Hatano-Nelson Chain: A Non-Hermitian Lattice Model

• The Hatano-Nelson chain is a one-dimensional non-Hermitian lattice model with asymmetric nearest-neighbor hopping that produces complex spectra and point-gap topology.
• It exhibits extreme boundary sensitivity where eigenstates localize at edges under open boundaries, manifesting the non-Hermitian skin effect with winding numbers.
• Generalizations include disorder-induced spectral transitions, nonlinear effects, many-body interactions, and experimental implementations in photonics and electric circuits.

The Hatano-Nelson chain is a paradigmatic one-dimensional non-Hermitian lattice model with asymmetric nearest-neighbor hopping. In its standard form, nonreciprocity is introduced either through unequal left- and right-hopping amplitudes or, equivalently, through an imaginary vector potential, and this produces complex spectra, point-gap topology, exceptional points, and the non-Hermitian skin effect. Across recent work, the chain has also become a reference setting for disorder-induced spectral transitions, interaction-driven many-body phenomena, nonlinear skin modes, frequency-dependent non-Markovian transport, and experimental realizations in photonics and electric circuits (Zhao et al., 14 Feb 2025, Zhang et al., 2022, Chen et al., 25 Mar 2026).

1. Canonical formulation

A common interacting form of the Hatano-Nelson Hamiltonian is

$$\hat{H}=\sum_{\ell}[(t+\gamma)\hat{c}_{\ell}^{\dagger}\hat{c}_{\ell+1}+(t-\gamma)\hat{c}_{\ell+1}^{\dagger}\hat{c}_{\ell}+U\hat{n}_{\ell}\hat{n}_{\ell+1}],$$

where $t$ is reciprocal hopping, $\gamma$ is the nonreciprocal hopping parameter, and $U>0$ is a repulsive nearest-neighbor interaction for spinless fermions (Zhang et al., 2022). An equivalent open-chain parameterization used in many-body studies is

$$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$

with $h$ the imaginary vector potential (Dóra et al., 2022).

In the non-interacting nearest-neighbor chain, boundary conditions immediately alter the spectrum. For the formulation with $t_r=1$ and $t_l=\alpha^2<1$, periodic boundary conditions give

$\sigma(H_0)= e^{ik} + \alpha^2 e^{-ik},$

which is an ellipse in the complex plane, while open boundary conditions give

$\sigma(H_0)= 2\alpha \cos k,$

a real line segment between $t$0 and $t$1 (Zhao et al., 14 Feb 2025). In the clean unidirectional limit, the periodic spectrum is a circle,

$t$2

and the eigenstates are extended (Ghosh et al., 12 Jan 2026).

These elementary spectral facts organize most subsequent generalizations. The Hatano-Nelson chain is therefore used as a prototypical one-dimensional non-Hermitian model exhibiting the non-Hermitian skin effect and complex energy-band topology, and as the fundamental non-Hermitian building block of the Su-Schrieffer-Heeger model (Chen et al., 25 Mar 2026, Ghosh et al., 12 Jan 2026).

2. Boundary sensitivity, point-gap topology, and skin localization

The defining feature of the Hatano-Nelson chain is extreme sensitivity to boundary conditions. Under open boundary conditions, bulk eigenstates in the non-interacting limit become exponentially localized at a boundary, producing the non-Hermitian skin effect; under periodic boundary conditions, the same system instead exhibits complex spectral loops (Dóra et al., 2022). In generalized settings, this boundary sensitivity is encoded by spectral winding numbers such as

$t$3

or equivalent momentum-space expressions used to diagnose point-gap topology (Ghosh et al., 12 Jan 2026).

In the unidirectional chain with binary diagonal disorder, the complex spectrum forms a single loop for weak disorder, bifurcates at a critical threshold, and becomes two separated loops in the strong-disorder regime. Correspondingly, the winding number changes from $t$4 in the weak-disorder phase, through $t$5 at criticality, to $t$6 in the strong-disorder limit (Ghosh et al., 12 Jan 2026). The same analysis shows that most states are exponentially localized, but at weak and critical disorder two special states remain completely delocalized, with diverging localization length. The divergence is directly correlated with non-trivial spectral winding (Ghosh et al., 12 Jan 2026).

Strictly open boundaries can eliminate the loop structure entirely in this disordered unidirectional setting. For $t$7, the spectrum collapses into two flat bands at $t$8, the winding number loses meaning, and the delocalized states disappear; for any $t$9, the thermodynamic spectrum and localization length recover the periodic-boundary result (Ghosh et al., 12 Jan 2026). This establishes that spectral topology in the Hatano-Nelson chain is not merely a bulk property in the Hermitian sense, but is inseparable from the boundary condition used to define the spectrum.

A frequent oversimplification is that the skin effect must always be unidirectional. In extensions with non-reciprocal next-nearest-neighbor hopping and structured onsite potentials, the open-boundary eigenstates can instead display a bidirectional non-Hermitian skin effect, localizing on both edges, and with a periodic onsite potential the direction of skin accumulation can reverse completely as the potential strength is varied (Padhi et al., 2024).

3. Disorder, quasiperiodicity, impurities, and structured potentials

The Hatano-Nelson chain supports several distinct routes to spectral reconstruction. For strictly ergodic potentials, the spectrum admits an explicit Lyapunov-exponent characterization,

$\gamma$0

with $\gamma$1 and $\gamma$2 (Wang et al., 2023). This yields a sharp real-complex transition: for $\gamma$3, the spectrum remains real; for $\gamma$4, it contains both real and complex parts; and for $\gamma$5, it is purely complex (Wang et al., 2023). If the Lyapunov exponent is continuous, the infinite-chain spectrum on $\gamma$6 is approximated by finite-interval truncations with periodic boundary conditions, a behavior explicitly contrasted with the random-potential case (Wang et al., 2023).

Adding a periodic modulation to the onsite energies introduces a second insulating mechanism besides disorder. In the period-2 model,

$\gamma$7

the periodic term opens a gap in the clean limit when $\gamma$8, with

$\gamma$9

while disorder promotes localization and can even close a previously opened gap before full localization sets in (Hébert et al., 2010). The resulting spectral topologies include $U>0$0, $U>0$1, $U>0$2, $U>0$3, $U>0$4, $U>0$5, and $U>0$6 (Hébert et al., 2010).

A single long-range impurity coupling is not a perturbatively small deformation in this model. For

$U>0$7

the thermodynamic spectrum becomes

$U>0$8

and the additional solutions generate tentacle-like spectral wings emerging from the Bloch ellipse or the open-boundary non-Bloch segment (Zhao et al., 14 Feb 2025). The number of impurity-induced bound states is $U>0$9, and their localization length scales with the impurity-coupling distance rather than with system size (Zhao et al., 14 Feb 2025). This makes long-range impurity coupling a nonperturbative control parameter for both spectrum and wavefunction structure.

4. Interactions and many-body Hatano-Nelson physics

With repulsive nearest-neighbor interactions at half filling, the interacting Hatano-Nelson chain displays two $$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$0 transitions as the interaction strength increases. The first transition occurs at an exceptional point between the first and second excited states in finite size and coincides with a first-order symmetry-breaking transition from a gapless phase into a gapped charge-density-wave regime; the persistent current abruptly vanishes at this point. A second transition at stronger interaction collapses all many-body eigenvalues onto the real axis, but its critical scale grows with system size and diverges in the thermodynamic limit, making it a finite-size effect (Zhang et al., 2022).

Away from half filling and at strong interaction, the many-body spectrum stratifies into clusters centered at interaction energies such as $$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$1. Under twisted boundary conditions, each cluster can exhibit a point gap with winding number

$$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$2

with $$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$3 for the first cluster (Zhang et al., 2022). These point-gap windings indicate a many-body skin effect under open boundaries. At half filling and strong $$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$4, by contrast, the spectrum collapses to open lines, the winding becomes ill-defined, and the many-body wave functions delocalize (Zhang et al., 2022).

Open-chain many-body observables do not simply mirror single-particle skin localization. In the interacting model with open boundaries, the ground-state density profile becomes only slightly tilted relative to the average filling, Friedel oscillations exhibit a beating pattern, and the full counting statistics of particle number over any finite interval is Gaussian. The mean scales with the imaginary vector potential, while the variance is symmetric with respect to the chain center and independent of $$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$5 (Dóra et al., 2022). This directly supports the statement that many-body effects can significantly alter and conceal the single-particle properties and the skin effect in non-Hermitian systems (Dóra et al., 2022).

Nonequilibrium many-body dynamics retains this tension between non-Hermitian driving and collective low-energy structure. In the Luttinger-liquid regime, switching the imaginary vector potential on or off produces spatio-temporal Friedel oscillations and ballistic light cones propagating from the open ends, with local currents of equal magnitude for both quench protocols; the long-wavelength continuity equation remains satisfied even under non-unitary evolution (Dóra et al., 2023). For a finite-time linear ramp of the imaginary vector potential, the excess energy becomes complex-valued although the instantaneous Hamiltonian retains the same real spectrum throughout; the approach to adiabaticity is slow, with a $$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$6 envelope and oscillation period $$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$7, while ramp durations commensurate with that period realize a shortcut to adiabaticity without auxiliary controls (Dupays et al., 2024).

5. Generalized Hatano-Nelson chains

Coupling two quasiperiodic Hatano-Nelson chains changes the character of the delocalization-localization transition. In the cross-coupled model, two critical potentials $$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$8 appear: $$H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$9 with all states delocalized below $h$0, all states localized above $h$1, and two mobility edges symmetrically placed about $h$2 in between (Samanta et al., 2024). In this intermediate regime, the mobility edges divide the spectrum in equal proportions, with $h$3 localized and $h$4 delocalized states (Samanta et al., 2024). The same work shows that the usual one-to-one correspondence between periodic-boundary states and open-boundary skin states breaks down in the coupled system (Samanta et al., 2024).

Longer-range asymmetric hopping yields another major extension. In the generalized model

$h$5

with $h$6 and $h$7, open boundaries can host exceptional points of arbitrary order that do not scale with system size (Gohsrich et al., 2024). Their order is controlled by the sublattice structure and by $h$8; the largest possible order is $h$9 for $t_r=1$0 (Gohsrich et al., 2024). These exceptional-point eigenstates can have support on only a subset of sites, still exhibit skin localization, and can be tuned to localize at the opposite edge from all remaining states. They are robust against generic hopping perturbations and against a specific class of on-site disorder (Gohsrich et al., 2024).

Further generalization to matrix-valued couplings leads to the non-Abelian Hatano-Nelson model with a nonreciprocal $t_r=1$1 gauge field. Its Bloch Hamiltonian,

$t_r=1$2

produces Hopf-link-shaped complex energy braiding with braiding degree $t_r=1$3 and a bipolar skin effect in which left- and right-localized skin modes coexist for the same parameters, a phenomenon impossible in the Abelian nearest-neighbor case (Chen et al., 25 Mar 2026). In a two-orbital interacting ladder with opposite non-Hermiticity on the two legs, Hermitian interchain hopping can restore a fully real non-interacting spectrum when $t_r=1$4, while interactions generate detached doublon branches and winding-number structures linking periodic-boundary doublons to open-boundary skin modes (Huang et al., 16 Apr 2026).

6. Nonlinear, non-Markovian, and experimental directions

Kerr nonlinearity does not destroy the skin effect but reorganizes it. In the nonlinear open-chain equation

$t_r=1$5

families of nonlinear skin modes bifurcate from the linear skin modes at any non-reciprocal strength (Manda et al., 2023). For focusing nonlinearity, these modes become more localized and connect continuously to discrete solitons in the anti-continuum limit; for defocusing nonlinearity, they broaden relative to the linear modes (Manda et al., 2023).

Under periodic boundary conditions, the nonlinear Hatano-Nelson model exhibits a different phenomenon. All nonlinear plane waves are modulationally unstable, and the total norm

$t_r=1$6

undergoes an initial growth followed by an abrupt halt: a dynamical growth blockade (Longhi, 2 Jan 2025). The paper interprets this as a stopping of convective motion caused by self-induced disorder, rather than by boundaries or external randomness (Longhi, 2 Jan 2025).

A non-Markovian generalization replaces constant dissipation by frequency-dependent dissipation and frequency-dependent nonreciprocal hopping,

$t_r=1$7

derived microscopically by integrating out dissipative auxiliary sites with non-equilibrium Green’s functions (Jana et al., 7 Nov 2025). This produces purely non-Markovian effects: unidirectional frequency blocking in the bosonic setting and a nonequilibrium dissipative quantum phase transition in the fermionic setting, both absent in Markovian theory (Jana et al., 7 Nov 2025).

Experimental realizations now access several of these phenomena directly. In integrated photonics, time-domain modulation in a lithium-niobate photonic molecule realizes dynamically tunable Hatano-Nelson couplings,

$t_r=1$8

reaches an exceptional point, surpasses previous asymmetry records, and yields giant $t_r=1$9 dB optical contrast together with non-reciprocal $t_l=\alpha^2<1$0-phase contrast or photonic gyration (Orsel et al., 2024). In electric circuits, the non-Abelian Hatano-Nelson model has been implemented experimentally, and the Hopf-link-shaped admittance spectra and bipolar skin admittance modes have been observed (Chen et al., 25 Mar 2026). Experimental relevance has also been noted for quantum dot arrays, cold atom systems, and monitored quantum circuits in interacting settings (Zhang et al., 2022).

Taken together, these developments establish the Hatano-Nelson chain not as a single isolated model but as a compact framework in which nonreciprocity, spectral topology, localization, many-body correlations, nonlinear self-organization, and open-system memory effects can all be formulated with unusual precision.