Odd-frequency superconducting pairing in one-dimensional systems

Annica M. Black-Schaffer; Christopher Triola; Jorge Cayao

REVIEW 2 major objections 4 minor 236 references

Odd-frequency superconducting pairing in one-dimensional systems

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.05466 v2 pith:WU3RFYZW submitted 2019-08-15 cond-mat.mes-hall cond-mat.supr-con

Jorge Cayao , Christopher Triola , Annica M. Black-Schaffer This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.supr-con

keywords odd-frequencysuperconductivityMajoranazeromodestopologicalRashbananowiresinsulatoredgesAndreevreflectionNambuGreen'sfunctionproximityeffect

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

A 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.

Overall, 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[Section 6, Eq. (10) and following paragraph] 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 7 versus Section 6] 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.

minor comments (4)

[Section 6, Eq. (11)] 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 5, protected zero-energy crossing] 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 3.1, Eq. (7)] 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.
[References] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central MZM–odd-omega relation is derived algebraically from the Majorana condition and the zero-energy propagator, not from a fitted input or a self-citation chain.

full rationale

I find no circular step in this review. The symmetry classification in Sec. 2 is derived from Fermi-Dirac statistics via Eq. (3), an independent constraint that does not presuppose the later conclusions. Sections 3–5 present results for NS/SNS junctions, Rashba nanowires, and helical edges that are computed from explicit BdG Hamiltonians (Eqs. (6), (8), (9)) using scattering Green's functions; the odd-frequency pair amplitudes are outputs of those calculations, not inputs chosen to reproduce the target claim. The central assertion of Sec. 6, that MZMs are always accompanied by odd-frequency pairing, rests on Eq. (10): for an isolated zero-energy Majorana mode, self-conjugacy gives g = f, and the zero-energy pole gives f(omega_m) = 1/(i omega_m), which is odd in Matsubara frequency. This is a direct algebraic consequence of the definition of a Majorana zero mode and of the standard propagator representation; it does not define odd-frequency pairing in terms of MZMs or vice versa. The paper explicitly notes, in the discussion of Ref. [145], that in realistic NS Kitaev junctions the OTE amplitude does not exhibit the 1/|omega| divergence because the MZM wavefunction has finite width beyond the interface. That is an important caveat about the strength of the 'any unambiguous MZM signature' corollary as an experimental diagnostic, but it is a robustness/validity concern, not a circularity: the claim that the isolated mode itself carries odd-frequency correlations still follows from Eq. (10). Self-citations to Refs. [110, 119, 145] are normal review summaries of prior peer-reviewed calculations; those papers are parameter-free model calculations whose stated assumptions do not include the MZM–odd-omega equivalence as an input. No fitted parameter is later renamed as a prediction, and no uniqueness theorem from the authors' prior work is imported to forbid alternatives. The derivation chain is therefore self-contained rather than circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are introduced in this review. The reviewed models contain parameters such as alpha, mu, Delta, and B, but the review does not fit them. The axioms listed are the background assumptions the review's synthetic claims rely on.

assumptions (6)

standard math The anomalous propagator f satisfies the antisymmetry constraint of Fermi-Dirac statistics, leading to Eq. (2) and the four symmetry classes in Table 1.
Foundation of the classification; standard many-body quantum mechanics.
domain assumption The superconducting order parameter is proportional to the anomalous propagator, Delta ~ f.
Used throughout to treat f as the Cooper pair amplitude; standard BCS/Green's function identification.
domain assumption An isolated Majorana zero mode has g = f = 1/(iω_m).
Section 6, Eq. (10); assumes exact gamma = gamma†, zero energy, and isolation.
domain assumption The scattering Green's function method with outgoing wave boundary conditions gives the physical pair amplitudes.
Sections 3-5; assumes non-interacting BdG quasiparticles and zero temperature.
domain assumption The 2D TI edge is perfectly helical, forbidding normal reflection.
Section 5; used to attribute all interface pairing to Andreev reflection; disorder breaks this.
domain assumption The symmetry classification is complete when only time, spin, and position are relevant degrees of freedom.
Section 2; the review acknowledges that band, lead, and dot indices enlarge the classification, as noted in Sections 2 and 7.

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Cite this review

Pith. "Pith review of Odd-frequency superconducting pairing in one-dimensional systems." pith.science (2026). https://pith.science/paper/WU3RFYZW

@misc{pith2026190805466,
  author       = {Pith},
  title        = {Pith review of: Odd-frequency superconducting pairing in one-dimensional systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WU3RFYZW}},
  note         = {Machine review of arXiv:1908.05466}
}

read the original abstract

Odd-frequency superconductivity represents a truly unconventional ordered state which, in contrast to conventional superconductivity, exhibits pair correlations which are odd in relative time and, hence, inherently dynamical. In this review article we provide an overview of recent advances in the study of odd-frequency superconducting correlations in one-dimensional systems. In particular, we focus on recent developments in the study of nanowires with Rashba spin-orbit coupling and metallic edges of two-dimensional topological insulators in proximity to conventional superconductors. These systems have recently elicited a great deal of interest due to their potential for realizing one-dimensional topological superconductivity whose edges can host Majorana zero modes. We also provide a detailed discussion of the intimate relationship between Majorana zero modes and odd-frequency pairing. Throughout this review, we highlight the ways in which odd-frequency pairing provides a deeper understanding of the unconventional superconducting correlations present in each of these intriguing systems and how the study and control of these states holds the potential for future applications.

Figures

Figures reproduced from arXiv: 1908.05466 by the authors.

**Figure 1.** Sketch of 1D systems with SOC which exhibit OTE pairing and are promising for 1D topological superconductivity. (a) The right region of a nanowire with Rashba SOC is placed in contact with a conventional spin-singlet s-wave superconductor (S), while its left region is in contact with either a normal (X=N) or superconducting lead (X=S). (b) A 2D TI is characterized by having an insulating bulk and a 1D metallic edge … view at source ↗

**Figure 2.** (a) Energy-momentum dispersion in a normal metal (N) with ∆ = 0, where solid red (dashed blue) parabola corresponds to electrons (holes). The bottom (top) of the electron (hole) band is indicated by dotted horizontal lines. (b) A finite order parameter ∆ 6= 0 in the superconductor (S) opens an energy gap at the Fermi momenta ±kF and mixes electron with hole bands, leading to quasielectron and quasihole bands with po… view at source ↗

**Figure 3.** (a) Energy versus momentum dispersion in the normal part (N) at ∆ = 0, where the Rashba SOC splits the normal bands around zero momentum k = 0. Solid (dashed) blue and red parabolas correspond to electrons (holes) with spin up and down, respectively. The bottom (top) of the electron (hole) band is indicated by dotted horizontal lines. (b) A finite order parameter ∆ 6= 0 in the superconductor (S) opens a gap at the F… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: (a,b) Induced interface pair magnitudes as function of distance from the interface in the N (a) and S (b) regions of NS junctions with Rashba SOC for x 0 = 0, ω = 0.1∆, ESOC = 0.5∆, and µN,S = 10∆. Magenta curve in (b) also shows the interface LDOS. (c,d) Induced inter…

**Figure 5.** Figure 5: (a) Energy versus momentum dispersion for the 1D metallic edge of a 2D TI in the normal part (N) at ∆ = 0, where solid (dashed) red and blue curves corresponds to electrons (holes). The chemical potential for electron (hole) band is indicated by dotted horizontal lines…

**Figure 6.** Figure 6: (a) Spatial dependence of the LDOS (top) and even-ω spin-singlet (ESE) and oddω spin-triplet (OTE) s-wave (local) pair magnitudes in a NS junction (bottom). In both panels, solid curves correspond to ω = 0 while dashed curves to ω = 0.5∆. (b) Frequency dependence of E…

**Figure 7.** Figure 7: (a) FS and FSF junctions at the helical edge of a 2D TI, with the F regions are located at x1 and x1,2, respectively. The length of F is 2δ, while the S region have length d. (b) Non-local conductance for FS (blue) and FSF (red) as a function of the bias between F and …

**Figure 8.** Figure 8: (a) Two pairs of 1D metallic edges of a 2D TI are coupled through a quantum point contact, with the left region containing proximity-induced conventional superconductivity (SC) and each edge is coupled to separate contacts 1-4. At the quantum point contact, t0 and tc c…

**Figure 9.** Figure 9: (a) 1D nanowire with Rasbha SOC and under a Zeeman field B in contact with a superconductor and an insulator. The left region of the nanowire contains a finite disordered section of length L. (b) The normal Green’s function g↑↑ (black solid curve) and equal-spin pair a…

**Figure 10.** Figure 10: (a,b) Odd-ω (OTE) and even-ω (ETO) (c,d) pair amplitudes as a function of the chemical potential in S, µ, and frequency for NS Kitaev junctions in both the N and S regions. In general, both pairings coexist but with larger OTE in the topological phase. Vertical red da…

**Figure 11.** Figure 11: (a,b) Even-ω (ETO) and odd-ω (OTE) pair magnitudes in the topological phase of short SNS Kitaev junctions as a function of frequency and phase difference φ. Bright areas in (a,b) coincide with the emergence of topological ABSs. (c,d) Frequency dependence of ETO (dashe…

**Figure 12.** Figure 12: Majorana STM tip tunnel coupled to a quantum dot (QD) via tσ which acquires pair correlations from an unknown superconductor SC. The setup only registers a current flow when the unknown SC has components of equal-spin OTE symmetry. Reprinted figure with permission fro…

**Figure 13.** Figure 13: (a) Array of MZMs (red filled spheres) coupled to a spin polarized nanowire (SPW). The separation between nearest MZMs is a. OTE correlations are induced in t he SPW and a 1D odd-ω bulk superconductor can emerge. (b) Unlike conventional superconductors, the odd-ω supe…

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