Fig. 1. Doubly degenerate topological flatbands in a strained HCL with twig boundaries. (a1) The 1D band structure of an unstrained HCL (δ=tb/ta=1) with twig boundaries, where the green lines represent the region of edge states. (a2) Winding loop for the twig boundary in the (σx,σy) plane at kx=π/3a. The orange dot marks the origin O. (b) Schematic diagram of the HCL with twig boundaries and periodic along the x-direction. The purple-shaded rhombus marks the unit cell with two sublattices (A and B), and the gray-dashed rectangle indicates the supercell corresponding to the twig boundary. The primitive vectors are a1=ay^ and a2=3a/2x^+a/2y^, where a is the lattice constant. Black and orange lines represent the weak couplings (ta) and strong couplings (tb) of the strained HCL, respectively, with the strain applied along the tb coupling direction indicated by two gray arrows. (c) Evolution of the density of states (DOS) as a function of the coupling ratio (δ). The DOS of the topological zero modes is highlighted by colors. (d1), (d2) Plots have the same layout as (a1), (a2) but for a strained HCL (δ=3), displaying two degenerate flatbands pinned to zero energy [green and red lines in panel (d1)]. In this latter case, the DOS of zero modes as well as the nontrivial winding number is doubled compared with those in panel (a).

Fig. 2. Topological origin of doubly degenerate flatbands in a strained HCL. (a1)–(c1) 1D band structure of the HCL with twig boundaries under different coupling ratios at δ=1 (a1), δ=1.5 (b1), and δ=3 (c1). The black dots mark the degenerate points, and the black arrows in panel (b1) indicate the movement of degenerate points under compression. The original twig edge states (edge state I) are highlighted by green lines, and the new edge states (edge state II) are shown in red lines. (a2)–(c2) The corresponding schematics for the winding number calculation, where the arrows point in varying directions of the vector h(k). Red and blue dots represent Dirac points with opposite Berry phases. The winding number takes the value of w=1 (w=2) in the green (red)-shaded region, which determines the number of edge states. The inset in the bottom-left corner of (c2) shows the merging of Dirac points at the M point when δ=2. (d) Illustration of the edge state distributions corresponding to edge states I (d1) and edge states II (d2) at kx=π/3a, denoted by the yellow star in (c1). Red and blue dots in panel (d) represent opposite phase distributions.

Fig. 3. Experimental observation of edge state II in strained photonic graphene. (a) A laser-written strained graphene lattice with twig boundary along the x-direction and the strain applied along d3 direction at δ=3. In this case, the distances between nearest-neighbor sites are d1=d2=40.5μm and d3=30.4μm. For a different strain at δ=1.5, d1=d2=40.5μm and d3=34.5μm. The gray arrows indicate the uniaxial strain direction. A and B are two sublattices within the unit cell. (b) Experimental outputs of the probe beam matching the eigenmode of edge state II at kx=0 (b1) and kx=π/3a (b2) under δ=1.5. (b3) Experimental outputs of in-phase probe beam. The corresponding phase distributions within the dashed squares and Fourier spectra of the input beam are shown in the insets. The solid (dashed) lines in the Fourier spectra mark the center (edge) of the 1D BZ. (c) Results presented in the same layout as panel (b), but they are for edge state II under δ=3. The red (white) arrows in panels (b) and (c) indicate the presence (absence) of light on the B sublattices. For all the experimental results, the propagation distance is 20 mm.

Fig. 4. Demonstration of compact edge states (CESs). (a1) Intensity and phase (inset on the right) distributions of the probe beam, consistent with the mode distribution of CES I. (a2) Spectrum distribution of CES I in the 1D BZ, where η(kx) is normalized. (a3) Output of the phase-structured probe beam shown in (a1) after 20 mm of propagation. (a4) Output of the in-phase (uniform without modulation) beam for comparison. (b), (c) Panels (b1)–(b4) and (c1)–(c4) have the same layout as (a1)–(a4), but they are for the demonstration of CES II (b) and the linear combination of CES I + CES II (c).