{"id":"40253587-1d58-4c10-b9f9-3dcd189962dd","arxiv_id":"2509.03774","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Proximity coupling to an s-wave superconductor plus interfacial Rashba spin-orbit coupling induces a generically nodal mixed singlet/triplet superconducting state in a thin altermagnetic film, with 8 Dirac nodes and a spin-current dynamo response.","lead":"A theory study shows that a thin altermagnet film on a normal superconductor turns into a superconductor with a mixed nodal pairing state, if the interface has spin-orbit coupling. The state can drive spin-polarized persistent currents, and the authors name specific material pairs with tiny lattice mismatch to test it.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'generically nodal' claim rests on a single parameter set; for weak altermagnet splitting or strong coupling the nodes can vanish, so the 8-node structure is not established as generic.","rationale":"The reader's weakest assumption concerned the required Rashba SOC and coherent tunneling, which the paper explicitly states as a condition. My concern is complementary: even when Rashba is present, the nodal structure—a key part of the advertised physics and experimental signatures—may not be generic across the parameter space of realistic altermagnets. The paper's own Effective theory and End Matter provide the condition for nodes (Eq. 18), and the numerical demonstration covers only a single η0 and a few g values. Since the central mechanism (triplet induction) is well supported by symmetry and numerics, and since the concern is about the extent of the claimed regime rather than an internal inconsistency, the appropriate verdict remains CONDITIONAL as the reader already concluded. No change to the verdict is needed, but the paper should be revised to map the nodal region and qualify the 'generic' statement. I agree partially with the reader: they focused on the Rashba assumption; I highlight the additional assumption that the altermagnetic splitting is large enough for nodes, which is equally load-bearing for the claim's advertised experimental consequences.","tokens_in":10183,"tokens_out":16156,"duration_ms":177988,"concrete_test":"Perform exact diagonalization of the 8x8 Hamiltonian Eq. (8) over a parameter scan: η0 from 0 to 0.5 in steps of 0.025, and g from 0.05 to 0.6 in steps of 0.05, keeping other parameters as in Fig. 2 (λ_R=0.1, Δ0=0.4, μ'=-3.05). For each point, compute the smallest positive eigenvalue along the ALM Fermi surface (ξ'_k=0) as a function of angle α, and record whether the minimum gap is zero (nodal) or positive (fully gapped). If a finite region with positive gap exists away from η0=0, the 'generically nodal' claim is parameter-dependent and needs qualification; if nodes persist throughout, the claim is robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the proximity-induced state is 'generically nodal with 8 Dirac nodes per BZ' relies on the Fermi surface intersecting the curve defined by |s_k|^2 = η_k^2 + |p_k|^2 (End Matter, Eq. (18)). This condition is demonstrated numerically for one parameter set (η0=0.2, λ_R=0.1, Δ0=0.4, μ'=-3.05, g up to 0.3), but the paper does not map the phase diagram. If η0 is smaller, the singlet component s_k (which is generally larger than the triplet component, |s_k|>|p_k|, per Eq. (11)) can satisfy |s_k|>|η_k| everywhere on the Fermi surface; then, as stated in the End Matter, the singlet gap dominates and the state is fully gapped. Similarly, for larger g, s_k grows like g^2 and may overwhelm the altermagnetic splitting, eliminating nodes. In that regime the advertised experimental fingerprints (cv ~ T^2, δρ_s^e ~ T) and part of the spin-current dynamo response would change or disappear. The mechanism of triplet induction is not questioned; the concern is specifically whether the nodal character is generic rather than a feature of the chosen parameters.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper argues that a thin-film metallic altermagnet (ALM) placed on a conventional s-wave superconductor acquires a proximity-induced superconducting state whose order parameter is a mixture of spin-singlet and spin-triplet components, provided Rashba spin-orbit coupling is present at the interface. The singlet component dominates near the Brillouin-zone diagonals, the triplet p↑− ⊗ p↓+ component dominates elsewhere, and the two regions are separated by Dirac nodal points. For the minimal d-wave ALM model the authors find eight nodes per BZ, leading to cv ∼ T^2 and δρ_s^e ∼ T, and they demonstrate a pure spin supercurrent for a phase gradient along the zone diagonal. The claims are supported by a symmetry analysis of the Ginzburg-Landau coupling, a second-order effective theory obtained by integrating out the superconductor, and numerical diagonalization of the full 8×8 BdG Hamiltonian. A table of candidate heterostructures with low lattice mismatch is also provided.","tokens_in":10491,"tokens_out":3058,"duration_ms":36252,"significance":"If established, the result is significant: it gives a concrete microscopic route to triplet pairing and nodal superconductivity starting from a conventional s-wave superconductor, and it proposes a device-oriented spin-current dynamo in a proximitized heterostructure. The symmetry argument connecting Jz = 0 equal-spin p-wave channels to a linear coupling with ψ0 is correct and well explained. The paper also gains credibility from the fact that the effective theory (Eqs. 9–11) and the full numerical solution agree, including the g-dependence of the nodal positions (Fig. 2c vs. Eq. 18), and from the explicit candidate materials list. The main weakness is that the central claim of generic nodality is demonstrated for a single parameter set, and the authors' own End Matter analysis shows that the nodes disappear when the singlet component dominates everywhere. Since the proposed experimental fingerprints (T^2 specific heat, T-linear superfluid-density suppression, and part of the spin-current response) all rely on the nodal structure, this regime question is load-bearing and needs to be addressed before the generic claims can be accepted.","major_comments":[{"comment":"The abstract and conclusions state that the resulting superconductor is 'generically nodal with 8 Dirac nodes per BZ.' The supporting analysis, however, establishes this only for the specific parameter set used in Fig. 2, namely (η0, λR, Δ0, μ′) = (0.2, 0.1, 0.4, −3.05), with g up to 0.3. The node condition derived in Eq. (18) is |s_k| = sqrt(η_k^2 + |p_k|^2) on the Fermi surface ξ′_k = 0. Since |s_k| > |p_k|, this requires a sufficiently large altermagnetic splitting relative to the induced singlet gap. For smaller η0, or for larger g where s_k ∼ g^2 grows, the inequality |s_k| > |η_k| can hold everywhere on the Fermi surface, in which case, as the End Matter itself notes, the singlet component opens a full gap and the nodes disappear. The claim of generic nodality is therefore not justified without a phase diagram over (η0, g, λR) or at least a parameter-regime discussion. This is a ma","section":"Conclusion and Abstract; End Matter, Eq. (18)"},{"comment":"The paper's advertised experimental signatures—cv(T) ∼ T^2, δρ_s^e(T) ∼ T, and the spin-current dynamo with js ≃ (η0/t)je—are all computed in the nodal regime. In the fully gapped regime that occurs for weak altermagnetic splitting or strong coupling, cv(T) becomes activated, δρ_s^e(T) becomes exponentially small, and the low-temperature spin-current response changes qualitatively. The paper should either state clearly that these predictions apply only in the nodal regime, or map out the regime boundaries. As it stands, the Conclusions overgeneralize the numerical results shown in Figs. 2 and 3.","section":"Figs. 2(d)–3 and Conclusions"},{"comment":"The inequality E^2_{k−} ≤ ξ′^2_k + (|s_k| − sqrt(η_k^2 + |p_k|^2))^2 is used to locate the nodes. This gives a sufficient condition for the right-hand side to vanish (ξ′_k = 0 and |s_k| = sqrt(η_k^2 + |p_k|^2)), but the paper does not show that these are necessary conditions for E_k− = 0 in the full model. The numerical agreement shown in Fig. 2(c) for one g-trajectory is encouraging, but a demonstration that the nodes are exactly at these points—or at least a statement of the approximation under which they are—would strengthen the derivation. Without this, the count of '8 Dirac nodes per BZ' remains partially numerical rather than fully analytic.","section":"End Matter, Eq. (17)"}],"minor_comments":[{"comment":"The notation Σk for the pairing matrix and its later decomposition in Eq. (13) into sk and pk is clear, but the text should state explicitly that pk in Eq. (13) is not the same as the momentum k but the triplet component; the notation is potentially confusing and should be flagged or renamed.","section":"Eq. (11)"},{"comment":"The horizontal axis of Fig. 2(c) is labeled kα, but the text says the gap is plotted as a function of the angle α. This is likely a typo or a missing definition of kα as a radial distance; please clarify. Also, the different curve styles for different g values are mentioned in the caption but the legend is not visible in the text; this should be fixed.","section":"Fig. 2(c)"},{"comment":"The table includes a hexagonal symmetry entry (Nb4Se8/FeBr3), but the text notes that spin-current dynamo effects are forbidden in hexagonal crystal symmetry. It would be helpful to state in the table caption that this candidate is included only to demonstrate low lattice mismatch, not for the spin-current application.","section":"Table I"},{"comment":"The sentence 'Ref. [13] studied proximitized altermagnets and found numerous interesting topological phases' is slightly vague; a brief specification of what Ref. [13] found (e.g., Majorana modes) would help the reader see the distinction between that work and the present one.","section":"Introduction, Ref. [13]"},{"comment":"The 8×8 Hamiltonian in Eq. (8) is written schematically with V = τ_z g. Since V is momentum-independent, this is fine, but it would be useful for reproducibility to state the full BdG basis and the sign conventions for the triplet components explicitly in the main text.","section":"Eq. (8) and general notation"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and the symmetry argument is sound. My main concern is not the mechanism but the strength of the claim 'generically nodal.' The paper itself contains the ingredients showing that the nodes can vanish (End Matter), so I would like the authors to either map the phases or soften the generic claim. I do not see a reason to reject: the model, the effective theory, and the numerics are consistent, and the direction is novel. The revision should focus on the parameter dependence of the nodal structure and the scope of the experimental predictions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful result here is that a conventional s-wave superconductor can induce spin-triplet pairing in a metallic altermagnet, provided the interface has Rashba SOC. The mechanism is clear and convincing: the symmetry selection (p↑− and p↓+ have Jz = 0, so they couple linearly to s-wave singlet order) plus the microscopic derivation show how singlet pairs from the substrate convert into equal-spin triplet pairs in the altermagnet. The second-order effective theory and the full 8×8 numerics agree, including the g-dependent node movement, which is a good sign. The thermodynamic signatures (cv ~ T², δρs ~ T) and the spin-current dynamo are natural consequences and are computed, not assumed. So the paper deserves a serious referee.\n\nSoft spots, in order of importance. First, the claim that the proximity-induced state is “generically nodal” rests on a single parameter set and the paper even states the condition for nodes: the singlet gap dominates when |s_k| > |η_k|, and since |s_k| > |p_k| always, nodes only appear for sufficiently large altermagnetic splitting relative to the induced singlet gap. The paper does not map the phase diagram, so the “generic” wording in the abstract and conclusions is stronger than what is demonstrated. This is a fixable overstatement, not a flaw in the mechanism. Second, the End Matter Eq. (16a) is likely a typo: the p_k=0 spectrum should be sqrt(ξ'² + s²) ± |η|, not ξ_k on the right, and that typo obscures the singlet-gap condition it is meant to illustrate. Third, the spin superfluid density calculation is sketched rather than shown; the reader is asked to trust that the phase gradient and expectation values were computed correctly. Fourth, the candidate materials list includes a hexagonal system even though the paper itself notes spin-current dynamo effects are forbidden there, which slightly muddies the experimental pitch.\n\nFor who this is for: anyone working on altermagnet-superconductor hybrids, proximity effects, or nodal superconductivity will get real value. I would cite it for the mechanism even if the nodal claim needs qualification. The paper is honest about its main limitations (it acknowledges the requirement for Rashba SOC and the rigid Δ0 assumption), the model is stated cleanly, and the numerical work is reproducible in principle. The stress-test concern about the nodal character depending on parameters is legitimate and should be addressed, but it does not undermine the core result.\n\nSend it to peer review. It will come back with requests for a phase diagram and a fuller spin-current derivation, but the physics is sound and worth the referees' time.","headline":"Solid proximity-effect paper: correct symmetry argument, consistent effective theory, and numerics; the nodal claim is real but its generality is a bit oversold.","tokens_in":11043,"tokens_out":673,"would_cite":true,"duration_ms":9788,"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":"A metallic altermagnet on a conventional s-wave superconductor picks up a mixed singlet/triplet superconducting order with eight Dirac nodes and a spin-current dynamo, provided Rashba coupling sits at the interface.","keywords":["altermagnetism","proximity effect","spin-triplet superconductivity","Rashba spin-orbit coupling","nodal superconductor","persistent spin current","s-wave superconductor","heterostructure"],"falsifier":"Measure the low-temperature specific heat and superfluid density of a clean Al/Rb1−δV2Te2O bilayer with a phase gradient along (1,1). The paper predicts cv ~ T², δρs^e ~ T, and a pure spin supercurrent perpendicular to the gradient; observing exponentially activated behavior and no spin supercurrent in the altermagnet layer would falsify the central claim. A simpler control is to remove or invert the interface Rashba symmetry: the nodal triplet state should vanish.","tokens_in":10014,"feed_emoji":"🧲","tokens_out":9909,"duration_ms":91455,"temperature":0.7,"pith_summary":"Altermagnets are magnetic metals whose spin-split bands make ordinary spin-singlet Cooper pairing unlikely, and none has yet been found to superconduct on its own. This paper asks whether superconductivity can be injected into a thin altermagnet film by placing it on a conventional s-wave superconductor, and what the induced state would be. It answers that, with Rashba spin-orbit coupling at the interface, the proximity effect creates a mixed singlet/triplet superconducting state: singlet pairing dominates near the Brillouin-zone diagonals, equal-spin p-wave triplet pairing dominates elsewhere, and the two regions are separated by eight Dirac nodal points (gapless points in the quasiparticle spectrum). The triplet component has the Jz = 0 structure p↑−/p↓+ and makes the bilayer a spin-current source: a charge supercurrent along the zone diagonal drives a pure spin supercurrent perpendicular to it. This matters because it offers a practical experimental route to the predicted topological and spintronic superconducting phases of altermagnets, with concrete low-temperature signatures and candidate heterostructures.","feed_headline":"Proximity effect creates spin-triplet superconductivity in altermagnets","feed_subtitle":"With Rashba coupling at the interface, the induced state has 8 Dirac nodes and can drive a pure spin supercurrent.","key_machinery":"The central object is the two-layer Bogoliubov–de Gennes Hamiltonian of Eq. (8), coupling a d-wave altermagnet (Eq. 4) to an s-wave superconductor with Rashba spin-orbit coupling (Eq. 3) through spin- and momentum-conserving tunneling (Eq. 5), together with its second-order effective theory (Eqs. 9–11) obtained by integrating out the superconductor. The load-bearing identity is the Jz = Sz + Lz = 0 selection rule: among the four degenerate equal-spin triplet channels of the altermagnet, p↑− and p↓+ transform trivially under the fourfold rotations of the square lattice and therefore couple linearly to a singlet s-wave order parameter once inversion is broken. The effective pairing matrix Σk h","core_discovery":"Contrary to the expectation that the spin-split Fermi surface of an altermagnet cannot accept spin-singlet Cooper pairs, the paper establishes that a thin metallic altermagnet proximitized by a conventional s-wave superconductor acquires a superconducting state with a mixed singlet/triplet order parameter. The key is inversion breaking at the interface, modeled as Rashba spin-orbit coupling: it permits the s-wave singlet order to couple linearly to the Jz = 0 equal-spin triplet channels p↑− and p↓+. In the induced pairing matrix, the singlet component sk gaps the Fermi surface only where the altermagnetic splitting is small (near the zone diagonals), while the triplet component pk gaps the r","pith_inferences":["I infer that the Jz = 0 selection rule is not specific to d-wave altermagnets: any compensated magnet with momentum-dependent spin splitting whose leading triplet channels contain Jz = 0 components should show the same proximity-induced triplet order, so the mechanism likely generalizes beyond the square-lattice model.","I infer that the nodal positions and the singlet/triplet balance are tunable through the interface Rashba strength and the tunneling amplitude; an electrically gated heterostructure could therefore sweep the system between mostly singlet, nodal, and triplet-dominated regimes in a single device, a knob the paper does not explore.","I infer that a fully gapped variant could be reached by using an altermagnet with spin-up and spin-down Fermi pockets centered at X and Y points, as the paper notes; such a state would be a natural platform for probing chiral or helical edge modes and Majorana zero modes in vortices.","I infer that allowing the superconducting substrate gap to respond self-consistently would renormalize the pair-breaking suppression of the charge superfluid density seen at intermediate tunneling, and could change the saturation value of the spin superfluid density; a self-consistent calculation would test the robustness of the few-percent estimate."],"forward_implications":["The predicted mixed singlet/triplet state can be searched for immediately in standard superconductor/altermagnet bilayers, without waiting for an intrinsically superconducting altermagnet; the paper proposes specific low-mismatch material pairs.","The 8 Dirac nodal points per Brillouin zone give distinctive low-temperature thermodynamics: electronic specific heat cv ~ T² and superfluid density δρs^e ~ T, plus flat band edge modes analogous to cuprates.","The bilayer is a persistent spin-current generator: a charge supercurrent along the zone diagonal produces a pure spin supercurrent perpendicular to it, with spin superfluid density up to a few percent of the charge superfluid density.","Removing or suppressing the interface Rashba spin-orbit coupling eliminates the triplet, the nodes, and the spin current, leaving only singlet pairing that gaps the altermagnet near the zone diagonals; hence the Rashba term is a switch for the whole effect.","The same nodal quasiparticles imply a T² specific heat and T-linear penetration-depth shift that distinguish this proximity state from a fully gapped conventional proximity superconductor."],"supporting_citations":[{"why":"Supplies the altermagnet's degenerate equal-spin triplet order parameters p↑±/p↓± and the spin-current dynamo effect that the proximitized bilayer reproduces.","marker":"[7, 9]"},{"why":"Earlier study of proximitized altermagnets that assumed superconducting order without a microscopic mechanism; the present paper supplies that mechanism.","marker":"[13]"},{"why":"Provides the unitary-transformation procedure used to derive the second-order effective Hamiltonian and the induced pairing matrix Σk.","marker":"[19]"},{"why":"Sets up the d-wave altermagnet lattice model and symmetry classification used in the microscopic Hamiltonian.","marker":"[3, 4, 18]"},{"why":"Establishes the nodal-superconductor phenomenology (gap nodes, quasiparticle signatures) used to characterize the proximitized state.","marker":"[14]"},{"why":"Supplies the T-linear superfluid-density signature of clean nodal superconductors used to identify the proximity-induced nodes.","marker":"[16, 17]"},{"why":"Documents the layered altermagnet Rb1−δV2Te2O and its properties, the basis of the proposed 0.04%-mismatch Al heterostructure.","marker":"[23, 31]"}],"fun_headline_variants":["Rashba-driven triplet superconductivity in proximitized altermagnets","Proximitized altermagnets host nodal singlet-triplet superconductivity","Rashba coupling turns altermagnets into nodal triplet superconductors","Induced superconductivity in altermagnets yields spin-polarized currents"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the superconductor/altermagnet interface has a non-negligible Rashba spin-orbit coupling and passes electrons coherently without flipping spin or momentum; if that coupling is absent, only singlet pairing is induced and the nodal triplet state and spin-current effect do not occur.","fun_headline_variants_meta":{"raw":{"variants":["Rashba-driven triplet superconductivity in proximitized altermagnets","Proximitized altermagnets host nodal singlet-triplet superconductivity","Rashba coupling turns altermagnets into nodal triplet superconductors","Induced superconductivity in altermagnets yields spin-polarized currents"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00054,"raw_usage":{"total_tokens":2393,"prompt_tokens":677,"completion_tokens":1716,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":421,"completion_tokens_details":{"reasoning_tokens":1634}},"tokens_in":421,"tokens_out":1716,"duration_ms":13248,"temperature":1.0,"reasoning_tokens":1634,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:43:38.276539+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the low-temperature specific heat and superfluid density of a clean Al/Rb1−δV2Te2O bilayer with a phase gradient along (1,1). The paper predicts cv ~ T², δρs^e ~ T, and a pure spin supercurrent perpendicular to the gradient; observing exponentially activated behavior and no spin supercurrent in the altermagnet layer would falsify the central claim. A simpler control is to remove or invert the interface Rashba symmetry: the nodal triplet state should vanish.","supporting_citations":[{"cited_title":"Paired states of fermions in two dimensions with breaking of parity and time- reversal symmetries and the fractional quantum hall ef- fect,","cited_arxiv_id":null,"evidence_quote":"Earlier study of proximitized altermagnets that assumed superconducting order without a microscopic mechanism; the present paper supplies that mechanism."},{"cited_title":"Minimal models for al- termagnetism,","cited_arxiv_id":null,"evidence_quote":"Provides the unitary-transformation procedure used to derive the second-order effective Hamiltonian and the induced pairing matrix Σk."},{"cited_title":"Altermagnetic routes to majorana modes in zero net magnetization,","cited_arxiv_id":null,"evidence_quote":"Establishes the nodal-superconductor phenomenology (gap nodes, quasiparticle signatures) used to characterize the proximitized state."}],"review_version":1}