Size-Dependent Skin Effect Transitions in Weakly Coupled Non-Reciprocal Chains

TL;DR

This work analyzes the size-dependent non-Hermitian skin effect in a ladder of two weakly coupled nonreciprocal chains, revealing how system size and interchain coupling steer spectral topology. By combining analytical non-Bloch band theory with numerical OBC/PBC spectra and winding-number diagnostics, it identifies a finite-size concurrent bipolar skin effect (CBSE) in the $W=0$ region that destabilizes as $N$ grows, yielding unipolar or conventional bipolar NHSE in the thermodynamic limit. The study characterizes three distinct regimes based on $\delta_b$ and $\delta_a$: nested, tangent, and intersecting spectral loops, each with unique CBSE persistence and transitions, and demonstrates that the size-dependent behavior survives beyond weak coupling. These insights advance understanding of NHSE mechanisms and provide practical guidelines for designing non-Hermitian topological phases in synthetic lattices and electrical-circuit implementations, including a concrete proposal using INIC-based circuits to realize the model.

Abstract

Non-Hermitian systems exhibit unique boundary phenomena absent in their Hermitian counterparts, most notably the non-Hermitian skin effect (NHSE). In this work, we explore a lattice model consisting of two coupled non-reciprocal chains, focusing on the interplay between system size, inter-chain coupling, and spectral topology. Using both analytical and numerical approaches, we systematically examine the evolution of the complex energy spectra and spectral winding numbers under periodic and open boundary conditions. Our results uncover a variety of size-dependent localization transitions, including the emergence and instability of concurrent bipolar skin effects in the $W=0$ region, and their crossover to unipolar and conventional bipolar NHSE as the system size increases. Notably, we demonstrate that these size-dependent behaviors persist even beyond the weak-coupling regime, highlighting their universality in non-Hermitian systems with complex spectral structures. This study provides insights into the mechanisms governing skin effects and offers practical guidelines for engineering non-Hermitian topological phases in synthetic lattices.

Figures (13)

• Figure 1: (a) Schematic illustration of two weakly coupled nonreciprocal chains, labeled $A$ and $B$, with couple amplitude $M$. The right- and left-directed hopping amplitudes of the $A$ chain are $J_1$ and $J_2$, respectively, while $J_3$ and $J_4$ denote the right- and left-directed hopping amplitudes of the $B$ chain. Energy spectra of the two decoupled chains under periodic boundary conditions (PBCs) for (b) $\delta_b=0.8$, (c) $\delta_b=0.5$, and (d) $\delta_b=0.4$. Here, $J_1=0.5$, $J_3=2$, $\delta_a=0.5$, and $M=0$.
• Figure 2: (a) Energy spectra for $\delta_{b}>\delta_{a}$ under PBCs and OBCs. Black curves represent the spectrum under PBCs, red crosses denote the OBC spectrum for $N = 30$, red dots correspond to $N = 150$, and purple dots indicate the thermodynamic-limit spectrum obtained from the non-Bloch band theory. The yellow (green) shading marks the $W = -1$ ($W = 0$) region. The spatial profiles $|\psi_{j,A}^{(E)}|$ and $|\psi_{j,B}^{(E)}|$ of all wave functions with eigenvalue $E$ from the complex energy loop under OBCs for (b) $N = 30$ and (c) $N = 150$. The quantities $I_S^{(E)}$ and $I_P^{(E)}$ for different eigenvalues for (d) $N = 30$ and (e) $N = 150$. Here, $J_1 = 0.5$, $J_3 = 2$, $\delta_a = 0.5$, $\delta_b = 0.8$, and $M = 0.01$.
• Figure 3: (a) $I_S^{(E)}$ and (b) $I_P^{(E)}$ as functions of $\mathrm{Re}(E)$ and system size $N$. (c) The values of $W_{\mathrm{OBC}}^{(E)}$ for OBC eigenstates and different system size $N$. Yellow dots mark the eigenenergies with $W_{\mathrm{OBC}}^{(E)}=0$, and red dots indicate the eigenenergies with $W_{\mathrm{OBC}}^{(E)}=-1$. (d) The number $N_W$ of different values of the winding number $W_{\mathrm{OBC}}^{(E)}$ as a function of system size $N$. The dashed lines in both (c) and (d) correspond to $N = 86$. Parameters are $J_1 = 0.5$, $J_3 = 2$, $\delta_a = 0.5$, $\delta_b = 0.8$, and $M = 0.01$.
• Figure 4: (a) Energy spectra for $\delta_{b}=\delta_{a}$ under PBCs and OBCs. Black curves represent the spectrum under PBCs, red crosses denote the OBC spectrum for $N = 30$, red dots correspond to $N = 150$, and purple dots indicate the thermodynamic-limit spectrum obtained from the non-Bloch band theory. The yellow (green) shading marks the $W = -1$ ($W = 0$) region. (b) The spatial profiles $|\psi_{j,A}^{(E)}|$ and $|\psi_{j,B}^{(E)}|$ of all wave functions with eigenvalue $E$ from the complex energy loop under OBCs for $N = 30$. (c) and (d) The spatial profiles $|\psi_{j,A}^{(E)}|$ and $|\psi_{j,B}^{(E)}|$ of all wave functions under OBCs for $N = 150$ with eigenvalue $E$ resided in the $W=-1$ and $W=0$ regions, respectively. The quantities $I_S^{(E)}$ and $I_P^{(E)}$ for different eigenvalues for (e) $N = 30$ and (f) $N = 150$. Here, $J_1 = 0.5$, $J_3 = 2$, $\delta_a = 0.5$, $\delta_b = 0.5$, and $M = 0.01$.
• Figure 5: (a) $I_S^{(E)}$ and (b) $I_P^{(E)}$ as functions of $\mathrm{Re}(E)$ and system size $N$. (c) The values of $W_{\mathrm{OBC}}^{(E)}$ for OBC eigenstates and different system size $N$. Yellow dots mark the eigenenergies with $W_{\mathrm{OBC}}^{(E)}=0$, and red dots indicates the eigenenergies with $W_{\mathrm{OBC}}^{(E)}=-1$. (d) The fitting of the transitions, where $W_{\mathrm{OBC}}^{(E)}$ changes from zero to nonzero, as a function of $1/N$. Here, $J_1 = 0.5$, $J_3 = 2$, $\delta_a = 0.5$, $\delta_b = 0.5$, and $M = 0.01$.