We investigate the critical non-Hermitian skin effect (CNHSE) and the coupling-driven morphing of scale-free localization (SFL) in two Hatano-Nelson chains coupled locally at two bulk sites separated by an inclusive distance d. At weak interchain coupling, competition between the opposite skin accumulations of the two chains generates a geometrically selected critical branch with a complex loop-like spectrum and scale-free skin states localized near the physical boundaries. Remarkably, this SFL branch persists as the coupling becomes strong, although its underlying mechanism and spatial profile change qualitatively. Strong local hybridization generates high-energy impurity states and effectively divides the low-energy Hilbert space into geometry-selected inner and outer segments, with the former supporting SFL states near the interchain-coupled sites and the latter retaining conventional skin states. The number and localization scale of the SFL states are controlled by the separation between the coupling sites. Consequently, increasing the interchain coupling continuously relocates the SFL states from the physical boundaries to the coupling-defined internal interfaces. Our results demonstrate that bilocal coupling can induce a CNHSE and further drive its scale-free states into a segmentation-induced localization regime.
We investigate the critical non-Hermitian skin effect (CNHSE) and the coupling-driven morphing of scale-free localization (SFL) in two Hatano-Nelson chains coupled locally at two bulk sites separated by an inclusive distance d. At weak interchain coupling, competition between the opposite skin accumulations of the two chains generates a geometrically selected critical branch with a complex loop-like spectrum and scale-free skin states localized near the physical boundaries. Remarkably, this SFL branch persists as the coupling becomes strong, although its underlying mechanism and spatial profile change qualitatively. Strong local hybridization generates high-energy impurity states and effectively divides the low-energy Hilbert space into geometry-selected inner and outer segments, with the former supporting SFL states near the interchain-coupled sites and the latter retaining conventional skin states. The number and localization scale of the SFL states are controlled by the separation between the coupling sites. Consequently, increasing the interchain coupling continuously relocates the SFL states from the physical boundaries to the coupling-defined internal interfaces. Our results demonstrate that bilocal coupling can induce a CNHSE and further drive its scale-free states into a segmentation-induced localization regime.
Abstract A critical non-Hermitian skin effect (CNHSE) was recently discovered in which the eigenenergy spectrum of a coupled system comprising two 1D non-Hermitian (NH) chains exhibits an abrupt change when the length of the system increases beyond a critical length. Here, we overturn the conventional wisdom that multiple chains are required for the CNHSE by showing that a similar abrupt transition in the eigenenergy spectrum occurs with increasing length in a single finite 1D chain when the two ends of the chain are connected by a weak terminal coupling to form a closed loop. We illustrate this phenomenon using the classic Hatano-Nelson and Su-Schrieffer-Heeger NH systems, and show that it can co-exist with the conventional CNHSE in coupled NH loops.
Spectral transitions in non-Hermitian lattices often arise from the competition between non-reciprocal skin accumulation and inter-component hybridization. In short-range systems formed by two coupled chains, this competition conventionally leads to the logarithmic critical-length law Nc∼lnD, where D is the transverse separation between the chains. Here we show that long-range hoppings fundamentally reorganizes this critical behavior, producing a hierarchy of distinct scaling laws. When only the hybridization couplings are power-law decaying with exponent α, the onset becomes algebraic, Nc∼Dα/3. When the hoppings within each chain are themselves also power-law decaying, in addition to the hybridization couplings, the system enters a scale-covariant regime for α<2, in which the criticality threshold equation depends only on the system aspect ratio Nc/D. At α=2 and beyond, this regime is followed by a marginal logarithmically corrected and algebraically corrected regimes, respectively. We identify two new non-local mechanisms that enable this unconventional critical hierarchy: a nonanalytic band-edge dispersion from long-range intra-chain hoppings, and parity-mixing hybridization induced by non-reciprocity. Our results show that nonlocality systematically removes the physical length scales i.e. skin depth underlying conventional critical non-Hermitian skin behavior, offering a platform-independent framework testable in programmable topoelectrical circuits, photonic lattices and digital quantum simulators.
In non-Hermitian open boundary systems, non-reciprocal hopping inevitably induces the non-Hermitian skin effect (NHSE). In this paper, we propose a purely geometric scheme to reconstruct this boundary accumulation pattern: by introducing a fixed step length k (where k < L), the system precisely decomposes into k independent sub-chains. Quantitative analysis shows that this geometric blockade strictly cuts off single-particle transport between sub-chains, and each sub-chain independently inherits the Hatano-Nelson type skin effect. Through boundary probability density and spatial distribution heatmaps, we demonstrate at the eigenstate ensemble level that the probability distribution is geometrically partitioned to the right ends of the k sub-chains. In the original coordinates, the probability peak is no longer dominated by a single lattice site but exhibits multi-site segmented accumulation. A single eigenstate only localizes in one of the sub-chains and does not distribute across the blockade. This isolation is strictly guaranteed by the block-diagonal structure of the Hamiltonian, independent of disorder or weak-coupling approximations. This mechanism provides a deterministic criterion for controlling geometric localization in non-Hermitian quantum networks.
Abstract We investigate impurity-induced spectral transitions in a non-Hermitian Hatano–Nelson chain with a non-local rank-one separable potential, providing a minimal yet analytically tractable framework for exploring the interplay between asymmetric hopping and directional impurities. By employing the Green’s function formalism in conjunction with the Sherman–Morrison identity, we derive a closed-form spectral equation for the impurity-induced eigenvalues and analyze their evolution in the complex energy plane. Under periodic boundary conditions, we demonstrate that the system undergoes a real-to-complex spectral transition, which is consistently associated with the contact between the impurity state and the bulk spectral boundary of the Hatano–Nelson model. In the weakly non-Hermitian regime, a real isolated bound state remains stable. However, as the asymmetric hopping strength increases, the impurity state hybridizes with the bulk spectrum, leading to the emergence of complex eigenvalues. In the strongly non-Hermitian regime, the imaginary parts of the impurity spectrum decrease again, revealing a tendency toward suppression of the imaginary component within the present directional separable impurity model. Furthermore, spectral phase diagrams in parameter space reveal the competition among the non-Hermiticity strength, impurity coupling, and impurity localization range. These results provide a simple and analytically transparent framework for understanding impurity-driven spectral transitions and directional effects in non-Hermitian systems.
Non-Hermitian skin effect is one of the most captivating phenomena in non-Hermitian systems, characterized by the extensive localization of eigenstates near open boundaries. In its conventional form, the localization length of a skin mode is independent of the system size. Remarkably, however, when the open boundaries are coupled to realize a generalized boundary condition, the localization length can undergo a drastic transformation, becoming proportional to the system length. This intriguing regime is known as the scale-free skin effect (SFSE). Although SFSE has been observed in numerous one-dimensional non-Hermitian models through case-by-case studies, a unified theoret ical framework capable of predicting its emergence and characteristic properties remains absent. In this work, we take a firm step forward by establishing a model-independent framework for the ana lytic determination of scale-free localization lengths in certain regimes. Our key insight is to treat generalized boundary conditions as a perturbation of the periodic boundary conditions, thereby circumventing the singular response-to-perturbation that would otherwise arise if they were pertur batively treated relative to open boundary conditions. Our work sheds new light on understanding SFSE in non-Hermitian systems.
Non-Hermitian dynamics is ubiquitous in various physical systems. While recent study shows that such a dynamics leads to an area-law scaling of the entanglement entropy due to the non-Hermitian skin effects, it remains unclear how disorder changes the behavior of the entanglement entropy in a non-Hermitian system with skin effects. Here we study the dynamics of a many-body state of free fermions in the paradigmatic Hatano-Nelson model with open boundaries, and find that the area-law behavior of the entanglement entropy in the pristine Hatano-Nelson model develops into a logarithmic scaling for small disorder strength. As we further increase the disorder strength, the system reenters an area-law regime through an entanglement phase transition. At the critical point, the entanglement entropy exhibits a universal algebraic scaling. We further demonstrate the absence of a conformal invariance in the log-law regime by examining the subsystem entanglement entropy, the connected correlation function and the mutual information. Finally, we show the existence of disorder induced entanglement phase transitions in the Hatano-Nelson model with periodic boundaries.