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REVIEW 3 major objections 6 minor 36 references

Strongly enlarged topological regime and enhanced superconducting gap in nanowires coupled to Ising superconductors

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that nanowires on Ising superconductors can keep their topological superconducting gap at in-plane magnetic fields near 10 T, roughly ten times the range for InSb/Al wires, because equal-spin triplet Cooper pairs from…

desk verdict The analytic decomposition is clean and the mechanism is plausible, but the headline 0.5–10 T window is extrapolated rather than computed: parent Zeeman and gap suppression are left out of both the tight-binding simulation and the self-energy derivation. read the letter →

arxiv 1908.11625 v1 pith:3AFTCH3B submitted 2019-08-30 cond-mat.supr-con

Ying-Ming Xie , Benjamin T. Zhou , T. K. Ng , K. T. Law This is my paper · ORCID
classification cond-mat.supr-con
keywords MajoranazeromodesIsingsuperconductorsequal-spintripletCooperpairstopologicalsuperconductinggapnanowireproximityeffectNbSe2in-planemagneticfieldKitaevchain
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

The paper argues that a semiconducting nanowire placed on an Ising superconductor—an atomically thin superconductor whose spin-orbit coupling locks electron spins out of plane—keeps its proximity-induced topological gap almost unchanged as an in-plane magnetic field grows. The reason is that the parent material supplies equal-spin triplet Cooper pairs, which align with the field instead of being broken by it. For a realistic InSb nanowire on monolayer NbSe2, the topological regime runs from about 0.5 T to about 10 T, roughly ten times the window seen in InSb wires on aluminum. If correct, this removes the main practical ceiling on Majorana-based qubits: the field that creates the topological phase no longer destroys the superconducting gap that protects it.

What carries the argument

The load-bearing object is the decomposition of the effective proximity-induced p-wave pairing in Eqs. (15)–(16). The total induced p-wave gap is split as $\Delta^{(p)}=\Delta^{(p)}_{s,\alpha}+\Delta^{(p)}_{s,\beta}+\Delta^{(p)}_t$: the first two terms come from singlet Cooper pairs combined with Rashba or induced Ising spin-orbit coupling and vanish when the Zeeman energy $V$ dominates the spin-orbit scales $\alpha,\beta$; the third term, $\Delta^{(p)}_t$, is the projection of the parent Ising superconductor's equal-spin triplet pairs, with $d_z(k_x)\propto k_x$ fixed by the $D_{3h}$ symmetry of monolayer NbSe2. In the strong-field limit $\Delta^{(p)}_t\approx -d_z(k_x)/2$, so the triplet channel keeps the wire topological even after the singlet channels have died. The self-energy calculation that produces this decomposition carries the argument from the parent superconductor's Gor'kov equations to the nanowire's effective Hamiltonian.

What would settle it

Measure the local density of states at the end of an InSb nanowire on monolayer NbSe2 as a function of in-plane field: if the induced gap fails to reopen or the zero-bias peak disappears well below 10 T—on the ~1 T scale characteristic of aluminum-contacted wires—then the triplet-pairing mechanism is not enough to protect the topological phase; a hard gap persisting past roughly 5 T would support the claim.

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Extended reading notes

Core claim

The central claim is that the effective p-wave pairing which makes the nanowire topological has three contributions, and only the two singlet-derived ones—one from the wire's Rashba spin-orbit coupling and one from the Ising spin-orbit coupling induced into the wire—are suppressed by the Zeeman field. The third contribution, $\Delta_t^{(p)}$, comes from equal-spin triplet Cooper pairs tunneling out of the Ising superconductor and, in the limit $V \gg \alpha,\beta$, approaches $-d_z(k_x)/2$, where $d_z(k_x) \propto k_x$ is a p-wave triplet pairing term inherited from the parent. Because this term does not rely on pairing opposite spins, the induced topological gap survives in-plane fields that would destroy singlet proximity pairing. In the specific InSb/NbSe2 heterostructure, the topological gap closes and reopens near 0.5 T, remains sizable to at least 5 T in the numerical local density of states, and persists until the parent NbSe2 itself reaches its Pauli-limited critical field near 10 T, giving a topological window about ten times wider than InSb/Al wires.

Load-bearing premise

The claim that the topological regime reaches about 10 T assumes that in-plane magnetic fields do not break apart the Cooper pairs in the NbSe2 itself or in the nanowire before that field strength, so the parent superconducting gap remains open and the only ceiling is the spin-alignment limit.

Editorial extensions

If this is right

  • In an InSb nanowire on monolayer NbSe2, the topological regime should extend from the gap closing near 0.5 T up to the Pauli-limited parent gap near 10 T, about ten times the range of InSb/Al wires.
  • A large Zeeman splitting is no longer a liability: the topological gap stays sizable as the field grows, so the Majorana qubit operating point can sit at high field where the single-mode window is wide.
  • The same equal-spin triplet proximity mechanism should transfer to other nanowire/Ising-superconductor combinations, so the protection is a property of the pairing channel rather than of NbSe2 specifically.
  • At the 10 T ceiling the parent NbSe2 becomes a nodal topological superconductor, so the wire loses its gapped topological phase only through the parent's own superconducting transition.

Reading between the lines

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

  • A direct experimental signature would be a hard induced gap that persists well past 1 T in tunneling spectroscopy at the wire end, with the zero-bias peak surviving as the field is raised; this is what distinguishes the triplet mechanism from the behavior of aluminum-contacted wires.
  • The same robustness is likely to extend to gated MoS2 and to Shiba chains on Ising superconductors, since both rely on the same equal-spin triplet pairing rather than on the specific NbSe2 band structure.
  • One could test the mechanism's field-insensitivity by measuring the gap at fixed chemical potential as a function of field angle: if the protection comes from in-plane equal-spin pairs, the gap should show the predicted flatness only for in-plane fields, not out-of-plane ones.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes that proximity coupling an InSb nanowire to an Ising superconductor such as monolayer NbSe2 creates a topological superconducting gap that is robust against in-plane magnetic fields, yielding a topological regime from about 0.5 T to about 10 T. The mechanism is attributed to equal-spin triplet Cooper pairs in the Ising superconductor, which are compatible with large in-plane Zeeman fields. The authors support this proposal with an analytic projection of the induced pairing in Appendix C, which separates the effective p-wave pairing into three contributions (Eq. 15-16), and with a numerical tight-binding model of an InSb nanowire on monolayer NbSe2, presenting the local density of states at the wire end as a function of magnetic field in Fig. 4.

Significance. If the claimed robustness holds, this would be a significant step toward practical Majorana-based qubits, as the topological gap would survive fields far beyond the ~1 T limit of InSb/Al heterostructures. The analytic decomposition of the induced pairing into singlet and triplet channels is instructive and provides a clear physical picture. The numerical model uses realistic parameters and directly compares with the conventional s-wave case. However, the central quantitative claim of a 0.5-10 T window is only partially computed, and the field dependence of the parent superconductor is not included in the main calculations; these gaps must be addressed for the conclusion to be fully supported.

major comments (3)
  1. [Sec. IIIB, Fig. 4(a)] The calculated LDOS is shown only for fields up to B=5 T, yet the text states that the topological regime extends to B~10 T, citing Ref. 32. Since the 10 T endpoint is not computed in this manuscript, the headline claim rests on an extrapolation. The authors should either compute the LDOS at higher fields (including a field-dependent parent gap) or explicitly label the 0.5-10 T window as an estimate based on the prior result of Ref. 32.
  2. [Sec. IIIB, Eq. (17); Appendix C, Eq. (C1)] The parent NbSe2 Hamiltonian in the tight-binding model contains no Zeeman term, and the analytic self-energy is derived from the zero-field Green's function. Consequently, the parent order parameter is fixed at its B=0 value over the entire computed range, which suppresses the field-induced reduction of the proximity gap. This is inconsistent with the self-consistent equation Eq. (B4) that shows Δ(Vx) decreasing monotonically. The robustness of the triplet term Δ(p)t in Eq. (16) is therefore partially an artifact of the zero-field parent. The authors should re-evaluate the induced gap using a field-dependent parent order parameter, e.g., by inserting the self-consistent Δ(Vx) into the effective model, and determine whether the topological gap remains sizable at B~10 T.
  3. [Sec. II and Sec. IIIB] The paper explicitly ignores orbital pair-breaking in the Ising superconductor, but the comparison with Al in Fig. 4(b) includes orbital pair-breaking in the parent via Δ(B) = Δ√(1-(B/Bc)²). The authors do not discuss orbital pair-breaking in the nanowire itself, nor whether the assumption of negligible orbital effects remains valid for the composite heterostructure at B~10 T. Given that the main quantitative claim concerns high fields, this neglect should be justified more thoroughly or shown to be numerically unimportant.
minor comments (6)
  1. [Sec. IIIA] There is a typo: 'nanaowire' should be 'nanowire'.
  2. [References] The same paper (He et al., Communications Physics 1, 40 (2018)) appears as both Ref. 32 and Ref. 33; the citations should be consolidated or renumbered.
  3. [Fig. 4 caption] The caption states that the topological gap 'persists up to conventional Pauli limiting fields B~10T', but the plotted data range ends at 5 T; the caption should indicate that this is an extrapolation based on Ref. 32.
  4. [Eq. (15)-(16)] The notation Δ(p)s,α and Δ(p)s,β is used before the terms are defined; consider providing brief definitions in the text for clarity.
  5. [Sec. IIIB, text after Fig. 4] The statement that 'for magnetic fields larger than 5T, the wire still remains a topological superconductor as long as the proximity gap is finite' would benefit from a numerical check, since the authors have the model at hand and could extend the simulation to verify the persistence.
  6. [General] The manuscript contains several grammatical errors (e.g., 'which results in a Kitaev chain' in the abstract, 'superductor' in Sec. IIIA); a careful proofreading pass is recommended.

Circularity Check

2 steps flagged · score 6.0 of 10

Robustness claim partially built in: the triplet d-vector is computed from the zero-field parent Green's function, and the 10 T endpoint is imported from a same-group citation.

  1. other [Appendix C, Eqs. (C1) and (C7); Section III A, Eqs. (15)-(16)]
    "The Green's function of a bulk Ising superconductor is G(k,iω ) = ( G(k, 0,iω ) −F (k, 0,iω ) ... ). (C1) ... d(kx) = −∆Γ2c/˜Z0 ∫ dk⊥/2π Ft(k,iω ) = −∆Γ2c/2˜Z0 ∫ dk⊥/2π (1/(ξ2+ + ∆2) − 1/(ξ2− + ∆2)) ˆg(k). (C7) ... in the limit V ≫ α,β, we have ∆(p)t ≈ −dz(kx)/2. Thus, ∆(p)t stays robust against Zeeman fields."

    Eq. (C7) evaluates the triplet d-vector from Ft(k,iω), and Eq. (C1) defines that Green's function with the parent Zeeman argument set to 0, namely G(k,0,iω) and F(k,0,iω). Consequently dz(kx) carries no field dependence, and Eq. (16)'s robust limit ∆(p)t ≈ −dz/2 simply restates this zero-field input as field-insensitivity. The field-dependent parent gap obtained in Eq. (B4) and Fig. 1(b) is not inserted into the self-energy, so the claimed near-independence of the triplet gap from in-plane Zeeman fields is partly constructed by evaluating the parent propagator at V = 0.

  2. self citation load bearing [Section III B, Fig. 4(a), text after Eq. (20); Refs. 32-33]
    "The proximity gap can eventually be destroyed if the parent superconducting gap in NbSe2 is closed by the applied magnetic field. This can indeed happen at a field strengthB∼ 10T corresponding to the conventional Pauli limit, where the parent superconducting NbSe2 becomes a nodal topological superconductor32. ... The value ofβso is taken from first-principle calculations in Ref.33."

    The tight-binding parent model in Eq. (17) has no Vx term, and the shown LDOS in Fig. 4(a) runs only to about 5 T; the stated 10 T upper endpoint of the topological window is imported from Ref. 32, while the key Ising SOC parameter is taken from the same paper cited as Ref. 33. The quantitative headline '0.5–10 T' therefore rests on a same-authors prior result rather than being produced by the calculation reported in this paper.

full rationale

The central effective-Hamiltonian construction is not wholly circular: Appendix A derives the singlet and triplet pairing correlations from Gor'kov equations, Appendix C integrates out the superconductor to obtain ψ(kx) and d(kx), and Eqs. (15)-(16) follow from projecting the wire Hamiltonian into the lowest band following Ref. 30. The equal-spin triplet mechanism is derived in the paper rather than merely assumed from Ref. 23, and Section II independently solves a self-consistent gap equation (Eq. B4) showing an enhanced in-plane critical field. However, two load-bearing steps introduce partial circularity. First, the analytic robustness of ∆(p)t is computed from the zero-field parent Green's function (C1), so the field-insensitivity of dz(kx) is an input to the self-energy calculation rather than a demonstrated finite-field result. Second, the numerical simulation omits Zeeman coupling in the parent NbSe2 Hamiltonian, the plotted LDOS ends near 5 T, and the 10 T endpoint of the topological window is taken from the same group's prior paper (Ref. 32, also cited as Ref. 33 for βso). These elements make the headline window and the V-insensitivity claim partially constructed from the paper's own inputs and citations, while the underlying equal-spin triplet proximity mechanism retains independent derivational content.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper's added value is the projection and numerical simulation; the physical inputs it leans on are the mean-field Ising superconductor model, the chosen tunnel coupling and chemical potentials, and an imported prior result for the NbSe2 endpoint. These are the main places where the central claim depends on material chosen upstream rather than derived in this work.

free parameters (2)
  • Nanowire-TMD tunnel coupling Gamma_c = 120 Delta with Delta = 0.5 meV
    Chosen by hand to model strong proximity; not derived from ab initio or measured. It sets the magnitude of the induced gap and the position of the topological phase boundary, though the qualitative robustness of the triplet term is less sensitive.
  • Chemical potentials mu (NbSe2) and mu_w (InSb wire) = not stated in Table I
    The numerical LDOS requires specifying where the Fermi level sits in each material, and the text only says the topological transition happens near the Gamma point. These choices affect single-band occupancy and the closing/reopening gap at B approximately 0.5 T.
assumptions (4)
  • domain assumption In-plane orbital pair-breaking is negligible in atomically thin NbSe2 and in the nanowire.
    Stated in Sec. II for the parent superconductor because thickness is far smaller than the coherence length; the nanowire orbital contribution is not separately justified, so the 10 T result rests on this.
  • domain assumption The mean-field description of the Ising superconductor with on-site s-wave pairing and Ising SOC captures the relevant proximity pairing.
    Used throughout Sec. II and Appendix C; ignores other pairing channels, disorder, and multi-band effects in NbSe2.
  • domain assumption The parent NbSe2 gap remains finite until about 10 T, where it becomes a nodal topological superconductor (Ref. 32).
    This sets the upper end of the claimed topological regime and is imported from a same-group prior calculation, not recomputed here.
  • standard math Gor'kov equations and the linear-in-omega expansion of the nanowire self-energy are valid.
    Standard Green's function techniques; the low-frequency approximation is conventional for effective low-energy Hamiltonians but is not explicitly justified.

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Pith. "Pith review of Strongly enlarged topological regime and enhanced superconducting gap in nanowires coupled to Ising superconductors." pith.science (2026). https://pith.science/paper/3AFTCH3B

@misc{pith2026190811625,
  author       = {Pith},
  title        = {Pith review of: Strongly enlarged topological regime and enhanced superconducting gap in nanowires coupled to Ising superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3AFTCH3B}},
  note         = {Machine review of arXiv:1908.11625}
}
read the original abstract

An external magnetic field is needed to drive a nanowire in proximity to an s-wave superconductor into a topological regime which supports Majorana end states. However, a magnetic field generally suppresses the proximity superconducting gap induced on the nanowire. In recent experiments using InSb nanowires coupled to Al, the induced proximity gap vanishes at magnetic fields B~1T. This results in a small superconducting gap on the wire and a narrow topological regime which is proportional to the strength of the magnetic field. In this work, we show that by placing nanowires in proximity to recently discovered Ising superconductors such as the atomically thin transition-metal dichalcogenide(TMD) NbSe2, the topological superconducting gap on the wire can maintain at a large magnetic field as strong as B~10T. This robust topological superconducting gap is induced by the unique equal-spin triplet Cooper pairs of the parent Ising superconductor. The strong magnetic field allows a topological regime ten times larger than those in InSb wires coupled to Al. Our work establishes a realistic platform for building robust Majorana-based qubits.

Figures

Figures reproduced from arXiv: 1908.11625 by the authors.

Figure 1
Figure 1. FIG. 1. (a) The superconducting spin susceptibility [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Fig.2. Expectedly, they can create a robust proximity [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic of a nanowire in proximity to an Ising [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Different [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison between topological superconduct [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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