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Anisotropic Anderson localization in higher-dimensional nonreciprocal lattices

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Pith's one-line read In the 2D Hatano–Nelson model, anisotropic nonreciprocity and disorder cooperate to create hybrid eigenstates that are skin-localized along one axis and Anderson-localized along the other, with an ALM–HM–ALM reentrant transition in…

desk verdict A clean exact similarity-transformation result and a new hybrid-mode phase diagram, with an overbroad claim about the reciprocal 2D model and underpowered scaling fits. read the letter →

arxiv 2507.14523 v1 pith:WWPHHP6B submitted 2025-07-19 cond-mat.dis-nn cond-mat.mes-hallcond-mat.stat-mechquant-ph

Jinyuan Shang , Haiping Hu This is my paper · ORCID
classification cond-mat.dis-nncond-mat.mes-hallcond-mat.stat-mechquant-ph
keywords Andersonlocalizationnon-HermitianskineffectHatano-NelsonmodelhybridmodesLyapunovexponentmobilityedgeBethelatticecomplexdisorder
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies a two-dimensional Hatano–Nelson lattice—a square lattice with directional nonreciprocal hopping plus random on-site disorder—and claims that disorder and nonreciprocity compete to produce three kinds of eigenstates: corner skin modes, bulk Anderson-localized states, and a previously unnoticed class of hybrid modes that are skin-localized along one axis and Anderson-localized along the other. Using transfer matrices and finite-size scaling of Lyapunov exponents, the authors derive a criterion for the skin–Anderson transition in each direction and map mobility edges in the complex energy plane. They further show that at fixed energy the system undergoes a reentrant ALM–HM–ALM transition as disorder strength grows, and that a skin–Anderson transition also survives on an infinite-dimensional nonreciprocal Bethe lattice. If correct, this establishes anisotropic nonreciprocity as a control knob for localization dimensionality.

What carries the argument

The central machinery is the similarity (imaginary gauge) transformation $c^\dagger_{x,y} \to e^{-xg_x-yg_y} c^\dagger_{x,y}$, $c_{x,y} \to e^{xg_x+yg_y} c_{x,y}$, which removes the nonreciprocal hoppings and reduces the Hamiltonian to a reciprocal complex Anderson model while shifting every Lyapunov exponent by $g_x$ (and similarly for $y$). This reduction converts Anderson localization into a comparison between the bare nonreciprocity $g_x$ and the inverse localization length $1/\tilde{\xi}_{X,\infty}$ of the reciprocal model, giving the transition criterion $g_x = 1/\tilde{\xi}_{X,\infty}$. Around this criterion, the transfer matrix and its essential Lyapunov exponent provide the finite-size scaling observable $\Lambda_X = 1/(\gamma_X(L_y) L_y)$ whose crossing points locate mobility edges. The same transfer-matrix construction is applied along each spatial direction, and the forward-scattering approximation supplies the Bethe-lattice extension.

What would settle it

Compute the rescaled Lyapunov exponent $\Lambda_X^{-1}$ for the 2D Hatano–Nelson model with $g_x=1$, $g_y=0.75$ at $E=0$ for $L_y = 16, 32, 48, 64$: the paper predicts a single crossing at $W_c \approx 10.69$ with $\nu \approx 1$, and at $E=3+3i$ two crossings at $W_{c1} \approx 4.506$ and $W_{c2} \approx 9.276$. A failure of these curves to intersect at those $W$ values, or an absence of the second crossing, would falsify the reentrant ALM–HM–ALM picture.

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Extended reading notes

Core claim

The central claim is that in the 2D Hatano–Nelson model with anisotropic nonreciprocity parameters $g_x$ and $g_y$ and complex box disorder of strength $W$, the eigenstates organize into three classes separated by directional mobility edges: skin modes pinned to a corner, Anderson-localized modes in the bulk, and hybrid modes whose wavefunction decays exponentially into the bulk along one direction while hugging a boundary along the other. The Anderson transition along each direction is governed by the essential Lyapunov exponent, and the paper derives the exact criterion $g_x = 1/\tilde{\xi}_{X,\infty}$—the nonreciprocity along $x$ equals the inverse localization length of the corresponding reciprocal complex Anderson model along that direction—with the analogous condition for $y$. Because the $x$- and $y$-mobility edges generally sit at different energies, a spectral window opens where hybrid modes exist; tracking a fixed energy as $W$ increases yields the ALM–HM–ALM reentrant sequence. The paper also extends the transfer-matrix criterion to arbitrary finite dimensions and uses the forward-scattering approximation on the Bethe lattice to argue that nonreciprocity, although it suppresses Anderson localization, does not eliminate the transition even in infinite dimensions.

Load-bearing premise

The derivation of the transition criterion assumes that the reciprocal (Hermitian-hopping) 2D Anderson model with complex box disorder is localized at every finite disorder strength, so its localization length $\tilde{\xi}_{X,\infty}$ stays finite as the transverse size grows; this absence of a metallic phase in 2D is asserted rather than proved in the paper.

Editorial extensions

If this is right

  • In any 2D nonreciprocal lattice whose nonreciprocity can be removed by a similarity transformation, the skin–Anderson transition along each axis is set solely by the ratio between the nonreciprocity parameter and the reciprocal-model localization length.
  • At fixed energy the three-regime sequence ALM–HM–ALM is a direct consequence of mobility-edge motion, so intermediate disorder acts as a directional filter that converts bulk-localized states into boundary-hugging hybrid states.
  • The mobility edges are symmetric about the real and imaginary axes in the complex energy plane, so the hybrid-mode window is bounded by two separate surfaces rather than a single curve.
  • A skin–Anderson transition persists on the infinite-dimensional Bethe lattice, where the critical disorder strength grows with nonreciprocity according to $W_c = 2 J e^h K \ln( W_c / (2 J e^h) )$.
  • The critical exponent $\nu$ remains close to 1 at all transition points studied, suggesting a common universality class for these directional transitions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: because the similarity transformation maps the nonreciprocal system to a reciprocal complex Anderson model, the same transition criterion should hold for any lattice whose nonreciprocity is a pure gauge, so the hybrid-mode window is controlled by the reciprocal localization length rather than by point-gap topology.
  • The reentrant sequence implies that at fixed disorder, sweeping the complex energy crosses two mobility-edge surfaces; this could be observed in photonic or electrical-circuit platforms by imaging the spatial profile of individual eigenmodes as the loss/gain term is tuned.
  • On the Bethe lattice, the forward-scattering estimate overestimates the critical disorder, so an exact self-consistent treatment should lower $W_c$; testing the predicted exponential growth of $W_c$ with nonreciprocity $h$ would show whether nonreciprocity merely rescales disorder or changes the universality class.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript studies the two-dimensional Hatano-Nelson model with anisotropic nonreciprocal hoppings (gx, gy) and complex box disorder. It identifies three eigenstate classes — corner skin modes, bulk Anderson localized modes, and hybrid modes localized at one edge along x and Anderson localized along y — and uses a similarity transformation plus transfer-matrix finite-size scaling of Lyapunov exponents to derive a transition criterion gx = 1/ξ~_{X,∞} (Eq. (9)) and to map mobility edges in the complex energy plane. It further reports an ALM-HM-ALM reentrant transition at fixed complex energy as W increases and uses the forward-scattering approximation to argue that skin-Anderson transitions persist on the nonreciprocal Bethe lattice.

Significance. If correct, the hybrid modes and the reentrant phase diagram add a new class of eigenstates and a genuinely two-dimensional phase diagram to non-Hermitian localization physics, going beyond the mostly 1D literature. The paper's exact similarity transformation establishing the Lyapunov-exponent shift, the explicit transfer-matrix setup, and the Bethe-lattice prediction are clear strengths. The main caveat is that the analytic transition criterion relies on the reciprocal 2D model being localized at the energy under study; the paper must reconcile this with the known Anderson transitions in 2D non-Hermitian disordered systems. The numerical evidence is suggestive but currently lacks error bars for the central critical points.

major comments (3)
  1. [Introduction; 'Anderson transition' section, Eqs. (8)-(9)] The derivation of Eq. (9) is conditional on the reciprocal (gx=gy=0) 2D model with complex box disorder having a finite asymptotic localization length ξ~_{X,∞}. The text nevertheless states unconditionally in the reciprocal-limit paragraph that any finite disorder strength W drives that model into an Anderson insulating phase. This assertion is in tension with Refs. [10,39], which report Anderson transitions in 2D non-Hermitian disordered systems; wherever the reciprocal model is delocalized, γ~_X(Ly) does not tend to a nonzero constant, the expansion used before Eq. (8) breaks down, and Eq. (9) is not a valid transition criterion. Please either restrict the analytic criterion to the localized energy/disorder windows and verify those windows by transfer-matrix scaling of the reciprocal model, or revise the unconditional phase-diagram claims in Figs. 2 and 3 accordingly.
  2. [Finite-size scaling analysis, Figs. 2(a) and 3(a)] All reported critical parameters are obtained from 100 disorder realizations and are quoted without error bars: Wc≈10.69 and ν≈1 at E=0, and Wc1≈4.506, Wc2≈9.276 at E=3+3i. The fits use NF=3 and Ny=2 with no sensitivity analysis. Because the reentrant transition and the mobility-edge surfaces are the paper's central quantitative claims, confidence intervals (for instance from bootstrap over disorder samples) and a statement of fit-order dependence are needed before the phase boundaries can be considered established.
  3. [Reentrance for intermediate disorder strength] The ALM-HM-ALM sequence is inferred from the two crossing points in Fig. 3(a) and is explained by a heuristic density-of-states argument, but no spatial profiles or inverse participation ratios at the fixed reference energy E=3+3i are shown for the three disorder windows. Please provide direct eigenstate or participation-ratio evidence for the classifications W<Wc1, Wc1<W<Wc2, and W>Wc2, or explicitly present the crossing-based inference as indirect.
minor comments (4)
  1. [Nonreciprocal Bethe lattices, Eq. (12)] Equation (12) uses h, while the model was defined with nonreciprocity parameter g; please define h or replace it with g for consistency.
  2. [Anderson transition, after Eq. (3)] The sentence defining ψx and Hx is duplicated ('where ψx is a length-Ly vector ... Hamiltonian.where ψx is a vector of length Ly ...'); please remove the repetition.
  3. [2D Hatano-Nelson model, Eq. (2)] Please state the normalization convention used for |ψ|^2 in the center-of-mass definition, especially whether right eigenstates are normalized in the usual L2 norm in the biorthogonal basis.
  4. [Finite-size scaling analysis, Fig. 2(b)] The text says the full mobility surfaces are symmetric with respect to the real or imaginary axes, but only the first quadrant is shown; a sentence explaining the symmetry (e.g., complex conjugation and chiral symmetries of the disorder distribution) would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central criterion follows from an exact similarity transformation and is checked by independent finite-size scaling, not fitted into the results.

full rationale

The paper's main derivation is the transition criterion gx = 1/ξ̃X,∞ (Eq. 9). This follows algebraically from the similarity transformation c†→e^{-xgx-ygy}c†, cx,y→e^{xgx+ygy}cx,y, which shifts the Lyapunov exponents by gx and maps the nonreciprocal model onto a reciprocal complex-disorder Anderson model. The beta-function calculation in Eq. (8) uses only this exact LE shift plus a localization-length ansatz for the reciprocal model; no parameter is fitted to obtain the form of the criterion. The existence of hybrid modes is established by exact diagonalization profiles (Fig. 1) and the phase boundaries by transfer-matrix finite-size scaling with data collapse (Figs. 2–3); the fitted Wc and ν are outputs of the scaling analysis, and the reentrant ALM–HM–ALM sequence is read off from crossings of rescaled Lyapunov exponents, not imposed by the input. Self-citations (Refs. [38,55,57,59]) are background or methodological and are not invoked as a uniqueness theorem or as the sole evidence for the central claim. The assumption that the reciprocal 2D model is localized for all finite W is an external physical premise; even if it is questionable, it is not an input that is renamed as a prediction, so it is a correctness risk rather than circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivation is a similarity transformation with no free parameters; the fitted quantities (Wc, nu) only locate transitions in the numerical phase diagram and do not feed back into the derivation of the HM criterion. The main domain assumption is the localization of the reciprocal 2D Anderson model. No new physical entities (particles, forces, dimensions) are postulated; 'hybrid modes' is a classification of eigenstates, not an invented entity.

free parameters (3)
  • Wc = 10.69 at E=0, gx=1, gy=0.75
    Critical disorder from finite-size scaling of Lyapunov exponents (Fig. 2a); determines transition point but central HM claim is qualitative.
  • critical exponent nu = approximately 1
    Fitted from data collapse in Fig. 2a; no error bars given.
  • Wc1, Wc2 = 4.506 and 9.276 at E=3+3i
    Two critical disorder strengths from crossings of rescaled LE curves (Fig. 3a), defining the reentrant transition.
assumptions (5)
  • standard math Oseledec's multiplicative ergodic theorem guarantees existence of Lyapunov exponents for the random transfer matrix product.
    Invoked in the Anderson transition section before Eq. (6).
  • domain assumption The reciprocal 2D Anderson model with complex box disorder is in the localized phase for every finite disorder strength W, so the essential LE \tilde{\gamma}_X(L_y) has a finite limit 1/\tilde{\xi}_{X,\infty}.
    Assumed in the derivation of the transition criterion following Eq. (9); not proved in the paper.
  • standard math A similarity (imaginary gauge) transformation with V_x = diag(e^{x g_x}) \otimes U exactly shifts all Lyapunov exponents by g_x without changing the spectrum.
    Exact for the Hatano-Nelson model with uniform nonreciprocity; used in Eq. (7) and the LE shift \tilde{\gamma}_i = \gamma_i - g_x.
  • domain assumption The forward-scattering approximation and the assumed large-deviation form P(ln|\psi|^2/n > -\alpha) ~ e^{-n \phi(\alpha)} correctly locate the Anderson transition on the Bethe lattice.
    Used in the Bethe lattice section; the paper notes FWA overestimates the critical disorder.
  • standard math For the Bethe lattice, sites at distance n are (K+1)K^{n-1} and the path from root to a site is unique.
    Tree geometry, used in Eq. (10) and the subsequent combinatorics.

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Cite this review

Pith. "Pith review of Anisotropic Anderson localization in higher-dimensional nonreciprocal lattices." pith.science (2026). https://pith.science/paper/WWPHHP6B

@misc{pith2026250714523,
  author       = {Pith},
  title        = {Pith review of: Anisotropic Anderson localization in higher-dimensional nonreciprocal lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WWPHHP6B}},
  note         = {Machine review of arXiv:2507.14523}
}
read the original abstract

Nonreciprocity breaks the symmetry between forward and backward propagation, giving rise to a range of peculiar wave phenomena. In this work, we investigate Anderson localization in higher-dimensional nonreciprocal lattices. Focusing on the two-dimensional Hatano-Nelson model, we uncover anisotropic hybrid modes (HMs) that exhibit skin localization along one direction and Anderson localization along the other. We determine the Anderson transition along different directions via the transfer matrix approach and finite-size scaling of Lyapunov exponents. This allows us to map out mobility edges that separate HMs from normal skin modes and Anderson localized modes (ALMs), revealing an ALM-HM-ALM reentrant transition. Our analysis extends to arbitrary dimensions, and we demonstrate the existence of skin-Anderson transitions on the infinite-dimensional nonreciprocal Bethe lattice using the forward-scattering approximation.

Figures

Figures reproduced from arXiv: 2507.14523 by the authors.

Figure 1
Figure 1. FIG. 1. Different types of eigenmodes revealed by the normal [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Phase diagram of the 2D Hatano–Nelson model ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Localization reentrance. (a) The rescaled LE, Λ [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Arbitrary Control of Non-Hermitian Skin Modes via Disorder and An Electric Field

    cond-mat.dis-nn 2025-11 conditional novelty 6.0 of 10

    Combining disorder with a static electric field lets the boundary accumulation direction of two-dimensional non-Hermitian skin modes be tuned by rotating the field relative to the nonreciprocal hopping.

Reference graph

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