Spatially inhomogeneous non-Hermitian topological metasurfaces

Abstract

Non-Hermitian physics provides a powerful route for controlling wave propagation in open systems, yet its role in spatially inhomogeneous topological structures remains insufficiently clarified. Here, we experimentally demonstrate a spatially inhomogeneous non-Hermitian topological metasurface on a microwave microstrip platform. A continuous geometric modulation introduces a position-dependent Dirac mass and forms a mass-domain-wall interface, while lumped resistive loading provides a controllable imaginary potential. This design transforms the conventional Hermitian degeneracy into a real-part degenerate region near the Γ point, accompanied by an imaginary-frequency bifurcation and mode-dependent attenuation. Tight-binding calculations, full-wave simulations, and microwave near-field measurements are used to characterize the complex bulk dispersion and lossy edge-state transport. We further show that the edge modes exhibit pseudospin-momentum-locked directional excitation, with non-Hermiticity providing an additional degree of freedom for attenuation and mode selectivity. These results clarify the distinct roles of geometric mass gradients and engineered dissipation, establishing a tunable microwave platform for loss-engineered topological wave transport.

Graphical abstract

Keywords

topological metasurfaces / non-Hermitian systems / chiral edge states / inhomogeneous lattices / complex band structures

1 Introduction

Extending wave physics to non-Hermitian systems has provided a powerful framework for describing and controlling open systems with gain, loss, radiation leakage, or dissipation [1−3]. In such systems, engineered loss can give rise to exceptional points, parity-time-symmetry breaking, non-Hermitian skin effects, and complex spectral topology [4−6]. The non-Hermitian skin effect has also become an important route for understanding boundary-sensitive spectra and unconventional bulk-boundary correspondence [7−9]. In parallel, photonic and microwave topological structures based on pseudospin or valley degrees of freedom have enabled robust interface transport and directional wave routing [10−15]. Recent developments in valley-Hall and circuit-based platforms further show that topological wave manipulation can be implemented in compact electromagnetic structures [16−18]. Nevertheless, most experimentally demonstrated topological photonic and microwave lattices remain periodic or piecewise homogeneous. Such homogeneous designs are essential for establishing topological band theory, but they offer limited freedom for tailoring wave confinement, attenuation, and routing in practical finite-size devices.

Spatial inhomogeneity offers a natural route to overcome this limitation by making the effective Hamiltonian position dependent. In Hermitian photonics and acoustics, gradient parameters and geometric deformation have been used to generate artificial gauge fields, pseudo-magnetic fields, and Landau-level-like spectra [19−23]. Strain-induced and gradient-induced photonic structures have further enabled photon trapping, chiral Landau modes, and robust edge transport [24−28]. Related valley-Hall and Dirac-type systems show that spatial modulation can also create interface states through band inversion or mass-domain-wall confinement [29−34]. Recent works on photonic Dirac cavities and adiabatic topological interfaces have clarified how a spatially varying mass term can confine modes without requiring a sharp boundary [35−39]. In parallel, recent studies have shown that non-Hermitian effects can enrich spatially structured topological platforms by reshaping modal spectra, modifying localization behaviour, and introducing additional loss- or coupling-controlled degrees of freedom [40−43]. However, the role of non-Hermiticity in a mass-gradient-induced topological interface remains less clear. In particular, it remains to be clarified how controlled dissipation modifies the complex spectrum, the attenuation contrast, and the chiral transport of domain-wall modes that are already localized by a spatially varying Dirac mass.

Here, we experimentally demonstrate a spatially inhomogeneous non-Hermitian topological metasurface on a microstrip platform [44, 45]. The lattice geometry is continuously varied to introduce a position-dependent mass term, while lumped resistive elements provide a controllable imaginary potential. This design separates the two physical roles: the geometric gradient forms the mass-domain-wall interface, whereas the engineered loss reconstructs the spectrum in the complex-frequency plane. As a result, the conventional Hermitian degeneracy is transformed into a real-part degenerate region near the Γ point, accompanied by an imaginary-frequency bifurcation that gives the edge modes distinct attenuation rates. We verify this complex-band reconstruction through tight-binding calculations, full-wave simulations, and microwave near-field measurements of bulk and edge dispersions [46−51]. We further examine confined edge transport and chiral excitation associated with pseudospin-momentum locking [52−54]. These results establish a tunable platform for studying the interplay between spatial inhomogeneity and non-Hermitian dissipation, and provide a route toward loss-engineered microwave devices with controllable attenuation and directionality.

2 Results and discussion

2.1 Design and complex band engineering of the inhomogeneous non-Hermitian metasurface

Figure 1(a) illustrates the architecture of a non-Hermitian topological metasurface realized on a microstrip platform. The fundamental building block is a hexagonal unit cell with C_6v symmetry [10], composed of six metal disks interconnected by microstrip lines. To introduce controlled non-Hermitian properties, we integrate lumped resistors into three of the six disks, while the remaining three remain pure metal. This resistive loading breaks local Hermitian symmetry and introduces intrinsic dissipation, thereby transforming the system’s effective Hamiltonian into a non-Hermitian quantity.

The topological properties of the lattice are governed by the geometric parameter r₁, defined as the distance from the unit cell center to each disk. A critical fourfold degeneracy emerges at the Γ point under the unperturbed condition r_1=\sqrt{3}{a}_0/3, where a₀ is the lattice constant. Starting from this unperturbed state, we construct a spatially inhomogeneous lattice by gradually modulating the unit cell geometry along the y-direction. The unit cells evolve continuously from a shrunken configuration to an expanded one with a gradient step of Δr = 0.3 mm. This geometric gradient effectively introduces a position-dependent mass term m(y) into the effective Hamiltonian, which can be expressed as [29, 35, 36, 55]

(1)H=v\left(k_x{\sigma }_x\pm k_y{\sigma }_y\right)+m\left(y\right){\sigma }_z,

where v is the Dirac velocity and σ_x,y,z are the Pauli matrices. The mass term m(y) determines the local bandgap size and dictates the topological phase transition across the domain wall.

Figure 1(b) presents the complex band structures for the shrunken, unperturbed, and expanded configurations. It should be emphasized that the formation of domain-wall edge states is not, by itself, unique to the non-Hermitian system, since analogous interface states can also occur in Hermitian spin-Hall or valley-Hall photonic lattices. The essential role of the engineered loss is to extend this inhomogeneous topological platform into the complex-frequency domain. Here, the term degenerate surface refers to a real-part degenerate region in the complex band structure: within a finite wave-vector range near the Γ point, the real parts of the relevant eigenfrequencies become nearly degenerate, whereas their imaginary parts bifurcate and give rise to distinct modal attenuation rates. This distinguishes the present non-Hermitian spectral feature from a conventional Hermitian degeneracy, where the eigenfrequencies are purely real and no imaginary-frequency splitting occurs.

We specifically selected a resistance of 43 Ω to optimize the wave-vector range associated with this degenerate surface [Figs. 1(c, d)]. To further confirm the bulk topology, we calculated the Wilson-loop spectrum of the tight-binding model for the first three-band subspace. The Wilson-loop spectra in both the Hermitian limit and a representative moderate-loss case exhibit the same qualitative topological feature, indicating that the non-Hermitian system remains adiabatically connected to the Hermitian topological phase before the selected band subspace closes with the upper bands (Fig. S4).

To further clarify the indispensable role of non-Hermiticity, we compared this lossy system with its Hermitian counterpart, in which the resistive elements are replaced by lossless metallic connections while the same lattice geometry is retained. In the Hermitian case, the geometric gradient modifies only the real-valued spectrum, and no imaginary-frequency bifurcation can occur. By contrast, in the non-Hermitian case, the engineered loss introduces an additional imaginary potential and generates a tunable splitting in the imaginary spectrum along the real-part degenerate region. We quantitatively characterize this non-Hermitian spectral reshaping by the normalized length of the real-part degenerate region L_dege and the maximum imaginary-frequency splitting ΔIm(f). As shown in Supplementary Figs. S1(d, e), both quantities evolve systematically with the loaded resistance, confirming that the degenerate surface and the associated loss-selective attenuation are controlled by the engineered non-Hermitian dissipation rather than by the geometric gradient alone. According to the continuum Dirac description and band-folding picture, the K point of the primitive cell is projected onto the Γ point of the supercell, leading to the following folded dispersion relation for n ≥ 1 [21, 55]:

(2){\omega }_n=\pm \sqrt{v^2{k}_y^2+2n\left|a\right|v}.

Here, |a| represents the spatial gradient of the Dirac mass term induced by the continuous geometric modulation of the metasurface. Rather than producing discrete Landau-level quantization [21, 42], this mass gradient controls the spatial evolution of the local bandgap and determines the confinement strength of the domain-wall interface modes. This mass-domain-wall framework lays the foundation for realizing robust edge transport in a lossy non-Hermitian environment. Detailed geometric parameters, dispersion characteristics, and tight-binding analyses are provided in the supplementary material.

It is worth noting that the step size of this geometric gradient Δr has a direct impact on the spatial confinement of the edge states. A larger gradient induces a steeper spatial transition of the mass term m(y), which tightly confines the edge modes and reduces their localization length. A detailed quantitative analysis of the edge state localization length as a function of the geometric gradient is provided in Supplementary Section S6 and Figs. S11 and S12.

2.2 Experimental observation of complex bulk band topology

To experimentally validate the designed non-Hermitian band structure, we fabricated a homogeneous metasurface prototype based on the unperturbed unit cell. Unlike ideal eigenmode simulations, experimentally retrieving the bulk dispersion requires a rigorous treatment of Brillouin zone (BZ) folding and mode selection in reciprocal space. Since the fundamental quasi-TEM mode of the microstrip line exhibits a monotonic dispersion, the topological modes of interest appear in higher-order BZs. Consequently, accurately mapping these modes requires redefining the high-symmetry paths to account for the supercell periodicity.

Figure 2(a) illustrates the relationship between the primitive and folded Brillouin zones. The primitive unit cell (blue contour) has an edge length of a₁ = 2a₀cos(π/6)/3, while the supercell (red contour) defines the folded BZ. To clearly identify the topological bands, we label the high-symmetry points as Γ_i, K_i, and M_i, where index i indicates the band order. The measured spectral response is analyzed along specific symmetry paths: K₁−Γ₂−M₁−K₁ for the first mode, followed by paths connecting Γ₂ to M_2,3,4 for higher modes. The recurrence of symmetry points in this sequence is a direct signature of band folding.

Figure 2(b) presents the experimentally mapped bulk dispersion obtained via two-dimensional Fourier transform (2D-FFT) of the measured near-field distribution. The results show excellent agreement with the simulated eigenmodes. We observe a distinct fourfold degeneracy at the Γ₂ point, where the four bands intersect. In the vicinity of Γ₂, the spectral intensity reveals a continuous degenerate line rather than a single discrete point. This feature, combined with the observed spectral broadening and attenuation contrast, serves as an experimental signature of the real-part degenerate line and the non-Hermitian spectral reshaping predicted in Fig. 1. The measured and simulated three-dimensional bulk bands are provided in Supplementary Section S3 and Figs. S5–S7.

2.3 Topological edge transport and non-Hermitian phase control

Having established the bulk topology, we now investigate the edge states emerging within the inhomogeneous metasurface. The spatial gradient illustrated in Fig. 1(a) effectively constructs a topological domain wall, where the effective mass term m(y) transitions across the lattice. This interface is predicted to support chiral edge modes that are robust against backscattering. To reveal their complex dispersion characteristics, we performed eigenmode calculations on the supercell structure, setting the representative resistive load to R₂ = 20 Ω. Here, R₂ denotes the representative resistance used in the edge-state supercell calculation, whereas R₁ = 43 Ω is used for the unit-cell design discussed above. The different resistance value is used as a representative moderate-loss case for resolving the edge-state dispersion, while the resistance-dependent evolution is summarized in Fig. 3(c).

Figures 3(a) and (b) present the calculated real and imaginary band structures of the edge states. In the real frequency domain, the edge modes exhibit a near-linear dispersion characteristic of massless Dirac fermions. However, a distinct signature of non-Hermiticity appears near the zero wave vector (k_x = 0): the intrinsic loss induces a real-part degenerate line rather than a discrete crossing point characteristic of Hermitian systems. The imaginary component of the eigenfrequency reveals a symmetry-breaking bifurcation along this degenerate line. The two edge branches split into modes with opposite imaginary extrema, corresponding to different attenuation rates. This splitting indicates that while the two modes are degenerate in frequency, they experience differential attenuation governed by the resistive loading.

We further explore the tunability of these non-Hermitian states by sweeping the resistance values, as shown in Fig. 3(c). As the resistance increases, the system undergoes a continuous phase evolution: the wave-vector range sustaining the real-part degeneracy widens, and the contrast in the imaginary eigenfrequencies becomes more pronounced. This tunable splitting offers a degree of freedom to engineer mode selectivity based on loss. Figure 3(d) confirms the spatial localization of these modes, showing the electric field energy tightly confined at the domain interface and decaying exponentially into the bulk, consistent with their topological interface localization. The corresponding Hermitian dispersion and tight-binding results are provided in Supplementary Section S4 and Figs. S8–S10.

Furthermore, considering practical microwave engineering where lumped resistors exhibit inherent manufacturing tolerances, we quantitatively evaluated the system’s robustness against random resistance deviations (e.g., ±10%). Since large-scale random disorder is computationally costly in full-wave simulations, we evaluated the tolerance effect using a tight-binding Monte Carlo analysis, following the common disorder-statistical strategy used in topological circuit and lattice models [56]. The implementation details and results are provided in Supplementary Section S7 and Figs. S13 and S14. The statistical results indicate that the random non-Hermitian scattering mainly induces localized spectral broadening. Within the tested ±10% tolerance range, the complex topological gap and the imaginary-frequency bifurcation remain distinguishable in our simulations, suggesting that the edge states are robust against typical resistance tolerances.

To further examine the defect tolerance of the non-Hermitian edge channel, we compared its scattering behavior with that of the corresponding Hermitian inhomogeneous lattice under the same local obstacle. The Hermitian counterpart was constructed by replacing the resistive elements with lossless metallic connections while keeping the same graded geometry. Full-wave simulations show that both systems preserve edge-channel propagation after the obstacle. However, the Hermitian system exhibits stronger defect-induced field spreading away from the domain wall, whereas the non-Hermitian system maintains a more compact transverse profile near the interface. We quantitatively evaluated this behavior using the bulk-leakage ratio extracted from the transverse field profiles. The non-Hermitian system shows a lower bulk-leakage ratio in the first few lattice periods after the obstacle, indicating that the engineered loss suppresses bulk-like scattered components and acts as a dissipation-assisted modal filter. Detailed field maps and quantitative comparisons are provided in Supplementary Section S9 and Fig. S17.

2.4 Experimental mapping of lossy edge transport and chiral excitation

To experimentally corroborate the existence of the predicted edge modes, we fabricated the non-Hermitian inhomogeneous metasurface shown in Fig. 4(a). A point source was positioned at the center of the domain wall to excite the topological edge modes. The spatial distribution of the electric field was mapped using a near-field scanning system. Figure 4(b) displays the measured field profile at 5.5 GHz, clearly revealing the propagation of edge states along the domain boundary. Due to the isotropic nature of the point source, both up-going and down-going modes are excited simultaneously.

Quantitative insight into the non-Hermitian dispersion is obtained by analyzing the measured complex fields along the propagation path. The phase constant is extracted via a one-dimensional Fourier transform (1D-FFT) of the spatial field distribution, mapping the spectral weight in reciprocal space. The attenuation constant was extracted from the spatial decay of the measured field amplitude. Specifically, we calculated the intensity ratio between a near-source position and a distal position along the domain wall. As shown in Figs. 4(c) and (d), the extracted spectra align well with the theoretical predictions. The observed spectral broadening in the FFT map, combined with the spatially measured attenuation rates, provides direct experimental evidence of the mode-dependent losses governing the above-defined real-part degenerate region, confirming the successful realization of the non-Hermitian Hamiltonian on the metasurface.

Finally, we numerically examine the spin-dependent transport properties of the chiral edge states arising from pseudospin-momentum locking. While the point source in the experiment excites bidirectional modes, full-wave simulations allow us to employ circularly polarized sources to selectively excite a single direction. As demonstrated in Figs. 5(a) and (b), a right-handed chiral source selectively launches the up-going edge state, whereas a left-handed source excites the down-going mode. In our full-wave numerical simulations, this chiral excitation was implemented using an array of six discrete ports arranged hexagonally at the center of the domain wall. To couple selectively to the topological pseudospin, this chiral source is generated by exciting these ports with equal signal amplitudes and maintaining a constant phase difference of 60° between adjacent ports. A detailed schematic of the six-port chiral source is provided in Supplementary Fig. S18. The phase distribution exhibits a distinct chiral winding along the excited boundary, a hallmark of the underlying topological spin texture. Although the non-Hermitian dissipation leads to a faster spatial decay compared to the Hermitian counterpart, the directional selectivity is retained despite the additional spatial decay. To experimentally support the chiral excitation, we performed a synthetic coherent multi-probe measurement. Three excitation probes were sequentially driven while the remaining probes were terminated with matched loads, and the measured complex responses were coherently superposed with equal amplitudes and a 120° phase difference. The synthesized transmission shows a stronger response at the right receiving port than at the left one, supporting the preferential excitation of the right-propagating edge mode. The synthetic coherent multi-probe measurement is described in Supplementary Section S10 and Fig. S19.

3 Conclusion

In summary, we have experimentally demonstrated a spatially inhomogeneous non-Hermitian topological metasurface that combines a geometric mass gradient with engineered resistive loss. This design produces a real-part degenerate region near the Γ point and an associated imaginary-frequency bifurcation, providing a complex-spectrum signature that is absent in the corresponding Hermitian structure. Microwave near-field measurements map the bulk dispersion and retrieve the lossy edge-state dispersion, supporting the topological confinement and mode-dependent attenuation of the edge modes. Full-wave simulations, supported by supplementary synthetic coherent multi-probe measurements, further support the chiral excitation associated with pseudospin-momentum locking. These results provide a tunable microwave platform for studying the interplay between spatial inhomogeneity and non-Hermitian dissipation, with potential applications in loss-engineered directional wave transport.

References

[1] Q. Yan , B. Zhao , R. Zhou , R. Ma , Q. Lyu , S. Chu , X. Hu , and Q. Gong , Advances and applications on non-Hermitian topological photonics, Nanophotonics 12(13), 2247 (2023)

[2] K. Ding , C. Fang , and G. Ma , Non-Hermitian topology and exceptional-point geometries, Nat. Rev. Phys. 4(12), 745 (2022)

[3] H. Wang , X. Zhang , J. Hua , D. Lei , M. Lu , and Y. Chen , Topological physics of non-Hermitian optics and photonics: A review, J. Opt. 23(12), 123001 (2021)

[4] H. Chen , J. L. Xie , H. S. Lai , X. C. Sun , C. He , and Y. F. Chen , Exceptional points of acoustic topological boundary states, Phys. Rev. Appl. 22(1), 014046 (2024)

[5] S. Dong , H. Wei , Z. Li , G. Cao , K. Xue , Y. Chen , and C. W. Qiu , Exceptional-point optics with loss engineering, Photonics Insights 4(1), R02 (2025)

[6] L. Zhou , R. Jing , and S. Wu , Topological characterization of phase transitions and critical edge states in one-dimensional non-Hermitian systems with sublattice symmetry, Front. Phys. (Beijing) 21(7), 075202 (2026)

[7] H. Xue , Q. Wang , B. Zhang , and Y. D. Chong , Non-Hermitian Dirac cones, Phys. Rev. Lett. 124(23), 236403 (2020)

[8] X. Zhang , T. Zhang , M. H. Lu , and Y. F. Chen, , A review on non-Hermitian skin effect, Adv. Phys. X 7(1), 2109431 (2022)

[9] R. Lin , T. Tai , M. Yang , L. Li , and C. H. Lee , Topological non-Hermitian skin effect, Front. Phys. (Beijing) 18(5), 53605 (2023)

[10] L. H. Wu and X. Hu , Scheme for achieving a topological photonic crystal by using dielectric material, Phys. Rev. Lett. 114(22), 223901 (2015)

[11] Y. Yang , Y. F. Xu , T. Xu , H. X. Wang , J. H. Jiang , X. Hu , and Z. H. Hang , Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials, Phys. Rev. Lett. 120(21), 217401 (2018)

[12] M. A. Gorlach , X. Ni , D. A. Smirnova , D. Korobkin , D. Zhirihin , A. P. Slobozhanyuk , P. A. Belov , A. Alù , and A. B. Khanikaev , Far-field probing of leaky topological states in all-dielectric metasurfaces, Nat. Commun. 9(1), 909 (2018)

[13] D. J. Bisharat and D. F. Sievenpiper , Electromagnetic‐dual metasurfaces for topological states along a 1D interface, Laser Photonics Rev. 13(10), 1900126 (2019)

[14] S. Xu , Y. Wang , and R. Agarwal , Absence of topological protection of the interface states in Z₂ photonic crystals, Phys. Rev. Lett. 131(5), 053802 (2023)

[15] J. Chen , Y. Zheng , S. Yang , F. Shi , Z. Y. Li , and C. W. Qiu , One-way valley-robust transport in edge-tailored photonic crystals, Phys. Rev. Lett. 134(20), 203803 (2025)

[16] Y. Yang , D. Zhu , Z. Hang , and Y. Chong , Observation of antichiral edge states in a circuit lattice, Sci. China Phys. Mech. Astron. 64(5), 257011 (2021)

[17] W. Wang , Y. J. Tan , T. C. Tan , A. Kumar , P. Pitchappa , P. Szriftgiser , G. Ducournau , and R. Singh , On-chip topological beamformer for multi-link terahertz 6G to XG wireless, Nature 632(8025), 522 (2024)

[18] R. Li , W. Wang , X. Kong , B. Lv , Y. Jia , H. Tao , P. Li , and Y. Liu , Realization of a non-Hermitian Haldane model in circuits, Front. Phys. (Beijing) 20(4), 44204 (2025)

[19] H. Jia , R. Zhang , W. Gao , Q. Guo , B. Yang , J. Hu , Y. Bi , Y. Xiang , C. Liu , and S. Zhang , Observation of chiral zero mode in inhomogeneous three-dimensional Weyl metamaterials, Science 363(6423), 148 (2019)

[20] X. Wen , C. Qiu , Y. Qi , L. Ye , M. Ke , F. Zhang , and Z. Liu , Acoustic Landau quantization and quantum-Hall-like edge states, Nat. Phys. 15(4), 352 (2019)

[21] H. Jia , M. Wang , S. Ma , R. Y. Zhang , J. Hu , D. Wang , and C. T. Chan , Experimental realization of chiral Landau levels in two-dimensional Dirac cone systems with inhomogeneous effective mass, Light Sci. Appl. 12(1), 165 (2023)

[22] W. Zhao , Y. Zheng , C. Lu , Z. Wang , Y. C. Liu , and S. Zhang , Landau rainbow induced by artificial gauge fields, Phys. Rev. Lett. 133(23), 233801 (2024)

[23] X. Zhang , L. Liang , K. Shao , F. Li , B. Jin , H. Wang , Y. Poo , W. Chen , and C. T. Chan , Photonic flat landau levels induced by antisymmetric nonuniform pseudomagnetic fields, Phys. Rev. Lett. 135(17), 176602 (2025)

[24] F. Guinea , M. I. Katsnelson , and A. K. Geim , Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering, Nat. Phys. 6(1), 30 (2010)

[25] M. C. Rechtsman , J. M. Zeuner , A. Tünnermann , S. Nolte , M. Segev , and A. Szameit , Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures, Nat. Photonics 7(2), 153 (2013)

[26] J. Guglielmon , M. C. Rechtsman , and M. I. Weinstein , Landau levels in strained two-dimensional photonic crystals, Phys. Rev. A 103(1), 013505 (2021)

[27] R. Barczyk , L. Kuipers , and E. Verhagen , Observation of Landau levels and chiral edge states in photonic crystals through pseudomagnetic fields induced by synthetic strain, Nat. Photonics 18(6), 574 (2024)

[28] M. Barsukova , F. Grisé , Z. Zhang , S. Vaidya , J. Guglielmon , M. I. Weinstein , L. He , B. Zhen , R. McEntaffer , and M. C. Rechtsman , Direct observation of Landau levels in silicon photonic crystals, Nat. Photonics 18(6), 580 (2024)

[29] A. Vakulenko , S. Kiriushechkina , D. Smirnova , S. Guddala , F. Komissarenko , A. Alù , M. Allen , J. Allen , and A. B. Khanikaev , Adiabatic topological photonic interfaces, Nat. Commun. 14(1), 4629 (2023)

[30] L. Ye , Q. Wang , Z. Fu , H. He , X. Huang , M. Ke , J. Lu , W. Deng , and Z. Liu , Hinge modes of surface arcs in a synthetic Weyl phononic crystal, Phys. Rev. Lett. 133(12), 126602 (2024)

[31] Z. Cui , C. Wu , Q. Wei , M. Yan , and G. Chen , On-chip elastic wave manipulations based on synthetic dimension, Phys. Rev. Lett. 133(25), 256602 (2024)

[32] Y. Chen , Z. Lan , S. Liang , L. Fan , J. Zhu , and Z. Su , Realization of spin‐locked acoustic helical Landau levels in both hexagonal and square lattices, Adv. Sci. (Weinh.) 12(36), e07059 (2025)

[33] F. Ding , J. Tang , X. Dai , and Y. Xiang , Valley modes separation of chiral Landau levels with in plane synthetic gauge fields in gyromagnetic photonic crystals, Laser Photonics Rev. 20(1), e01170 (2026)

[34] H. K. Ke , R. Li , and J. F. Liu , Band gap and quantum valley Hall interface state in uniaxial strain superlattice of graphene, Front. Phys. (Beijing) 21(2), 25202 (2026)

[35] S. Kiriushechkina , A. Vakulenko , D. Smirnova , S. Guddala , Y. Kawaguchi , F. Komissarenko , M. Allen , J. Allen , and A. B. Khanikaev , Spin-dependent properties of optical modes guided by adiabatic trapping potentials in photonic Dirac metasurfaces, Nat. Nanotechnol. 18(8), 875 (2023)

[36] K. Chen , F. Komissarenko , D. Smirnova , A. Vakulenko , S. Kiriushechkina , I. Volkovskaya , S. Guddala , V. Menon , A. Alù , and A. B. Khanikaev , Photonic Dirac cavities with spatially varying mass term, Sci. Adv. 9(12), eabq4243 (2023)

[37] Y. Yang , J. Zhang , B. Yang , S. Liu , W. Zhang , X. Shen , L. Shi , and Z. H. Hang , Photonic Dirac waveguide in inhomogeneous spoof surface plasmonic metasurfaces, Nanophotonics 13(20), 3847 (2024)

[38] S. Liu , Y. Yang , M. Li , B. Yang , X. Shen , L. Shi , W. Zhao , and Z. H. Hang , Coupled pseudo-magnetic field and valley spin in photonic crystals, APL Photonics 10(3), 036112 (2025)

[39] H. X. Wang , W. Li , J. Hu , and J. H. Jiang , Photonic Dirac cavity on a gradient dislocation, Front. Phys. (Beijing) 21(3), 032302 (2026)

[40] M. L. N. Chen , L. J. Jiang , S. Zhang , R. Zhao , Z. Lan , and W. E. I. Sha , Comparative study of Hermitian and non-Hermitian topological dielectric photonic crystals, Phys. Rev. A 104(3), 033501 (2021)

[41] X. Wang , Y. Li , X. Hu , R. Gu , Y. Ao , P. Jiang , and Q. Gong , Non-Hermitian high-quality-factor topological photonic crystal cavity, Phys. Rev. A 105(2), 023531 (2022)

[42] Z. Wang , J. Jiang , Y. Zheng , W. Zhao , C. Wang , Z. Guo , Y. C. Liu , S. Zhang , and C. Lu , Non-Hermitian reshaping of high-order landau modes, Sci. Bull. (Beijing) 71(8), 1949 (2026)

[43] W. Zhang , Y. Yang , X. Shen , L. Shi , and Z. H. Hang , Non-Hermitian topological electric circuits with projective symmetry, Front. Phys. (Beijing) 21(2), 25200 (2026)

[44] Y. Li , Y. Sun , W. Zhu , Z. Guo , J. Jiang , T. Kariyado , H. Chen , and X. Hu , Topological LC-circuits based on microstrips and observation of electromagnetic modes with orbital angular momentum, Nat. Commun. 9(1), 4598 (2018)

[45] Y. Shu , M. Ren , X. Qi , Z. Guo , H. Jiang , Y. Yang , H. Chen , and Y. Sun , Observation of coexisting large-area topological pseudospin and valley waveguide states in a planar microstrip heterostructure based on topological LC circuits, Photon. Res. 13(8), 2213 (2025)

[46] S. Peng , N. J. Schilder , X. Ni , J. Van De Groep , M. L. Brongersma , A. Alù , A. B. Khanikaev , H. A. Atwater , and A. Polman , Probing the band structure of topological silicon photonic lattices in the visible spectrum, Phys. Rev. Lett. 122(11), 117401 (2019)

[47] A. Vakulenko , S. Kiriushechkina , M. Wang , M. Li , D. Zhirihin , X. Ni , S. Guddala , D. Korobkin , A. Alù , and A. B. Khanikaev , Near‐field characterization of higher‐order topological photonic states at optical frequencies, Adv. Mater. 33(18), 2004376 (2021)

[48] Z. Xu , X. Kong , J. Chang , D. F. Sievenpiper , and T. J. Cui , Topological flat bands in self-complementary plasmonic metasurfaces, Phys. Rev. Lett. 129(25), 253001 (2022)

[49] Z. Xu , X. Sun , H. Wu , Z. Xiong , X. Zhou , H. Yu , X. Yin , D. F. Sievenpiper , and T. J. Cui , Terminal‐matched topological photonic substrate‐integrated waveguides and antennas for microwave systems, Adv. Sci. (Weinh.) 11(33), 2404163 (2024)

[50] X. F. Guo , J. W. Liu , H. X. Chen , F. L. Shi , X. D. Chen , and J. W. Dong , Experimental probe of band structures of bilayer valley photonic crystals, Phys. Rev. B 111(2), 024203 (2025)

[51] L. He , S. Liu , W. Zhai , Y. Yang , Q. Ren , Y. Yang , and J. Yao , Experimental observation of valley Hall topological waveguide based on interface engineering, Front. Phys. (Beijing) 21(10), 103203 (2026)

[52] M. F. Picardi , A. V. Zayats , and F. J. Rodríguez-Fortuño , Janus and Huygens dipoles: Near-field directionality beyond spin-momentum locking, Phys. Rev. Lett. 120(11), 117402 (2018)

[53] Z. Xu , J. Chang , J. Tong , D. F. Sievenpiper , and T. J. Cui , Near-field chiral excitation of universal spin-momentum locking transport of edge waves in microwave metamaterials, Adv. Photonics 4(4), 046004 (2022)

[54] X. X. Wang , Z. Guo , J. Song , H. Jiang , H. Chen , and X. Hu , Unique Huygens-Fresnel electromagnetic transportation of chiral Dirac wavelet in topological photonic crystal, Nat. Commun. 14(1), 3040 (2023)

[55] M. I. Shalaev , W. Walasik , A. Tsukernik , Y. Xu , and N. M. Litchinitser , Robust topologically protected transport in photonic crystals at telecommunication wavelengths, Nat. Nanotechnol. 14(1), 31 (2019)

[56] S. Imhof , C. Berger , F. Bayer , J. Brehm , L. W. Molenkamp , T. Kiessling , F. Schindler , C. H. Lee , M. Greiter , T. Neupert , and R. Thomale , Topolectrical-circuit realization of topological corner modes, Nat. Phys. 14(9), 925 (2018)

RIGHTS & PERMISSIONS

Higher Education Press