{"id":"8a2a4a8c-5fbb-4766-b0a0-2a373d6cc989","arxiv_id":"2412.19096","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"An in-plane magnetic-field vortex (x-vortex) in iron-based superconductors supports a simple topological phase with unpaired Majorana zero modes over a wide parameter range, unlike the previously studied z-vortex.","lead":"This paper studies vortices in iron-based superconductors when the magnetic field lies in the plane of the iron layers, a geometry most prior work ignored. It finds that this 'x-vortex' can host stable Majorana zero modes in a simple, strain-tunable phase diagram, which could make these materials more practical for topological quantum computing.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The simple x-vortex phase is established only at kx=0; for a vortex along x, MZMs require comparing the class-D invariants at kx=0 and kx=π, and A'_3 = t'_{3x} cos kx − t'_{3y} cos ky flips sign between these planes, so the missing kx=π check could invalidate the headline phase diagram.","rationale":"The reader's verdict is CONDITIONAL, and my concern does not change that: the result should remain conditional pending a check of the full vortex-line topology. The reader's weakest assumption focused on experimental realizability of low SOC and the strain sign, and mentioned the kx=0-only phase boundaries in the rationale. I agree that the experimental mapping is an open question, but the more immediate and load-bearing problem is internal: the paper's own MZM criterion requires information at both kx=0 and kx=π, and the kx-dependence of A'_3 makes it impossible to infer the kx=π sector from the kx=0 diagrams. This is not a disagreement with external consensus; it is a missing step in the derivation. If the proposed test shows a finite full gap and distinct invariants, the central claim would be substantially supported. If not, the simple robust phase is an artifact of the slice used. Either way the supplementary material and code would be needed for full verification; as written, the preprint does not yet establish the headline claim.","tokens_in":12872,"tokens_out":10446,"duration_ms":102970,"concrete_test":"Take a representative point inside the claimed low-SOC A'_3>0 topological region (e.g., λ=A3 and a chemical potential in the topological phase of Fig. 3(b)). Construct the x-vortex BdG Hamiltonian on a lattice with Nx=120, Ny=60, Nz=30 and compute, for each kx in [-π,π], the lowest positive eigenvalue of the transverse (y,z) problem; verify the minimum over all kx is nonzero. Independently compute the class-D Z2 invariant at kx=0 and kx=π from the mirror-x eigenspaces (e.g., via Mx=±i Chern numbers or the Pfaffian invariant) on the two planes. If the finite-kx gap closes, or if the kx=0 and kx=π invariants are equal, the simple topological phase does not survive the full quasi-1D criterion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that a low-SOC x-vortex with A'_3>0 supports stable, unpaired MZMs in a simple phase diagram, immune to multiband entanglement. The load-bearing step is the reduction of the vortex-line topology to the kx=0 mirror plane. For a vortex along x, the BdG Hamiltonian is a quasi-1D system in class D parameterized by kx, and the authors themselves state that MZMs require (i) a sign change of the Z2 index between the mirror-invariant planes kx=0 and kx=π, and (ii) a fully gapped quasi-1D spectrum. However, all phase diagrams and Berry-phase analyses (Figs. 1(b), 3(b,c), 4) are computed only at kx=0; the kx=π plane is never analyzed. This is not a minor omission because the strain term is momentum dependent: A'_3 = t'_{3x} cos kx − t'_{3y} cos ky. With t'_{3x} > t'_{3y}, A'_3 is positive near Γ (kx=0, ky=0) but negative at (kx=π, ky=0), placing the kx=π plane in the A'_3<0 regime that the paper itself identifies with the complex, z-vortex-like phase diagram. Thus the simple phase could be an artifact of inspecting only one slice of the vortex-line Brillouin zone. The full quasi-1D gap condition (ii) is also asserted rather than demonstrated for all kx; the open-boundary DOS in Fig. 3(f,g) covers only selected line cuts. This is an internal gap in the argument, independent of whether real FeSCs reach the low-SOC regime.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies the topology of vortex lines in a six-band BdG model of iron-based superconductors. It contrasts vortices along the z-axis (z-vortex) with vortices along the x-axis (x-vortex) and identifies a low-spin-orbit-coupling regime in which the x-vortex phase diagram contains only trivial and topological superconducting phases, supporting a single unpaired Majorana vortex despite the coexistence of Dirac and topological-insulator bands. With strain-induced C4z breaking, the sign of the inter-layer hopping A'_3 selects either this simple diagram or a complex z-vortex-like diagram. The authors explain the phase boundaries using Berry-phase and mirror-Chern-number arguments and propose x-vortex nanowires as a platform for Majorana-based quantum devices.","tokens_in":13267,"tokens_out":23804,"duration_ms":393388,"significance":"The central result is potentially significant: it identifies a vortex orientation that may circumvent the multiband entanglement that has complicated Majorana searches in FeSCs, and it predicts strain control of the vortex topology. The paper has genuine strengths: the model parameters are anchored to ARPES data (Ref. [55]), the phase diagrams are direct numerical outputs of the BdG Hamiltonian rather than fits, and the Berry-phase/Chern-number arguments provide testable organizing principles. The main reservation is that the decisive x-vortex invariant is only analyzed at the kx=0 mirror plane; until the kx=pi plane is shown to be trivial or included in the invariant, the headline 'simple' phase diagram is not yet established. The practical reach also depends on the physical realizability of the low-SOC regime and on the strain-hopping relation, which are assumed rather than demonstrated.","major_comments":[{"comment":"The authors state that MZMs in an x-vortex require both a sign change of the Z2 index between the mirror-invariant planes kx=0 and kx=pi and a fully gapped quasi-1D spectrum, but every phase diagram and Berry-phase calculation in the paper is performed only at kx=0. This is not a minor omission: the strain term defined in the text, A'_3 = t'_{3x} cos kx - t'_{3y} cos ky, changes sign between the two planes, so under the stated assumption t'_{3x}>t'_{3y}>0 the kx=pi plane satisfies A'_3(kx=pi,ky) = -t'_{3x} - t'_{3y} cos ky < 0 for all ky. The kx=pi plane thus lies entirely in the A'_3<0 regime that the paper associates with the complex z-vortex-like phase diagram. The kx=pi invariant and the full-kx gap structure must be computed before the simple A'_3>0 phase diagram can be claimed as the vortex-line topology. If this analysis exists in the supplementary material, it should be brought into the main text.","section":"Vortex topology with Dirac nodes / Vortex topology without C4z symmetry; Figs. 1(b), 3(b,c), 4"},{"comment":"The assertion that the lowest band in the Mx=+i subspace always carries Chern number c3=-1, and the consequent result (c1,c2)=(1,0) for A'_3>0, is deferred to the supplementary material. This result is load-bearing: without it, the Berry-phase argument that the A'_3>0 diagram has only two phases (while A'_3<0 has two phase transitions in the second band) has no foundation. The main text should either prove this statement or reproduce the supplementary derivation in sufficient detail for the reader to verify it.","section":"Vortex topology without C4z symmetry; Fig. 4"},{"comment":"The physical realization of the predicted simple phase depends on two assumptions that are stated but not demonstrated: that the low-SOC regime (lambda < A3, with A3=1) is accessible in FeSCs, and that uniaxial strain reverses the sign of t'_{3x}-t'_{3y} in the manner assumed. The second assumption is particularly important because the entire 'tunable' claim rests on it, yet no microscopic derivation or estimate of the strain-hopping coupling is provided. I would like to see at least a quantitative estimate of the required strain, or a discussion of which candidate materials could plausibly reach lambda < A3, or both.","section":"Theoretical model / Vortex topology without C4z symmetry"}],"minor_comments":[{"comment":"The values of alpha, beta, t'_{3x}, t'_{3y}, and the coherence length xi are never specified, even though they enter Eq. (3) and the strain term; please list all parameters used in the figures.","section":"Theoretical model"},{"comment":"The expression TD(k) = A(sz sin kx + i sin ky) + A'_3 sx sin kz has a term 'i sin ky' with no Pauli matrix, which is dimensionally inconsistent as written; please correct this typo.","section":"Theoretical model, Eq. (3)"},{"comment":"There are several typos: 'stabablity' in the introduction, 'lambda A3 o smaller' in the discussion of Fig. 1(b), and 'the lower ratio of MZMs evidence' should probably be 'the low ratio'.","section":"Introduction and discussion of Fig. 1"},{"comment":"References [55] and [85] are identical, and references [79] and [80] are identical; please deduplicate them.","section":"References"},{"comment":"The text refers to 'A'_3 = t'_x cos kx - t'_y cos ky' and later 'A'_3 = t'_{3x} cos kx - t'_{3y} cos ky'; please use one consistent notation for the hopping amplitudes.","section":"Vortex topology without C4z symmetry"},{"comment":"The statement that the two phase transitions for A'_3<0 are 'placed in different C2z sections' seems to be a typo: for the x-vortex the relevant rotational symmetry is C2x, and the paper elsewhere assigns the sections by C2x.","section":"Vortex topology without C4z symmetry"},{"comment":"The caption says the green dots indicate positions on the Fermi surface, but panels (c,d) plot Berry phase versus chemical potential; please clarify what is actually marked.","section":"Fig. 4 caption"},{"comment":"The DOS calculation reports lattice sizes Nx=120, Ny=60, Nz=30 but does not specify which directions use open versus periodic boundary conditions; please state this explicitly.","section":"Fig. 3(f,g)"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to interest the topological-superconductivity community, and the numerical work appears internally careful. The primary obstacle is the missing kx=pi analysis: if the supplementary material contains the invariant and gap calculations for that plane, the central claim may be salvageable and the revision could be limited to moving those results into the main text. If the supplementary material does not contain them, the headline phase diagram is not supported. I also note that the paper's device-oriented language in the abstract and conclusion goes beyond what the single-model analysis demonstrates; this should be tempered during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Xin's group has a genuinely new idea here. Prior vortex work in FeSCs has been almost all z-vortices; they show that an in-plane x-vortex has a different phase diagram, and that in the low-SOC limit, with a particular sign of the strain-induced hopping A'_3, the diagram simplifies to just trivial and topological phases. If true, that is a useful guide for experiment, because grown FeSC nanowires lie in the a-b plane. The six-band model is standard, parameters anchored to ARPES, and the symmetry arguments tying the phase transition to π Berry phase and mirror Chern numbers are coherent. Credit where due: this is not a fitting exercise, the phase boundaries come from direct diagonalization, and the sign duality between A'_3>0 and A'_3<0 is a nice observation.\n\nBut the stress-test concern is real and lands on the load-bearing step. For a vortex along x, the topological invariant is a 1D class D invariant that must be compared at the mirror-invariant planes kx=0 and kx=π. The paper computes phase diagrams only at kx=0. The strain term A'_3 = t'_{3x} cos kx - t'_{3y} cos ky flips sign between those planes, so the kx=π plane will sit in the opposite A'_3 regime — the one the paper itself identifies with the complex, z-vortex-like diagram. That means the simple phase seen at kx=0 does not by itself determine the vortex-line topology; the paper has to show the invariant at kx=π and the full quasi-1D gap. It also defers the proof that the lowest band's Chern number stays -1 to the supplementary materials, which are not in the preprint. No code or data is shipped, so the numerics can't be checked directly.\n\nThe claims in the abstract that the x-vortex 'supports unpaired Majorana vortices across a wide parameter range' and 'introduce a novel paradigm' outrun the evidence as presented. The 'low-SOC' regime itself is a physical assumption that may or may not hold in real FeSCs, though that is a legitimate question for experimentalists rather than a flaw in the calculation.\n\nThis deserves a serious referee, but not in its current form. The missing kx=π analysis is the key to the whole story, and the authors need to either produce it or soften the headline claim. I would encourage you to send it out, with referees asked specifically about the quasi-1D classification and the supplementary derivations.","headline":"The x-vortex idea is fresh and plausibly important, but the central phase diagram is only computed at kx=0, leaving the required kx=π check undone.","tokens_in":13854,"tokens_out":2630,"would_cite":false,"duration_ms":24683,"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":"The paper claims that an x-oriented vortex in a low-spin-orbit iron-based superconductor hosts stable, unpaired Majorana zero modes over a wide chemical-potential range, even with Dirac nodes present and multiband entanglement, and that…","keywords":["Majorana zero modes","iron-based superconductors","vortex topology","multiband superconductivity","topological crystalline superconductivity","uniaxial strain","Dirac nodes","mirror Chern number"],"falsifier":"Angle-resolved photoemission on an iron-based superconductor could measure $\\lambda$ and $A_3$ directly; if $\\lambda$ is never found below $A_3$ in any accessible composition, the predicted low-SOC x-vortex phase cannot be realized. A second test is strain-tuned scanning tunneling microscopy on an a-axis-oriented FeSC nanowire: if uniaxial strain with $A'_3 > 0$ does not produce a zero-bias conductance peak at both ends of the wire across a broad chemical-potential range, the simple phase diagram would be contradicted.","tokens_in":12630,"feed_emoji":"🌀","tokens_out":13075,"duration_ms":105798,"temperature":0.7,"pith_summary":"This paper argues that in multiband iron-based superconductors, the direction of the vortex line controls whether Majorana zero modes are stable. It shows that an x-vortex, oriented perpendicular to the Dirac axis, can have a phase diagram containing only conventional and topological superconducting phases when the spin-orbit coupling is low ($\\lambda < A_3$), in contrast to the z-vortex's alternating gapless, topological-crystalline, and topological phases. In that simple regime, one unpaired Majorana zero mode sits in the vortex core across a wide chemical-potential range even while Dirac nodes remain in the electronic bands, and multiband entanglement does not destroy it. Uniaxial strain flips the sign of the anisotropy term $A'_3$ and switches between this simple diagram and a z-vortex-like complex diagram, providing a controllable knob. The paper's payoff is a plausible route to stable Majorana zero modes in iron-based nanowires and quantum devices.","feed_headline":"X-vortices give iron-based superconductors stable Majorana modes","feed_subtitle":"At low spin-orbit coupling, the x-vortex hosts unpaired Majorana modes across a wide chemical-potential range.","key_machinery":"The load-bearing mechanism is the vortex line treated as a quasi-1D class D superconductor, with a vortex-phase-transition plane (VPTP) at the mirror-invariant $k_x = 0$ plane. Majorana zero modes appear when the $Z_2$ index changes sign across this plane and the quasi-1D system is fully gapped; the sign change is governed by a $\\pi$ Berry phase on Fermi surfaces in the $M_x = +i$ mirror subspace. Mirror-x symmetry block-diagonalizes the six-band Hamiltonian on the VPTP, and the paper relies on the lowest band's Chern number being fixed at $-1$ in that subspace. The anisotropy is traced to how the Dirac nodes interact with the VPTP: for the x-vortex they sit inside it, and the transition from type I to type II Dirac nodes at $\\lambda = A_3$ removes the $\\pi$ Berry phase that would otherwise create gapless vortex phases. With strain, the mirror Chern numbers $(c_1, c_2) = (1,0)$ for $A'_3 > 0$ versus $(-1,2)$ for $A'_3 < 0$ determine whether the Berry phase crosses $\\pi$ once or twice, producing the simple versus complex phase diagrams.","core_discovery":"The central claim is that the x-vortex configuration—a vortex line running along the x-direction, perpendicular to the $\\Gamma$–$Z$ Dirac axis—has two distinct topological phase diagrams, and the simpler one is unique to this orientation. With $C_{4z}$ symmetry intact and low spin-orbit coupling ($\\lambda < A_3$), the Dirac nodes become type II, their Fermi surfaces become non-closed, and the $\\pi$ Berry phase that would normally generate gapless vortex phases becomes ill-defined; the phase diagram then collapses to a conventional region and a topological superconducting region. The topological region hosts one unpaired Majorana zero mode per vortex in the $C_{2x} = -1$ mirror subspace, stable even when Dirac and topological-insulator bands coexist. When strain breaks $C_{4z}$, the sign of $A'_3 = t'_{3x}\\cos k_x - t'_{3y}\\cos k_y$ selects which diagram appears: $A'_3 > 0$ preserves the simple Fu–Kane-like phase structure in the low-SOC limit, while $A'_3 < 0$ reproduces the z-vortex-like sequence of conventional, topological crystalline, and topological superconducting phases. The paper further shows that the simple phase diagram is resilient to the multiband entanglement that destabilizes z-vortex Majorana modes.","pith_inferences":["Editorial inference: The same mirror-plane Berry-phase criterion suggests that other multiband superconductors with mirror symmetry and Dirac nodes on the vortex-phase-transition plane could exhibit the same clean x-vortex phase; the mechanism is not obviously limited to iron-based materials.","Editorial inference: The type-I-to-type-II Dirac transition at the crossover scale is a sharp feature that could serve as a smoking-gun signature; a band-structure study of Dirac-cone tilt under strain or doping would locate the simple phase without waiting for a full Majorana measurement.","Editorial inference: Since the simple phase requires weaker spin-orbit coupling and a sign-stable anisotropy term, materials among the iron-based family with intrinsically lighter pnictogen or chalcogen atoms are the most promising near-term targets for the proposed x-vortex devices."],"forward_implications":["In the low-spin-orbit x-vortex regime, unpaired Majorana zero modes appear over a wide chemical-potential range even when Dirac nodes coexist with topological-insulator bands, making the phase diagram insensitive to multiband entanglement.","Uniaxial strain flips the sign of A'_3 and switches the x-vortex between the simple two-phase diagram and the z-vortex-like complex diagram, adding a strain knob to Majorana physics in iron-based superconductors.","The x-vortex geometry matches the orientation of grown iron-based nanowires and nanoribbons, so Majorana zero modes should appear at both ends of such wires, amenable to tunneling-conductance correlation measurements.","In the high-spin-orbit regime with C4z symmetry, the x-vortex still shows gapless and Majorana vortex states split across C2x = +1 and -1 subspaces, connecting it to topological crystalline superconducting physics."],"supporting_citations":[{"why":"supplies the Fu–Kane topological-insulator/superconductor paradigm that the low-SOC x-vortex phase reproduces.","marker":"[33]"},{"why":"identifies the C4z-protected Dirac nodes on the Γ–Z line, the band feature whose position in the VPTP drives the anisotropy.","marker":"[78]"},{"why":"establishes the quasi-1D class D viewpoint and the two conditions for Majorana zero modes in vortex lines.","marker":"[79]"},{"why":"provides the π Berry phase criterion for vortex topological phase transitions used to locate the phase boundaries.","marker":"[80]"},{"why":"gives the single-Dirac-band vortex phase diagram that the paper contrasts with its multiband result.","marker":"[81]"},{"why":"supplies experimental evidence that uniaxial strain breaks C4z symmetry and controls the sign of A'_3.","marker":"[68–71]"},{"why":"contains the supplementary proof that the lowest band Chern number is fixed at −1 and the details behind the phase diagrams.","marker":"[77]"}],"fun_headline_variants":["X-vortex yields stable Majorana modes in iron superconductors","Strain control of x-vortex topology for Majorana qubits","Unique x-vortex phase diagram enables robust Majorana vortices","Iron-based x-vortex hosts unpaired Majorana modes even with Dirac nodes","X-vortex topology: simpler phase diagram, stable Majoranas"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results stand on the assumption that real iron-based superconductors can be placed in the low spin-orbit coupling regime ($\\lambda < A_3$, where $A_3$ sets the topological-insulator gap) and that uniaxial strain flips the sign of the anisotropy term $A'_3$ as modeled; if materials always sit at high spin-orbit coupling, or strain does not flip that sign, the clean x-vortex Majorana phase does not occur in any known material. A second, deferred premise is that the lowest band's Chern number is fixed at $-1$ in the $M_x = +i$ subspace.","fun_headline_variants_meta":{"raw":{"variants":["X-vortex yields stable Majorana modes in iron superconductors","Strain control of x-vortex topology for Majorana qubits","Unique x-vortex phase diagram enables robust Majorana vortices","Iron-based x-vortex hosts unpaired Majorana modes even with Dirac nodes","X-vortex topology: simpler phase diagram, stable Majoranas"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000221,"raw_usage":{"total_tokens":1508,"prompt_tokens":1063,"completion_tokens":445,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":355}},"tokens_in":679,"tokens_out":445,"duration_ms":4436,"temperature":1.0,"reasoning_tokens":355,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:57:59.988744+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Angle-resolved photoemission on an iron-based superconductor could measure $\\lambda$ and $A_3$ directly; if $\\lambda$ is never found below $A_3$ in any accessible composition, the predicted low-SOC x-vortex phase cannot be realized. A second test is strain-tuned scanning tunneling microscopy on an a-axis-oriented FeSC nanowire: if uniaxial strain with $A'_3 > 0$ does not produce a zero-bias conductance peak at both ends of the wire across a broad chemical-potential range, the simple phase diagram would be contradicted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"identifies the C4z-protected Dirac nodes on the Γ–Z line, the band feature whose position in the VPTP drives the anisotropy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the single-Dirac-band vortex phase diagram that the paper contrasts with its multiband result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"contains the supplementary proof that the lowest band Chern number is fixed at −1 and the details behind the phase diagrams."}],"review_version":1}