Compact localized states and flat-band generators in one dimension

Flat bands (FB) are strictly dispersionless bands in the Bloch spectrum of a periodic lattice Hamiltonian, recently observed in a variety of photonic and dissipative condensate networks. FB Hamiltonians are fine-tuned networks, still lacking a comprehensive generating principle. We introduce a FB generator based on local network properties. We classify FB networks through the properties of compact localized states (CLS) which are exact FB eigenstates and occupy $U$ unit cells. We obtain the complete two-parameter FB family of two-band $d=1$ networks with nearest unit cell interaction and $U=2$. We discover a novel high symmetry sawtooth chain with identical hoppings in a transverse dc field, easily accessible in experiments. Our results pave the way towards a complete description of FBs in networks with more bands and in higher dimensions.

Figure 1

(a) Top: canonical $\nu =2$ chain for $U=2$. Circles—lattices sites (different sizes correspond to different onsite energies). Lines—hopping connections (different lines correspond to different hopping strengths). Filled circles—location of a CLS. Bottom: generalized sawtooth chain after basis rotation (see text for details). Signs indicate signs of the CLS amplitudes. (b) The known sawtooth ST1 chain. (c) New sawtooth ST2 chain. Top of (b), (c): lattice structure. Bottom of (b), (c): band structure.

Figure 2

Schematics of the compact localized state with $m_c=1$ and $\nu =2$.

Figure 3

The flat-band energy $E_{\text{FB}}\left(\theta ,\phi \right)$ for $m_c=1,\nu =2,U=2$. The colored squares host FB networks, while the white ones do not. The color code shows the energy of the FB. Dashed-dotted lines—same onsite energies in gST chain. Dotted lines—$t_1=t_2$ in gST chain. Dashed lines—$t_2=t_3$ in gST chain. Solid lines—$t_1=t_3$ in gST chain. Filled circles: ST1 chain in Fig. 1. Filled squares: ST2 chain in Fig. 1.

Figure 4

The ratio of the dispersive bandwidth to the distance between the flat band and the dispersive band center ${\mathrm{\Delta }}_w/\mathrm{\Delta }E$ versus $\left(\theta ,\phi \right)$ for $m_c=1,\nu =2,U=2$ in one irreducible quadrant. The color code shows the value of the ratio.