Linear and nonlinear compact modes in quasi-one-dimensional flatband systems

We examine analytically and numerically the spectral properties of three quasi-one-dimensional lattices, namely, kagome, Lieb, and stub lattices, which are characterized for having flatbands in their spectrum. It is observed that the degenerate eigenmodes modes of these flatbands form a Starklike ladder where each mode is shifted by one lattice site. Their combination can give rise to compact modes that do not diffract due to a geometrical phase cancellation. For all three cases we computed the stability of the fundamental band mode against perturbation of their amplitude and phase, the effect of possible anisotropy of the couplings, and the presence of small random perturbations of the coupling. For the Lieb and stub ribbon, the compact mode turns out to be quite robust and the flatband survives, while for the kagome ribbon, the compact mode is destroyed and the flatband is lost. When adding nonlinear effects, the compact mode turns out to be also a nonlinear eigenvector, with a power curve that is proportional to the eigenvalue and exists for any eigenvalue, in marked contrast to the usual case of discrete solitons, which can exist only outside the linear bands. These properties look promising for a future design of a robust system for long-distance propagation of information.

Figure 1

Quasi-one-dimensional arrays. (a) Kagome ribbon, (b) Lieb ribbon, (c) Stub ribbon. The lines between dots connect nearest neighbor sites.

Figure 2

The five bands of the kagome ribbon, $V=1$.

Figure 3

Top: Eigenvalues of the kagome ribbon, arranged in order of increasing value. The flatband is clearly appreciable at $\mathrm{\Omega }=-2V$. Bottom: Density plot of the eigenvectors of the kagome ribbon, arranged in order of increasing eigenvalue ($N=150$).

Figure 4

Top: Fundamental compact ring mode belonging to the flatband. Every site on the ring possesses equal amplitude, but their phases change by $\pi$ when moving from site to site along the ring. Bottom: Participation ratio $R$ of the ring mode as a function of the evolution coordinate $z$.

Figure 5

Participation ratio $R$ of the fundamental ring as a function of $z$ for a random perturbation of the initial conditions. The width of the noise is $w=0.1$ (top), $w=0.5$ (middle), and $w=1.0$ (bottom).

Figure 6

Eigenvalues $\mathrm{\Omega }$ for a large ($N=120$) kagome ribbon for different values of the anisotropy parameter $\delta$. The eigenvalues are plotted in order of increasing value and just the flatband region is shown.

Figure 7

Kagome ribbon. Evolution of the participation ratio $R$ for different values of the anisotropy parameter $\delta$ in the couplings.

Figure 8

Kagome ribbon with coupling disorder. Top: Eigenvalues of the kagome ribbon, ordered according to increasing value. Bottom: Evolution of the participation ratio of the compact mode, for different disorder widths.

Figure 9

Power vs propagation constant diagram for the fundamental compact kagome mode. Continuous (dashed) line denotes stable (unstable) behavior.

Figure 10

Kagome ribbon. Top: Participation ratio as a function of the power content of the nonlinear mode depicted in Fig. 9. Bottom: Stability gain vs the nonlinear mode propagation constant. The onset of instability is clearly visible around $\mathrm{\Omega }\sim 4.6$.

Figure 11

Kagome ribbon. Power vs propagation constant diagram for the fundamental compact kagome mode. Continuous (dashed) line denotes stable (unstable) behavior.

Figure 12

Top: The five bands of the Lieb ribbon. Bottom: Density plot of the eigenvectors of the Lieb ribbon, arranged in order of increasing eigenvalue.

Figure 13

Evolution of the participation ratio of the fundamental compact ring of the Lieb ribbon.

Figure 14

The first four upper plots show the eigenvalue ($\mathrm{\Omega }$) and the participation ratio of the Lieb ribbon for two different anisotropies $\delta$ and zero disorder ($w=0$), while the lower four plots show $\mathrm{\Omega }$ and $R$ for two different disorder widths $w$ and zero anisotropy ($\delta =0$).

Figure 15

Power vs propagation constant diagram for the fundamental compact Lieb mode. Continuous (dashed) line denotes stable (unstable) behavior.

Figure 16

Power vs propagation constant diagram for a mode centered on a single site in the middle of the Lieb ribbon. Continuous (dashed) line denote stable (unstable) behavior.

Figure 17

The three bands of the stub ribbon.

Figure 18

Stub ribbon. Top: Evolution of the participation ratio of the fundamental stub compact mode, at long evolution distances. Inset shows the geometrical disposition of the fundamental compact mode. Bottom: Density plot of the eigenvectors of the stub ribbon, arranged in order of increasing eigenvalue.

Figure 19

Power $P$ vs propagation constant $\mathrm{\Omega }$ for the nonlinear fundamental mode for the stub ribbon. Solid (dashed) line denotes stable (unstable) portions.