Solution of Wave Acceleration and Non-Hermitian Jump in Nonreciprocal Lattices

TL;DR

The study analyzes wave-packet dynamics in the discrete Hatano–Nelson lattice with nonreciprocal couplings, revealing a time-dependent center-of-mass motion that accelerates, decelerates, and then attains uniform motion while the amplitude grows exponentially. A continuum approximation yields a higher-order PDE with coefficients depending on the initial momentum $k_i$ and nonreciprocity $g$, enabling closed-form expressions for the center-of-mass trajectory $x_{\rm com}(t)$ and acceleration $a_{\rm com}(t)$, and identifying three evolution stages separated by a characteristic time $T^*$. The authors predict a non-Hermitian jump—an abrupt shift between velocity regimes—and a non-Hermitian boomerang effect, including a velocity reversal driven by time-dependent nonreciprocal forcing; these predictions match lattice simulations. They show momentum-dependent amplitude growth, largest near $k_i=\pi/2$, arising from mode-space evolution $\varphi_k(t)$ with growth $\propto t(g-1)\sin k$, and explain the jump as a competition between $k=0$ and $k=\pi/2$ components, with disorder being unnecessary. The work provides a framework for controlling waves in nonreciprocal, non-Hermitian metamaterials and discusses connections to related recent results.

Abstract

The time evolution of initially localized wavepackets in the discrete Hatano-Nelson lattice displays a rich dynamical structure shaped by the interplay between dispersion and nonreciprocity. Our analysis reveals a characteristic evolution of the wave-packet center of mass, which undergoes an initial acceleration, subsequently slows down, and ultimately enters a regime of uniform motion, accompanied throughout by exponential amplification of the wave-packet amplitude. To capture this behavior, we develop a continuum approximation that incorporates higher-order dispersive and nonreciprocal effects and provides accurate analytical predictions across all relevant time scales. Building on this framework, we then demonstrate the existence of a non-Hermiticity-induced jump - an abrupt spatial shift of the wave-packet center even in the absence of disorder - and derive its underlying analytical foundation. The analytical predictions are in excellent agreement with direct numerical simulations of the Hatano-Nelson chain. Our results elucidate the interplay between dispersion and nonreciprocity in generating unconventional transport phenomena, and pave the way for controlling wave dynamics in nonreciprocal and non-Hermitian metamaterials.

Figures (4)

• Figure 1: (a) Schematic of the nonreciprocal Hatano–Nelson (HN) model. The nonreciprocity is introduced through asymmetric couplings: $g$ to the right and $1$ to the left. (b) Real part of the dispersion of the HN model as a function of momentum $k$. (c) Distinct velocity dynamics of a localized wavepacket in the Hatano–Nelson model, shown in the velocity–time plane for initial momenta $k = 0$, $\pi/4$, and $\pi/2$. For $k = \pi/2$, the wavepacket propagates with a constant group velocity $1+g$, exhibiting zero acceleration. For the smaller $k$ a non-Hermitian jump is observed.
• Figure 2: Normalized spatiotemporal response $|\Psi_{n}(t)|$ of the initial wavepacket $\Psi_{\rm in}$ in the Hatano–Nelson chain, as given by Eq. (\ref{['initialwavepacket']}): Panels (a1), (b1), and (c1) correspond to initial momenta $k_i = 0$, $\pi/4$, and $\pi/2$, respectively. The inset in (a1) shows the initial profile $\Psi_{\rm in}$ at $t = 0$ and the time-evolved wavepacket $\Psi_f$ at $t = t_f$. The corresponding final profiles for $k_i = \pi/4$ and $\pi/2$ are shown in the insets of (b1) and (c1). The progression from (a1) to (c1) highlights that the amplitude of $|\Psi_f|$ is largest for $k_i = \pi/2$. Group velocity and acceleration from lattice simulations and PDE theory: Panels (b1) and (c1) correspond to $k_i=0$, panels (b2) and (c2) to $k_i=\pi/4$, and panel (c3) shows the acceleration phase diagram in the $(t, k)$ plane. In all the panels we chose $\sigma_0=4$, $g=3$.
• Figure 3: Non-Hermitian Boomerang Effect: (a) Spatiotemporal response of the wave dynamics under Hermitian ($g = 1$) evolution for $k_i = -\pi/4$. (b) Nonreciprocal ($g = 2$) dynamics for the same initial momentum, $k_i = -\pi/4$, revealing a boomerang-like reversal in the wave's motion. The insets in (a) and (b) shows the wave-packet group velocity $v_g(t)$ obtained from simulations of Eq.( \ref{['schrodinger']}) (white) and the analytical expression in Eq.( \ref{['eqboomerang']}) (orange). Specifically, $v_g(t)$ in (b) show a sign reversal of the group velocity from negative to positive over time. For both panels, we choose $\sigma_0 = 3.5$.
• Figure 4: Non-Hermitian Jump: (a) Normalized spatiotemporal evolution $|\Psi_n(t)|$ for $g$=2 of the initial wavepacket $\Psi_{\rm in}$, with the corresponding momentum-space distribution $|\Psi_k(t)|$ shown at $t = 0$, $45$, and $60$. (b) Time dependence of the mode weights $\varphi_k(t)$ during the evolution. (c) Jump time $t_{\mathrm{jump}}$ as a function of nonreciprocity $g$ for $\delta = 45$. All panels correspond to $k_i = 0$, and $\sigma_0 = 8$.