Interacting Hatano-Nelson Model

• The interacting Hatano–Nelson model is a non-Hermitian framework characterized by asymmetric hopping and repulsive interactions, leading to rich many-body dynamics.
• It reveals quantum phase transitions, unusual spectral topologies, and transport phenomena such as superdiffusive behavior and the non-Hermitian skin effect.
• Experimental implementations in photonics and acoustics, alongside synthetic gauge fields, demonstrate its potential for simulating nonreciprocal quantum systems.

The interacting Hatano–Nelson model describes systems with asymmetric (nonreciprocal) hopping and particle interactions, placing it at the forefront of non-Hermitian quantum many-body physics. Originally conceived to paper delocalization-localization transitions and the non-Hermitian skin effect in disordered fermion chains, recent extensions and rigorous analyses have established a diverse phenomenology in interacting and disordered settings. These include novel quantum phase transitions, non-trivial spectral topology, exceptional and point-gap physics, distinctive entanglement scaling, and dynamical effects such as growth blockade, superdiffusive transport, and anomalous work statistics.

1. Hamiltonians and Model Structure

The standard interacting Hatano–Nelson model for spinless fermions on a chain is

$$H = \sum_\ell \left[ (t+\gamma) c^\dagger_\ell c_{\ell+1} + (t-\gamma) c^\dagger_{\ell+1} c_\ell + U n_\ell n_{\ell+1} \right],$$

where $t$ is the symmetric hopping, $\gamma$ parameterizes hopping asymmetry (nonreciprocity; $t\pm\gamma$), and $U$ is the strength of a repulsive nearest-neighbor interaction. In bosonic settings and generalizations, the Hamiltonian may incorporate density-dependent hopping and nonlinearities, e.g., density-dependent gauge fields, Kerr or cubic nonlinearities, and higher-dimensional extensions (Zhang et al., 2022, Faugno et al., 2022, Manda et al., 2023, Longhi, 2 Jan 2025, Naichuk et al., 8 Apr 2025).

Non-Hermiticity arises through asymmetric hopping: nonzero $\gamma$ breaks time-reversal and ensures the spectrum and eigenvectors fundamentally differ from Hermitian cases. Interactions $U$ drive many-body effects, quantum phase transitions, and alter the non-Hermitian skin effect.

2. Quantum Phase Transitions and Spectral Topology

Interacting Hatano–Nelson chains display unconventional many-body phase diagrams. At half-filling, increasing $U$ induces two distinct $\mathcal{PT}$ symmetry transitions (Zhang et al., 2022):

• First $\mathcal{PT}$ transition: The lowest excited states, initially forming a complex conjugate pair, coalesce on the real axis at a critical $U_c$ (an exceptional point), marking a first-order transition to a charge-density wave (CDW) state with spontaneously broken translation symmetry. Persistent current (real or imaginary, reflecting nonreciprocal transport) vanishes abruptly at this point.
• Second $\mathcal{PT}$ transition: At a larger $U_{c, \rm all}$, which diverges in the thermodynamic limit, all eigenvalues become real—this is a finite-size effect.

At strong interaction, and away from half-filling, the spectrum organizes into distinct energy clusters corresponding to many-body Fock sectors (“point-gap” clusters) with non-trivial winding numbers under spectral flow, directly connected to the topological properties and appearance of many-body skin effects (Zhang et al., 2022, Lu et al., 2023, Manda et al., 2023). The spectral topology depends on the ordering of real and complex eigenvalues, the existence of gaps, and the mixing of extended and localized regions; disorder, non-reciprocity, and periodic potentials each generate diverse topologies, reentrant transitions, and mobility edge physics (Hébert et al., 2010, Samanta et al., 6 Sep 2024, Dóra et al., 2023).

3. Localization, Skin Effect, and Disorder

The skin effect, a hallmark of non-Hermitian systems, refers to the exponential localization of a macroscopic fraction of eigenstates at one boundary under open boundary conditions. For weak or absent interactions and disorder, the Hatano–Nelson model’s eigenstates are entirely “skinned” (Dóra et al., 2022, Shang et al., 6 Apr 2025). Interactions coexist with the skin effect in several regimes; strong interactions can preserve or reorganize boundary localization via the cluster topology of the many-body spectrum, enabling an “interaction-induced skin effect” for repulsively bound pairs (doublons) (Brighi et al., 15 Mar 2024).

Disorder introduces competition between Anderson localization (random potential-induced eigenstate localization) and the skin effect. This leads to a “skin–Anderson transition,” where the Lyapunov exponent signals the crossover between boundary-localized and Anderson-localized modes. The dynamical scaling of a wave packet distinguishes regimes: ballistic (clean, skin-dominated), superdiffusive ($\Delta x \sim t^{2/3}$, coexistence), and diffusive ($\Delta x \sim t^{1/2}$, strong disorder), with eventual crossover back to superdiffusive behavior at long times due to density-of-states singularities (Shang et al., 6 Apr 2025). In coupled chain geometries, the interaction between chains introduces additional critical points and precisely divides states into delocalized and localized sectors with sharp mobility edges (Samanta et al., 6 Sep 2024).

4. Entanglement, Quantum-Classical Crossover, and Many-Body Steady States

The “random singlet” entanglement log-scaling ($S \sim \ln L$) of disordered Hermitian chains breaks down in non-Hermitian interacting systems. Strong-disorder renormalization group (SDRG) analyses of the non-Hermitian disordered XXZ chain (equivalent to the Hatano–Nelson chain) reveal that non-Hermiticity ($\gamma$) is a relevant perturbation, driving the ground state into a network of strongly coupled pairs (SCPs) that are not pure singlets but admixtures of singlet and $M=0$ triplet. As the RG flow increases $\gamma$, SCPs become “classical” (entanglement vanishes, EE saturates), and Curie-like contributions to susceptibility become negative and divergent in the transverse plane, reflecting non-Hermitian physics (Mattiello et al., 19 Sep 2025).

In clean interacting systems, the ground-state entanglement entropy in the gapless phase exhibits conformal scaling $S \sim (c/3)\log L$ with central charge $c = 1$ (robust to non-Hermiticity), before an abrupt drop to area law at the $\mathcal{PT}$ transition into the CDW phase (Lu et al., 2023). Out-of-equilibrium dynamics further highlight the fragility of volume-law entanglement: non-Hermitian evolution “filters out” all but maximally amplified/attenuated eigenstates, leading to a collapse of entanglement entropy that only scales logarithmically in subsystem size in the steady state (Orito et al., 2023).

5. Dynamical Effects: Quench, Adiabaticity, Growth Blockade

Quantum quenches—rapid changes to model parameters such as sudden switching of the asymmetric hopping or interaction strength—show unique dynamical behavior. In the Hatano–Nelson Luttinger liquid, quenching the imaginary vector potential generates propagating Friedel oscillations and light-cone dynamics, with the continuity equation holding for the long-wavelength sector even for non-unitary evolution (Dóra et al., 2023). Under finite-time ramps, the approach to adiabaticity is anomalously slow (decay $\sim 1/\tau$) and oscillatory, revealing the breakdown of instantaneous eigenstate tracking due to intrinsic non-orthogonality; but exact “shortcuts to adiabaticity” can occur for special ramp durations matching the system’s finite-size recurrence time (Dupays et al., 13 Aug 2024).

Nonlinear extensions (cubic/Kerr interaction) give rise to universal modulational instability for all plane waves—unlike in Hermitian discrete nonlinear Schrödinger equations where MI is selective—and, in periodic boundary systems, to a “growth blockade” where total norm growth due to non-Hermitian amplification is arrested by self-induced disorder that dynamically produces effective on-site potential randomness, leading to saturation of the norm (Longhi, 2 Jan 2025). Nonlinear skin modes and skin solitons persist, bridging the linear and strongly localized limits; interactions modulate both localization and dynamical stability (Manda et al., 2023).

6. Topological Features, Synthetic Gauge Fields, and Experimental Realizations

A central insight is that interactions in the Hatano–Nelson setting can induce topologically nontrivial many-body spectra characterized by point gaps and quantized winding numbers (from spectral flow with twisted boundary conditions), directly generalizing single-particle non-Hermitian topology to the many-body context (Zhang et al., 2022, Faugno et al., 2022). In density-dependent gauge field extensions, non-Hermitian topological phases only emerge for multi-particle sectors, with the doublon effective model (a non-Hermitian SSH chain) capturing the essential spectral winding and topological invariants.

Synthetic gauge fields—associating non-reciprocal hopping with engineered Peierls phases—allow the simulation of “non-Hermitian Aharonov–Bohm effect,” where both real and imaginary persistent currents are observed as flux derivatives of the ground-state energy in non-Hermitian HN rings. Correlated disorder (e.g., Aubry–André–Harper or Fibonacci models) can anomalously amplify persistent currents, and bond-resolved analysis shows strict separation between bonds carrying real (interdimer) and imaginary (intradimer) currents (Ganguly et al., 19 Dec 2024).

Experimental mapping of the HN model has been achieved in continuous acoustic waveguides using two-port transfer matrices and active feedback to implement non-reciprocal hopping; the correspondence to the discrete model is exact, enabling direct observation of the skin effect and exponential boundary sensitivity (Maddi et al., 2023). In photonics, asymmetric Hatano–Nelson couplings can be dynamically engineered using modulated resonator frequencies, enabling tuning through exceptional points and realization of giant non-reciprocity and gyration with high contrasts (Orsel et al., 14 Oct 2024).

7. Higher-Dimensional, Multi-Chain, and Open Directions

Generalizations of the Hatano–Nelson model as building blocks for correlated two-dimensional non-Hermitian flux lattices reveal that nonreciprocal hopping and onsite Hubbard interactions produce antiferromagnetic order with unique properties: $\mathcal{PT}$-symmetry-protected regions with real spectra and non-Hermitian “spin wave” modes of a dissipative d-wave character. The metal-insulator transition coincides with the transition between $\mathcal{PT}$-unbroken and broken symmetry regions (Naichuk et al., 8 Apr 2025).

In coupled multi-chain systems, cross-coupling introduces additional criticality: two mobility edge transitions, equal division of localized and delocalized states in the intermediate regime, and unconventional breakdown of bulk-boundary correspondence under open boundaries. Asymmetric cross-hopping can localize half the spectrum at arbitrarily weak quasiperiodic potential (Samanta et al., 6 Sep 2024).

These developments underline the ubiquity and richness of non-Hermitian many-body physics, with the interacting Hatano–Nelson model remaining a paradigmatic testbed for discovering and examining novel phases, dynamics, topological effects, and experimental implementations across quantum and classical platforms.