{"id":"3fa1f4d7-140f-4b0b-840f-66700409bda7","arxiv_id":"2507.21446","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A helical magnetic adatom chain on an s-wave superconductor plus a d-wave altermagnet is predicted to host a topological Fulde-Ferrell superconducting phase with Majorana end states and a field-free superconducting diode efficiency over 45%.","lead":"The paper proposes a new magnetic atomic chain setup that produces a superconducting diode effect and hosts Majorana zero modes without needing an external magnetic field. A smart generalist would read it because it unites two sought-after quantum technologies, topological Majorana states and dissipationless current control, in one field-free device architecture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper never demonstrates that the η≈45% SDE parameters and the topological FF parameters coexist; topology figures use g=π/2, T=0.01 while SDE figures use g=2π/3, T=0.1, so the 'same phase' claim is unsupported.","rationale":"The reader's verdict correctly flags the J_A modeling assumption as a risk: a real d-wave altermagnet interface may not produce the uniform spin-dependent hopping of Eq. (2), and that idealization deserves a microscopic check. I partially agree with that concern. However, the more immediate load-bearing gap is internal to the paper's own calculations: the 'same topological FF phase' that supports the 45% diode efficiency is never shown to be topological at a common parameter set. The topology calculations and the SDE calculations use different g, t, U, and temperature, so the abstract's unification claim outruns the presented evidence. This is not a disagreement with the consensus or a demand for a different model; it is a missing consistency check within the authors' own framework. The proposed test is specific, inexpensive, and decisive: compute P_x and δ_m at the SDE-optimal parameters. If the phase there is trivial or gapless, the paper should be reframed as two independent effects. If it is topological, the central claim is substantially supported. The self-consistent BdG machinery, the use of P_x=0.5, and the LDOS signatures are legitimate internal evidence; the 40% versus 45% efficiency discrepancy is a reporting issue but secondary to the coexistence question. A conditional verdict is appropriate, with the coexistence calculation as the explicit condition.","tokens_in":20068,"tokens_out":19437,"duration_ms":262536,"concrete_test":"Run a single BdG calculation at the exact Fig. 5(c) helical parameters (t/Δ0=1.0, g=2π/3, U/Δ0=1.574, μ/Δ0=1.0, (βΔ0)−1=0.1, J/Δ0=0.5, J_A/Δ0=0.4), compute Δ(q), P_x(q), and δ_m(q) over the full q range used for I(q), and locate the q values of ±I_c. If P_x≠0.5 or δ_m=0 at those q values, the same-phase claim fails; if P_x=0.5 with a finite minigap, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central abstract claim is that the same topological FF phase supports both MZMs and a 45% diode efficiency. In the body, topology is established only for the helical case with (t/Δ0, g, U/Δ0, μ/Δ0, (βΔ0)−1, J/Δ0) = (0.5, π/2, 1.38, 1.0, 0.01, 0.65) in Figs. 3–4, while the SDE is computed at (t/Δ0, g, U/Δ0, μ/Δ0, (βΔ0)−1) = (1.0, 2π/3, 1.574, 1.0, 0.1) with J and J_A varied in Fig. 5. These parameter sets differ in t, g, U, and temperature, and no P_x or δ_m is reported at the SDE-optimal parameters. It is therefore possible that the high-efficiency regime is trivial or gapless, so the headline coexistence would be an artifact of comparing different regimes. The conclusion's assertion that 'the same topological FF phase supports strong nonreciprocal supercurrents' is currently a conjecture rather than a demonstrated result of the calculations. This is the most load-bearing gap because the paper's novelty is the unification of topology and diode functionality in one field-free phase; if the two effects live in disjoint parameter regions, the central claim is materially weakened even if each effect is individually correct.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a one-dimensional Shiba chain with helical or conical spin texture on an s-wave superconductor proximitized by a d-wave altermagnet, and studies it with a self-consistent real-space BdG mean-field approach using an FF pairing ansatz Δ_n ∝ e^{iqn}. The central claims are that (i) a field-free topological FF superconducting phase with bulk polarization P_x = 0.5 and finite minigap hosts tunable end Majorana zero modes, with the Cooper pair momentum controlled by an injected supercurrent, and (ii) this same phase supports a field-free superconducting diode effect with efficiencies exceeding 45% in the helical case and about 35% in the conical case. The authors support these claims with self-consistent gap and condensation-energy calculations, open-boundary spectra, LDOS profiles, a bulk-polarization phase diagram, and supercurrent-versus-q curves, supplemented by a gauge-transformed momentum-space analysis in the Supplementary Material.","tokens_in":20383,"tokens_out":9090,"duration_ms":113337,"significance":"If the coexistence claim is established, the proposal would be a notable step toward unifying field-free topological superconductivity and the superconducting diode effect in a single junction-free architecture, which is relevant for scalable Majorana-based and low-dissipation quantum devices. The paper's strengths include the self-consistent BdG treatment, the explicit use of P_x and δ_m to characterize the topological phase, open-boundary and LDOS checks of the Majorana modes, and a clear symmetry discussion distinguishing helical from conical spin textures. The model produces concrete quantitative predictions such as q0(J_A), P_x=0.5 regions, and diode efficiencies that are falsifiable by scanning-tunneling and transport experiments. The significance is currently limited by two internal gaps: the topological and SDE calculations are not performed in the same parameter regime, and the abstract's 45% efficiency is not reconciled with the body's reported 40% maximum from the momentum-space analysis.","major_comments":[{"comment":"The headline claim that the same topological FF phase supports both Majorana zero modes and a ~45% diode efficiency is not demonstrated, because the topological and SDE calculations use disjoint parameter sets. Topology is established only for the helical chain at (t/Δ0, g, U/Δ0, μ/Δ0, (βΔ0)^{-1}) = (0.5, π/2, 1.38, 1.0, 0.01) in Figs. 3–4, whereas the SDE in Fig. 5 is computed at (1.0, 2π/3, 1.574, 1.0, 0.1) with J and J_A varied. No P_x or δ_m is reported in the Fig. 5 parameter region, so the high-efficiency regime could be topologically trivial or gapless. The same mismatch exists for the conical case between SM S3 and Fig. 7. Please compute and report P_x, δ_m, and the zero-energy LDOS at the SDE-optimal parameters, and state explicitly whether the topological FF phase and the high-η regime overlap in the (J, J_A, q) phase diagram.","section":"Abstract; §III B; SM S2"},{"comment":"The abstract and introduction claim diode efficiencies exceeding 45% for the helical texture, but §III B states that the momentum-space analysis yields a maximum of 40%, and SM S2 reports a maximum of about 40%. Figure 5's color scale saturates at 45% but no concrete maximum value is given in the text. The abstract's 'exceeding 45%' claim is not supported unless the real-space Fig. 5 reaches a value strictly above 45%, which is not reported. Please give the exact maximum η and the parameters at which it occurs for both the real-space and momentum-space calculations, then adjust the abstract and conclusion to the supported value.","section":null}],"minor_comments":[{"comment":"The text repeatedly uses 'ad-wave' where 'a d-wave' is intended; similar typos appear in the Fig. 3 caption ('T opological') and the SM S3 heading ('W ave').","section":"Abstract, Introduction"},{"comment":"The spin-dependent hopping term containing J_A is written without its Hermitian conjugate; as displayed the Hamiltonian is not manifestly Hermitian. Please add the conjugate explicitly or state that it is implied.","section":"Eq. (2)"},{"comment":"The conclusion reports 'η ≳ 35%' for the conical texture, but Fig. 7(b,c) shows negative-valued efficiencies. Please state explicitly whether the quoted number is |η| and define the sign convention consistently with Eq. (11).","section":"§IV, Fig. 7"},{"comment":"Reference [61] already studies topological Majorana zero modes and the superconducting diode effect driven by FFLO pairing in a helical Shiba chain. The introduction should state explicitly that the new element here is the altermagnetic proximity and the resulting field-free operation, rather than the FFLO-SDE phenomenon itself.","section":"Introduction, Ref. [61]"},{"comment":"The momentum-space SM uses U/Δ0 = 0.2985 and 0.279, which differ by roughly a factor of five from the real-space U/Δ0 = 1.574 used in Fig. 5. Please clarify whether the gauge-transformed momentum-space model is meant to be quantitatively equivalent to Eq. (2) or is used only for qualitative insight.","section":"SM S2 vs §III B"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of cond-mat.supr-con and the modeling is standard for this field. The main gap is a missing parameter-regime calculation rather than a fundamental error, so I view major revision as appropriate rather than rejection. I would recommend the editor ask for the coexistence calculation (P_x, δ_m, LDOS at the SDE-optimal parameters) and a reconciliation of the 45% versus 40% efficiency figures before reconsideration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know one thing up front: the basic mechanism here is sound, and the paper is worth a careful read. The combination of a helical Shiba chain with a d-wave altermagnet is new in the cited literature, and the symmetry logic is clear: the altermagnet breaks time-reversal without a net field, and the helical texture plus altermagnetic spin-splitting breaks inversion, so a junction-free SDE becomes plausible. The self-consistent BdG treatment, the gauge transformation to an effective nanowire, the bulk polarization P_x=0.5, and the LDOS with a finite minigap are all standard and competently done.\n\nThe soft spot is the headline claim, and the stress-test note has it right. Topology is established at one parameter set (t/Δ0=0.5, g=π/2, U/Δ0=1.38, T/Δ0=0.01), while the diode efficiency is computed at a substantially different set (t/Δ0=1.0, g=2π/3, U/Δ0=1.574, T/Δ0=0.1). The paper never reports P_x or the minigap at the SDE-optimal parameters, so the sentence in the conclusion that \"the same topological FF phase supports strong nonreciprocal supercurrents\" is currently a conjecture, not a result. If the high-η regime is actually trivial, or gapless, the central novelty of a unified topological diode platform collapses even though each individual effect might be real.\n\nTwo smaller issues. First, the abstract and conclusion quote η≳45% for the helical case, but the momentum-space calculation in the SM reports a maximum of about 40%. That discrepancy needs a reconciliation or a clear statement about which calculation, parameter set, and definition of efficiency the abstract uses. Second, the altermagnet is modeled as a uniform spin-dependent hopping that does not reorient the classical spin spiral; that is a reasonable starting assumption, but a real interface could give a local spin-splitting or a spatially varying J_A, and the paper does not argue why the qualitative results would survive those details.\n\nThis is a paper for people working on Majorana chains, altermagnet-superconductor hybrids, and field-free SDEs. They will find a clean model and a useful symmetry analysis. It deserves a serious referee, but the referee should ask for the coexistence check: compute P_x and the minigap at the diode-optimal parameters, and reconcile the efficiency numbers. If that check holds, the paper is a solid contribution; if not, the claims need to be scaled back to a field-free SDE mechanism without the topological unification.","headline":"A plausible symmetry-based model for a field-free Shiba-chain diode, but the advertised coexistence of topological FF pairing and >45% efficiency is not actually demonstrated because the two effects are computed in different parameter regimes.","tokens_in":20945,"tokens_out":2580,"would_cite":true,"duration_ms":33094,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A helical Shiba chain proximitized by a d-wave altermagnet hosts a field-free topological Fulde–Ferrell superconducting state with tunable Majorana zero modes and nonreciprocal supercurrents that reach diode efficiencies above 45%.","keywords":["superconducting diode effect","altermagnet","Shiba chain","Majorana zero modes","Fulde-Ferrell superconductivity","finite-momentum pairing","topological superconductivity","Bogoliubov-de Gennes"],"falsifier":"Measure the zero-field critical currents of a helical Fe/Co/Mn Shiba chain on a superconductor proximitized by a $d$-wave altermagnet such as RuO$_2$; if $|I_c^+|$ equals $|I_c^-|$ within experimental error while both $J$ and $J_A$ are finite, the predicted field-free diode effect is absent. On the theory side, a first-principles electronic-structure calculation of the altermagnet/superconductor interface that yields an effective coupling different from the uniform hopping of Eq. (2) would break the mechanism that sets the finite Cooper-pair momentum $q_0$ and with it the topological FF phase.","tokens_in":19821,"feed_emoji":"⚛️","tokens_out":8435,"duration_ms":92760,"temperature":0.7,"pith_summary":"The paper claims that a one-dimensional chain of magnetic adatoms on an s-wave superconductor, when brought into proximity with a d-wave altermagnet, forms a Fulde–Ferrell superconductor with finite Cooper-pair momentum without any applied magnetic field. In that state the chain is topologically nontrivial: it has bulk polarization $P_x = 0.5$, a finite minigap, and Majorana zero modes localized at its ends, all tunable through the injected supercurrent. The same topological phase carries direction-dependent critical currents, giving a field-free, junction-free superconducting diode effect with efficiencies above 45% for a helical spin texture and above 35% for a conical one. If true, this would unify topological Majorana physics and supercurrent rectification in one zero-field device, removing the magnetic-field bottlenecks that plague scalable superconducting electronics.","feed_headline":"Field-free superconducting diode hits 45% efficiency","feed_subtitle":"A d-wave altermagnet lets one Shiba chain host Majorana modes and a 45% diode effect at zero field.","key_machinery":"The load-bearing object is the real-space Bogoliubov–de Gennes Hamiltonian of Eq. (2), in which the altermagnet proximity effect is encoded as a uniform spin-dependent hopping $(J_A/2)\\sigma_z$ along the chain, and magnetism enters as a classical spin spiral $\\mathbf{S}_n=\\{\\sin\\theta\\cos[g n], \\sin\\theta\\sin[g n], \\cos\\theta\\}$. A local spin-dependent gauge transformation maps this to a translation-invariant nanowire model with an effective spin–orbit coupling proportional to $g\\,k\\,\\sigma_z$ and an altermagnetic correction to the kinetic energy, making the symmetry structure transparent. The superconducting order parameter is solved self-consistently for each Cooper-pair momentum $q$ from the gap equation, and the physical ground state minimizes the condensation energy $\\Omega(q,\\Delta)$, selecting the finite momentum $q_0$ of the FF state. Topology is diagnosed by the bulk polarization $P_x$, with $P_x=0.5$ marking the Majorana phase, and by the minigap $\\delta_m$; the diode effect is quantified by the supercurrent $I(q)=-2e\\,\\partial\\Omega/\\partial q$ and the efficiency $\\eta=(|I_c^-|-|I_c^+|)/(|I_c^+|+|I_c^-|)$.","core_discovery":"The central discovery is that a d-wave altermagnet, through its momentum-dependent spin splitting, can simultaneously supply the two symmetry breakings that a helical Shiba chain otherwise lacks. Time-reversal symmetry is broken by the altermagnetic exchange even at zero net magnetization, and inversion symmetry is broken when this altermagnetic coupling coexists with the spin-spiral exchange, i.e., when both $J$ and $J_A$ are finite. The resulting self-consistent Bogoliubov–de Gennes ground state is a Fulde–Ferrell superconductor whose Cooper-pair momentum $q_0$ grows with $J_A$; this state is gapped, topologically nontrivial with $P_x = 0.5$, and supports Majorana zero modes at the chain ends. The same FF state makes the supercurrent–momentum relation nonreciprocal, $|I_c^+| \\neq |I_c^-|$, producing diode efficiencies that reach about 45% for the helical texture and about 35% for the conical one. The conical texture already breaks inversion on its own, so it shows the diode effect even without the altermagnet, while the helical texture requires both $J$ and $J_A$.","pith_inferences":["If the uniform $(J_A/2)\\sigma_z$ proximity coupling is replaced by a more realistic interface model with position-dependent $J_A$ or local spin splitting, the predicted $q_0$ and diode efficiency would likely shift; computing that coupling from first principles is the natural next step and would turn the present prediction into a quantitative materials forecast.","The conical-chain result, with a diode effect from $J$ alone and a topological phase even without $J_A$, suggests that the altermagnet's special role is to make the helical texture field-free as well, and that other noncollinear textures or spin-orbit-coupled magnets might achieve similar field-free diodes without altermagnets.","Extending the calculation to a two-dimensional Shiba lattice could produce chiral FF phases with chiral Majorana edge modes and possibly a field-free perfect diode ($\\eta \\to 1$), an outcome hinted at by the strengthening of the one-dimensional efficiency as $J_A$ and $J$ grow.","A zero-field critical-current measurement on an existing Fe-on-Pb(110) chain capped with RuO$_2$ would be a cheap falsifier and could be done before any Majorana detection, since the diode signal is a DC transport measurement."],"forward_implications":["Cooper-pair momentum $q_0$ becomes an in-situ tuning knob: changing the injected supercurrent shifts the topological phase boundaries, allowing the Majorana phase to be switched on and off in a single device.","The helical-texture diode requires both magnetic exchange $J$ and altermagnet coupling $J_A$; the conical texture works with $J$ alone, so the two textures offer complementary field-free diode mechanisms with opposite signs of $\\eta$.","Because no Zeeman field is needed, the proximity-induced superconducting gap is not suppressed, so the Majorana minigap $\\delta_m$ and the diode efficiency should survive in parameter regimes where conventional field-driven proposals fail.","Quantitatively, efficiencies above 45% (helical) and above 35% (conical) place this mechanism in the range of practical superconducting diodes, and the model uses weak-coupling BCS parameters ($\\Delta_0 \\sim 1$ meV) compatible with existing adatom chains on Pb, Nb, or Re surfaces."],"supporting_citations":[{"why":"Establishes that helical Shiba chains host a topological superconducting phase, the platform this paper modifies.","marker":"[16]"},{"why":"Shows that d-wave altermagnet proximity yields field-free topological superconductivity in nanowires, the basis of the zero-field Majorana mechanism used here.","marker":"[17]"},{"why":"Provides the intrinsic superconducting diode effect framework and the efficiency definition used for $\\eta$.","marker":"[25]"},{"why":"Supplies the self-consistent BdG method and the supercurrent formula $I(q)=-2e\\,\\partial\\Omega/\\partial q$ used to compute critical currents.","marker":"[57]"},{"why":"Connects finite-momentum superconductors to the supercurrent diode effect, grounding the FF-state-as-diode interpretation.","marker":"[59]"},{"why":"Provides the spin-dependent gauge transformation used to map the spin spiral to an effective nanowire model.","marker":"[71]"},{"why":"Shows that a supercurrent can induce topological phase transitions, supporting the $q$-tunable topological FF phase.","marker":"[73]"},{"why":"Predicted topological FF states in spin-orbit-coupled Fermi gases, the prior context for the tunable topological FF phase realized here in a solid-state chain.","marker":"[77]"},{"why":"Demonstrates Majorana modes in ferromagnetic atomic chains on a superconductor, the experimental template for Shiba-chain devices.","marker":"[80]"}],"fun_headline_variants":["Altermagnet enables field-free diode and Majorana modes","Field-free diode and Majorana modes from altermagnetic Shiba chain","Altermagnet produces 45% field-free diode in Shiba chain","Topological FF state from altermagnet drives field-free diode","Shiba chain plus altermagnet: 45% diode, Majorana modes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the altermagnet proximity effect taking the form of a uniform spin-dependent hopping $(J_A/2)\\sigma_z$ along the chain (Eq. 2) and on that coupling not reorienting the classical spin spiral; if a real $d$-wave altermagnet interface instead induces a local on-site spin splitting or a spatially varying coupling, the finite FF momentum, the Majorana phase, and the diode effect could all disappear.","fun_headline_variants_meta":{"raw":{"variants":["Altermagnet enables field-free diode and Majorana modes","Field-free diode and Majorana modes from altermagnetic Shiba chain","Altermagnet produces 45% field-free diode in Shiba chain","Topological FF state from altermagnet drives field-free diode","Shiba chain plus altermagnet: 45% diode, Majorana modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000728,"raw_usage":{"total_tokens":3319,"prompt_tokens":1065,"completion_tokens":2254,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":681,"completion_tokens_details":{"reasoning_tokens":2160}},"tokens_in":681,"tokens_out":2254,"duration_ms":18841,"temperature":1.0,"reasoning_tokens":2160,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:46:20.726338+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the zero-field critical currents of a helical Fe/Co/Mn Shiba chain on a superconductor proximitized by a $d$-wave altermagnet such as RuO$_2$; if $|I_c^+|$ equals $|I_c^-|$ within experimental error while both $J$ and $J_A$ are finite, the predicted field-free diode effect is absent. On the theory side, a first-principles electronic-structure calculation of the altermagnet/superconductor interface that yields an effective coupling different from the uniform hopping of Eq. (2) would break the mechanism that sets the finite Cooper-pair momentum $q_0$ and with it the topological FF phase.","supporting_citations":[{"cited_title":"Majorana quasiparticles in atomic spin chains on superconductors,","cited_arxiv_id":null,"evidence_quote":"Establishes that helical Shiba chains host a topological superconducting phase, the platform this paper modifies."},{"cited_title":"Colloquium: Atomic spin chains on surfaces,","cited_arxiv_id":null,"evidence_quote":"Shows that d-wave altermagnet proximity yields field-free topological superconductivity in nanowires, the basis of the zero-field Majorana mechanism used here."},{"cited_title":"Josephson diode effect in supercurrent inter- ferometers,","cited_arxiv_id":null,"evidence_quote":"Supplies the self-consistent BdG method and the supercurrent formula $I(q)=-2e\\,\\partial\\Omega/\\partial q$ used to compute critical currents."},{"cited_title":"Optimizing one dimensional superconducting diodes: interplay of rashba spin-orbit coupling and magnetic fields,","cited_arxiv_id":null,"evidence_quote":"Connects finite-momentum superconductors to the supercurrent diode effect, grounding the FF-state-as-diode interpretation."},{"cited_title":"Prevalence of trivial zero-energy subgap states in nonuniform helical spin chains on the surface of superconductors,","cited_arxiv_id":null,"evidence_quote":"Shows that a supercurrent can induce topological phase transitions, supporting the $q$-tunable topological FF phase."},{"cited_title":"Many-body electric multipole operators in ex- tended systems,","cited_arxiv_id":null,"evidence_quote":"Predicted topological FF states in spin-orbit-coupled Fermi gases, the prior context for the tunable topological FF phase realized here in a solid-state chain."},{"cited_title":"Topo- logical superfluids with finite-momentum pairing and ma- jorana fermions,","cited_arxiv_id":null,"evidence_quote":"Demonstrates Majorana modes in ferromagnetic atomic chains on a superconductor, the experimental template for Shiba-chain devices."}],"review_version":1}