Simple method to construct flat-band lattices

We develop a simple and general method to construct arbitrary flat-band lattices. We identify the basic ingredients behind zero-dispersion bands and develop a method to construct extended lattices based on a consecutive repetition of a given miniarray. The number of degenerated localized states is defined by the number of connected miniarrays times the number of modes preserving the symmetry at a given connector site. In this way, we create one or more (depending on the lattice geometry) complete degenerated flat bands for quasi-one- and two-dimensional systems. We probe our method by studying several examples and discuss the effect of additional interactions such as anisotropy or nonlinearity. We test our method by studying numerically a ribbon lattice using a continuous description.

Figure 1

(a) Dimer and its linear modes. (b) Two miniarrays connected by a connector (dark) site. (c) Rhombic lattice. (d) Linear spectrum for $V=\overline{V}=1$.

Figure 2

Shown on the left is a three-site miniarray and on the right are linear modes.

Figure 3

(a) Three-site miniarray and one of its modes. (b) Two miniarrays connected by a connector (dark) site. (c) Four linear modes of the composed system. (d) Cross lattice. (e) Linear spectrum for $V_1=V_2=1$.

Figure 4

(a) Three-site miniarray and one of its linear modes. (b) Two miniarrays connected by a connector (dark) site. (c) Sawtooth lattice. (d) Linear spectrum for $\delta =0.75$ (thin), $\sqrt{2}$ (thick), and 2 (dashed).

Figure 5

(a) Eight-site miniarray. (b) Two linear modes.

Figure 6

(a) Two miniarrays connected by a connector (dark) site. (b) Lieb lattice. (c) Linear spectrum for $V_x=V_y=1$.

Figure 7

(a) Two miniarrays connected by a connector (dark) site. (b) Lieb 2 lattice. (c) Linear spectrum for $V_x=V_y=1$.

Figure 8

(a) Four-site miniarray and two of their linear modes. (b) Two miniarrays connected by a connector (dark) site. (c) B2-ribbon lattice. (d) Linear spectrum for $V_x=V_y=V_d=1$.

Figure 9

(a) Composed aperiodic system and its (b) linear spectrum. Insets in (b) show the amplitude profile for some localized modes of this system (the color scale increases from red to yellow, with orange being equal to zero).

Figure 10

(a) Six-site miniarray and (b) three of their linear modes. (c) Two miniarrays connected by a connector (dark) site. (d) B3-ribbon lattice. (e) Linear spectrum for $V_x=V_y=V_d=1$.

Figure 11

Transversal intensity profile ${\left|\psi (x,y,L)\right|}^2$, after propagating a distance $L=10\mathrm{cm}$, in a B3-ribbon lattice geometry. (a)–(c) Different output profiles for the input conditions sketched in each figure. Density plots are normalized to one, as indicated by the color map.