customizable propagation path designs. In addition, the team also provides evidence for the existence of the valley Hall effect in the proposed chiral system.

The programmable metamaterials have been proposed for the manipulation of elastic and electromagnetic waves (Qi et al., 2022; Xiu et al., 2022; On et al., 2024; Abadal et al., 2020; Yi et al., 2023; Darabi et al., 2020; Zhang et al., 2019). Darabi et al. (2020) proposed breaking time-reversal symmetry through time modulation, inducing synthetic angular momentum bias and opening a topological bandgap with protected edge states. The reconfigurable topological metamaterials enable robust acoustic wave transmission and are applicable to devices such as acoustic emitters and mechanical logic circuits. These systems outperform their electronic counterparts under harsh operating conditions. Zhang et al. (2019) innovatively proposed using programmable ferrofluid distributions within the unit cells of elastic lattices to break inversion symmetry and create tunable bandgaps. This approach allows for easy reconfiguration of propagation paths by controlling the ferrofluid distribution. Numerical simulations and experimental tests demonstrate the ability to guide bending waves along different interface paths, highlighting the potential applications of this programmable elastic valley-Hall insulator.

By introducing a difference in elastic stiffness, the Dirac cone can be opened, resulting in the formation of a topologically nontrivial bandgap. With the assistance of a programmable external electric field, the elastic stiffness distribution within each unit cell can be independently modified. This allows for the creation of topologically protected interface paths with arbitrary, programmable shapes in the elastic lattice, leveraging the valley degrees of freedom. Then, the distinct localized interface state is discussed. Finally, the paper discusses the application of programmable topological interface states in the domain of haptic feedback.

2 Model and theory

2.1 Topological structures

As shown in Figure 1c, the proposed model consists of periodically arranged hexagonal unit cell. Each unit cell consists of two parts as seen in Figure 1b. One part is the Aluminum hexagonal thin plate as shown in the green part of the picture. The other part consists of three raised fan-shaped structures made of Lead material, which are present on both the front and back sides of the hexagonal aluminum plate as shown in the orange part of the picture. The central star-shaped part with a length c of 13.27 mm and a width b of 6 mm is cut out, as indicated by the white color. The red dotted lines outline the unit cell with in-plane lattice constant of $a=30mm$. When expressed using lattice vectors, the unit cell can be described as a₁ = (0, a) and a₂=(-a/2, $\sqrt{3}a/2$). As shown in Figure 1c, the thickness of the aluminum hexagonal thin plate and the height of the lead fan-shaped structure are ${\mathrm{h}}_l=1mm$ and ${\mathrm{h}}_a=5mm$, respectively. The elastic modulus of the aluminum hexagonal thin plate is E = 73 GPa, with a density of ρ = 2,590 kg/m³ and a Poisson’s ratio of ν = 0.23. The elastic modulus of Lead fan-shaped structure is E = 16 GPa. The density and Poisson’s ratio are ρ = 11340 kg/m³ and ν = 0.42, respectively.

Figure 1

Figure 1. (a) Illustration of the metamaterial with lattice vector $a_1=\left(a/2,\sqrt{3}a/2\right)$ and $a_2=\left(-\frac{a}{2},\sqrt{3}a/2\right)$; (b) unit cell structure, the green hexagons represents the thin aluminum plate, while the orange parts represent the fan-shaped structures made of lead material; (c) the section view of the unit cell, the heights of the aluminum hexagonal thin plate and the lead fan-shaped structure are h_a and h_l, respectively; (d) The Brillouin zone of the unit cell; (e) The mesh distribution in the unit cell, the minimum element in the unit cell is 0.012 mm, the maximum element is 1.2 mm, and the model adopts Quadratic serendipity and triangular prism elements.

2.2 Theory

The propagation of elastic waves through the structure is governed (Huo et al., 2021):

$\rho \ddot{\mathbf{u}}=\nabla \left(\lambda \nabla \mathbf{u}\right)+\nabla \left[\mu \left(\mathbf{u}\nabla +\nabla \mathbf{u}\right)\right](1)$

where $\rho$ represents the material density and $\mathit{u}$ is the displacement vector, $\lambda$ and $\mu$ are lame constant.

The dynamic problem in periodic structures is addressed using Bloch-Floquet wave theory (Kittel and McEuen, 2018). The boundary condition can be expressed as in Equation 2:

$\mathbf{u}\left(\mathbf{r},t\right)={\mathbf{u}}_{\mathbf{k}}\left(\mathbf{r}\right)e^{i\left(\mathbf{k}·\mathbf{r}+\omega t\right)}(2)$

where $\mathit{k}$ is the Bloch vector, r is spatial position, $\omega$ is the eigen frequency. Applying the Bloch-Floquet boundary condition, the dynamic problem in Equation 1 can be simplified to the following eigenvalue problem in Equation 3:

$\left(\mathbf{K}-{\omega }^2\mathbf{M}\right)\mathbf{u}=0(3)$

where the M and K represent the mass matrix and stiffness matrix of the model, respectively. Then, COMSOL Multiphysics is used to calculate the band structure by scanning the first Brillouin zone with respect to the wavevector $\mathit{k}$, followed by the analysis of elastic wave propagation properties.

3 Valley-protected elastic wave properties

3.1 Dispersion analysis and band inversion

As depicted in Figure 2, the unit cell exhibits three distinct configurations, corresponding to those in Figure 2a the state with zero rotation angle, where the structure maintains symmetry; (b) the case with a positive rotation angle, indicating that the internal circular sector of the hexagon has undergone clockwise rotation relative to state (a); and (c) the case with a negative rotation angle, where the internal circular sector has rotated counterclockwise relative to state (a). Then, a parameter $\Delta$φ is defined to characterize the angular deviation relative to the symmetric configuration.

Figure 2

Figure 2. Illustration of the unit cell structure with (a) φ = 0, (b) φ > 0 and (c) φ < 0.

When $\Delta \phi =0$, the model simultaneously satisfies both time-reversal and spatial inversion symmetry. Subsequently, the band structure is calculated and presented in Figure 3a. It can be seen that the 4th and 5th branches degenerate at K point, which is known as Dirac degeneracy. To explore the topological properties of the model at this point, a perturbation with $\Delta \phi =1°$ is introduced to break the spatial symmetry of the unit cell. As shown in Figure 3b, after the symmetry is broken, the Dirac degeneracy is lifted and forming a bandgap.

Figure 3

Figure 3. Band structure corresponds to the unit cell with (a) $\Delta \phi =0$ (solid lines) and (b) $\Delta \phi \ne 0$.

To elucidate the topological properties of elastic waves, the band inversion process at the degenerate point is presented. Figure 4 visually depicts how the topological phase evolves as Δφ changes continuously. It can be distinctly observed that as $\Delta \phi$ continuously increases, the bandgap first closes and then reopens.

Figure 4

Figure 4. The phase transition at the K point under the continuous variation of $\Delta \phi$.

The eigenstates of the two bands above and below the bandgap are denoted as ${\psi }^{+}$ and ${\psi }^{-}$, respectively, with the sign indicating the direction of the eigenmode. When $\Delta \phi <0$, eigenstates ${\psi }^{+}$ is located above ${\psi }^{-}$, whereas for $\Delta \phi >0$, the positions are reversed. This indicates that a topological phase transition takes place when the symmetry is broken. Combined with the dispersion curves in Figure 3, we can infer that the breaking of symmetry leads to non-trivial topological states.

Based on the kp perturbation theory (Chen et al., 2018), when the system satisfies the condition of perturbation energy difference ΔE ≠ 0, it indicates that the equivalent model corresponds to a non-degenerate or weakly degenerate system obtained after perturbation treatment. For such perturbative treated systems, the effective Hamiltonian can be expressed as in Equations 4, 5:

$\Delta \mathbf{H}\psi =\Delta \omega \psi (4)$

$\Delta \mathbf{H}=v_{\mathrm{D}}\delta k_x{\mathbf{\sigma }}_x+v_{\mathrm{D}}\delta k_y{\mathbf{\sigma }}_y+m{v}_{\mathrm{D}}^2{\mathbf{\sigma }}_z(5)$

where ${\sigma }_x$, ${\sigma }_y$ and ${\sigma }_z$ are the Pauli matrices, $v_D$ is the Dirac velocity, $m=\left({\omega }_{{\psi }^{+}}-{\omega }_{{\psi }^{-}}\right)/2v_D^2$ is the Dirac mass. $\delta \mathit{k}$ represents the deviation from the Dirac point. The Berry curvature can be expressed as in Equation 6

$\mathrm{\Omega }\left(\delta \mathbf{k}\right)=\frac{1}{2}m{\nu }_{\mathrm{D}}{\left(\delta k^2+m^2{\nu }_{\mathrm{D}}^2\right)}^{-3/2}(6)$

The Valley Chern number is calculated in Equation 7

$C_v=\frac{1}{2\pi }\int \mathrm{\Omega }\left(\delta \mathit{k}\right)dS(7)$

The Berry curvature plays a central role in the analysis of topological properties, with its physical significance lying in quantifying the local curvature of the wave-function in momentum space. This physical quantity serves as a key basis for determining topological invariants (such as the valley Chern number) and identifying topologically protected states, directly guiding the research direction of a system’s topological properties. From the perspective of spatial distribution characteristics, the Berry curvature exhibits opposite signs in the upper and lower regions of the Brillouin zone (positive in the upper region and negative in the lower region). This feature provides the foundation for integration over partitioned regions. Taking the distribution corresponding to the blue line as seen in Figure 3b, integration over the upper and lower regions yields the numerical results $C_k=0.5$ and $C_{k^{\prime }}=-0.5$, respectively. Based on the definition of the valley Chern number, $C_v=C_k-C_{k^{\prime }}$, the calculation gives a nonzero integer value of $C_v=1$ According to the criterion for topological phase transitions, this result unambiguously confirms the occurrence of a valley-dependent topological phase transition in the system.

3.2 1D topological protected interface state analysis

To verify the existence of topologically protected interface states within the bandgap induced by symmetry breaking at the K point in this metamaterial, this study further investigates the dispersion characteristics of a supercell consisting of 20 unit cells. As illustrated in Figure 5, the interface is defined at the boundary between two types of structural units. The supercell model employs free boundary conditions at both ends, with Bloch periodic boundary conditions applied along the $k_x$ as shown in Figure 5A.

Figure 5

Figure 5. (a) Schematic diagram of the 1D supercell structure, the geometrical parameter on the two side of the interface is opposite; (b,c) Band structure of the model with perturbation parameter $\Delta \phi =15°$ and $\Delta \phi =-15°$; (d) The modal displacement corresponds to the interface state represented by the red solid line in (b); (e) The modal displacement corresponds to the interface state represented by the blue solid line in (c); (f,g) the edge state corresponds to the green dashed line and red solid line in (c).

As shown in Figures 5b,c, the edge-bulk band structure of the 1D supercell correspond to $\Delta \phi =15°$ and $\Delta \phi =-15°$, respectively, are calculated. As shown in Figure 5b, when $\Delta \phi =15°$, a dispersion branch associated with the interface state appears within the bandgap. Figure 5d illustrates the eigenmode on this dispersion branch at $f=1797Hz$. The displacement distribution is concentrated near the interface. When $\Delta \phi =-15°$, a dispersion branch related to the interface state, along with two dispersion branches associated with boundary states, appears within the bandgap, as illustrated by the blue solid line, red solid line and green dashed line in Figure 5c, respectively. Figure 5e shows the eigenmode corresponding the dashed line in Figure 5c at $f=1779Hz$. The displacement distribution is localized near the interface. Figure 5f and (g) present the eigenmodes on the green dashed and red solid branches at $f=1775Hz$ and $f=1770Hz$, respectively. The modal displacement is primarily localized at the two ends of the 1D supercell. Obviously, the numerical simulation results show that the proposed model exhibits topological interface states, which can be realized by introducing a geometry perturbation with $\Delta \phi \ne 0$.

3.3 2D topological protected local interface state

Figure 6 illustrates the numerical results of elastic wave propagation in the frequency domain for interfaces with distinct topological phases. As shown in Figures 6a,b, perturbation parameters for the models are $\Delta \phi =15°$. In both models, the interface path is configured in a Z-shape. The interface divides the model into two regions, R1 and R2. As is shown in Figure 5a, the perturbation parameters in R1 is $\Delta \phi =-15°$ and the perturbation parameters in R2 is $\Delta \phi =15°$. The perturbation parameter distribution in Figure 6b is opposite. Figure 6c,d illustrate the frequency-domain displacement field corresponding to point excitations at locations A and B in Figures 6a,b, respectively. It can be observed that under point excitation, the interface state is confined to a localized region and does not propagate into the inner of region R1 and R2. It should be pointed out that the displacement on both sides in Figure 6d are caused by the edge state in Figures 5f,g. Then we set a local protrusion structure in the middle of the Z-shaped path, which can be equivalent to a local defect of the path. Figures 7c,d and Figure 8 show the displacement field distribution after introducing this local defect. The amplitude of the point excitation is 15 μm, and as can be seen from the figure, the vibration amplitude remains approximately 15 μm along the entire path up to the end. It can be can be concluded that the topologically protected interface states exhibit significant robustness against the defect—elastic waves can still penetrate the defect and propagate continuously.

Figure 6

Figure 6. (a,b) Schematic of the Z-shaped interface path with different perturbation parameters; (c,d) Displacement field distribution of the model in (a,b) respectively.

Figure 7

Figure 7. (a,b) Schematic of the Z-shaped interface path with bends local defects; (c,d) Displacement field distribution of the model in (a,b) respectively.

Figure 8

Figure 8. (a,b) Schematic of the Z-shaped interface path with local defects of discontinuous angles; (c,d) Displacement field distribution of the model in (a,b) respectively.

4 Conclusion

This paper presents a topological metamaterial design based on the introduction of geometrical rotation parameter, which subsequently breaks the spatial inversion symmetry. The topological band inversion can be realized by continuously changing the geometrical rotation parameter $\Delta \phi$. The physical mechanism of the topological phase transition is characterized by the kp perturbation theory. Both the theoretical and numerical analysis indicate that the introduction of a geometrical rotation can break the spatial reversion symmetry, thereby facilitating the design of a topologically nontrivial state. Also, the interface state characteristics of the metamaterial within the Dirac bandgap are investigated through numerical simulation. The interface states of the metamaterial are localized in a small region near the interface and can propagate along the interface. This localized characteristic enables the realization of complex local vibration patterns within the plane.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

YW: Writing – original draft. SY: Data curation, Writing – original draft. CJ: Software, Writing – review and editing. JH: Validation, Writing – review and editing.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This work was funded by Jiangsu Provincial Qinglan Engineering Project for Higher Education Institutions (Grant No. 2023QL003) and Major Projects of Basic Science (Natural Science) Research in Colleges and Universities of Jiangsu Province (Grant No. 22KJA460008).

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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