{"id":"ba16a617-57ed-4519-8461-015f64b49fed","arxiv_id":"2505.15745","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A heterogeneous mixture of topological and trivial superconducting domains in FeSe1-xTex explains why only some vortices host Majorana zero modes, with a domain-wall Majorana edge mode as the distinguishing signature.","lead":"The authors model FeSe1-xTex as a patchwork of topological and trivial superconducting domains and show that vortices in the two domain types host or lack Majorana zero modes. This gives a concrete, testable explanation for why only some vortex cores in this material show zero-energy states, with a predicted signature at domain walls.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central explanation depends on unverified presence of adjacent topological and trivial domains in FeSe1-xTex; the paper's domain-wall linecut is a prediction, not evidence for this premise.","rationale":"The reader identified as weakest assumption the lack of evidence for real adjacent strong-topological and trivial domains. I agree that this is the load-bearing premise. The static calculations are internally consistent, and the STM linecut prediction is a strong falsifiable consequence, so no internal inconsistency is alleged. However, the explanation's applicability to FeSe1-xTe depends entirely on whether such domains exist; the paper only assumes them. The time-dependent rigid-vortex approximation is a secondary concern because the abstract's main explanatory claim and the domain-wall prediction do not depend on the dynamical transfer result. The proposed STM test would settle the domain-existence question by checking the paper's own necessary consequence: a domain-wall MEM must appear between a topological and a trivial vortex. Since this is the reader's conditional requirement and no new objection emerges, the verdict remains CONDITIONAL (UNCHANGED).","tokens_in":9677,"tokens_out":15685,"duration_ms":152145,"concrete_test":"Re-analyze existing high-resolution STM maps of FeSe1-xTe (e.g., Machida et al., Nat. Mater. 18, 811 (2019)) to mark every vortex core with and without a zero-bias peak, and generate dI/dV linecuts connecting vortices of opposite type. If no zero-energy feature appears at a boundary between the two populations, the proposed adjacent topological/trivial domain structure is falsified. If such features appear, the scenario is supported. This directly tests the paper's necessary-condition prediction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that the puzzling coexistence of vortex MZMs and non-MZM vortices in FeSe1-xTex is resolved by a heterogeneous mixture of strong-topological (odd-Chern) and trivial (C=0) domains. This explanation holds only if the real material actually contains adjacent domains of both types, separated by a Chern-number-changing boundary. The paper imposes such domains by hand—spatial variations of mu_r and J_r in Eq. (1), Fig. 1(d)—but provides neither measurement nor citation establishing that FeSe1-xTex realizes these variations at the required length scale. The predicted zero-energy domain-wall MEM in Fig. 2(d) is falsifiable, but it is a consequence of the assumed domain structure, not independent evidence that the structure exists. The paper's own Discussion weakens the claim further: vortices without zero modes can also arise in phases with even Chern number, so the experimental observation of non-ubiquitous MZMs does not uniquely require the proposed topological/trivial domain scenario; other mechanisms (Refs. [18-20]) remain viable. Thus the central claim is conditional on an unverified material premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes that the puzzling observation of vortices with and without Majorana zero modes (MZMs) in FeSe1−xTex can be explained by a heterogeneous mixture of strong-topological and trivial superconducting domains. Using a two-dimensional Rashba ferromagnet/s-wave superconductor model with self-consistently computed order parameters, the authors find that vortices in topological domains carry MZMs while those in trivial domains do not, and that a domain wall between the two types of domain hosts a Majorana edge mode (MEM). They predict a falsifiable STM signature: a dI/dV zero-bias linecut between a topological and a trivial vortex shows a peak at the topological vortex, no peak at the trivial vortex, and a peak at the intervening domain wall. In addition, using a non-equilibrium Green's function formalism with a moving vortex whose profile is prescribed by Eqs. (20)–(22), they show that a MEM is transferred to the vortex as an MZM when the vortex moves from a trivial into a topological domain. The manuscript argues that this explains the non-ubiquitous vortex MZMs and distinguishes the scenario from earlier topological-domain proposals.","tokens_in":9950,"tokens_out":4802,"duration_ms":44795,"significance":"If the central premise is correct, the paper provides a concrete, falsifiable experimental prediction — a zero-bias STM linecut with a distinct domain-wall MEM peak — that could settle the origin of non-ubiquitous vortex MZMs in FeSe1−xTex. The static BdG calculations are self-consistent, and the domain-wall MEM follows from the Chern number change at the interface, so that part of the argument is internally sound and not fitted to experiment. The time-dependent transfer simulation is also a clear and explicit demonstration of a conceptually interesting process, namely MEM-to-MZM transfer during vortex motion. The main weakness is that the existence of adjacent topological and trivial domains in the real material is assumed rather than evidenced, and several robustness claims, including the W-independence and the rigid-vortex approximation, are not fully demonstrated. The paper is therefore a valuable theoretical proposal with testable consequences, but its central explanatory claim is conditional on an unverified material premise.","major_comments":[{"comment":"The central premise of the paper — that FeSe1−xTex realizes adjacent strong-topological and trivial superconducting domains — is imposed by hand through spatial variations of μ_r and J_r in Eq. (1); no experimental measurement or citation is provided that establishes such domains at the required length scale. Moreover, the paper's own Discussion (final paragraph) concedes that vortices without zero modes can also occur in phases with even Chern number, so the experimental observation of non-ubiquitous MZMs does not uniquely require the proposed topological/trivial domain structure. This weakens the abstract's claim that the observation \"can be explained\" by the heterogeneous-domain scenario. The authors should either present supporting evidence or arguments for domain formation, or explicitly reframe the result as a scenario with falsifiable predictions and discuss how it can be distinguished from the alternative mechanisms in Refs. [18]–[20].","section":"Theoretical Methods / Discussion"},{"comment":"The statement in Theoretical Methods that the qualitative results are \"unaffected by changes in W\" is not substantiated: all static simulations use W = 4a0 (Figs. 1–3), and no data for other domain-wall widths are shown. This matters because W controls the hybridization and energy discretization of the MEM, which directly affects the height and position of the predicted zero-bias domain-wall peak in Fig. 2(d). The authors should provide a W-dependence study, at least for Fig. 2's linecut, and quantify the MEM energy ϵ0 as a function of W and system size.","section":"Theoretical Methods / Fig. 1(d)"},{"comment":"The time-dependent MEM-to-MZM transfer shown in Fig. 4 rests on the rigid-vortex approximation of Eqs. (20)–(22), in which the order-parameter magnitude and phase move with the vortex core without self-consistent relaxation, and on an artificially increased Δ to shorten the coherence length. The paper cites Refs. [27,28] for robustness of MZM existence to the spatial profile, but those references do not address dynamic profile relaxation during vortex motion. Since the transfer process depends on the instantaneous gap profile and on the vortex crossing the domain wall at finite velocity, the authors should either demonstrate that the result persists when the order parameter is allowed to relax self-consistently during the motion, or state as a clear limitation that the transfer is an approximation within the rigid-profile model.","section":"Appendix C / Fig. 4"}],"minor_comments":[{"comment":"The word \"Majoarana\" in the abstract is a typo and should read \"Majorana.\"","section":"Abstract"},{"comment":"\"Heterogenous\" should be \"Heterogeneous.\"","section":"Title"},{"comment":"The phrase \"hards ±-wave superconducting gap\" should be \"hard s±-wave superconducting gap.\"","section":"Introduction"},{"comment":"The notation e_{r'−r} for the unit vector in the Rashba term is not defined; please define it explicitly and clarify the sign convention for the cross product.","section":"Eq. (1)"},{"comment":"The parameter sets for points 1–4 in Fig. 1(c) are not fully specified in the caption; only the values for Fig. 4 are given. Please list the (μ, J) values for all points used in Figs. 1–3, along with lattice size and boundary conditions.","section":"Fig. 1 caption / Results"},{"comment":"The text acknowledges that the MEM has finite energy discretization ϵ0 due to finite size, but the zero-bias linecut in Fig. 2(d) shows a peak at the domain wall. Please clarify whether ϵ0 is small enough relative to the STM thermal broadening for the peak to be observable at zero bias in a realistic experiment, and specify how ϵ0 scales with the domain-wall length.","section":"Fig. 2(d)"},{"comment":"The displayed expression for ρ(r, σ, t) appears typeset incorrectly (with tilde symbols over V(t) and V†); please correct the notation or define the matrices involved.","section":"Appendix D, Eq. (24)"},{"comment":"\"we previous predicted\" should be \"we previously predicted.\"","section":"Discussion"},{"comment":"Ref. [21] is an arXiv preprint from 2025; if a published version has appeared, please update the citation.","section":"References"},{"comment":"Supplementary Movie 1 is referenced but its content and accessibility are not described; please ensure it is provided with the submission or clearly state where it can be obtained.","section":"Supplementary Material"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious theoretical proposal with a falsifiable experimental prediction and appears to fit the journal's scope. The main concern is that the central explanatory claim rests on an assumed domain structure that is not evidenced, and the authors' own Discussion admits alternative explanations for vortices without MZMs. If the authors can either strengthen the experimental grounding or carefully reframe the claim as a scenario, the paper could be appropriate for publication. I do not see issues with citation practice or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe takeaway: this is a respectable numerical paper that turns the non-ubiquitous vortex MZM puzzle in FeSe1-xTex into a concrete, falsifiable STM prediction. The new bit is not the idea of domains per se (Refs 18-19 already did patches), but the specific claim that trivial (C=0) superconducting domains host MZM-free vortices, and that the Chern-number-changing domain wall between topological and trivial regions necessarily carries a Majorana edge mode visible in dI/dV linecuts. That's a clean, testable consequence.\n\nWhat the paper does well: the static BdG calculations are self-consistent; the domain-wall MEM follows from bulk-boundary correspondence and shows up exactly where expected; and the control calculations in Fig 3 (both domains topological, both trivial) convincingly show that a MEM appears only when the Chern number changes. The time-dependent transfer of a MEM into a vortex MZM (Fig 4) is a fresh result, though it's a demonstration using their non-equilibrium formalism and a rigid vortex profile, not a quantitative prediction for a specific experiment.\n\nSoft spots, in proportion: the load-bearing premise—that real FeSe1-xTex actually contains adjacent strong-topological and trivial domains realized by spatial variations in chemical potential or exchange—is assumed, not evidenced. The paper doesn't measure or cite direct evidence for such domains at the required length scale; it imposes them by hand. The paper's own Discussion admits that even-Chern phases can also produce MZM-free vortices, so the experimental puzzle doesn't uniquely select this scenario. The time-dependent transfer uses a prescribed, non-self-consistent order parameter; the authors argue robustness with respect to static profile from prior work, but the transfer dynamics themselves could be sensitive to relaxation. The W-robustness claim is stated baldly without shown data—minor, since it's not the core.\n\nWho it's for: people working on iron-chalcogenide superconductors and vortex Majoranas; it gives STM a concrete thing to look for (zero-bias peak at domain walls between vortices with and without MZMs). It deserves a serious referee. The central scenario is plausible and testable, even if the material evidence is missing. I'd send it to review with a request for the authors to either strengthen the case for domain formation or soften the claim of explaining the experiments.","headline":"A solid, testable proposal for the mixed vortex-MZM puzzle in FeSeTe, but its material premise is assumed, not shown.","tokens_in":10468,"tokens_out":2016,"would_cite":true,"duration_ms":17846,"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 proposes that vortices with and without Majorana zero modes in FeSe1-xTex reside in separate topological and trivial superconducting domains, with a Majorana edge mode necessarily localized at each intervening domain wall.","keywords":["Majorana zero modes","FeSe1-xTex","topological superconductivity","magnetic vortices","domain walls","bulk-boundary correspondence","scanning tunneling microscopy","iron chalcogenide superconductor"],"falsifier":"Perform an STM zero-bias dI/dV linecut between a vortex that shows a zero-energy mode and one that does not. The scenario predicts a zero-energy peak at the domain wall between them; if no such peak appears while the two vortices still differ in their zero-mode content, the heterogeneous-domain explanation is ruled out. Conversely, imaging a domain wall between two zero-mode-hosting vortices should show no edge-mode peak.","tokens_in":9474,"feed_emoji":"🧲","tokens_out":5202,"duration_ms":42182,"temperature":0.7,"pith_summary":"This paper proposes an explanation for a longstanding puzzle in the iron-chalcogenide superconductor FeSe1-xTex: some magnetic vortices host a zero-energy Majorana mode while others, in the same material, do not. The explanation is that the material contains a heterogeneous mixture of strong topological and trivial superconducting domains, and only vortices in the topological domains carry a Majorana zero mode. Because the Chern number changes at the boundary between a topological and a trivial domain, bulk-boundary correspondence requires a Majorana edge mode at every such domain wall. The paper works out the spectroscopic fingerprints of this picture and predicts that a scanning-tunneling linecut between a vortex with and without a zero mode will show a third zero-bias peak at the domain wall. It also simulates, with a non-equilibrium method, a vortex being dragged from a trivial into a topological domain, showing the domain-wall Majorana mode transferred into the vortex as a localized Majorana zero mode.","feed_headline":"FeSe1-xTex's missing Majorana modes traced to trivial domains","feed_subtitle":"An STM linecut between the two vortex types should reveal a Majorana edge-mode peak at the domain wall between them.","key_machinery":"The central machinery is a two-dimensional lattice model of FeSe1-xTex in which topological superconductivity emerges from the interplay of s-wave pairing, Rashba spin-orbit coupling, and ferromagnetic exchange. Spatial variations of the chemical potential or the magnetic exchange energy split the system into strong topological and trivial domains separated by a domain wall of width W. Vortices are introduced through Peierls phases and the superconducting order parameter is computed self-consistently. The load-bearing identity is the bulk-boundary correspondence: the Chern number changes across a topological-to-trivial domain wall, forcing a Majorana edge mode localized at the wall. The time-dependent part of the argument uses a non-equilibrium Green's-function formalism to follow the motion of a vortex and the transfer of this edge mode into a zero mode in the vortex core.","core_discovery":"We demonstrate that a heterogeneous structure of topological and trivial superconducting domains in FeSe1-xTex explains why only some vortices host Majorana zero modes. We show that vortices in topological domains host an MZM while those in trivial domains do not, and that the two types of domains are necessarily separated by a domain wall carrying a Majorana edge mode. We predict that an STM zero-bias linecut between a topological and a trivial vortex will reveal zero-energy peaks at the topological vortex and at the domain wall, and none at the trivial vortex. Using a non-equilibrium formalism, we show that moving a vortex in real time from a trivial into a topological domain transfers a Majorana edge mode from the domain wall to the vortex as a localized Majorana zero mode.","pith_inferences":["If the scenario holds, the predetermined position of domain walls in FeSe1-xTex could be used as a bottom-up template for arranging Majorana zero modes in arrays, since vortices in topological domains automatically inherit the modes.","The prediction that the domain-wall peak accompanies every topological-trivial pair could be tested against existing STM datasets of Fe(Se,Te) samples, which already contain linecuts through zero and nonzero vortex cores.","Because the model attributes the domains to spatial variation in chemical potential or magnetic exchange, local manipulation of these parameters, for example by a gate or by magnetic adatom deposition, should be able to engineer where MZMs appear and disappear.","The same heterogeneous-domain mechanism may operate in other iron-chalcogenide superconductors in which ferromagnetism coexists with superconductivity, not only FeSe1-xTex."],"forward_implications":["If the heterogeneous-domain picture is right, an STM zero-bias linecut between a vortex with a zero mode and one without will see a zero-energy peak at the domain wall between them, a signature absent in previous all-topological domain proposals.","Vortices inside topological domains always carry a Majorana zero mode; vortices inside trivial domains never do, so the fraction of zero-mode vortices directly maps the spatial distribution of topological versus trivial regions.","The observed increase in the number of trivial vortices with increasing magnetic field is explained by the ferromagnetic ordering driving the system from an odd-Chern topological phase into a trivial or even-Chern phase.","Dragging a vortex across a domain wall moves a Majorana edge mode into the vortex core, providing a controlled way to create or erase a localized Majorana zero mode in real time.","No domain-wall Majorana peak should appear between vortices that lie in the same kind of domain, even if their chemical potential or magnetic moment differ."],"supporting_citations":[{"why":"Supplies the founding observation of topological surface superconductivity on FeSeTe.","marker":"[1]"},{"why":"Reports zero-energy states in vortex cores attributed to Majorana zero modes.","marker":"[5]"},{"why":"Reports that zero-energy vortex bound states are not present in all vortices, the puzzle this paper addresses.","marker":"[6]"},{"why":"Provides the nearly quantized conductance plateau evidence for a vortex zero mode.","marker":"[8]"},{"why":"Provides evidence for dispersing 1D Majorana channels at domain walls, which the paper builds on.","marker":"[9]"},{"why":"Supplies the model of topological superconductivity from ferromagnetism, Rashba spin-orbit coupling, and s-wave pairing that the domain picture is based on.","marker":"[15]"},{"why":"Presents a previous domain-based explanation in which both domains are topological, the scenario the paper distinguishes itself from.","marker":"[18]"},{"why":"Reports bulk ferromagnetism in FeSe1-xTex, motivating the ferromagnetism-based scenario.","marker":"[21]"},{"why":"Supplies the non-equilibrium formalism used to simulate vortex motion and the MEM-to-MZM transfer.","marker":"[22]"}],"fun_headline_variants":["Heterogeneous domains explain which FeSeTe vortices host Majoranas","Trivial domains block Majorana modes in FeSeTe vortices","Moving vortices across domain walls transfers Majorana modes","Domain-wall Majorana edge modes migrate to moving vortices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The explanation assumes that FeSe1-xTex actually contains adjacent strong-topological and trivial superconducting domains, a scenario the paper posits through spatial variations in chemical potential or magnetic exchange; it does not present or cite direct experimental evidence that such domains exist in the material.","fun_headline_variants_meta":{"raw":{"variants":["Heterogeneous domains explain which FeSeTe vortices host Majoranas","Trivial domains block Majorana modes in FeSeTe vortices","Moving vortices across domain walls transfers Majorana modes","Domain-wall Majorana edge modes migrate to moving vortices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00072,"raw_usage":{"total_tokens":3167,"prompt_tokens":812,"completion_tokens":2355,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":2284}},"tokens_in":428,"tokens_out":2355,"duration_ms":17002,"temperature":1.0,"reasoning_tokens":2284,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:11:43.832727+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform an STM zero-bias dI/dV linecut between a vortex that shows a zero-energy mode and one that does not. The scenario predicts a zero-energy peak at the domain wall between them; if no such peak appears while the two vortices still differ in their zero-mode content, the heterogeneous-domain explanation is ruled out. Conversely, imaging a domain wall between two zero-mode-hosting vortices should show no edge-mode peak.","supporting_citations":[{"cited_title":"Machida, Y","cited_arxiv_id":null,"evidence_quote":"Reports that zero-energy vortex bound states are not present in all vortices, the puzzle this paper addresses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides evidence for dispersing 1D Majorana channels at domain walls, which the paper builds on."},{"cited_title":"Mascot, S","cited_arxiv_id":null,"evidence_quote":"Supplies the model of topological superconductivity from ferromagnetism, Rashba spin-orbit coupling, and s-wave pairing that the domain picture is based on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents a previous domain-based explanation in which both domains are topological, the scenario the paper distinguishes itself from."},{"cited_title":"Bedow, E","cited_arxiv_id":null,"evidence_quote":"Supplies the non-equilibrium formalism used to simulate vortex motion and the MEM-to-MZM transfer."}],"review_version":1}