--- title: 'Hatano-Nelson Chain: A Non-Hermitian Lattice Model' url: https://www.emergentmind.com/topics/hatano-nelson-chain type: topic --- # Hatano-Nelson Chain: A Non-Hermitian Lattice Model The Hatano-Nelson chain is a paradigmatic one-dimensional non-Hermitian lattice model with asymmetric nearest-neighbor hopping. In its standard form, nonreciprocity is introduced either through unequal left- and right-hopping amplitudes or, equivalently, through an imaginary vector potential, and this produces complex spectra, point-gap topology, exceptional points, and the non-Hermitian skin effect. Across recent work, the chain has also become a reference setting for disorder-induced spectral transitions, interaction-driven many-body phenomena, nonlinear skin modes, frequency-dependent non-Markovian transport, and experimental realizations in photonics and electric circuits [2502.10494] [2201.12653] [2603.24642]. ## 1. Canonical formulation A common interacting form of the Hatano-Nelson Hamiltonian is \[ \hat{H}=\sum_{\ell}[(t+\gamma)\hat{c}_{\ell}^{\dagger}\hat{c}_{\ell+1}+(t-\gamma)\hat{c}_{\ell+1}^{\dagger}\hat{c}_{\ell}+U\hat{n}_{\ell}\hat{n}_{\ell+1}], \] where \(t\) is reciprocal hopping, \(\gamma\) is the nonreciprocal hopping parameter, and \(U>0\) is a repulsive nearest-neighbor interaction for spinless fermions [2201.12653]. An equivalent open-chain parameterization used in many-body studies is \[ H = \sum_{n=1}^{N-1} \frac{J}{2} e^{ah} c_n^\dagger c_{n+1} + \frac{J}{2} e^{-ah} c_{n+1}^\dagger c_n + U c_n^\dagger c_n c_{n+1}^\dagger c_{n+1}, \] with \(h\) the imaginary vector potential [2208.12017]. In the non-interacting nearest-neighbor chain, boundary conditions immediately alter the spectrum. For the formulation with \(t_r=1\) and \(t_l=\alpha^2<1\), periodic boundary conditions give \[ \sigma(H_0)= e^{ik} + \alpha^2 e^{-ik}, \] which is an ellipse in the complex plane, while open boundary conditions give \[ \sigma(H_0)= 2\alpha \cos k, \] a real line segment between \(-2\alpha\) and \(2\alpha\) [2502.10494]. In the clean unidirectional limit, the periodic spectrum is a circle, \[ E(k)= t_1 - t_2 e^{ik}, \] and the eigenstates are extended [2601.07236]. These elementary spectral facts organize most subsequent generalizations. The Hatano-Nelson chain is therefore used as a prototypical one-dimensional non-Hermitian model exhibiting the non-Hermitian skin effect and complex energy-band topology, and as the fundamental non-Hermitian building block of the Su-Schrieffer-Heeger model [2603.24642] [2601.07236]. ## 2. Boundary sensitivity, point-gap topology, and skin localization The defining feature of the Hatano-Nelson chain is extreme sensitivity to boundary conditions. Under open boundary conditions, bulk eigenstates in the non-interacting limit become exponentially localized at a boundary, producing the non-Hermitian skin effect; under periodic boundary conditions, the same system instead exhibits complex spectral loops [2208.12017]. In generalized settings, this boundary sensitivity is encoded by spectral winding numbers such as \[ \nu(E_0) = \frac{1}{2\pi i}\int_0^{2\pi} \frac{d}{d\Phi}\ln\det[H(\Phi)-E_0]\,d\Phi, \] or equivalent momentum-space expressions used to diagnose point-gap topology [2601.07236]. In the unidirectional chain with binary diagonal disorder, the complex spectrum forms a single loop for weak disorder, bifurcates at a critical threshold, and becomes two separated loops in the strong-disorder regime. Correspondingly, the winding number changes from \(\nu=1\) in the weak-disorder phase, through \(\nu=1/2\) at criticality, to \(\nu=0\) in the strong-disorder limit [2601.07236]. The same analysis shows that most states are exponentially localized, but at weak and critical disorder two special states remain completely delocalized, with diverging localization length. The divergence is directly correlated with non-trivial spectral winding [2601.07236]. Strictly open boundaries can eliminate the loop structure entirely in this disordered unidirectional setting. For \(\alpha=0\), the spectrum collapses into two flat bands at \(E=\pm h\), the winding number loses meaning, and the delocalized states disappear; for any \(\alpha>0\), the thermodynamic spectrum and localization length recover the periodic-boundary result [2601.07236]. This establishes that spectral topology in the Hatano-Nelson chain is not merely a bulk property in the Hermitian sense, but is inseparable from the boundary condition used to define the spectrum. A frequent oversimplification is that the skin effect must always be unidirectional. In extensions with non-reciprocal next-nearest-neighbor hopping and structured onsite potentials, the open-boundary eigenstates can instead display a bidirectional non-Hermitian skin effect, localizing on both edges, and with a periodic onsite potential the direction of skin accumulation can reverse completely as the potential strength is varied [2403.05382]. ## 3. Disorder, quasiperiodicity, impurities, and structured potentials The Hatano-Nelson chain supports several distinct routes to spectral reconstruction. For strictly ergodic potentials, the spectrum admits an explicit Lyapunov-exponent characterization, \[ \Sigma(g)=\mathcal{E}_0 \cup (\Sigma(0)\cap \mathcal{E}_+), \] with \(\mathcal{E}_0=\{E:L(E)=g\}\) and \(\mathcal{E}_+=\{E:L(E)>g\}\) [2311.09899]. This yields a sharp real-complex transition: for \(g\le g_{cr}\), the spectrum remains real; for \(g_{cr}g_{er}\), it is purely complex [2311.09899]. If the Lyapunov exponent is continuous, the infinite-chain spectrum on \(\ell^2(\mathbb{Z})\) is approximated by finite-interval truncations with periodic boundary conditions, a behavior explicitly contrasted with the random-potential case [2311.09899]. Adding a periodic modulation to the onsite energies introduces a second insulating mechanism besides disorder. In the period-2 model, \[ \mathcal{H} = \sum_{x=1}^L \left[-\frac{t}{2}\left( e^h c^\dagger_{x+1} c_x + e^{-h} c^\dagger_{x} c_{x+1} \right) + \mu_x n_x + V \cos(\pi x) n_x \right], \] the periodic term opens a gap in the clean limit when \(V>V_c^{\rm o}=t\sinh(h)\), with \[ W_{\rm G}=2\sqrt{V^2-t^2\sinh^2(h)}, \] while disorder promotes localization and can even close a previously opened gap before full localization sets in [1009.1946]. The resulting spectral topologies include \(E\), \(LEL\), \(EGE\), \(LELGLEL\), \(LELEL\), \(L\), and \(LGL\) [1009.1946]. A single long-range impurity coupling is not a perturbatively small deformation in this model. For \[ H = H_0 + t_{pq} c_p^\dagger c_q, \] the thermodynamic spectrum becomes \[ \lim_{L\rightarrow\infty} \sigma(H) = \sigma(H_0) \cup \left\{ \lambda \in \mathbb{C} \setminus \sigma(H_0) : d_{qp}(\lambda) = -\frac{1}{t_{pq}} \right\}, \] and the additional solutions generate tentacle-like spectral wings emerging from the Bloch ellipse or the open-boundary non-Bloch segment [2502.10494]. The number of impurity-induced bound states is \(|p-q|\), and their localization length scales with the impurity-coupling distance rather than with system size [2502.10494]. This makes long-range impurity coupling a nonperturbative control parameter for both spectrum and wavefunction structure. ## 4. Interactions and many-body Hatano-Nelson physics With repulsive nearest-neighbor interactions at half filling, the interacting Hatano-Nelson chain displays two \(\mathcal{PT}\) transitions as the interaction strength increases. The first transition occurs at an exceptional point between the first and second excited states in finite size and coincides with a first-order symmetry-breaking transition from a gapless phase into a gapped charge-density-wave regime; the persistent current abruptly vanishes at this point. A second transition at stronger interaction collapses all many-body eigenvalues onto the real axis, but its critical scale grows with system size and diverges in the thermodynamic limit, making it a finite-size effect [2201.12653]. Away from half filling and at strong interaction, the many-body spectrum stratifies into clusters centered at interaction energies such as \(\varepsilon_s=(N-s)U\). Under twisted boundary conditions, each cluster can exhibit a point gap with winding number \[ \nu_s=\frac{1}{2\pi i}\int_0^{2\pi} d\phi \sum_j \partial_\phi \log\left[E_j(\phi)-\varepsilon_s\right], \] with \(\nu_1=\mathrm{sgn}(\gamma)N\) for the first cluster [2201.12653]. These point-gap windings indicate a many-body skin effect under open boundaries. At half filling and strong \(U\), by contrast, the spectrum collapses to open lines, the winding becomes ill-defined, and the many-body wave functions delocalize [2201.12653]. Open-chain many-body observables do not simply mirror single-particle skin localization. In the interacting model with open boundaries, the ground-state density profile becomes only slightly tilted relative to the average filling, Friedel oscillations exhibit a beating pattern, and the full counting statistics of particle number over any finite interval is Gaussian. The mean scales with the imaginary vector potential, while the variance is symmetric with respect to the chain center and independent of \(h\) [2208.12017]. This directly supports the statement that many-body effects can significantly alter and conceal the single-particle properties and the skin effect in non-Hermitian systems [2208.12017]. Nonequilibrium many-body dynamics retains this tension between non-Hermitian driving and collective low-energy structure. In the Luttinger-liquid regime, switching the imaginary vector potential on or off produces spatio-temporal Friedel oscillations and ballistic light cones propagating from the open ends, with local currents of equal magnitude for both quench protocols; the long-wavelength continuity equation remains satisfied even under non-unitary evolution [2304.09688]. For a finite-time linear ramp of the imaginary vector potential, the excess energy becomes complex-valued although the instantaneous Hamiltonian retains the same real spectrum throughout; the approach to adiabaticity is slow, with a \(\tau^{-1}\) envelope and oscillation period \(2L/v\), while ramp durations commensurate with that period realize a shortcut to adiabaticity without auxiliary controls [2408.07122]. ## 5. Generalized Hatano-Nelson chains Coupling two quasiperiodic Hatano-Nelson chains changes the character of the delocalization-localization transition. In the cross-coupled model, two critical potentials \(V_{c1}