Hatano–Nelson Model: Non-Hermitian Phenomena

• The Hatano–Nelson model is a 1D non-Hermitian quantum system defined by asymmetric (nonreciprocal) hopping, leading to unusual transport and localization phenomena.
• Its Hamiltonian structure produces a complex energy spectrum with distinctive topological winding and skin effects, underpinning phenomena such as exceptional points.
• Generalizations incorporate disorder, interactions, nonlinearities, and higher-dimensional extensions, providing deep insights into non-Hermitian quantum and many-body physics.

The Hatano–Nelson model (HN model) is a paradigmatic example of a one-dimensional (1D) non-Hermitian quantum system characterized by asymmetric (nonreciprocal) hopping, leading to distinctive transport, localization, and topological properties. Originally formulated to study non-Hermitian delocalization transitions in the presence of disorder, the model has become central in the development of the theory of non-Hermitian skin effects, exceptional points, and non-Hermitian many-body physics. Its variants span lattice fermion and boson systems, continuous wave media, models with interactions and disorder, and recent extensions to higher dimensions and spinful settings.

1. Model Definition, Fundamental Properties, and Variants

The prototypical HN Hamiltonian for spinless fermions on an N-site chain is

$$H = \sum_{j=1}^{N-1} \left[ t_R c^\dagger_{j+1}c_j + t_L c^\dagger_j c_{j+1} \right]$$

where $t_R, t_L > 0$ are the right/left hopping amplitudes, typically parameterized as $t_R = t e^{g}$, $t_L = t e^{-g}$ with $g \in \mathbb{R}$ the "imaginary vector potential" (non-Hermitian "flux"). The system is non-Hermitian for $t_R \neq t_L$, breaking time-reversal and reciprocity.

Key generalizations include:

• On-site potentials (random, periodic, or quasiperiodic): $V_j n_j$
• Interactions: $U n_j n_{j+1}$ or Hubbard-like terms
• Longer-range hoppings: $t_{l}, t_{-r}$ for generalized exceptional points (Gohsrich et al., 2024)
• Nonlinear terms: Kerr/cubic nonlinearity $g|\psi_n|^2 \psi_n$ (Manda et al., 2023, Longhi, 2 Jan 2025)
• Spin and synthetic gauge coupling: spin-dependent Peierls phases (Sanahal et al., 8 May 2025)
• Continuous analogues: e.g., waveguides mapped to HN via transfer matrices (Maddi et al., 2023)
• Non-Markovian dissipation: frequency-dependent bath-induced parameters (Jana et al., 7 Nov 2025)

Distinct boundary conditions (periodic or open) reveal starkly different spectral and eigenstate properties.

2. Spectral Theory, Skin Effect, and Topology

Complex Spectra and Non-Hermitian Topology

Under periodic boundary conditions (PBC), the single-particle spectrum is

$$E(k) = t_R e^{ik} + t_L e^{-ik} = 2t \cos k \cosh g + 2i t \sin k \sinh g,$$

tracing an ellipse in the complex plane. The spectrum under PBC exhibits a nontrivial point-gap, with a bulk spectral winding number

$$\nu = \frac{1}{2\pi i} \int_{0}^{2\pi} \frac{d}{dk} \ln[E(k) - E_b] dk,$$

characterizing the intrinsic non-Hermitian topology (Zhang et al., 2022, Sanahal et al., 8 May 2025).

Non-Hermitian Skin Effect (NHSE)

Under open boundary conditions (OBC), all eigenstates collapse exponentially to one edge—the non-Hermitian skin effect. For $g > 0$,

$|\psi_n|^2 \propto e^{-2g n}$

with localization length $\xi = 1/|g|$. The OBC spectrum collapses to a real segment, exhibiting boundary sensitivity exponentially large in system size (Maddi et al., 2023).

The "bulk–boundary correspondence" is generalized in non-Hermitian settings: a nonzero spectral winding in PBC implies skin modes under OBC, but strong breakdowns can occur, especially in models with spin-dependent or unconventional gauge structure (Sanahal et al., 8 May 2025).

Exceptional Points and Higher-order Defectiveness

Generalized HN models with longer-range asymmetric hoppings exhibit robust exceptional points (EPs) of arbitrary finite order. These high-order EPs correspond to coalescence of multiple eigenvectors into non-diagonalizable Jordan blocks, with eigenstates localized according to the associated spectral winding (Gohsrich et al., 2024). The location and order of EPs are topologically protected by the graph structure and generalized chiral symmetry.

3. Localization, Disorder, and Spectral Transitions

Disorder-driven Phase Diagram

The interplay between nonreciprocal hopping and disorder (random and correlated) yields a rich phase diagram:

• Delocalized (skin) regime: For weak disorder ($W < W_c$), skin-localized bands survive, leading to ballistic or superdiffusive spreading of wave packets (Shang et al., 6 Apr 2025).
• Anderson-localized regime: For strong disorder ($W > W_c$), all states are spatially localized and the skin effect is suppressed.
• Skin–Anderson transition: The critical point is determined via the Lyapunov exponent:

$$\gamma(E, g) = \gamma_0(E) + g, \quad \text{mobility edge:} \quad \gamma(E, g) = 0.$$

• Quasiperiodic and periodic potentials introduce bidirectional and direction-reversed skin effects, splitting the delocalization–localization and complex–real transitions, with a unique bidirectionality parameter ($\eta_B$) and explicit winding number jumps (Padhi et al., 2024).

Spectral Transitions in Strictly Ergodic and Random Potentials

For strictly ergodic potentials, the HN spectrum is sharply characterized in terms of the Lyapunov exponent $L(E)$: $$\Sigma(g) = \big\{E \mid (E \in \Sigma(0) \wedge L(E) \leq g)\ \vee\ (E \notin \Sigma(0) \wedge L(E) \geq g)\big\}.$$ Critical thresholds separate purely real, mixed, and purely complex spectra (Wang et al., 2023).

In i.i.d. random potentials, the infinite-volume spectrum is always two-dimensional in the complex plane, and no real–complex transition survives in the thermodynamic limit.

4. Interactions, Many-body Phenomena, and Bosonization

Many-body Effects and Interaction-induced Phases

Interacting HN chains (e.g., nearest-neighbor repulsion, or hard-core bosons) support:

• Charge-density wave (CDW) transitions with first-order symmetry breaking, exceptional points, and abrupt persistent current quenching (Zhang et al., 2022, Lu et al., 2023).
• Many-body skin effect: Clusters of many-body eigenstates (indexed by, e.g., number of occupied bonds) under OBC show large spatial polarization set by nonzero many-body winding numbers (Zhang et al., 2022, Lu et al., 2023).
• Spectral clusterization: In strong interaction regimes, the spectrum fragments into elliptic (or real, if filled) clusters, each with universal scaling for major and minor axes (Lu et al., 2023).

Bosonization, Luttinger Liquid Structure, and Non-Hermiticity

In the weakly interacting, low-energy regime, bosonization maps the HN chain to a Luttinger liquid (LL) in an imaginary vector potential: $$H = \int_0^L \frac{dx}{2\pi} v \left[ K (\pi\Pi(x) - i h)^2 + \frac{1}{K}\left(\partial_x\phi(x)\right)^2 \right],$$ with LL parameter $K$ and velocity $v$ set by the interaction (Dóra et al., 2023, Dóra et al., 2022). Notable consequences:

• Density tilt: Average density profile tilts logarithmically across the chain,

$$n_0(x) = -\frac{K h}{\pi^2} \sum_{\pm} \ln \left| \tan \frac{\pi(x \pm v t)}{2L} \right|$$

• Suppression of single-particle skin effect: In the many-body ground state, the exponential pile-up is replaced by a mild tilt and modulated Friedel oscillations.
• Gaussian full counting statistics: Particle-number fluctuations over intervals are normally distributed, with variance independent of the non-Hermitian field (Dóra et al., 2022).

5. Nonequilibrium Dynamics, Entanglement, and Thermodynamic Statistics

Quantum Quenches and Non-Hermitian Dynamics

Quenches of the imaginary vector potential (sudden "on" or "off") induce:

• Ballistic density and current propagation: Light-cone-like fronts and Friedel oscillations emerge from chain ends.
• Continuity equations: Long-wavelength density and current obey standard continuity even under non-unitary time evolution (Dóra et al., 2023).
• Comparison with numerics: Analytic bosonization results accurately predict density and current evolution at long wavelengths (MPS/DMRG numerics) (Dóra et al., 2023).

Entanglement Growth and Non-Hermitian Many-body Orthogonality

The entanglement entropy, after a quench in a disordered interacting HN model, exhibits distinctive features:

• Non-monotonicity in the delocalized phase: Initial linear growth, peak, and subsequent decay to a suppressed value (Orito et al., 2023).
• Final state: The system "post-selects" the eigenstate with maximal imaginary energy, leading to entanglement entropy scaling logarithmically with subsystem size.
• Generalized quasiparticle picture: Describes entanglement dynamics, incorporating the decay/amplification of mode occupations due to non-Hermiticity (Orito et al., 2023).
• persistence of logarithmic entanglement only in the delocalized phase; area-law scaling in the localized/strong disorder regime.

Work Statistics and Loschmidt Echo

The biorthogonal Loschmidt echo $G(t)$ after introducing nonreciprocity decays as a stretched power law with a time-dependent exponent. Key universal features:

• Survival probability at zero work is Gaussian-suppressed in system size and non-Hermitian field ($\ln P(0) \sim -L^2 h^2$)
• High-energy tail is universally $P(W) \sim W^{-3}$
• Mean and variance of work scale as $\sim L$ and $\sim \ln L$, higher cumulants are non-extensive (Dóra et al., 2023). These statistics are signatures of non-unitary quantum thermodynamics and are relevant for monitored quantum simulations and non-Hermitian field theories.

Adiabaticity, Ramping, and Shortcut Protocols

For a finite-time ramp of the imaginary vector potential, the system approaches adiabaticity slowly: observables such as excess energy and Loschmidt echo decay as $\tau^{-1}$ (contrasting with $\tau^{-2}$ for Hermitian quenches). Remarkably, when the ramp duration is commensurate with the ballistic oscillation period ($\tau = n \cdot 2L/v$), the protocol constitutes a shortcut to adiabaticity without auxiliary controls (Dupays et al., 2024).

6. Extended Phenomena: Nonlinearities, Spin, and Higher Dimensions

Nonlinear and Non-Markovian Effects

• Nonlinear HN models with Kerr-type terms admit "nonlinear skin modes" and soliton-like states, whose localization is tunable by the interaction sign and amplitude. Nonlinearity can enhance or suppress the skin effect and stabilize or destabilize nonlinear stationary states, breaking the linear paradigm (Manda et al., 2023, Longhi, 2 Jan 2025).
• Dynamical growth blockade: Modulational instability of nonlinear plane waves induces a rapid self-induced localization (self-generated disorder), arresting the convective amplification typical of the linear HN model (Longhi, 2 Jan 2025).
• Non-Markovian environments: Frequency-dependent baths induce complex, frequency-selective NHSE and phenomena such as unidirectional frequency blocking and dissipative quantum phase transitions, impossible in Markovian settings (Jana et al., 7 Nov 2025).

Spinful and Multicomponent HN Models

Spin-dependent Abelian gauges and Zeeman fields enrich the HN phase diagram:

• Unconventional skin effect: Coexistence of spin-polarized scale-restricted skin modes (with zero total winding) and conventional skin modes. Critical Zeeman fields drive transitions between bidirectional, critical, and unidirectional skin effects, unifying the Z₂ and critical NHSE under a synthetic gauge framework (Sanahal et al., 8 May 2025).

Higher-dimensional Generalizations

In two dimensions, the "Hatano–Nelson flux model" features maximally unidirectional hopping per plaquette and Hubbard interactions:

• Antiferromagnetic metal–insulator transition: Coincides with the $\mathcal{PT}$-breaking transition and emergence of purely real spectrum (Naichuk et al., 8 Apr 2025).
• Spin-wave excitations: Gain–loss pairs of diffusive modes with $d$-wave structure arise, a direct consequence of non-Hermiticity.

7. Experimental Realizations and Physical Significance

Continuous nonreciprocal systems in photonics, acoustics, and electronics can be mapped exactly to the HN model using transfer-matrix methods. Experimental demonstrations include:

• Acoustic waveguides: Nonreciprocal transfer elements implement skin effect and boundary-sensitive spectral flows; boundary condition perturbations yield exponentially large spectral response (Maddi et al., 2023).
• Optical, cold atom, electric-circuit, and synthetic-dimension platforms: All can realize controlled nonreciprocal hopping, synthetic gauge fields, and skin modes, as well as model spinful or nonlinear extensions.

The HN model and its extensions provide robust settings for exploiting and investigating edge amplification, parameter-sensitive exceptional points, fast-adaptation protocols (shortcuts to adiabaticity), and topological sensing, with direct consequences for next-generation quantum devices, non-Hermitian signal processing, and open system quantum thermodynamics.

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Key References

• Quench dynamics and Luttinger liquid mapping (Dóra et al., 2023)
• Nonlinear skin modes and modulational instability (Manda et al., 2023, Longhi, 2 Jan 2025)
• Generalizations to disorder, extended hopping, and potential (Padhi et al., 2024, Hébert et al., 2010)
• Interacting and many-body spectral structure (Zhang et al., 2022, Lu et al., 2023, Orito et al., 2023, Dóra et al., 2022)
• Spectral theory for strictly ergodic and random potentials (Wang et al., 2023)
• Exceptional points and skin effect (Gohsrich et al., 2024)
• Non-Markovian generalizations (Jana et al., 7 Nov 2025)
• Spinful and unconventional skin effect (Sanahal et al., 8 May 2025)
• Photonic/acoustic implementations (Maddi et al., 2023)

This synthesis encapsulates the multifaceted research landscape of the Hatano–Nelson model and its profound influence on non-Hermitian condensed matter theory, topology, and transport.