2D Hatano–Nelson Models in Non-Hermitian Systems

• The two-dimensional Hatano–Nelson model is a non-Hermitian lattice generalization characterized by asymmetric hopping in planar geometries, realized through both flux and cylinder-type constructions.
• The flux-model construction uses bipartite square-lattice plaquettes with alternating nonreciprocal hoppings and Hubbard interactions to produce distinct phases such as AF metal, semimetal, and insulator with exceptional lines.
• The cylinder-type SU(2) model leverages non-Abelian gauge fields to induce complex-energy braiding and diverse skin modes, with analyses via the generalized Brillouin zone elucidating topological transitions.

Searching arXiv for papers on the two-dimensional Hatano–Nelson model and closely related extensions. arxiv_search(query="^^^^^^^^4\4 Hatano-Nelson^^^^^^^^4\4 OR ^^^^^^^^4\4 OR \4^^^^D Hatano-Nelson^^^^^^^^4\4 OR ^^^^^^^^4\4 flux model^^^^^^^^4\4 OR ^^^^^^^^4\4 Gauge Effect for ^^^^4 OR \4^^^^-D Non-Hermitian Hatano-Nelson Model in Cylinder Type^^^^^^^^4\4 max_results=^^^^4 OR \4\4^^^^, sort_by="submittedDate") Refining the arXiv search to include general Hatano–Nelson higher-dimensional extensions and related non-Hermitian skin-effect papers. arxiv_search(query="Hatano-Nelson higher-dimensional non-Hermitian skin effect two-dimensional", max_results=^^^^4 OR \4\4^^^^, sort_by="relevance") The two-dimensional Hatano–Nelson model denotes genuinely two-dimensional non-Hermitian lattice generalizations of the one-dimensional Hatano–Nelson chain, retaining asymmetric hopping as the basic non-Hermitian mechanism while embedding it into a planar geometry. In the recent literature, two concrete realizations have been developed: a PRESERVED_PLACEHOLDER_^^^^4\4^^^^-symmetric two-dimensional Hatano–Nelson flux model with on-site Hubbard interaction, designed from plaquettes with clockwise and anticlockwise non-reciprocal hopping (&&&^^^^4\4^^^^&&&), and a cylinder-type two-dimensional Hatano–Nelson model with an SU(^^^^4 OR \4^^^^) non-Abelian gauge field, where nonreciprocity along one direction and gauge structure jointly control complex-energy braiding and the non-Hermitian skin effect (&&&^^^^4 OR \4^^^^&&&). These constructions extend Hatano–Nelson physics beyond one-dimensional skin accumulation into settings with exceptional lines, interaction-driven antiferromagnetism, diffusive spin modes, generalized Brillouin zones, and left-, right-, and bipolar boundary localization.

^^^^4 OR \4^^^^. Definition and principal realizations

The modern two-dimensional Hatano–Nelson literature is built around two explicit lattice constructions rather than a single universal Hamiltonian. One construction uses the one-dimensional Hatano–Nelson chain as a plaquette-level building block and promotes it to a bipartite square-lattice flux model with non-reciprocal hoppings circulating clockwise or anticlockwise around elementary plaquettes. The second construction considers a spinful cylinder geometry with nonreciprocal hopping along PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^, reciprocal hopping along PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^, and an SU(^^^^4 OR \4^^^^) non-Abelian gauge structure that acts differently on opposite PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^-direction hoppings (&&&^^^^4\4^^^^&&&).

Realization	Key ingredients	Principal phenomena
^^^^4 OR \4^^^^D Hatano–Nelson flux model	Bipartite square lattice, clock-anticlockwise non-reciprocal plaquettes, Hubbard $U$	$\mathcal{PT}$-symmetry breaking/restoration, exceptional lines, AF metal/semimetal/insulator, diffusive $d$-wave spin modes
^^^^4 OR \4^^^^D cylinder-type SU(^^^^4 OR \4^^^^) model	Spin-$\tfrac12$, nonreciprocal $x$-hopping, reciprocal $y$-hopping, PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^, PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^	Hopf-link bulk braiding, GBZ polarization, left/right/bipolar skin modes, zero-imaginary-energy degeneracy

Both models preserve the Hatano–Nelson emphasis on nonreciprocal transport, but they organize it differently. In the flux model, non-Hermiticity is encoded in the plaquette pattern and combined with local Hubbard repulsion. In the cylinder model, non-Hermiticity is intertwined with internal spin structure and noncommuting gauge generators. This suggests that, in two dimensions, Hatano–Nelson physics is less a single lattice prescription than a family of nonreciprocal design principles.

^^^^4 OR \4^^^^. Flux-model construction on the square lattice

In the two-dimensional non-Hermitian Hatano–Nelson flux model, the one-dimensional Hatano–Nelson unit with maximally asymmetric hopping is used as a building block for a square plaquette network. The resulting correlated Hamiltonian is

PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^

with

PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^

The lattice is bipartite, with sublattices PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^ and PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5, and the model is formulated so that the nonreciprocal hopping pattern alternates in a clock-anticlockwise manner from plaquette to plaquette (&&&^^^^4\4^^^^&&&).

The local vertex structure makes the plaquette organization explicit: PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7

A central feature of this construction is that it is non-Hermitian without introducing explicit gain/loss terms. The non-Hermiticity arises because the hopping amplitudes fail to satisfy the Hermitian relation PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8. At the same time, the model is explicitly PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9-symmetric, which permits both real-spectrum and complex-spectrum regions. This makes the flux construction a natural two-dimensional analogue of Hatano–Nelson nonreciprocity in a correlated lattice setting.

^^^^4 OR \4^^^^. Band structure, exceptional lines, and interaction-driven phases

For the noninteracting flux model, the energy bands are

PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^

Because of the square-root structure, the spectrum is real where PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^ and imaginary where PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^. The exceptional lines are therefore determined by

PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^

that is,

PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^

At half-filling, PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5, these exceptional lines coincide with the Fermi surface. For finite PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6, only one band crosses the Fermi level, while for PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7 the noninteracting system is a band insulator (&&&^^^^4\4^^^^&&&).

With on-site Hubbard interaction, the main ordered state considered is long-range antiferromagnetism with ordering vector

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8

After Hubbard–Stratonovich decoupling and the mean-field ansatz

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9

one defines PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^, and the quasiparticle spectrum becomes

PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^

This produces a three-regime phase structure. For PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^, the system is an AF metal, with real and complex regions coexisting in momentum space. At PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^, it becomes an AF semimetal with Dirac points at PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^ and PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5 in the sublattice Brillouin zone. For PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6, it is an AF insulator, and the entire Brillouin zone has real energies. In this model, the transition from broken to unbroken PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7 symmetry coincides with the metal-insulator transition. Under the change of variables

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8

the dispersion becomes

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9

At $U$^^^^4\4^^^^, this matches the familiar $U$^^^^4 OR \4^^^^-flux spectrum, and near the four special points the dispersion is linear,

$U$^^^^4 OR \4^^^^

The same work also notes that long-range Coulomb interactions can dynamically generate a gap. In the truncated Schwinger–Dyson treatment, the critical coupling is $U$^^^^4 OR \4^^^^, above which a gap opens. This places the two-dimensional Hatano–Nelson flux model in direct contact with correlated Dirac physics as well as non-Hermitian spectral transitions.

4. Spin response and diffusive $U$4-wave collective modes

The spin sector of the flux model is analyzed through the dynamical susceptibilities

$U$5

with $U$6, and within random phase approximation,

$U$7

The transverse Goldstone condition is

$U$8

which is equivalent to the self-consistent mean-field equation for $U$9 (&&&^^^^4\4^^^^&&&).

The characteristic result is that the low-energy transverse spin excitations are not conventional propagating magnons. Expanding near $\mathcal{PT}$^^^^4\4^^^^ and $\mathcal{PT}$^^^^4 OR \4^^^^ gives

$\mathcal{PT}$^^^^4 OR \4^^^^

with $\mathcal{PT}$^^^^4 OR \4^^^^. The poles occur at

$\mathcal{PT}$4

so the mode frequencies are purely imaginary.

These collective excitations are therefore interpreted as diffusive $\mathcal{PT}$5-wave modes, with one branch corresponding to loss and the other to gain. The coefficients $\mathcal{PT}$6 and $\mathcal{PT}$7 are well defined only in the regime $\mathcal{PT}$8, where the single-particle spectrum is fully real and $\mathcal{PT}$9 symmetry is unbroken. The longitudinal channel, by contrast, is gapped up to roughly twice the antiferromagnetic gap. Numerically, the spin-wave dispersion is nearly flat along the $d$^^^^4\4^^^^ and $d$^^^^4 OR \4^^^^ directions, which the paper associates with frustration of $d$^^^^4 OR \4^^^^ order and possible tendencies toward stripe ordering at $d$^^^^4 OR \4^^^^ or $d$4. In the two-dimensional Hatano–Nelson setting, this establishes that gain/loss signatures can survive in the collective sector even when the quasiparticle spectrum is globally real.

5. Cylinder geometry with SU(^^^^4 OR \4^^^^) non-Abelian gauge structure

A second two-dimensional Hatano–Nelson construction introduces an SU(^^^^4 OR \4^^^^) non-Abelian gauge field in a spinful cylinder geometry. The lattice Hamiltonian is

$d$5

where $d$6, $d$7, and the non-Abelian gauge components are

$d$8

The paper focuses on the non-Abelian case $d$9, specifically

$\tfrac12$^^^^4\4^^^^

In cylinder form, the $\tfrac12$^^^^4 OR \4^^^^-direction hoppings are

$\tfrac12$^^^^4 OR \4^^^^

$\tfrac12$^^^^4 OR \4^^^^

while the $\tfrac12$4-direction contribution is reciprocal,

$\tfrac12$5

(&&&^^^^4 OR \4^^^^&&&).

Under periodic boundary conditions in both directions, the Bloch Hamiltonian is

$\tfrac12$6

with

$\tfrac12$7

Its eigenvalues are

$\tfrac12$8

The same study contrasts this non-Abelian model with a gauge-free Hatano–Nelson case and an Abelian-gauge case in which both opposite-direction hoppings use the same generator. The noncommutativity of $\tfrac12$9 and $x$^^^^4\4^^^^ is the key new ingredient. The reported consequence is that non-Abelian gauge structure can generate Hopf-link bulk braiding with only nearest-neighbor couplings and can create left-, right-, and bipolar skin modes that are not fully characterized by exceptional-point data alone.

6. Generalized Brillouin zone, skin localization, and Hopf-link braiding

For the cylinder model, open-system bulk physics is formulated through the generalized Brillouin zone. One replaces $x$^^^^4 OR \4^^^^ by a complex variable $x$^^^^4 OR \4^^^^ and solves

$x$^^^^4 OR \4^^^^

If the characteristic equation has order $x$4, the roots are arranged by modulus,

$x$5

and the GBZ is defined by the matching condition

$x$6

To quantify skin accumulation, the paper introduces the polarization parameter

$x$7

with

$x$8

The interpretation is explicit: $x$9 indicates maximal right-skinned localization, $y$^^^^4\4^^^^ maximal left-skinned localization, and $y$^^^^4 OR \4^^^^ a bipolar skin regime (&&&^^^^4 OR \4^^^^&&&).

The real-space diagnosis is based on the averaged density

$y$^^^^4 OR \4^^^^

and its weighted center

$y$^^^^4 OR \4^^^^

Eigenstates are then qualitatively encoded as $y$4 for bipolar skin, $y$5 for left skin, $y$6 for right skin, $y$7 for nonlocalized or flat, and $y$8 for other. The GBZ analysis and real-space eigenstate encoding are reported to agree.

The same model exhibits a topological transition controlled by the exceptional-point condition

$y$9

The corresponding braiding number is

PRESERVED_PLACEHOLDER_^^^^4 OR \4\4\4^^^^

which reduces to

PRESERVED_PLACEHOLDER_^^^^4 OR \4\4 OR \4^^^^

The two phases have PRESERVED_PLACEHOLDER_^^^^4 OR \4\4 OR \4^^^^, corresponding to distinct Hopf-link braidings in PRESERVED_PLACEHOLDER_^^^^4 OR \4\4 OR \4^^^^ space. At the critical boundary, the system develops highly degenerate zero-imaginary-energy states; when PRESERVED_PLACEHOLDER_^^^^4 OR \4\44^^^^, about half of the eigenstates become degenerate in both real and imaginary spectra. These states show bipolar localization, and the IPR peaks at the critical boundary. The reported persistence of the most localized bipolar state from PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^5 to PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^6 indicates that the effect is not a finite-size artifact.

7. Relation to one-dimensional Hatano–Nelson physics and common scope errors

A recurrent issue in the Hatano–Nelson literature is that many influential papers remain strictly one-dimensional even when they motivate higher-dimensional extensions. The disordered spreading study "Spreading dynamics in the Hatano-Nelson model" explicitly analyzes the one-dimensional nonreciprocal chain

PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^7

with random imaginary onsite potential, and does not define or analyze a genuinely two-dimensional lattice Hamiltonian (Shang et al., 6 Apr 2025). The paper on imaginary boost deformation likewise reformulates Hatano–Nelson non-Hermiticity as a broader one-dimensional or PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^8D integrable deformation and states that higher-dimensional extension is a future direction rather than a completed construction (Guo et al., 2023). The interacting full-counting-statistics work also treats an open one-dimensional chain, not a two-dimensional model (&&&^^^^4 OR \4\4^^^^&&&).

This distinction matters because some phenomena that appear “higher-dimensional” by analogy are not, in those works, derived in a two-dimensional geometry. Bidirectional skin accumulation, for example, already occurs in an extended one-dimensional Hatano–Nelson model with non-reciprocal next-nearest-neighbour hopping and quasiperiodic or periodic onsite modulation (&&&^^^^4 OR \4 OR \4^^^^&&&). A plausible implication is that the two-dimensional literature should not be reduced to simple dimensional uplift of the one-dimensional chain: the two-dimensional flux model and the cylinder-type non-Abelian model both introduce genuinely new organizing structures—plaquette chirality, exceptional lines in a two-dimensional Brillouin zone, interaction-driven antiferromagnetism, Hopf-link braiding, and GBZ-resolved bipolar skin modes—that are specific to planar nonreciprocal lattices.

Within that perspective, the two-dimensional Hatano–Nelson model is best understood as a non-Hermitian platform in which asymmetric hopping is promoted from directional drift in one dimension to a richer interplay among lattice geometry, symmetry, topology, and correlations. The current arXiv record establishes two especially developed forms of this program: the PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^9-symmetric flux model with Hubbard-driven antiferromagnetism and diffusive PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4\4^^^^-wave spin excitations (&&&^^^^4\4^^^^&&&), and the SU(^^^^4 OR \4^^^^) cylinder model with Hopf-link braiding, GBZ polarization, and robust bipolar non-Hermitian skin effect (&&&^^^^4 OR \4^^^^&&&).