---
title: Nonlinear Hatano-Nelson Model Extensions
url: https://www.emergentmind.com/topics/nonlinear-extension-of-the-hatano-nelson-model
type: topic
---

# Nonlinear Hatano-Nelson Model Extensions

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 [2310.07599, 2201.12653].

- **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 [2311.09139, 2501.01226].

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 [2310.07599].

- **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 [2310.07599]. 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 [2201.12653].

## 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 [2311.09139].

- **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 [2311.09139].

- **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 [2311.09139].

- **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 [2311.09139].

## 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 [2310.07599, 2201.12653].

- **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 [2310.07599, 2201.12653].

- **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 [2310.07599].

## 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 [2501.01226].

- **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 [2501.01226].

- **Marginal stability:** The resultant, time-averaged Bogoliubov analysis reveals purely real spectra, confirming the absence of further instability after the MI-driven blockade [2501.01226].

## 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 [2201.12653].

- **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 [2201.12653].

## 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 [2310.07599, 2311.09139, 2501.01226, 2201.12653].

Source: https://www.emergentmind.com/topics/nonlinear-extension-of-the-hatano-nelson-model