Topological insulators represent a unique quantum state in condensed matter systems, revealing nontrivial phases [1–6]. Their electronic properties, such as Dirac fermions [7–9] and valley polarization [10–12], are determined by atomic constituents and bonding arrangements. Photonic crystals (PCs), analogous to conventional crystals, exhibit band gaps due to Bragg diffraction [13–15]. In these systems, periodicity and symmetry play a pivotal role, making topological photonic crystals (TPCs) a burgeoning field in the realm of topological states, ensured by inversion (P) and time-reversal (T) symmetries [16–24]. When time-reversal symmetry is conserved, classical analogues of the quantum spin-Hall effect [25–27] and the valley-Hall effect [23,24,28,29] rely on broken inversion symmetry, with edge states protected by pseudospin conservation. Gapless edge states at interfaces of different topological phases are stable and non-local [25–30]. In addition, besides the conventional descriptions of bulk-edge correspondence, higher-order topological insulators have been introduced [31–38]. These topological states are relatively insensitive to external disturbances, offering a stable and reliable avenue for the manipulation of information processing.

A topological photonics insulator based on graphene-like structures has Dirac cones typically in hexagonal lattices of C3v or C6 symmetry. Topological phenomena in lower-symmetry systems are rare. Recently, symmetry breaking in electronics has induced novel nonlinear quantum phenomena by modifying the energy band structure through the introduction of anisotropy [39–42]. Chen et al. have achieved terahertz emission in PtTe2 films through symmetry breaking, uncovering a universal pathway for efficient nonlinear transport [40]. Duan et al. have disrupted the rotational symmetry of WSe2 materials, clearly presenting a novel academic concept for realizing Berry curvature dipole moments in energy bands by exploiting lattice symmetry breaking at heterogeneous interfaces [41]. Yuan et al. have demonstrated that FeSe films exhibit anisotropy at low temperatures, triggering nonlinear phenomena [42]. In addition, disorder can induce or alter topological states, leading to unique phase transition phenomena, which are also crucial for understanding and designing materials with desired topological properties [43–47].

As an additional degree of freedom, anisotropy has profound implications. Achieving in-plane anisotropy in PCs typically involves breaking and reducing rotational symmetry, a feature rarely reported in low-symmetry TPCs. In this work, we propose a triangular lattice composed of elliptical metal scatterers. By rotating these scatterers, we induce strong in-plane anisotropy in a C2v symmetric triangular PC, allowing flexible control over the band structure. We investigate the interplay between topological edge dispersion and anisotropy at the interface. Notably, we theoretically examine anisotropy-induced domain walls and observe a significant flat band in the edge state dispersion as the interface’s geometric orientation changes. Furthermore, we find novel multi-type localized states, which are associated with higher-order topological phenomena in corners formed by anisotropic domain walls. The result challenges conventional views on space inversion symmetrical interfaces, which are compatible with the supercells for flattened edge dispersion and higher-order corner states. Our findings highlight that anisotropy has a critical role in topological effects stemming from the reduction of structural symmetry. It can be expected that our findings will provide new insights into flat band dispersion and higher-order topology.

To induce in-plane anisotropy in the crystal structure, we start with a C2v symmetric elliptical metal scatterer while keeping the unit cell hexagonal, as depicted in the right panel of Fig. 1(a). The proposed PC consists of a parallel-plate metal waveguide filled with a periodically arranged triangular array of elliptical metal copper pillars connected to the top and bottom metal plates, with a height of h=4mm. Each unit cell of the PC contains a pillar with an ellipticity ρ=4.5, minor axis lengths l=4mm, and the lattice constant a=20mm. In contrast, as the triangular lattice of uniform circular pillars shown in the left panel of Fig. 1(a), the air channels surrounding them can form a graphene-like honeycomb lattice; therefore, such valley degree of freedom is existent [48]. However, as the pillars are transformed into asymmetric triangular sublattices such as ellipses, strong C2v symmetric anisotropic scatterers are induced.

Figure 1(b) illustrates the crystal structure of the circular pillars elongated along the horizontal b1 direction, with the symmetry reduced from C3v to C2v or from C6 to C2v. The corresponding complex first Brillouin zone is presented in the right panel in Fig. 1(b). In fact, the choice of elliptical metal pillars as a unit cell is a visualization of the anisotropic primitive cell. Our design strategy for anisotropy is exemplified by the artificial modification of band dispersion in different directions. By solving Maxwell equations, we explicitly reveal the harmonic transverse magnetic modes (TM modes with nonvanishing field components Ez, Hx, and Hy) hosted by the C2v symmetry triangular PC. By retaining the lowest-order transverse modes, we can assume the field decomposition as follows [49]: EZTM(r,t)=∑n,k⊥amn(k⊥)ezn,k⊥(r⊥)eik⊥r⊥−iωn(k⊥)t+c.c.,(1)where r⊥=(x,y), k⊥=(kx,ky) are the Bloch wave numbers inside the Brillouin zone, ezn,k⊥(r⊥) are the normalized periodic field profiles, and n index refers to lower (upper) propagation bands.

By setting up boundaries and electromagnetic wave modes corresponding to Eq. (1), we use the commercial software ANSYS HFSS to calculate the band dispersion diagram of the proposed crystal. Figure 1(c) presents the numerical results of photonic bands along the specific direction of the lattice. Significantly, we observe an inconsistency in the dispersion relation corresponding to the ΓK and ΓK′ paths in Fig. 1(c), which is certainly absent in the case of conventional photonic Dirac systems. Hence, as the symmetry of the triangular photonic lattice is reduced from C3v to C2v or from C6 to C2v in the triangular photonic lattice, the structure exhibits strong in-plane anisotropy. The ellipticity factor ρ characterizes the shape deviation from a perfect circle to an ellipse, which determines the strength of the crystal structure anisotropy (see Appendix A for supporting content). Furthermore, the anisotropic phase diagrams of elliptical PCs are presented in Appendix B for supporting content.

We examine the spectral characteristics of a two-dimensional (2D) PC arranged in a triangular lattice, influenced by anisotropy. Conversely, we can implement the different arrangements of the PC lattice with different crystal basis vector directions, which also introduces anisotropy. The upper panel of Fig. 2(a) represents the arrangement of the PC relative to various crystal orientation vectors as φ=0°. The orange, blue, and dark gray ellipses represent the lattices aligned with the b1, b2, and b3 basis vectors, respectively. The calculated band structure of the unit cell is presented in the lower panel of Fig. 2(a), where a pair of Dirac points is generated in a specific direction. The emergence of the Dirac points at M and M′ is not deterministic, and the position of these degenerate points is influenced by the spatial filling rate γ of the crystal structure, which in turn depends on the ellipticity ρ in our proposed structure. A detailed evolution is provided in Appendix C to support content.

A Dirac mass m2D is introduced to characterize the distinct bulk topology gap in our 2D triangular lattice [50]. As φ=±30°, a topological gap emerges, as depicted in the lower panels of Figs. 2(b) and 2(c). There are two opposite chiral vortex phases: the pseudospin topological insulator with m2D>0 and m2D<0. We can understand the doubly degenerate dispersions around the Dirac point utilizing the k⋅p theory: H(k)=VD(k^xσy+k^yσx)+m2Dσ [36]. The effective Hamiltonian for a gapped Dirac point is written as H=SHS+=(M+NN*M−), where S is the unitary transformation matrix, N is the coupling matrix, and M+ and M− refer to the pseudospin-up and pseudospin-down matrices, respectively [50]. We have the expressions of M±=((ωD+12|Δω|)212(Akx±iBky)12[Akx±(−iBky)](ωD−12|Δω|)2),(2)where ωD is the Dirac degeneracy frequency, |Δω| is the value of the band gap, and A and B are determined by the first-order and second-order perturbation terms, respectively. The parity inversion at the M/M′ point leads to the formation of the two opposite chiral vortex phases. The sign of the Dirac mass is controlled by the band flipping of the first and second bands, and this flipping can be regulated by the rotation angle φ of the ellipse scattering. Topological edge states can emerge at the interface between two PCs with opposite signs of the Dirac mass. When anisotropy is not considered, the detailed process of the dispersion relation of the edge states is proved in Appendix D for supporting content. Similarly, the crystal structure induced by crystal anisotropy can exhibit markedly different arrangement shapes based on different basis vector considerations, as shown by the permutation of orange, blue, and dark gray shapes in the upper panels of Figs. 2(b) and 2(c). Although the presence of anisotropy does not affect bulk-boundary correspondence, it provides new possibilities for the arrangement of the structure domain wall interfaces.

To investigate the topological edge states changing with the transformation of the anisotropy interface, we construct three types of zigzag domain walls along lattice directions corresponding to the b1, b2, and b3 basis vectors as shown in the upper panels of Figs. 3(a)–3(c), named as e1, e2, and e3 interfaces, respectively. We numerically simulate the corresponding dispersion of these edge states, as depicted in the lower panels in Figs. 3(d)–3(e). At the zigzag interface along the b1 basis vector, the edge states display a pair of almost linear lines, and near-gapless dispersion curves are observed in Fig. 3(d). Due to the chiral vortex locking, the topologically projected edge modes exhibit chirality, implying that the two counter-propagating modes can be selectively excited by chiral sources with different chiralities. The underlying physics of this phenomenon mimics that of quantum valley-Hall insulators in photonic systems.

However, a mode gap emerges between the upper and lower bulk modes for the zigzag domain walls along the b2 (b3) basis vector, as illustrated in Fig. 3(e). It is notable that the group velocity of the edge states decreases and a flat band dispersion is observed, which creates a slow light region within the Brillouin zone. In particular, the slow light dispersions of the domain walls constructed along the lattice directions of b2 and b3 are the same due to mirror symmetry. The simulated electric field amplitude components Ez for the edge modes at 14.47 GHz are shown in Figs. 3(a)–3(c), where the edge mode is strongly localized at the interface. We investigate that anisotropy-induced interfaces profoundly affect edge states and can achieve slow light dispersion. Numerical simulations reveal a transition from gapless chiral edge modes to the gapped flat band dispersion. This is due to the strong anisotropy in the crystal plane induced by the low symmetry of the structure, which provides a new way to reconfigure the lattice for the topological waveguide interface. It is also the key factor to achieve the flat dispersion. Although the topological phases of the two structures forming the topological interface remain essentially unchanged, the interface configuration is changed. The interface geometry supporting the valley-Hall dispersion relation is mirror-symmetric, where the momentum space patterns along the ΓK and ΓK′ paths are consistent. In contrast, the interface geometry responsible for flat dispersion relations lacks this regularity. This inconsistency of the dispersion relations along the ΓK and ΓK′ paths explains the difference.

To reveal the flatness of the dispersion relation in the 2D crystal structure, we calculate the density of states (DOS). The right panel of Fig. 3(f) shows the DOS values for the e1 and e2 zigzag topological interface dispersion relations. The e2 interface has a very high DOS value around 14.47 GHz, indicating better electromagnetic wave localization and a flatter band. In contrast, the e1 interface has a lower DOS value, suggesting that it does not exhibit a flat band dispersion. Similarly, the left panel of Fig. 3(f) shows the normalized transmission spectra for different interfaces, which reveals a sharp transmission peak for the e2 interface in the frequency range of 14.45–14.50 GHz, while the e1 interface shows a high transmission across a broad frequency range of 12.40–17.26 GHz. This further highlights the difference between valley-Hall edge states and flat band dispersions. Additionally, we present the calculated group velocities of the edge states e1 and e2 in Fig. 3(g), which clearly demonstrates that the anisotropy interface can induce a slow light waveguide.

Closer examination of the flat band dispersion curve is shown in Fig. 3(h), which involves a local zoom into the energy bands depicted in Fig. 3(e). The amplified dispersion curve exhibits a monotonically increasing trend in the range of −π/a<kx<0. In addition, the slope of the dispersion curve nearly remains constant within the range of −3π/4a<kx<−π/4a. The slow light properties of the waveguide modes on the flat band dispersion curve are further characterized quantitatively by using the group velocity vg=dωdkx, group refractive index ng=cvg, and group velocity dispersion GVD=d(1/vg)dω=d2kxdω2. As the dispersion curve closely shows a straight line within the range of −3π/4a<kx<−π/4a, the group velocity dispersion is approximately equal to zero (GVD≈0). To this end, we calculate the intrinsic fields at the frequencies of 14.462 GHz, 14.482 GHz, and 15.502 GHz, indicated by the three blue dots in Fig. 3(h), as well as the electric field intensity distribution perpendicular to the domain wall interface. Figure 3(i) shows that the electric field intensity distributions at these three eigenfrequencies are almost completely overlapped within the PC waveguide channel. The results intuitively demonstrate that the slow light exhibits near-zero group velocity dispersion within the frequency range of 14.462–15.502 GHz.

To demonstrate the stability of the flat bands, we introduce cavity and disorder defects into the e2 domain walls. The results are depicted in Fig. 4(a), which shows that electromagnetic waves can still propagate along the interfaces, implying the robustness of the transmission. In addition, in order to confirm that anisotropy impacts the topological edge states, we investigate the evolution of e2 topological states with different ellipticities ρ, as shown in Fig. 4(b). When both sides of the domain wall interface have the same anisotropy strength ρ, it is found that the bulk gap of the supercell increases with the increasing of ρ, but the frequency ranges of the edge states decrease. Lower panels of Fig. 4(b) show the calculated band dispersion diagrams with ρ=1.5, 3, and 5.5. Only as ρ reaches a certain level will the anisotropy induced edge states show a flat dispersion. Moreover, a larger ρ results in a narrower dispersion frequency range and a flatter dispersion. We demonstrate a new method to generate stable flat bands via lattice anisotropy induced by low symmetry. Note that the anisotropy ρ is limited to a maximum of 5.5 by the constraint of the structural physical size. In Fig. 4(c), we set φ=−30° on one side of the domain interface wall with the constant anisotropy strength ρ=4.5. The results show that the bulk gap and band dispersion evolution for different ρ have the same trend as those in Fig. 4(b), except for minor changes in the e2 edge mode dispersion.

The anisotropy interface has induced slow light flattened dispersion and a wide edge band gap, offering an opportunity to study higher-order topological corner states. We construct a 60° zigzag edge corner structure, as depicted in Fig. 5(a). The frequencies of the eigenmodes are presented in Fig. 5(b), where two types of higher-order topological corner states are proved within the edge band gap. The Ez profiles for the two corner modes at the frequencies of 13.33 GHz and 14.29 GHz are displayed in Fig. 5(c). Notably, the Zak phases of the two gapped 1D edge states formed at the corner structures are quantized by mirror symmetries, providing an alternative perspective on understanding topological corner states. The Dirac masses of the one-dimensional edge states exhibit opposite signs for the edges along the x and y directions, as shown in Fig. 5(a), resulting in in-gap Jackiw–Rebbi soliton states localized at the corners where these edges intersect [50]. We measure the transmission by using four different antenna-probe methods to visualize two corner states and a flat band. A sample used in the experiment is shown in the left panel of Fig. 5(d). The transmission spectra are presented in the right panel of Fig. 5(d), featuring three sharp peaks around 13.33 GHz, 14.29 GHz, and 14.45 GHz. These peaks clearly indicate the presence of the edge and corner modes within the bulk band gap, which is in good agreement with the simulations. Additionally, the appearance of two corner states with distinct local field distributions is attributed to the localization of the Jackiw–Rebbi soliton state at different lattice positions in the corner structure. The result is due to the reduced symmetry. As depicted in Fig. 5(a), there are three distinct zigzag domain walls (e1,e2,e3) at the corner structure induced by anisotropy. The energy of the Jackiw–Rebbi soliton state is localized within the angular lattice for type-I higher-order states, influenced solely by the coupling structure of boundary e1. In contrast, the electromagnetic wave energy is localized at the lattice position adjacent to the corner for type-II higher-order states, influenced by the coupling of boundaries e2 and e3.

It is crucial to note that our system can bear close resemblance to symmetry-mismatched heterointerfaces in electronic systems [51–53], and these symmetry-mismatched heterointerfaces exemplify a novel academic concept of utilizing lattice symmetry breaking at the heterojunction interface to achieve a Berry curvature dipole moment within the band structure. Anisotropy introduces an additional degree of freedom; it can support topological flattened edge dispersion and multi-type higher-order states on anisotropic crystal structures with reduced symmetry, enabling the investigation of novel topological phenomena. It is worth mentioning that the exploration of topological similarities and differences induced by anisotropy requires a well-established physical mechanism that explicitly considers irregular geometries. This remains an open question and necessitates further research.

In conclusion, our work reveals crystal anisotropy on topological edge states and higher-order topological states in the C2v symmetric triangular lattice. Anisotropy is induced through the deformation of the interface, and flat band dispersion can be achieved. Additionally, our research demonstrates the presence of multi-type higher-order states in anisotropy-induced corner structures. We have theoretically and experimentally observed that the edge states can be from chiral gapless to gapped flattened dispersion, all while maintaining topological protection. Anisotropy emerges as a pivotal factor in the engineering of topological states. These results offer valuable insights and contribute significantly to the exploration of photonics topological phenomena, highlighting the advancement of our understanding in this field.