Robust flat bands of the honeycomb wire network

Abstract

We show that periodic honeycomb networks of ballistic conducting channels generically host exact flat bands spanning the entire Brillouin zone. These flat bands are independent of microscopic vertex scattering, persist for any number of transverse modes, and occur in a universal $1\colon 2$ ratio with dispersive bands. Their existence is enforced by local $D_3$ vertex symmetry and lattice translations. We construct compact localized states obeying a Bohr-Sommerfeld-type quantization condition and demonstrate that flat bands survive in realistic antidot lattices, establishing honeycomb wire networks as a robust flat band platform relevant to gated high-mobility 2D electron gases and molecule-patterned metallic surfaces.

Figures (8)

• Figure 1: (a) The single channel honeycomb wire network, with translation vectors ${\boldsymbol{e}}_{1,2}$. Each wire contains two counter-propagating plane waves with amplitude $\alpha_{m,n,i}$ and $\beta_{m,n,i}$. Elastic scattering happens at the nodes, described by scattering matrices $S_A$ for sublattice $A$ (open red) and $S_B$ for sublattice $B$ (filled green). (b) Energy spectrum of the honeycomb wire network as a function of the Bloch momentum ${\boldsymbol{k}}$ in the Brillouin zone (dashed blue hexagon). Flat bands (green) are observed for any $S_A$ and $S_B$, the particular scattering parameters being here $(\theta_v,\phi_v) = (\pi/9,4\pi/5)$ for both $v=A,B$, see Eq. \ref{['Ssinglebis']}.
• Figure 2: Data of the lowest energy compact localized states (CLSs): (a) The value of the 1D wave vector $q$ (shown as color) of the CLSs as a function of $\phi_{A,B}$: $q/\pi \equiv - \frac{\phi_A+\phi_B}{2 \pi}~ \mathrm{mod}~1$, and (b) Two representative CLSs corresponding to the values $(\phi_A,\phi_B) = \pm (3 \pi/10,7\pi/10)$, where the amplitude of the wave function is shown as the height, and the argument as the color. A CLS in the lower left (upper right) triangle region of $(\phi_A,\phi_B)$ parameter space in (a) has zero (one) node: this is manifest in the CSLs shown in (b).
• Figure 3: The spectrum along high symmetry path in the Brillouin zone for the $N$-channel honeycomb quantum graph with $N = N_++N_-$: (a) $N_\pm=2$, (b) $N_\pm=5$. We take arbitrary parametrization following Eq. \ref{['eq:Ss_N_chanenel']} for the scattering matrices $S_{A/B}$ at the nodes, and assume the $\mu$-th channel has the energy-momentum relation $E = q_\mu^2+(\pi \mu /w)^2$. We take $w=0.2$.
• Figure 4: (a) The antidots potential profile (see Eq. \ref{['eq:Kronig_Penney_V']}); the narrow potential-free region ($V=0$) appears as trenches between barrier regions ($V=V_0$), forming a quasi-1D hexagonal network of width $w$. (b) The 2D compact localized state (CLS) around one hexagon solved from the antidot potential in (a). The amplitude and phase of the wave function are shown as the height and color, respectively; the latter follows the colorbar in Fig. \ref{['fig2']}(b). (c) Low energy spectrum of the antidot model along high symmetry path in the Brillouin zone. All panels are plotted with parameters $(V_0,w) = (400,\sqrt{3}/20)$.
• Figure 5: The argument of the eigenvalues of $e^{iQ(E)} S_{B,0} e^{i Q(E)} S_{A,0}$ as a function of $E$: we choose arbitrary $S_{A/B,0} \in U(N)$ with $N=N_p+N_n=16$, and $w=2$. Whenever one of the arguments reaches multiples of $2\pi$ (shown as horizontal dashed gray line) an eigenvalue of $\lambda=1$ is resulted.