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arXiv:2604.14533

The Two Orbital, Interacting Hatano-Nelson Model

Jonah Huang, Rubem Mondaini, Nancy Aggarwal, Richard Scalettar

cond-mat.str-elcond-mat.mes-hallcond-mat.quant-gasquant-phSubmitted Apr 16, 2026• Announced Apr 17, 2026• Revised Jun 4, 2026 (v2)
v1 · Apr 16, 2026v2 · Jun 4, 2026
arXivHTMLPDF

Abstract

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 $E_α$ 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 $E_α$. 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.

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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 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).
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.
Abstract Many-body localized phases retain memory of their initial conditions in disordered interacting systems with unitary dynamics. The stability of the localized phase due to the breakdown of unitarity is of relevance to experiment in the presence of dissipation. Here we investigate the impact of non-Hermitian perturbations on many-body localization. We focus on the interacting Hatano-Nelson model which breaks unitarity via asymmetric hopping. We explore the phase diagram for the mid-spectrum eigenstates as a function of the interaction strength and the non-Hermiticity. In contrast to the non-interacting case, our findings are consistent with a two-step approach to the localized regime. We also study the dynamics of the particle imbalance. We show that the distribution of relaxation time scales differs qualitatively between the localized and ergodic phases. Our findings suggest the possibility of an intermediate dynamical regime in disordered open systems.
The static and dynamical properties of a one-dimensional quantum system described by a non-Hermitian Hamiltonian of the so-called Hatano-Nelson type; a tight-binding model with asymmetric (or non-reciprocal) hopping, are studied.The static properties of the model have been much studied; the complex spectrum, skin effect and its topological interpretation, etc. Effects of disorder and interparticle interaction, especially, when they coexist, may be less understood.Here, we will also focus on its dynamical properties and reveal some unique features in the wave-packet and entanglement (entropy) dynamics.For that some latest (original) results based on improved numerics (with this a system of larger system size L becomes accessible) are shown.
Using grand canonical quantum Monte Carlo (QMC) simulations combined with maximum entropy analytic continuation, as well as analytical methods, we examine the one- and two-particle dynamical properties of the Hubbard model on two coupled chains at half-filling. The one-particle spectral weight function, A(k,\ensuremath{\omega}), undergoes a qualitative change with interchain hopping ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ associated with a transition from a four-band insulator to a two-band insulator. A simple analytical model based on the propagation of exact rung singlet states gives a good description of the features at large ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$. For smaller ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$, A(k,\ensuremath{\omega}) is similar to that of the one-dimensional model, with a coherent band of width the effective antiferromagnetic exchange J which is reasonably well described by renormalized spin-wave theory. The coherent band rides on a broad background of width several times the parallel hopping integral t, an incoherent structure similar to that found in calculations on both the one- and two-dimensional models. We also present QMC results for the two-particle spin and charge excitation spectra, and relate their behavior to the rung singlet picture for large ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ and to the results of spin-wave theory for small ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$. \textcopyright{} 1996 The American Physical Society.