{"id":"28fc1a0b-e642-4887-8615-d16c0ea3bb5e","arxiv_id":"1908.03303","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"In disordered QAH-superconductor junctions, a single chiral edge mode keeps perfect zero-bias transmission, while multiple chiral modes or an added metallic mode produce strong Andreev conversion and vanishing charge transmission.","lead":"This paper simulates how disorder affects electrical transport in junctions between a quantum anomalous Hall insulator and a superconductor, depending on how many edge channels exist. It predicts that extra channels sharply change the conductance and could help distinguish competing explanations of recent experiments.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Zero-bias robustness in the single-CEM case is only checked for chemical-potential disorder; pairing disorder or full-model disorder could lift T13 from unity.","rationale":"The reader's weakest assumption identifies the same load-bearing gap: the single-CEM zero-energy robustness is demonstrated only for chemical-potential disorder in the effective edge model, and unchecked pairing or full-model disorder could invalidate it. My stress-test neither confirms nor refutes the central claim; it sharpens the condition under which the claim holds and proposes a concrete numerical test. The clean-limit agreement with the full model is real support for the effective model, but it does not cover the disordered regime, which is the paper's main new content. The reader's CONDITIONAL verdict already reflects this uncertainty, so no adjustment is needed.","tokens_in":12553,"tokens_out":16488,"duration_ms":183053,"concrete_test":"In the microscopic Hamiltonian (1), add an uncorrelated random on-site singlet-pairing fluctuation δΔ0(x) with zero mean and amplitude comparable to μ_imp in region II, keeping the same geometry and parameters as in Fig. 1c/d. Compute the disorder-averaged zero-bias T13 using the recursive Green's function method over at least 100 configurations. If ⟨T13(ε=0)⟩ remains unity within error bars, the s-wave pairing-disorder concern is refuted; if it drops visibly below unity, the single-CEM zero-bias robustness claim fails for generic interface disorder.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central contrast—zero-bias T13 remains unity for a single CEM under disorder, while multiple CEMs or non-chiral modes yield exponential decay and half-electron/half-hole transmission—depends on the effective edge Hamiltonian (2) containing only the p-wave pairing `-v_Δ k`. The disordered transport calculation then averages only over local chemical-potential fluctuations μ_i (Section 'Model Hamiltonian and Transport of a single CEM', Fig. 2). This leaves the zero-energy robustness claim narrower than stated: nothing in the manuscript checks whether the full microscopic model (1), which contains both singlet Δ0 and triplet Δz pairing, preserves T13=1 when disorder is included at the microscopic level, nor whether spatial fluctuations in the pairing amplitudes themselves are harmful. Appendix B argues that a constant singlet term is absent in the clean single-CEM projection by particle-hole symmetry, but disorder-induced coupling to evanescent modes could in principle produce an effective constant pairing in the edge subspace; if that happens, the zero-bias perfect transmission and the resistance divergence in Fig. 4a cease to be generic. The clean-limit agreement between Eq. (3) and full-model numerics (Fig. 1c,d) gives good support for the effective model in the ballistic case, but it does not test the disordered case.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies disordered quantum transport through a planar junction between a quantum anomalous Hall insulator and a superconductor, using effective edge models. For a single chiral edge mode (CEM), described by a k-linear p-wave pairing Hamiltonian, the authors derive an analytic transmission formula T13(ε,L) and validate it against numerical transport simulations of a full BdG lattice model in the clean limit. They then add on-site chemical-potential disorder along the junction and find that the zero-energy transmission stays at unity, so the four-terminal resistance diverges at zero bias. In contrast, for two CEMs or a CEM coexisting with non-chiral metallic modes, disorder causes the averaged transmission to decay exponentially with junction length, giving equal electron and hole transmission; the authors propose resistance sum rules R24,14+R24,34 = −h/e², −h/2e², or a non-quantized value to distinguish the four physical cases.","tokens_in":12830,"tokens_out":18943,"duration_ms":194606,"significance":"If the central results hold, the paper makes a useful contribution to the ongoing debate about the e2/2h conductance kink attributed to chiral Majorana modes. It provides concrete, falsifiable signatures: the zero-bias resistance divergence for a single CEM, the exponential decay and half-electron/half-hole conversion for multiple or non-chiral modes, and resistance sum rules that distinguish the cases. The clean-limit analytic formula (Eq. 3) is a strength, as it is checked against full-model numerics, and the disorder-averaged results use a stated number of configurations with standard errors. The main uncertainties are the restricted disorder model and the sign convention of the proposed sum rules; both are addressable.","major_comments":[{"comment":"The zero-bias robustness result for the single CEM is established only against on-site chemical-potential disorder μ_i in the effective edge Hamiltonian (2), although Eq. (1) states that μ, Δ0, and Δz are all in principle spatially dependent to incorporate disorder-induced variations. The particle-hole symmetry argument in Appendix B rules out a constant singlet pairing term only in the clean projection; disorder-induced coupling to evanescent modes could generate an effective constant s-wave pairing in the edge subspace, which would lift T13(ε=0) from unity and remove the resistance divergence in Fig. 4a. The clean-limit agreement in Fig. 1 does not test this. Please add disorder simulations of the full BdG model, or at least disorder in Δ0/Δz within the effective model, or provide a concrete symmetry argument that excludes such an induced term; otherwise the headline claim 'even in the presence of disorders' is narrower than stated.","section":"Model Hamiltonian and Transport of a single CEM (Eq. (2), Fig. 2, Appendix B)"},{"comment":"The sign of the resistance sum rules is inconsistent between the main text and Appendix F. Solving V=T^{-1}I with I=(0,I0,0) for the transmission matrices (F.2) and (F.5) gives R24,14+R24,34=+h/e^2 for the single CEM and +h/2e^2 for the two-CEM case, when R24,34≡−V34/I24 with V34=V3−V4; the main text and Fig. 4 report −h/e^2 and −h/2e^2. Please reconcile the definition of V34 or I24 so that the stated sum rules and the plotted curves correspond to the same convention. Since these sum rules are the proposed experimental discriminators, the inconsistency must be fixed.","section":"Experimental Relevance and Appendix F (Eqs. (F.2)–(F.13))"}],"minor_comments":[{"comment":"Please state how vf and vΔ are obtained from the microscopic model parameters used in the fits of Fig. 1; without this, the good agreement between Eq. (3) and the full lattice calculation is a fit of the functional form rather than an ab initio validation of the effective parameters.","section":"Model Hamiltonian and Transport of a single CEM"},{"comment":"In the multiple-CEM case, 'half-electron half-hole transmission' is inferred from T13→0; please define T13 and explicitly state the current-conservation relation (e.g., Tee+Teh=N for all-chiral incoming modes) that converts T13=0 into equal electron and hole probabilities.","section":"Disordered transport of multiple modes"},{"comment":"The typesetting of the sine argument is ambiguous; it should be clear that L multiplies the entire prefactor sqrt(...)/(vf²-vΔ²).","section":"Eq. (3)"},{"comment":"'MEM' appears to be a typo for 'CEM' in the phrases 'Single MEM case' and 'Multiple MEMs case'.","section":"Fig. 2 caption"},{"comment":"The stray 'T' after 'linear in k' in the paragraph containing Eq. (B.3) should be removed.","section":"Appendix B"},{"comment":"'free mean path' should read 'mean free path'.","section":"Abstract and main text"},{"comment":"The notation for the non-chiral mode velocity is inconsistent: Eq. (6) uses v1 while the parameters are quoted as vf1=-vf2=4.5; please align the notation.","section":"Appendix E"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a compact communication and the core physics is likely correct, but the two issues above (narrow disorder model for the single-CEM claim, and sign inconsistency in the resistance sum rules) must be resolved before publication. I do not see grounds for rejection; the clean-limit validation and effective-model derivation are solid."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWhat you should know: this is a solid effective-model study of disorder in QAH/SC junctions. The main new results are the disorder-averaged exponential decay of transmission for multiple chiral edge modes and for a CEM coexisting with non-chiral modes, plus resistance sum rules that distinguish four experimental scenarios. The single-CEM zero-bias robustness to chemical-potential disorder is real, but it is narrower than the abstract suggests.\n\nThe paper earns its keep. Eq. (3) gives an analytic clean-limit transmission that matches full BdG numerics in Fig. 1, and the disorder averages use 1000 (and 100) configurations with standard errors. The derivation of the effective edge Hamiltonian in Appendix B is clear; the particle-hole symmetry argument for why only p-wave pairing survives in the single-CEM projection is standard and correctly applied. The multiple-CEM and CEM-plus-metallic-mode models are minimal, and the qualitative contrast—T13 = 1 at zero bias for one CEM versus exponential decay with half-electron/half-hole transmission when extra modes are present—is concrete and testable.\n\nThe soft spot is exactly what the stress-test note flags: zero-energy robustness is only checked for on-site chemical-potential disorder in the effective model, not for disorder in the pairing term nor for disorder treated in the full microscopic model. The clean-limit validation of the effective model does not automatically cover the disordered case. Coupling to evanescent modes could in principle generate an effective s-wave pairing in the edge subspace, which would kill the perfect transmission. That is a legitimate caveat, not a fatal flaw: the central contrast between single and multiple modes relies on the p-wave/s-wave distinction, and the paper should have either run full-model disorder or explicitly limited the claim. Missing error bars on the resistance curves and a few typos are minor.\n\nThis paper deserves a serious referee. The predictions are sharp enough that experimentalists will care, and the resistance sum rules give a clean way to distinguish the cases. I would send it out; the main request in review should be to test the zero-bias robustness against pairing disorder or to state the limitation clearly.\n\nBring it to the next group meeting.","headline":"Disorder-averaged transport in QAH/SC junctions is a genuinely useful addition, but the zero-bias robustness for a single chiral mode is only proven for chemical-potential disorder.","tokens_in":13389,"tokens_out":1750,"would_cite":true,"duration_ms":17946,"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":"This paper claims that disorder leaves the zero-bias transmission of a single chiral edge mode through a quantum anomalous Hall/superconductor junction at unity, while multiple edge modes or coexisting metallic modes make the transmission…","keywords":["quantum anomalous Hall effect","chiral edge modes","Andreev conversion","disorder transport","superconductor proximity effect","Majorana modes","Landauer-Büttiker formalism","resistance sum rules"],"falsifier":"Numerically simulate the lattice QAH/SC junction with disorder applied to the pairing order parameter as well as to the chemical potential (e.g., random on-site $\\Delta_0$ with zero mean), and compute $T_{13}$ at $\\epsilon=0$ for a junction many times the elastic mean free path; if $\\bar{T}_{13}$ decays exponentially with $L$, the single-mode p-wave protection is not robust to pairing disorder, and an experimental probe would be a zero-bias two-terminal conductance of $e^2/2h$ rather than $e^2/h$ in the single-mode limit.","tokens_in":12317,"feed_emoji":"⚛️","tokens_out":9050,"duration_ms":90388,"temperature":0.7,"pith_summary":"This paper asks what happens to electron transmission through a quantum anomalous Hall/superconductor junction when elastic scattering is so strong that the junction length is much longer than the mean free path. It claims that with exactly one chiral edge mode, the zero-bias transmission $T_{13}$ remains one in the presence of disorder, because particle-hole symmetry permits only a p-wave pairing term in the edge basis and that term cannot convert electrons into holes at zero energy. When a second chiral mode or a non-chiral metallic mode enters, an s-wave pairing component appears, and disorder then makes the averaged transmission decay exponentially with length; the output becomes a balanced mixture of electrons and holes. The paper also derives resistance relations that distinguish the single-mode, multi-mode, and coexisting-mode regimes in a realistic four-lead measurement, which is why the result matters for interpreting experiments that look for topological-superconductor signatures.","feed_headline":"Disorder can't break zero-bias transmission of one chiral edge mode","feed_subtitle":"Extra modes make disorder push the junction to equal electron and hole output—a measurable signature.","key_machinery":"The analysis is carried by the effective edge Hamiltonian $H_{\\mathrm{eff}}$ for the chiral edge modes, with only a p-wave (linear-in-$k$) pairing term; in the single-mode basis particle-hole symmetry forbids a constant s-wave pairing term, so the Hamiltonian reads $H_{\\mathrm{eff}} = (v_f k - \\mu)\\tau_z - v_\\Delta k \\tau_x$ in the electron-hole basis. This linear pairing is what makes the zero-energy transmission exactly one, because at $\\epsilon = 0$ the two eigenvectors of the transfer matrix acquire equal phase and no particle-hole conversion develops. Disorder is modeled as independent random local chemical potentials distributed uniformly on $[-\\mu_{\\mathrm{imp}}/2, \\mu_{\\mathrm{imp}}/2]$, and the quoted transmission is the ensemble average over many such configurations.","core_discovery":"The central claim, stated on the paper's own terms, is that the diffusive transport regime of a QAH/SC junction is controlled by the number and character of the conducting channels. In the single chiral edge mode case, the effective edge Hamiltonian contains only a p-wave (linear-in-momentum) pairing term, and this is the reason that the electron-electron transmission $T_{ee}$ stays at one while the Andreev-conversion transmission $T_{eh}$ stays at zero as the incident energy $\\epsilon$ approaches zero. Disorder, modeled as random local chemical potentials, does not change that zero-bias result because the p-wave pairing term has no $k$-independent component to scatter the electron-hole pair at zero energy. For multiple chiral modes, or for a chiral mode coexisting with a non-chiral metallic mode, an $s$-wave pairing component between different modes appears, and disorder then dephases the electron-hole oscillations, so the averaged transmission $\\bar{T}_{13}$ decays exponentially with length and the junction acts as a half-electron half-hole converter. The paper further shows that the four-terminal resistances satisfy $R_{24,14}+R_{24,34}=-h/e^2$ for one chiral mode and for the fully disordered coexisting case, and $-h/2e^2$ for two chiral modes, providing a measurable fingerprint of the transport regime.","pith_inferences":["A natural extension is to let disorder act on the pairing term as well as the chemical potential; if interface roughness generates an s-wave component in the edge basis, the zero-bias plateau in the single-mode case should disappear, and the paper's central contrast would be weakened.","The exponential decay in the multi-mode case suggests that the crossover from clean oscillatory conductance to the diffusive half-electron/half-hole limit could be used as a length-dependent diagnostic of how many transport channels actually reach a superconducting contact.","The resistance sum rule offers a practical channel-counting probe that could be checked on existing QAH/SC devices before attempting Majorana interpretation of conductance kinks."],"forward_implications":["In the single chiral edge mode case, the zero-bias transmission $T_{13}$ stays at 1 even when the elastic scattering length is much shorter than the junction, so Andreev conversion is completely suppressed at zero bias.","Away from zero bias, disorder makes $\\bar{T}_{13}$ decay exponentially with junction length, with a decay length that grows as energy goes to zero, so finite-bias measurements lose the protection.","When two chiral modes are present, $\\bar{T}_{13}$ decays exponentially and saturates near zero for long junctions, so the outgoing current becomes half electrons and half holes.","When a chiral mode coexists with a non-chiral metallic mode and the QAH region is disordered, the helical mode Anderson-localizes but still mediates Andreev conversion, so $\\bar{T}_{13}$ decays to zero even at zero energy.","The resistance combinations $R_{24,14}+R_{24,34}$ equal $-h/e^2$ for one CEM and $-h/2e^2$ for two CEMs, giving a measurable distinction between the regimes."],"supporting_citations":[{"why":"Supplies the effective edge Hamiltonian with only a p-wave pairing term and the zero-bias unit transmission in the clean limit.","marker":"[20]"},{"why":"Gives the clean-limit Landauer-Buttiker prediction of quantized conductance that the disorder results are compared against.","marker":"[31]"},{"why":"The experiment claiming e^2/2h conductance kink for chiral Majorana modes that motivates the disorder study.","marker":"[11]"},{"why":"Previous calculation of ballistic conductance oscillation in QAH/SC junctions that the paper's clean limit reproduces.","marker":"[43]"},{"why":"Another ballistic oscillation calculation whose length and energy dependence is fitted by the effective model.","marker":"[44]"},{"why":"Theoretical proposal of non-chiral modes coexisting with a chiral edge mode in magnetically doped topological insulators.","marker":"[47]"},{"why":"Experimental demonstration of those non-chiral modes, providing the physical situation modeled in the coexistence case.","marker":"[48]"}],"fun_headline_variants":["Single chiral mode immune to disorder; extra modes yield half-electron half-hole","One chiral edge survives disorder; extra modes give 50/50 electron-hole output","Disorder plus extra modes gives equal electron-hole output","Extra modes turn disordered junction into 50/50 electron-hole splitter","Disorder kills zero-bias protection only when extra modes exist"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that disorder in the junction only changes the local chemical potential and never generates a constant s-wave pairing term in the edge-mode basis; if interface roughness couples to the pairing field itself, the p-wave-only protection of the zero-bias transmission collapses.","fun_headline_variants_meta":{"raw":{"variants":["Single chiral mode immune to disorder; extra modes yield half-electron half-hole","One chiral edge survives disorder; extra modes give 50/50 electron-hole output","Disorder plus extra modes gives equal electron-hole output","Extra modes turn disordered junction into 50/50 electron-hole splitter","Disorder kills zero-bias protection only when extra modes exist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001012,"raw_usage":{"total_tokens":4283,"prompt_tokens":963,"completion_tokens":3320,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":3228}},"tokens_in":579,"tokens_out":3320,"duration_ms":24974,"temperature":1.0,"reasoning_tokens":3228,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:17:37.190032+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically simulate the lattice QAH/SC junction with disorder applied to the pairing order parameter as well as to the chemical potential (e.g., random on-site $\\Delta_0$ with zero mean), and compute $T_{13}$ at $\\epsilon=0$ for a junction many times the elastic mean free path; if $\\bar{T}_{13}$ decays exponentially with $L$, the single-mode p-wave protection is not robust to pairing disorder, and an experimental probe would be a zero-bias two-terminal conductance of $e^2/2h$ rather than $e^2/h$ in the single-mode limit.","supporting_citations":[{"cited_title":"Van Ostaay, A","cited_arxiv_id":null,"evidence_quote":"Supplies the effective edge Hamiltonian with only a p-wave pairing term and the zero-bias unit transmission in the clean limit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the clean-limit Landauer-Buttiker prediction of quantized conductance that the disorder results are compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The experiment claiming e^2/2h conductance kink for chiral Majorana modes that motivates the disorder study."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous calculation of ballistic conductance oscillation in QAH/SC junctions that the paper's clean limit reproduces."},{"cited_title":"Gamayun, J","cited_arxiv_id":null,"evidence_quote":"Another ballistic oscillation calculation whose length and energy dependence is fitted by the effective model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Theoretical proposal of non-chiral modes coexisting with a chiral edge mode in magnetically doped topological insulators."},{"cited_title":"Disordered Quantum Transport in Quantum Anomalous Hall Insulator-Superconductor Junctions","cited_arxiv_id":"1908.03303","evidence_quote":"Experimental demonstration of those non-chiral modes, providing the physical situation modeled in the coexistence case."}],"review_version":1}