A two-scale effective model for defect-induced localization transitions in non-Hermitian systems

Abstract

We illuminate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localization in the bulk. We study a Hamiltonian with non-reciprocal couplings that exhibits the skin effect (the localization of all eigenvectors at one edge) and add an on-site defect in the center. Using a two-scale asymptotic method, we characterize the long-scale growth and decay of the eigenvectors and derive a simple and intuitive effective model for the transition that occurs when the defect is sufficiently large that one of the modes is localized at the defect site, rather than at the edge of the system.

Figures (3)

• Figure 1: Hamiltonians with non-reciprocal couplings exhibit the non-Hermitian skin effect whereby eigenvalues falling within a specified region $\mathbb{W}$ are all localized at one edge of the system, as shown in (c), (d) and (e). All the eigenvalues for this 29-site system are shown in the complex plane in (a), along with the region $\mathbb{W}$. When a defect $d$ is added to the potential at one of the sites, this can cause an eigenvalue to leave $\mathbb{W}$ and be localized at the defect site, as shown in (b). We propose a two-scale effective model for this transition.
• Figure 2: The maximal or minimal eigenvectors of a 29-site system with a defect in the center. (a)-(e) show the eigenvector with the largest eigenvalue in a system with a positive defect $d=2\sinh\gamma+\alpha\epsilon$. The effective amplitude predicted by our two-scale model \ref{['eq:effective']} is shown in a dotted line. Inset shows the eigenvalues in the complex plane with $\mathbb{W}$ shaded and the eigenvalue of the plotted eigenvector circled in red. (f)-(j) show the analogous eigenpairs for a negative defect $d=-(2\sinh\gamma+\alpha\epsilon)$. $\gamma=0.4$ and $v=1$ are used throughout and the values of $\alpha\in\{-1,1\}$ and $\epsilon>0$ are shown above each plot.
• Figure 3: The maximal or minimal eigenvalues for a system with either (a) a positive defect or (b) a negative defect, respectively. Numerical values for a 29-site system with a single defect $d=\pm(2\sinh\gamma+\alpha\epsilon)$ at the center are compared to the asymptotic formula \ref{['eq:Eval']} for $\gamma=0.4$ and $v=1$.