Quasiperiodicity-induced non-Hermitian skin effect from the breakdown of scale-free localization

TL;DR

The work investigates how quasiperiodicity interacts with non-Hermitian boundary sensitivity in a non-reciprocal lattice with a tunable boundary impurity. By interpolating between open and periodic boundaries via a generalized boundary condition parameter $\mu$ and introducing a quasiperiodic onsite potential $\lambda$, the authors map regimes using the condition number, a non-normality ratio $\kappa_{\mathrm{R}}$, and entanglement entropy. They find a quasiperiodicity-induced breakdown of scale-free localization, producing an NHSE regime embedded between SFL and localized phases, with the NHSE region collapsing to the bulk transition $\lambda_c=2J e^{\alpha}$ in the thermodynamic limit; near PBC, a quasiperiodicity-assisted delocalization to an extended regime is also observed. Additionally, quasiperiodicity can effectively disconnect the impurity bond, decoupling boundary links from bulk hoppings and driving a breakdown of SFL before localization. Together, these results illuminate the intricate interplay of quasiperiodicity, non-Hermiticity, and boundary conditions, and point to controllable realizations in circuit-like platforms.

Abstract

Non-reciprocal systems exhibit extreme sensitivity to boundary conditions, typically manifesting as the non-Hermitian skin effect (NHSE) under open boundaries. By bridging the boundaries with a tunable impurity bond, one can access intermediate regimes where scale-free localization (SFL) can emerge. Here, we investigate the competition between such boundary coupling and quasiperiodic disorder in a non-reciprocal lattice. Our analyses reveal a quasiperiodicity-induced breakdown of the SFL regime, which evolves into either the NHSE or an extended regime, depending on boundary conditions. These results uncover the crucial role of quasiperiodicity in non-Hermitian systems.

Figures (8)

• Figure 1: Schematic illustration of the system, with non-reciprocal hopping strength $J e^{ \pm \alpha}$, quasiperiodic onsite potential $\lambda_{j}$, and an impurity bond with hopping strength tunable by $\mu$.
• Figure 2: (a-1) Non-normality ratio-based regime diagram on the $(\ln\mu,\lambda)$ plane for $L = 89$ and $\phi = 0$. The black-gray dashed line represents the NHSE-SFL boundary. Specific values on the $\ln \mu / (\alpha L)$ axis are indicated by an open star ($\approx -0.56$) and a solid star ($\approx -0.01$). The corresponding parameters for $L = 89$ in (b-1) and (c-1) are indicated by the same symbols. (a-2) $\mu$-dependence of $\log_{10}\kappa_{\mathrm{R}}$ corresponding to cuts in (a-1). (b-1, c-1) Regime diagrams on the $(\lambda,L)$ plane for (b-1) $\ln \mu = -25$ and (c-1) $\ln \mu = -0.5$. The dashed curves correspond to the boundary shown in (a-1). (b-2, c-2) Scaling behavior of $\kappa_{\mathrm{R}}$ for the parameters used in (b-1) and (c-1), respectively.
• Figure 3: Characteristics of various regimes for $\phi = 0$ and representative $\lambda$ and $\mu$ values. (a,b) Energy spectra for $L = 89$. The closed (open) symbols represent the spectra for $\ln \mu = {-25}$ ($\mu = 1$). (c) Spatial distributions of the density $|\psi(x)|^2$ for $L = 89$ and $\ln \mu = {-25}$. (d) Localization lengths, $\xi_{0}$ (for $\lambda = 0$) and $\xi_{1}$ (for $\lambda = 2.7 J$), for $\ln \mu = {-25}$.
• Figure 4: Spatial distributions of the density $|\psi(x)|^2$ for $L = 89$, $\ln \mu = {-0.5}$ and $\phi = 0$ (a) for SFL and extended states and (b) for extended and localized states.
• Figure S1: (a) Regime diagram of the model \ref{['Seq:imag. gauge transformed H']} for $\lambda = 0$ in the $(\ln \mu, L)$ plane. (b) Spatial distributions of the density in various regimes for $\alpha = 0.5$ and $L = 50$.