{"id":"44ff6da0-047e-4f55-a49d-9f962bbbf2a1","arxiv_id":"2507.20713","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"A single random interlayer hopping defect creates an in-gap bound state that crosses mid-gap only in the trivial phase, and random interlayer tunneling can close the gap and degrade quantized edge conductance.","lead":"This paper analyzes what happens when the tunneling between layers of a multilayer topological insulator is random. It reports a rule for using a single defect to tell apart topological and trivial phases, and models how such disorder erodes quantized edge transport.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The two DOS methods explicitly disagree in the topological phase at weak disorder; because this self-admitted discrepancy is unresolved, the paper's finite-disorder claims are not established.","rationale":"The reader's weakest assumption correctly identifies the unresolved DOS discrepancy. My independent read finds the same issue and considers it the single most load-bearing concern because the paper's abstract and conclusion claim the two methods agree, while Section V.A explicitly reports a qualitative disagreement in the topological phase at weak disorder. The single-defect bound-state crossing rule is exact and unaffected, so the paper retains substantial value; however, the finite-disorder DOS conclusions (gap closure, Weyl robustness, AQH fragility, Hall plateau narrowing) all flow from these approximate methods and are not established in the disputed regime. The proposed numerical test would settle which approximation, if either, captures the gap renormalization. No additional concerns change the verdict; the paper remains CONDITIONAL pending this resolution.","tokens_in":26119,"tokens_out":7903,"duration_ms":86652,"concrete_test":"Perform a disorder-averaged kernel-polynomial or exact-diagonalization calculation of the DOS for a finite multilayer Burkov-Balents stack (e.g., 200 layers, k⊥ grid, η0 ≈ 5 meV, ΔS < ΔD) and compare the gap-edge shift to the clean gap. If the numerical gap narrows, the locator scheme is correct in this regime; if it widens, the Bloch-state expansion is correct; if it does neither, both approximations fail. Repeating at η0 = 0.1, 5, 13, 22 meV maps the full discrepancy range and settles which DOS curves in Fig. 9 are reliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section V.A states that in the topological phase at weak disorder the locator scheme predicts gap narrowing while the Bloch-state (t-matrix+RPA) expansion predicts gap widening, and \"the origin of this discrepancy is not yet understood.\" This contradicts the abstract's claim that the two approaches give matching densities of states and undermines every finite-disorder conclusion that uses the DOS: gap closure, Lifshitz tails, Weyl robustness, AQH fragility, and Hall-plateau shrinkage. The single-defect crossing rule (Sec. IV) is independent and likely correct, but the finite-disorder central claims depend on a DOS that is method-dependent in exactly the regime where the phase is topological. Until the discrepancy is resolved, one cannot tell whether weak off-diagonal disorder stabilizes or destabilizes the topological gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript analyzes off-diagonal disorder—random interlayer tunneling—in the Burkov–Balents multilayer topological insulator model. The central single-defect claim is that a Hermitian tunneling defect produces an in-gap bound state whose energy crosses zero only in the trivial phase and never in the topological phase, with the same crossing rule preserved under non-Hermitian perturbations, so that the bound state serves as a local marker of topology. The paper also develops two diagrammatic Green-function schemes (self-consistent locator and t-matrix+RPA in the Bloch basis) and uses them to compute the density of states in the normal, topological, Weyl, and anomalous quantum Hall regimes, concluding that off-diagonal disorder fills the gap with bulk states, closes the gap for strong disorder, leaves the Weyl phase robust, and shrinks the anomalous Hall plateau. Finally, the manuscript studies edge-mode localization lengths for uniform, Gaussian, and Lorentzian disorder and derives a correction to the longitudinal edge conductance from inter-edge tunneling. The single-defect analysis is analytically self-contained and likely independent of the finite-disorder approximations, but several load-bearing statements in the finite-disorder and localization parts are not currently supported.","tokens_in":26321,"tokens_out":9644,"duration_ms":103736,"significance":"If established, the single-defect crossing rule would be a useful and elegant local diagnostic of topology in multilayer topological insulators, extending the Green-function-zeros approach to off-diagonal defects. The manuscript also makes concrete, falsifiable predictions about Hall-plateau narrowing and disorder-dependent edge conductance. Its strengths include exact or analytic treatments of the single-defect problem, analytic localization-length expressions for three disorder distributions, and a self-contained derivation of the diagrammatic expansions without fitting to the target results. However, the finite-disorder conclusions currently rest on two approximations that the paper itself shows to disagree in the topological phase at weak disorder, and the localization-length section contains a sign inconsistency. These issues must be resolved before the finite-disorder claims can be considered reliable.","major_comments":[{"comment":"The delocalization condition has the wrong logarithmic argument. From Eq. (7) the transfer factor is (Delta_S/Delta_D) exp(A-1), so the edge mode becomes extended when A-1 > ln(Delta_D/Delta_S), not A-1 > ln(Delta_S/Delta_D). As printed, the inequality is automatically satisfied in the topological phase (Delta_S < Delta_D) even for infinitesimal disorder, which contradicts the text's own discussion and the finite critical line shown in Fig. 2. The derivation and the phase diagram must be corrected.","section":"Section II, Eq. (9)"},{"comment":"The two self-consistent schemes do not give matching densities of states in the topological phase at weak disorder: the locator method predicts gap narrowing while the Bloch-state t-matrix+RPA expansion predicts gap widening, a discrepancy the authors explicitly state is not yet understood. This contradicts the abstract's claim of matching densities of states and leaves the subsequent gap-closure, Weyl-robustness, AQH-fragility, and Hall-plateau conclusions dependent on which approximation is used. Please resolve the discrepancy or substantially qualify the finite-disorder claims.","section":"Section V.A, Figs. 8-9"},{"comment":"Eq. (62) defines L_c^{-1} as the absolute value of the difference of mean logarithms, so it cannot be negative. Yet Fig. 15(b) plots negative values and the text describes L_c^{-1} approaching zero from below and changing sign. The sign convention and the plotted quantity must be stated consistently. The claimed Gaussian divergence of L_c and the resulting inter-edge tunneling correction in Eq. (66) depend on this sign, so the current presentation is internally inconsistent.","section":"Section V.B, Eq. (62) and Fig. 15"},{"comment":"The root-selection statement is incorrect for the in-gap case. For epsilon_perp^2 < (Delta_S - Delta_D)^2, the dimensionless combination x = (epsilon_perp^2 - Delta_S^2 - Delta_D^2)/(2 Delta_S Delta_D) is less than -1, and the root inside the unit circle is z1 = x + sqrt(x^2 - 1), not the 'minus' root as stated. Since Eq. (B3) feeds into the single-defect pole condition Eq. (31) and the eigenvalue analysis Eq. (35), the sign/root convention should be fixed and the crossing rule re-verified.","section":"Appendix B, Eqs. (B2)-(B3)"}],"minor_comments":[{"comment":"The manuscript contains numerous missing spaces and broken formatting, for example the opening words 'Westudymultilayertopologicalinsulators' and similar throughout; the text should be properly typeset.","section":"Throughout"},{"comment":"Reference [12] is incomplete ('R. J. Slager and et. al.'); the full author list and title should be supplied.","section":"Reference [12]"},{"comment":"The printed form of Eq. (66) appears to omit the normalization: Appendix D, Eq. (D15), defines delta_sigma_xx/(pi e^2/4h), so Eq. (66) should be written as the normalized correction rather than as 'sigma_xx = sigma_xx^Delta pi e^2/4h = ...'.","section":"Eq. (66)"},{"comment":"Section numbers are inconsistent: the text refers to 'Sec. 3' and 'Sec. 5', while the paper is organized into Sections III and V; please unify the references.","section":"Section V, notation"}],"recommendation":"major_revision","confidential_remarks":"The manuscript overlaps heavily with the authors' own Ref. [42], and the incremental novelty should be delineated more carefully. The unresolved locator-versus-Bloch discrepancy in the topological phase is a serious correctness issue; if it cannot be resolved, I would recommend restricting the paper's claims to the single-defect result, which appears to be the most robust part. The Eq. (9) sign error and the localization-length sign inconsistency also need to be corrected before the paper can be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe one result worth taking away is the single-defect crossing rule: for off-diagonal (interlayer-tunneling) disorder in the Burkov–Balents multilayer, a bound state from a single tunneling defect crosses mid-gap only in the trivial phase, never in the topological phase. That extends Refs. [12,13] from diagonal to off-diagonal disorder and looks right, despite the sloppy signs in Appendix B. The non-Hermitian extension is a reasonable bonus. If I were working on impurity-bound-state diagnostics, I would cite the single-defect part after a cleanup.\n\nWhat the paper does not deliver is the advertised agreement between the two DOS methods. In Section V.A the authors state explicitly that in the topological phase at weak disorder the locator scheme predicts gap narrowing while the Bloch-state (t-matrix+RPA) scheme predicts gap widening, and that the origin is not understood. That directly contradicts the abstract's \"matching densities of states\" and the conclusion's \"mutually consistent results.\" The stress-test note is on target here, with one qualification: the discrepancy is specific to the topological phase at weak disorder. Trivial-phase, Weyl, and AQH DOS curves do agree, so the broad claims about Weyl robustness and AQH plateau shrinkage rest on firmer ground than the topological-phase claims. Still, the gap-closure and plateau-narrowing narrative is sold as if the two methods cross-check everything, and that is not true.\n\nThere are two concrete technical errors a referee should insist on. Equation (9) states the delocalization condition as A > ln(ΔS/ΔD); from Eq. (7) the threshold is A > ln(ΔD/ΔS). That is a sign error in the central condition for destroying the topological phase. Second, Eq. (62) uses |⟨ln ΔS⟩ − ⟨ln ΔD⟩|, while the localization-length plots show signed values, especially Fig. 15(b) where the inverse length approaches zero “from below.” For Gaussian and Lorentzian distributions the argument ΔS+η has support across zero, so the real logarithm is not defined; the formula needs |ΔS+η|. The Gaussian divergence and the edge-conductance correction therefore need to be re-examined. These are fixable, but they are load-bearing for the localization/transport section.\n\nWho is this for? A theory reader interested in exact single-defect signatures of topology will get value from Sec. IV. The rest is a promising but unfinished treatment of off-diagonal disorder. It deserves a serious referee, because the core defect result is new and likely correct, and the open problems are specific. I would send it to review with the expectation of major revision, not desk-reject it.","headline":"The single-defect crossing rule is a genuine result, but the paper's finite-disorder DOS claims are overstated and the localization-length section has sign errors that need fixing before publication.","tokens_in":26781,"tokens_out":7667,"would_cite":false,"duration_ms":77911,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In a multilayer topological insulator, a single interlayer-tunneling defect creates an in-gap bound state that crosses zero energy only in the trivial phase, never in the topological phase, making the defect a local marker of bulk topology.","keywords":["off-diagonal disorder","multilayer topological insulator","Burkov-Balents model","interlayer tunneling","bound states","density of states","Weyl semimetal","anomalous quantum Hall effect"],"falsifier":"A numerically exact calculation on a finite multilayer with random interlayer tunneling would settle the central claims: if a single defect's in-gap level crosses zero in the topological phase for any finite defect strength, or if the disorder-averaged density of states at moderate disorder does not fill and close the gap, the picture fails. The same calculation can test the Gaussian delocalization by measuring the inverse participation ratio of zero-energy edge states as the fluctuation width grows.","tokens_in":2102,"feed_emoji":"🧲","tokens_out":3034,"duration_ms":121736,"temperature":0.7,"pith_summary":"This paper asks what random interlayer tunneling - off-diagonal disorder - does to a multilayer topological insulator built from alternating topological and trivial layers. Its central claim is that a single tunneling defect creates an in-gap bound state whose energy crosses the middle of the gap only when the stack is in the trivial phase, never in the topological phase, so the defect acts as a local marker of bulk topology. For many defects, the paper argues, disorder fills the gap with bulk states and can close it: the Weyl semimetal phase survives strong disorder, the anomalous quantum Hall phase does not, and the added bulk states shrink the Hall plateau. It also claims that the disorder statistics control edge-mode localization, with Gaussian disorder able to delocalize the edges and thereby pull the longitudinal conductance away from its quantized value without breaking time-reversal symmetry. A sympathetic reader would care because these effects supply testable signatures - a narrowed Hall plateau and a suppressed edge conductance - for off-diagonal disorder in real van der Waals stacks.","feed_headline":"Bound state crosses zero only in the trivial phase","feed_subtitle":"Random interlayer hopping fills the gap and can delocalize edge modes, eroding quantized conductance.","key_machinery":"The load-bearing object is the chiral, off-block-diagonal structure of the Burkov-Balents Hamiltonian in the layer basis, combined with the pole condition for a single defect. The defect perturbation enters as $\\delta\\hat h = \\tau^x\\,\\delta\\Delta_S$, and because the clean Hamiltonian is chiral ($KHK^{-1}=-H$, e.g. $K=\\tau^z\\otimes\\sigma^z$), every eigenstate has a partner at opposite energy. In-gap bound states are poles of the Dyson-resummed Green function, $\\det(1-G^0\\,\\delta\\hat h)=0$; inside the gap the local Green operator has eigenvalues $\\lambda_\\pm = G_x \\pm |G_0|$, and the condition $1/\\delta\\Delta_S = \\lambda_\\pm(\\varepsilon)$ controls whether a bound-state branch crosses zero. For the disordered density of states the paper uses two complementary self-consistent schemes: the locator expansion in localized states (each layer carrying a local propagator) and the $t$-matrix+RPA expansion in Bloch states. At zero energy the projector structure of the locators makes every diagram that returns to its starting layer vanish, so the Green function factorizes into a product of strictly forward locators, giving the inverse localization length $L_c^{-1}=|\\langle\\ln\\Delta_S^i\\rangle-\\langle\\ln\\Delta_D^i\\rangle|$.","core_discovery":"The central discovery is that off-diagonal disorder - random interlayer tunneling - acts on the Burkov-Balents multilayer in a phase-selective way. A single Hermitian tunneling defect, a change $\\delta\\Delta_S$ in one layer's tunneling amplitude, creates an in-gap bound state whose energy crosses zero only in the trivial (normal-insulator) phase, at $\\delta\\Delta_S = -\\Delta_S$; in the topological phase the pole condition $\\delta\\Delta_S/(\\Delta_S+\\delta\\Delta_S)=1$ has no finite-strength solution, so the state never reaches zero energy, and a non-Hermitian defect splits the level without changing the crossing rule. For a finite density of defects, the paper argues, disorder fills the gap with bulk states and can close it: the Weyl semimetal phase survives strong fluctuations, while the anomalous quantum Hall phase is fragile and its Hall plateau shrinks as disorder grows. In the topological phase at weak disorder the two self-consistent schemes (locator and $t$-matrix+RPA) disagree on whether the gap narrows or widens, a discrepancy the paper reports as not yet understood. At zero energy each Green-function diagram containing a return path vanishes, leaving only forward paths, which yields a disorder-dependent localization length for edge modes; for Gaussian fluctuations the inverse localization length $L_c^{-1}$ can vanish, so opposite edges overlap and the longitudinal conductance drops below $\\pi e^2/4h$ even though time-reversal symmetry and chirality are preserved.","pith_inferences":["Beyond the paper, the zero-crossing rule could be tested locally with scanning tunneling spectroscopy: tune a single junction between two layers and look for a zero-bias in-gap resonance only when the clean stack is trivial.","Beyond the paper, the unresolved weak-disorder discrepancy in the topological phase - gap narrowing versus widening - is a concrete target for exact diagonalization or transfer-matrix numerics, and whichever approximation survives would change how disorder renormalizes the topological mass.","Beyond the paper, the predicted divergence of the Gaussian localization length suggests a disorder-driven loss of helical protection in the thermodynamic limit; finite-size transport simulations could look for a threshold in fluctuation variance where two-terminal conductance departs from quantization.","Beyond the paper, applying the same forward-path analysis to related layered models, such as antiferromagnetic or superconducting stacks, would test whether the zero-energy diagrammatic cancellation generalizes to other chiral multilayer Hamiltonians."],"forward_implications":["A single off-diagonal defect can serve as a local, topologically selective probe: its zero-energy crossing occurs only in the trivial phase, not in the topological phase, and adding non-Hermitian loss or gain or asymmetric hopping preserves that rule.","Finite off-diagonal disorder generates bulk in-gap states and can close the gap; in the anomalous quantum Hall regime this shrinks the Hall plateau, which the paper proposes as an explanation for experimental deviations from quantized Hall behavior.","The Weyl semimetal phase is robust against strong off-diagonal disorder, while the anomalous quantum Hall phase is not, so disorder of this type produces a phase-selective stability.","The edge-mode penetration depth is distribution-dependent; Gaussian or Lorentzian disorder enlarges it and in the Gaussian case can make it diverge, enabling inter-edge tunneling and a correction that lowers the longitudinal conductance below $\\pi e^2/4h$ without breaking time-reversal symmetry.","In the gapless Dirac or Weyl phase the Green-function series never converges, implying that off-diagonal disorder cannot localize zero-energy states or open a gap there."],"supporting_citations":[{"why":"Supplies the clean Burkov-Balents multilayer Hamiltonian and its phase diagram, which the disordered model extends.","marker":"[43]"},{"why":"Establishes the locator expansion in localized states and the earlier study of disorder-driven phase transitions that this work builds on.","marker":"[42]"},{"why":"Provides the Green-function-zero criterion for impurity bound states as local signatures of topology, which the single-defect analysis adapts to off-diagonal disorder.","marker":"[12]"},{"why":"Supplies the t-matrix cancellation trick and the localization criterion used for the Bloch-state expansion and the zero-energy convergence test.","marker":"[48]"},{"why":"Derives the self-consistent locator equations that form one of the two diagrammatic schemes.","marker":"[49]"},{"why":"Provides the generalized locator-coherent-potential formulation that underlies the self-consistent treatment of the interlayer hopping.","marker":"[50]"},{"why":"Supplies the earlier theory of localization with off-diagonal disorder against which the multilayer edge-mode localization is compared.","marker":"[23]"},{"why":"Gives the exact one-dimensional treatment of off-diagonal disorder whose mid-gap singularity the paper contrasts with its own regular density of states at zero energy.","marker":"[14]"},{"why":"Provides the localization-length estimate for off-diagonal disorder that the paper applies to the edge-mode penetration depth.","marker":"[15]"}],"fun_headline_variants":["Off-diagonal disorder: gap-filling and edge delocalization","Disorder closes gap, delocalizes edges in topological multilayers","Bound state zero-crossing marks trivial phase","Random tunneling shrinks Hall plateau, delocalizes edges","Weyl robust, quantum Hall fragile under off-diagonal disorder"],"cache_read_input_tokens":29056,"weakest_assumption_plain":"The load-bearing premise is that the two approximate calculation schemes used for the disordered density of states are valid in every regime studied; this is strained because at weak disorder in the topological phase one scheme says the gap narrows while the other says it widens, and the paper leaves that disagreement unexplained.","fun_headline_variants_meta":{"raw":{"variants":["Off-diagonal disorder: gap-filling and edge delocalization","Disorder closes gap, delocalizes edges in topological multilayers","Bound state zero-crossing marks trivial phase","Random tunneling shrinks Hall plateau, delocalizes edges","Weyl robust, quantum Hall fragile under off-diagonal disorder"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000465,"raw_usage":{"total_tokens":2377,"prompt_tokens":1057,"completion_tokens":1320,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":673,"completion_tokens_details":{"reasoning_tokens":1236}},"tokens_in":673,"tokens_out":1320,"duration_ms":11146,"temperature":1.0,"reasoning_tokens":1236,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T13:22:55.686912+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A numerically exact calculation on a finite multilayer with random interlayer tunneling would settle the central claims: if a single defect's in-gap level crosses zero in the topological phase for any finite defect strength, or if the disorder-averaged density of states at moderate disorder does not fill and close the gap, the picture fails. The same calculation can test the Gaussian delocalization by measuring the inverse participation ratio of zero-energy edge states as the fluctuation width grows.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the clean Burkov-Balents multilayer Hamiltonian and its phase diagram, which the disordered model extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the locator expansion in localized states and the earlier study of disorder-driven phase transitions that this work builds on."},{"cited_title":"Zhang, P","cited_arxiv_id":null,"evidence_quote":"Provides the Green-function-zero criterion for impurity bound states as local signatures of topology, which the single-defect analysis adapts to off-diagonal disorder."},{"cited_title":"Gao, Y.-F","cited_arxiv_id":null,"evidence_quote":"Supplies the t-matrix cancellation trick and the localization criterion used for the Bloch-state expansion and the zero-energy convergence test."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the self-consistent locator equations that form one of the two diagrammatic schemes."},{"cited_title":"Theodorou and M","cited_arxiv_id":null,"evidence_quote":"Supplies the earlier theory of localization with off-diagonal disorder against which the multilayer edge-mode localization is compared."},{"cited_title":"Jiang, Z","cited_arxiv_id":null,"evidence_quote":"Gives the exact one-dimensional treatment of off-diagonal disorder whose mid-gap singularity the paper contrasts with its own regular density of states at zero energy."},{"cited_title":"Jiang, H","cited_arxiv_id":null,"evidence_quote":"Provides the localization-length estimate for off-diagonal disorder that the paper applies to the edge-mode penetration depth."}],"review_version":1}