{"schemaVersion":"1.0","revision":"fa961ab24277a91737c3","article":{"id":"scwiki-b3","topicId":"scwiki-topic-b3","code":"B3","slug":"spin-fluctuation-pairing","title":"Antiferromagnetic spin fluctuations: pairing and evidence","summary":"How repulsive interactions can favor sign-changing gaps, why magnetic signals do not prove a mechanism, and how probe limits and published corrections constrain the evidence.","aliases":["spin fluctuation pairing","antiferromagnetic pairing","AFM","paramagnon","spin resonance","Hubbard pairing","NMR hyperfine"],"evidenceTypes":["theory","experiment","hypothesis"],"relatedCodes":["B1","B2","B12","C2","D5"],"editorialStatus":"scoped-check","contentState":"pilot","modifiedAt":"2026-10-19","review":{"checkedAt":"2026-10-19","scope":"Selected pairing-vertex and NMR response equations, primary abstracts and published erratum text; independent two-patch and square-lattice checks. Follow-up selected Scalapino pp. 39-42 and p. 41 equation image, explicit local-derivation mapping, and 432 weighted-kernel cases with zero-coupling boundary.","limitations":["Complete scientific acceptance, raw-data checks and material-model reproduction remain open.","Review summaries of phase-sensitive evidence do not replace a primary-experiment audit. No sample-specific Tc is extracted.","Kuroki erratum text was read; its figure and underlying band calculation were not independently reproduced.","Author-manuscript passage checks do not establish final-version equivalence. Model eigenvalues are not measured transitions."]},"provenance":{"repository":"JackZH26/jzis-superconductivity","commit":"0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf","path":"B_mechanisms/B3_spin_fluctuation_AFM.md","sha256":"8bbed91652c4ebfe4d30d74844bd39c4f40d77e788d6f2a4670db256ae92f249","sourceId":"file-6b9b8d3dff0a6241"},"sourceMap":[{"startLine":1,"endLine":5,"targetSections":["scope"],"decision":"edited-with-scoped-evidence","note":"Candidate mechanism and causality boundary."},{"startLine":6,"endLine":15,"targetSections":["mathematical-assumptions","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Sign-changing kernels retained with independent non-uniqueness checks and original theoretical proposals."},{"startLine":16,"endLine":23,"targetSections":["scope","experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Three evidence levels retained; corrected NMR interpretation; resonance feedback separated."},{"startLine":24,"endLine":31,"targetSections":["experimental-criteria","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Cuprate symmetry evidence and material-specific iron/nickelate boundaries."},{"startLine":32,"endLine":39,"targetSections":["discovery-implications","experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Normalization, electronic structure, self-energy and competing channels."},{"startLine":40,"endLine":50,"targetSections":["limits-and-disputes","related-topics"],"decision":"edited-with-scoped-evidence","note":"All five source anchors retained with scoped checks and added erratum/probe sources."}],"references":[{"id":"source-eaa3beeed4368f46","doi":"10.1103/RevModPhys.84.1383","title":"A common thread: The pairing interaction for unconventional superconductors","year":2012,"url":"https://doi.org/10.1103/RevModPhys.84.1383","publicationStatus":"journal-article","verification":{"scope":"author-review-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1207.4093v1","note":"Selected manuscript text pp. 4, 18–20, 39–46; p. 41 Eq. (13) visually checked. Full 87-page review and final-version equivalence not audited. Independent two-patch algebra is an illustration, not reproduction of a Hubbard calculation."}},{"id":"source-7755cb2d156620a8","doi":"10.1103/PhysRevB.49.4261","title":"Spin-fluctuation-induced superconductivity and normal-state properties of YBa2Cu3O7","year":1994,"url":"https://doi.org/10.1103/PhysRevB.49.4261","publicationStatus":"journal-article","verification":{"scope":"primary-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prb/abstract/10.1103/PhysRevB.49.4261","note":"Monthoux–Pines publisher abstract: model calculations and parameter/impurity sensitivity. Full derivation, input spectrum and numerical data not reanalyzed."}},{"id":"source-7c2b55d128348e6c","doi":"10.1103/PhysRevLett.101.057003","title":"Unconventional Superconductivity with a Sign Reversal in the Order Parameter of LaFeAsO1−xFx","year":2008,"url":"https://doi.org/10.1103/PhysRevLett.101.057003","publicationStatus":"journal-article","verification":{"scope":"publisher-passages-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.057003","note":"Mazin et al. abstract and published note 8 checked. The note qualifies the initial identification of the magnetic instability; full electronic-structure calculation not reproduced."}},{"id":"source-f80a18149666bd31","doi":"10.1103/PhysRevLett.101.087004","title":"Unconventional Pairing Originating from the Disconnected Fermi Surfaces of Superconducting LaFeAsO1−xFx","year":2008,"url":"https://doi.org/10.1103/PhysRevLett.101.087004","publicationStatus":"journal-article","verification":{"scope":"primary-abstract-with-erratum-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.087004","note":"Kuroki et al. five-orbital RPA abstract checked alongside 2009 erratum 10.1103/PhysRevLett.102.109902. The original gap plot must not be used without its correction; full model not reproduced."}},{"id":"source-e933e3bbd4beade6","doi":"10.1103/RevModPhys.72.969","title":"Pairing symmetry in cuprate superconductors","year":2000,"url":"https://doi.org/10.1103/RevModPhys.72.969","publicationStatus":"journal-article","verification":{"scope":"primary-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.72.969","note":"Tsuei–Kirtley publisher abstract: phase-sensitive evidence for predominantly d-wave in specified optimally doped cuprates. Original experiments, sample data and full review not audited here."}},{"id":"source-6570d4401413e026","doi":"10.1103/PhysRevLett.102.109902","title":"Erratum: Unconventional Pairing Originating from the Disconnected Fermi Surfaces of Superconducting LaFeAsO1−xFx [Phys. Rev. Lett. 101, 087004 (2008)]","year":2009,"url":"https://doi.org/10.1103/PhysRevLett.102.109902","publicationStatus":"journal-article","verification":{"scope":"primary-erratum-text-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.102.109902","note":"Complete one-page text read via web: orbital-to-band transformation correction, fully open diagonal gap for displayed parameters, parameter-dependent nodal competition retained. Direct download returned 403; figure was not visually audited or numerically reproduced."}},{"id":"source-e5c4ccac05133124","doi":"10.1103/PhysRevB.52.R9831","title":"NMR relaxation rates for the spin-1/2 Heisenberg chain","year":1995,"url":"https://doi.org/10.1103/PhysRevB.52.R9831","publicationStatus":"journal-article","verification":{"scope":"author-manuscript-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/cond-mat/9509118v1","note":"Sandvik manuscript pp. 2–3; p. 3 Eq. (2) and response conventions visually checked. Used only for hyperfine-weighted low-frequency NMR relation, not as superconducting-material evidence or reproduction of QMC data."}},{"id":"source-824b2cddc30b3c2a","doi":"10.1103/PhysRevB.75.115118","title":"Ultrashort lifetime expansion for indirect resonant inelastic x-ray scattering","year":2007,"url":"https://doi.org/10.1103/PhysRevB.75.115118","publicationStatus":"journal-article","verification":{"scope":"author-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/abs/cond-mat/0609767","note":"Ament, Forte and van den Brink author abstract and linked journal identity. Indirect-RIXS core-hole approximation and mixed charge/spin contributions; full derivation and applicability to direct RIXS not audited."}}],"claims":[{"id":"scwiki-b3-claim-scope-1","sectionId":"scope","statement":"Antiferromagnetic spin fluctuations are a candidate pairing interaction with substantial weight at a nonzero momentum transfer. A repulsive interaction can favor a gap that changes sign between the states it connects. This is a model-dependent mechanism proposal: observing magnetism, or finding superconductivity near an antiferromagnetic phase, does not establish causation.","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-scope-2","sectionId":"scope","statement":"Keep three questions separate: are magnetic fluctuations present; do they change systematically with superconductivity; and can a consistently specified interaction reproduce pairing and other observables? An affirmative answer to the first two does not by itself answer the third. The relevant Fermi surfaces, orbitals, frequency dependence and competing channels must be specified.","evidenceType":"hypothesis","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-mathematical-assumptions-1","sectionId":"mathematical-assumptions","statement":"A dressed particle-particle eigenproblem can be written, with $k_B=1$ and $k$ including momentum and Matsubara frequency, as","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"Scalapino author v1 p. 41 Eq. (13) supplies the dressed particle-particle eigenproblem, with p. 39 Eq. (12) distinguishing vertex channels. The entry uses Lambda for the eigenvalue and explicitly limits the instability interpretation to that treatment; no material transition is measured."},{"id":"scwiki-b3-claim-mathematical-assumptions-2","sectionId":"mathematical-assumptions","statement":"$$-\\frac{T}{N}\\sum_{k'}\\Gamma^{pp}(k,k')G_\\uparrow(k')G_\\downarrow(-k')\\phi(k')=\\Lambda(T)\\phi(k).$$","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"Scalapino author v1 p. 41 Eq. (13) supplies the dressed particle-particle eigenproblem, with p. 39 Eq. (12) distinguishing vertex channels. The entry uses Lambda for the eigenvalue and explicitly limits the instability interpretation to that treatment; no material transition is measured."},{"id":"scwiki-b3-claim-mathematical-assumptions-3","sectionId":"mathematical-assumptions","statement":"Here $N$ is the number of lattice sites and the Green functions and irreducible pairing vertex must belong to the same approximation. A leading eigenvalue reaching one signals a pairing instability within that treatment; it is not a measurement of a material transition. The selected review gives this equation and distinguishes spin, charge and fully irreducible contributions.","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"Scalapino author v1 p. 41 Eq. (13) supplies the dressed particle-particle eigenproblem, with p. 39 Eq. (12) distinguishing vertex channels. The entry uses Lambda for the eigenvalue and explicitly limits the instability interpretation to that treatment; no material transition is measured."},{"id":"scwiki-b3-claim-mathematical-assumptions-4","sectionId":"mathematical-assumptions","statement":"A local static illustration, checked by separate matrix calculations, uses two patches with positive single-spin density-of-states weights $N_h,N_e$, a common pairing cutoff, constant gaps and real reciprocal interactions $U_h,U_e,V\\ge0$. After the common energy logarithm is factored out, define $\\eta\\Delta_i=-\\sum_jV_{ij}N_j\\Delta_j$, with $V_{he}=V_{eh}=V$. The rescaling $x_i=\\sqrt{N_i}\\Delta_i$ gives a symmetric matrix with diagonal elements $-U_iN_i$ and off-diagonal element $-V\\sqrt{N_hN_e}$. This is a separate static model, not a reduction validated against the review's dynamical calculation.","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling."},{"id":"scwiki-b3-claim-mathematical-assumptions-5","sectionId":"mathematical-assumptions","statement":"Its larger eigenvalue is","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling."},{"id":"scwiki-b3-claim-mathematical-assumptions-6","sectionId":"mathematical-assumptions","statement":"$$\\eta_+=-\\frac{U_hN_h+U_eN_e}{2}+\\sqrt{\\frac{(U_hN_h-U_eN_e)^2}{4}+V^2N_hN_e}.$$","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling."},{"id":"scwiki-b3-claim-mathematical-assumptions-7","sectionId":"mathematical-assumptions","statement":"For nonnegative intrapatch repulsions it is positive precisely when $V^2>U_hU_e$. The trace is nonpositive and the determinant is $(U_hU_e-V^2)N_hN_e$, establishing this threshold within the two-dimensional model. If both $U_i$ vanish and $V>0$, $\\eta_+=V\\sqrt{N_hN_e}$ and $\\Delta_h/\\Delta_e=-\\sqrt{N_e/N_h}$. If all interactions vanish, the kernel is zero and no preferred gap ratio is selected. A 432-case check compares direct and symmetrized eigensolvers, exact rational determinants and threshold cases. These are illustrative pairing strengths, not the phonon spectral coupling $\\lambda$, a calculated transition temperature, or a fit to any material.","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling."},{"id":"scwiki-b3-claim-mathematical-assumptions-8","sectionId":"mathematical-assumptions","statement":"A second independent check shows why a wavevector alone cannot select the gap symmetry. With lattice spacing one, both $\\cos k_x-\\cos k_y$ and $\\cos k_x+\\cos k_y$ reverse sign under $\\mathbf Q=(\\pi,\\pi)$. Under a ninety-degree rotation the former changes sign while the latter does not. Their values on the actual Fermi surface and their interaction-weighted eigenvalues still differ. A sign-reversal condition is therefore insufficient to select a unique representation.","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling."},{"id":"scwiki-b3-claim-experimental-criteria-1","sectionId":"experimental-criteria","statement":"Neutron, RIXS and NMR measurements have different response functions, selection rules and energy windows. In the magnetic low-frequency limit, NMR relaxation has the schematic dependence $1/(T_1T)\\propto\\sum_{\\mathbf q,\\alpha\\perp H}|A_\\alpha(\\mathbf q)|^2\\lim_{\\omega\\to0}\\chi''_\\alpha(\\mathbf q,\\omega)/\\omega$. The hyperfine weights and perpendicular spin components matter. The cited spin-chain paper supplies the response formula; it does not make that chain a superconducting-material example. NMR is not a momentum-resolved magnetic spectrum.","evidenceType":"theory","sourceIds":["source-e5c4ccac05133124"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-experimental-criteria-2","sectionId":"experimental-criteria","statement":"RIXS intensities also require a scattering model. Ament and colleagues' indirect-RIXS treatment contains charge and spin response contributions whose relative weights depend on incident energy and the stated core-hole approximation. A measured peak should not automatically be identified with the pairing kernel. This abstract-scoped example is not a derivation for every direct-RIXS geometry.","evidenceType":"theory","sourceIds":["source-824b2cddc30b3c2a"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-experimental-criteria-3","sectionId":"experimental-criteria","statement":"A superconducting spin resonance can reflect feedback of a sign-changing gap through coherence factors. Its interpretation still needs the normal-state response and a susceptibility calculation; it is not independent proof that the resonance caused pairing.","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-experimental-criteria-4","sectionId":"experimental-criteria","statement":"The Tsuei–Kirtley review describes phase-sensitive evidence for predominantly d-wave pairing in specified optimally doped cuprates. Such symmetry evidence constrains mechanisms without uniquely identifying their microscopic origin. No sample-specific temperature or raw-data reanalysis is added here.","evidenceType":"experiment","sourceIds":["source-e933e3bbd4beade6"],"conditions":"Review abstract reports phase-sensitive evidence in specified optimally doped cuprates; not a fresh primary-experiment or sample-temperature audit.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-limits-and-disputes-1","sectionId":"limits-and-disputes","statement":"Mazin and colleagues proposed a compound-specific sign-changing extended-s state for fluorine-doped LaFeAsO. Their published note 8 revises the initial identification of the magnetic instability involved in the doping argument. Reading only the abstract can miss that qualification. Kuroki and colleagues' five-orbital RPA study likewise has parameter-dependent s- and d-wave competition. Neither result assigns one state to every iron-based material.","evidenceType":"theory","sourceIds":["source-7c2b55d128348e6c","source-f80a18149666bd31"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-limits-and-disputes-2","sectionId":"limits-and-disputes","statement":"Kuroki's 2009 erratum corrects an orbital-to-band transformation error affecting the original gap plot. For the displayed parameter set, the corrected diagonal band gap is fully open on the Fermi surface. The authors retain strong anisotropy and parameter-dependent nodal or competing d-wave possibilities. Cite the original together with the erratum; the original nodal plot is not the corrected result. This check reads the erratum text, not a reproduction of its band calculation.","evidenceType":"theory","sourceIds":["source-f80a18149666bd31","source-6570d4401413e026"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-limits-and-disputes-3","sectionId":"limits-and-disputes","statement":"Monthoux–Pines report strong-coupling calculations for a YBCO model, including sensitivity to the spin spectrum and impurity treatment. Their calculated temperature is a theoretical result under chosen parameters. Agreement with selected data is a test of that model, not a mechanism proof that can be transferred to unrelated samples.","evidenceType":"theory","sourceIds":["source-7755cb2d156620a8"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-limits-and-disputes-4","sectionId":"limits-and-disputes","statement":"A kernel validated for one cuprate must not be transferred unchanged to iron-based compounds or nickelates. Band occupation, orbital weights, interlayer coupling and pressure or strain can alter the relevant low-energy problem. Nickelate material records remain separate; this entry makes no new nickelate gap assignment. Compare normal self-energy, gap structure, magnetic response and competing channels using compatible parameters.","evidenceType":"hypothesis","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."},{"id":"scwiki-b3-claim-discovery-implications-1","sectionId":"discovery-implications","statement":"Record the measured composition, phase and conditions; spectral normalization and coverage; electronic structure; and the approximation linking response to pairing. Compare alternative channels and report the prediction that could discriminate them. Numerical convergence, agreement with one observable and uniqueness of a mechanism are separate checks. This article's finite-model calculations establish algebraic consistency only.","evidenceType":"hypothesis","sourceIds":["source-eaa3beeed4368f46"],"conditions":"Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.","verificationScope":"Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open."}],"corrections":[{"id":"scwiki-b3-correction-1","description":"Distinguished hyperfine-weighted low-frequency NMR relaxation from a momentum-resolved magnetic spectrum.","sourceId":"source-e5c4ccac05133124","scope":"Measurement interpretation correction to inherited summary line 18; no new experiment."},{"id":"scwiki-b3-correction-2","description":"Linked Kuroki 2008 proposal to its 2009 orbital-to-band transformation erratum; retained conditional nodal competition.","sourceId":"source-6570d4401413e026","scope":"Published erratum text, not a new correction to the research paper."}],"equationChecks":[{"targetSectionId":"mathematical-assumptions","tex":"\\eta_+","check":"Five two-patch eigenvalue, original/symmetric residual and threshold checks; unequal density weights and zero boundary. Follow-up: 432 direct/symmetric matrix cases, 324 rational determinant checks and explicit all-zero coupling degeneracy; not an experimental or dynamical-model reproduction."},{"targetSectionId":"mathematical-assumptions","tex":"\\cos k_x\\pm\\cos k_y","check":"41 by 41 grid verifies Q=(pi,pi) sign reversal for both basis functions and their different C4 behavior."}],"materialCases":[],"body":"## Scope\n\nAntiferromagnetic spin fluctuations are a candidate pairing interaction with substantial weight at a nonzero momentum transfer. A repulsive interaction can favor a gap that changes sign between the states it connects. This is a model-dependent mechanism proposal: observing magnetism, or finding superconductivity near an antiferromagnetic phase, does not establish causation. [R1](#ref-source-eaa3beeed4368f46)\n\nKeep three questions separate: are magnetic fluctuations present; do they change systematically with superconductivity; and can a consistently specified interaction reproduce pairing and other observables? An affirmative answer to the first two does not by itself answer the third. The relevant Fermi surfaces, orbitals, frequency dependence and competing channels must be specified.\n\n## Mathematical assumptions\n\nA dressed particle-particle eigenproblem can be written, with $k_B=1$ and $k$ including momentum and Matsubara frequency, as\n\n$$-\\frac{T}{N}\\sum_{k'}\\Gamma^{pp}(k,k')G_\\uparrow(k')G_\\downarrow(-k')\\phi(k')=\\Lambda(T)\\phi(k).$$\n\nHere $N$ is the number of lattice sites and the Green functions and irreducible pairing vertex must belong to the same approximation. A leading eigenvalue reaching one signals a pairing instability within that treatment; it is not a measurement of a material transition. The selected review gives this equation and distinguishes spin, charge and fully irreducible contributions. [R1](#ref-source-eaa3beeed4368f46)\n\nA local static illustration, checked by separate matrix calculations, uses two patches with positive single-spin density-of-states weights $N_h,N_e$, a common pairing cutoff, constant gaps and real reciprocal interactions $U_h,U_e,V\\ge0$. After the common energy logarithm is factored out, define $\\eta\\Delta_i=-\\sum_jV_{ij}N_j\\Delta_j$, with $V_{he}=V_{eh}=V$. The rescaling $x_i=\\sqrt{N_i}\\Delta_i$ gives a symmetric matrix with diagonal elements $-U_iN_i$ and off-diagonal element $-V\\sqrt{N_hN_e}$. This is a separate static model, not a reduction validated against the review's dynamical calculation.\n\nIts larger eigenvalue is\n\n$$\\eta_+=-\\frac{U_hN_h+U_eN_e}{2}+\\sqrt{\\frac{(U_hN_h-U_eN_e)^2}{4}+V^2N_hN_e}.$$\n\nFor nonnegative intrapatch repulsions it is positive precisely when $V^2>U_hU_e$. The trace is nonpositive and the determinant is $(U_hU_e-V^2)N_hN_e$, establishing this threshold within the two-dimensional model. If both $U_i$ vanish and $V>0$, $\\eta_+=V\\sqrt{N_hN_e}$ and $\\Delta_h/\\Delta_e=-\\sqrt{N_e/N_h}$. If all interactions vanish, the kernel is zero and no preferred gap ratio is selected. A 432-case check compares direct and symmetrized eigensolvers, exact rational determinants and threshold cases. These are illustrative pairing strengths, not the phonon spectral coupling $\\lambda$, a calculated transition temperature, or a fit to any material.\n\nA second independent check shows why a wavevector alone cannot select the gap symmetry. With lattice spacing one, both $\\cos k_x-\\cos k_y$ and $\\cos k_x+\\cos k_y$ reverse sign under $\\mathbf Q=(\\pi,\\pi)$. Under a ninety-degree rotation the former changes sign while the latter does not. Their values on the actual Fermi surface and their interaction-weighted eigenvalues still differ. A sign-reversal condition is therefore insufficient to select a unique representation.\n\n## Experimental criteria\n\nNeutron, RIXS and NMR measurements have different response functions, selection rules and energy windows. In the magnetic low-frequency limit, NMR relaxation has the schematic dependence $1/(T_1T)\\propto\\sum_{\\mathbf q,\\alpha\\perp H}|A_\\alpha(\\mathbf q)|^2\\lim_{\\omega\\to0}\\chi''_\\alpha(\\mathbf q,\\omega)/\\omega$. The hyperfine weights and perpendicular spin components matter. The cited spin-chain paper supplies the response formula; it does not make that chain a superconducting-material example. NMR is not a momentum-resolved magnetic spectrum. [R7](#ref-source-e5c4ccac05133124)\n\nRIXS intensities also require a scattering model. Ament and colleagues' indirect-RIXS treatment contains charge and spin response contributions whose relative weights depend on incident energy and the stated core-hole approximation. A measured peak should not automatically be identified with the pairing kernel. This abstract-scoped example is not a derivation for every direct-RIXS geometry. [R8](#ref-source-824b2cddc30b3c2a)\n\nA superconducting spin resonance can reflect feedback of a sign-changing gap through coherence factors. Its interpretation still needs the normal-state response and a susceptibility calculation; it is not independent proof that the resonance caused pairing. [R1](#ref-source-eaa3beeed4368f46)\n\nThe Tsuei–Kirtley review describes phase-sensitive evidence for predominantly d-wave pairing in specified optimally doped cuprates. Such symmetry evidence constrains mechanisms without uniquely identifying their microscopic origin. No sample-specific temperature or raw-data reanalysis is added here. [R5](#ref-source-e933e3bbd4beade6)\n\n## Limits and disputes\n\nMazin and colleagues proposed a compound-specific sign-changing extended-s state for fluorine-doped LaFeAsO. Their published note 8 revises the initial identification of the magnetic instability involved in the doping argument. Reading only the abstract can miss that qualification. Kuroki and colleagues' five-orbital RPA study likewise has parameter-dependent s- and d-wave competition. Neither result assigns one state to every iron-based material. [R3](#ref-source-7c2b55d128348e6c) [R4](#ref-source-f80a18149666bd31)\n\nKuroki's 2009 erratum corrects an orbital-to-band transformation error affecting the original gap plot. For the displayed parameter set, the corrected diagonal band gap is fully open on the Fermi surface. The authors retain strong anisotropy and parameter-dependent nodal or competing d-wave possibilities. Cite the original together with the erratum; the original nodal plot is not the corrected result. This check reads the erratum text, not a reproduction of its band calculation. [R4](#ref-source-f80a18149666bd31) [R6](#ref-source-6570d4401413e026)\n\nMonthoux–Pines report strong-coupling calculations for a YBCO model, including sensitivity to the spin spectrum and impurity treatment. Their calculated temperature is a theoretical result under chosen parameters. Agreement with selected data is a test of that model, not a mechanism proof that can be transferred to unrelated samples. [R2](#ref-source-7755cb2d156620a8)\n\nA kernel validated for one cuprate must not be transferred unchanged to iron-based compounds or nickelates. Band occupation, orbital weights, interlayer coupling and pressure or strain can alter the relevant low-energy problem. Nickelate material records remain separate; this entry makes no new nickelate gap assignment. Compare normal self-energy, gap structure, magnetic response and competing channels using compatible parameters.\n\n## Discovery implications\n\nRecord the measured composition, phase and conditions; spectral normalization and coverage; electronic structure; and the approximation linking response to pairing. Compare alternative channels and report the prediction that could discriminate them. Numerical convergence, agreement with one observable and uniqueness of a mechanism are separate checks. This article's finite-model calculations establish algebraic consistency only.\n\n## Related topics\n\nContinue with [BCS theory](/scwiki/bcs-weak-coupling), [Eliashberg theory](/scwiki/eliashberg-strong-coupling), [multiband pairing](/scwiki/multiband-pairing), [s± pairing](/scwiki/s-plus-minus), and [nickelates](/scwiki/nickelates).","revision":"5e7b6442f74d58ee63d5","sections":[{"id":"scope","title":"Scope","markdown":"## Scope\n\nAntiferromagnetic spin fluctuations are a candidate pairing interaction with substantial weight at a nonzero momentum transfer. A repulsive interaction can favor a gap that changes sign between the states it connects. This is a model-dependent mechanism proposal: observing magnetism, or finding superconductivity near an antiferromagnetic phase, does not establish causation. [R1](#ref-source-eaa3beeed4368f46)\n\nKeep three questions separate: are magnetic fluctuations present; do they change systematically with superconductivity; and can a consistently specified interaction reproduce pairing and other observables? An affirmative answer to the first two does not by itself answer the third. The relevant Fermi surfaces, orbitals, frequency dependence and competing channels must be specified."},{"id":"mathematical-assumptions","title":"Mathematical assumptions","markdown":"## Mathematical assumptions\n\nA dressed particle-particle eigenproblem can be written, with $k_B=1$ and $k$ including momentum and Matsubara frequency, as\n\n$$-\\frac{T}{N}\\sum_{k'}\\Gamma^{pp}(k,k')G_\\uparrow(k')G_\\downarrow(-k')\\phi(k')=\\Lambda(T)\\phi(k).$$\n\nHere $N$ is the number of lattice sites and the Green functions and irreducible pairing vertex must belong to the same approximation. A leading eigenvalue reaching one signals a pairing instability within that treatment; it is not a measurement of a material transition. The selected review gives this equation and distinguishes spin, charge and fully irreducible contributions. [R1](#ref-source-eaa3beeed4368f46)\n\nA local static illustration, checked by separate matrix calculations, uses two patches with positive single-spin density-of-states weights $N_h,N_e$, a common pairing cutoff, constant gaps and real reciprocal interactions $U_h,U_e,V\\ge0$. After the common energy logarithm is factored out, define $\\eta\\Delta_i=-\\sum_jV_{ij}N_j\\Delta_j$, with $V_{he}=V_{eh}=V$. The rescaling $x_i=\\sqrt{N_i}\\Delta_i$ gives a symmetric matrix with diagonal elements $-U_iN_i$ and off-diagonal element $-V\\sqrt{N_hN_e}$. This is a separate static model, not a reduction validated against the review's dynamical calculation.\n\nIts larger eigenvalue is\n\n$$\\eta_+=-\\frac{U_hN_h+U_eN_e}{2}+\\sqrt{\\frac{(U_hN_h-U_eN_e)^2}{4}+V^2N_hN_e}.$$\n\nFor nonnegative intrapatch repulsions it is positive precisely when $V^2>U_hU_e$. The trace is nonpositive and the determinant is $(U_hU_e-V^2)N_hN_e$, establishing this threshold within the two-dimensional model. If both $U_i$ vanish and $V>0$, $\\eta_+=V\\sqrt{N_hN_e}$ and $\\Delta_h/\\Delta_e=-\\sqrt{N_e/N_h}$. If all interactions vanish, the kernel is zero and no preferred gap ratio is selected. A 432-case check compares direct and symmetrized eigensolvers, exact rational determinants and threshold cases. These are illustrative pairing strengths, not the phonon spectral coupling $\\lambda$, a calculated transition temperature, or a fit to any material.\n\nA second independent check shows why a wavevector alone cannot select the gap symmetry. With lattice spacing one, both $\\cos k_x-\\cos k_y$ and $\\cos k_x+\\cos k_y$ reverse sign under $\\mathbf Q=(\\pi,\\pi)$. Under a ninety-degree rotation the former changes sign while the latter does not. Their values on the actual Fermi surface and their interaction-weighted eigenvalues still differ. A sign-reversal condition is therefore insufficient to select a unique representation."},{"id":"experimental-criteria","title":"Experimental criteria","markdown":"## Experimental criteria\n\nNeutron, RIXS and NMR measurements have different response functions, selection rules and energy windows. In the magnetic low-frequency limit, NMR relaxation has the schematic dependence $1/(T_1T)\\propto\\sum_{\\mathbf q,\\alpha\\perp H}|A_\\alpha(\\mathbf q)|^2\\lim_{\\omega\\to0}\\chi''_\\alpha(\\mathbf q,\\omega)/\\omega$. The hyperfine weights and perpendicular spin components matter. The cited spin-chain paper supplies the response formula; it does not make that chain a superconducting-material example. NMR is not a momentum-resolved magnetic spectrum. [R7](#ref-source-e5c4ccac05133124)\n\nRIXS intensities also require a scattering model. Ament and colleagues' indirect-RIXS treatment contains charge and spin response contributions whose relative weights depend on incident energy and the stated core-hole approximation. A measured peak should not automatically be identified with the pairing kernel. This abstract-scoped example is not a derivation for every direct-RIXS geometry. [R8](#ref-source-824b2cddc30b3c2a)\n\nA superconducting spin resonance can reflect feedback of a sign-changing gap through coherence factors. Its interpretation still needs the normal-state response and a susceptibility calculation; it is not independent proof that the resonance caused pairing. [R1](#ref-source-eaa3beeed4368f46)\n\nThe Tsuei–Kirtley review describes phase-sensitive evidence for predominantly d-wave pairing in specified optimally doped cuprates. Such symmetry evidence constrains mechanisms without uniquely identifying their microscopic origin. No sample-specific temperature or raw-data reanalysis is added here. [R5](#ref-source-e933e3bbd4beade6)"},{"id":"limits-and-disputes","title":"Limits and disputes","markdown":"## Limits and disputes\n\nMazin and colleagues proposed a compound-specific sign-changing extended-s state for fluorine-doped LaFeAsO. Their published note 8 revises the initial identification of the magnetic instability involved in the doping argument. Reading only the abstract can miss that qualification. Kuroki and colleagues' five-orbital RPA study likewise has parameter-dependent s- and d-wave competition. Neither result assigns one state to every iron-based material. [R3](#ref-source-7c2b55d128348e6c) [R4](#ref-source-f80a18149666bd31)\n\nKuroki's 2009 erratum corrects an orbital-to-band transformation error affecting the original gap plot. For the displayed parameter set, the corrected diagonal band gap is fully open on the Fermi surface. The authors retain strong anisotropy and parameter-dependent nodal or competing d-wave possibilities. Cite the original together with the erratum; the original nodal plot is not the corrected result. This check reads the erratum text, not a reproduction of its band calculation. [R4](#ref-source-f80a18149666bd31) [R6](#ref-source-6570d4401413e026)\n\nMonthoux–Pines report strong-coupling calculations for a YBCO model, including sensitivity to the spin spectrum and impurity treatment. Their calculated temperature is a theoretical result under chosen parameters. Agreement with selected data is a test of that model, not a mechanism proof that can be transferred to unrelated samples. [R2](#ref-source-7755cb2d156620a8)\n\nA kernel validated for one cuprate must not be transferred unchanged to iron-based compounds or nickelates. Band occupation, orbital weights, interlayer coupling and pressure or strain can alter the relevant low-energy problem. Nickelate material records remain separate; this entry makes no new nickelate gap assignment. Compare normal self-energy, gap structure, magnetic response and competing channels using compatible parameters."},{"id":"discovery-implications","title":"Discovery implications","markdown":"## Discovery implications\n\nRecord the measured composition, phase and conditions; spectral normalization and coverage; electronic structure; and the approximation linking response to pairing. Compare alternative channels and report the prediction that could discriminate them. Numerical convergence, agreement with one observable and uniqueness of a mechanism are separate checks. This article's finite-model calculations establish algebraic consistency only."},{"id":"related-topics","title":"Related topics","markdown":"## Related topics\n\nContinue with [BCS theory](/scwiki/bcs-weak-coupling), [Eliashberg theory](/scwiki/eliashberg-strong-coupling), [multiband pairing](/scwiki/multiband-pairing), [s± pairing](/scwiki/s-plus-minus), and [nickelates](/scwiki/nickelates)."},{"id":"material-cases","title":"Material cases","markdown":"## Material cases\n\nNo sample-specific material records are included in this article. Model results are not experimental transition temperatures."}],"markdown":"# Antiferromagnetic spin fluctuations: pairing and evidence\n\nHow repulsive interactions can favor sign-changing gaps, why magnetic signals do not prove a mechanism, and how probe limits and published corrections constrain the evidence.\n\n## Scope\n\nAntiferromagnetic spin fluctuations are a candidate pairing interaction with substantial weight at a nonzero momentum transfer. A repulsive interaction can favor a gap that changes sign between the states it connects. This is a model-dependent mechanism proposal: observing magnetism, or finding superconductivity near an antiferromagnetic phase, does not establish causation. [R1](#ref-source-eaa3beeed4368f46)\n\nKeep three questions separate: are magnetic fluctuations present; do they change systematically with superconductivity; and can a consistently specified interaction reproduce pairing and other observables? An affirmative answer to the first two does not by itself answer the third. The relevant Fermi surfaces, orbitals, frequency dependence and competing channels must be specified.\n\n## Mathematical assumptions\n\nA dressed particle-particle eigenproblem can be written, with $k_B=1$ and $k$ including momentum and Matsubara frequency, as\n\n$$-\\frac{T}{N}\\sum_{k'}\\Gamma^{pp}(k,k')G_\\uparrow(k')G_\\downarrow(-k')\\phi(k')=\\Lambda(T)\\phi(k).$$\n\nHere $N$ is the number of lattice sites and the Green functions and irreducible pairing vertex must belong to the same approximation. A leading eigenvalue reaching one signals a pairing instability within that treatment; it is not a measurement of a material transition. The selected review gives this equation and distinguishes spin, charge and fully irreducible contributions. [R1](#ref-source-eaa3beeed4368f46)\n\nA local static illustration, checked by separate matrix calculations, uses two patches with positive single-spin density-of-states weights $N_h,N_e$, a common pairing cutoff, constant gaps and real reciprocal interactions $U_h,U_e,V\\ge0$. After the common energy logarithm is factored out, define $\\eta\\Delta_i=-\\sum_jV_{ij}N_j\\Delta_j$, with $V_{he}=V_{eh}=V$. The rescaling $x_i=\\sqrt{N_i}\\Delta_i$ gives a symmetric matrix with diagonal elements $-U_iN_i$ and off-diagonal element $-V\\sqrt{N_hN_e}$. This is a separate static model, not a reduction validated against the review's dynamical calculation.\n\nIts larger eigenvalue is\n\n$$\\eta_+=-\\frac{U_hN_h+U_eN_e}{2}+\\sqrt{\\frac{(U_hN_h-U_eN_e)^2}{4}+V^2N_hN_e}.$$\n\nFor nonnegative intrapatch repulsions it is positive precisely when $V^2>U_hU_e$. The trace is nonpositive and the determinant is $(U_hU_e-V^2)N_hN_e$, establishing this threshold within the two-dimensional model. If both $U_i$ vanish and $V>0$, $\\eta_+=V\\sqrt{N_hN_e}$ and $\\Delta_h/\\Delta_e=-\\sqrt{N_e/N_h}$. If all interactions vanish, the kernel is zero and no preferred gap ratio is selected. A 432-case check compares direct and symmetrized eigensolvers, exact rational determinants and threshold cases. These are illustrative pairing strengths, not the phonon spectral coupling $\\lambda$, a calculated transition temperature, or a fit to any material.\n\nA second independent check shows why a wavevector alone cannot select the gap symmetry. With lattice spacing one, both $\\cos k_x-\\cos k_y$ and $\\cos k_x+\\cos k_y$ reverse sign under $\\mathbf Q=(\\pi,\\pi)$. Under a ninety-degree rotation the former changes sign while the latter does not. Their values on the actual Fermi surface and their interaction-weighted eigenvalues still differ. A sign-reversal condition is therefore insufficient to select a unique representation.\n\n## Experimental criteria\n\nNeutron, RIXS and NMR measurements have different response functions, selection rules and energy windows. In the magnetic low-frequency limit, NMR relaxation has the schematic dependence $1/(T_1T)\\propto\\sum_{\\mathbf q,\\alpha\\perp H}|A_\\alpha(\\mathbf q)|^2\\lim_{\\omega\\to0}\\chi''_\\alpha(\\mathbf q,\\omega)/\\omega$. The hyperfine weights and perpendicular spin components matter. The cited spin-chain paper supplies the response formula; it does not make that chain a superconducting-material example. NMR is not a momentum-resolved magnetic spectrum. [R7](#ref-source-e5c4ccac05133124)\n\nRIXS intensities also require a scattering model. Ament and colleagues' indirect-RIXS treatment contains charge and spin response contributions whose relative weights depend on incident energy and the stated core-hole approximation. A measured peak should not automatically be identified with the pairing kernel. This abstract-scoped example is not a derivation for every direct-RIXS geometry. [R8](#ref-source-824b2cddc30b3c2a)\n\nA superconducting spin resonance can reflect feedback of a sign-changing gap through coherence factors. Its interpretation still needs the normal-state response and a susceptibility calculation; it is not independent proof that the resonance caused pairing. [R1](#ref-source-eaa3beeed4368f46)\n\nThe Tsuei–Kirtley review describes phase-sensitive evidence for predominantly d-wave pairing in specified optimally doped cuprates. Such symmetry evidence constrains mechanisms without uniquely identifying their microscopic origin. No sample-specific temperature or raw-data reanalysis is added here. [R5](#ref-source-e933e3bbd4beade6)\n\n## Limits and disputes\n\nMazin and colleagues proposed a compound-specific sign-changing extended-s state for fluorine-doped LaFeAsO. Their published note 8 revises the initial identification of the magnetic instability involved in the doping argument. Reading only the abstract can miss that qualification. Kuroki and colleagues' five-orbital RPA study likewise has parameter-dependent s- and d-wave competition. Neither result assigns one state to every iron-based material. [R3](#ref-source-7c2b55d128348e6c) [R4](#ref-source-f80a18149666bd31)\n\nKuroki's 2009 erratum corrects an orbital-to-band transformation error affecting the original gap plot. For the displayed parameter set, the corrected diagonal band gap is fully open on the Fermi surface. The authors retain strong anisotropy and parameter-dependent nodal or competing d-wave possibilities. Cite the original together with the erratum; the original nodal plot is not the corrected result. This check reads the erratum text, not a reproduction of its band calculation. [R4](#ref-source-f80a18149666bd31) [R6](#ref-source-6570d4401413e026)\n\nMonthoux–Pines report strong-coupling calculations for a YBCO model, including sensitivity to the spin spectrum and impurity treatment. Their calculated temperature is a theoretical result under chosen parameters. Agreement with selected data is a test of that model, not a mechanism proof that can be transferred to unrelated samples. [R2](#ref-source-7755cb2d156620a8)\n\nA kernel validated for one cuprate must not be transferred unchanged to iron-based compounds or nickelates. Band occupation, orbital weights, interlayer coupling and pressure or strain can alter the relevant low-energy problem. Nickelate material records remain separate; this entry makes no new nickelate gap assignment. Compare normal self-energy, gap structure, magnetic response and competing channels using compatible parameters.\n\n## Discovery implications\n\nRecord the measured composition, phase and conditions; spectral normalization and coverage; electronic structure; and the approximation linking response to pairing. Compare alternative channels and report the prediction that could discriminate them. Numerical convergence, agreement with one observable and uniqueness of a mechanism are separate checks. This article's finite-model calculations establish algebraic consistency only.\n\n## Related topics\n\nContinue with [BCS theory](/scwiki/bcs-weak-coupling), [Eliashberg theory](/scwiki/eliashberg-strong-coupling), [multiband pairing](/scwiki/multiband-pairing), [s± pairing](/scwiki/s-plus-minus), and [nickelates](/scwiki/nickelates).\n\n## Material cases\n\nNo sample-specific material records are included in this article. Model results are not experimental transition temperatures.\n\n## Sources\n\n### Ref source-eaa3beeed4368f46\n\n[R1] A common thread: The pairing interaction for unconventional superconductors (2012). [DOI 10.1103/RevModPhys.84.1383](https://doi.org/10.1103/RevModPhys.84.1383). Publication: journal-article. Check: author-review-scoped-check. Checked: 2026-10-18. Selected manuscript text pp. 4, 18–20, 39–46; p. 41 Eq. (13) visually checked. Full 87-page review and final-version equivalence not audited. Independent two-patch algebra is an illustration, not reproduction of a Hubbard calculation.\n### Ref source-7755cb2d156620a8\n\n[R2] Spin-fluctuation-induced superconductivity and normal-state properties of YBa2Cu3O7 (1994). [DOI 10.1103/PhysRevB.49.4261](https://doi.org/10.1103/PhysRevB.49.4261). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. Monthoux–Pines publisher abstract: model calculations and parameter/impurity sensitivity. Full derivation, input spectrum and numerical data not reanalyzed.\n### Ref source-7c2b55d128348e6c\n\n[R3] Unconventional Superconductivity with a Sign Reversal in the Order Parameter of LaFeAsO1−xFx (2008). [DOI 10.1103/PhysRevLett.101.057003](https://doi.org/10.1103/PhysRevLett.101.057003). Publication: journal-article. Check: publisher-passages-scoped-check. Checked: 2026-10-18. Mazin et al. abstract and published note 8 checked. The note qualifies the initial identification of the magnetic instability; full electronic-structure calculation not reproduced.\n### Ref source-f80a18149666bd31\n\n[R4] Unconventional Pairing Originating from the Disconnected Fermi Surfaces of Superconducting LaFeAsO1−xFx (2008). [DOI 10.1103/PhysRevLett.101.087004](https://doi.org/10.1103/PhysRevLett.101.087004). Publication: journal-article. Check: primary-abstract-with-erratum-check. Checked: 2026-10-18. Kuroki et al. five-orbital RPA abstract checked alongside 2009 erratum 10.1103/PhysRevLett.102.109902. The original gap plot must not be used without its correction; full model not reproduced.\n### Ref source-e933e3bbd4beade6\n\n[R5] Pairing symmetry in cuprate superconductors (2000). [DOI 10.1103/RevModPhys.72.969](https://doi.org/10.1103/RevModPhys.72.969). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. Tsuei–Kirtley publisher abstract: phase-sensitive evidence for predominantly d-wave in specified optimally doped cuprates. Original experiments, sample data and full review not audited here.\n### Ref source-6570d4401413e026\n\n[R6] Erratum: Unconventional Pairing Originating from the Disconnected Fermi Surfaces of Superconducting LaFeAsO1−xFx [Phys. Rev. Lett. 101, 087004 (2008)] (2009). [DOI 10.1103/PhysRevLett.102.109902](https://doi.org/10.1103/PhysRevLett.102.109902). Publication: journal-article. Check: primary-erratum-text-scoped-check. Checked: 2026-10-18. Complete one-page text read via web: orbital-to-band transformation correction, fully open diagonal gap for displayed parameters, parameter-dependent nodal competition retained. Direct download returned 403; figure was not visually audited or numerically reproduced.\n### Ref source-e5c4ccac05133124\n\n[R7] NMR relaxation rates for the spin-1/2 Heisenberg chain (1995). [DOI 10.1103/PhysRevB.52.R9831](https://doi.org/10.1103/PhysRevB.52.R9831). Publication: journal-article. Check: author-manuscript-scoped-check. Checked: 2026-10-18. Sandvik manuscript pp. 2–3; p. 3 Eq. (2) and response conventions visually checked. Used only for hyperfine-weighted low-frequency NMR relation, not as superconducting-material evidence or reproduction of QMC data.\n### Ref source-824b2cddc30b3c2a\n\n[R8] Ultrashort lifetime expansion for indirect resonant inelastic x-ray scattering (2007). [DOI 10.1103/PhysRevB.75.115118](https://doi.org/10.1103/PhysRevB.75.115118). Publication: journal-article. Check: author-abstract-scoped-check. Checked: 2026-10-18. Ament, Forte and van den Brink author abstract and linked journal identity. Indirect-RIXS core-hole approximation and mixed charge/spin contributions; full derivation and applicability to direct RIXS not audited.\n\n## Claim evidence and conditions\n\n### scwiki-b3-claim-scope-1\n\nAntiferromagnetic spin fluctuations are a candidate pairing interaction with substantial weight at a nonzero momentum transfer. A repulsive interaction can favor a gap that changes sign between the states it connects. This is a model-dependent mechanism proposal: observing magnetism, or finding superconductivity near an antiferromagnetic phase, does not establish causation.\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-scope-2\n\nKeep three questions separate: are magnetic fluctuations present; do they change systematically with superconductivity; and can a consistently specified interaction reproduce pairing and other observables? An affirmative answer to the first two does not by itself answer the third. The relevant Fermi surfaces, orbitals, frequency dependence and competing channels must be specified.\n\nEvidence: hypothesis. Sources: source-eaa3beeed4368f46.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-mathematical-assumptions-1\n\nA dressed particle-particle eigenproblem can be written, with $k_B=1$ and $k$ including momentum and Matsubara frequency, as\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nSource relationship: mixed-source-and-local-derivation. Scalapino author v1 p. 41 Eq. (13) supplies the dressed particle-particle eigenproblem, with p. 39 Eq. (12) distinguishing vertex channels. The entry uses Lambda for the eigenvalue and explicitly limits the instability interpretation to that treatment; no material transition is measured.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-mathematical-assumptions-2\n\n$$-\\frac{T}{N}\\sum_{k'}\\Gamma^{pp}(k,k')G_\\uparrow(k')G_\\downarrow(-k')\\phi(k')=\\Lambda(T)\\phi(k).$$\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nSource relationship: mixed-source-and-local-derivation. Scalapino author v1 p. 41 Eq. (13) supplies the dressed particle-particle eigenproblem, with p. 39 Eq. (12) distinguishing vertex channels. The entry uses Lambda for the eigenvalue and explicitly limits the instability interpretation to that treatment; no material transition is measured.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-mathematical-assumptions-3\n\nHere $N$ is the number of lattice sites and the Green functions and irreducible pairing vertex must belong to the same approximation. A leading eigenvalue reaching one signals a pairing instability within that treatment; it is not a measurement of a material transition. The selected review gives this equation and distinguishes spin, charge and fully irreducible contributions.\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nSource relationship: mixed-source-and-local-derivation. Scalapino author v1 p. 41 Eq. (13) supplies the dressed particle-particle eigenproblem, with p. 39 Eq. (12) distinguishing vertex channels. The entry uses Lambda for the eigenvalue and explicitly limits the instability interpretation to that treatment; no material transition is measured.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-mathematical-assumptions-4\n\nA local static illustration, checked by separate matrix calculations, uses two patches with positive single-spin density-of-states weights $N_h,N_e$, a common pairing cutoff, constant gaps and real reciprocal interactions $U_h,U_e,V\\ge0$. After the common energy logarithm is factored out, define $\\eta\\Delta_i=-\\sum_jV_{ij}N_j\\Delta_j$, with $V_{he}=V_{eh}=V$. The rescaling $x_i=\\sqrt{N_i}\\Delta_i$ gives a symmetric matrix with diagonal elements $-U_iN_i$ and off-diagonal element $-V\\sqrt{N_hN_e}$. This is a separate static model, not a reduction validated against the review's dynamical calculation.\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nSource relationship: context-for-local-derivation. Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-mathematical-assumptions-5\n\nIts larger eigenvalue is\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nSource relationship: context-for-local-derivation. Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-mathematical-assumptions-6\n\n$$\\eta_+=-\\frac{U_hN_h+U_eN_e}{2}+\\sqrt{\\frac{(U_hN_h-U_eN_e)^2}{4}+V^2N_hN_e}.$$\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nSource relationship: context-for-local-derivation. Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-mathematical-assumptions-7\n\nFor nonnegative intrapatch repulsions it is positive precisely when $V^2>U_hU_e$. The trace is nonpositive and the determinant is $(U_hU_e-V^2)N_hN_e$, establishing this threshold within the two-dimensional model. If both $U_i$ vanish and $V>0$, $\\eta_+=V\\sqrt{N_hN_e}$ and $\\Delta_h/\\Delta_e=-\\sqrt{N_e/N_h}$. If all interactions vanish, the kernel is zero and no preferred gap ratio is selected. A 432-case check compares direct and symmetrized eigensolvers, exact rational determinants and threshold cases. These are illustrative pairing strengths, not the phonon spectral coupling $\\lambda$, a calculated transition temperature, or a fit to any material.\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nSource relationship: context-for-local-derivation. Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-mathematical-assumptions-8\n\nA second independent check shows why a wavevector alone cannot select the gap symmetry. With lattice spacing one, both $\\cos k_x-\\cos k_y$ and $\\cos k_x+\\cos k_y$ reverse sign under $\\mathbf Q=(\\pi,\\pi)$. Under a ninety-degree rotation the former changes sign while the latter does not. Their values on the actual Fermi surface and their interaction-weighted eigenvalues still differ. A sign-reversal condition is therefore insufficient to select a unique representation.\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nSource relationship: context-for-local-derivation. Scalapino pp. 39-42 provide pairing-vertex/eigenvalue context. The reciprocal two-patch static model, DOS-weighted similarity transform, trace/determinant criterion, pure-interpatch ratio and square-lattice sign/rotation examples are local illustrations, not quoted results or a reproduction of the review's dynamic Hubbard calculation. Positive DOS weights, a common cutoff and constant gaps are explicit assumptions; the gap ratio needs nonzero interpatch coupling.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-experimental-criteria-1\n\nNeutron, RIXS and NMR measurements have different response functions, selection rules and energy windows. In the magnetic low-frequency limit, NMR relaxation has the schematic dependence $1/(T_1T)\\propto\\sum_{\\mathbf q,\\alpha\\perp H}|A_\\alpha(\\mathbf q)|^2\\lim_{\\omega\\to0}\\chi''_\\alpha(\\mathbf q,\\omega)/\\omega$. The hyperfine weights and perpendicular spin components matter. The cited spin-chain paper supplies the response formula; it does not make that chain a superconducting-material example. NMR is not a momentum-resolved magnetic spectrum.\n\nEvidence: theory. Sources: source-e5c4ccac05133124.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-experimental-criteria-2\n\nRIXS intensities also require a scattering model. Ament and colleagues' indirect-RIXS treatment contains charge and spin response contributions whose relative weights depend on incident energy and the stated core-hole approximation. A measured peak should not automatically be identified with the pairing kernel. This abstract-scoped example is not a derivation for every direct-RIXS geometry.\n\nEvidence: theory. Sources: source-824b2cddc30b3c2a.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-experimental-criteria-3\n\nA superconducting spin resonance can reflect feedback of a sign-changing gap through coherence factors. Its interpretation still needs the normal-state response and a susceptibility calculation; it is not independent proof that the resonance caused pairing.\n\nEvidence: theory. Sources: source-eaa3beeed4368f46.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-experimental-criteria-4\n\nThe Tsuei–Kirtley review describes phase-sensitive evidence for predominantly d-wave pairing in specified optimally doped cuprates. Such symmetry evidence constrains mechanisms without uniquely identifying their microscopic origin. No sample-specific temperature or raw-data reanalysis is added here.\n\nEvidence: experiment. Sources: source-e933e3bbd4beade6.\n\nConditions: Review abstract reports phase-sensitive evidence in specified optimally doped cuprates; not a fresh primary-experiment or sample-temperature audit.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-limits-and-disputes-1\n\nMazin and colleagues proposed a compound-specific sign-changing extended-s state for fluorine-doped LaFeAsO. Their published note 8 revises the initial identification of the magnetic instability involved in the doping argument. Reading only the abstract can miss that qualification. Kuroki and colleagues' five-orbital RPA study likewise has parameter-dependent s- and d-wave competition. Neither result assigns one state to every iron-based material.\n\nEvidence: theory. Sources: source-7c2b55d128348e6c, source-f80a18149666bd31.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-limits-and-disputes-2\n\nKuroki's 2009 erratum corrects an orbital-to-band transformation error affecting the original gap plot. For the displayed parameter set, the corrected diagonal band gap is fully open on the Fermi surface. The authors retain strong anisotropy and parameter-dependent nodal or competing d-wave possibilities. Cite the original together with the erratum; the original nodal plot is not the corrected result. This check reads the erratum text, not a reproduction of its band calculation.\n\nEvidence: theory. Sources: source-f80a18149666bd31, source-6570d4401413e026.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-limits-and-disputes-3\n\nMonthoux–Pines report strong-coupling calculations for a YBCO model, including sensitivity to the spin spectrum and impurity treatment. Their calculated temperature is a theoretical result under chosen parameters. Agreement with selected data is a test of that model, not a mechanism proof that can be transferred to unrelated samples.\n\nEvidence: theory. Sources: source-7755cb2d156620a8.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-limits-and-disputes-4\n\nA kernel validated for one cuprate must not be transferred unchanged to iron-based compounds or nickelates. Band occupation, orbital weights, interlayer coupling and pressure or strain can alter the relevant low-energy problem. Nickelate material records remain separate; this entry makes no new nickelate gap assignment. Compare normal self-energy, gap structure, magnetic response and competing channels using compatible parameters.\n\nEvidence: hypothesis. Sources: source-eaa3beeed4368f46.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n### scwiki-b3-claim-discovery-implications-1\n\nRecord the measured composition, phase and conditions; spectral normalization and coverage; electronic structure; and the approximation linking response to pairing. Compare alternative channels and report the prediction that could discriminate them. Numerical convergence, agreement with one observable and uniqueness of a mechanism are separate checks. This article's finite-model calculations establish algebraic consistency only.\n\nEvidence: hypothesis. Sources: source-eaa3beeed4368f46.\n\nConditions: Specified model, response function and source scope only. Independent two-patch and lattice examples are editorial algebra, not a published material prediction or measured Tc.\n\nVerification scope: Selected source locations; independent models do not certify a microscopic mechanism. Full claim-level scientific acceptance remains open.\n\n## Revision and review\n\nArticle ID: scwiki-b3. Revision: 5e7b6442f74d58ee63d5. Updated: 2026-10-19.\n\nReview scope: Selected pairing-vertex and NMR response equations, primary abstracts and published erratum text; independent two-patch and square-lattice checks. Follow-up selected Scalapino pp. 39-42 and p. 41 equation image, explicit local-derivation mapping, and 432 weighted-kernel cases with zero-coupling boundary.\n\n- Complete scientific acceptance, raw-data checks and material-model reproduction remain open.\n- Review summaries of phase-sensitive evidence do not replace a primary-experiment audit. No sample-specific Tc is extracted.\n- Kuroki erratum text was read; its figure and underlying band calculation were not independently reproduced.\n- Author-manuscript passage checks do not establish final-version equivalence. Model eigenvalues are not measured transitions.\n\nSource: [B_mechanisms/B3_spin_fluctuation_AFM.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/B_mechanisms/B3_spin_fluctuation_AFM.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 8bbed91652c4ebfe4d30d74844bd39c4f40d77e788d6f2a4670db256ae92f249.\n","wordCount":921}}