Double-Chain Hatano-Nelson Model

• The double-chain Hatano-Nelson model is a two-leg non-Hermitian lattice featuring asymmetric intrachain hopping and interchain coupling via matrix‐valued links.
• Methodologies include formulating nonreciprocal Hubbard ladders and non-Abelian two-component chains, revealing spectral transitions, exceptional points, and point-gap topologies.
• Key implications are the emergence of bipolar skin effects and interaction-induced doublon states, which inform insights into non-Hermitian boundary and bulk behaviors.

Searching arXiv for 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 (Huang et al., 16 Apr 2026), 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 (Chen et al., 25 Mar 2026). 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 (Huang et al., 16 Apr 2026). 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 $U$ (Huang et al., 16 Apr 2026). 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 (Chen et al., 25 Mar 2026).

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 $H^\dagger$ (Ghosh et al., 12 Jan 2026). 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 $U(2)$0, Hermitian rung hopping $U(2)$1, onsite $U(2)$2	$U(2)$3 symmetry, real-spectrum threshold, doublon winding, opposite-edge skinning
Non-Abelian two-component chain	Two-component sites with nonreciprocal $U(2)$4 link matrices	Hopf-link braiding, bipolar skin effect, Abelian/non-Abelian distinction
SSH-type doubled construction	Off-diagonal block form built from $U(2)$5 and $U(2)$6	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

$U(2)$7

with

$U(2)$8

and

$U(2)$9

The asymmetry is chosen to be opposite on the two chains,

$\alpha=A,B$0

so the ladder is not merely a duplication of the single-chain Hatano-Nelson model (Huang et al., 16 Apr 2026).

In the noninteracting limit $\alpha=A,B$1, the Bloch Hamiltonian in each spin sector is

$\alpha=A,B$2

with

$\alpha=A,B$3

Using $\alpha=A,B$4 and $\alpha=A,B$5,

$\alpha=A,B$6

and the two bands are

$\alpha=A,B$7

This ladder is genuinely $\alpha=A,B$8-symmetric: parity is defined as spatial inversion plus leg exchange, so eigenvalues are either real or in complex-conjugate pairs (Huang et al., 16 Apr 2026).

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 $\alpha=A,B$9. The transformation

$V_0$0

symmetrizes intraleg hoppings but converts the rung term into

$V_0$1

so the ladder is not mapped to a simple uniform Hermitian model when $V_0$2 (Huang et al., 16 Apr 2026).

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,

$V_0$3

with real-space Hamiltonian

$V_0$4

and Bloch Hamiltonian

$V_0$5

Its eigenvalues are

$V_0$6

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 (Chen et al., 25 Mar 2026).

For the explicit choice

$V_0$7

one has

$V_0$8

In ladder language, $V_0$9 produces opposite-sign nearest-neighbor hopping on the two legs, $U$0 produces interchain hopping between neighboring cells, and $U$1 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 (Chen et al., 25 Mar 2026).

The paper describes these bond matrices as a nonreciprocal $U$2 gauge field. The non-Abelian condition is

$U$3

whereas the Abelian condition is

$U$4

For Pauli matrices,

$U$5

so non-Abelianity is equivalent to $U$6 and $U$7 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 (Chen et al., 25 Mar 2026).

4. Spectrum, topology, exceptional points, and skin accumulation

For the two-chain Hubbard ladder, exceptional points occur when

$U$8

This yields three noninteracting regimes: if $U$9, the spectrum is partly complex; if $|n,\uparrow\rangle$0, exceptional points occur at $|n,\uparrow\rangle$1; and if $|n,\uparrow\rangle$2, the full single-particle spectrum is entirely real. The threshold

$|n,\uparrow\rangle$3

is the noninteracting criterion for a purely real spectrum, and the paper interprets this as a $|n,\uparrow\rangle$4-broken to $|n,\uparrow\rangle$5-unbroken transition (Huang et al., 16 Apr 2026).

For the non-Abelian formulation with

$|n,\uparrow\rangle$6

the eigenvalues reduce to

$|n,\uparrow\rangle$7

The complex-energy braiding is characterized by

$|n,\uparrow\rangle$8

For this traceless two-band model, $|n,\uparrow\rangle$9, and the phase diagram in the $|n,\downarrow\rangle$0-$|n,\downarrow\rangle$1 plane contains

$|n,\downarrow\rangle$2

The key topological result is Hopf-link-shaped complex energy braiding: the two trajectories $|n,\downarrow\rangle$3 and $|n,\downarrow\rangle$4, viewed in $|n,\downarrow\rangle$5 space, are linked once like the standard Hopf link. The paper gives the examples

$|n,\downarrow\rangle$6

while $|n,\downarrow\rangle$7 corresponds to an unlink (Chen et al., 25 Mar 2026).

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

$|n,\downarrow\rangle$8

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

$|n,\downarrow\rangle$9

the PBC spectrum contains loops with opposite orientations and gives

$$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$0

The localization diagnostic is

$$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$1

with $$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$2 for left localization, $$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$3 for right localization, and $$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$4 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 (Chen et al., 25 Mar 2026).

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 $$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$5, the cylinder geometry reduces exactly to a 1D two-component chain,

$$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$6

with

$$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$7

The transition condition is

$$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$8

and the model supports $$H_{\mathrm{SSH}}= \begin{pmatrix} 0 & H^\dagger\ H & 0 \end{pmatrix},$$9 Hopf-link braiding and a GBZ-based polarization parameter $H$0 that distinguishes left-, right-, and bipolar skin modes (Zhao et al., 10 Jun 2025). 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

$H$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

$H$2

The paper finds that finite $H$3 generally increases imaginary parts of eigenvalues, the noninteracting threshold $H$4 no longer suffices, and the high-energy doublon-like branch can show non-monotonic dependence on $H$5 (Huang et al., 16 Apr 2026).

The reported phase diagrams are organized by

$H$6

for $H$7. At fixed weak interaction $H$8, the real-spectrum region is reduced relative to the noninteracting reference line $H$9, and for weakly interacting ladders the threshold tends toward

$H^\dagger$0

in the small-$H^\dagger$1 regime. At fixed $H^\dagger$2, the threshold jumps upward toward

$H^\dagger$3

for small but finite $H^\dagger$4, and for strong $H^\dagger$5 the entire spectrum becomes real only when

$H^\dagger$6

The scale $H^\dagger$7 is traced to the separation of the three Hermitian-limit two-particle continua

$H^\dagger$8

$H^\dagger$9

$U(2)$00

each of width $U(2)$01, which separate only if $U(2)$02 (Huang et al., 16 Apr 2026).

Point-gap topology in the interacting ladder is probed by opposite flux insertion,

$U(2)$03

and the many-body winding number

$U(2)$04

For the detached doublon branch around $U(2)$05, the paper finds

$U(2)$06

A spin-resolved definition gives

$U(2)$07

At larger $U(2)$08, the isolated doublon point gap is lost and this winding becomes trivial (Huang et al., 16 Apr 2026).

Under OBC, doublon-branch states become skin modes with opposite accumulation on the two legs,

$U(2)$09

The localization length $U(2)$10 decreases as $U(2)$11 increases and also decreases as $U(2)$12 increases. The same work further shows that a Lindblad no-jump effective Hamiltonian,

$U(2)$13

realizes the ladder up to a uniform imaginary shift, and that full Lindblad dynamics exhibits transient edge accumulation before eventual decay to the vacuum (Huang et al., 16 Apr 2026).

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

$U(2)$14

with

$U(2)$15

At the operating frequency

$U(2)$16

one has $U(2)$17, so

$U(2)$18

Up to the uniform shift $U(2)$19, this is topologically equivalent to

$U(2)$20

under

$U(2)$21

The reported observables are Hopf-link-shaped admittance spectra and bipolar skin admittance modes (Chen et al., 25 Mar 2026).

The broader context is shaped by several adjacent results. A disordered unidirectional Hatano-Nelson block with binary onsite disorder,

$U(2)$22

has exact eigenvalues

$U(2)$23

and a winding transition

$U(2)$24

In the weak and critical regimes, it hosts two completely delocalized states at $U(2)$25 with

$U(2)$26

Because this chain is used as the irreducible block of the doubled SSH Hamiltonian, it provides a block-level picture of how spectral winding, disorder, and boundary conditions can enter doubled non-Hermitian constructions (Ghosh et al., 12 Jan 2026).

A different extension, the generalized long-range Hatano-Nelson model

$U(2)$27

does not itself define a double-chain ladder, but its block-cyclic decomposition and generalized chiral symmetry provide a reusable framework for exceptional points of order not scaling with system size (Gohsrich et al., 2024). A plausible implication is that similar block-structured reasoning can be adapted to ladders with internal leg space. Likewise, the metric-operator construction for Hatano-Nelson variants,

$U(2)$28

does not directly study a double-chain Hatano-Nelson model, but it clarifies how boundary conditions, pseudo-Hermiticity, and biorthogonal metrics reorganize non-Hermitian chain families (Bagarello et al., 2021).

Taken together, these works establish the double-chain Hatano-Nelson model as a family of coupled-chain non-Hermitian lattices in which interleg structure is not an auxiliary perturbation but a primary topological ingredient. In the commuting or Abelian limit, the system reduces to two scalar Hatano-Nelson channels in an appropriate basis; in the genuinely non-Abelian or interacting ladder regimes, it supports spectral phases, braiding classes, and skin-effect patterns that are inaccessible to two independent chains (Chen et al., 25 Mar 2026).