Discrete Hatano–Nelson Lattice

Updated 24 December 2025

Discrete Hatano–Nelson lattice is a one-dimensional non-Hermitian tight-binding system characterized by asymmetric hopping and complex spectral topologies.
It exhibits key phenomena such as the non-Hermitian skin effect, exceptional points, and localization transitions, with demonstrations in photonic and atomic systems.
Advanced implementations using dynamic modulation and nonlinear effects enable unidirectional transport, phase gyration, and controlled disorder-induced transitions.

A discrete Hatano–Nelson lattice is a one-dimensional non-Hermitian tight-binding system characterized by asymmetric nearest-neighbor (and sometimes long-range) hopping. The resulting spatial non-reciprocity produces complex spectral topologies, the non-Hermitian skin effect, and a range of topological, dynamical, and localization phenomena fundamentally inaccessible to Hermitian lattices. While originally theoretical, the core features of Hatano–Nelson (HN) physics have been realized in integrated photonics, cold atom arrays, driven acoustic metamaterials, and other synthetic lattice platforms (Orsel et al., 14 Oct 2024, Maddi et al., 2023). This article provides a comprehensive technical account, with emphasis on the underlying Hamiltonians, spectral structures, implementation in photonics, nontrivial dynamical regimes, and topological exceptional points.

1. Lattice Hamiltonian, Asymmetry, and Non-Hermiticity

The archetypal discrete HN lattice is defined on $N$ sites, with fermionic (or bosonic) annihilation and creation operators $c_n, c_n^\dagger$ . The standard nearest-neighbor Hamiltonian is

$H = \sum_{n=1}^{N-1} \left( t_R\, c_{n+1}^\dagger c_n + t_L\, c_n^\dagger c_{n+1} \right) + \sum_{n=1}^N V_n c_n^\dagger c_n$

where $t_R$ and $t_L$ are right- and left-hopping amplitudes, and $V_n$ is a (possibly absent) on-site potential. For $t_R \ne t_L$ , Hermiticity is broken and the model supports non-reciprocal transport. In photonic molecules, asymmetric couplings are realized by time-dependent modulation of cavity resonances, leading to effective $t_{12} \neq t_{21}$ between neighboring sites (Orsel et al., 14 Oct 2024).

Under periodic boundary conditions (PBC), Bloch's theorem yields the single-particle dispersion

$E(k) = t_R e^{-ik} + t_L e^{ik}$

which traces an ellipse in the complex energy plane as $k$ traverses $[0,2\pi)$ . The degree of non-Hermiticity is set by $\alpha = (t_R - t_L)/(t_R + t_L)$ . Open boundary conditions (OBC) transform the spectrum to a real segment, while eigenstates accumulate at a boundary—this is the non-Hermitian skin effect (Maddi et al., 2023).

Generalized models introduce longer-range hopping, $H_{\ell r} = \sum_n \left( t_\ell\, c_n^\dagger c_{n+\ell} + t_{-r}\, c_n^\dagger c_{n - r} \right)$ , producing higher-order exceptional points and star-like spectral features (Gohsrich et al., 18 Mar 2024).

2. Real-Space Implementations and Modulation-Induced Couplings

Recent progress enables direct realization of a discrete Hatano–Nelson lattice in integrated photonics by dynamic, time-domain modulation of uncoupled resonators (Orsel et al., 14 Oct 2024). Consider two identical resonators, each with resonance frequency $\omega_0$ and (half-)loss $\gamma_0$ , modulated electro-optically: $\omega_j(t) = \omega_0 + \beta \cos(\omega_m t + \theta_j), \quad j=1,2$ where the phase offset $\Delta\theta$ controls the non-reciprocity. Projecting out sidebands produces the effective (zeroth subband) Hamiltonian

$H_\text{eff} = \begin{pmatrix} - i \gamma & i t_{12} \ i t_{21} & -i \gamma \end{pmatrix}$

with couplings

$t_{12,21} = \lambda \left(1 \pm \alpha\right), \quad \alpha = \frac{\lambda \beta^2}{(4 \lambda^2 + \gamma_0^2)\gamma_0}$

Critically, $\alpha$ can be tuned dynamically via the modulation amplitude $\beta$ . At $|\alpha|=1$ , one coupling vanishes, yielding an exceptional point (EP) with fully unidirectional energy exchange—the first direct photonic realization of this regime.

This platform achieves isolation/contrast ratios $A = |t_{12}/t_{21}|^2$ up to 60 dB at the EP, and beyond the EP, sign reversal of $t_{21}$ produces a $\pi$ phase shift in through-chain transport (gyration) (Orsel et al., 14 Oct 2024).

The scheme generalizes to $N$ resonators by spatiotemporal engineering of modulation phases and frequencies, enabling arbitrary real-space HN chains. The effective intersite couplings $t_{R,L}$ on each bond satisfy $t_R = \lambda(1+\alpha)$ , $t_L = \lambda(1-\alpha)$ , thus supporting the skin effect and tunable higher-order non-Hermitian phenomena throughout the chain.

3. Spectral Structure: Skin Effect, Exceptional Points, and Topological Invariants

The HN lattice under PBC exhibits a complex energy loop (ellipse if only nearest-neighbor hopping). The spectral winding number,

$w(E_0) = \frac{1}{2\pi i} \int_{0}^{2\pi} \frac{d}{dk} \ln [E(k) - E_0]$

counts the number of times $E(k)$ encircles a reference point $E_0$ and determines the presence and direction of skin accumulation under OBC. A nonzero $w$ implies that OBC eigenstates are exponentially localized at one edge; the localization length is $\xi = 1/|\ln \sqrt{t_R / t_L}|$ .

For the photonic two-ring system, the effective Hamiltonian admits exceptional points (EPs) whenever $t_{12} t_{21}=0$ ( $|\alpha|=1$ ). At the EP, eigenvalues and eigenvectors coalesce; the system supports only unidirectional transport. Beyond the EP, one of the intersite couplings turns negative, yielding $\pi$ -phase gyration—a hallmark of non-Hermitian topology in photonics (Orsel et al., 14 Oct 2024).

Generalized HN models with longer-range asymmetric hopping support EPs of arbitrary order, not scaling with system size, with their order controlled by the range parameters and chain length modulo the total range. Associated eigenstates exhibit skin localization tunable to either boundary, and the defectiveness of the EP is robust to generic variations in hopping strengths and certain kinds of disorder (Gohsrich et al., 18 Mar 2024).

4. Nonlinear Effects, Dynamical Blockade, and Modulational Instability

Nonlinear extensions to the HN lattice, particularly with onsite Kerr nonlinearity, yield rich dynamical regimes (Longhi, 2 Jan 2025, Manda et al., 2023):

Nonlinear HN equation (mean field):

$i \frac{d\psi_n}{dt} = \kappa e^h \psi_{n+1} + \kappa e^{-h} \psi_{n-1} + \chi |\psi_n|^2 \psi_n$

All exact plane wave solutions are modulationally unstable, contrasting with the conservative DNLSE where only bands of modes are unstable.
In the linear regime ( $\chi=0$ ), initial wavepackets grow unboundedly in norm due to convective (point-gap) instability.
With nonlinearity, growth is dynamically arrested as MI generates self-induced disorder, which blocks further secular amplification—the "growth blockade" (Longhi, 2 Jan 2025).
- Nonlinear skin modes (NLSMs):
For open boundaries, exact solutions interpolate continuously between linear skin modes and highly localized edge-pinned solitons.
Focusing ( $\chi>0$ ) nonlinearity enhances localization, driving NLSMs into single-site boundary solitons in the anti-continuum limit; defocusing ( $\chi<0$ ) broadens modes into extended states.
Stability regimes are analytically tractable near both linear and anti-continuum limits; intermediate regimes host oscillatory instabilities related to the rich spectral structure of the nonlinear problem (Manda et al., 2023).

Relevant photonic, cold atom, and polariton platforms already realize such nonlinear HN-type lattices (Longhi, 2 Jan 2025).

5. Disorder, Delocalization Transitions, and Bound States

The interplay of asymmetric hopping and disorder produces distinct localization-delocalization transitions and novel bound states (Samanta et al., 6 Sep 2024, Hébert et al., 2010, Wang et al., 2023, Zhao et al., 14 Feb 2025):

Localization Transition:
- For i.i.d random or (ergodic/quasiperiodic) potentials, the critical disorder for the collapse from complex to real spectrum is determined by the localization length $\xi$ : the transition occurs at $\xi = 1/g$ , with $g$ the asymmetry parameter (Hébert et al., 2010, Wang et al., 2023).
- In quasiperiodic lattices (Aubry-André type), the critical potential is $V_c = 2\sqrt{t_R t_L}$ , separating extended and localized regimes (Samanta et al., 6 Sep 2024).
- Coupled HN chains produce multiple critical points and 50/50 localized-extended mixed regimes, with mobility edges sharply delineating spectral regions (Samanta et al., 6 Sep 2024).
Bound States and Tentacle Spectra:
- Introduction of local or long-range impurity couplings ( $g_\ell$ ) generates “tentacle”-like spectral branches extending from the original Bloch or non-Bloch spectrum (Zhao et al., 14 Feb 2025). These new spectral features correspond to impurity-bound states, the number and localization length of which are set by the impurity coupling range.
- Such impurity-induced states can be engineered for targeted state manipulation in non-Hermitian platforms.

6. Topological and Many-Body Phenomena

The discrete HN lattice also exhibits robust topological invariants (point-gap winding numbers), exceptional points of arbitrary (system-size independent) order, and novel behavior in the presence of interactions and many-body effects:

Exceptional Points (EPs):
- In generalized HN models, the characteristic polynomial structure and sublattice decomposition analytically predict EPs of arbitrary finite order. The size and spatial support of coalescent Jordan chains depend on chain length and hopping ranges (Gohsrich et al., 18 Mar 2024).
- EPs yield robust defectiveness under hopping disorder and targeted on-site potentials, providing an experimentally controllable platform for higher-order non-Hermitian singularities.
Many-Body Extensions:
- Non-Hermitian interacting HN models, e.g., with fermion-fermion repulsion, exhibit finite-size and thermodynamic spectral phase transitions, $\mathcal{PT}$ -symmetry breaking, and cluster formation with well-defined point-gap topology (Zhang et al., 2022).
- Under OBC, these many-body states undergo extensive skin effects—macroscopically many eigenstates localize at a boundary, a phenomenon unavailable in conventional Hermitian systems.
Spin-Polarized and Unconventional Skin Effects:
- Spinful extensions with Abelian gauge fields enable coexistence of conventional (winding-based) and unconventional (zero-winding but boundary-localized) spin-resolved skin effects, as well as transitions to system-size-dependent, “critical” NHSE regimes under Zeeman coupling (Sanahal et al., 8 May 2025).

7. Experimental Realizations and Applications

Discrete Hatano–Nelson lattices have been realized in diverse platforms:

Integrated photonics: Dynamically modulated lithium niobate rings achieve real-space nonreciprocal couplings, giant isolation ( $>$ 60 dB), and photonic gyration (Orsel et al., 14 Oct 2024).
Acoustic waveguides: Exact transfer matrices map continuous nonreciprocal structures to HN chains, observing bulk-like spectral ellipses under periodic boundaries and pronounced exponential skin profiles under open ones (Maddi et al., 2023).
Waveguide, exciton–polariton, and ultracold atom arrays: Asymmetric coupling and on-site nonlinearities or interactions yield the full range of HN, nonlinear, and localization physics (Longhi, 2 Jan 2025).
Future directions include non-Markovian extensions with frequency-dependent effective hoppings and dissipation (Jana et al., 7 Nov 2025), quantum walks and force sensing (Anisur et al., 5 Nov 2025), and engineered impurity states (Zhao et al., 14 Feb 2025).