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Physica Scripta

Spectral transitions in the non-Hermitian Hatano–Nelson chain induced by a nonlocal rank-one separable potential

Sara Movahedmanesh, Ali Maghari

Published Aug 27, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ae9f9c

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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.
We study the Hatano-Nelson model, i.e., a one-dimensional non-Hermitian chain of spinless fermions with nearest-neighbor nonreciprocal hopping, in the presence of repulsive nearest-neighbor interactions. At half filling, we find two PT\mathcal{PT}PT transitions, as the interaction strength increases. The first transition is marked by an exceptional point between the first and the second excited state in a finite-size system and is a first-order symmetry-breaking transition into a charge-density wave regime. Persistent currents characteristic of the Hatano-Nelson model abruptly vanish at the transition. The second transition happens at a critical interaction strength that scales with the system size and can thus only be observed in finite-size systems. It is characterized by a collapse of all energy eigenvalues onto the real axis. We further show that in a strong interaction regime, but away from half filling, the many-body spectrum shows point gaps with nontrivial winding numbers, akin to the topological properties of the single-particle spectrum of the Hatano-Nelson chain, which indicates the skin effect of extensive many-body eigenstates under open boundary conditions. Our results can be applied to other models such as the non-Hermitian Su-Schrieffer-Heeger-type model and contribute to an understanding of fermionic many-body systems with non-Hermitian Hamiltonians.
At present, non-Hermitian topological systems continue to be actively studed. In a rigorous approach, we study one of the key non-Hermitian systems — the Hatano–Nelson model HHH. We find the Green function for this Hamiltonian. Using the Green function, we analytically obtain the eigenvalues and eigenfunctions of HHH for finite and semi-infinite chains, as well as for an infinite chain with a local potential. We discuss the non-Hermitian skin effect for the models mentioned above. We also describe the boundary between localized and resonant eigenfunctions (for the zero spectral parameter, this is the boundary between non-Hermitian topological phases).
The single orbital, one-dimensional, Hatano-Nelson Hamiltonian provides deep insight into the physics of non-Hermiticity, resulting from asymmetric left/right hopping, and its connections to localization. In the absence of disorder, its single particle eigenvalues EαE_αEα​ lie on an ellipse in the complex plane whose extent in the imaginary direction is controlled by the degree of asymmetry. When randomness is introduced, two sets of real eigenvalues emerge at the extremes of the largest and smallest real part of EαE_αEα​. These real eigenvalues are associated with localized eigenvectors. For spinless fermions, increasing near-neighbor interactions first cause a transition to a charge density wave phase, and ultimately, on finite lattices, a collapse of all eigenvalues to the real axis. In this paper, we explore the presence of real eigenvalues in the interacting, two-particle sector for the spinful case (Hubbard model) in a two-chain (two-band) geometry with a Hermitian interchain hopping. Our key results are to obtain the ``phase" diagrams for the existence of a purely real spectrum, as a function of the interaction strength, degree of non-Hermiticity, and interchain hopping. We study the sensitivity to boundary conditions of the spectral properties of our two-chain model with winding number analysis and explore the relationship between PBC doublon states and OBC skin modes. To address the question of stability in such non-equilibrium systems, we solve the dynamics at low filling according to Lindbladian evolution and find that the non-Hermitian description is able to qualitatively describe such systems.
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.
The non-Hermitian systems exhibit extreme sensitivity to the boundary conditions. The change in the eigenspectrum with tuning boundary parameter is intimately connected to the non-Hermitian skin effect. The single-particle systems are affected by the boundary perturbations; however, the interplay of a random disorder potential and nonreciprocal hopping under boundary perturbations of an interacting many-body system is not yet clear. In this work, we examine the boundary sensitivity of a non-Hermitian interacting fermionic system in the presence of a random disorder potential. A nonzero boundary parameter results in real-complex spectral transitions with nonreciprocal (or unidirectional) hopping at weak disorder, while the many-body localization at strong disorder washes away real-complex transitions leading to dynamical stability and real eigenvalue spectrum. We show that the boundary-driven real-complex spectral transitions of the non-Hermitian chain are accompanied by the corresponding changes in the level statistics and nearest level-spacing distributions. The intriguing features of nonreciprocity and boundary sensitivity are further revealed using the averaged inverse participation ratios. Finally, we find distinct behavior in the quench dynamics of local particle density, population imbalance, and entanglement entropy of charge-density-wave ordered state that corroborate the real-complex and localization transitions. Our results provide a route to understanding disordered many-body systems under a generalized boundary.
Non-Hermitian systems exhibit unique spectral properties, including the non-Hermitian skin effect and exceptional points, often influenced by boundary conditions. The modulation of these phenomena by generalized boundary conditions remains unexplored and not understood. Here, we analyze the Hatano-Nelson model with generalized boundary conditions induced by complex hopping amplitudes at the boundary. Using similarity transformations, we determine the conditions yielding real energy spectra and skin effect, and identify the emergence of exceptional points where spectra transition from real to complex. We demonstrate that tuning the boundary hopping amplitudes precisely controls the non-Hermitian skin effect, i.e., the localization of eigenmodes at the lattice edges. These findings reveal the sensitivity of spectral and localization properties to boundary conditions, providing a framework for engineering quantum lattice models with tailored spectral and localization features, with potential applications in quantum devices.