Building flat-band lattice models from Gram matrices

We propose a powerful and convenient method to systematically design flat-band lattice models, which overcomes the difficulties underlying the previous method. Especially, our method requires no elaborate calculations, applies to arbitrary spatial dimensions, and guarantees to result in a completely flat ground band. We use this method to generate several classes of lattice models, including models with both short- and long-range hoppings, both topologically trivial and nontrivial flat bands. Some of these models were previously known. Our method, however, provides important insight. For example, we have reproduced and generalized the Kapit-Mueller model [Kapit and Mueller, Phys. Rev. Lett. 105, 215303 (2010)] and demonstrated a universal scaling rule between the flat-band degeneracy and the magnetic flux that was not noticed in previous studies. We show that the flat band of this model results from the (over-)completeness properties of coherent states.

Figure 1

Tasaki lattice in one dimension (a) and two dimensions (b) governed by Hamiltonian (2). Bipartite lattice in one dimension (c) and two dimensions (d) governed by Hamiltonian (3). Rectangular boxes represent unit cells, with internal sites labeled by circled numbers. Solid lines represent nonzero hoppings starting from a single cell. Duplicated hopping terms are marked by dashed lines.

Figure 2

(a)–(d) Spectrum of $H^S$ for a square lattice (a), triangular lattice (b), honeycomb lattice (c), and random lattice (d). The color map represents $(D+1)$ where $D\left(E\right)={\sum }_n\delta (E-E_n)$ is the density of states, with $E_n$ being the $n^{\mathrm{th}}$ eigenvalue of $H^S$. In the calculation, the Dirac $\delta$ function is replaced by a smooth narrow distribution function. $S$ is the averaged area per state. (e) We count $N_0$, the number of eigenenergy that is less than ${10}^{-5}$, and compare the ratio $N_0/N$ with the theoretical value $\rho$ marked by the line, where $\rho =max(1-S/\pi ,0)$.

Figure 3

(a) Evolution of the density profile of a wave packet on an $L\times L=81\times 81$ square lattice with a linear potential along the $y$ axis with gradient $U$. The evolution operator is given by $exp\{-2\pi it(G^S+Uy)\}$, where we take $U=3.75\times {10}^{-4}$. The positions $x$ and $y$ are renormalized by the lattice constant. The initial wave packet is obtained by projecting a completely localized state to the lowest (zero-energy) band. $t_0=5000$. (b) The evolution of the central position $\overline{x}$ and the half width $\sigma$ along the $x$ direction of the wave packet in (a). (c) The same evolution as that in (a) except that now $S=5\pi /4$ such that the lowest band is not a flat band.

Figure 4

The spectrum of the generalized Kapit-Mueller model given by the $T$ matrix Eq. (13). The color map has the same meaning as that in Fig. 2.