--- title: Double-Chain Hatano-Nelson Model url: https://www.emergentmind.com/topics/double-chain-hatano-nelson-model type: topic --- # Double-Chain Hatano-Nelson Model Searching arXiv for recent papers on the Double-Chain Hatano-Nelson model and closely related non-Abelian / two-orbital generalizations. The double-chain Hatano-Nelson model is a two-leg, two-orbital, or two-component generalization of the one-dimensional Hatano-Nelson chain, in which nonreciprocal left/right hopping acts together with interchain hybridization, matrix-valued link couplings, or onsite interaction. In recent arXiv literature, two complementary formulations are especially prominent: a non-Hermitian Hubbard ladder with opposite nonreciprocal hopping on the two legs and Hermitian rung tunneling [2604.14533], and a two-component non-Abelian Hatano-Nelson chain that is naturally reinterpreted as a double-chain or two-leg ladder with nonreciprocal \(2\times 2\) \(U(2)\)-valued bond matrices [2603.24642]. Across these formulations, the central themes are spectral reality versus complexification, exceptional points, point-gap topology, non-Hermitian skin accumulation, and the distinction between Abelian and non-Abelian interchannel structure. ## 1. Terminology and representative realizations The recent literature uses “two-chain,” “two-leg ladder,” and “two-orbital/two-band” language for coupled one-dimensional Hatano-Nelson systems [2604.14533]. In one formulation, the two chains are literal ladder legs labeled \(\alpha=A,B\), each carrying asymmetric intrachain hopping and connected by a Hermitian interchain hopping \(V_0\) together with onsite Hubbard interaction \(U\) [2604.14533]. In another, each unit cell carries two pseudospin components, and the model is written as a single chain with matrix-valued hoppings; under the identification of \(|n,\uparrow\rangle\) and \(|n,\downarrow\rangle\) as two legs of a ladder, it becomes a double-chain Hatano-Nelson generalization [2603.24642]. A related but conceptually distinct strand treats the unidirectional Hatano-Nelson chain as the “fundamental non-Hermitian building block” of a doubled SSH Hamiltonian, \[ H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\\ H & 0 \end{pmatrix}, \] so that doubled-chain structure enters through the pairing of a non-Hermitian block \(H\) with its adjoint \(H^\dagger\) [2601.07236]. This does not by itself define a two-leg Hatano-Nelson ladder, but it clarifies how doubled non-Hermitian constructions inherit block-level spectral winding and boundary sensitivity. | Realization | Basic structure | Characteristic phenomena | |---|---|---| | Two-chain Hubbard ladder | Opposite leg asymmetry \( \delta_A=-\delta_B\equiv\delta \), Hermitian rung hopping \(V_0\), onsite \(U\) | \(\mathcal{PT}\) symmetry, real-spectrum threshold, doublon winding, opposite-edge skinning | | Non-Abelian two-component chain | Two-component sites with nonreciprocal \(2\times2\) link matrices | Hopf-link braiding, bipolar skin effect, Abelian/non-Abelian distinction | | SSH-type doubled construction | Off-diagonal block form built from \(H\) and \(H^\dagger\) | Disorder-driven winding transition in the block, strong boundary sensitivity | ## 2. Canonical two-leg ladder Hamiltonian The two-chain Hatano-Nelson-Hubbard ladder is defined by \[ \hat{\mathcal H}=\hat{\mathcal K}+\hat{\mathcal U}, \] with \[ \hat{\mathcal K} = -\sum_{j,\alpha,\sigma} \Big[ (t+\delta_\alpha)\, \hat c_{j+1,\alpha\sigma}^\dagger \hat c_{j,\alpha\sigma} + (t-\delta_\alpha)\, \hat c_{j,\alpha\sigma}^\dagger \hat c_{j+1,\alpha\sigma} \Big] -V_0\sum_{j,\sigma} \left( \hat c^\dagger_{j,A\sigma}\hat c_{j,B\sigma} + \hat c^\dagger_{j,B\sigma}\hat c_{j,A\sigma} \right), \] and \[ \hat{\mathcal U} = U\sum_{j,\alpha} \hat n_{j,\alpha\uparrow}\hat n_{j,\alpha\downarrow},\qquad \hat n_{j,\alpha\sigma}=\hat c^\dagger_{j,\alpha\sigma}\hat c_{j,\alpha\sigma}. \] The asymmetry is chosen to be opposite on the two chains, \[ \delta_A=-\delta_B\equiv \delta, \] so the ladder is not merely a duplication of the single-chain Hatano-Nelson model [2604.14533]. In the noninteracting limit \(U=0\), the Bloch Hamiltonian in each spin sector is \[ H_k= \begin{pmatrix} \varepsilon_A(k) & -V_0 \\ -V_0 & \varepsilon_B(k) \end{pmatrix}, \] with \[ \varepsilon_\alpha(k) = -(t+\delta_\alpha)e^{ik}-(t-\delta_\alpha)e^{-ik}. \] Using \(\delta_A=+\delta\) and \(\delta_B=-\delta\), \[ \varepsilon_A(k)=-2t\cos k - i\,2\delta\sin k,\qquad \varepsilon_B(k)=-2t\cos k + i\,2\delta\sin k, \] and the two bands are \[ E_\pm(k)= -2t\cos k \pm \sqrt{V_0^2-4\delta^2\sin^2 k}. \] This ladder is genuinely \(\mathcal{PT}\)-symmetric: parity is defined as spatial inversion plus leg exchange, so eigenvalues are either real or in complex-conjugate pairs [2604.14533]. A recurrent misconception is that open boundaries should always remove Hatano-Nelson asymmetry by similarity transformation. That is true for a single chain, but not generically for the ladder when \(V_0\neq 0\). The transformation \[ \hat c_{j,A\sigma}=e^{gj}\hat d_{j,A\sigma},\qquad \hat c_{j,B\sigma}=e^{-gj}\hat d_{j,B\sigma}, \] symmetrizes intraleg hoppings but converts the rung term into \[ -V_0\sum_{j,\sigma} \Big( e^{-2gj}\hat d^\dagger_{j,A\sigma}\hat d_{j,B\sigma} +e^{2gj}\hat d^\dagger_{j,B\sigma}\hat d_{j,A\sigma} \Big), \] so the ladder is not mapped to a simple uniform Hermitian model when \(V_0\neq 0\) [2604.14533]. ## 3. Non-Abelian double-chain formulation A more general double-chain Hatano-Nelson system is obtained by promoting each site to a two-component spinor, \[ a_n^\dagger = [a_{n,\uparrow}^\dagger,\; a_{n,\downarrow}^\dagger],\qquad a_n = [a_{n,\uparrow},\; a_{n,\downarrow}]^T, \] with real-space Hamiltonian \[ H=\sum_{n=1}^{N-1} a_n^\dagger\, t_0\, \mathbf d_R\!\cdot\!\boldsymbol{\sigma}\, a_n +\sum_{n=1}^{N-1}\left( a_n^\dagger\, t_L\, \mathbf d_L\!\cdot\!\boldsymbol{\sigma}\, a_{n+1} + a_{n+1}^\dagger\, t_R\, \mathbf d_R\!\cdot\!\boldsymbol{\sigma}\, a_n \right), \] and Bloch Hamiltonian \[ H(k)= t_0\,\mathbf d_R\!\cdot\!\boldsymbol{\sigma} + t_L e^{ik}\,\mathbf d_L\!\cdot\!\boldsymbol{\sigma} + t_R e^{-ik}\,\mathbf d_R\!\cdot\!\boldsymbol{\sigma}. \] Its eigenvalues are \[ E_\pm(k)=\pm\sqrt{ \bigl(t_0+t_R e^{-ik}\bigr)^2 +t_L^2 e^{2ik} +2\bigl(t_0+t_R e^{-ik}\bigr)t_L e^{ik}\,\mathbf d_L\!\cdot\!\mathbf d_R }. \] Because each site carries two pseudospin components, this is naturally read as a two-leg ladder: diagonal matrix elements give intrachain hopping, off-diagonal matrix elements give interchain hopping between neighboring cells, and an off-diagonal onsite term gives intracell rung coupling [2603.24642]. For the explicit choice \[ \mathbf d_L=(0,0,1),\qquad \mathbf d_R=(1,0,0), \] one has \[ H(k)= t_0 \sigma_x + t_L e^{ik}\sigma_z + t_R e^{-ik}\sigma_x. \] In ladder language, \(t_L\sigma_z\) produces opposite-sign nearest-neighbor hopping on the two legs, \(t_R\sigma_x\) produces interchain hopping between neighboring cells, and \(t_0\sigma_x\) is an intracell rung coupling. This is therefore not merely “two chains plus a separate rung”; the nearest-neighbor bond itself carries a matrix structure that rotates or mixes the two legs [2603.24642]. The paper describes these bond matrices as a nonreciprocal \(U(2)\) gauge field. The non-Abelian condition is \[ [\mathbf d_L\!\cdot\!\boldsymbol{\sigma},\,\mathbf d_R\!\cdot\!\boldsymbol{\sigma}] \neq 0, \] whereas the Abelian condition is \[ [\mathbf d_L\!\cdot\!\boldsymbol{\sigma},\,\mathbf d_R\!\cdot\!\boldsymbol{\sigma}] = 0. \] For Pauli matrices, \[ [\mathbf d_L\!\cdot\!\boldsymbol{\sigma},\,\mathbf d_R\!\cdot\!\boldsymbol{\sigma}] =2i(\mathbf d_L\times \mathbf d_R)\cdot\boldsymbol{\sigma}, \] so non-Abelianity is equivalent to \(\mathbf d_L\) and \(\mathbf d_R\) not being parallel. In the Abelian limit the two matrices can be simultaneously diagonalized, and the Hamiltonian decomposes into two uncoupled scalar Hatano-Nelson-type channels [2603.24642]. ## 4. Spectrum, topology, exceptional points, and skin accumulation For the two-chain Hubbard ladder, exceptional points occur when \[ V_0^2-4\delta^2\sin^2 k = 0, \qquad \sin k_{\rm EP} = \pm \frac{V_0}{2\delta}. \] This yields three noninteracting regimes: if \(V_0<2\delta\), the spectrum is partly complex; if \(V_0=2\delta\), exceptional points occur at \(k=\pm \pi/2\); and if \(V_0>2\delta\), the full single-particle spectrum is entirely real. The threshold \[ V_0\ge 2\delta \] is the noninteracting criterion for a purely real spectrum, and the paper interprets this as a \(\mathcal{PT}\)-broken to \(\mathcal{PT}\)-unbroken transition [2604.14533]. For the non-Abelian formulation with \[ H(k)= t_L e^{ik}\sigma_z + \bigl(t_0+t_R e^{-ik}\bigr)\sigma_x, \] the eigenvalues reduce to \[ E_\pm(k)=\pm \sqrt{t_L^2 e^{2ik}+\bigl(t_0+t_R e^{-ik}\bigr)^2}. \] The complex-energy braiding is characterized by \[ \nu=\frac{1}{2\pi i}\int_{-\pi}^{\pi} dk\, \frac{d}{dk} \ln \det\!\left[H(k)-\frac{1}{2}\operatorname{Tr}H(k)\right]. \] For this traceless two-band model, \(\operatorname{Tr}H(k)=0\), and the phase diagram in the \(t_L\)-\(t_R\) plane contains \[ \nu=0,\quad \nu=\pm 2. \] The key topological result is Hopf-link-shaped complex energy braiding: the two trajectories \(E_+(k)\) and \(E_-(k)\), viewed in \((\Re E,\Im E,k)\) space, are linked once like the standard Hopf link. The paper gives the examples \[ P_1:(t_L,t_R)=(1,3)\ \Rightarrow\ \nu=-2,\qquad P_2:(t_L,t_R)=(3,1)\ \Rightarrow\ \nu=2, \] while \(\nu=0\) corresponds to an unlink [2603.24642]. The skin physics is correspondingly richer than in the single-chain model. In an ordinary Hatano-Nelson chain, all OBC bulk modes pile up at the same boundary. In the non-Abelian two-component model, the spectral winding number \[ w(E_0)= -\frac{i}{2\pi}\int_{-\pi}^{\pi} dk\, \partial_k \ln \det[H(k)-E_0] \] can take both orientations in the same spectrum, and the paper reports a bipolar skin effect: under OBC, some eigenstates localize to the left and others to the right. At \[ P_3=(t_L,t_R)=(1.2,0.9), \] the PBC spectrum contains loops with opposite orientations and gives \[ w=\pm 1. \] The localization diagnostic is \[ \gamma=\frac{D_L-D_R}{D_L+D_R}, \qquad D_{L,R}=\sum_x\in L,R |w(x)|^2, \] with \(\gamma\approx 1\) for left localization, \(\gamma\approx -1\) for right localization, and \(\gamma\approx 0\) for coexistence of both. A central statement is that the bipolar skin effect cannot emerge under the Abelian condition and is only possible under the non-Abelian condition [2603.24642]. A closely related two-component reduction also appears in a two-dimensional non-Hermitian Hatano-Nelson model with SU(2) non-Abelian gauge. For fixed \(k_y\), the cylinder geometry reduces exactly to a 1D two-component chain, \[ H(k_y)=\sum_x \Big[ c_x^\dagger (2t\cos k_y\,\sigma_0)c_x +c_x^\dagger T_L c_{x+1} +c_{x+1}^\dagger T_R c_x \Big], \] with \[ T_L=J_L(\cos\alpha\,\sigma_0+i\sin\alpha\,\sigma_y),\qquad T_R=J_R(\cos\beta\,\sigma_0+i\sin\beta\,\sigma_x). \] The transition condition is \[ J_R^2\sin^2\beta=J_L^2\sin^2\alpha, \] and the model supports \(\nu=\pm2\) Hopf-link braiding and a GBZ-based polarization parameter \(I_p\) that distinguishes left-, right-, and bipolar skin modes [2506.08714]. This suggests that directional matrix hopping is a generic route to double-chain Hatano-Nelson behavior beyond scalar rung-coupled ladders. ## 5. Interactions, doublons, and many-body winding The interacting two-chain ladder is studied mainly in the dilute sector \[ N_\uparrow=N_\downarrow=1, \] where doublons can form. A doublon is a state in which the up and down particles occupy the same site on the same leg, producing a detached branch near \[ \operatorname{Re}E \sim U. \] The paper finds that finite \(U\) generally increases imaginary parts of eigenvalues, the noninteracting threshold \(V_0\ge 2\delta\) no longer suffices, and the high-energy doublon-like branch can show non-monotonic dependence on \(V_0\) [2604.14533]. The reported phase diagrams are organized by \[ \max_n \big(\operatorname{Im}E_n\big) \] for \(L=100\). At fixed weak interaction \(U=1\), the real-spectrum region is reduced relative to the noninteracting reference line \(V_0=2\delta\), and for weakly interacting ladders the threshold tends toward \[ V_0 \simeq 4t \] in the small-\(\delta\) regime. At fixed \(\delta=0.5\), the threshold jumps upward toward \[ V_0 \gtrsim 4t \] for small but finite \(U\), and for strong \(U\) the entire spectrum becomes real only when \[ V_0 \gg U. \] The scale \(V_0>4t\) is traced to the separation of the three Hermitian-limit two-particle continua \[ E_{--}(K,q) = -4t\cos\!\left(\frac{K}{2}\right)\cos q -2V_0, \] \[ E_{+-}(K,q) = -4t\cos\!\left(\frac{K}{2}\right)\cos q, \] \[ E_{++}(K,q) = -4t\cos\!\left(\frac{K}{2}\right)\cos q +2V_0, \] each of width \(8t\), which separate only if \(V_0>4t\) [2604.14533]. Point-gap topology in the interacting ladder is probed by opposite flux insertion, \[ \phi_A=-\phi_B\equiv \phi, \] and the many-body winding number \[ W(E)=\oint_{0}^{2\pi}\frac{d\phi}{2\pi i}\, \frac{d}{d\phi}\ln\det[\hat{\mathcal H}(\phi)-E]. \] For the detached doublon branch around \(E\simeq U\), the paper finds \[ W(E=U)=4. \] A spin-resolved definition gives \[ W_\uparrow(E)=W_\downarrow(E)=2,\qquad W(E)=W_\uparrow+W_\downarrow=4. \] At larger \(V_0\), the isolated doublon point gap is lost and this winding becomes trivial [2604.14533]. Under OBC, doublon-branch states become skin modes with opposite accumulation on the two legs, \[ \langle \hat n_{i_x,0}\rangle \propto e^{-i_x/\xi},\qquad \langle \hat n_{i_x,1}\rangle \propto e^{-(L-1-i_x)/\xi}. \] The localization length \(\xi\) decreases as \(\delta\) increases and also decreases as \(U\) increases. The same work further shows that a Lindblad no-jump effective Hamiltonian, \[ \hat H_{\rm eff} = \hat{\mathcal H}-2i\delta\,\hat N, \] realizes the ladder up to a uniform imaginary shift, and that full Lindblad dynamics exhibits transient edge accumulation before eventual decay to the vacuum [2604.14533]. ## 6. Experimental realization and broader theoretical context The non-Abelian double-chain formulation has been implemented in a topolectrical circuit in which each lattice site contains two circuit nodes representing the two pseudospin components or two chains. The Bloch admittance matrix is \[ J_0 = i\omega\Big[ m_0 \sigma_0 + m_1(\sigma_0-\sigma_z)e^{ik} + C_0 \sigma_x + C_1 e^{ik}\sigma_z + C_2 e^{-ik}\sigma_x \Big], \] with \[ m_0=-C_2-2C_0+\frac{1}{\omega^2 L_0}-\frac{1}{i\omega R_0}-C_1+\frac{1}{\omega^2 L_1}, \qquad m_1=\frac{1}{2}\left(C_1-\frac{1}{\omega^2 L_1}\right). \] At the operating frequency \[ \omega=\omega_0=\frac{1}{\sqrt{L_1 C_1}}, \] one has \(m_1=0\), so \[ J_0 = i\omega_0 \left[m_0\sigma_0 + C_0\sigma_x + C_1 e^{ik}\sigma_z + C_2 e^{-ik}\sigma_x\right]. \] Up to the uniform shift \(m_0\sigma_0\), this is topologically equivalent to \[ H(k)= t_0\sigma_x + t_L e^{ik}\sigma_z + t_R e^{-ik}\sigma_x, \] under \[ t_0\leftrightarrow C_0,\qquad t_L\leftrightarrow C_1,\qquad t_R\leftrightarrow C_2. \] The reported observables are Hopf-link-shaped admittance spectra and bipolar skin admittance modes [2603.24642]. The broader context is shaped by several adjacent results. A disordered unidirectional Hatano-Nelson block with binary onsite disorder, \[ H = \sum_{n=1}^{N} \left(t_{1n}\, a_n^\dagger a_n - t_2\, a_n^\dagger a_{n+1} \right), \qquad t_{1n}\in\{+h,-h\}, \] has exact eigenvalues \[ E_\pm = \pm \sqrt{h^2+t_2^2 e^{iq}}, \] and a winding transition \[ \nu(0)= \begin{cases} 0, & h>t_2,\\[4pt] \frac12, & h=t_2,\\[4pt] 1, & h