Anomalous Scaling Behaviors of the Green's Function in Critical Skin Effects

Yifei Yi; Zhesen Yang

REVIEW 3 major objections 5 minor 54 references

Anomalous Scaling Behaviors of the Green's Function in Critical Skin Effects

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Green's functions obey generalized Brillouin zone theory in the bulk but diverge at boundaries in the anomalous regions of the critical skin effect, and the paper traces the divergence to residual contributions from the zero-coupling GBZ.

desk verdict Solid numerics and a genuinely new scaling observation, but the anomalous-region mechanism is asserted rather than derived. read the letter →

arxiv 2507.20843 v1 pith:UA7ULSGU submitted 2025-07-28 cond-mat.mes-hall

Yifei Yi , Zhesen Yang This is my paper

classification cond-mat.mes-hall

keywords non-HermitianskineffectcriticalGreen'sfunctiongeneralizedBrillouinzonespectralwindingnumberHatano-Nelsonmodelopenboundaryconditionsperturbationtheory

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

The paper asks whether the standard generalized Brillouin zone (GBZ) formula for open-boundary Green's functions survives the critical non-Hermitian skin effect, where the GBZ jumps discontinuously as an inter-chain coupling is switched on. Using a double-chain Hatano-Nelson model with small coupling $\Delta=1/10000$, it shows that in regions where each band's open-boundary spectral winding number is trivial, first-order perturbation theory reproduces the observed zigzag scaling structure and gives a precise formula for the transition length. In regions where individual bands carry nonzero winding numbers, the Green's function matches GBZ predictions in the bulk but diverges near boundaries, and the paper attributes that divergence to residual contributions from the $\Delta=0$ GBZ. If this explanation is right, GBZ theory captures the bulk response of the critical skin effect but systematically misses finite-size boundary dynamics that complex-frequency Green's-function measurements could expose.

What carries the argument

The central object is the open-boundary Green's function $G(\omega)=(\omega-H)^{-1}$ for the double-chain Hatano-Nelson Hamiltonian with Bloch form $H(\beta)=\begin{pmatrix} H_A(\beta) & \Delta \\ \Delta & H_B(\beta)\end{pmatrix}$, where $H_A(\beta)=t_0+t_{-1}/\beta+t_1\beta$ and $H_B(\beta)=w_0+w_{-1}/\beta+w_1\beta$. The argument is carried by the generalized Brillouin zone (GBZ), the contour or contours in the complex $\beta$ plane fixed by the characteristic-root condition $|\beta_2|=|\beta_3|$, which determine the open-boundary spectrum and enter the Green's function through the multiband integral formula (Eq. 4). The paper combines that formula with the argument theorem: for a frequency $\omega$, the number of roots of $\omega-H_\alpha(\beta)=0$ enclosed by a sub-GBZ contour is fixed by the open-boundary spectral winding number $\nu_\alpha^{OBC}$. In anomalous regions the $\Delta=0+$ contours enclose zero roots for band A and two roots for band B, so the $\Delta=0+$ formula has exact zero sectors and cannot produce the amplification seen numerically. First-order perturbation theory in $\Delta$ (Eqs. 7-9) supplies the zigzag competition and the transition-length formula, while the missing boundary amplification is supplied by roots that lie between the $\Delta=0$ and $\Delta=0+$ GBZ contours.

What would settle it

Compute the exact open-boundary Green's function $G_{BB}$ at $\omega=-0.28i$ for $N=500$ and again for $N=2000$, first with $\Delta=1/10000$ and then with $\Delta=0.5/N$; if the boundary divergence comes from residual $\Delta=0$ GBZ roots, its onset should follow the $L_c$ formula and persist when $\Delta N$ is held fixed, whereas the $\Delta=0+$ formula alone predicts an exact zero in the $i<j$ sector of $G_{BB}$.

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Extended reading notes

Core claim

The paper's central claim is that the open-boundary Green's function $G(\omega)=(\omega-H)^{-1}$ of the double-chain Hatano-Nelson model at small inter-chain coupling has two regimes. In conventional regions (1, 2, 6), where the open-boundary spectral winding numbers of both bands are trivial, the Green's function shows a universal asymptotic scaling and a zigzag pattern near the excitation, both captured by first-order perturbation theory, with transition points given by a closed formula whose dominant dependence is $L_c\sim 2\log\Delta/\log(|\beta_A|/|\beta_B|)$. In anomalous regions (3, 4, 5), where individual bands carry nonzero open-boundary spectral winding numbers, the Green's function agrees with the multiband GBZ formula in the bulk but diverges near boundaries. The paper identifies the divergence as a residual contribution from the $\Delta=0$ GBZ: for example, for $G_{BB}$ at $\omega=-0.28i$, the root $\beta_2^B$ lies outside the $\Delta=0$ sub-GBZ for band B but inside the $\Delta=0+$ sub-GBZ, so it generates a finite boundary layer even though $\Delta$ is tiny. The finite-size Green's function in these regions is thus a coexistence of two GBZ limits rather than a small perturbative correction to either one.

Load-bearing premise

The argument rests on the premise that, for a 500-site chain with a tiny inter-chain coupling, the exact finite-size Green's function separates cleanly into a bulk part governed by the near-zero-coupling generalized Brillouin zone and a boundary part governed by zero-coupling roots, without a derived scaling condition that links the coupling size to the chain length.

Editorial extensions

If this is right

In the conventional regions with trivial OBC spectral winding, first-order perturbation theory reproduces both the universal asymptotic scaling and the zigzag structure of the Green's-function components, with transition points given by Eq. (10).
In the anomalous regions with nontrivial individual-band winding, the multiband GBZ formula correctly captures the bulk behavior but misses the boundary layer, so GBZ-based response calculations for finite systems are incomplete there.
The boundary discrepancy traces to residual contributions from the $\Delta=0$ GBZ, meaning the finite-size Green's function carries information about both the $\Delta=0$ and $\Delta=0+$ limits at once.
The presence or absence of boundary divergence is selected by the open-boundary spectral winding numbers summarized in Tab. I, giving a Green's-function signature of that winding structure.
Complex-frequency Green's-function measurements on a realization of this model should see these region-dependent boundary divergences directly.

Reading between the lines

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

If the same boundary-residual mechanism governs time-domain dynamics, a pulse launched near a boundary in an anomalous region should show a slow tail controlled by the $\Delta=0$ GBZ root even at tiny coupling; the paper does not test this directly.
The decomposition into $\Delta=0+$ bulk and $\Delta=0$ boundary contributions implies an unexplored scaling window $\Delta\sim 1/N$ where the two contributions have comparable strength; the paper fixes $\Delta=1/10000$ and $N=500$, so it does not locate that crossover.
The same logic predicts boundary Green's-function divergences in other multi-band non-Hermitian models whose sub-GBZs change enclosed roots discontinuously at a critical coupling, making the phenomenon generic rather than confined to this model.
Because the transition length grows as $\log\Delta$, a tunable-coupling experiment could observe the zigzag boundary layer expand or contract at fixed frequency as $\Delta$ is varied.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Short version: this paper reports a genuinely new observation—Green's functions in a critical NHSE model show a zigzag scaling structure in conventional regions and a boundary divergence in anomalous winding regions—and backs the zigzag part with a first-order perturbative formula that matches exact numerics. The anomalous boundary divergence, however, is explained by a residual-Δ=0-GBZ mechanism that is asserted but never computed. That is the load-bearing soft spot.

What is solid: the numerics are exact matrix inversions for N=500, the perturbative formula Eq. (10) is checked against simulation and correctly predicts Lc scaling with Δ and N, and the claim that Δ=0 versus Δ=0+ GBZs enclose different roots is clearly presented. The observation that GBZ theory fails near boundaries in anomalous regions is credible and new. I also think the citation pattern is fine; refs 42/43 did eigenstates, and the earlier GBZ Green's function papers did not identify this boundary effect.

The soft spot: the residual contribution is identified only by root location—β_B^2 lies outside C_B^0 but inside C_B^0+—and then declared to dominate near the boundary. No amplitude is computed, no expression is derived, and there is no scaling analysis in Δ and N. The paper also needs the clean split G_finite = G(Δ=0+) + (Δ=0 residual), but the finite-Δ sub-GBZ differs from C_α^0+ by O(Δ) corrections, so the 'bulk' prediction has uncontrolled finite-Δ corrections. Root location is not a residue calculation. I'd want a derivation of the boundary contribution, or at minimum a scaling argument showing when the residual term survives, before I'd call the mechanism established. The observation stands regardless.

Bottom line: this is a useful paper for people working on non-Hermitian response and critical skin effect. The conventional-region part is publishable as is; the anomalous-region part needs more work but is worth referee time. I would send it to review. My own verdict would be conditional.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the open-boundary Green's function of a two-chain Hatano-Nelson model with inter-chain coupling Δ, in the context of the critical non-Hermitian skin effect. For Δ=0, the decoupled single-chain Green's functions are compared with the generalized Brillouin zone (GBZ) integral formula. For small Δ=1/10000, the complex frequency plane is divided into six regions by spectral winding numbers; in conventional regions (1, 2, 6) the authors observe a zigzag scaling structure and explain it by first-order perturbation theory, deriving a transition distance Lc in Eq. (10) that is checked against numerics. In anomalous regions (3, 4, 5) they report that the numerical Green's function agrees with the Δ=0+ GBZ prediction in the bulk but deviates near boundaries, and they attribute this boundary divergence to residual contributions from the Δ=0 GBZ. The latter attribution is presented qualitatively, without a quantitative derivation or scaling analysis.

Significance. The paper contains several solid elements: the numerical Green's functions are exact inversions of the finite-size Hamiltonian, the comparison with GBZ predictions involves no fitted parameters, and the first-order perturbation formula Eq. (10) is verified directly against numerical data for the conventional regions. The distinction between conventional and anomalous regions based on OBC spectral winding numbers is clear and useful. If the residual-Δ=0-GBZ mechanism for the anomalous-region boundary divergence were made quantitative, the paper would be a valuable contribution to understanding finite-size and boundary effects that go beyond GBZ theory, especially in view of recent interest in complex-frequency Green's function probes. As submitted, the central explanatory claim for the anomalous regions is not established by the evidence presented.

major comments (3)

[Anomalous regions, paragraph after Eq. (11) and Fig. 3] The central claim that residual contributions from the Δ=0 GBZ cause the boundary divergence is supported only by root-location arguments: a root is said to lie outside C0_B but inside C0+_B, and this is taken to indicate dominance near the left boundary. No expression for the residual contribution is derived, no residue or amplitude is computed, and no scaling with Δ or N is provided. Because the exact finite-size Green's function at Δ=1/10000 is determined by the coupled open-boundary eigenproblem, whose sub-GBZ differs from the Δ=0+ contour by O(Δ) corrections, root positions relative to C0 and C0+ do not by themselves establish the decomposition G_finite = G(Δ=0+) + residual(Δ=0). A quantitative derivation of the residual term, or an explicit numerical construction of it, together with the condition under which the decomposition holds, is needed before the statement that residual contributions 'account for' the deviations can be accepted.
[Anomalous regions, finite-size scaling] The boundary divergence is demonstrated for a single lattice size N=500 and a single coupling Δ=1/10000. The paper does not analyze how the boundary layer or the deviation from the GBZ prediction scales with N or Δ, so it is unclear whether this is a genuine thermodynamic-limit anomaly of critical NHSE or a finite-size transient. A scaling analysis of the boundary deviation, for example the width of the boundary region as a function of N and Δ, is required to support the claim that GBZ theory fails near boundaries in anomalous regions.
[Residue-theorem discussion after Eq. (11)] The statement 'GAA_{i>j,j}(ω,Δ=0+)=GBB_{i<j,j}(ω,Δ=0+)=0 exactly' is not justified as written. For GBB with i<j, the contour C0+B encloses two roots of ω-HB(β)=0, and the integral in Eq. (2) has a pole at β=0 of order j-i; both sets of residues contribute, so the residue theorem does not give an exact zero. Please clarify the intended index convention or correct the statement. This is not merely cosmetic, because the paper uses this exact-zero claim to argue that the Δ=0+ limit cannot generate the observed zigzag structure.

minor comments (5)

[Table I caption] The caption writes 'νΔ=0 α and νΔ=0 α' with the same superscript twice; presumably the second symbol should be ν^{Δ=0+}_α, matching the column headings.
[Eq. (7)] The quantities labeled E_A^(1) and E_B^(1) are the zeroth-order (unperturbed) energies, not first-order corrections; using E_A^(0) and E_B^(0) would avoid confusion.
[Fig. 3 and text] The text cites Fig. 3(c) when discussing that βB2 is outside C0_B but inside C0+_B; since Fig. 3(c) appears to show the Δ=0 contours, the relevant comparison between the two contours requires referring to both panels (c) and (d).
[Eq. (10)] The definition of Lc as 'i-j' should be stated more carefully for the left and right transition points, since distances are positive quantities and the left transition involves i<j.
[Eq. (8)] The perturbative expansion retains O(Δ²) cross-branch terms but drops the O(Δ²) normalization correction ⟨L|R⟩^{-1}=1-Δ²/H_AB² for the same branch; stating this as an explicit approximation would clarify the order of the expansion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central comparisons are between exact numerical Green's functions and parameter-free GBZ/perturbation formulas, with the residual Δ=0 GBZ contribution offered as a proposed mechanism rather than an input.

full rationale

The paper's derivation chain is self-contained against direct numerical computation. For Δ=0, Eq. (2) is the standard GBZ integral formula from Ref. [47], and the resulting asymptotic expressions are compared with direct numerical solutions of G(ω)=(ω−H)^{-1} in Fig. 2(a)-(b); no parameter is fitted to the Green's function data. For small Δ in conventional regions, Eqs. (7)-(9) are a first-order perturbative expansion with no adjustable parameters, and the transition-point formula Eq. (10) is derived from amplitude matching between the two decoupled-chain contributions and then checked against numerical Lc values in Fig. 2(e)-(f). This is a genuine parameter-free prediction. In anomalous regions, the discontinuity statement Eq. (11) follows from the argument theorem applied to the winding numbers listed in Table I, again with no fitted input. The observed boundary divergence is extracted from exact finite-size numerics for Δ=1/10000, while the attribution of that divergence to 'residual contributions from the Δ=0 GBZ' is an interpretive hypothesis based on root locations relative to C_α^0 and C_α^{0+}; this hypothesis is under-supported because no amplitude or scaling for the residual term is computed, but under-support is a correctness concern, not circularity. The GBZ theory is used both to identify anomalous regions and to explain its own boundary failure, but that is conceptual tension rather than a reduction of the prediction to the input. Self-citations (e.g., Refs. [42], [46], [52]) supply background and standard formulas but are not load-bearing in a way that forces the paper's numerical-observation claims. No circular step can be exhibited, so the score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. The central results rest on known GBZ formulas and on the critical NHSE property of the chosen model. The main unstated assumptions are the validity of perturbation theory for all relevant β and the qualitative separation of bulk and boundary contributions.

assumptions (4)

domain assumption The GBZ integral formulas for OBC Green's functions, Eqs. (2) and (4), are valid for the model.
The paper adopts these formulas from refs. [47,52] without re-deriving them; the central comparisons rely on them as benchmarks.
domain assumption The two-chain model with t0=1, t_{-1}=2, t1=1, w0=-1, w_{-1}=1, w1=2 exhibits critical NHSE, with a discontinuous GBZ at Δ=0 versus Δ=0+.
This is taken from refs. [42,46,52]; the paper uses it to define the anomalous regions.
domain assumption First-order perturbative eigenstates in Eq. (7) are accurate at Δ=1/10000, and the denominators HAB(β) never become small enough to invalidate the expansion.
The derivation of Eqs. (8)-(10) assumes this; the paper does not check the perturbation condition across all β.
standard math The argument theorem and residue theorem can be applied to the contour integrals with the stated pole order.
Used in the derivation of Eqs. (3), (6), and (11).

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Cite this review

Pith. "Pith review of Anomalous Scaling Behaviors of the Green's Function in Critical Skin Effects." pith.science (2026). https://pith.science/paper/UA7ULSGU

@misc{pith2026250720843,
  author       = {Pith},
  title        = {Pith review of: Anomalous Scaling Behaviors of the Green's Function in Critical Skin Effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UA7ULSGU}},
  note         = {Machine review of arXiv:2507.20843}
}

abstract

We study the Green's functions in non-Hermitian systems exhibiting the critical non-Hermitian skin effect (critical NHSE) using a double-chain Hatano-Nelson model with inter-chain coupling $\Delta$. For $\Delta=0$, the system decouples into two independent chains, and the Green's functions follow predictable patterns based on the GBZ theory. For small $\Delta$ ($\Delta=1/10000$), in conventional regions with trivial OBC spectral winding numbers, inter-chain coupling induces a zigzag scaling structure in Green's functions due to competition between the two chains, explainable by first-order perturbation theory. In anomalous regions with non-trivial winding numbers, Green's functions match GBZ predictions in the bulk but diverge near boundaries, with residual contributions from the $\Delta=0$ GBZ accounting for the deviations. These results reveal the unique non-perturbative features of critical NHSE and highlight the limitations of GBZ theory in capturing finite-size and boundary effects, emphasizing the need to consider both bulk and boundary dynamics in such systems.

Figures

Figures reproduced from arXiv: 2507.20843 by the authors.

**Figure 2.** FIG. 2. Green’s functions calculated numerically (light color) and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Green’s functions and root distributions for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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