Nonlinear Hatano-Nelson Model Extensions

• The nonlinear extension of the Hatano-Nelson model is a framework that adds interaction and Kerr-type nonlinear terms to nonreciprocal hopping, resulting in unprecedented spectral and topological effects.
• It demonstrates that incorporating many-body interactions and amplitude-dependent terms leads to phenomena like nonlinear skin modes, modulational instabilities, and spectral cluster formation.
• These insights pave the way to explore quantum phase transitions, boundary localization, and novel topological signatures absent in traditional Hermitian frameworks.

The nonlinear extension of the Hatano-Nelson model refers to classes of non-Hermitian lattice systems in which the linear, nonreciprocal hopping of the canonical Hatano-Nelson chain is modified by adding interaction or nonlinear terms. These extensions produce qualitatively new spectral, topological, and dynamical phenomena without Hermitian or single-particle analogues—especially in settings involving Kerr-type nonlinearities, nearest-neighbor interactions, or hard-core constraints. Major research directions include the study of nonlinear skin modes, modulational instabilities, many-body spectral clustering, $\mathcal{PT}$-breaking quantum phase transitions, and emergent disorder-like growth blockades.

1. Model Variants and Definitions

Nonlinear variants of the Hatano-Nelson model replace or augment its linear, asymmetric hopping Hamiltonian with additional many-body or amplitude-dependent terms:

• Interacting many-body forms: For hardcore bosons or spinless fermions, a prototypical Hamiltonian with nonreciprocal hopping $t\pm\gamma$ and nearest-neighbor interaction $V$ reads

$$H = \sum_{l=1}^L \big[ (t+\gamma) b_l^\dagger b_{l+1} + (t-\gamma) b_{l+1}^\dagger b_{l} + V n_l n_{l+1} \big ]$$

where $b_l, b_l^\dagger$ are annihilation/creation operators and $n_l = b_l^\dagger b_l$, with $n_l \in \{0,1\}$ in the hard-core limit (Lu et al., 2023, Zhang et al., 2022).

• Kerr-nonlinear (DNLS-type) forms: The nonlinear Hatano-Nelson equation with Kerr nonlinearity is

$$i \frac{d\psi_n}{dt} = C [\psi_{n+1} + t \psi_{n-1}] + \sigma |\psi_n|^2 \psi_n,$$

where $C$ is the hopping strength, $t$ is the nonreciprocity parameter ($t\neq1$), and $\sigma$ determines the nature of the nonlinearity (focusing or defocusing). Open or periodic boundary conditions are considered (Manda et al., 2023, Longhi, 2 Jan 2025).

Interactions $V$, Kerr coefficients $\chi$, or density-type terms induce nonlinearity, fundamentally altering the spectrum, eigenstates, and evolution.

2. Spectral Structure and Cluster Formation

Nonlinear interactions fragment the many-body energy spectrum and modify the spatial and spectral localization properties:

• Spectral clustering (hardcore bosons): For $V \gg t, \gamma$, the $N$-particle spectrum splits into $N$ clusters, each forming an ellipse in the complex-energy plane. The $n_s$-th cluster is centered at $E_c \simeq (n_s-1)V + C(t^2-\gamma^2)/V$, with the axes obeying universal power-law scalings: $2a = C_a V^{p_a}$, $2b = C_b V^{p_b}$, with filling- and cluster-index–dependent exponents (Lu et al., 2023).
• Level counting: The number of states per cluster/ellipse is governed by combinatorial rules, with $N_q=L$ for extremal clusters, and $N_q=(n_s+L/2-N-1)L$ for intermediates at half-filling (Lu et al., 2023). Exact formulas for level multiplicities reflect emergent many-body regularity.
• Clustered point-gap topology: For large $V$ in the fermionic model, spectral clusters acquire point-gap windings, generalizing the single-particle non-Hermitian topology to many-body settings. Each cluster's eigenvalues exhibit nontrivial winding with respect to boundary phase, which is linked to bulk-boundary correspondence and skin effect (Zhang et al., 2022).

3. Nonlinear Skin Modes and Boundary Localization

Non-Hermitian models with nonreciprocal hopping exhibit the skin effect—exponential localization of bulk eigenstates at the system boundary under open boundary conditions. The inclusion of nonlinearities radically restructures this scenario:

• Nonlinear skin mode (NLSM) families: Every linear skin mode bifurcates into a continuous family of nonlinear counterparts parameterized by amplitude and coupling. Focusing nonlinearity ($\sigma=+1$) enhances edge localization, producing sharper skin modes; defocusing ($\sigma=-1$) broadens modes, which can become nearly extended (Manda et al., 2023).
• Perturbative and anti-continuum analyticity: For weak nonlinearity, families branch smoothly from the linear limit; in the anti-continuum limit ($C\to 0$), skin solitons localize on $M=q$ sites, and NLSM branches connect these extremes analytically (Manda et al., 2023).
• Stability regimes: Stability windows are found near both linear and fully nonlinear limits using Bogoliubov–de Gennes analysis, with intermittent instability windows depending on amplitude and cluster. The $q=1$ (ground state) branch with defocusing nonlinearity is always stable (Manda et al., 2023).
• Boundary sensitivity: The basin of attraction for nonlinear skin modes is nontrivial: large boundary perturbations at the unfavored edge can collapse a BdG-stable NLSM, due to nonlinear amplification and skin migration (Manda et al., 2023).

4. Quantum Phase Transitions and Entanglement Scaling

The introduction of nonlinear interactions induces sharp quantum phase transitions distinguished by their spectral, entanglement, and symmetry characteristics:

• CDW transition: Both bosonic and fermionic models manifest a ground-state transition from a gapless phase (with Luttinger-liquid–like features and central charge $c=1$) to a symmetry-broken charge-density wave (CDW) phase at critical $V_c$. This is signaled by the crossing of a biorthogonal CDW order parameter and the opening of a gap (Lu et al., 2023, Zhang et al., 2022).
• PT-symmetry transitions: At half-filling, the first excited state undergoes a $\mathcal{PT}$ phase transition at $V_{PT}$. The excitation remains real up to $V_{PT}$, beyond which it becomes a complex-conjugate pair. This is accompanied by a discontinuous jump in excited-state entanglement entropy (Lu et al., 2023, Zhang et al., 2022).
• Entanglement scaling: In the gapless regime, entanglement entropy displays logarithmic scaling $S_{L/2} \sim (c/3)\ln L$, with $c=1$. In the CDW phase, this crosses over to an area law—saturating with subsystem size (Lu et al., 2023).

5. Dynamical Properties: Modulational Instability and Growth Blockade

Nonlinear Hatano-Nelson models support new classes of dynamical phenomena absent from both Hermitian and linear non-Hermitian analogues:

• Modulational instability (MI): All nonlinear plane waves in the discrete nonlinear Hatano–Nelson model are modulationally unstable. Both focusing and defocusing Kerr-type nonlinearities, combined with nonreciprocal hopping, lead to rogue growth of superimposed perturbations on top of plane-wave backgrounds (Longhi, 2 Jan 2025).
• Dynamical growth blockade: Despite the secular exponential amplification found in the linear model under periodic boundary conditions, the nonlinear system exhibits a striking growth blockade. After an initial period of norm growth, MI-induced self-disorder leads to the arrest and saturation of the total norm. This is interpreted as a feedback effect in which nonlinearity generates an Anderson-like disorder potential that traps and localizes excitation, halting further convective transport (Longhi, 2 Jan 2025).
• Marginal stability: The resultant, time-averaged Bogoliubov analysis reveals purely real spectra, confirming the absence of further instability after the MI-driven blockade (Longhi, 2 Jan 2025).

6. Topological and Bulk–Boundary Effects

Nonlinear extensions preserve, enrich, or generalize familiar topological features of the Hatano-Nelson model:

• Many-body point-gap topology: Strong interaction regimes fragment the spectrum into clusters, each carrying a winding number with respect to a synthetic boundary phase twist. These many-body windings generalize the notion of single-particle point-gap topology, translating into robust signatures of the non-Hermitian skin effect and exotic spectral features under open boundary conditions (Zhang et al., 2022).
• Bulk-boundary correspondence: Nontrivial cluster windings ensure that open-chain spectra are constrained to the interiors of the corresponding closed loops in the complex plane. This correspondence extends to nonlinear/extensively many-body versions of the skin effect, in which many-body eigenstates concentrate at the edge of a Fock-space or real-space chain (Zhang et al., 2022).

7. Experimental Relevance and Theoretical Significance

Nonlinear Hatano-Nelson physics is relevant to a range of synthetic and engineered quantum platforms, including nonreciprocal photonic lattices with Kerr media, cold atom arrays with tunable dissipation or density-dependent hoppings, and electronic circuits with gain/loss asymmetry. The various qualitative phenomena—nonlinear skin modes, growth blockade, many-body spectral clustering, and interaction-driven quantum phase transitions—provide new benchmarks for the study of non-Hermitian many-body theory, $\mathcal{PT}$-transitions in extended lattices, and the interplay of nonlinearity with nonreciprocity (Lu et al., 2023, Manda et al., 2023, Longhi, 2 Jan 2025, Zhang et al., 2022).