{"id":"e62efb77-4c59-44a6-96ae-63f1693e19dd","arxiv_id":"1908.05466","paper_version":2,"verdict":"UNVERDICTED","confidence":"MODERATE","novelty_score":1.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A review consolidating how odd-frequency Cooper pairing arises in 1D superconductor junctions and why Majorana zero modes always come with odd-frequency correlations.","lead":"This paper is a review of odd-frequency superconductivity in one-dimensional nanowires and topological insulator edges, explaining how such pairing arises and how it connects to Majorana zero modes. It is a useful entry point for researchers who want the state of this subfield without reading dozens of primary papers.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The universal MZM-odd-omega link rests on a bare 1/(iω_m) pole that the review's own cited results show is not a robust observable; the 'any unambiguous MZM signature' corollary overstates the evidence.","rationale":"The reader's weakest_assumption and my analysis converge on the same point: the idealized relation f = g = 1/(iω_m) for an isolated MZM is the load-bearing step, and the review's own summary of Ref. [145] admits the low-frequency divergence is not universal in realistic NS junctions. This is not a fatal flaw for the review's descriptive content, but it does mean the strong corollary — that any unambiguous MZM signature doubles as an odd-frequency signature — is overstated. Since this is a review article with no new primary result, the correct disposition remains UNVERDICTED; the concern qualifies the headline conceptual claim but does not turn the manuscript into an accept/reject target. I also note the symmetry-based derivation of odd-frequency pairing from Fermi-Dirac statistics (Section 2) is parameter-free and internally consistent, and the review clearly discloses that odd-ω pairing can occur without MZMs, which gives the paper independent support. The concrete test I propose would place the MZM-odd-ω relation on firmer footing by separating the idealized pole from what a finite, coupled system actually exhibits.","tokens_in":31468,"tokens_out":1491,"duration_ms":12839,"concrete_test":"Evaluate Eq. (10) in a minimal but finite system: compute the Matsubara anomalous Green's function f(iω_m) for a finite Kitaev chain with coupling to a normal lead, at the MZM location and averaged over the MZM decay length, and compare with the isolated-mode prediction 1/(iω_m). If the low-frequency divergence is absent or the frequency dependence is non-universal for realistic parameters, the 'any unambiguous signature' corollary fails. This directly tests whether the isolated-MZM relation survives coupling to external degrees of freedom, as the review's discussion of Ref. [145] suggests it may not.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Section 6) is that MZMs are always accompanied by odd-frequency pairing and that any unambiguous MZM signature also identifies odd-frequency pairing. The supporting argument is Eq. (10): for an isolated MZM, f = g = 1/(iω_m), which is odd and divergent at low frequency. The load-bearing assumption is that this idealized isolated-MZM relation survives contact with realistic junctions. The review itself undercuts this: in the discussion of Ref. [145], the OTE amplitude in NS Kitaev junctions does not exhibit the 1/|ω| divergence because the MZM wavefunction has finite width beyond the interface. So the low-frequency divergence — the sharpest, most diagnostic odd-frequency feature — can be absent even when a MZM is present. What remains is that a MZM contributes odd-frequency correlations somewhere in the system, which is a much weaker statement than 'any unambiguous signature of MZMs can also be used to identify odd-ω pairing.' The paper also says explicitly that odd-ω pairing exists without MZMs (Section 7), so the reverse implication is admittedly not claimed; but the forward implication for observable discrimination is precisely what is threatened. The argument is internally consistent but the universal diagnostic claim is not established by Eq. (10) alone because experiment probes the system, not the isolated mode.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is a review article on odd-frequency (odd-ω) superconducting correlations in one-dimensional systems, with a focus on Rashba nanowires and the metallic edges of two-dimensional topological insulators proximitized to conventional s-wave superconductors. The review develops the Fermi-Dirac symmetry classification of pair amplitudes (Table 1), presents a scattering Green's function approach for NS and SNS junctions, and shows how Rashba spin-orbit coupling and helicity generate spin-triplet odd-ω pairing. In Section 6 the authors argue that Majorana zero modes (MZMs) are always accompanied by odd-ω pairing, using the identity f = g = 1/(iω_m) for an isolated zero-energy Majorana mode (Eq. (10)), and infer that any unambiguous MZM signature can also be used to identify odd-ω pairing. The review concludes with device proposals and a survey of experimental probes, while noting in Section 7 that existing probes are system-dependent and do not provide an unambiguous direct measurement of all odd-ω pair amplitudes in a generic system.","tokens_in":31568,"tokens_out":21662,"duration_ms":216700,"significance":"The review is a timely and readable synthesis of an active field. Its strengths are the clear pedagogical presentation of the symmetry classification, the explicit connection between Andreev reflection processes and odd-ω pair amplitudes, the distinction between local and nonlocal pairing, and the honest reporting of cases where the 1/ω divergence is absent. The claimed universal link between MZMs and odd-ω pairing is an important organizing principle, but the quantitative results are largely taken from the authors' own prior work and that of close collaborators, so the review's independent evidential weight is limited. The logical corollary that any unambiguous MZM signature identifies odd-ω pairing needs to be stated more carefully, in light of the system-dependent caveats the authors themselves enumerate. With that qualification, the review will be a useful reference for both specialists and newcomers to the field.","major_comments":[{"comment":"The statement that 'any unambiguous signature of MZMs can also be used to identify odd-ω pairing' is stronger than the evidence presented. Eq. (10) describes an isolated, exactly zero-energy Majorana mode, whereas the discussion of Ref. [145] later in the same section states that in NS Kitaev junctions the OTE amplitude does not exhibit the 1/|ω| divergence because the MZM wavefunction has finite width beyond the interface. The subsequent qualification 'although not always with a divergent behavior' mitigates this, but the universal identification corollary is not similarly qualified. Please reformulate the central claim as the more defensible statement that a system hosting an MZM necessarily also hosts odd-ω pair correlations, and explicitly specify which observables, if any, remain robust identifiers of odd-ω pairing when the divergent 1/ω feature is absent.","section":"Section 6, Eq. (10) and following paragraph"},{"comment":"There is a tension between Section 6 and the final section. Section 7 states that none of the discussed tools provides an unambiguous direct measurement of all kinds of odd-ω pair amplitudes in a generic system and that the signatures are system-dependent, while Section 6 suggests that any unambiguous MZM signature can be used to identify odd-ω pairing. The authors should explicitly distinguish an indirect model-level implication from an experimental diagnostic, or reconcile the two statements by specifying the conditions under which a given MZM signature constitutes a reliable identifier of odd-ω pairing.","section":"Section 7 versus Section 6"}],"minor_comments":[{"comment":"Equation (11) is attributed to Ref. [229], which is listed as 'in preparation' in the reference list. Since this equation is used to support the claim that a single MZM coupled to a quantum dot induces purely OTE pairing, please replace the citation with a published reference or include a derivation in the review.","section":"Section 6, Eq. (11)"},{"comment":"The text says 'the zero-energy ABS is its own charge-conjugate state and corresponds to a MZM which is twofold degenerate.' A single Majorana zero mode is not twofold degenerate; the zero-energy crossing at φ=π in a time-reversal-invariant junction hosts a Kramers pair of Majorana modes, with the twofold degeneracy referring to the fermion parity ground state. Please correct the terminology for clarity.","section":"Section 5, protected zero-energy crossing"},{"comment":"The prefactor in Eq. (7) is typeset in a way that is easy to misread (η 2i ...). Please typeset it consistently with the subsequent expressions f^r,O = -(r_eh η/2) ... so that the factor of 2 is unambiguous.","section":"Section 3.1, Eq. (7)"},{"comment":"Several references (e.g., Refs. [102,103,132,143,145,150,216,219,229]) are arXiv preprints with no published version noted. For a review, it would be helpful to update these to the final journal versions where available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The quantitative centerpieces of Section 6, including Eq. (10)-(12) and the Kitaev-junction analysis, are drawn heavily from the authors' own work and from close collaborators. This is acceptable for a review, but the universal MZM-odd-ω identification claim would carry more weight if it were also grounded in independent published derivations. The editor may also wish to check the balance of self-citation relative to the review's stated scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. This is a review and should be judged as one. It contains no new derivations or data; its contribution is a careful, useful synthesis of how odd-frequency pairing arises in 1D Rashba nanowires and helical-edge systems, plus a focused discussion of the connection to Majorana zero modes. The symmetry classification in Section 2 and the scattering Green's-function treatment of NS/SNS junctions in Sections 3-5 are clear and pedagogically effective. I would happily give this to a student entering the field: it maps the territory, collects the relevant results, and points to recent work in a way that would take months to assemble from primary sources.\n\nThe main soft spot is Section 6. The claim that MZMs are always accompanied by odd-frequency pairing is correct in the formal sense: for an isolated Majorana zero mode, f = g = 1/(i omega_m), which is odd in frequency. But the follow-on sentence, that any unambiguous MZM signature can also be used to identify odd-omega pairing, overstates what the argument buys. A zero-bias conductance peak or other MZM probe tells you a MZM exists, and from that you can infer odd-omega correlations exist somewhere in the system. That is not the same as having identified the odd-omega pairing itself, and the review's own cited results show the sharpest signature, the 1/(i omega_m) divergence, can be absent in an NS Kitaev junction even with a MZM present. The \"always accompanied\" result is real; the \"any signature identifies odd-omega pairing\" corollary is too strong as stated and should be qualified.\n\nA couple of minor items. The review leans heavily on the authors' own papers, which is normal for a review on this topic; the in-preparation reference [229] is an uncomfortable citation in a final manuscript. There is also a small reference glitch - Ref. [86] cites a 1994 Nature paper for work that is actually from 2015 - which should be corrected.\n\nOverall, this deserves a serious referee and likely publication after moderate revision. The core physics is sound, the synthesis is useful, and the overreach is confined to the concluding implications, not the underlying calculations. I'd bring it to reading group and would cite it as a reference review for odd-frequency pairing in 1D systems.","headline":"A well-organized review of odd-frequency pairing in 1D systems whose central MZM-diagnostic claim overreaches: MZMs imply odd-omega correlations, but the paper's own cited results show the sharpest signature is not universal.","tokens_in":32204,"tokens_out":3173,"would_cite":true,"duration_ms":33264,"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 review establishes a necessary link: Majorana zero modes are always accompanied by odd-frequency superconducting pairing, making any unambiguous Majorana signature also a signature of odd-frequency correlations.","keywords":["odd-frequency superconductivity","Majorana zero modes","topological superconductivity","Rashba nanowires","topological insulator edges","Andreev reflection","Nambu Green's function","proximity effect"],"falsifier":"One concrete check: in a well-localised junction believed to host a Majorana zero mode, measure the low-frequency (low-temperature) anomalous response or the frequency dependence of the induced pairing. The claimed link predicts a $1/|\\omega_m|$ divergence (or a clear odd-frequency enhancement) that must be present whenever standard Majorana signatures are present; observing a zero-energy mode with all standard signatures but an even-frequency pair amplitude, or no $1/\\omega$ enhancement at any frequency, would falsify the universality claim. A numerical version: compute the OTE amplitude in a short SNS Kitaev junction at $\\varphi=\\pi$ with a finite-width Majorana wavefunction; if the divergence disappears entirely rather than merely softening, then the link holds only in the idealised isolated-mode limit.","tokens_in":31117,"feed_emoji":"⚛️","tokens_out":7561,"duration_ms":68819,"temperature":0.7,"pith_summary":"This review argues that odd-frequency (odd-$\\omega$) superconducting pairing is not a rare exotic state but a generic feature of one-dimensional superconductor hybrids, and that it is deeply tied to Majorana zero modes. The paper's central claim is that the appearance of Majorana zero modes is always accompanied by odd-$\\omega$ pairing: for an isolated Majorana mode the anomalous propagator equals the normal one, $f(\\omega_m)=g(\\omega_m)=1/(i\\omega_m)$, which is odd in Matsubara frequency. If correct, any unambiguous experimental signature of a Majorana zero mode can also be read as evidence for odd-frequency pair correlations. The review supports this by showing that Andreev reflection at interfaces, Rashba spin-orbit coupling, helical edge states of two-dimensional topological insulators, and Kitaev-type junctions all generically generate odd-$\\omega$ amplitudes, with the odd-frequency channel enhanced in the topological phase.","feed_headline":"Majorana zero modes always carry odd-frequency pairing","feed_subtitle":"If the paper is right, any Majorana signature doubles as evidence for odd-frequency superconductivity.","key_machinery":"The load-bearing identity is $f(\\omega_m)=g(\\omega_m)=1/(i\\omega_m)$ for an isolated Majorana zero mode: the equality of normal and anomalous propagators follows from $\\gamma=\\gamma^\\dagger$, and the $1/(i\\omega_m)$ form follows from the mode being pinned to zero energy, making the pair amplitude necessarily odd in Matsubara frequency. This identity converts the abstract Majorana condition into a concrete statement about Cooper-pair correlations, and it is what lets the paper claim that Majorana zero modes always come with odd-$\\omega$ pairing. A second piece of machinery is the Nambu Green's function framework together with the Fermi-Dirac antisymmetry constraint $f^t_{\\sigma\\sigma'}(x,x';\\omega)=-f^t_{\\sigma'\\sigma}(x',x;-\\omega)$, which organises pair amplitudes into the four symmetry classes ESE, OSO, ETO, and OTE and identifies odd-$\\omega$ amplitudes by their frequency and spatial parity. The review repeatedly uses a scattering-state Green's function method in which Andreev reflection amplitudes directly build up the anomalous propagator, which is why conductance and local density of states measurements can serve as practical probes of odd-frequency pairing.","core_discovery":"The central discovery claimed by the paper is a necessary connection between Majorana zero modes and odd-frequency superconductivity. Because a Majorana operator obeys $\\gamma=\\gamma^\\dagger$ and sits at zero energy, its normal and anomalous propagators coincide and take the form $f(\\omega_m)=g(\\omega_m)=1/(i\\omega_m)$, a function that is odd under $\\omega_m\\to-\\omega_m$ and divergent at low frequency. The paper therefore asserts that whenever a Majorana zero mode exists in a proximitized one-dimensional system, odd-frequency pair correlations exist in the same system, and that any unambiguous signature of Majorana zero modes can also be used to identify odd-$\\omega$ pairing. This is illustrated in Rashba nanowire and two-dimensional topological insulator edge junctions, where scattering calculations show coexistence of all four symmetry classes (even-frequency singlet even-parity, odd-frequency singlet odd-parity, even-frequency triplet odd-parity, odd-frequency triplet even-parity) at interfaces, with the odd-frequency triplet-even-parity (OTE) amplitude enhanced and often dominant in the topological phase. In short SNS Kitaev junctions the OTE amplitude diverges as $\\sim 1/|\\omega|$ exactly at the phase difference $\\varphi=\\pi$ where zero-energy Majorana modes appear, while in NS junctions the divergence is softened because the Majorana wavefunction has finite width beyond the interface.","pith_inferences":["If the claimed link is exact, experiments already interpreted as Majorana signatures carry an untapped odd-frequency signature; one testable extension is to re-analyse zero-bias peak data for the predicted $1/\\omega$ spectral weight and its temperature dependence.","The identity $f=g$ suggests that odd-frequency pairing is not a separate condensate but the same spectral weight as the normal density of states; a direct way to test this is to compare the low-frequency anomalous response with the single-particle spectral function in a topological junction.","The softened divergence in NS junctions points to a quantitative diagnostic: as the junction transparency or Majorana localisation length is tuned, the odd-frequency peak shape should track the Majorana wavefunction extent, a prediction the review does not itself state.","The 'any unambiguous signature' statement, taken at face value, inverts the usual search logic: instead of looking for odd-frequency pairing as a byproduct of topology, one could deliberately use engineered Majorana modes as sources of odd-frequency correlations in hybrid devices."],"forward_implications":["A zero-bias conductance peak or any other accepted Majorana signature also counts as evidence for odd-frequency pair correlations in the same system.","Odd-frequency pairing can arise without Majorana modes, for instance at any normal-superconductor interface, but the presence of Majorana modes enhances the odd-frequency amplitude; in short SNS Kitaev junctions the OTE amplitude diverges as $\\sim 1/|\\omega|$ at the $\\varphi=\\pi$ Majorana crossing.","In Rashba nanowire and topological insulator edge junctions, all four symmetry classes coexist at interfaces, so odd-$\\omega$ pairing is a generic part of the proximity effect rather than a separate exotic phase.","Because Andreev reflection coefficients are measurable through conductance, existing nanowire and edge-state devices can be used to characterise induced odd-frequency amplitudes.","Proposed Majorana-based devices, including a Majorana STM tip and an array of Majorana modes coupled to a spin-polarized wire, inherit the odd-$\\omega$ correlations, including a paramagnetic Meissner effect with negative superfluid density."],"supporting_citations":[{"why":"Supplies the Kitaev chain model in which Majorana zero modes emerge as self-conjugate, zero-energy end modes, the canonical setting for the whole discussion.","marker":"[97]"},{"why":"Provides the result that an isolated Majorana zero mode has the anomalous propagator $f(\\omega_m)=1/(i\\omega_m)$, the load-bearing identity of the review.","marker":"[217]"},{"why":"Shows in a disordered nanowire junction that the normal Green's function and equal-spin odd-frequency pair amplitude both sharply increase when the system enters the topological phase, grounding the Majorana-odd-$\\omega$ link in a realistic model.","marker":"[221]"},{"why":"Computes pair amplitudes in ballistic NS and SNS Kitaev junctions, showing the OTE amplitude is enhanced in the topological phase and that the $1/|\\omega|$ divergence is softened in NS junctions by the finite Majorana wavefunction width.","marker":"[145]"},{"why":"Establishes that equal-spin OTE correlations survive over long distances in disordered normal regions and coincide with the zero-energy local density of states peak from the Majorana mode.","marker":"[219]"},{"why":"Supplies the spectral bulk-boundary correspondence $f\\approx Z/\\omega+B\\omega$, separating the divergent Majorana contribution from the regular part and characterising the odd-$\\omega$ pairs accumulated at the boundary.","marker":"[220]"},{"why":"Provides the scattering Green's function calculation of pair amplitudes at the helical edges of two-dimensional topological insulators, the basis for Section 5 of the review.","marker":"[110]"},{"why":"Provides the full analytic pair amplitudes in Rashba nanowire NS and SNS junctions, the basis for the Section 4 discussion of odd-$\\omega$ generation by spin-orbit coupling.","marker":"[119]"}],"fun_headline_variants":["Odd-frequency pairing is a universal Majorana fingerprint","Majorana zero modes always signal odd-frequency pairing","No Majorana without odd-frequency pairing","Where Majoranas appear, odd-frequency pairing is there","Majorana zero modes dictate odd-frequency correlations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument presupposes that a Majorana zero mode can be treated as perfectly isolated and exactly at zero energy, with particle and antiparticle propagators exactly equal; if real Majorana modes are slightly delocalised, hybridised, or shifted from zero energy, the clean $1/(i\\omega_m)$ odd-frequency signature may be smeared or absent.","fun_headline_variants_meta":{"raw":{"variants":["Odd-frequency pairing is a universal Majorana fingerprint","Majorana zero modes always signal odd-frequency pairing","No Majorana without odd-frequency pairing","Where Majoranas appear, odd-frequency pairing is there","Majorana zero modes dictate odd-frequency correlations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1499,"prompt_tokens":978,"completion_tokens":521,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":452}},"tokens_in":594,"tokens_out":521,"duration_ms":4906,"temperature":1.0,"reasoning_tokens":452,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:12:45.387941+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check: in a well-localised junction believed to host a Majorana zero mode, measure the low-frequency (low-temperature) anomalous response or the frequency dependence of the induced pairing. The claimed link predicts a $1/|\\omega_m|$ divergence (or a clear odd-frequency enhancement) that must be present whenever standard Majorana signatures are present; observing a zero-energy mode with all standard signatures but an even-frequency pair amplitude, or no $1/\\omega$ enhancement at any frequency, would falsify the universality claim. A numerical version: compute the OTE amplitude in a short SNS Kitaev junction at $\\varphi=\\pi$ with a finite-width Majorana wavefunction; if the divergence disappears entirely rather than merely softening, then the link holds only in the idealised isolated-mode limit.","supporting_citations":[{"cited_title":"Huang, P","cited_arxiv_id":null,"evidence_quote":"Provides the result that an isolated Majorana zero mode has the anomalous propagator $f(\\omega_m)=1/(i\\omega_m)$, the load-bearing identity of the review."},{"cited_title":"Asano, Y","cited_arxiv_id":null,"evidence_quote":"Shows in a disordered nanowire junction that the normal Green's function and equal-spin odd-frequency pair amplitude both sharply increase when the system enters the topological phase, grounding the Majorana-odd-$\\omega$ link in a realistic model."},{"cited_title":"Odd-frequency superconducting pairing in Kitaev-based junctions","cited_arxiv_id":"1905.01171","evidence_quote":"Computes pair amplitudes in ballistic NS and SNS Kitaev junctions, showing the OTE amplitude is enhanced in the topological phase and that the $1/|\\omega|$ divergence is softened in NS junctions by the finite Majorana wavefunction width."},{"cited_title":"Odd-frequency pairing and proximity effect in Kitaev chain systems including topological critical point","cited_arxiv_id":"1809.09324","evidence_quote":"Establishes that equal-spin OTE correlations survive over long distances in disordered normal regions and coincide with the zero-energy local density of states peak from the Majorana mode."},{"cited_title":"Tamura, S","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral bulk-boundary correspondence $f\\approx Z/\\omega+B\\omega$, separating the divergent Majorana contribution from the regular part and characterising the odd-$\\omega$ pairs accumulated at the boundary."}],"review_version":1}