Finite-size Effects on The Edge Loss Probability in Non-Hermitian Quantum Walks

TL;DR

This work examines how finite-size boundary effects shape the edge loss probability in non-Hermitian quantum walks with on-site dissipation. By combining Green's-function techniques, GBZ analysis, and Lyapunov-exponent considerations, it shows that boundary scattering can either quench or enable non-Hermitian edge bursts, depending on the strength of the NHSE and the dissipation regime. The study reveals that even when imaginary-gap conditions predict edge bursts in the infinite limit, finite chains exhibit rich behavior where edge bursts depend sensitively on boundary scattering and can reemerge under extreme dissipation. These insights advance understanding of dynamical bulk–edge relations in realistic, finite systems and have implications for experimental implementations of lossy non-Hermitian lattices.

Abstract

A dynamical bulk-edge relation in quantum walks has been theoretically proposed and experimentally observed, in which a power-law dependence of the bulk loss probability is associated with a pronounced peak of loss probability at the edge. This behavior has been proven to arise from imaginary gap closing and the non-Hermitian skin effect in the infinite limit without boundary effects. However, in a finite-size chain, we find that boundary scattering can suppress this edge burst. Meanwhile, imaginary gap opening together with the non-Hermitian skin effect, can also induce a large loss probability at the edge. Our results provide insights into finite-size quantum dynamics.

Figures (5)

• Figure 1: (a) Coupled two chains with $t_1,t_2$ being the coupling parameters and $i\gamma$ representing the on-site dissipation on chain B. The variation of edge loss probability with $\gamma$ under the condition of imaginary gap closing (b) and opening (c) with parameters $t_2 = 0.5$, size $L=100$ and excitation location $x_0=90$. Here, the gap closing and opening correspond to whether the PBC spectrum at $\gamma=0.5$ touch the real axis or not, as shown in the insets.
• Figure 2: (a) Comparison between the analytical results obtained from Eq. \ref{['eq-edge']} (dashed lines) and the numerical results of $P_1$ for different $t_1$. (b) Radius of GBZ $r_G$ as the function of $\gamma$. Under weak scattering (c) and strong scattering (d), the distribution of loss probability. The evolution of wavepacket temporally and spatially is shown in the insets. Common parameters: $t_2=0.5,L=100,x_0=90$.
• Figure 3: Lyapunov exponent for (a) negative velocity for different $t_1$ and fixed $\gamma=1$ before the wavepacket reaches the boundary and (c) positive velocity for different $\gamma$ and fixed $t_1=0.3$ after it arrives at the boundary. The extreme point under $t_1=0.2$ in (a) corresponds to $\omega_0$ on the PBC spectrum (red lines) in (b). (d) The variation of $\left| \frac{d \lambda (v)}{d v }|_{v=0} \right|$ with $\gamma$ for different $t_1$.
• Figure 4: (a) Under the condition of imaginary gap opening, comparison of edge loss probability obtained from numerical results (dots), and analytic results (dashed lines) and estimated based on an expansion around $\omega=0$ (squares). Variation of (b) $\left| \frac{d \lambda (v)}{d v }|_{v=0} \right|$ and (c) $\text{min}[|\beta_L(w)|]$ with $\gamma$. (d) The appearance of edge burst under a large $\gamma=10$ under strong scattering. Common parameters: $t_2=0.5,L=100,x_0=90$.
• Figure 5: Scale of bulk loss probability with respect to relative distance at (a, d) $\gamma=0.02$, (b, e) $\gamma=1.4$ and (c, f) $\gamma=30$, respectively. Double logarithmic coordinate is used in (a, b, c) and single logarithmic coordinate in (d, e, f). Other parameters: $t_1=0.7,t_2=0.5,L=100,x_0=90.$