{"schemaVersion":"1.0","revision":"fa961ab24277a91737c3","article":{"id":"scwiki-b4","topicId":"scwiki-topic-b4","code":"B4","slug":"ferromagnetic-fluctuations","title":"Ferromagnetic fluctuations and odd-parity pairing","summary":"Separate triplet-pairing models from magnetic evidence, compare uranium compounds under stated conditions, and trace the limits of Knight shifts and critical fields.","aliases":["FM spin fluctuations","odd parity pairing","equal spin pairing","ferromagnetic superconductivity","UTe2","URhGe","UGe2","UCoGe","Knight shift"],"evidenceTypes":["theory","experiment","hypothesis"],"relatedCodes":["B1","B2","B3","A5","A6"],"editorialStatus":"scoped-check","contentState":"pilot","modifiedAt":"2026-10-18T12:19:32.196903+00:00","review":{"checkedAt":"2026-10-18","scope":"Selected primary abstracts, Sr2RuO4/Ran manuscript passages,2021 neutron extended-data text and2026 UTe2 publisher passages; independent spin, parity and angular-projection checks.","limitations":["Full scientific acceptance, raw-data review and a comprehensive current literature survey remain open.","Author-manuscript checks do not establish final-version equivalence; several historical references are abstract-scoped.","Independent illustrations do not predict a material transition or establish a pairing kernel.","UTe2 4 K is a normal-state measurement temperature, not a superconducting transition. Different field/probe windows cannot be conflated."]},"provenance":{"repository":"JackZH26/jzis-superconductivity","commit":"0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf","path":"B_mechanisms/B4_spin_fluctuation_FM.md","sha256":"0440ac552dfaad93ae3a3e94517ee603ba14626f3af2745ec52dfd14479b39f7","sourceId":"file-429480e8c9d9c129"},"sourceMap":[{"startLine":1,"endLine":5,"targetSections":["scope"],"decision":"edited-with-scoped-evidence","note":"Magnetic correlations versus state identification."},{"startLine":6,"endLine":11,"targetSections":["mathematical-assumptions","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Conditional FM model; independent spin/parity/angular illustrations."},{"startLine":12,"endLine":17,"targetSections":["experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Uranium material evidence; two inherited DOI identities corrected."},{"startLine":18,"endLine":23,"targetSections":["experimental-criteria","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Sr2RuO4 and UTe2 caveats extended with scoped primary passages."},{"startLine":24,"endLine":31,"targetSections":["experimental-criteria","discovery-implications"],"decision":"edited-with-scoped-evidence","note":"Probe conditions, symmetry and same-sample volume criteria."},{"startLine":32,"endLine":43,"targetSections":["limits-and-disputes","related-topics"],"decision":"edited-with-scoped-evidence","note":"All inherited source anchors retained with corrected links and review limits."}],"references":[{"id":"source-82c04c9061dbca4c","doi":"10.1103/PhysRevB.22.3173","title":"Coexistence of p-state superconductivity and itinerant ferromagnetism","year":1980,"url":"https://doi.org/10.1103/PhysRevB.22.3173","publicationStatus":"journal-article","verification":{"scope":"primary-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prb/abstract/10.1103/PhysRevB.22.3173","note":"Fay–Appel abstract: longitudinal-fluctuation equal-spin model and nonmonotonic calculated Tc. Full calculation not read or reproduced. The article two-spin/angular examples are independent editorial illustrations, not equations attributed to this paper."}},{"id":"source-db2917242488a101","doi":"10.1038/35020500","title":"Superconductivity on the border of itinerant-electron ferromagnetism in UGe2","year":2000,"url":"https://doi.org/10.1038/35020500","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://www.nature.com/articles/35020500","note":"Saxena et al. publisher indexed abstract and Crossref identity checked. Restricted-pressure superconductivity; original curves and microscopic coexistence not audited. Replaces inherited DOI35050500 whose actual target remained unresolved."}},{"id":"source-15aec45a868e9fb9","doi":"10.1038/35098048","title":"Coexistence of superconductivity and ferromagnetism in URhGe","year":2001,"url":"https://doi.org/10.1038/35098048","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://www.nature.com/articles/35098048","note":"Aoki et al. publisher indexed abstract: ambient-pressure thermodynamic evidence, with qualified triplet interpretation. Corrects inherited35065045, which identifies unrelated semiconductor melting research. Full data not reanalyzed."}},{"id":"source-e917adb251d93c59","doi":"10.1126/science.1115498","title":"Magnetic Field-Induced Superconductivity in the Ferromagnet URhGe","year":2005,"url":"https://doi.org/10.1126/science.1115498","publicationStatus":"journal-article","verification":{"scope":"primary-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://pubmed.ncbi.nlm.nih.gov/16123293/","note":"Levy et al. original abstract as indexed by PubMed: field along b, spin reorientation12T, second superconducting pocket. Full text and phase boundaries not audited; associated commentary1117436 is not the original research."}},{"id":"source-a4695ba5bfa789f3","doi":"10.1103/PhysRevLett.99.067006","title":"Superconductivity on the Border of Weak Itinerant Ferromagnetism in UCoGe","year":2007,"url":"https://doi.org/10.1103/PhysRevLett.99.067006","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.99.067006","note":"Huy et al. original abstract: ambient-pressure bulk signatures. Curie3K and best-sample resistive0.8K are distinct; no sample Tc extracted here. Full curves not reviewed."}},{"id":"source-c7f8b99a795aba41","doi":"10.1038/s41586-019-1596-2","title":"Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance","year":2019,"url":"https://doi.org/10.1038/s41586-019-1596-2","publicationStatus":"journal-article","verification":{"scope":"author-manuscript-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1904.00047v1","note":"Pustogow et al. v1 selected pp.4–7; p.7 RF-energy and heating passage visually checked. Zero-strain0.7107T versus strained1.9980T conditions distinguished. Published identity checked; full final-version equivalence and subsequent literature survey not completed."}},{"id":"source-ca6322be2694e430","doi":"10.1126/science.aav8645","title":"Nearly ferromagnetic spin-triplet superconductivity","year":2019,"url":"https://doi.org/10.1126/science.aav8645","publicationStatus":"journal-article","verification":{"scope":"primary-passages-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://vector.umd.edu/images/publications/Science365_6842019.pdf","note":"Ran et al. published paper pp.1–2; p.2 visually checked: maximum measured20T versus extrapolated b-axis40T. Historical spin/pairing interpretations are not adopted as settled consensus. Supplement and raw data not fully audited."}},{"id":"source-df199ac9453d6985","doi":"10.1038/s41586-021-04151-5","title":"Resonance from antiferromagnetic spin fluctuations for superconductivity in UTe2","year":2021,"url":"https://doi.org/10.1038/s41586-021-04151-5","publicationStatus":"journal-article","verification":{"scope":"publisher-passages-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://www.nature.com/articles/s41586-021-04151-5","note":"Duan et al. publisher indexed abstract and Extended Data Fig.7 text checked: AFM resonance and no FM signal above background within specified window. Complete spectra and raw data not reproduced; not an exclusion of all FM fluctuations."}},{"id":"source-15cfc4a2f9587af0","doi":"10.1038/s41467-026-71899-7","title":"Giant transverse magnetic fluctuations at the edge of re-entrant superconductivity in UTe2","year":2026,"url":"https://doi.org/10.1038/s41467-026-71899-7","publicationStatus":"journal-article","verification":{"scope":"publisher-passages-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://www.nature.com/articles/s41467-026-71899-7","note":"Zambra et al., published29April2026. Selected introduction/results, Eq.1 and Fig.1 caption checked:4K normal state, pulsedfields to60T, transverse relative to applied field;300mK phase boundaries reproduced from earlier work. No raw-data or complete spectral/pairing-kernel reconstruction."}}],"claims":[{"id":"scwiki-b4-claim-scope-1","sectionId":"scope","statement":"Ferromagnetic fluctuations describe enhanced magnetic response near zero momentum transfer. They can support odd-parity pairing in specified models, but neither a large uniform susceptibility nor a large upper critical field identifies a superconducting spin state. Magnetic order, correlations, gap symmetry and the pairing interaction are separate questions. Fay and Appel calculated equal-spin p-state pairing from longitudinal fluctuations in an itinerant-electron model; this is a theoretical possibility, not a universal material classification.","evidenceType":"theory","sourceIds":["source-82c04c9061dbca4c"],"conditions":"Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.","verificationScope":"Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition."},{"id":"scwiki-b4-claim-mathematical-assumptions-1","sectionId":"mathematical-assumptions","statement":"For a simple effective exchange $V=-J\\boldsymbol{\\sigma}_1\\cdot\\boldsymbol{\\sigma}_2$ with $J>0$, the spin operator has eigenvalues $-3$ in the singlet and $+1$ in the triplet sector. The corresponding interaction eigenvalues are $+3J$ and $-J$. This independent two-spin illustration assumes a ferromagnetic effective interaction; it does not derive a material's exchange constant or its frequency dependence.","evidenceType":"theory","sourceIds":["source-82c04c9061dbca4c"],"conditions":"Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.","verificationScope":"Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition."},{"id":"scwiki-b4-claim-mathematical-assumptions-2","sectionId":"mathematical-assumptions","statement":"For even-frequency, single-orbital pairing, Fermi antisymmetry requires $\\Delta(\\mathbf k)=-\\Delta^T(-\\mathbf k)$. Writing a triplet matrix as $\\Delta=(\\mathbf d\\cdot\\boldsymbol\\sigma)i\\sigma_y$ gives a symmetric spin matrix, so $\\mathbf d(-\\mathbf k)=-\\mathbf d(\\mathbf k)$. Odd parity here follows from these assumptions. Strong spin–orbit coupling requires a suitable pseudospin description; additional orbital or frequency structure changes this elementary argument.","evidenceType":"theory","sourceIds":["source-82c04c9061dbca4c"],"conditions":"Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.","verificationScope":"Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition."},{"id":"scwiki-b4-claim-mathematical-assumptions-3","sectionId":"mathematical-assumptions","statement":"A second independent illustration uses a circular two-dimensional Fermi surface and an attractive angular kernel $V(\\theta)=-g/(a-b\\cos\\theta)$, with $g>0$ an energy scale and dimensionless $a>b>0$. Its dimensionless Fourier coefficients are","evidenceType":"theory","sourceIds":["source-82c04c9061dbca4c"],"conditions":"Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.","verificationScope":"Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition."},{"id":"scwiki-b4-claim-mathematical-assumptions-4","sectionId":"mathematical-assumptions","statement":"$$C_\\ell=\\int_0^{2\\pi}\\frac{d\\theta}{2\\pi}\\frac{\\cos(\\ell\\theta)}{a-b\\cos\\theta}=\\frac{r^\\ell}{\\sqrt{a^2-b^2}},\\qquad r=\\frac{a-\\sqrt{a^2-b^2}}{b},\\quad \\ell\\geq0.$$","evidenceType":"theory","sourceIds":["source-82c04c9061dbca4c"],"conditions":"Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.","verificationScope":"Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition."},{"id":"scwiki-b4-claim-mathematical-assumptions-5","sectionId":"mathematical-assumptions","statement":"The odd-parity projection $[V(\\theta)-V(\\theta+\\pi)]/2$ retains odd angular harmonics. Since $0<r<1$, the strongest attractive odd harmonic in this example is $\\ell=1$. Numerical quadrature checks the coefficients and projection. Crystal anisotropy, competing interactions and self-energy are absent: this example neither selects a universal p-wave state nor predicts a transition temperature.","evidenceType":"theory","sourceIds":["source-82c04c9061dbca4c"],"conditions":"Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.","verificationScope":"Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition."},{"id":"scwiki-b4-claim-experimental-criteria-1","sectionId":"experimental-criteria","statement":"UGe2 superconductivity was reported in a restricted pressure range within an itinerant-ferromagnetic setting. URhGe supplied ambient-pressure thermodynamic evidence for coexistence; UCoGe supplied ambient-pressure superconducting and ferromagnetic signatures. These primary abstracts support the historical observations, while microscopic volume overlap and a particular gap representation require more than proximity in a phase diagram. Corrected original links replace two inherited citation errors.","evidenceType":"experiment","sourceIds":["source-db2917242488a101","source-15aec45a868e9fb9","source-a4695ba5bfa789f3"],"conditions":"Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.","verificationScope":"Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance."},{"id":"scwiki-b4-claim-experimental-criteria-2","sectionId":"experimental-criteria","statement":"Lévy and colleagues reported a second superconducting pocket in URhGe around spin-axis reorientation near 12 T for field parallel to the crystal b axis, after low-field superconductivity had already disappeared. This is an orientation-specific re-entrant phenomenon. The original abstract suggests a connection to spin rotation; it does not uniquely establish a microscopic pairing kernel.","evidenceType":"experiment","sourceIds":["source-e917adb251d93c59"],"conditions":"Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.","verificationScope":"Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance."},{"id":"scwiki-b4-claim-experimental-criteria-3","sectionId":"experimental-criteria","statement":"Pustogow and colleagues' 2019 Sr2RuO4 author manuscript reports a reduced oxygen-17 Knight shift and strong pulse-energy sensitivity. Its unstrained example uses an in-plane b-axis field of 0.7107 T; 20 mK is the base temperature, not a transition temperature. Selected methods explain why RF heating and different electronic/nuclear relaxation times can make a relaxation check insufficient. These observations challenge older unchanged-susceptibility interpretations; this entry does not infer a unique order parameter from them.","evidenceType":"experiment","sourceIds":["source-c7f8b99a795aba41"],"conditions":"Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.","verificationScope":"Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance."},{"id":"scwiki-b4-claim-experimental-criteria-4","sectionId":"experimental-criteria","statement":"UTe2's discovery paper reported unusual anisotropic critical fields in a paramagnetic normal state. Its page 2 distinguishes a highest measured field of 20 T from an estimated zero-temperature b-axis value of 40 T. The estimate is not a measured endpoint. Historical triplet and ferromagnetic-pairing interpretations remain interpretations; the paper's schematic does not establish one continuous phase diagram shared by different uranium compounds.","evidenceType":"experiment","sourceIds":["source-ca6322be2694e430"],"conditions":"Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.","verificationScope":"Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance."},{"id":"scwiki-b4-claim-experimental-criteria-5","sectionId":"experimental-criteria","statement":"Record field direction, pressure, composition, purity and temperature for each probe. Separate orbital/background contributions and pulse heating in Knight shifts; retain coupling, g-factor, multiband and orbital-limiting assumptions in critical-field comparisons. Combine symmetry-sensitive measurements with magnetic and superconducting volume evidence from the same sample. A bulk signal alone does not identify the mediating fluctuation.","evidenceType":"hypothesis","sourceIds":["source-82c04c9061dbca4c"],"conditions":"Editorial inference criteria; no unique gap symmetry or pairing causation inferred.","verificationScope":"Pinned chapter edited with explicit probe and source boundaries."},{"id":"scwiki-b4-claim-limits-and-disputes-1","sectionId":"limits-and-disputes","statement":"In UTe2, Duan and colleagues reported a neutron resonance associated with antiferromagnetic wavevectors. Their extended-data comparison found no ferromagnetic signal above background in the stated measurement window. This is not absence of every possible small-momentum fluctuation at every field, energy and temperature; a resonance coupled to superconductivity also does not alone prove the direction of causation.","evidenceType":"experiment","sourceIds":["source-df199ac9453d6985"],"conditions":"Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.","verificationScope":"Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance."},{"id":"scwiki-b4-claim-limits-and-disputes-2","sectionId":"limits-and-disputes","statement":"A study published on 29 April 2026 measured UTe2 magnetotropic response in pulsed fields up to 60 T at 4 K, where the studied superconducting phases were suppressed. It inferred enhanced transverse, near-zero-momentum susceptibility at high fields. Here transverse means perpendicular to the applied field, not an off-diagonal tensor component. These normal-state measurements differ from zero/low-field neutron conditions. The displayed 300 mK superconducting boundaries came from earlier work, and 4 K must not be recorded as this experiment's superconducting transition.","evidenceType":"experiment","sourceIds":["source-15cfc4a2f9587af0"],"conditions":"Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.","verificationScope":"Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance."},{"id":"scwiki-b4-claim-limits-and-disputes-3","sectionId":"limits-and-disputes","statement":"Fluctuations can enhance attraction while also renormalizing quasiparticles and affecting pair breaking. Fay–Appel's model gives a maximum followed by a decrease of calculated transition temperature toward the magnetic transition. Neither that model nor the elementary angular projection proves a universal optimum at a ferromagnetic quantum critical point. Longitudinal/transverse spectral weight, spin–orbit coupling and band structure must be specified.","evidenceType":"theory","sourceIds":["source-82c04c9061dbca4c"],"conditions":"Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.","verificationScope":"Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition."},{"id":"scwiki-b4-claim-discovery-implications-1","sectionId":"discovery-implications","statement":"Treat small-momentum magnetic response as a candidate-screening observation. Test whether a proposed kernel simultaneously explains symmetry constraints, normal-state responses and field/pressure trends under matched conditions. Keep negative observations and alternative channels. The recent UTe2 comparison illustrates why distinct experiments should retain their conditions instead of being reduced to a single ferromagnetic or antiferromagnetic label. Selected source passages and model checks here leave full scientific acceptance open.","evidenceType":"hypothesis","sourceIds":["source-82c04c9061dbca4c"],"conditions":"Editorial inference criteria; no unique gap symmetry or pairing causation inferred.","verificationScope":"Pinned chapter edited with explicit probe and source boundaries."}],"corrections":[{"id":"scwiki-b4-correction-1","description":"Replaced inherited UGe2 DOI35050500 with publisher-verified35020500; incorrect target remains unresolved.","sourceId":"source-db2917242488a101","scope":"Bibliographic correction to source lines14/35; no claim about the unresolved DOI target."},{"id":"scwiki-b4-correction-2","description":"Replaced inherited URhGe DOI35065045, an unrelated semiconductor paper, with35098048.","sourceId":"source-15aec45a868e9fb9","scope":"Publisher identity correction to source lines14/36."},{"id":"scwiki-b4-correction-3","description":"Qualified UTe2 magnetic/pairing labels by field and probe, and separated40T estimate from20T measured range.","sourceId":"source-ca6322be2694e430","scope":"Selected historical primary passage; contemporary2021/2026 conditions separately described."}],"equationChecks":[{"targetSectionId":"mathematical-assumptions","tex":"-J\\boldsymbol{\\sigma}_1\\cdot\\boldsymbol{\\sigma}_2","check":"Four explicit spin-sector eigenvectors."},{"targetSectionId":"mathematical-assumptions","tex":"\\Delta(\\mathbf k)=-\\Delta^T(-\\mathbf k)","check":"Three complex triplet matrices satisfy transpose and odd-momentum antisymmetry."},{"targetSectionId":"mathematical-assumptions","tex":"C_\\ell","check":"20 angular coefficient and odd-projection quadratures against analytic values."}],"materialCases":[{"id":"scwiki-b4-material-1","formula":"UTe2","sample":"Zambra et al.2026 single-crystal magnetotropic response","recordType":"experiment","tcK":null,"reportedTemperatureK":4,"temperatureKind":"normal-state","temperatureNote":"4 K normal-state measurement;300mK superconducting boundaries in comparison figure are from earlier work","criterion":"magnetotropic-response","pressureRegime":"unknown","pressureGPa":null,"equilibrium":"unknown","probe":"Pulsed fields up to60T; field rotations in ac/bc planes; transverse susceptibility inferred from magnetotropic response","sourceId":"source-15cfc4a2f9587af0","aliases":["heavy fermion","heavy fermions"]}],"body":"## Scope\n\nFerromagnetic fluctuations describe enhanced magnetic response near zero momentum transfer. They can support odd-parity pairing in specified models, but neither a large uniform susceptibility nor a large upper critical field identifies a superconducting spin state. Magnetic order, correlations, gap symmetry and the pairing interaction are separate questions. Fay and Appel calculated equal-spin p-state pairing from longitudinal fluctuations in an itinerant-electron model; this is a theoretical possibility, not a universal material classification. [R1](#ref-source-82c04c9061dbca4c)\n\n## Mathematical assumptions\n\nFor a simple effective exchange $V=-J\\boldsymbol{\\sigma}_1\\cdot\\boldsymbol{\\sigma}_2$ with $J>0$, the spin operator has eigenvalues $-3$ in the singlet and $+1$ in the triplet sector. The corresponding interaction eigenvalues are $+3J$ and $-J$. This independent two-spin illustration assumes a ferromagnetic effective interaction; it does not derive a material's exchange constant or its frequency dependence.\n\nFor even-frequency, single-orbital pairing, Fermi antisymmetry requires $\\Delta(\\mathbf k)=-\\Delta^T(-\\mathbf k)$. Writing a triplet matrix as $\\Delta=(\\mathbf d\\cdot\\boldsymbol\\sigma)i\\sigma_y$ gives a symmetric spin matrix, so $\\mathbf d(-\\mathbf k)=-\\mathbf d(\\mathbf k)$. Odd parity here follows from these assumptions. Strong spin–orbit coupling requires a suitable pseudospin description; additional orbital or frequency structure changes this elementary argument.\n\nA second independent illustration uses a circular two-dimensional Fermi surface and an attractive angular kernel $V(\\theta)=-g/(a-b\\cos\\theta)$, with $g>0$ an energy scale and dimensionless $a>b>0$. Its dimensionless Fourier coefficients are\n\n$$C_\\ell=\\int_0^{2\\pi}\\frac{d\\theta}{2\\pi}\\frac{\\cos(\\ell\\theta)}{a-b\\cos\\theta}=\\frac{r^\\ell}{\\sqrt{a^2-b^2}},\\qquad r=\\frac{a-\\sqrt{a^2-b^2}}{b},\\quad \\ell\\geq0.$$\n\nThe odd-parity projection $[V(\\theta)-V(\\theta+\\pi)]/2$ retains odd angular harmonics. Since $0<r<1$, the strongest attractive odd harmonic in this example is $\\ell=1$. Numerical quadrature checks the coefficients and projection. Crystal anisotropy, competing interactions and self-energy are absent: this example neither selects a universal p-wave state nor predicts a transition temperature.\n\n## Experimental criteria\n\nUGe2 superconductivity was reported in a restricted pressure range within an itinerant-ferromagnetic setting. URhGe supplied ambient-pressure thermodynamic evidence for coexistence; UCoGe supplied ambient-pressure superconducting and ferromagnetic signatures. These primary abstracts support the historical observations, while microscopic volume overlap and a particular gap representation require more than proximity in a phase diagram. Corrected original links replace two inherited citation errors. [R2](#ref-source-db2917242488a101) [R3](#ref-source-15aec45a868e9fb9) [R5](#ref-source-a4695ba5bfa789f3)\n\nLévy and colleagues reported a second superconducting pocket in URhGe around spin-axis reorientation near 12 T for field parallel to the crystal b axis, after low-field superconductivity had already disappeared. This is an orientation-specific re-entrant phenomenon. The original abstract suggests a connection to spin rotation; it does not uniquely establish a microscopic pairing kernel. [R4](#ref-source-e917adb251d93c59)\n\nPustogow and colleagues' 2019 Sr2RuO4 author manuscript reports a reduced oxygen-17 Knight shift and strong pulse-energy sensitivity. Its unstrained example uses an in-plane b-axis field of 0.7107 T; 20 mK is the base temperature, not a transition temperature. Selected methods explain why RF heating and different electronic/nuclear relaxation times can make a relaxation check insufficient. These observations challenge older unchanged-susceptibility interpretations; this entry does not infer a unique order parameter from them. [R6](#ref-source-c7f8b99a795aba41)\n\nUTe2's discovery paper reported unusual anisotropic critical fields in a paramagnetic normal state. Its page 2 distinguishes a highest measured field of 20 T from an estimated zero-temperature b-axis value of 40 T. The estimate is not a measured endpoint. Historical triplet and ferromagnetic-pairing interpretations remain interpretations; the paper's schematic does not establish one continuous phase diagram shared by different uranium compounds. [R7](#ref-source-ca6322be2694e430)\n\nRecord field direction, pressure, composition, purity and temperature for each probe. Separate orbital/background contributions and pulse heating in Knight shifts; retain coupling, g-factor, multiband and orbital-limiting assumptions in critical-field comparisons. Combine symmetry-sensitive measurements with magnetic and superconducting volume evidence from the same sample. A bulk signal alone does not identify the mediating fluctuation.\n\n## Limits and disputes\n\nIn UTe2, Duan and colleagues reported a neutron resonance associated with antiferromagnetic wavevectors. Their extended-data comparison found no ferromagnetic signal above background in the stated measurement window. This is not absence of every possible small-momentum fluctuation at every field, energy and temperature; a resonance coupled to superconductivity also does not alone prove the direction of causation. [R8](#ref-source-df199ac9453d6985)\n\nA study published on 29 April 2026 measured UTe2 magnetotropic response in pulsed fields up to 60 T at 4 K, where the studied superconducting phases were suppressed. It inferred enhanced transverse, near-zero-momentum susceptibility at high fields. Here transverse means perpendicular to the applied field, not an off-diagonal tensor component. These normal-state measurements differ from zero/low-field neutron conditions. The displayed 300 mK superconducting boundaries came from earlier work, and 4 K must not be recorded as this experiment's superconducting transition. [R9](#ref-source-15cfc4a2f9587af0)\n\nFluctuations can enhance attraction while also renormalizing quasiparticles and affecting pair breaking. Fay–Appel's model gives a maximum followed by a decrease of calculated transition temperature toward the magnetic transition. Neither that model nor the elementary angular projection proves a universal optimum at a ferromagnetic quantum critical point. Longitudinal/transverse spectral weight, spin–orbit coupling and band structure must be specified. [R1](#ref-source-82c04c9061dbca4c)\n\n## Discovery implications\n\nTreat small-momentum magnetic response as a candidate-screening observation. Test whether a proposed kernel simultaneously explains symmetry constraints, normal-state responses and field/pressure trends under matched conditions. Keep negative observations and alternative channels. The recent UTe2 comparison illustrates why distinct experiments should retain their conditions instead of being reduced to a single ferromagnetic or antiferromagnetic label. Selected source passages and model checks here leave full scientific acceptance open.\n\n## Related topics\n\nCompare [antiferromagnetic spin fluctuations](/scwiki/spin-fluctuation-pairing), [Eliashberg theory](/scwiki/eliashberg-strong-coupling) and the keyword catalogue for gap symmetry, critical fields and uranium material families.","revision":"f1debd312a212d40bd97","sections":[{"id":"scope","title":"Scope","markdown":"## Scope\n\nFerromagnetic fluctuations describe enhanced magnetic response near zero momentum transfer. They can support odd-parity pairing in specified models, but neither a large uniform susceptibility nor a large upper critical field identifies a superconducting spin state. Magnetic order, correlations, gap symmetry and the pairing interaction are separate questions. Fay and Appel calculated equal-spin p-state pairing from longitudinal fluctuations in an itinerant-electron model; this is a theoretical possibility, not a universal material classification. [R1](#ref-source-82c04c9061dbca4c)"},{"id":"mathematical-assumptions","title":"Mathematical assumptions","markdown":"## Mathematical assumptions\n\nFor a simple effective exchange $V=-J\\boldsymbol{\\sigma}_1\\cdot\\boldsymbol{\\sigma}_2$ with $J>0$, the spin operator has eigenvalues $-3$ in the singlet and $+1$ in the triplet sector. The corresponding interaction eigenvalues are $+3J$ and $-J$. This independent two-spin illustration assumes a ferromagnetic effective interaction; it does not derive a material's exchange constant or its frequency dependence.\n\nFor even-frequency, single-orbital pairing, Fermi antisymmetry requires $\\Delta(\\mathbf k)=-\\Delta^T(-\\mathbf k)$. Writing a triplet matrix as $\\Delta=(\\mathbf d\\cdot\\boldsymbol\\sigma)i\\sigma_y$ gives a symmetric spin matrix, so $\\mathbf d(-\\mathbf k)=-\\mathbf d(\\mathbf k)$. Odd parity here follows from these assumptions. Strong spin–orbit coupling requires a suitable pseudospin description; additional orbital or frequency structure changes this elementary argument.\n\nA second independent illustration uses a circular two-dimensional Fermi surface and an attractive angular kernel $V(\\theta)=-g/(a-b\\cos\\theta)$, with $g>0$ an energy scale and dimensionless $a>b>0$. Its dimensionless Fourier coefficients are\n\n$$C_\\ell=\\int_0^{2\\pi}\\frac{d\\theta}{2\\pi}\\frac{\\cos(\\ell\\theta)}{a-b\\cos\\theta}=\\frac{r^\\ell}{\\sqrt{a^2-b^2}},\\qquad r=\\frac{a-\\sqrt{a^2-b^2}}{b},\\quad \\ell\\geq0.$$\n\nThe odd-parity projection $[V(\\theta)-V(\\theta+\\pi)]/2$ retains odd angular harmonics. Since $0<r<1$, the strongest attractive odd harmonic in this example is $\\ell=1$. Numerical quadrature checks the coefficients and projection. Crystal anisotropy, competing interactions and self-energy are absent: this example neither selects a universal p-wave state nor predicts a transition temperature."},{"id":"experimental-criteria","title":"Experimental criteria","markdown":"## Experimental criteria\n\nUGe2 superconductivity was reported in a restricted pressure range within an itinerant-ferromagnetic setting. URhGe supplied ambient-pressure thermodynamic evidence for coexistence; UCoGe supplied ambient-pressure superconducting and ferromagnetic signatures. These primary abstracts support the historical observations, while microscopic volume overlap and a particular gap representation require more than proximity in a phase diagram. Corrected original links replace two inherited citation errors. [R2](#ref-source-db2917242488a101) [R3](#ref-source-15aec45a868e9fb9) [R5](#ref-source-a4695ba5bfa789f3)\n\nLévy and colleagues reported a second superconducting pocket in URhGe around spin-axis reorientation near 12 T for field parallel to the crystal b axis, after low-field superconductivity had already disappeared. This is an orientation-specific re-entrant phenomenon. The original abstract suggests a connection to spin rotation; it does not uniquely establish a microscopic pairing kernel. [R4](#ref-source-e917adb251d93c59)\n\nPustogow and colleagues' 2019 Sr2RuO4 author manuscript reports a reduced oxygen-17 Knight shift and strong pulse-energy sensitivity. Its unstrained example uses an in-plane b-axis field of 0.7107 T; 20 mK is the base temperature, not a transition temperature. Selected methods explain why RF heating and different electronic/nuclear relaxation times can make a relaxation check insufficient. These observations challenge older unchanged-susceptibility interpretations; this entry does not infer a unique order parameter from them. [R6](#ref-source-c7f8b99a795aba41)\n\nUTe2's discovery paper reported unusual anisotropic critical fields in a paramagnetic normal state. Its page 2 distinguishes a highest measured field of 20 T from an estimated zero-temperature b-axis value of 40 T. The estimate is not a measured endpoint. Historical triplet and ferromagnetic-pairing interpretations remain interpretations; the paper's schematic does not establish one continuous phase diagram shared by different uranium compounds. [R7](#ref-source-ca6322be2694e430)\n\nRecord field direction, pressure, composition, purity and temperature for each probe. Separate orbital/background contributions and pulse heating in Knight shifts; retain coupling, g-factor, multiband and orbital-limiting assumptions in critical-field comparisons. Combine symmetry-sensitive measurements with magnetic and superconducting volume evidence from the same sample. A bulk signal alone does not identify the mediating fluctuation."},{"id":"limits-and-disputes","title":"Limits and disputes","markdown":"## Limits and disputes\n\nIn UTe2, Duan and colleagues reported a neutron resonance associated with antiferromagnetic wavevectors. Their extended-data comparison found no ferromagnetic signal above background in the stated measurement window. This is not absence of every possible small-momentum fluctuation at every field, energy and temperature; a resonance coupled to superconductivity also does not alone prove the direction of causation. [R8](#ref-source-df199ac9453d6985)\n\nA study published on 29 April 2026 measured UTe2 magnetotropic response in pulsed fields up to 60 T at 4 K, where the studied superconducting phases were suppressed. It inferred enhanced transverse, near-zero-momentum susceptibility at high fields. Here transverse means perpendicular to the applied field, not an off-diagonal tensor component. These normal-state measurements differ from zero/low-field neutron conditions. The displayed 300 mK superconducting boundaries came from earlier work, and 4 K must not be recorded as this experiment's superconducting transition. [R9](#ref-source-15cfc4a2f9587af0)\n\nFluctuations can enhance attraction while also renormalizing quasiparticles and affecting pair breaking. Fay–Appel's model gives a maximum followed by a decrease of calculated transition temperature toward the magnetic transition. Neither that model nor the elementary angular projection proves a universal optimum at a ferromagnetic quantum critical point. Longitudinal/transverse spectral weight, spin–orbit coupling and band structure must be specified. [R1](#ref-source-82c04c9061dbca4c)"},{"id":"discovery-implications","title":"Discovery implications","markdown":"## Discovery implications\n\nTreat small-momentum magnetic response as a candidate-screening observation. Test whether a proposed kernel simultaneously explains symmetry constraints, normal-state responses and field/pressure trends under matched conditions. Keep negative observations and alternative channels. The recent UTe2 comparison illustrates why distinct experiments should retain their conditions instead of being reduced to a single ferromagnetic or antiferromagnetic label. Selected source passages and model checks here leave full scientific acceptance open."},{"id":"related-topics","title":"Related topics","markdown":"## Related topics\n\nCompare [antiferromagnetic spin fluctuations](/scwiki/spin-fluctuation-pairing), [Eliashberg theory](/scwiki/eliashberg-strong-coupling) and the keyword catalogue for gap symmetry, critical fields and uranium material families."},{"id":"material-cases","title":"Material cases","markdown":"## Material cases\n\nValues describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.\n\n| Material and sample | Reported temperature | Criterion | Probe | Pressure | State | Evidence |\n|---|---|---|---|---|---|---|\n| UTe2 (aliases: heavy fermion, heavy fermions); Zambra et al.2026 single-crystal magnetotropic response | 4 K; 4 K normal-state measurement;300mK superconducting boundaries in comparison figure are from earlier work | normal-state; magnetotropic-response | Pulsed fields up to60T; field rotations in ac/bc planes; transverse susceptibility inferred from magnetotropic response | unknown; unknown GPa | unknown | experiment; [source](#ref-source-15cfc4a2f9587af0) |"}],"markdown":"# Ferromagnetic fluctuations and odd-parity pairing\n\nSeparate triplet-pairing models from magnetic evidence, compare uranium compounds under stated conditions, and trace the limits of Knight shifts and critical fields.\n\n## Scope\n\nFerromagnetic fluctuations describe enhanced magnetic response near zero momentum transfer. They can support odd-parity pairing in specified models, but neither a large uniform susceptibility nor a large upper critical field identifies a superconducting spin state. Magnetic order, correlations, gap symmetry and the pairing interaction are separate questions. Fay and Appel calculated equal-spin p-state pairing from longitudinal fluctuations in an itinerant-electron model; this is a theoretical possibility, not a universal material classification. [R1](#ref-source-82c04c9061dbca4c)\n\n## Mathematical assumptions\n\nFor a simple effective exchange $V=-J\\boldsymbol{\\sigma}_1\\cdot\\boldsymbol{\\sigma}_2$ with $J>0$, the spin operator has eigenvalues $-3$ in the singlet and $+1$ in the triplet sector. The corresponding interaction eigenvalues are $+3J$ and $-J$. This independent two-spin illustration assumes a ferromagnetic effective interaction; it does not derive a material's exchange constant or its frequency dependence.\n\nFor even-frequency, single-orbital pairing, Fermi antisymmetry requires $\\Delta(\\mathbf k)=-\\Delta^T(-\\mathbf k)$. Writing a triplet matrix as $\\Delta=(\\mathbf d\\cdot\\boldsymbol\\sigma)i\\sigma_y$ gives a symmetric spin matrix, so $\\mathbf d(-\\mathbf k)=-\\mathbf d(\\mathbf k)$. Odd parity here follows from these assumptions. Strong spin–orbit coupling requires a suitable pseudospin description; additional orbital or frequency structure changes this elementary argument.\n\nA second independent illustration uses a circular two-dimensional Fermi surface and an attractive angular kernel $V(\\theta)=-g/(a-b\\cos\\theta)$, with $g>0$ an energy scale and dimensionless $a>b>0$. Its dimensionless Fourier coefficients are\n\n$$C_\\ell=\\int_0^{2\\pi}\\frac{d\\theta}{2\\pi}\\frac{\\cos(\\ell\\theta)}{a-b\\cos\\theta}=\\frac{r^\\ell}{\\sqrt{a^2-b^2}},\\qquad r=\\frac{a-\\sqrt{a^2-b^2}}{b},\\quad \\ell\\geq0.$$\n\nThe odd-parity projection $[V(\\theta)-V(\\theta+\\pi)]/2$ retains odd angular harmonics. Since $0<r<1$, the strongest attractive odd harmonic in this example is $\\ell=1$. Numerical quadrature checks the coefficients and projection. Crystal anisotropy, competing interactions and self-energy are absent: this example neither selects a universal p-wave state nor predicts a transition temperature.\n\n## Experimental criteria\n\nUGe2 superconductivity was reported in a restricted pressure range within an itinerant-ferromagnetic setting. URhGe supplied ambient-pressure thermodynamic evidence for coexistence; UCoGe supplied ambient-pressure superconducting and ferromagnetic signatures. These primary abstracts support the historical observations, while microscopic volume overlap and a particular gap representation require more than proximity in a phase diagram. Corrected original links replace two inherited citation errors. [R2](#ref-source-db2917242488a101) [R3](#ref-source-15aec45a868e9fb9) [R5](#ref-source-a4695ba5bfa789f3)\n\nLévy and colleagues reported a second superconducting pocket in URhGe around spin-axis reorientation near 12 T for field parallel to the crystal b axis, after low-field superconductivity had already disappeared. This is an orientation-specific re-entrant phenomenon. The original abstract suggests a connection to spin rotation; it does not uniquely establish a microscopic pairing kernel. [R4](#ref-source-e917adb251d93c59)\n\nPustogow and colleagues' 2019 Sr2RuO4 author manuscript reports a reduced oxygen-17 Knight shift and strong pulse-energy sensitivity. Its unstrained example uses an in-plane b-axis field of 0.7107 T; 20 mK is the base temperature, not a transition temperature. Selected methods explain why RF heating and different electronic/nuclear relaxation times can make a relaxation check insufficient. These observations challenge older unchanged-susceptibility interpretations; this entry does not infer a unique order parameter from them. [R6](#ref-source-c7f8b99a795aba41)\n\nUTe2's discovery paper reported unusual anisotropic critical fields in a paramagnetic normal state. Its page 2 distinguishes a highest measured field of 20 T from an estimated zero-temperature b-axis value of 40 T. The estimate is not a measured endpoint. Historical triplet and ferromagnetic-pairing interpretations remain interpretations; the paper's schematic does not establish one continuous phase diagram shared by different uranium compounds. [R7](#ref-source-ca6322be2694e430)\n\nRecord field direction, pressure, composition, purity and temperature for each probe. Separate orbital/background contributions and pulse heating in Knight shifts; retain coupling, g-factor, multiband and orbital-limiting assumptions in critical-field comparisons. Combine symmetry-sensitive measurements with magnetic and superconducting volume evidence from the same sample. A bulk signal alone does not identify the mediating fluctuation.\n\n## Limits and disputes\n\nIn UTe2, Duan and colleagues reported a neutron resonance associated with antiferromagnetic wavevectors. Their extended-data comparison found no ferromagnetic signal above background in the stated measurement window. This is not absence of every possible small-momentum fluctuation at every field, energy and temperature; a resonance coupled to superconductivity also does not alone prove the direction of causation. [R8](#ref-source-df199ac9453d6985)\n\nA study published on 29 April 2026 measured UTe2 magnetotropic response in pulsed fields up to 60 T at 4 K, where the studied superconducting phases were suppressed. It inferred enhanced transverse, near-zero-momentum susceptibility at high fields. Here transverse means perpendicular to the applied field, not an off-diagonal tensor component. These normal-state measurements differ from zero/low-field neutron conditions. The displayed 300 mK superconducting boundaries came from earlier work, and 4 K must not be recorded as this experiment's superconducting transition. [R9](#ref-source-15cfc4a2f9587af0)\n\nFluctuations can enhance attraction while also renormalizing quasiparticles and affecting pair breaking. Fay–Appel's model gives a maximum followed by a decrease of calculated transition temperature toward the magnetic transition. Neither that model nor the elementary angular projection proves a universal optimum at a ferromagnetic quantum critical point. Longitudinal/transverse spectral weight, spin–orbit coupling and band structure must be specified. [R1](#ref-source-82c04c9061dbca4c)\n\n## Discovery implications\n\nTreat small-momentum magnetic response as a candidate-screening observation. Test whether a proposed kernel simultaneously explains symmetry constraints, normal-state responses and field/pressure trends under matched conditions. Keep negative observations and alternative channels. The recent UTe2 comparison illustrates why distinct experiments should retain their conditions instead of being reduced to a single ferromagnetic or antiferromagnetic label. Selected source passages and model checks here leave full scientific acceptance open.\n\n## Related topics\n\nCompare [antiferromagnetic spin fluctuations](/scwiki/spin-fluctuation-pairing), [Eliashberg theory](/scwiki/eliashberg-strong-coupling) and the keyword catalogue for gap symmetry, critical fields and uranium material families.\n\n## Material cases\n\nValues describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.\n\n| Material and sample | Reported temperature | Criterion | Probe | Pressure | State | Evidence |\n|---|---|---|---|---|---|---|\n| UTe2 (aliases: heavy fermion, heavy fermions); Zambra et al.2026 single-crystal magnetotropic response | 4 K; 4 K normal-state measurement;300mK superconducting boundaries in comparison figure are from earlier work | normal-state; magnetotropic-response | Pulsed fields up to60T; field rotations in ac/bc planes; transverse susceptibility inferred from magnetotropic response | unknown; unknown GPa | unknown | experiment; [source](#ref-source-15cfc4a2f9587af0) |\n\n## Sources\n\n### Ref source-82c04c9061dbca4c\n\n[R1] Coexistence of p-state superconductivity and itinerant ferromagnetism (1980). [DOI 10.1103/PhysRevB.22.3173](https://doi.org/10.1103/PhysRevB.22.3173). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. Fay–Appel abstract: longitudinal-fluctuation equal-spin model and nonmonotonic calculated Tc. Full calculation not read or reproduced. The article two-spin/angular examples are independent editorial illustrations, not equations attributed to this paper.\n### Ref source-db2917242488a101\n\n[R2] Superconductivity on the border of itinerant-electron ferromagnetism in UGe2 (2000). [DOI 10.1038/35020500](https://doi.org/10.1038/35020500). Publication: journal-article. Check: publisher-abstract-scoped-check. Checked: 2026-10-18. Saxena et al. publisher indexed abstract and Crossref identity checked. Restricted-pressure superconductivity; original curves and microscopic coexistence not audited. Replaces inherited DOI35050500 whose actual target remained unresolved.\n### Ref source-15aec45a868e9fb9\n\n[R3] Coexistence of superconductivity and ferromagnetism in URhGe (2001). [DOI 10.1038/35098048](https://doi.org/10.1038/35098048). Publication: journal-article. Check: publisher-abstract-scoped-check. Checked: 2026-10-18. Aoki et al. publisher indexed abstract: ambient-pressure thermodynamic evidence, with qualified triplet interpretation. Corrects inherited35065045, which identifies unrelated semiconductor melting research. Full data not reanalyzed.\n### Ref source-e917adb251d93c59\n\n[R4] Magnetic Field-Induced Superconductivity in the Ferromagnet URhGe (2005). [DOI 10.1126/science.1115498](https://doi.org/10.1126/science.1115498). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. Levy et al. original abstract as indexed by PubMed: field along b, spin reorientation12T, second superconducting pocket. Full text and phase boundaries not audited; associated commentary1117436 is not the original research.\n### Ref source-a4695ba5bfa789f3\n\n[R5] Superconductivity on the Border of Weak Itinerant Ferromagnetism in UCoGe (2007). [DOI 10.1103/PhysRevLett.99.067006](https://doi.org/10.1103/PhysRevLett.99.067006). Publication: journal-article. Check: publisher-abstract-scoped-check. Checked: 2026-10-18. Huy et al. original abstract: ambient-pressure bulk signatures. Curie3K and best-sample resistive0.8K are distinct; no sample Tc extracted here. Full curves not reviewed.\n### Ref source-c7f8b99a795aba41\n\n[R6] Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance (2019). [DOI 10.1038/s41586-019-1596-2](https://doi.org/10.1038/s41586-019-1596-2). Publication: journal-article. Check: author-manuscript-scoped-check. Checked: 2026-10-18. Pustogow et al. v1 selected pp.4–7; p.7 RF-energy and heating passage visually checked. Zero-strain0.7107T versus strained1.9980T conditions distinguished. Published identity checked; full final-version equivalence and subsequent literature survey not completed.\n### Ref source-ca6322be2694e430\n\n[R7] Nearly ferromagnetic spin-triplet superconductivity (2019). [DOI 10.1126/science.aav8645](https://doi.org/10.1126/science.aav8645). Publication: journal-article. Check: primary-passages-scoped-check. Checked: 2026-10-18. Ran et al. published paper pp.1–2; p.2 visually checked: maximum measured20T versus extrapolated b-axis40T. Historical spin/pairing interpretations are not adopted as settled consensus. Supplement and raw data not fully audited.\n### Ref source-df199ac9453d6985\n\n[R8] Resonance from antiferromagnetic spin fluctuations for superconductivity in UTe2 (2021). [DOI 10.1038/s41586-021-04151-5](https://doi.org/10.1038/s41586-021-04151-5). Publication: journal-article. Check: publisher-passages-scoped-check. Checked: 2026-10-18. Duan et al. publisher indexed abstract and Extended Data Fig.7 text checked: AFM resonance and no FM signal above background within specified window. Complete spectra and raw data not reproduced; not an exclusion of all FM fluctuations.\n### Ref source-15cfc4a2f9587af0\n\n[R9] Giant transverse magnetic fluctuations at the edge of re-entrant superconductivity in UTe2 (2026). [DOI 10.1038/s41467-026-71899-7](https://doi.org/10.1038/s41467-026-71899-7). Publication: journal-article. Check: publisher-passages-scoped-check. Checked: 2026-10-18. Zambra et al., published29April2026. Selected introduction/results, Eq.1 and Fig.1 caption checked:4K normal state, pulsedfields to60T, transverse relative to applied field;300mK phase boundaries reproduced from earlier work. No raw-data or complete spectral/pairing-kernel reconstruction.\n\n## Claim evidence and conditions\n\n### scwiki-b4-claim-scope-1\n\nFerromagnetic fluctuations describe enhanced magnetic response near zero momentum transfer. They can support odd-parity pairing in specified models, but neither a large uniform susceptibility nor a large upper critical field identifies a superconducting spin state. Magnetic order, correlations, gap symmetry and the pairing interaction are separate questions. Fay and Appel calculated equal-spin p-state pairing from longitudinal fluctuations in an itinerant-electron model; this is a theoretical possibility, not a universal material classification.\n\nEvidence: theory. Sources: source-82c04c9061dbca4c.\n\nConditions: Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.\n\nVerification scope: Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition.\n\n### scwiki-b4-claim-mathematical-assumptions-1\n\nFor a simple effective exchange $V=-J\\boldsymbol{\\sigma}_1\\cdot\\boldsymbol{\\sigma}_2$ with $J>0$, the spin operator has eigenvalues $-3$ in the singlet and $+1$ in the triplet sector. The corresponding interaction eigenvalues are $+3J$ and $-J$. This independent two-spin illustration assumes a ferromagnetic effective interaction; it does not derive a material's exchange constant or its frequency dependence.\n\nEvidence: theory. Sources: source-82c04c9061dbca4c.\n\nConditions: Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.\n\nVerification scope: Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition.\n\n### scwiki-b4-claim-mathematical-assumptions-2\n\nFor even-frequency, single-orbital pairing, Fermi antisymmetry requires $\\Delta(\\mathbf k)=-\\Delta^T(-\\mathbf k)$. Writing a triplet matrix as $\\Delta=(\\mathbf d\\cdot\\boldsymbol\\sigma)i\\sigma_y$ gives a symmetric spin matrix, so $\\mathbf d(-\\mathbf k)=-\\mathbf d(\\mathbf k)$. Odd parity here follows from these assumptions. Strong spin–orbit coupling requires a suitable pseudospin description; additional orbital or frequency structure changes this elementary argument.\n\nEvidence: theory. Sources: source-82c04c9061dbca4c.\n\nConditions: Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.\n\nVerification scope: Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition.\n\n### scwiki-b4-claim-mathematical-assumptions-3\n\nA second independent illustration uses a circular two-dimensional Fermi surface and an attractive angular kernel $V(\\theta)=-g/(a-b\\cos\\theta)$, with $g>0$ an energy scale and dimensionless $a>b>0$. Its dimensionless Fourier coefficients are\n\nEvidence: theory. Sources: source-82c04c9061dbca4c.\n\nConditions: Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.\n\nVerification scope: Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition.\n\n### scwiki-b4-claim-mathematical-assumptions-4\n\n$$C_\\ell=\\int_0^{2\\pi}\\frac{d\\theta}{2\\pi}\\frac{\\cos(\\ell\\theta)}{a-b\\cos\\theta}=\\frac{r^\\ell}{\\sqrt{a^2-b^2}},\\qquad r=\\frac{a-\\sqrt{a^2-b^2}}{b},\\quad \\ell\\geq0.$$\n\nEvidence: theory. Sources: source-82c04c9061dbca4c.\n\nConditions: Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.\n\nVerification scope: Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition.\n\n### scwiki-b4-claim-mathematical-assumptions-5\n\nThe odd-parity projection $[V(\\theta)-V(\\theta+\\pi)]/2$ retains odd angular harmonics. Since $0<r<1$, the strongest attractive odd harmonic in this example is $\\ell=1$. Numerical quadrature checks the coefficients and projection. Crystal anisotropy, competing interactions and self-energy are absent: this example neither selects a universal p-wave state nor predicts a transition temperature.\n\nEvidence: theory. Sources: source-82c04c9061dbca4c.\n\nConditions: Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.\n\nVerification scope: Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition.\n\n### scwiki-b4-claim-experimental-criteria-1\n\nUGe2 superconductivity was reported in a restricted pressure range within an itinerant-ferromagnetic setting. URhGe supplied ambient-pressure thermodynamic evidence for coexistence; UCoGe supplied ambient-pressure superconducting and ferromagnetic signatures. These primary abstracts support the historical observations, while microscopic volume overlap and a particular gap representation require more than proximity in a phase diagram. Corrected original links replace two inherited citation errors.\n\nEvidence: experiment. Sources: source-db2917242488a101, source-15aec45a868e9fb9, source-a4695ba5bfa789f3.\n\nConditions: Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.\n\nVerification scope: Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance.\n\n### scwiki-b4-claim-experimental-criteria-2\n\nLévy and colleagues reported a second superconducting pocket in URhGe around spin-axis reorientation near 12 T for field parallel to the crystal b axis, after low-field superconductivity had already disappeared. This is an orientation-specific re-entrant phenomenon. The original abstract suggests a connection to spin rotation; it does not uniquely establish a microscopic pairing kernel.\n\nEvidence: experiment. Sources: source-e917adb251d93c59.\n\nConditions: Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.\n\nVerification scope: Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance.\n\n### scwiki-b4-claim-experimental-criteria-3\n\nPustogow and colleagues' 2019 Sr2RuO4 author manuscript reports a reduced oxygen-17 Knight shift and strong pulse-energy sensitivity. Its unstrained example uses an in-plane b-axis field of 0.7107 T; 20 mK is the base temperature, not a transition temperature. Selected methods explain why RF heating and different electronic/nuclear relaxation times can make a relaxation check insufficient. These observations challenge older unchanged-susceptibility interpretations; this entry does not infer a unique order parameter from them.\n\nEvidence: experiment. Sources: source-c7f8b99a795aba41.\n\nConditions: Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.\n\nVerification scope: Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance.\n\n### scwiki-b4-claim-experimental-criteria-4\n\nUTe2's discovery paper reported unusual anisotropic critical fields in a paramagnetic normal state. Its page 2 distinguishes a highest measured field of 20 T from an estimated zero-temperature b-axis value of 40 T. The estimate is not a measured endpoint. Historical triplet and ferromagnetic-pairing interpretations remain interpretations; the paper's schematic does not establish one continuous phase diagram shared by different uranium compounds.\n\nEvidence: experiment. Sources: source-ca6322be2694e430.\n\nConditions: Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.\n\nVerification scope: Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance.\n\n### scwiki-b4-claim-experimental-criteria-5\n\nRecord field direction, pressure, composition, purity and temperature for each probe. Separate orbital/background contributions and pulse heating in Knight shifts; retain coupling, g-factor, multiband and orbital-limiting assumptions in critical-field comparisons. Combine symmetry-sensitive measurements with magnetic and superconducting volume evidence from the same sample. A bulk signal alone does not identify the mediating fluctuation.\n\nEvidence: hypothesis. Sources: source-82c04c9061dbca4c.\n\nConditions: Editorial inference criteria; no unique gap symmetry or pairing causation inferred.\n\nVerification scope: Pinned chapter edited with explicit probe and source boundaries.\n\n### scwiki-b4-claim-limits-and-disputes-1\n\nIn UTe2, Duan and colleagues reported a neutron resonance associated with antiferromagnetic wavevectors. Their extended-data comparison found no ferromagnetic signal above background in the stated measurement window. This is not absence of every possible small-momentum fluctuation at every field, energy and temperature; a resonance coupled to superconductivity also does not alone prove the direction of causation.\n\nEvidence: experiment. Sources: source-df199ac9453d6985.\n\nConditions: Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.\n\nVerification scope: Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance.\n\n### scwiki-b4-claim-limits-and-disputes-2\n\nA study published on 29 April 2026 measured UTe2 magnetotropic response in pulsed fields up to 60 T at 4 K, where the studied superconducting phases were suppressed. It inferred enhanced transverse, near-zero-momentum susceptibility at high fields. Here transverse means perpendicular to the applied field, not an off-diagonal tensor component. These normal-state measurements differ from zero/low-field neutron conditions. The displayed 300 mK superconducting boundaries came from earlier work, and 4 K must not be recorded as this experiment's superconducting transition.\n\nEvidence: experiment. Sources: source-15cfc4a2f9587af0.\n\nConditions: Source-specific temperature, pressure, field direction, frequency/momentum and sample limits; measured endpoints separated from extrapolations.\n\nVerification scope: Selected primary abstracts/passages as detailed in references; no raw-data reproduction or full scientific acceptance.\n\n### scwiki-b4-claim-limits-and-disputes-3\n\nFluctuations can enhance attraction while also renormalizing quasiparticles and affecting pair breaking. Fay–Appel's model gives a maximum followed by a decrease of calculated transition temperature toward the magnetic transition. Neither that model nor the elementary angular projection proves a universal optimum at a ferromagnetic quantum critical point. Longitudinal/transverse spectral weight, spin–orbit coupling and band structure must be specified.\n\nEvidence: theory. Sources: source-82c04c9061dbca4c.\n\nConditions: Effective ferromagnetic two-spin exchange; even-frequency single-orbital pairing; circular 2D Fermi surface a>b>0. Fay–Appel statements separately abstract-scoped.\n\nVerification scope: Independent spin/parity/quadrature checks; not reproduction of a material calculation or measured transition.\n\n### scwiki-b4-claim-discovery-implications-1\n\nTreat small-momentum magnetic response as a candidate-screening observation. Test whether a proposed kernel simultaneously explains symmetry constraints, normal-state responses and field/pressure trends under matched conditions. Keep negative observations and alternative channels. The recent UTe2 comparison illustrates why distinct experiments should retain their conditions instead of being reduced to a single ferromagnetic or antiferromagnetic label. Selected source passages and model checks here leave full scientific acceptance open.\n\nEvidence: hypothesis. Sources: source-82c04c9061dbca4c.\n\nConditions: Editorial inference criteria; no unique gap symmetry or pairing causation inferred.\n\nVerification scope: Pinned chapter edited with explicit probe and source boundaries.\n\n## Revision and review\n\nArticle ID: scwiki-b4. Revision: f1debd312a212d40bd97. Updated: 2026-10-18T12:19:32.196903+00:00.\n\nReview scope: Selected primary abstracts, Sr2RuO4/Ran manuscript passages,2021 neutron extended-data text and2026 UTe2 publisher passages; independent spin, parity and angular-projection checks.\n\n- Full scientific acceptance, raw-data review and a comprehensive current literature survey remain open.\n- Author-manuscript checks do not establish final-version equivalence; several historical references are abstract-scoped.\n- Independent illustrations do not predict a material transition or establish a pairing kernel.\n- UTe2 4 K is a normal-state measurement temperature, not a superconducting transition. Different field/probe windows cannot be conflated.\n\nSource: [B_mechanisms/B4_spin_fluctuation_FM.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/B_mechanisms/B4_spin_fluctuation_FM.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 0440ac552dfaad93ae3a3e94517ee603ba14626f3af2745ec52dfd14479b39f7.\n","wordCount":862}}