{"id":"98a672f3-cb9e-4d2b-8e2e-5f352cfb23c4","arxiv_id":"2608.09758","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A new 'chiral overlap' quantity bounds how much disorder can split a near-zero-energy state, explaining how topologically trivial Andreev states can mimic Majorana robustness.","lead":"This paper finds that the two components of a low-energy nanowire state determine how strongly disorder can push it away from zero energy. If those components sit in different regions, a topologically ordinary state can stay near zero energy even under strong disorder, so robustness alone is not proof of Majorana topology.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"First-order chiral-overlap bound is extrapolated to strong full-wire disorder; numerical evidence uses only left-half disorder, so the broad claim is not yet established.","rationale":"The reader's weakest assumption correctly identifies that Eq. (22) is only a first-order bound and that the strong-disorder demonstration is limited to left-half disorder. I agree that this is the most load-bearing gap: the paper's headline claim is that chiral overlap controls disorder robustness, yet the only analytic statement is first-order and the only strong-disorder numerics use an asymmetric disorder profile. The proposed full-wire disorder test would settle whether the numerical robustness is generic or an artifact of the placement; the moderate-V0 comparison would test whether the first-order bound has any exact counterpart. If the full-wire test passes, the mechanism is supported empirically, but the abstract should still be qualified to first-order or to the specific disordered-eigenstate computation. If it fails, the central demonstration collapses. Since the reader already requested similar checks and conditionally accepted the paper, my read does not change that verdict.","tokens_in":17796,"tokens_out":12165,"duration_ms":111685,"concrete_test":"Repeat the strong-disorder simulation of Fig. 4(j,k,l) with V0 = 50 mu_R applied to (i) the entire wire and (ii) the right half only, with at least 100 disorder realizations, and record the lowest positive energy and the disordered-eigenstate chiral overlap Omega for B_c1 < B < B_c2. If the lowest positive energy rises above 10^-3 mu_R or Omega becomes O(1) in either placement, the claimed robustness is not generic; if the mode remains pinned and Omega stays small for all placements, the mechanism is independent of where the disorder acts. As a secondary check, at moderate V0 = 1 mu_R compare the exact |E(V0)-E0| with V_max Omega_clean to test whether the first-order bound actually controls the exact splitting.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bound in Eq. (22), |delta E^(1)| <= V_max Omega, is derived for the first-order shift of a single clean eigenstate under a perturbation V. The abstract and Sec. II state without qualification that 'disorder-induced splitting is bounded by their spatial overlap,' but no exact, non-perturbative version of this bound is proven. For V0 = 50 mu_R, V is not a small perturbation, and the exact eigenvalue of H_clean + V can in principle acquire contributions from all other states, uncontrolled by the clean-state Omega. The numerical support in Sec. V is also narrower than the claim: disorder is applied only to the left half of the wire (Eq. 29 and text), i.e., only to the spatial region of one chiral component. If disorder were present on the right half or across the whole wire, the chiral components could be moved together or coupled elsewhere, and the apparent robustness of the doubly smooth case (Fig. 4j-l) might be an artifact of the asymmetric disorder placement. Because the paper's central demonstration of 'disorder-robust trivial Majorana-like states' rests on this simulation, this gap is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the stability of near-zero-energy states in chiral-symmetric Bogoliubov–de Gennes nanowires against chiral-preserving local disorder. It introduces a chiral overlap Ω defined from the two opposite-chirality components of a low-energy wavefunction and proves a first-order bound |δE^(1)| ≤ V_max Ω (Eq. 22) for the energy shift induced by a local perturbation that anticommutes with the chiral operator. The paper then applies this criterion to a finite Rashba nanowire with smoothly varying chemical potential and pairing profiles, showing numerically that a topologically trivial Andreev bound state with small Ω remains near zero energy under strong scalar disorder applied to the left half of the wire. The central conclusion is that disorder robustness in this system is controlled by the real-space overlap of the chiral components rather than by the bulk topological invariant.","tokens_in":17992,"tokens_out":10490,"duration_ms":90228,"significance":"The proposed chiral-overlap diagnostic is conceptually appealing and potentially useful for interpreting zero-bias conductance peaks in Majorana nanowire experiments: it provides a simple, parameter-free measure that can be computed from the clean wavefunction and that quantitatively predicts first-order sensitivity to chiral-preserving disorder. The derivation of Eq. (22) is correct and transparent, using only Cauchy–Schwarz and locality. The numerical observation of a disorder-robust trivial regime in the doubly smooth inhomogeneous wire is interesting and, if confirmed, would strengthen the case that zero-energy pinning alone is not a topological fingerprint. However, the paper's central claim is stated more strongly than what is proven: the bound is first-order, while the numerical demonstration uses V0 = 50 μ_R, far beyond the perturbative regime, and only for disorder on the left half. This limits the current support for the advertised 'disorder-robust trivial Majorana-like states' conclusion.","major_comments":[{"comment":"The bound |δE^(1)| ≤ V_max Ω is derived for the first-order shift of a single clean eigenstate under a perturbation V. The abstract and Sec. II nonetheless state without qualification that 'disorder-induced splitting is bounded by their spatial overlap' and that a state with small chiral overlap 'is insensitive to any local perturbation that preserves the chiral symmetry.' For the strong disorder used in Sec. V (V0 = 50 μ_R, corresponding to 25 meV for the chosen μ_R = 0.5 meV, i.e., 25 times the hopping t = 1 meV), V is far from a small perturbation, and the exact eigenvalue of H0 + V can receive contributions from all other states that are not controlled by the clean-state Ω. A non-perturbative version of the bound would require controlling the exact eigenstate's overlap, which is not provided. Please either (i) prove such a bound under explicit assumptions, or (ii) carefully qualify the central claim to 'first-order' or 'weakly perturbative' and adjust the abstract, Sec. II, and the concluding remarks accordingly.","section":"§II, Eq. (22)"},{"comment":"The numerical evidence for the robustness of the trivial near-zero mode is obtained with disorder applied only to the left half of the wire. In the doubly smooth case (Figs. 4(j–l) and 8(d)), the left chiral component is initially localized at the left end, so left-half disorder effectively cuts off that component but does not perturb the Gaussian right component at the NS interface. This asymmetric placement does not probe the scenario in which disorder on the right half or across the entire wire couples the two chiral components in other spatial regions. Because the paper's broad conclusion—'disorder-robust trivial Majorana-like states'—rests on this simulation, the authors should provide additional numerical results with right-half or full-wire disorder, or a theoretical argument explaining why left-half disorder is sufficient to establish the general claim.","section":"§III–V, Eq. (29), Figs. 4 and 6"},{"comment":"The overlap Ω shown in Fig. 6 under strong disorder is computed from the eigenstates of the disordered Hamiltonian, not from the clean wavefunctions used in the bound Eq. (22). The observation that the disordered Ω remains small in the robust regime is evidence that the system's own chiral components stay separated, but it does not prove that the disorder-induced splitting is bounded by the clean Ω. The logical status of Fig. 6 should be clarified: it is a diagnostic of the disordered eigenstate, not a verification of Eq. (22).","section":"§V, Fig. 6"}],"minor_comments":[{"comment":"The figure captions for Figs. 3 and 4 use inconsistent panel labels relative to the text. For example, the text refers to 'Fig. 3(a,b,c)' for the case W_μ = L/2, W_Δ = 0, but the caption labels the spectra as (a), (c), (e) and the wavefunctions as (b), (d), (f). Please unify the labeling (e.g., (a), (b,c) for the first case, (d), (e,f) for the second, and (g), (h,i) for the third) or revise the text.","section":"Figs. 3 and 4 captions"},{"comment":"The equality δE^(1) = ⟨ψ_E|V|ψ_E⟩ is the standard non-degenerate first-order result. If the near-zero pair is exactly degenerate at E = 0, the first-order splitting should be obtained from degenerate perturbation theory; the manuscript should justify that the non-degenerate formula remains the relevant one for the near-zero pair considered.","section":"§II, Eq. (13)"},{"comment":"Consider stating explicitly that V0 = 50 μ_R corresponds to 25 meV for the parameters used, which is 25 times the hopping t = 1 meV, to make the non-perturbative nature of the numerical test transparent.","section":"§III"},{"comment":"The statement 'We have confirmed that increasing disorder realizations does not affect our results' would be more convincing with a plot of the disorder-averaged splitting including statistical error bars, especially because only 10 realizations are used.","section":"§V"},{"comment":"There are minor typographical issues: 'we use W μ =L/2, W Δ = 0' has inappropriate spaces; 'To end this section, we note that Symmetry-based arguments' has an unnecessary capital 'S'; and the sentence structure around Eq. (25) could be smoothed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The first-order bound in Eq. (22) is sound and the numerical observation is potentially interesting, but the central claim overreaches the proof: the abstract and Sec. II state an unqualified bound on the disorder-induced splitting, while the proof covers only first-order shifts of a clean eigenstate. The authors can likely address the main concerns by restricting the claim to first order and by adding full-wire or right-half disorder simulations to support the strong-disorder robustness. No ethical concerns; the work fits the journal scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper's core is a simple inequality, and it is correct as far as it goes. For a chiral-symmetric Hamiltonian, any local chiral-symmetric perturbation V with local blocks bounded by V_max gives |<phi+|V|phi->| <= V_max Omega, where Omega is the spatial overlap of the two chiral components. That is a clean first-order bound, and using it to explain why trivial near-zero modes in smooth-confinement nanowires can look robust is a genuinely useful framing. The diagnostic is distinct from the usual Majorana-component overlap, and the paper does not fit any parameter to make the correlation work: Omega is computed from clean eigenstates, and the disorder splittings come from independent realizations.\n\nThe numerical study is competent. The sharp NS junction is fragile under disorder, the doubly smooth case shows a broad field window where the trivial mode stays pinned, and the Bc1 crossover from interface-like to chiral-separated wavefunctions is visible in the local chiral density. The authors also place their result relative to earlier Majorana-polarization measures.\n\nThe soft spots are two, and they are connected. First, the abstract and Sec. II state that disorder-induced splitting is bounded by the spatial overlap, without the first-order qualifier. Only the first-order energy shift is proven. The numerics use V0 = 50 mu_R, which is not a small perturbation, so the exact splitting could in principle receive contributions from other states that the clean Omega does not control. The paper needs either a non-perturbative argument or a numerical check that the bound tracks the exact splitting over a range of V0.\n\nSecond, the disorder is applied only to the left half of the wire. That is exactly the region where one chiral component sits in the doubly smooth case. Full-wire disorder, or disorder on the right half, might couple the components differently. The mechanism would probably still suppress the first-order matrix element because the components are spatially separated everywhere, but the current evidence does not establish the broad disorder-robust claim as stated. A quick simulation with disorder across the whole wire would settle this.\n\nOverall, a solid contribution to the Majorana-nanowire subfield. It does not resolve the detection problem, and the headline claim overreaches, but the bound is correct and the diagnostic is useful. I would send it to peer review and ask for revision: add the first-order qualifier, and either extend the numerics to full-wire disorder or restrict the claims accordingly.\n\nRecommendation: engage with it, but with the above caveats.","headline":"A correct first-order chiral-overlap bound, over-sold as an exact disorder-robustness guarantee, with numerics that only test disorder on one half of the wire.","tokens_in":18522,"tokens_out":4787,"would_cite":true,"duration_ms":41329,"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":"Disorder robustness of a near-zero mode is set by the real-space overlap of its two chiral components, not by the bulk topological invariant.","keywords":["Majorana bound states","Andreev bound states","chiral symmetry","disorder robustness","chiral overlap","smooth confinement","Rashba nanowire","topologically trivial zero modes"],"falsifier":"A decisive numerical check is to compute the exact disorder-averaged splitting of the near-zero pair as a function of disorder strength $V_0$ for a smooth-profile wire above $B_{c1}$ while holding $\\Omega$ fixed: if the splitting grows linearly with $V_0$ although $\\Omega$ is exponentially small, the bound is not controlling the robustness, and if it stays exponentially small up to $V_0\\approx 50\\mu_R$, the mechanism is confirmed. A complementary experiment is to replace scalar disorder with chiral-symmetry-breaking magnetic disorder of the same strength, which should immediately lift the trivial near-zero pinning if the protection is chiral in origin.","tokens_in":17565,"feed_emoji":"⚛️","tokens_out":8719,"duration_ms":69574,"temperature":0.7,"pith_summary":"This paper argues that a near-zero-energy state in a chiral-symmetric superconductor can survive symmetry-preserving local disorder even when the system is topologically trivial. The mechanism is a wavefunction property: the state decomposes into two components of opposite chirality, and any chiral-symmetric perturbation can split the state only by coupling those components, with the splitting bounded by their spatial overlap $\\Omega$. In a finite Rashba nanowire with smoothly varying chemical potential and pairing, smooth confinement separates the two chiral components below the bulk topological transition, producing trivial Majorana-like Andreev bound states that stay exponentially close to zero energy under strong scalar disorder. The result means that zero-energy pinning under disorder is not by itself evidence of a topological phase, and it proposes the chiral overlap as a concrete diagnostic for classifying robust low-energy states.","feed_headline":"Chiral separation pins trivial zero modes despite strong disorder","feed_subtitle":"The chiral overlap of the low-energy wavefunction, not the bulk invariant, sets how disorder splits the mode.","key_machinery":"The central object is the chiral overlap $\\Omega=\\sum_x\\sqrt{\\rho_+(x)\\rho_-(x)}$, formed from the two normalized chiral densities of a low-energy eigenstate obtained by projecting the state onto the $\\pm1$ eigenspaces of the chiral operator $\\Gamma$. The load-bearing identity is the bound $|\\delta E^{(1)}|\\le V_{\\max}\\Omega$: chiral symmetry forces any local symmetry-preserving perturbation to be off-diagonal in the chiral basis, so its effect on the energy is exactly the inter-chirality matrix element, whose magnitude is controlled by the real-space overlap of the two chiral components. The machinery also includes the smooth step profiles for the chemical potential and pairing (tanh profiles with widths $W_\\mu$, $W_\\Delta$) that realize partial chiral separation, and the crossover field $B_{c1}$ defined by the onset of small $\\Omega$, which is distinct from the bulk topological transition field $B_{c2}$.","core_discovery":"For a Bogoliubov–de Gennes Hamiltonian with a unitary chiral symmetry $\\Gamma$ ($\\Gamma^2=1$, $\\{H,\\Gamma\\}=0$), the paper decomposes any nonzero-energy eigenstate into normalized chiral components $\\phi_+$, $\\phi_-$ and shows that a local perturbation $V$ preserving chiral symmetry has no diagonal matrix elements in the chiral basis. The first-order energy shift of the state is $\\delta E^{(1)}=\\mathrm{Re}\\langle\\phi_+|V|\\phi_-\\rangle$, which is bounded by $V_{\\max}\\Omega$, where $\\Omega=\\sum_x\\sqrt{\\rho_+(x)\\rho_-(x)}$ is the spatial overlap of the two chiral densities. The central statement is that a state with small $\\Omega$ is insensitive to any local chiral-symmetric perturbation even without a global topological invariant. In a finite Rashba nanowire with smooth chemical-potential and pairing profiles, the lowest state develops exponentially small $\\Omega$ above a crossover field $B_{c1}$ but below the bulk topological transition $B_{c2}$; numerical simulations with scalar disorder of strength $V_0=50\\mu_R$ show these trivial Majorana-like states remain pinned near zero energy, whereas sharp-interface Andreev bound states are lifted. The paper concludes that disorder robustness is controlled primarily by the wavefunction's chiral overlap rather than by the bulk topology.","pith_inferences":["Beyond the paper's own claims, the chiral-overlap criterion should apply to other chiral-symmetric platforms, such as nodal superconductors, topological insulator edges, or engineered cold-atom wires, wherever disorder robustness of zero modes is debated.","A testable extension is to extract $\\Omega$ experimentally from spin-resolved local density-of-states measurements or from the response of a zero-bias peak to a local gate that shifts the chemical potential, without having to vary the disorder itself.","The paper demonstrates the bound numerically for disorder on the left half of the wire with ten realizations; an open extension is to check whether the suppression of splitting persists for full-wire disorder, larger disorder strengths, or higher-order perturbation theory, where renormalization of $\\Omega$ may become important.","The $B_{c1}$ versus $B_{c2}$ distinction suggests a third category beyond simply 'topological' or 'trivial' robust zero modes: locally Majorana-like but globally trivial states may be generic in inhomogeneous devices, which could matter for interpreting future braiding or fusion-rule experiments."],"forward_implications":["Robust zero-bias conductance peaks appearing below the estimated bulk topological transition should not be read as evidence of a global topological phase; they can come from chiral-separated trivial Andreev bound states.","The chiral overlap $\\Omega$ provides a direct diagnostic: computing it for the low-energy state tells whether the state will survive chiral-symmetric local disorder, independent of the bulk topological invariant.","The crossover field $B_{c1}$, where $\\Omega$ becomes small, marks the onset of disorder robustness and does not coincide with the bulk gap-closing field $B_{c2}$.","Chiral-symmetry-preserving disorder can reshape and move the low-energy chiral components without splitting the near-zero pair; the same bound applies to any local perturbation that anticommutes with $\\Gamma$, including certain magnetic impurity configurations.","In sharp-interface or single-profile-smooth wires, $\\Omega$ stays large in the trivial regime, and the same strong disorder lifts the near-zero modes, consistent with the known fragility of sharp-interface Andreev bound states."],"supporting_citations":[{"why":"Supplies the experimental parameter values for InSb and InAs nanowires used in the numerical model.","marker":"[11]"},{"why":"Establishes the Andreev–Majorana crossover and partially separated Andreev bound states that this paper's chiral-separated states build on.","marker":"[12]"},{"why":"Provides the prior result that sharp-interface trivial Andreev bound states in these nanowires are fragile to small disorder, the contrast case for the smooth-profile robustness.","marker":"[20]"},{"why":"Shows that smooth confinement creates near-zero-energy end states in topologically trivial spin-orbit-coupled superconducting nanowires, the starting point for the demonstration.","marker":"[21]"},{"why":"Introduces the Majorana/Andreev crossover in inhomogeneous nanowires and the real-space separation that motivates the chiral overlap.","marker":"[22]"},{"why":"Quantifies wave-function overlaps in inhomogeneous Majorana nanowires, the measure to which this paper's $\\Omega$ is compared.","marker":"[23]"},{"why":"Provides the chiral-symmetry argument for zero-energy edge states in superconductors that underpins the decomposition into chiral sectors.","marker":"[53]"},{"why":"Shows stability of flat zero-energy states at dirty surfaces of nodal superconductors, the similar symmetry-based mechanism referenced in the discussion.","marker":"[55]"}],"fun_headline_variants":["Chiral overlap dictates disorder robustness of trivial Majoranas","Smooth confinement yields disorder-robust trivial zero modes","Disorder-proof Majorana-like states from smooth confinement","Chiral separation explains disorder-tolerant zero modes","Trivial Majorana modes survive disorder via chiral separation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's load-bearing premise is that a bound on the energy shift computed by treating disorder as a small perturbation continues to control the exact splitting of the near-zero pair at the extremely strong disorder used in the simulations, which are run with disorder on the left half of the wire only and averaged over just ten realizations.","fun_headline_variants_meta":{"raw":{"variants":["Chiral overlap dictates disorder robustness of trivial Majoranas","Smooth confinement yields disorder-robust trivial zero modes","Disorder-proof Majorana-like states from smooth confinement","Chiral separation explains disorder-tolerant zero modes","Trivial Majorana modes survive disorder via chiral separation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000664,"raw_usage":{"total_tokens":3054,"prompt_tokens":989,"completion_tokens":2065,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":1990}},"tokens_in":605,"tokens_out":2065,"duration_ms":11996,"temperature":1.0,"reasoning_tokens":1990,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:18:26.555831+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive numerical check is to compute the exact disorder-averaged splitting of the near-zero pair as a function of disorder strength $V_0$ for a smooth-profile wire above $B_{c1}$ while holding $\\Omega$ fixed: if the splitting grows linearly with $V_0$ although $\\Omega$ is exponentially small, the bound is not controlling the robustness, and if it stays exponentially small up to $V_0\\approx 50\\mu_R$, the mechanism is confirmed. A complementary experiment is to replace scalar disorder with chiral-symmetry-breaking magnetic disorder of the same strength, which should immediately lift the trivial near-zero pinning if the protection is chiral in origin.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Andreev–Majorana crossover and partially separated Andreev bound states that this paper's chiral-separated states build on."},{"cited_title":"Ahmed, Y","cited_arxiv_id":null,"evidence_quote":"Shows that smooth confinement creates near-zero-energy end states in topologically trivial spin-orbit-coupled superconducting nanowires, the starting point for the demonstration."},{"cited_title":"Marra and A","cited_arxiv_id":null,"evidence_quote":"Quantifies wave-function overlaps in inhomogeneous Majorana nanowires, the measure to which this paper's $\\Omega$ is compared."},{"cited_title":"Kaladzhyan, J","cited_arxiv_id":null,"evidence_quote":"Provides the chiral-symmetry argument for zero-energy edge states in superconductors that underpins the decomposition into chiral sectors."},{"cited_title":"Mizushima, Y","cited_arxiv_id":null,"evidence_quote":"Shows stability of flat zero-energy states at dirty surfaces of nodal superconductors, the similar symmetry-based mechanism referenced in the discussion."}],"review_version":1}