Wave-packet dynamics in pseudo-Hermitian lattices: Coexistence of Hermitian and non-Hermitian wavefronts

TL;DR

The paper reveals that in pseudo-Hermitian lattices, notably the Hatano–Nelson model with open boundaries, wave-packet dynamics feature coexisting Hermitian and non-Hermitian fronts. By employing a local similarity transformation, the non-Hermitian evolution is linked to Hermitian propagators, exposing dual dynamics and explaining boundary reflections, emergent packets, and disorder-driven transitions. It provides analytic expressions and saddle-point analyses for both single-site and Gaussian initial conditions, and identifies critical widths governing transitions. The findings extend to other pseudo-Hermitian systems (e.g., non-Hermitian SSH), offering a framework to understand topology and experimental consequences in non-Hermitian physics.

Abstract

This paper investigates wave-packet dynamics in non-Hermitian lattice systems and reveals a surprising phenomenon: The simultaneous propagation of two distinct wavefronts, one traveling at the non-Hermitian velocity and the other at the Hermitian velocity. We show that this dual-front behavior arises naturally in systems governed by a pseudo-Hermitian Hamiltonian. Using the paradigmatic Hatano-Nelson model as our primary example, we demonstrate that this coexistence is essential for understanding a wide array of unconventional dynamical effects, including abrupt ``non-Hermitian reflections'', sudden shifts of Gaussian wave-packets, and disorder-induced emergent packets seeded by the small initial tails. We present analytic predictions that closely match numerical simulations. These results may offer new insight into the topology of non-Hermitian systems and point toward measurable experimental consequences.

Figures (16)

• Figure 1: Snapshot of the non-Hermitian dynamics with and without the transformation (the arrows indicate the direction of propagation of the wave-packets). See Video_1 in the Supplementary Material for an animation of the dynamics. Blue dataset: A wave-packet with initial conditions $\left\langle n\right.\left|\psi(t=0)\right\rangle =\delta_{n,x_{0}}$, where $x_{0}=N/2$, evolved numerically to time $t=60$ with the non-Hermitian Hamiltonian, with $t_{l}=2,t_{r}=0.2$ (OBC). For convenience, the wave-packet is normalized to 1 (otherwise, it would present an exponential growth). Orange dataset: the same data after the transformation $y_{n}\rightarrow y_{n}r^{-n}$ (and normalization). The green/red dotted lines are the predicted positions of the fronts of the Hermitian and non-Hermitian wave-packets: $x_{0}-v_{\mathrm{nh}}t$ (green) and $x_{0}-v_{h}t$ (red). The number of sites is $N=200$, and the lattice spacing is $a=1$.
• Figure 2: Numerical results of the amplitude at the leftmost site as a function of time in the same setup as Fig. \ref{['fig:delta_dynamics']}. Recalling that $x_{0}=N/2$, three timestamps are marked by $t_{1}=\frac{x_{0}}{v_{\mathrm{nh}}}\approx45$ (green), indicating the time when the non-Hermitian wave-packet hits the edge, $t_{2}=\frac{x_{0}}{v_{\mathrm{h}}}\approx79$ (red), when the Hermitian wave-packet hits the edge, and $t_{3}=\frac{2N-x_{0}}{v_{\mathrm{nh}}}\approx237$ (purple), when the Hermitian wave-packet (that went to the other side) hits the edge after being reflected from the other end.
• Figure 3: Numerical results comparing the non-Hermitian dynamics of a Gaussian wave-packet on the lattice versus the continuum limit. While the initial acceleration is similar in both cases, only the lattice dynamics eventually saturates to a maximum velocity. See Video_2 in the Supplementary Material for an animation of the dynamics. The initial conditions are $n_{0}=300,$$k_{0}=0$, $\sigma=3$, the lattice spacing is $a=1$, and the parameter of the Hamiltonian are $t_{l}=2,$$t_{r}=1.5$. For the ease of presentation, the wave-function is normalized such that $\max\left(|\psi_{n}|^{2}\right)=1$ for any $t$. The black and red dotted-lines represent the approximation and the continuum limit given by Eq. (\ref{['eq:max_position_on_lattice']}), (\ref{['eq:max_position_continuum']}) respectively.
• Figure 4: Numerical results of a Gaussian wave-packet, with initial parameters identical to those of Fig. \ref{['fig:gaussian_dynamics']}, except for $k_{0}$ which is now $\pi/4$ instead of $0$. of propagation of the wave-packets). See Video_3 in the Supplementary Material for an animation of the dynamics. The behavior of the wave-packets in the continuum and on the lattice is similar until the time when the Hermitian wavepacket (marked in cyan) hits the wall: $t_{\mathrm{hit}}=\frac{100}{v_{0}}\approx37$ (marked in green), where $v_{0}=2\sqrt{t_{r}t_{l}}k_{0}$. The approximation (marked in black) was calculated using the image method, and is in agreement with the numerical results.
• Figure 5: A Hermitian Gaussian wave-packet (in blue) and its image (in red) moving towards a wall at $n=400$, on the lattice (left column) and in the continuum (right column). The rows represent the timestamps: $t_{\mathrm{hit}}-10$, $t_{\mathrm{hit}}$, $t_{\mathrm{hit}}+10$ (top, middle, bottom, respectively), where $t_{\mathrm{hit}}$ is the time where the packet hits the wall ($t_{\mathrm{hit}}=\frac{d}{2\sqrt{t_{r}t_{l}}\sin(k_{0})}$ on the lattice and $t_{\mathrm{hit}}=\frac{d}{2\sqrt{t_{r}t_{l}}k_{0}}$ in continuum, where $d=100$). The dashed lines present the results of the transformation (\ref{['eq:Transformation']}), normalized to 1, which gives the non-Hermitian wave-packets that is derived from the incident and image Hermitian counterpart, respectively. See Video_4 in the Supplementary Material for an animation in the case of the lattice. The setup is the same as in Fig. \ref{['fig:gaussian_dynamics_reflection_hermitian']} (note that $a=1$ is used, so that $n$ on the lattice is equivalent to $x$ in continuum).