Flatband systems are characterized by their dispersionless energy spectra and vanishing group velocity1^,2—features that enable the emergence of strongly correlated phenomena such as the fractional quantum Hall effect, Mott insulating states, and unconventional superconductivity.3^–7 These unique properties arise from the macroscopic degeneracy of states at a given energy level, enhancing interaction effects and enabling exotic quantum phases. A variety of fundamental phenomena, including Anderson localization,8^–10 Landau–Zener Bloch oscillations,11 and compact localized states,12^–17 have been explored to deepen the understanding of flatband physics. However, the susceptibility of flatband states to disorder and perturbations presents a significant challenge, limiting their applications. By contrast, topological states, which are protected by global topological invariants, exhibit enhanced robustness and transport properties, even in the presence of disorder.18^–21 Such inherent robustness has already been leveraged in various advanced applications, including topological lasers,22^–26 terahertz transmission,27^–29 and quantum photonics,30^–32 where stability and coherence are critical. Recent experiments have demonstrated that the synergy between flatbands and topological phases can be realized through engineered edges in honeycomb lattices33 or through embedded flux,34^,35 greatly enhancing the robustness of flatband modes. Nevertheless, multiple degeneracy of topological flatbands remains unexplored, which could offer immense potential for advancements in areas such as slow light effect,36^,37 enhanced light-matter interaction,38 and dispersionless image transmission.39^,40 Achieving flatband degeneracy with desired features, however, requires precise control over both flatband conditions and topological invariants, posing a significant challenge.

Graphene, a single layer of carbon atoms arranged in a honeycomb lattice (HCL), has long been considered a paradigm of materials due to its remarkable electronic properties.41 Apart from carbon-based real materials, engineered HCL structures have been realized in optics,42^–51 with photonic graphene emerging as a versatile platform to emulate the electronic behavior of graphene, particularly the edge-related effects. The four fundamental boundary conditions—bearded, armchair, zigzag, and twig—have been proposed and experimentally demonstrated using photonic graphene.33^,43^,45 Notably, graphene edge states, characterized by the nontrivial winding number,52 can be found under the bearded, zigzag, and twig boundary conditions, where they degenerate to exhibit flat dispersion relations. Therefore, by tailoring boundary configurations and introducing external modulations, graphene offers a promising platform to investigate the interplay between flatband physics, topology, and degeneracy.

In this work, we unveil the existence of doubly degenerate topological flatbands of edge states in strained yet chiral-symmetric graphene. The first flatband arises from the twig boundary in the HCL with uniform coupling (referred to here as “unstrained graphene”). The second flatband emerges by applying a uniaxial strain to the HCL along a specific direction, thereby tuning the couplings (referred to as “strained graphene”). Both flatbands consist of topological edge states and are characterized by a nontrivial winding of lattice Hamiltonian. This flatband degeneracy leads to a doubling of the density of states (DOS) for topological zero-energy modes, enabling the formation of two distinct sets of elongated edge states along with two sets of compact localized edge states (CESs) at the boundary. Experimentally, we observe such edge states in photonic graphene with the twig boundary condition (TBC), where coupling ratios (controlled by the relative positions between sites) are tuned by judiciously applying a strain on the HCL. Supported by numerical simulations, the existence of doubly degenerate topological flatbands is conclusively verified. These findings open new possibilities for enhancing the DOS of topological states, which can be particularly advantageous in applications such as flatband lasing.

We begin by considering an HCL with the TBC, as depicted in Fig. 1(b). Under the tight-binding approximation, there are two sites (black and white filled circles) in each unit cell and the nearest-neighbor couplings are denoted as $t_a$ and $t_b$ [black and orange lines in Fig. 1(b)]. The bulk Hamiltonian can be expressed as $$H(\mathbf{k})=\left(\begin{array}{cc}0& h(\mathbf{k})\\ h^{*}(\mathbf{k})& 0\end{array}\right),$$(1)with $h(\mathbf{k})=t_a+t_b{e}^{i{\mathbf{ka}}_1}+t_a{e}^{i{\mathbf{ka}}_2}$, where ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the basis vectors of the unit cell [highlighted by purple rhombus in Fig. 1(b)]. In the case of an unstrained HCL, three couplings are equal and typically set to $t_0$. The twig boundary supports edge states (edge states I) with eigenvalue $\beta =0$, forming a complete flatband that spans the entire one-dimensional Brillouin zone (1D BZ),33 as shown by the green line in Fig. 1(a1). In contrast to the 2D Chern insulators, which are characterized by the Chern number, the nontrivial topology of these degenerate edge states aligns with that of the SSH model.53 It originates from the bulk topological properties and is characterized by the winding number, expressed as $$w=\frac{1}{2\pi }\oint \frac{\text{d}}{\text{d}\mathbf{k}}\arg [h(\mathbf{k})]\mathrm{d}\mathbf{k},$$(2)where $h(\mathbf{k})$ is the off-diagonal term of the bulk Hamiltonian $H(\mathbf{k})$. For the twig boundary along the x-direction [Fig. 1(b)], the integral is performed along a closed loop in $k_y$-direction for a fixed $k_x$. The winding loop at the boundary of the 1D BZ ($k_x=\pi /\sqrt{3}a$) is plotted in the (${\sigma }_x,{\sigma }_y$) plane in Fig. 1(a2). As $k_y$ completes one period, the loop completes two circles. However, only the larger circle encircles the origin (denoted by an origin dot) in (${\sigma }_x,{\sigma }_y$) plane, resulting in $w=1$ and the presence of edge states I. By contrast, for the other three boundary conditions (zigzag, bearded, and armchair), the winding can form only a single loop in the $({\sigma }_x,{\sigma }_y)$ plane. In other words, in an unstrained HCL with any edge termination, the winding number is always restricted to either $w=1$ or $w=0$, but not larger. Further details are provided in the Supplementary Material. Because the twig boundary condition offers the possibility for doubling the winding, one may wonder if it is possible to achieve doubly degenerate topological states by enabling the winding to encircle the origin twice (i.e., to realize $w=2$). We show that this is indeed possible and can be realized by tuning the coupling ratio $\delta$, defined as $\delta =t_b/t_a$ [Fig. 1(b)], i.e., in a strained HCL. When uniaxial strain (compression) is applied along the coupling direction of $t_b$, the couplings become highly non-uniform, resulting in $\delta >1$ and leading to changes in both the bulk and edge properties. The DOS for the ribbon under TBC is shown in Fig. 1(c) for different $\delta$ values, with the peaks corresponding to the zero-energy flatband of edge states, and the gray region representing the bulk bands. For $\delta <2$, the DOS of zero-energy modes increases significantly, indicating the emergence of new edge states (edge states II) that degenerate with edge states I at the zero energy. For $\delta >2$, the bulk band gap opens, and the DOS of zero-energy modes remains constant but has a doubled value as compared with that for the unstrained HCL ($\delta =1$). During this process, the smaller circle of the winding loop under TBC begins to grow larger, and eventually, both loops encircle the origin [Fig. 1(d2)], thus yielding $w=2$ at any given $k_x$. The nontrivial winding implies the existence of edge states II across the entire 1D BZ, forming a new flatband [red line in Fig. 1(d1)] after the gap opens, which becomes fully degenerate with edge states I.

To directly illustrate the formation of the doubly degenerate topological flatbands, the distributions of the vector field of $h(\mathbf{k})$ under different coupling ratios are shown in Fig. 2. For the two-band HCL with chiral symmetry, the integral of the vector field along the $k_y$-path determines the winding number at a fixed $k_x$. For the unstrained HCL ($\delta =1$), the inequivalent Dirac points [red and blue dots in Fig. 2(a2)] reside at the corners of the first BZ (orange hexagon), and the path taken by the integral follows the green vertical line in Fig. 2(a2). The green-shaded region corresponds to the first 1D BZ for the twig boundary. It is evident that the green arrows of the vector $h(\mathbf{k})$ make a complete loop in this region, resulting in $w=1$. According to the principle of “bulk-edge correspondence,” the twig edge states (edge states I) appear across the entire 1D BZ, with their degenerate energies pinned at zero due to chiral symmetry, thus forming a topological flatband [green lines in Fig. 2(a1)].

When $\delta >1$, the Dirac points move along the boundary of the 2D BZ, as indicated by the red and blue arrows in Fig. 2(b2). This shift creates a red-shaded region where the vector $h(\mathbf{k})$ winds twice, resulting in $w=2$. Consequently, the winding-2 implies the presence of an additional set of topological edge states (edge states II) in the red-shaded region, as shown by the red line in the projected 1D band structure [Fig. 2(b1)]. Importantly, tuning the coupling ratios by straining an HCL preserves its chiral symmetry, keeping edge states II pinned at zero energy while edge states I remain intact. As the coupling ratio $\delta$ reaches a critical value ($\delta =2$), the Dirac points merge at the M-point [inset in Fig. 2(c2)]. Further increasing of $\delta$ results in the gap opening [Fig. 2(c1)], leading to a semimetal-to-insulator transition. This transition establishes the winding-2 region extending across the entire 1D BZ [red-shaded region in Fig. 2(c)], where the nontrivial topology at each momentum $k_x$ features the flatband edge modes. The edge modes are pinned at the “zero-energy” due to the preserved chiral symmetry, leading to the formation of doubly degenerate topological flatbands of edge states [green and red lines in Fig. 2(c1)]. Furthermore, flatbands with more than two-fold degeneracy can be achieved by introducing long-range A-B sublattice couplings (see the Supplementary Material). We note that, different from the chiral edge modes in Chern insulators, here the bulk structure preserves the time-reversal symmetry, and the edge states forming the flatbands exhibit topological properties originating from the SSH topology. The flatness and degeneracy of the flatband are protected by chiral symmetry, ensuring robustness against any perturbations that preserve this symmetry (see the Supplementary Material). The degeneracy not only doubles the DOS at zero energy but also supports two distinct sets of topological edge states at any $k_x$. The two edge states at the boundary of the 1D BZ ($k_x=\pi /\sqrt{3}a$) are shown in Figs. 2(d1) and 2(d2), and the edge states at other $k_x$ are provided in the Supplementary Material. Although both edge states are localized on the same sublattice (black circles) due to chiral symmetry, they differ in their phase distributions: edge state I exhibits opposite phase relation at the outermost A sublattices (marked by 1 and 3) of each supercell, whereas edge state II exhibits identical phase relation. In addition, the doubly degenerate topological flatbands give rise to two distinct sets of CESs. It is important to note that this double degeneracy of topological flatbands can only be achieved under TBC in the HCL [Fig. 1(b)]. Tuning the coupling ratios under other boundary conditions, such as zigzag, bearded, and armchair edges, results in either a single flatband or a completely destroyed flatband when the strain is introduced.43^,54 The presence of a single flatband or doubly degenerate flatbands crossing the entire first 1D BZ in strained graphene can also be confirmed by the relation between edge states and supercells (see the Supplementary Material).

In the experiment, we demonstrate the existence of the degenerate topological flatbands in photonic graphene by observing the edge states at different $k_x$. As analyzed above, the topological flatband associated with edge state I is always present, whereas the fully flatband of edge state II emerges only after the gap opens at $k_x=\pi /\sqrt{3}a$ ($\delta >2$). Here, we focus on the demonstration of the evolution of edge state II, whereas the results for edge state I are provided in the Supplementary Material. The photonic HCL with TBC is fabricated using the continuous-wave laser-writing technique in a nonlinear crystal (SBN).55 By applying a strain (compression) to the photonic lattice, the distance between lattice sites is altered, thereby tuning the coupling ratio. Experimentally, two HCLs with different coupling ratios ($\delta =1.5$ and $\delta =3$) are established. An example of the strained HCL with $\delta =3$ is shown in Fig. 3(a), where the nearest-neighbor distances are $d_1=d_2=40.5\mu \mathrm{m}$ and $d_3=30.4\mu \mathrm{m}$, and the uniaxial strain direction is indicated by the gray arrow. The probe beams with specific amplitude and phase distributions, matching the theoretically calculated edge modes (see the Supplementary Material) are generated using a spatial light modulator and sent into the strained HCL. Under different coupling ratios, the probe beams are configured with two different transverse momenta to match the eigenmode distributions of edge states II at $k_x=0$ and $k_x=\pi /\sqrt{3}a$, all exhibiting identical phases at the outermost A sublattices (marked by 1 and 3) of each supercell [Figs. 3(b1), 3(b2), 3(c1), and 3(c2)]. The Fourier spectra and phase distributions of the input beams are shown in the insets. The corresponding outputs after 20 mm propagation exhibit distinct differences [Figs. 3(b) and 3(c)]. Specifically, at $\delta =1.5$, the probe beam corresponding to edge state II remains localized at $k_x=0$ [Fig. 3(b1)] but couples into neighboring sites at $k_x=\pi /\sqrt{3}a$ [Fig. 3(b2)], indicating that the structure cannot support a fully flatband of edge states II before the gap opens. After the gap opens ($\delta =3$), the probe beam remains localized at the edge and in one sublattice for both $k_x=0$ and $k_x=\pi /\sqrt{3}a$ after 20 mm of propagation [Figs. 3(c1) and 3(c2)], suggesting the formation of a fully flatband. For comparison, in-phase mixed bulk modes are also considered, where light spreads into the bulk after propagation under both strain conditions of $\delta =1.5$ [Fig. 3(b3)] and $\delta =3$ [Fig. 3(c3)]. Simulation results for longer propagation distance are provided in the Supplementary Material. These results confirm the formation of edge states II throughout the 1D BZ once the gap opens.

In addition to the two sets of elongated edge states, two compact localized edge states (CES I and CES II) arising from the doubly degenerate flatbands are also supported in the strained HCL. The real-space distributions of the two distinct CESs are shown in Fig. 4(a1) (CES I) and Fig. 4(b1) (CES II), with each holding the characteristic phase distribution of their corresponding edge states. CES I exhibits opposite phases at the outermost A sublattices, whereas CES II exhibits in-phase characteristics [see insets in Figs. 4(a1) and 4(b1)]. In momentum space, the occupation of CESs at different momenta is quantified as $\eta (k_x)={|⟨{\phi }_{k_x}|\mathrm{\Psi }⟩|}^2$, where $\mathrm{\Psi }$ represents the mode of CES I or CES II and ${\phi }_{k_x}$ is the eigenstate of all elongated edge states at $k_x$. The occupation of CES I and CES II across the entire 1D BZ [Figs. 4(a2) and 4(b2)] results in their compact form along the boundaries in real space. Experimentally, probe beams matching the CES I and CES II are launched into the lattice to demonstrate their compactness. Both probe beams remain intact after 20 mm of propagation [Figs. 4(a3) and 4(b3)]. By contrast, the in-phase probe beams fail to sustain compactness, with light spreading into the B sublattices [Figs. 4(a4) and 4(b4)]. Furthermore, the degeneracy of all edge states enables any linear combination of the two CESs to serve as eigenmodes of the structure. Here, we show a linear combination of CES I + CES II, where the energy is nearly localized at a single site at the boundary [Fig. 4(c1)]. This highly compact edge state remains localized at the initially excited A sublattice sites after propagation [Fig. 4(c3)], whereas under in-phase excitation, it becomes distorted and spreads into the neighboring B sublattices [Fig. 4(c4)]. These experimental results, along with corresponding simulations (see the Supplementary Material) confirm the formation of doubly degenerate flatbands. The degeneracy and superposition of the two CESs enable precise tuning of the intensity distribution of the compact edge states.

We have demonstrated the existence of doubly degenerate topological flatbands in strained photonic graphene under the twig boundary condition. In contrast to the zigzag, bearded, and armchair boundary conditions, only the twig boundary condition facilitates unique double degeneracy of flatband edge states, which in turn doubles the DOS of edge modes. The flatband topology is characterized by nontrivial double winding in momentum space. The topological flatbands give rise to two distinct sets of robust edge states and CESs, which are experimentally observed in our laser-written photonic graphene. Experimental results agree well with numerical simulations. Our work highlights the interplay between lattice geometry, symmetry, and topology, presenting a new approach for controlling topological flatbands. This framework opens exciting possibilities for advancements in non-Abelian physics,56^–58 topological lasing,59 and robust energy confinement.60^–62 By leveraging doubly degenerate flatband structures, our findings bring about new methods for enhancing light-matter interactions, paving the way for novel designs of compact and efficient photonic devices. The increased DOS facilitates stronger optical responses, which can be harnessed for low-threshold lasing, enhanced nonlinear effects, and robust light confinement. Moreover, our results expand the frontier of topological phases in engineered materials, offering new opportunities for exploring exotic quantum and classical wave phenomena in photonics and beyond.

Yongsheng Liang is a PhD student working under supervision of Prof. Zhigang Chen in the TEDA Institute of Applied Physics and School of Physics at Nankai University. His research interests include topological edge states and Bloch oscillation.

Jingyan Zhan is a PhD student working under supervision of Prof. Zhigang Chen in the TEDA Institute of Applied Physics and School of Physics at Nankai University. Her research interests include nonlinear optics and optical manipulation.

Shiqi Xia is a professor in the TEDA Institute of Applied Physics and School of Physics at Nankai University. He received his PhD from Nankai University in 2021. His research interests include topological photonics, non-Hermitian optics, and synthetic dimensions.

Daohong Song is a professor in the TEDA Institute of Applied Physics and School of Physics at Nankai University. He received his PhD from Nankai University in 2009. His research interests include topological photonics, pseudospin, and vortex beams.

Zhigang Chen received his PhD from Bryn Mawr College and was a research staff member at Princeton University before joining the faculty at San Francisco State University, where he became a full professor in 2006. He is currently a chair professor at Nankai University. His research interests include nonlinear optics, topological photonics, and optical manipulation. He has served as an editor for several journals and as a chair for numerous conferences, including CLEO Fundamental Science. He is a fellow of Optica and APS.