Energy transport and chaos in a one-dimensional disordered nonlinear stub lattice

TL;DR

This work investigates energy transport in a one-dimensional stub lattice with onsite disorder and nonlinearity, focusing on how flat-band structure influences chaotic spreading. By mapping the disorder strength $W$ and nonlinear shift $\delta$ to dynamical regimes, the authors identify weak chaos, strong chaos, and self-trapping, observing subdiffusive spreading with $m_2 \propto t^{0.33}$ in weak chaos and $m_2 \propto t^{0.5}$ in strong chaos, along with corresponding ftMLE decays. The analysis reveals that small frequency gaps near gap closing have little effect on spreading, and that the observed behaviors closely mirror those seen in disordered DNLS and KG lattices, supporting universality across network geometries. The results extend nonlinear disordered-lattice characterizations to flat-band networks like the stub lattice, with implications for energy transport in engineered materials and photonic systems.

Abstract

We investigate energy propagation in a one-dimensional stub lattice in the presence of both disorder and nonlinearity. In the periodic case, the stub lattice hosts two dispersive bands separated by a flat band; however, we show that sufficiently strong disorder fills all intermediate band gaps. By mapping the two-dimensional parameter space of disorder and nonlinearity, we identify three distinct dynamical regimes (weak chaos, strong chaos, and self-trapping) through numerical simulations of initially localized wave packets. When disorder is strong enough to close the frequency gaps, the results closely resemble those obtained in the one-dimensional disordered discrete nonlinear Schrödinger equation and Klein-Gordon lattice model. In particular, subdiffusive spreading is observed in both the weak and strong chaos regimes, with the second moment $m_2$ of the norm distribution scaling as $m_2 \propto t^{0.33}$ and $m_2 \propto t^{0.5}$, respectively. The system's chaotic behavior follows a similar trend, with the finite-time maximum Lyapunov exponent $Λ$ decaying as $Λ\propto t^{-0.25}$ and $Λ\propto t^{-0.3}$. For moderate disorder strengths, i.e., near the point of gap closing, we find that the presence of small frequency gaps does not exert any noticeable influence on the spreading behavior. Our findings extend the characterization of nonlinear disordered lattices in both weak and strong chaos regimes to other network geometries, such as the stub lattice, which serves as a representative flat-band system.

Figures (16)

• Figure 1: Schematic representation of the stub lattice model composed of $N$ unit cells (one of which is indicated by a dashed box). Each cell contains three sites, which are denoted by circles labeled A, B and C. The solid lines connecting the sites represent the intra- and inter-unit cell hoppings.
• Figure 2: The dispersion relation of the linear ($\beta=0$), ordered $\left ( \epsilon_{n}^{(K)} = 0 \right )$ stub lattice model \ref{['H_SL_wave']}, consisting of three distinct bands: a FB (orange curve, $\omega = 0$) and two symmetric dispersive ones (green and blue curves, $\omega = \pm \sqrt{3+2\cos q}$). A gap (indicated as a gray region) of size $\alpha=1$ exists between the FB and the maximum (minimum) propagating frequency of the lower (upper) dispersive band.
• Figure 3: Averaged (over $50$ disorder realizations) frequency spectra of the linear ($\beta=0$) disordered stub lattice model \ref{['H_SL_wave']} with $N=1000$ unit cells for different disorder strengths: (a) $W=0.5$, (b) $W=1.0$, (c) $W=1.5$, (d) $W=2.0$ and (e) $W=2.5$. In each panel the $3N=3000$ normal mode frequencies ($\omega$) are ordered in increasing value, with $\nu$ being the related index of this arrangement. The lowest $1000$ frequencies are colored in blue, the next $1000$$\omega$ values are colored in orange and the $1000$ largest frequencies are shown in green. We note that error bars denoting one standard deviation from the average frequency value are not present in the plots as they are too small to be visible.
• Figure 4: The profiles, i.e. $\left| U_{\nu,n}^{(K)} \right|$ values, versus site index $n$ (with K denoting the A, B or C sites), of mode $\nu=291$ (blue), mode $\nu = 1291$ (orange), and mode $\nu = 2291$ (green) for a specific disorder realization for the linear ($\beta = 0$) stub lattice \ref{['H_SL_wave']} with $N=1000$ and $W=1$. Results are presented for sites (a) A, (b) B, and (c) C.
• Figure 5: Averaged (over 50 disorder realizations) values of the localization volume $\langle V^{(K)} \rangle$ on A (black curve), B (gray curve) and C sites (red curve) of normal modes with mean position at the center of the lattice, versus the disorder strength $W$. The black dashed line denotes a function proportional to $70/W^2$.