{"id":"794e5a6b-34ab-4ee2-bf14-81414bf154ff","arxiv_id":"2507.20843","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a critical non-Hermitian skin-effect model, small inter-chain coupling produces anomalous zigzag scaling of Green's functions, with boundary deviations that GBZ theory cannot capture.","lead":"This paper shows that in a non-Hermitian two-chain model with critical skin effect, tiny inter-chain coupling creates an unusual zigzag pattern in the Green's function, with edge behavior that standard theory misses. The result matters because it pinpoints where the generalized Brillouin zone framework fails for finite systems.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Residual Δ=0 GBZ contributions are invoked but never computed; root-location arguments alone cannot establish that they cause the boundary divergence in anomalous regions.","rationale":"The paper's most valuable result is the unambiguous numerical observation of boundary divergence in anomalous regions, which is independent of the proposed mechanism. My stress test nevertheless converges on the same weak point the reader identified: the residual Δ=0 GBZ explanation is a root-location argument, not a calculation. I could not find a derivation of the residual amplitude, nor a scaling law, nor a justification for separating the finite-Δ Green's function into a Δ=0+ bulk part and a Δ=0 boundary part. Because this mechanism is the central explanatory claim, the paper is correctly CONDITIONAL. The concrete test would settle it by comparing the exact boundary excess to the predicted Δ=0 residue with a fitted amplitude and Δ-scaling.","tokens_in":10618,"tokens_out":7492,"duration_ms":86832,"concrete_test":"Compute exact OBC G(i,j=251) for N=500, Δ=10^-4, ω=-0.28i. Let δG(i)=G_exact(i,j)-G_GBZ(i,j), where G_GBZ is Eq.5. Fit δG for i<j to C[β_B^2]^{i-j}; extract C. Compare C to the Δ=0 residue coefficient A_B^0(β_B^2) times Δ^2/[H_AB(β_B^2)]^2 (from Eq.8-style perturbation). Repeat for Δ=10^-3,10^-5 and N=500,1000. The residual mechanism requires C ∝ Δ^2 and a boundary length L=1/|ln|β_B^2|| independent of N. If the fit or scaling fails, the attribution to Δ=0 GBZ residues is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for anomalous regions is that the boundary divergence of the Green's function at Δ=1/10000 is caused by residual contributions from the Δ=0 GBZ. The evidence offered is root-location: for ω=-0.28i, β_B^2 lies outside C_B^0 but inside C_B^0+ (Fig. 3(c)-(d)), and the authors assert this root dominates near the left boundary. But no expression for the residual contribution is derived, no amplitude is computed, and no scaling with Δ or N is given. A root's position relative to Δ=0 and Δ=0+ contours is not by itself a calculation of the exact finite-size Green's function, whose residues at finite Δ are set by the full OBC eigenproblem, not by the Δ=0 contour. The paper also does not justify the clean separation G_finite = G(Δ=0+ bulk) + (residual Δ=0 boundary) that the explanation requires; the actual sub-GBZ for Δ=1/10000 differs from C_α^0+ by O(Δ) corrections, so the 'bulk' prediction itself has uncontrolled finite-Δ corrections. Since this residual mechanism is the paper's stated explanation for why GBZ theory fails near boundaries, the explanation is under-supported even though the numerical observation of the divergence is credible.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10811,"tokens_out":13208,"duration_ms":158924,"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":[{"comment":"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.","section":"Anomalous regions, paragraph after Eq. (11) and Fig. 3"},{"comment":"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.","section":"Anomalous regions, finite-size scaling"},{"comment":"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.","section":"Residue-theorem discussion after Eq. (11)"}],"minor_comments":[{"comment":"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.","section":"Table I caption"},{"comment":"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.","section":"Eq. (7)"},{"comment":"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).","section":"Fig. 3 and text"},{"comment":"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.","section":"Eq. (10)"},{"comment":"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.","section":"Eq. (8)"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the conventional-region analysis is solid, but the anomalous-region explanation is the main advertised result and it is currently qualitative. I would be willing to reconsider after the authors provide a quantitative construction or verification of the residual Δ=0 GBZ contribution and a scaling analysis in N and Δ, and after they correct the apparent residue-theorem error regarding GBB_{i<j}=0."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nShort 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.\n\nWhat 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.\n\nThe 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.\n\nBottom 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.","headline":"Solid numerics and a genuinely new scaling observation, but the anomalous-region mechanism is asserted rather than derived.","tokens_in":11364,"tokens_out":2746,"would_cite":true,"duration_ms":27913,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["non-Hermitian skin effect","critical skin effect","Green's function","generalized Brillouin zone","spectral winding number","Hatano-Nelson model","open boundary conditions","perturbation theory"],"falsifier":"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 $G_{BB}$.","tokens_in":10383,"feed_emoji":"⚛️","tokens_out":17977,"duration_ms":171691,"temperature":0.7,"pith_summary":"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.","feed_headline":"Green's functions diverge at boundaries in critical skin effect","feed_subtitle":"Generalized Brillouin zone theory misses finite-size boundary effects; complex-frequency probes can expose them.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the single-band GBZ integral formula (Eq. 2) that gives the $\\Delta=0$ Green's functions.","marker":"[47]"},{"why":"Provides the multiband GBZ Green's-function formula (Eq. 4) and the sub-GBZ construction used for $\\Delta=0+$.","marker":"[52]"},{"why":"Establishes the auxiliary GBZ theory behind the discontinuous $\\Delta=0$ to $\\Delta=0+$ GBZ jump that defines the critical skin effect here.","marker":"[46]"},{"why":"Introduced the critical non-Hermitian skin effect whose thermodynamic-limit discontinuity and finite-size scaling the paper tests with Green's functions.","marker":"[42]"},{"why":"Gives the scaling rule for critical-NHSE eigenstates that motivates the anomalous scaling analysis in finite systems.","marker":"[43]"}],"fun_headline_variants":["Critical skin effect: boundary divergence in Green's function","Coexisting GBZ limits explain Green's function anomaly","Tiny coupling, big boundary effect: Green's function diverges","Anomalous Green's function scaling challenges GBZ theory","Non-perturbative critical skin effect: boundary layers from decoupled limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Critical skin effect: boundary divergence in Green's function","Coexisting GBZ limits explain Green's function anomaly","Tiny coupling, big boundary effect: Green's function diverges","Anomalous Green's function scaling challenges GBZ theory","Non-perturbative critical skin effect: boundary layers from decoupled limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001085,"raw_usage":{"total_tokens":4572,"prompt_tokens":1018,"completion_tokens":3554,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":3469}},"tokens_in":634,"tokens_out":3554,"duration_ms":29193,"temperature":1.0,"reasoning_tokens":3469,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T13:13:07.547049+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $G_{BB}$.","supporting_citations":[{"cited_title":"Hu and Z","cited_arxiv_id":null,"evidence_quote":"Provides the multiband GBZ Green's-function formula (Eq. 4) and the sub-GBZ construction used for $\\Delta=0+$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the auxiliary GBZ theory behind the discontinuous $\\Delta=0$ to $\\Delta=0+$ GBZ jump that defines the critical skin effect here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the critical non-Hermitian skin effect whose thermodynamic-limit discontinuity and finite-size scaling the paper tests with Green's functions."},{"cited_title":"Yokomizo and S","cited_arxiv_id":null,"evidence_quote":"Gives the scaling rule for critical-NHSE eigenstates that motivates the anomalous scaling analysis in finite systems."}],"review_version":1}