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

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.

💡 Research Summary

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The paper investigates the dynamics of initially localized Gaussian wave packets in the discrete Hatano‑Nelson (HN) lattice, a prototypical non‑reciprocal (non‑Hermitian) system where the hopping to the right, (g), differs from the hopping to the left (set to 1). The authors focus on three intertwined phenomena: (i) a time‑dependent acceleration of the wave‑packet centre of mass (COM), (ii) exponential amplification of the packet amplitude, and (iii) a non‑Hermitian “jump” – an abrupt spatial shift of the COM that occurs even without disorder.

Continuum Approximation with Higher‑Order Dispersion
Starting from the lattice Schrödinger‑type equation (-i\dot\Psi_n = g\Psi_{n-1} + \Psi_{n+1}) and a Gaussian initial condition (\Psi_n(0)) characterised by width (\sigma_0), centre (n_0) and momentum (k_i), the authors replace the discrete field by a continuous function (\Psi(x,t)). By expanding the lattice dispersion around the chosen momentum (k_i) they derive a fourth‑order partial differential equation (PDE): \