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Superconductivity via paramagnon and magnon exchange in a 2D near-ferromagnetic full metal and ferromagnetic half-metal

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.00158 v2 pith:Q5U6I6QH submitted 2025-06-30 cond-mat.supr-con

Zachary M Raines , Andrey V Chubukov This is my paper · ORCID
classification cond-mat.supr-con
keywords unconventionalsuperconductivityparamagnonmagnon-mediatedpairingferromagnetichalf-metalGoldstonemodesodd-parityStonertransitiontwo-dimensionalelectrongas
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $T^* \sim \mu_0 e^{-1/\lambda_{sc}}$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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 $\delta k_0$.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Sec. IV, Eq. (58) and the following paragraph] 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.
  2. [Sec. IV, Eqs. (51)-(55) and footnote [70]] 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.
minor comments (4)
  1. [Sec. III, between Eqs. (29) and (30)] 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).
  2. [Footnote [70]] There is a typo: 'Matsubata' should be 'Matsubara'.
  3. [Appendix C, Fig. 17 caption] 'Thin ertical lines' should read 'Thin vertical lines'.
  4. [Sec. IV, Eq. (46)] 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.

Circularity Check

1 steps flagged · score 4.0 of 10

Qualitative two-magnon attraction is derived independently, but the FM-phase pairing magnitude is fixed by an uncomputed momentum scale δk(0); the 'sizable fraction of EF' conclusion follows only if δk(0) ~ kF is assumed.

  1. other [Section IV, Eqs. (48)-(59), especially Eq. (58) and footnote [71]]
    "To proceed with this approach, we conjecture that ¯Γ2,tot(δk) = ¯Γ2,tot(0)Ψ(δk), where Ψ(δk) is a decreasing function of |δk| with a characteristic scale δk(0) ≤ kF. ... The upper limit of this integration qmax is comparable to δk(0) (one has to compute ¯Γ2,tot(δk) to see this) [71]. ... We recall that δk(0)/kF ≤ 1, hence the dimensionless coupling has no parametric smallness. This implies that the attraction is rather strong, i.e., the pairing scale T* in the ferromagnetically ordered state is a sizable fraction of μ0 = EF."

    The central quantitative prediction, λ_sc = (4/(πc))(δk(0)/kF)^5 in Eq. (58), is obtained by substituting the uncomputed scale δk(0) for both the momentum-space width of ¯Γ2,tot(δk) (via the conjectured factorization) and the upper cutoff qmax of the q-integral in Eq. (57). Since T* ~ μ0 exp(-1/λ_sc), the claim that T* is a sizable fraction of μ0 holds only if δk(0) is near kF. For δk(0)/kF = 0.5, λ_sc ≈ 0.04 and T*/μ0 ~ 10^-11. Footnote [71] explicitly defers the q-dependent calculation needed to determine δk(0), so the magnitude of the 'prediction' is an input assumption rather than a derived first-principles result. The qualitative sign and odd-parity character of the interaction are independently derived, hence the circularity is only partial.

full rationale

The core derivation of the odd-parity attraction in the FM phase is self-contained: it follows from the Adler-zero cancellation between γ2mag and the fermion propagators (Eqs. (51)-(56)), leaving the S_b^2 term, whose subleading iΩm piece gives Γ̄2,tot(0) < 0. The paramagnetic-phase scaling T* ~ (bk_F^2)^{1/2} is also a genuine functional derivation; b is an input parameter taken from prior work, but the dependence on it is derived, not assumed. The self-citations (Refs. [37], [51], [72]) are not load-bearing for the pairing mechanism: the FM-state equations are given in the paper, and [72] is only a pointer to competing scenarios. The flagged limitation is the uncomputed scale δk(0): Eq. (58) makes λ depend on its fifth power, footnote [71] explicitly defers its calculation, and footnote [70] conjectures the cancellation of the S_a^2 term at finite δk. Consequently the qualitative mechanism has independent content, but the headline quantitative statement that T* is a sizable fraction of EF is an output of the assumption δk(0) ~ kF rather than a derived first-principles result. Score 4 reflects this partial assumption-to-conclusion reduction, not a fully circular derivation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central quantitative claims rely on two assumed scales: the small gradient coefficient b and the momentum width delta_k0 of the magnon-mediated interaction. Neither is computed self-consistently in this paper; both are taken from prior work or conjectured.

free parameters (3)
  • b k_F^2 = assumed small (crossover at O(1))
    Coefficient of the q^2 gradient term in the paramagnon propagator; taken as an input from Ref. [51], controls the PM-phase T* scaling T* ~ (b k_F^2)^{1/2}.
  • delta_k0/k_F = assumed O(1), not computed
    Characteristic momentum scale of the two-magnon pairing interaction; lambda_sc is proportional to (delta_k0/k_F)^5, so the claimed large T* in the FM phase depends on this being order one.
  • q_max = comparable to delta_k0
    Upper cutoff in the momentum integration for barGamma_2tot(0); the paper notes in footnote [71] that a full evaluation is beyond the approximations used.
assumptions (5)
  • domain assumption Ladder approximation sums only particle-hole bubble and ladder diagrams for the pairing interaction.
    Neglects vertex corrections beyond the ladder series; the paper acknowledges that the approximation overestimates the Stoner instability and that numerics suggest the paramagnet may remain stable for c > 1.
  • domain assumption The 2D Stoner transition is first order into a fully spin-polarized (half-metal) state at c = 1.
    Based on prior work by the authors (Ref. [37]); the FM-phase analysis relies on the existence of this half-metal state.
  • ad hoc to paper The non-analytic -a|zeta|^3 term in the Landau energy is small and can be neglected.
    The authors argue that a is numerically small and does not affect the pairing scales.
  • ad hoc to paper The pairing interaction in the FM phase factorizes as barGamma_2tot(delta_k) = barGamma_2tot(0) Psi(delta_k) with a decreasing Psi of scale delta_k0.
    This conjecture, stated in Sec. IV after Eq. (48), is used to reduce the gap equation to a BCS form and to identify delta_k0.
  • standard math At T = 0, the Mermin-Wagner theorem does not forbid long-range ferromagnetic order in 2D.
    The study is restricted to T = 0 to allow ferromagnetic order.

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Pith. "Pith review of Superconductivity via paramagnon and magnon exchange in a 2D near-ferromagnetic full metal and ferromagnetic half-metal." pith.science (2026). https://pith.science/paper/Q5U6I6QH

@misc{pith2026250700158,
  author       = {Pith},
  title        = {Pith review of: Superconductivity via paramagnon and magnon exchange in a 2D near-ferromagnetic full metal and ferromagnetic half-metal},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5U6I6QH}},
  note         = {Machine review of arXiv:2507.00158}
}
read the original abstract

We study superconductivity in paramagnetic and ferromagnetically-ordered phases in a two-dimensional electron system with parabolic fermionic dispersion and short-range repulsive interaction. In the paramagnetic phase, we find that a weak momentum dependence of a paramagnon propagator parametrically reduces the onset temperature for the pairing compared to that in phenomenological theories which assume a strong dispersion of a paramagnon and also changes the topology of the gap function. In the ferromagnetic phase, we show that the order instantly polarizes low-energy fermionic excitations. We derive the fully renormalized pairing interaction between low-energy fermions, mediated by two transverse Goldstone modes and show that it is attractive in a spatially-odd channel. The pairing temperature in the ferromagnetic phase is found to be a fraction of the Fermi energy, significantly larger than in the paramagnetic phase near the transition. Our results are relevant for understanding superconductivity in proximity to itinerant ferromagnetism in multi-valley graphene systems, particularly the ones with full valley and spin polarization.

Figures

Figures reproduced from arXiv: 2507.00158 by the authors.

Figure 1
Figure 1. FIG. 1. Ladder and bubble diagrams contributing to the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Linearized equation for the pairing vertex for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color Online) The pairing scale [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (11 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Bethe-Salpeter equation for the renormalized trans [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Top: Eigenfunction of the kernel matrix [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Effective 4-fermion interaction mediated by a single [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Effective magnon-mediated interaction between two [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Vertex for the interaction between two spin-up [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Contribution Γ [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. (Color Online) Upper panel: Schematic depiction [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The full set of ladder and bubble diagrams con [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 16
Figure 16. Figure 16: FIG. 16. The pairing interaction [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Log-log plot of the pairing interaction [PITH_FULL_IMAGE:figures/full_fig_p018_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. The gap function [PITH_FULL_IMAGE:figures/full_fig_p019_18.png]

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  1. Stoner Transition at Finite Temperature in a 2D Isotropic Fermi Liquid

    cond-mat.str-el 2025-07 conditional novelty 7.0 of 10

    In a mean-field model of 2D fermions with flat dispersion, ferromagnetic order can appear as temperature increases (reentrant Stoner transition) once the dispersion is flat enough (alpha > 1.4).

Reference graph

Works this paper leans on

77 extracted references · 47 canonical work pages · cited by 1 Pith paper

  1. [70]

    We checked the cancellation between Sa [q, Ωm] and χ↑↓(q, Ωm) at δk = 0

    A word of caution. We checked the cancellation between Sa [q, Ωm] and χ↑↓(q, Ωm) at δk = 0. We conjecture that 16 this holds also at a finite δk. One argument for such a cancellation is that integration over Matsubata fre- quency in Eq. (51) runs over an infinite range (−∞, +∞), hence R dΩmS2 a [q, Ωm] χ2 ↑↓(q, Ωm) is formally infinite, while the true pai...

    work page

  2. [71]

    However, this calculation goes beyond the level of our approximations

    The momentum integral can, in principle, be explic- itly evaluated if in S2 b we keep the q-dependence of the Green’s functions next to 2µ0c. However, this calculation goes beyond the level of our approximations

    work page

  3. [1]

    N. F. Berk and J. R. Schrieffer, Effect of Ferromagnetic Spin Correlations on Superconductivity, Phys. Rev. Lett. 17, 433 (1966)

    1966

  4. [2]

    Layzer and D

    A. Layzer and D. Fay, Spin-fluctuation exchange mech- anism for P-wave pairing in liquid 3He, Int. J. Magn 1, 135 (1971)

    work page 1971

  5. [3]

    A. J. Leggett, A theoretical description of the new phases of liquid 3He, Rev. Mod. Phys. 47, 331 (1975)

    1975

  6. [4]

    H. v. L¨ ohneysen, A. Rosch, M. Vojta, and P. W¨ olfle, Fermi-liquid instabilities at magnetic quantum phase transitions, Rev. Mod. Phys. 79, 1015 (2007); A. V. Chubukov, A. Abanov, Y. Wang, and Y.-M. Wu, The interplay between superconductivity and non-Fermi liq- uid at a quantum-critical point in a metal, Ann. Phys. 417, 168142 (2020)

    work page 2007

  7. [5]

    Monthoux and G

    P. Monthoux and G. G. Lonzarich, p-wave and d-wave su- perconductivity in quasi-two-dimensional metals, Phys. Rev. B 59, 14598 (1999)

    work page 1999

  8. [6]

    Z. Wang, W. Mao, and K. Bedell, Superconductivity near itinerant ferromagnetic quantum criticality, Phys. Rev. Lett. 87, 257001 (2001)

    work page 2001

Show all 77 references
  1. [7]

    Roussev and A

    R. Roussev and A. J. Millis, Quantum critical effects on transition temperature of magnetically mediated p -wave superconductivity, Phys. Rev. B 63, 140504 (2001)

    2001

  2. [8]

    A. V. Chubukov, A. M. Finkel’stein, R. Haslinger, and D. K. Morr, First-Order Superconducting Transition near a Ferromagnetic Quantum Critical Point, Phys. Rev. Lett. 90, 077002 (2003)

    2003

  3. [9]

    A. A. Abrikosov and L. P. Gor’kov, Contribution to the theory of superconducting alloys with paramagnetic im- purities, Sov. Phys. JETP V ol: 12, 1243

  4. [10]

    P. B. Littlewood and C. M. Varma, Phenomenology of the superconductive state of a marginal Fermi liquid, Phys. Rev. B 46, 405 (1992)

    1992

  5. [11]

    Belitz, T

    D. Belitz, T. R. Kirkpatrick, and T. Vojta, How generic scale invariance influences quantum and classical phase transitions, Rev. Mod. Phys. 77, 579 (2005)

    2005

  6. [12]

    A. V. Chubukov, C. P´ epin, and J. Rech, Instability of the Quantum-Critical Point of Itinerant Ferromagnets, Phys. Rev. Lett. 92, 147003 (2004); D. L. Maslov and A. V. Chubukov, Nonanalytic paramagnetic response of itinerant fermions away and near a ferromagnetic quan- tum ph...

    2004

  7. [13]

    Brando, D

    M. Brando, D. Belitz, F. M. Grosche, and T. R. Kirkpatrick, Metallic quantum ferromagnets, Rev. Mod. Phys. 88, 025006 (2016)

    2016

  8. [14]

    Fay and J

    D. Fay and J. Appel, Coexistence of p -state supercon- ductivity and itinerant ferromagnetism, Phys. Rev. B22, 3173 (1980)

    1980

  9. [15]

    N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker, D. M. Freye, R. K. W. Haselwimmer, and G. G. Lonzarich, Magnetically mediated superconductivity in heavy fermion compounds, Nature 394, 39 (1998)

    1998

  10. [16]

    S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxleyß, I. Sheikinß, D. Braithwaiteß, and J. Flou- quetß, Superconductivity on the border of itinerant- electron ferr...

    2000

  11. [17]

    D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J.-P. Brison, E. Lhotel, and C. Paulsen, Coexistence of superconductivity and ferromagnetism in URhGe, Nature 413, 613 (2001)

    2001

  12. [18]

    N. T. Huy, A. Gasparini, D. E. De Nijs, Y. Huang, J. C. P. Klaasse, T. Gortenmulder, A. De Visser, A. Hamann, T. G¨ orlach, and H. V. L¨ ohneysen, Super- conductivity on the Border of Weak Itinerant Ferromag- netism in UCoGe, Phys. Rev. Lett. 99, 067006 (2007)

    2007

  13. [19]

    Kozii, M

    V. Kozii, M. P. Zaletel, and N. Bultinck, Spin-triplet superconductivity from intervalley Goldstone modes in magic-angle graphene, Phys. Rev. B 106, 235157 (2022)

    2022

  14. [20]

    A. M. Seiler, F. R. Geisenhof, F. Winterer, K. Watanabe, T. Taniguchi, T. Xu, F. Zhang, and R. T. Weitz, Quan- tum cascade of correlated phases in trigonally warped bilayer graphene, Nature 608, 298 (2022)

    2022

  15. [21]

    H. Zhou, L. Holleis, Y. Saito, L. Cohen, W. Huynh, C. L. Patterson, F. Yang, T. Taniguchi, K. Watanabe, and A. F. Young, Isospin magnetism and spin-polarized su- perconductivity in Bernal bilayer graphene, Science 375, 774 (2022)

    2022

  16. [22]

    H. Zhou, T. Xie, A. Ghazaryan, T. Holder, J. R. Ehrets, E. M. Spanton, T. Taniguchi, K. Watanabe, E. Berg, M. Serbyn, et al., Half-and quarter-metals in rhombohe- dral trilayer graphene, Nature 598, 429 (2021)

    2021

  17. [23]

    S. C. de la Barrera, S. Aronson, Z. Zheng, K. Watanabe, T. Taniguchi, Q. Ma, P. Jarillo-Herrero, and R. Ashoori, Cascade of isospin phase transitions in Bernal bilayer 14 graphene at zero magnetic field, Nat. Phys. 18, 771 (2022), arXiv:2110.13907 [cond-mat]

    2022 arXiv

  18. [24]

    A. M. Seiler, M. Statz, I. Weimer, N. Jacobsen, K. Watanabe, T. Taniguchi, Z. Dong, L. S. Levitov, and R. T. Weitz, Interaction-Driven Quasi-Insulating Ground States of Gapped Electron-Doped Bilayer Graphene, Phys. Rev. Lett. 133, 066301 (2024), arXiv:2308.00827 [cond-mat.str-el]

    2024 arXiv

  19. [25]

    T. Arp, O. Sheekey, H. Zhou, C. L. Tschirhart, C. L. Patterson, H. M. Yoo, L. Holleis, E. Re- dekop, G. Babikyan, T. Xie, J. Xiao, Y. Vituri, T. Holder, T. Taniguchi, K. Watanabe, M. E. Huber, E. Berg, and A. F. Young, Intervalley coherence and intrinsic spin–orbit coupling in...

    2024 arXiv

  20. [26]

    Holleis, C

    L. Holleis, C. L. Patterson, Y. Zhang, H. M. Yoo, H. Zhou, T. Taniguchi, K. Watanabe, S. Nadj-Perge, and A. F. Young, Ising Superconductivity and Nematicity in Bernal Bilayer Graphene with Strong Spin Orbit Cou- pling (2023), arXiv:2303.00742 [cond-mat.supr-con]

    2023 arXiv

  21. [27]

    Zhang, R

    Y. Zhang, R. Polski, A. Thomson, ´E. Lantagne- Hurtubise, C. Lewandowski, H. Zhou, K. Watan- abe, T. Taniguchi, J. Alicea, and S. Nadj-Perge, En- hanced superconductivity in spin–orbit proximitized bi- layer graphene, Nature 613, 268 (2023)

    2023

  22. [28]

    Huang, T

    C. Huang, T. M. R. Wolf, W. Qin, N. Wei, I. V. Blinov, and A. H. MacDonald, Spin and orbital metallic mag- netism in rhombohedral trilayer graphene, Phys. Rev. B 107, L121405 (2023); I. V. Blinov, C. Huang, N. Wei, Q. Wei, T. Wolf, and A. H. MacDonald, Partial conden- sation o...

    2023 arXiv

  23. [29]

    C. L. Patterson, O. I. Sheekey, T. B. Arp, L. F. W. Holleis, J. M. Koh, Y. Choi, T. Xie, S. Xu, E. Re- dekop, G. Babikyan, H. Zhou, X. Cheng, T. Taniguchi, K. Watanabe, C. Jin, E. Lantagne-Hurtubise, J. Alicea, and A. F. Young, Superconductivity and spin canting in spin-orbit ...

    2024 arXiv

  24. [30]

    T. Han, Z. Lu, Z. Hadjri, L. Shi, Z. Wu, W. Xu, Y. Yao, A. A. Cotten, O. S. Sedeh, H. Weldeyesus, J. Yang, J. Seo, S. Ye, M. Zhou, H. Liu, G. Shi, Z. Hua, K. Watan- abe, T. Taniguchi, P. Xiong, D. M. Zumb¨ uhl, L. Fu, and L. Ju, Signatures of chiral superconductivity in rhombo...

    2025

  25. [31]

    Levin and O

    K. Levin and O. T. Valls, Strong-coupling theory of su- perfluid transition temperatures for paramagnon models: Application to 3He, Phys. Rev. B 17, 191 (1978)

    1978

  26. [32]

    K. B. Blagoev, J. R. Engelbrecht, and K. S. Bedell, Effect of ferromagnetic spin correlations on superconductivity in ferromagnetic metals, Phys. Rev. Lett. 82, 133 (1999)

    1999

  27. [33]

    T. R. Kirkpatrick, D. Belitz, T. Vojta, and R. Narayanan, Strong Enhancement of Superconducting T c in Ferro- magnetic Phases, Phys. Rev. Lett. 87, 127003 (2001)

    2001

  28. [34]

    Dong, ´E

    Z. Dong, ´E. Lantagne-Hurtubise, and J. Alicea, Super- conductivity from spin-canting fluctuations in rhombo- hedral graphene (2024), arXiv:2406.17036 [cond-mat]

    2024 arXiv

  29. [35]

    Y.-Z. Chou, J. Zhu, and S. Das Sarma, Intravalley spin- polarized superconductivity in rhombohedral tetralayer graphene, Phys. Rev. B 111, 174523 (2025); M. Geier, M. Davydova, and L. Fu, Chiral and topological su- perconductivity in isospin polarized multilayer graphene (2024...

    2025 arXiv

  30. [36]

    Mæland, S

    K. Mæland, S. Abnar, J. Benestad, and A. Sudbø, Topo- logical superconductivity mediated by magnons of helical magnetic states, Phys. Rev. B 108, 224515 (2023)

    2023

  31. [37]

    Z. M. Raines, L. I. Glazman, and A. V. Chubukov, Unconventional discontinuous transitions in a two- dimensional system with spin and valley degrees of free- dom, Phys. Rev. B 110, 155402 (2024), arXiv:2406.04416 [cond-mat]; Unconventional Discontinuous Transitions in Isospin S...

    2024 arXiv

  32. [38]

    Kanamori, Electron Correlation and Ferromagnetism of Transition Metals, Prog

    J. Kanamori, Electron Correlation and Ferromagnetism of Transition Metals, Prog. Theor. Phys. 30, 275 (1963)

    1963

  33. [39]

    L. Liu, H. Yao, E. Berg, S. R. White, and S. A. Kivelson, Phases of the infinite U hubbard model on square lattices, Phys. Rev. Lett. 108, 126406 (2012)

    2012

  34. [40]

    Valenti, V

    A. Valenti, V. Calvera, S. A. Kivelson, E. Berg, and S. D. Huber, Nematic Metal in a Multivalley Electron Gas: Variational Monte Carlo Analysis and Applica- tion to AlAs, Phys. Rev. Lett. 132, 266501 (2024), arXiv:2307.15119 [cond-mat.str-el]

    2024 arXiv

  35. [41]

    Calvera, A

    V. Calvera, A. Valenti, S. D. Huber, E. Berg, and S. A. Kivelson, Theory of Coulomb driven nematicity in a multi-valley two-dimensional electron gas (2024), arXiv:2406.12825 [cond-mat]

    2024 arXiv

  36. [42]

    Shayegan, E

    M. Shayegan, E. P. De Poortere, O. Gunawan, Y. P. Shkolnikov, E. Tutuc, and K. Vakili, Two-dimensional electrons occupying multiple valleys in AlAs, Phys. Sta- tus Solidi B 243, 3629 (2006)

    2006

  37. [43]

    Gunawan, Y

    O. Gunawan, Y. P. Shkolnikov, K. Vakili, T. Gokmen, E. P. De Poortere, and M. Shayegan, Valley Susceptibil- 15 ity of an Interacting Two-Dimensional Electron System, Phys. Rev. Lett. 97, 186404 (2006)

    2006

  38. [44]

    M. S. Hossain, M. K. Ma, K. A. V. Rosales, Y. J. Chung, L. N. Pfeiffer, K. W. West, K. W. Baldwin, and M. Shayegan, Observation of spontaneous ferromagnetism in a two-dimensional electron system, Proc. Natl. Acad. Sci. 117, 32244 (2020)

    2020

  39. [45]

    Klein and A

    A. Klein and A. Chubukov, Superconductivity near a nematic quantum critical point: Interplay between hot and lukewarm regions, Phys. Rev. B 98, 220501 (2018); A. Klein, Y.-M. Wu, and A. V. Chubukov, Multiple in- tertwined pairing states and temperature-sensitive gap anisotropy...

    2018

  40. [46]

    Abanov and A

    A. Abanov and A. V. Chubukov, Interplay between su- perconductivity and non-Fermi liquid at a quantum criti- cal point in a metal. I. Theγ model and its phase diagram at T = 0 : The case 0 < γ <1, Phys. Rev. B 102, 024524 (2020)

    2020

  41. [47]

    D. J. Scalapino, A common thread: The pairing inter- action for unconventional superconductors, Rev. Mod. Phys. 84, 1383 (2012)

    2012

  42. [48]

    Maiti and A

    S. Maiti and A. V. Chubukov, Superconductivity from repulsive interaction, AIP Conf. Proc. 1550, 3 (2013)

    2013

  43. [49]

    Z. Dong, A. V. Chubukov, and L. Levitov, Transformer spin-triplet superconductivity at the onset of isospin or- der in bilayer graphene, Phys. Rev. B107, 174512 (2023); Z. Dong, L. Levitov, and A. V. Chubukov, Superconduc- tivity near spin and valley orders in graphene multila...

    2023

  44. [50]

    A. V. Chubukov, Kohn-Luttinger effect and the instabil- ity of a two-dimensional repulsive Fermi liquid at T=0, Phys. Rev. B 48, 1097 (1993)

    1993

  45. [51]

    D. L. Maslov, P. Sharma, D. Torbunov, and A. V. Chubukov, Gradient terms in quantum-critical theories of itinerant fermions, Phys. Rev. B 96, 085137 (2017)

    2017

  46. [52]

    T. R. Kirkpatrick and D. Belitz, Coexistence of ferromag- netism and superconductivity, Phys. Rev. B 67, 024515 (2003)

    2003

  47. [53]

    Dell’Anna and W

    L. Dell’Anna and W. Metzner, Fermi surface fluctuations and single electron excitations near Pomeranchuk insta- bility in two dimensions, Phys. Rev. B73, 045127 (2006)

    2006

  48. [54]

    Abanov, A

    A. Abanov, A. V. Chubukov, and A. M. Finkel’stein, Coherent vs . incoherent pairing in 2D systems near magnetic instability, Europhys. Lett. EPL 54, 488 (2001); A. Abanov, A. V. Chubukov, and J. Schmalian, Quantum-critical theory of the spin-fermion model and its application t...

    2001

  49. [55]

    A. L. Fitzpatrick, S. Kachru, J. Kaplan, S. Raghu, G. Torroba, and H. Wang, Enhanced pairing of quantum critical metals near d=3+1, Phys. Rev. B 92, 045118 (2015); H. Wang, S. Raghu, and G. Torroba, Non-Fermi- liquid superconductivity: Eliashberg approach versus the renormaliz...

    2015

  50. [56]

    Lederer, Y

    S. Lederer, Y. Schattner, E. Berg, and S. A. Kivel- son, Enhancement of superconductivity near a ne- matic quantum critical point, Phys. Rev. Lett. 114, 097001 (2015); Y. Schattner, S. Lederer, S. A. Kivelson, and E. Berg, Ising nematic quantum crit- ical point in a metal: A m...

    2015

  51. [57]

    W¨ olfle and E

    P. W¨ olfle and E. Abrahams, Quasiparticles beyond the Fermi liquid and heavy fermion criticality, Phys. Rev. B 84, 041101 (2011)

    2011

  52. [58]

    A. V. Chubukov, A. Abanov, I. Esterlis, and S. A. Kivel- son, Eliashberg theory of phonon-mediated superconduc- tivity — When it is valid and how it breaks down, Annals of Physics Eliashberg Theory at 60: Strong-coupling Su- perconductivity and Beyond, 417, 168190 (2020)

    2020

  53. [59]

    Zhang, Z

    S.-S. Zhang, Z. M. Raines, and A. V. Chubukov, Applicability of Eliashberg theory for systems with electron-phonon and electron-electron interaction: A comparative analysis, Phys. Rev. B 109, 245132 (2024), arXiv:2404.11820 [cond-mat.str-el]

    2024 arXiv

  54. [60]

    Chubukov, N

    A. Chubukov, N. V. Prokof’ev, and B. V. Svistunov, Im- plicit renormalization approach to the problem of Cooper instability, Phys. Rev. B100, 064513 (2019); D. Pimenov and A. V. Chubukov, Twists and turns of superconduc- tivity from a repulsive dynamical interaction, Annals of...

    2019

  55. [61]

    M. H. Christensen and A. V. Chubukov, Dynamical vor- tices in electron-phonon superconductors, Phys. Rev. B 104, L140501 (2021)

    2021

  56. [62]

    J. R. Schrieffer, X. G. Wen, and S. C. Zhang, Dy- namic spin fluctuations and the bag mechanism of high- ${T} {c}$ superconductivity, Phys. Rev. B 39, 11663 (1989)

    1989

  57. [63]

    J. R. Schrieffer, Ward’s identity and the suppression of spin fluctuation superconductivity, J. Low Temp. Phys. 99, 397 (1995)

    1995

  58. [64]

    Sachdev, A

    S. Sachdev, A. V. Chubukov, and A. Sokol, Crossover and scaling in a nearly antiferromagnetic Fermi liquid in two dimensions, Phys. Rev. B 51, 14874 (1995); A. V. Chubukov, P. Monthoux, and D. K. Morr, Vertex cor- rections in antiferromagnetic spin-fluctuation theories, Phys. ...

    1995

  59. [65]

    Flambaum, M

    V. Flambaum, M. Kuchiev, and O. Sushkov, Hole-hole superconducting pairing in the t-J model induced by long-range spin-wave exchange, Phys. C Supercond. 227, 267 (1994); M.Yu. Kuchiev and O. Sushkov, Large-size two-hole bound states in the t-J model, Phys. C Super- cond. 218, ...

    1994

  60. [66]

    Ismer, I

    J.-P. Ismer, I. Eremin, E. Rossi, D. K. Morr, and G. Blumberg, Theory of multiband superconductivity in spin-density-wave metals, Phys. Rev. Lett. 105, 037003 (2010)

    2010

  61. [67]

    Watanabe and A

    H. Watanabe and A. Vishwanath, Criterion for stability of Goldstone modes and Fermi liquid behavior in a metal with broken symmetry, Proc. Natl. Acad. Sci.111, 16314 (2014)

    2014

  62. [68]

    S. L. Adler, Consistency Conditions on the Strong Inter- actions Implied by a Partially Conserved Axial-Vector Current. II, Phys. Rev. 139, B1638 (1965)

    1965

  63. [69]

    Vasiliou, Y

    K. Vasiliou, Y. He, and N. Bultinck, Electrons interacting with Goldstone modes and the rotating frame, Phys. Rev. B 109, 045155 (2024)

    2024

  64. [72]

    Raines and A

    Z. Raines and A. Chubukov, To Appear (2025)

    2025

  65. [73]

    Stern, Polarizability of a Two-Dimensional Electron Gas, Phys

    F. Stern, Polarizability of a Two-Dimensional Electron Gas, Phys. Rev. Lett. 18, 546 (1967)

    1967

  66. [74]

    Giuliani and G

    G. Giuliani and G. Vignale, Quantum Theory of the Elec- tron Liquid (Cambridge University Press, 2008)

    2008

  67. [75]

    Mihaila, Lindhard function of a d-dimensional Fermi gas (2011), arXiv:1111.5337 [cond-mat, physics:math- ph]

    B. Mihaila, Lindhard function of a d-dimensional Fermi gas (2011), arXiv:1111.5337 [cond-mat, physics:math- ph]. Appendix A: Paramagnon Propagator In the paramagnetic state, the spin polarization bubble is given by Π(q) = −T X k 1 iωn + iΩm − ξk+q 1 iωn − ξk = X k nk+q − nk iΩ...

    2011 arXiv

  68. [76]

    (D1) by contour inte- gration with the change of variables z = eiθ

    I[u] We can perform the integral Eq. (D1) by contour inte- gration with the change of variables z = eiθ. Then, I[u] ≡ I dθ 2π 1 u − cos θ = I γ dz 2πiz 1 u − z+z−1 2 = −2 I γ dz 2πi 1 z2 − 2uz + 1, (D3) where the path γ is the unit circle which has poles at z = u ± √ u2 − 1. C...

  69. [77]

    (D2) we rewrite J[r] = Z kdk√ r − k √ r + k = Z dk r + k − (r − k) 2 √ r − k √ r + k = Z dk √ r + k√ r − k − √ r − k√ r + k = − √ r + k √ r − k + C (D5)

    J[r] For Eq. (D2) we rewrite J[r] = Z kdk√ r − k √ r + k = Z dk r + k − (r − k) 2 √ r − k √ r + k = Z dk √ r + k√ r − k − √ r − k√ r + k = − √ r + k √ r − k + C (D5)

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