Flat band states: Disorder and nonlinearity

We study the critical behavior of Anderson localized modes near intersecting flat and dispersive bands in the quasi-one-dimensional diamond ladder with weak diagonal disorder $W$. The localization length $\xi$ of the flat band states scales with disorder as $\xi \sim W^{-\gamma }$, with $\gamma \approx 1.3$, in contrast to the dispersive bands with $\gamma =2$. A small fraction of dispersive modes mixed with the flat band states is responsible for the unusual scaling. Anderson localization is therefore controlled by two different length scales. Nonlinearity can produce qualitatively different wave spreading regimes, from enhanced expansion to resonant tunneling and self-trapping.

Figure 1

(a) Lattice structure with $a$, $b$, $c$ sublattices, unit cell marked by dashed line. (b) Band structure. Disorder $W$ smears out the flat band to a width $W$ (shaded region).

Figure 2

Mode properties: (a) mean localization length $\xi$, (b) participation ratio $P$, (c) second moment $m_2$, and (d) compactness index $\zeta$ as a function of energy $E$ for different disorder strengths $W=0.5$ [(r)ed], 1 [(b)lue], 2 [(g)reen], 4 [(m)agenta]. Statistical errors do not exceed the spot size. Vertical bars indicate the cutoff energy $\left|E\right|=W/2$.

Figure 3

(a) Localization length $\xi \left(W\right)$ for low ($E=0$) and high ($E=\sqrt{2}$) energy modes, following the power laws $\xi \sim W^{-1.3}$ and $\xi \sim W^{-2}$, respectively. The $E=0.1$ curve shows the crossover between these laws at $W\sim \left|E\right|$. $\xi (E=0)$ remains finite in the gapped cases $m=0.5$ (lower green curve) and $m=0.05$ (upper green curve). (b) Mode profile of a sparse ($\zeta =0.1$) low energy mode $W=0.5$, and the effective coupling $|C_n|$ defined by Eq. (6).

Figure 4

(a) Participation ratio of a flat band excitation after evolution for $t=400$ as a function of nonlinearity strength $\beta$. For each point we obtain 500 realizations of disorder, and show the mean $\langle P\rangle$ [(b)lue], standard deviation [(r)ed], and typical value $exp(\langle lnP\rangle )$ [(g)reen]. Error bars indicate 95% confidence intervals obtained via bootstrap method. (b) Same, but second moment instead. (c) Intensity $I_n=|a_n{|}^2+|b_n{|}^2+{\left|c_n\right|}^2$ during linear propagation when sparse modes are strongly excited. (d) Same, but linear scale. In all panels, $W=1$.

Figure 5

Effect of nonlinearity on spreading of a single site excitation $W=1$. (a) Linear propagation $\beta =0$. (b) Intermediate nonlinearity $\beta =2$. (c) Strong nonlinearity $\beta =8$. (d) The intensity at the initially excited unit cell in all three cases.