Disordered Interacting Fermionic Hatano-Nelson Model

• The disordered interacting fermionic Hatano-Nelson model is a one-dimensional non-Hermitian system where asymmetric hopping, disorder, and fermionic interactions yield distinctive spectral topology and localization effects.
• Analytical (bosonization, Luttinger liquid) and numerical (SDRG) methods reveal that interactions can suppress or reshape the skin effect while altering transport and entanglement dynamics.
• Results show that weak disorder may enhance spreading and entanglement before strong disorder induces localization and a quantum-to-classical crossover marked by random strongly coupled pairs.

Searching arXiv for the specified Hatano–Nelson papers and closely related work to ground the article in current literature. The disordered interacting fermionic Hatano-Nelson model denotes a family of one-dimensional non-Hermitian lattice problems in which asymmetric hopping is combined with disorder and fermionic interactions. In these systems, nonreciprocity produces complex spectra and the non-Hermitian skin effect, disorder drives localization phenomena related to Anderson physics, and interactions reshape both static and dynamical observables in ways that do not reduce to single-particle intuition. The resulting problems have been studied in quasi-periodic chains with nearest-neighbor repulsion, in open interacting chains analyzed by bosonization and numerics, and in fully random chains treated by strong-disorder renormalization group, together forming a technically coherent but physically diverse body of work on skin accumulation, spectral topology, entanglement, and transport (Orito et al., 2022, Orito et al., 2023, Mattiello et al., 19 Sep 2025).

1. Canonical formulations and core observables

A widely studied interacting disordered realization is the one-dimensional tight-binding chain with non-reciprocal hopping, nearest-neighbor interaction, and quasi-periodic Aubry-André disorder,

$$H = \sum_{j} \left[ - (e^{g} \hat{c}_j^\dagger \hat{c}_{j+1} + e^{-g} \hat{c}_{j+1}^\dagger \hat{c}_{j}) + V \hat{n}_j \hat{n}_{j+1} + W_j \hat{n}_j \right], \qquad W_j = W \cos(2\pi \theta j + \theta_0).$$

Here $g$ is the asymmetry parameter, $V$ the nearest-neighbor repulsion, and $W_j$ the quasi-periodic potential. Closely related work also uses the open interacting Hatano-Nelson chain

$$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$

which provides the clean interacting baseline for low-energy and quench analyses. At the strong-disorder end, the disordered interacting fermionic Hatano-Nelson chain has also been formulated as a non-Hermitian spin-$1/2$ XXZ chain with random couplings, random interaction strengths, and asymmetric hopping parameterized by site-dependent non-Hermitian couplings $\gamma_i$ (Orito et al., 2022, Dóra et al., 2022, Mattiello et al., 19 Sep 2025).

Several observables recur across these formulations. Single-particle localization is tracked by the inverse participation ratio

$\mathrm{IPR} = \sum_j |\psi_j|^4,$

with $\mathrm{IPR}\simeq 1$ for skin or localized states and $\mathrm{IPR}\simeq 0$ for delocalized states. Spectral topology is encoded by a winding number defined from twisted boundary conditions,

$g$0

For dynamics, the wave-packet width, density imbalance, and entanglement entropy are central, with the bipartite entropy often written as

$g$1

Formulation	Main ingredients	Typical observables
Quasi-periodic interacting chain	$g$2, $g$3, $g$4	spectrum, IPR, wave-packet spreading, entanglement
Clean interacting OBC chain	$g$5, $g$6	density tilt, Friedel oscillations, current, FCS
Fully random interacting chain	random couplings, $g$7	SDRG flow, susceptibility, entanglement saturation

2. Spectral structure, localization, and many-body skin physics

In the quasi-periodic model, the noninteracting single-particle spectrum under periodic boundary conditions is generally complex for weak disorder and becomes real beyond the critical disorder strength $g$8. The same threshold separates delocalized and localized single-particle states. Under open boundary conditions, by contrast, the spectrum is always real, while the wavefunctions display the non-Hermitian skin effect. This sharp dependence of static spectral properties on boundary conditions is a defining feature of Hatano-Nelson physics, and in the interacting problem the transition point $g$9 is renormalized while a many-body localization regime emerges at strong disorder (Orito et al., 2022).

Interactions substantially modify the skin effect at the many-body level. Under open boundary conditions, a many-body skin effect appears as a density accumulation toward one edge, but it is markedly less extreme than in the single-particle problem because Pauli exclusion and repulsive interactions inhibit macroscopic edge piling-up. The many-body inverse participation ratio is correspondingly reduced relative to the single-particle skin states. At larger repulsion, charge-density-wave order develops; the charge-density-wave order parameter increases sharply for $V$0, and this increase coincides with the vanishing of the winding number $V$1. In this regime, interactions suppress rather than amplify visible skin accumulation in static density observables (Orito et al., 2022).

A complementary clean interacting analysis at half filling identifies two $V$2 transitions as the nearest-neighbor repulsion increases. The first is a first-order symmetry-breaking transition into a charge-density-wave regime, signaled in finite size by an exceptional point between the first and second excited states and by the abrupt disappearance of persistent currents characteristic of the Hatano-Nelson model. The second occurs only in finite-size systems and is characterized by a collapse of all many-body eigenvalues onto the real axis. Away from half filling and in the strong-interaction regime, the many-body spectrum develops point gaps with nontrivial winding numbers, implying a many-body skin effect of extensive eigenstates under open boundary conditions (Zhang et al., 2022).

These results exclude a common simplification according to which the many-body skin effect is merely the single-particle skin effect replicated in Fock space. In the interacting chain, the spectral topology can persist while density profiles become only weakly tilted or while charge order suppresses skin signatures in local observables.

3. Disorder, transport, and entanglement dynamics

Non-Hermitian time evolution is naturally expressed in a biorthogonal eigenbasis,

$V$3

with right and left eigenstates entering through biorthogonal normalization. In the quasi-periodic Hatano-Nelson problem, an initially localized single-particle wave packet slides in one direction when disorder is absent, reflecting the asymmetric hopping. With disorder present, the dynamics differ sharply from the Hermitian expectation: disorder can enhance spreading in the non-Hermitian regime, especially near $V$4. The same work finds that dynamical observables are largely insensitive to boundary conditions, except at long times when the propagating front encounters a boundary; this stands in clear contrast to the static boundary sensitivity of the spectrum and eigenstates (Orito et al., 2022).

For many-body dynamics, a standard protocol starts from a domain-wall state at half filling,

$V$5

In the delocalized phase, the entanglement entropy initially grows as the domain wall melts, then reaches a plateau, and subsequently decreases. The non-monotonicity arises because the long-time evolution becomes dominated by eigenstates with maximal imaginary parts, so the many-body state collapses toward a low-entangled skin mode. In the localized phase, $V$6 instead grows logarithmically, resembling many-body localization in Hermitian systems while retaining non-Hermitian spectral control of the dynamics. The same disorder-and-interaction setting therefore supports both enhanced spreading and late-time entanglement suppression, depending on the spectral sector that dominates the biorthogonal evolution (Orito et al., 2022).

A more detailed study of the many-body Hatano-Nelson model with onsite disorder and nearest-neighbor interaction shows that the entanglement entropy $V$7 is qualitatively different from its Hermitian counterpart in the delocalized weak-disorder regime. Instead of monotone growth toward a volume law, $V$8 rises, overshoots, and then decreases to a strongly suppressed asymptotic value. In the long-time limit,

$V$9

or, at fixed system size,

$W_j$0

In this non-Hermitian setting, weak disorder first increases the saturated entropy by disrupting the collapse toward a single low-entropy sector, while stronger disorder eventually reduces the entropy as localization takes over. With interactions present, the same logarithmic asymptotic form survives but with a reduced coefficient, described there as an effective central charge (Orito et al., 2023).

The resulting picture is not that disorder simply suppresses transport and entanglement. In the non-Hermitian interacting Hatano-Nelson problem, weak disorder can enhance spreading and entanglement, whereas stronger disorder drives localization and logarithmic growth.

4. Luttinger-liquid description and low-energy observables

For the interacting open chain without strong disorder, the low-energy long-wavelength sector is described by a Luttinger liquid with an imaginary vector potential,

$W_j$1

A similarity transformation removes the explicit non-Hermitian term from this low-energy Hamiltonian for purposes of evaluating eigenvalues and correlation functions, mapping the problem to a standard Hermitian Luttinger liquid description of the long-wavelength sector. Within this framework, the smooth density profile is

$W_j$2

Repulsive interactions flatten the density tilt, whereas attractive interactions enhance it. The full density contains Friedel oscillations with a spatially dependent wavenumber, producing a beating pattern not present in the Hermitian or noninteracting case, while preserving the power-law decay of oscillations (Dóra et al., 2022).

The same bosonized framework yields explicit statements about full counting statistics. For the number of particles in any finite interval, the distribution is Gaussian. Its mean and variance are

$W_j$3

The mean is linear in the imaginary vector potential, whereas the variance is independent of $W_j$4 at the low-energy level and remains symmetric under reflection about the chain center. Thus the static fluctuations measured by full counting statistics are much less sensitive to the skin effect than single-particle eigenfunctions are (Dóra et al., 2022).

Quench dynamics in the weakly interacting open chain provide a complementary baseline for the interacting problem. When the imaginary vector potential is switched on or off, both density and current develop ballistic light cones propagating from the two boundaries. For the switch-off protocol,

$W_j$5

$W_j$6

Both density and current exhibit spatio-temporal Friedel oscillations, and the long-wavelength continuity equation remains satisfied even though the time evolution is non-unitary. These clean-chain results do not address strong disorder directly, but they provide the reference Luttinger-liquid structure against which disorder-driven deviations are interpreted (Dóra et al., 2023).

5. Strong-disorder renormalization and the random strongly coupled pair phase

The strongest available statement on the disordered interacting fermionic Hatano-Nelson chain comes from a strong-disorder renormalization group treatment that is asymptotically exact. In this formulation, the model is equivalent to a non-Hermitian spin-$W_j$7 XXZ chain with random couplings, random interaction strengths, and non-Hermitian asymmetric hopping. The key RG result is that non-Hermitian couplings are relevant perturbations to the Hermitian problem. Their flow is additive,

$W_j$8

so the distribution of $W_j$9 broadens without bound and $$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$0 is an unstable fixed point. This drives a quantum-to-classical crossover at long length scales (Mattiello et al., 19 Sep 2025).

The ground state is not the Hermitian random-singlet state. Instead it consists of random strongly coupled pairs located at arbitrary positions. The two-spin ground state of a pair is a mixture of the singlet and the $$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$1 triplet,

$$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$2

As $$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$3, the pair becomes classical and separable, so long pairs no longer contribute entanglement. The paper identifies the corresponding phase as a random strongly coupled pair phase rather than a random singlet phase (Mattiello et al., 19 Sep 2025).

This altered ground-state structure produces striking thermodynamic signatures. The susceptibility in the $$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$4 and $$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$5 directions becomes negative and diverges at a finite small temperature, whereas the $$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$6-susceptibility behaves as in the Hermitian case. For an individual strongly coupled pair,

$$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$7

After summing over decimated pairs, the divergence occurs at

$$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$8

The work also emphasizes strong sample-to-sample fluctuations, so the low-temperature response is non-self-averaging (Mattiello et al., 19 Sep 2025).

Entanglement scaling is equally altered. In the Hermitian random XXZ chain one has $$H_{HN} = \sum_{n=1}^{N-1} \left[\frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n \right] + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1},$$9, but in the non-Hermitian disordered chain the entanglement entropy saturates at large block size because the long pairs become classical. For one pair, the right-right entanglement entropy is

$1/2$0

which vanishes as $1/2$1. The saturation of $1/2$2 is therefore the entanglement signature of the quantum-to-classical crossover (Mattiello et al., 19 Sep 2025).

6. Field-theoretic interpretation and noninteracting benchmarks

At the level of disordered single-particle field theory, non-Hermitian disordered systems admit a fermionic replica nonlinear sigma model obtained by Hermitization. For the one-dimensional Hatano-Nelson problem, the continuum Hamiltonian is written as

$1/2$3

and the corresponding sigma model has the form

$1/2$4

The crucial point is that the Anderson transition unique to nonreciprocal disordered systems in one dimension, including the Hatano-Nelson model, originates from the competition between the kinetic term and the topological term. More broadly, the universality class of a non-Hermitian disordered system is identified with that of the Hermitized system with an additional chiral symmetry (Chen et al., 2024).

This field-theoretic result is not itself a many-body solution, but it clarifies why nonreciprocity qualitatively changes localization in one dimension. A plausible implication is that interacting extensions should inherit some of this topological competition in renormalized form, although the explicit interacting treatment requires additional machinery beyond the single-particle replica sigma model.

A noninteracting dynamical benchmark further sharpens the disorder problem. In the disordered Hatano-Nelson chain, the wave-packet width $1/2$5 obeys distinct scaling laws across disorder regimes: $1/2$6 in the clean limit,

$1/2$7

at long times in the weak-disorder coexistence regime after an early ballistic stage, and

$1/2$8

in the deeply Anderson-localized regime. The analysis relates these exponents to the imaginary density of states near the band edge, with a linear iDOS drop yielding the $1/2$9 law. That work is explicitly noninteracting, but it also states that the universal features of the scaling exponents and the interplay of nonreciprocity with localization are expected to persist in the presence of interactions; this suggests a reference scenario for future studies of the fully interacting disordered chain rather than an established many-body theorem (Shang et al., 6 Apr 2025).

Taken together, the current literature supports three general conclusions. First, interactions can suppress or qualitatively reshape visible skin accumulation even when non-Hermitian spectral topology remains present. Second, disorder in non-Hermitian fermionic chains is not reducible to the Hermitian narrative of monotone transport suppression: it can enhance spreading or entanglement before localization dominates. Third, in the strongly disordered interacting limit, non-Hermiticity is RG-relevant and drives a genuine quantum-to-classical crossover, replacing the Hermitian random-singlet phenomenology by a random strongly coupled pair phase.