{"id":"801c6a8a-18b2-4c58-9637-62a93a51cd81","arxiv_id":"2507.00158","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In a 2D model with a short-range repulsion, the fully spin-polarized half-metal phase has an attractive odd-parity pairing interaction from two-magnon exchange, with a pairing scale possibly reaching a substantial fraction of the Fermi energy.","lead":"A theory paper derives pairing interactions for superconductivity in a 2D metal near a ferromagnetic transition, finding that the fully spin-polarized phase should superconduct more strongly than the paramagnetic phase. This matters for understanding superconductivity in graphene multilayers, where experiments see superconductivity inside ferromagnetic states.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"FM-phase pairing scale depends on an uncomputed momentum scale δk(0) to the fifth power; without a direct calculation of Γ_{2,tot}(δk), the claim that T* is a sizable fraction of EF is not established.","rationale":"The paper's strongest claim, as identified by the reader, is that the two-magnon-mediated interaction between spin-up fermions in the ferromagnetic half-metal gives a pairing scale T* that is a sizable fraction of the Fermi energy. The reader's weakest-assumption analysis correctly locates the fragile point: the characteristic momentum δk(0) of the pairing interaction is never computed, and λ_sc is proportional to (δk(0)/k_F)^5. My independent reading of Sec. IV confirms this. Equations (48)-(50) reduce the pairing problem to λ_sc = -ν ¯Γ_{2,tot}(0) δk(0)/(π k_F), but only ¯Γ_{2,tot}(0) is evaluated. The scale δk(0) enters through a conjecture after Eq. (48), and Eq. (57) takes q_max ~ δk(0) without a derivation. The paper is honest about this, flagging the conjecture and deferring the q-dependent calculation to a footnote, but the omission is load-bearing because the claimed 'sizable fraction of EF' depends exponentially on λ_sc and hence quintically on δk(0)/k_F. A numeric factor of two in δk(0) changes λ_sc by a factor of 32 and can turn a strong-coupling-scale T* into an exponentially small one. I also note that footnote [70] contains a second conjecture, that the S_a^2 contribution cancels at finite δk; this is not verified and could affect the sign of the effective interaction. These are concerns about missing quantitative control, not about the internal logic of the ladder summations or the Adler-principle argument, both of which are presented carefully. The paramagnetic-phase analysis is more complete and internally consistent, with explicit asymptotic forms and numerical solution of the gap equation, though it also relies on the smallness of bk_F^2. Given that the qualitative mechanism is derived but the quantitative scale in the FM phase is not controlled, the conditional verdict is appropriate. My stress-test does not identify a reason to change that verdict, so I recommend leaving it unchanged.","tokens_in":26045,"tokens_out":3743,"duration_ms":44653,"concrete_test":"Compute ¯Γ_{2,tot}(δk) directly from Eqs. (44)-(45) for δk along the Fermi surface, without invoking the factorization conjecture after Eq. (48). Keep the full q-dependence in S_b and the finite-δk corrections to S_a that are dropped in Eqs. (54)-(55) and deferred in footnote [71]; perform the angle-averaged (q, Ω_m) integrals at T = 0. Extract δk(0) as the half-width or first zero of the normalized function ¯Γ_{2,tot}(δk)/¯Γ_{2,tot}(0), then evaluate λ_sc from Eq. (50) with this computed value. If δk(0) ≲ k_F/2, the fifth power suppresses λ_sc and T* is not a sizable fraction of EF; if δk(0) ~ k_F, the central claim is supported. As part of the same check, test the footnote [70] cancellation by computing the contribution of S_a^2 at finite δk and verifying that it does not change the sign of the odd-parity pairing interaction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Sec. IV is that the two-magnon pairing coupling in the ordered phase is λ_sc = (4/πc)(δk(0)/k_F)^5, giving T* ~ μ0 exp(-1/λ_sc) as a sizable fraction of μ0. The derivation fixes only the δk = 0 value of the angle-subtracted interaction, ¯Γ_{2,tot}(0), in Eqs. (51)-(58). The momentum scale δk(0) is introduced after Eq. (48) by conjecturing a factorization ¯Γ_{2,tot}(δk) = ¯Γ_{2,tot}(0)Ψ(δk), with Ψ a decreasing function having characteristic scale δk(0) ≤ k_F. The paper never computes δk(0). In Eq. (57) the upper limit q_max of the q-integral is stated to be 'comparable to δk(0)', and footnote [71] explicitly defers the required q-dependent calculation. Because λ_sc scales as the fifth power of δk(0)/k_F, an O(1) uncertainty in δk(0) changes λ_sc by an order of magnitude and e^{-1/λ_sc} by many orders of magnitude. The conclusion that T* is a fraction of EF therefore requires δk(0) to be near k_F, not merely ≤ k_F. In addition, the conjecture in footnote [70] that the S_a^2 contribution cancels at finite δk is unverified; if this cancellation fails, the sign and shape of ¯Γ_{2,tot}(δk) could differ from the assumed attractive form. This is a missing quantitative step rather than an internal inconsistency: the qualitative two-magnon mechanism may survive, but the paper's headline quantitative claim rests on an undetermined scale.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper analyzes superconductivity in a two-dimensional electron gas with parabolic dispersion and short-range repulsion, treating the paramagnetic and ferromagnetically ordered sides of a Stoner transition within the ladder approximation. In the paramagnetic phase, the authors compute the paramagnon-mediated triplet pairing interaction with a small gradient coefficient b k_F^2, derive T* ~ µ0 (b k_F^2)^{1/2} for small b, and find a gap function with two sign changes on the Matsubara axis associated with dynamical vortices. In the ferromagnetic half-metal phase, they derive an effective interaction between spin-up fermions mediated by two transverse Goldstone magnons, show that the fermion-magnon vertex obeys an Adler zero, obtain an attractive odd-parity interaction with coupling λ_sc = (4/(πc))(δk0/k_F)^5, and conclude that T* is a sizable fraction of the Fermi energy, much larger than in the paramagnetic phase.","tokens_in":26486,"tokens_out":9977,"duration_ms":105311,"significance":"The paramagnetic-phase analysis is a solid and useful correction to semi-phenomenological spin-fermion theories: it derives the small gradient coefficient from the microscopic model, verifies the T* scaling numerically (Fig. 3), and makes a falsifiable prediction about the sign-change structure of the gap. The ferromagnetic-phase derivation of the two-magnon exchange interaction, including the Adler zero at long wavelengths, is an interesting and original contribution that strengthens the case for odd-parity pairing inside the half-metal state. However, the headline quantitative claim that T* is a sizable fraction of E_F rests on an uncomputed momentum scale δk0, so the paper's central conclusion is not yet established to the standard of the journal.","major_comments":[{"comment":"The coupling λ_sc = (4/(πc))(δk0/k_F)^5 depends on δk0, which is introduced after Eq. (48) through the conjectured factorization arΓ_{2,tot}(δk) = arΓ_{2,tot}(0)Ψ(δk) and is never computed. The statement that δk0/k_F ≤ 1 implies the coupling 'has no parametric smallness' is not logically valid, since δk0/k_F can be much smaller than unity while still satisfying the inequality. Because T* ~ µ0 exp(-1/λ_sc), a modest reduction of δk0/k_F from O(1) to, say, 0.3 changes λ_sc by two orders of magnitude and makes T* exponentially small. Footnote [71] explicitly defers the required computation of arΓ_{2,tot}(δk). The qualitative two-magnon mechanism may well be correct, but the quantitative claim that pairing in the ferromagnetic phase is much stronger than in the paramagnetic phase is not established by the present calculation.","section":"Sec. IV, Eq. (58) and the following paragraph"},{"comment":"The evaluation of arΓ_{2,tot}(0) assumes that the S_a^2 contribution to Γ_{2,tot}(δk), which is constant at δk=0, is exactly cancelled by the subtraction of the momentum-independent part for all δk. Footnote [70] states that this cancellation at finite δk is a conjecture, supported only by the argument that the frequency integral of S_a^2 χ^2 would otherwise be formally infinite. That argument does not prove the cancellation, and if it fails the sign and momentum profile of arΓ_{2,tot}(δk) could differ from the assumed attractive form. This is a load-bearing assumption for the existence and scale of odd-parity pairing in the ferromagnetic phase and should be checked by a direct calculation of the q- and δk-dependent integrand, at least in the limit of small δk.","section":"Sec. IV, Eqs. (51)-(55) and footnote [70]"}],"minor_comments":[{"comment":"The sentence beginning 'The two expressions for T*' cites 'Eq. (29) and Eq. (29)'; the first should be Eq. (25), and the attribution of Eq. (29) to previous semi-phenomenological studies appears to refer to Eq. (25).","section":"Sec. III, between Eqs. (29) and (30)"},{"comment":"There is a typo: 'Matsubata' should be 'Matsubara'.","section":"Footnote [70]"},{"comment":"'Thin ertical lines' should read 'Thin vertical lines'.","section":"Appendix C, Fig. 17 caption"},{"comment":"The pairing equation (46) is written for zero frequency; the derivation would benefit from a statement that the frequency dependence of Γ_{2,tot} is neglected in this estimate, since the two-magnon propagators are gapless and the frequency sum in Eq. (38) uses the factorized form.","section":"Sec. IV, Eq. (46)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest about its conjectures, but the abstract and conclusion present the FM-phase T* as a definitive result. The missing computation of δk0 is the single most important issue. I would recommend that the authors either compute δk0 (or bound it from below) or substantially soften the quantitative claim, e.g., by stating that T* could be a fraction of E_F if δk0 is O(k_F) while a small δk0 would suppress it. The paper would then be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe part of this paper worth taking seriously is the paramagnetic-side analysis. The authors compute the paramagnon-mediated pairing interaction microscopically, find that the gradient coefficient b k_F^2 is small, and show both analytically and numerically that T* scales as (b k_F^2)^{1/2} rather than the phenomenological (g/2πμ0)^2. They also identify a topological transition of the gap function on the Matsubara axis. That section is internally consistent, the numerics back the asymptotics, and the result has real implications for spin-fermion models.\n\nThe ferromagnetic-side mechanism is genuinely new: pairing between spin-up fermions mediated by two transverse Goldstone magnons, with an Adler-zero vertex, giving attraction in an odd-parity channel. The derivation of the sign from the Adler zero is a real step, and the λ_sc ∼ (δk0/k_F)^5 dependence is a clean takeaway.\n\nThe soft spot is exactly where the reader put it. The quantitative claim that T* in the FM phase is a sizable fraction of E_F rests on δk0 being near k_F, and δk0 is never computed. The factorization after Eq. (48) is a conjecture; footnote [70] admits the S_a^2 cancellation is unverified at finite δk; footnote [71] defers the q-dependent integral that would fix q_max. Since λ_sc enters exponentially, an O(1) uncertainty in δk0 changes T* by orders of magnitude. The paper says this itself, but that does not make the headline quantitative claim established. On the other hand, this is a missing calculation, not an internal contradiction; the two-magnon mechanism may well survive.\n\nThe stress-test note is right, and I do not think the reader over- or under-scored. The paper is worth a serious referee: the paramagnetic section is solid, the mechanism is novel, and the FM section can be fixed by computing δk0 or by reframing the claim as qualitative. I would engage with it and would cite the paramagnetic T* scaling.","headline":"Paramagnetic-side analysis is solid and the two-magnon mechanism is new, but the ferromagnetic-phase T* claim rests on an uncomputed scale and should be softened.","tokens_in":26951,"tokens_out":1592,"would_cite":true,"duration_ms":16865,"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 argues that in a fully spin-polarized two-dimensional metal, two transverse Goldstone magnons mediate an attractive odd-parity pairing interaction between same-spin fermions, with a pairing scale that is a sizable fraction of…","keywords":["unconventional superconductivity","paramagnon","magnon-mediated pairing","ferromagnetic half-metal","Goldstone modes","odd-parity pairing","Stoner transition","two-dimensional electron gas"],"falsifier":"A numerical computation of the $\\delta k$-dependence of $\\bar{\\Gamma}_{2,\\mathrm{tot}}$ would settle the claim: if $\\delta k_0/k_F$ turns out to be, say, $0.1$, then $\\lambda_{sc} \\sim 10^{-5}$ and $T^*$ is exponentially small rather than a sizable fraction of $E_F$. An experimental check would be a measured superconducting $T_c$ inside the half-metal phase of a graphene multilayer that is comparable to the Fermi energy, not orders of magnitude below it.","tokens_in":25847,"feed_emoji":"🧲","tokens_out":10626,"duration_ms":99675,"temperature":0.7,"pith_summary":"This paper asks where superconductivity appears around a ferromagnetic quantum-critical point in a two-dimensional metal, and how strong it is. In the paramagnetic side, it finds that the spin-fluctuation (paramagnon) propagator's weak momentum dependence suppresses the pairing temperature relative to earlier phenomenological estimates and changes the gap function's frequency structure. In the ferromagnetically ordered half-metal, where only one spin band has a Fermi surface, it derives the pairing interaction between same-spin fermions mediated by two transverse Goldstone magnons and shows it is attractive in an odd-parity channel. The associated dimensionless coupling is $\\lambda_{sc} = (4/(\\pi c))(\\delta k_0/k_F)^5$, with no parametric smallness if the momentum scale $\\delta k_0$ is of order $k_F$, giving a pairing temperature $T^*$ that is a sizable fraction of the Fermi energy, much larger than on the paramagnetic side. If correct, this explains why superconductivity in graphene-based systems would sit inside the ferromagnetic phase rather than dome symmetrically around it.","feed_headline":"Two-magnon exchange yields odd-parity pairing in a ferromagnet","feed_subtitle":"In a fully spin-polarized 2D metal, the pairing scale is predicted to be a sizable fraction of the Fermi energy.","key_machinery":"The central object is the two-magnon exchange interaction between two spin-up fermions, $\\Gamma_{2,\\mathrm{tot}}(\\delta k)$, built from two single-magnon scatterings, a direct two-magnon/two-fermion vertex, and their cross-terms. The combination $S_a^2 - S_b^2$ cancels the Adler-principle zero (the vanishing of the fermion–magnon vertex at $q=0$) and leaves a subleading frequency-linear term that produces an attractive, momentum-dependent interaction after the frequency integration. The load-bearing identity is the dimensionless coupling $\\lambda_{sc} = \\frac{4}{\\pi c}\\left(\\frac{\\delta k_0}{k_F}\\right)^5$, where $\\delta k_0$ is the characteristic momentum scale over which the interaction falls off along the Fermi surface; this $\\lambda_{sc}$ enters the BCS-like condition $1 = \\lambda_{sc} \\log(\\Lambda/T^*)$ and gives $T^* \\sim \\mu_0 e^{-1/\\lambda_{sc}}$.","core_discovery":"The paper's central claim is that in the fully spin-polarized (half-metallic) state, superconductivity is carried by pairs of spin-up fermions interacting via two transverse Goldstone magnons. Although each fermion–magnon vertex vanishes at long wavelength (the Adler principle for a Goldstone boson), the squared vertex is not zero, and the subleading frequency-dependent term in the two-magnon exchange produces an attractive interaction in the odd-parity (p-wave-type) channel. The resulting dimensionless coupling is $\\lambda_{sc} = (4/(\\pi c))(\\delta k_0/k_F)^5$, where $\\delta k_0$ is the characteristic momentum scale of the momentum-dependent part of the interaction, conjectured to be of order $k_F$, and hence a pairing scale $T^* \\sim \\mu_0 e^{-1/\\lambda_{sc}}$, a sizable fraction of the Fermi energy $\\mu_0$. In the same paper, the paramagnetic side is shown to have a parametrically smaller $T^*$ because the weak momentum dependence of the paramagnon propagator (small $bk_F^2$) reduces the p-wave component of the interaction; this also produces a topologically nontrivial gap with sign changes on the Matsubara axis.","pith_inferences":["A testable extension of the paper's ladder calculation would be to compute the full momentum dependence of $\\bar{\\Gamma}_{2,\\mathrm{tot}}(\\delta k)$; the resulting value of $\\delta k_0/k_F$ would determine whether the predicted pairing scale is truly a fraction of $E_F$ or is exponentially suppressed.","If the two-magnon mechanism is generic, fully spin-polarized Fermi surfaces in other two-dimensional systems, such as AlAs quarter-metals, should also be intrinsically superconducting, which could be tested in transport experiments.","An implicit consequence of the paper's comparison is that the superconducting region around a two-dimensional ferromagnetic quantum-critical point should be asymmetric, with high $T_c$ only on the ordered side; in three dimensions a continuous Stoner transition would smooth this into a sharp peak near $c = 1$, as sketched in the paper's lower panel of Fig. 13."],"forward_implications":["Inside the ferromagnetic half-metal, the pairing scale $T^*$ is predicted to be a sizable fraction of the Fermi energy, much larger than the paramagnetic-phase scale near the transition, so superconductivity near a two-dimensional ferromagnetic quantum-critical point should be concentrated on the ordered side of the Stoner transition.","The paired fermions have the same spin projection, so the superconducting condensate is spin-triplet (odd-parity), with the p-wave channel likely the leading instability.","In two-valley systems such as rhombohedral multilayer graphene, pairing of same-spin fermions on one side of the Fermi surface implies a pair-density-wave (finite-momentum) superconducting order inside the half-metal or quarter-metal state.","In the paramagnetic phase, the small gradient coefficient $bk_F^2$ of the paramagnon propagator reduces the pairing scale relative to phenomenological estimates and produces a gap function with two sign changes on the Matsubara axis, i.e., a topologically nontrivial state with dynamical vortices."],"supporting_citations":[{"why":"Introduced coexistence of p-state superconductivity and itinerant ferromagnetism, the problem this paper analyzes in two dimensions.","marker":"[14]"},{"why":"Argued that coupling to transverse Goldstone modes strongly enhances Tc in the ferromagnetic phase; here that enhancement is derived microscopically via two-magnon exchange.","marker":"[33]"},{"why":"Showed that the two-dimensional Stoner transition is first order into a fully spin-polarized half-metal, the ground state on which the ferromagnetic-side pairing calculation is based.","marker":"[37]"},{"why":"Derived the magnon propagator and non-diagonal magnon-mediated interaction in spin-canting rhombohedral graphene, supplying the form of the single-magnon vertex used here.","marker":"[34]"},{"why":"Supplies the gradient (b q^2) correction to the paramagnon propagator that controls the suppression of pairing in the paramagnetic phase.","marker":"[50]"},{"why":"Computes the small gradient coefficient bk_F^2 in a parabolic two-dimensional electron gas, the input for the paramagnetic T* results.","marker":"[51]"},{"why":"Criterion for stability of Goldstone modes in a metal with broken symmetry, justifying the Adler-principle cancellation of the two-magnon vertex at long wavelength.","marker":"[67]"},{"why":"Adler's consistency conditions establishing the vanishing of the Goldstone-boson vertex at long wavelength, which the two-magnon calculation must overcome.","marker":"[68]"},{"why":"Modern formulation of electrons interacting with Goldstone modes in a rotating frame; fixes the structure of the two-magnon vertex used in the calculation.","marker":"[69]"}],"fun_headline_variants":["Odd-parity pairing from two-magnon exchange in ferromagnet","Two Goldstone magnons mediate superconductivity in half-metal","Ferromagnetic pairing via magnons exceeds paramagnetic onset","Magnon-exchange mechanism yields fraction-of-Fermi-energy T_c","P-wave superconductivity from transverse magnon exchange in 2D"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the momentum scale $\\delta k_0$ of the two-magnon pairing interaction is of order $k_F$, so that the dimensionless coupling $\\lambda_{sc} = (4/(\\pi c))(\\delta k_0/k_F)^5$ has no parametric smallness; the paper conjectures this factorization but does not compute $\\delta k_0$.","fun_headline_variants_meta":{"raw":{"variants":["Odd-parity pairing from two-magnon exchange in ferromagnet","Two Goldstone magnons mediate superconductivity in half-metal","Ferromagnetic pairing via magnons exceeds paramagnetic onset","Magnon-exchange mechanism yields fraction-of-Fermi-energy T_c","P-wave superconductivity from transverse magnon exchange in 2D"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001188,"raw_usage":{"total_tokens":4933,"prompt_tokens":1003,"completion_tokens":3930,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":619,"completion_tokens_details":{"reasoning_tokens":3843}},"tokens_in":619,"tokens_out":3930,"duration_ms":33030,"temperature":1.0,"reasoning_tokens":3843,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:22:21.453848+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A numerical computation of the $\\delta k$-dependence of $\\bar{\\Gamma}_{2,\\mathrm{tot}}$ would settle the claim: if $\\delta k_0/k_F$ turns out to be, say, $0.1$, then $\\lambda_{sc} \\sim 10^{-5}$ and $T^*$ is exponentially small rather than a sizable fraction of $E_F$. An experimental check would be a measured superconducting $T_c$ inside the half-metal phase of a graphene multilayer that is comparable to the Fermi energy, not orders of magnitude below it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Computes the small gradient coefficient bk_F^2 in a parabolic two-dimensional electron gas, the input for the paramagnetic T* results."},{"cited_title":"Watanabe and A","cited_arxiv_id":null,"evidence_quote":"Criterion for stability of Goldstone modes in a metal with broken symmetry, justifying the Adler-principle cancellation of the two-magnon vertex at long wavelength."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Adler's consistency conditions establishing the vanishing of the Goldstone-boson vertex at long wavelength, which the two-magnon calculation must overcome."},{"cited_title":"Vasiliou, Y","cited_arxiv_id":null,"evidence_quote":"Modern formulation of electrons interacting with Goldstone modes in a rotating frame; fixes the structure of the two-magnon vertex used in the calculation."}],"review_version":1}