Evolving disorder in non-Hermitian lattices

TL;DR

This work investigates how evolving disorder modifies wave transport in one-dimensional non-Hermitian lattices, contrasting static disorder in Hermitian systems with time-renewed randomness. It analyzes two representative models—symmetric-coupled lattices with evolving complex on-site disorder and Hatano–Nelson lattices with evolving real disorder—by varying the disorder strength $W$, the renewal period $\\ell$, and, in the Hatano–Nelson case, the asymmetry $h$. The main findings show that rapid disorder updates suppress localization and induce diffusion-like spreading, while long periods restore Anderson localization with abrupt jumps in the symmetric case; in Hatano–Nelson lattices, evolving disorder tunes the competition between the non-Hermitian skin effect and localization, producing monotonic or non-monotonic velocity responses depending on $(W,h)$. The results establish evolving disorder as a versatile knob for controlling non-Hermitian transport with potential applications in photonics and quantum simulations, and open directions for extending to higher dimensions and nonlinear settings.

Abstract

The impact of disorder on wave transport has been extensively studied in Hermitian systems, where static randomness gives rise to Anderson localization. In non-Hermitian lattices, static disorder can lead to peculiar transport features, including jumpy wave evolution. By contrast, much less is known about how transport is modified when the on-site disorder evolves during propagation. Here we address this problem by investigating two pertinent non-Hermitian lattice models with disorder altered at regular intervals, characterized by a finite disorder period. In lattices with symmetric couplings and complex on-site disorder, short disorder periods suppress localization and give rise to diffusion-like spreading, while longer periods allow the emergence of jumps. In Hatano-Nelson lattices with real on-site disorder, the non-Hermitian skin effect asymptotically dominates regardless of the disorder strength, while the disorder period reshapes the drift velocity and modulates its competition with Anderson localization. These results establish evolving disorder as a novel way of tuning non-Hermitian transport.

Figures (9)

• Figure 1: Weak evolving disorder ($W=1$): Normalized intensity evolution for four disorder periods: (a) $\ell=1$, (b) $\ell=10$, (c) $\ell=25$, (d) $\ell=100$. Each panel corresponds to a single disorder realization. For rapid updates ($\ell=1$), Anderson localization is suppressed and the wave packet spreads throughout the lattice. As $\ell$ increases, partial localization develops, and, for large disorder periods ($\ell=100$), distinct Anderson jumps become visible.
• Figure 2: Intermediate evolving disorder ($W=3$): Same system as Fig. \ref{['fig:W1']}, but for $W=3$. Each panel corresponds to a single disorder realization. For short disorder periods, the wave packet forms a branching structure of successive jumps across the lattice. Increasing $\ell$ suppresses these fluctuations and enhances confinement, while at longer periods Anderson jumps dominate the dynamics.
• Figure 3: Strong evolving disorder ($W=6$): Same system as Fig. \ref{['fig:W1']}, but for $W=6$. Each panel corresponds to a single disorder realization. Although localization dominates, rapid updates ($\ell=1$) partially disrupt confinement, producing a branching pattern of successive jumps. As $\ell$ increases, localization strengthens, and, for $\ell=25$ and $\ell=100$, well-defined Anderson jumps emerge.
• Figure 4: Long-term uncertainty map: Ensemble-averaged position uncertainty $\overline{\Delta x}(z_{\max})$ as a function of disorder strength $W$ and disorder period $\ell$ at $z_{\max}=400$. For weak disorder and rapid updates, transport is diffusion-like with large $\overline{\Delta x}$. At intermediate disorder, $\overline{\Delta x}$ depends strongly on $\ell$, reflecting the competition between Anderson localization and evolving disorder. For strong disorder, transport is largely suppressed, although very frequent updates still partially disrupt localization.
• Figure 5: Hatano–Nelson lattice with intermediate evolving disorder ($W=3$): Normalized intensity evolution for asymmetry parameter $h=0.05$ and four disorder periods: (a) $\ell=1$, (b) $\ell=10$, (c) $\ell=25$, (d) $\ell=100$. Each panel corresponds to a single disorder realization. The red line indicates the mean position $\langle x(z)\rangle$. For $\ell=1$, the skin effect dominates and the wave packet is confined near the boundary. As $\ell$ increases, localization gradually re-emerges, broadening the intensity profile over roughly half of the lattice.