Photonic Realization of a Generic Type of Graphene Edge States Exhibiting Topological Flat Band

Cutting a honeycomb lattice (HCL) ends up with three types of edges (zigzag, bearded, and armchair), as is well known in the study of graphene edge states. Here, we propose and demonstrate a distinctive twig-shaped edge, thereby observing new edge states using a photonic platform. Our main findings are (i) the twig edge is a generic type of HCL edge complementary to the armchair edge, formed by choosing the right primitive cell rather than simple lattice cutting or Klein edge modification; (ii) the twig edge states form a complete flat band across the Brillouin zone with zero-energy degeneracy, characterized by nontrivial topological winding of the lattice Hamiltonian; (iii) the twig edge states can be elongated or compactly localized at the boundary, manifesting both flat band and topological features. Although realized here in a photonic graphene, such twig edge states should exist in other synthetic HCL structures. Moreover, our results may broaden the understanding of graphene edge states, as well as new avenues for realization of robust edge localization and nontrivial topological phases based on Dirac-like materials.

Figure 1

Illustration of HCLs with four different edge conditions. (a) Schematic of an HCL with four distinct types of edges. Two shaded rectangles illustrate supercells for bearded or zigzag (horizontal) and twig or armchair (vertical) edges. A probe beam aiming at single site of the twig (zigzag) edge will be compact (elongated) along the edge, indicated by bright yellow dots in (b). (b) HCL structure with two sublattices ($A$, $B$) in $(x,y)$ plane. The white dashed line is merely used to separate the two parts in the HCL that are described by different unit cells and Bloch Hamiltonian. The unit cell enclosed by a red (yellow) rhombus is for bearded or armchair (zigzag or twig) edges. ${\mathit{a}}_{\mathbf{1}},{\mathit{a}}_{\mathbf{2}}$ and ${\mathit{a}}_{\mathbf{3}},{\mathit{a}}_{\mathbf{4}}$ are the corresponding basis vectors when different unit cells are selected. ${\mathit{a}}_{\mathbf{1}}=\sqrt{3}a/2\widehat{\mathit{x}}+a/2\widehat{\mathit{y}}$, ${\mathit{a}}_{\mathbf{2}}=\sqrt{3}a/2\widehat{\mathit{x}}-a/2\widehat{\mathit{y}}$ and ${\mathit{a}}_{\mathbf{3}}=\sqrt{3}a/2\widehat{\mathit{x}}-a/2\widehat{\mathit{y}}$, ${\mathit{a}}_{\mathbf{4}}=-a\widehat{\mathit{y}}$, $a$ is the lattice constant. Double headed arrows for $t_1$, $t_2$, $t_3$ mark the three nearest couplings from any chosen $B$ site.

Figure 2

Theoretical analysis of the topological property of zigzag or twig edge states in HCLs. (a) 1D band structures for HCL with (a1) zigzag and (a2) twig edges, where the red lines represent the regions of edge states. (b) Vector fields of ${\mathit{h}}_{ZT}(\mathit{k})$. Red and blue dots represent two inequivalent Dirac points $K$ and $K^{\prime }$ at corners of the first BZ. The values of $k_y$ at the shaded regions indicate the zigzag edge with nontrivial winding, where blue arrows indicate the direction of the vector field along $k_y=\pi /a$. Twig edge states exist for any $k_x$, and orange arrows are for that of the twig edge at $k_x=\pi /\sqrt{3}a$. (c) Normalized intensity distributions of twig edge states at (c1) $k_x=\pi /\sqrt{3}a$ and (c2) $k_x=\pi /2\sqrt{3}a$ denoted by blue and green dots in (a2), respectively. (d) Winding loops for the zigzag edge in the $({\sigma }_x,{\sigma }_y)$ plane at different $k_y$. The red dot marks the origin point $\mathcal{O}$. (e) has the same layout as (d) but for the twig edge at different $k_x$.

Figure 3

Experimental and numerical demonstration of the twig edge states in photonic graphene. (a) An HCL with twig edges along the $x$ direction established in the experiment. $d=32\mathrm{\mu }\mathrm{m}$ is the distance between nearest sites. (b1),(b2) Experimental outputs of the probe beams which match the eigenmode of twig edge states at (b1) $k_x=\pi /\sqrt{3}a$ and (b2) $k_x=\pi /2\sqrt{3}a$. (b3) The output for the mixed mode excitation at $k_x=0$. The corresponding simulation results for a longer propagation distance ($z=50\mathrm{mm}$) are shown in (c), where the solid white circles mark the topmost sites of the twig edge. (d) Fourier spectra of the input beams corresponding to (b), in which solid (dashed) lines mark the center (edge) of the 1D BZ.

Figure 4

Demonstration of compact edge states at the twig edge of photonic graphene. (a) Theoretical results of a compact edge state for the twig edge. (a1) The intensity distribution of the modified edge state. The initial selected site is marked by a green arrow. (a2) The percentage of energy projected on supercells ($N$) along the boundary. (a3) The spectrum distribution of the compact edge state in $k$ space, where edge states exist in the red shaded region. (b) Experimental results of a compact edge state. (b1) A probe beam matching the compact edge state at the input. (b2) The output of the probe beam after 20 mm propagation. (c),(d) have the same layout as (a),(b) but obtained from the zigzag edge for a direct comparison.