{"schemaVersion":"1.0","revision":"fa961ab24277a91737c3","article":{"id":"scwiki-c6","topicId":"scwiki-topic-c6","code":"C6","slug":"f-wave","title":"f-wave pairing: crystal representations and nodal evidence","summary":"Compare higher-order odd pairing, UPt3 candidate states and the limits of Kerr, nodal and multicomponent evidence.","aliases":["f-wave","f wave","E2u","E1u","UPt3","UTe2","Kerr rotation","tropical line nodes","higher angular momentum"],"evidenceTypes":["theory","experiment"],"relatedCodes":["C5","C4","C3","B4","B11"],"editorialStatus":"scoped-check","contentState":"pilot","modifiedAt":"2026-10-19","review":{"checkedAt":"2026-10-19","scope":"Selected UPt3 Kerr and planar-state manuscripts, historical transport/review abstracts, UTe2 contrasting Kerr reports, and independent harmonic/nodal/free-energy checks. Follow-up: version-pinned Tsutsumi II.A/Table I text, explicit local derivation attribution, real/complex admixture and endpoint checks. Selected Vorontsov-Vekhter model/Results and Micklitz-Norman accepted manuscript I/II.A-B; three frozen claims reassessed with explicit model and space-group limits.","limitations":["Full scientific acceptance, complete final-version comparisons and raw-data review remain open.","Spherical harmonics and chosen free-energy coefficients are not a multiband material calculation.","Null and trained Kerr signals are scoped to sample,protocol and sensitivity; no universal exclusion claimed.","No definite f-wave assignment or bulk topological invariant follows from these local checks."]},"provenance":{"repository":"JackZH26/jzis-superconductivity","commit":"0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf","path":"C_pairing_symmetry/C6_f_wave.md","sha256":"788c392c4ae92a9c8c716666621564e27ca9690a53c336507be4f70358d766c4","sourceId":"file-bb58c569bf3bf5d6"},"sourceMap":[{"startLine":1,"endLine":15,"targetSections":["scope"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; candidate symmetry and evidence boundaries expanded with primary checks."},{"startLine":16,"endLine":28,"targetSections":["mathematical-assumptions"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; candidate symmetry and evidence boundaries expanded with primary checks."},{"startLine":29,"endLine":38,"targetSections":["experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; candidate symmetry and evidence boundaries expanded with primary checks."},{"startLine":39,"endLine":46,"targetSections":["limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; candidate symmetry and evidence boundaries expanded with primary checks."},{"startLine":47,"endLine":58,"targetSections":["experimental-criteria","discovery-implications"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; candidate symmetry and evidence boundaries expanded with primary checks."},{"startLine":59,"endLine":72,"targetSections":["related-topics","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; candidate symmetry and evidence boundaries expanded with primary checks."}],"references":[{"id":"source-8d1b859d5ffcad9f","doi":"10.1103/RevModPhys.63.239","title":"Phenomenological theory of unconventional superconductivity","year":1991,"url":"https://doi.org/10.1103/RevModPhys.63.239","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.63.239","note":"Identity and abstract scope checked. Full review not obtained; explicit angular illustrations are independently evaluated, not attributed as a reproduction of this review."}},{"id":"source-a329f7df31c7d3e9","doi":"10.1103/RevModPhys.74.235","title":"The superconducting phases of UPt3","year":2002,"url":"https://doi.org/10.1103/RevModPhys.74.235","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.74.235","note":"Historical E1g/E2u comparison and unresolved magnetic coupling; full review not obtained."}},{"id":"source-d3edee404071ac81","doi":"10.1126/science.1248552","title":"Observation of broken time-reversal symmetry in the heavy-fermion superconductor UPt3","year":2014,"url":"https://doi.org/10.1126/science.1248552","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1410.1482v1","note":"Selected pp.1–6, Fig.2 on p.11 visually checked, and sample/apparatus passages on p.13. Published July2014; author upload October2014. Heating and training limits retained; full final-version and raw-data comparison open."}},{"id":"source-31577c7cb2ae0ac8","doi":"10.1103/PhysRevLett.108.157002","title":"Twofold Spontaneous Symmetry Breaking in the Heavy-Fermion Superconductor UPt3","year":2012,"url":"https://doi.org/10.1103/PhysRevLett.108.157002","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.157002","note":"Journal abstract B/C coverage differs from arxiv1107.3082v1 A/B/C wording. E1u is the authors interpretation; full version/data comparison open."}},{"id":"source-34df31af237b76a7","doi":"10.1143/JPSJ.81.074717","title":"A Spin Triplet Superconductor UPt3","year":2012,"url":"https://doi.org/10.1143/JPSJ.81.074717","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1202.1894v2","note":"Selected pp.1–3, Fig.1 and basis table; p.2 visually checked. Version2 July2012 marked published version by authors. Weak-SOC planar construction and competing scenarios retained; Eilenberger results not reproduced."}},{"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-5ff091aacae25bb0","doi":"10.1126/science.abb0272","title":"Multicomponent superconducting order parameter in UTe2","year":2021,"url":"https://doi.org/10.1126/science.abb0272","publicationStatus":"journal-article","verification":{"scope":"selected-first-release-text-check","checkedAt":"2026-10-18","evidenceUrl":"https://vector.umd.edu/images/publications/Science_abb02722021.pdf","note":"Selected first-release15July2021 pp.1–2: reported Kerr,training and calorimetric interpretation. Final pagination/version and full supplement not checked."}},{"id":"source-dd4e897d833ce69c","doi":"10.1103/PhysRevX.13.041019","title":"Fate of Time-Reversal Symmetry Breaking in UTe2","year":2023,"url":"https://doi.org/10.1103/PhysRevX.13.041019","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/2305.00589v1","note":"Selected pp.2–3 and methods pp.5–6; Fig.1 p.3 visually checked. Kerr sample set distinguished from susceptibility set. Final publisher abstract checked; full version comparison open."}},{"id":"source-6c82ca23c0fda096","doi":"10.1103/PhysRevLett.96.237001","title":"Nodal Structure of Quasi-Two-Dimensional Superconductors Probed by a Magnetic Field","year":2006,"url":"https://doi.org/10.1103/PhysRevLett.96.237001","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/cond-mat/0601126v1","note":"Selected pp. 1–3 and Fig. 2 visually checked: finite-energy vortex scattering, anisotropy reversal and BPT/low-field/Zeeman limitations. Publisher identity checked. Full journal comparison and vortex calculation not reproduced."}},{"id":"source-591b5b95b965e1af","doi":"10.1103/PhysRevB.95.024508","title":"Nodal lines and nodal loops in nonsymmorphic odd-parity superconductors","year":2017,"url":"https://doi.org/10.1103/PhysRevB.95.024508","publicationStatus":"journal-article","verification":{"scope":"selected-accepted-manuscript-check","checkedAt":"2026-10-19","evidenceUrl":"https://link.aps.org/accepted/10.1103/PhysRevB.95.024508","note":"Accepted manuscript dated23Dec2016, publisher identity17Jan2017; selected printedpp1-3 (PDFpp2-4), introduction andII.A-B. Space-group/fractional-phase and spin-orbit/time-reversal limits checked in text; full band/topology derivations and final-version comparison not reproduced."}}],"claims":[{"id":"scwiki-c6-claim-scope-1","sectionId":"scope","statement":"In a rotationally invariant continuum, f-wave means orbital angular momentum $l=3$. In a crystal, specify the point group, spin or pseudospin structure and basis functions; nonsymmorphic materials also require the fractional translations of the space group. An odd cubic polynomial is not automatically a pure $l=3$ harmonic, and a node inferred from a response is not an irreducible-representation label. The discussion below assumes even-frequency pairing unless stated otherwise. Exchange symmetry must include momentum, spin, orbital indices and frequency before drawing parity conclusions in a multiorbital material.","evidenceType":"theory","sourceIds":["source-34df31af237b76a7","source-591b5b95b965e1af"],"conditions":"Continuum l assumes rotational invariance; crystal representations include spin/pseudospin and, when relevant, nonsymmorphic fractional translations. Even-frequency default; full index exchange required.","verificationScope":"Selected versioned primary text and local algebra checked 2026-10-19; full material theory, final-version comparison and independent scientific review remain open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"Tsutsumi supplies the harmonic/classification context; Micklitz-Norman supplies space-group and band-pairing context. Harmonic decomposition and separable exchange-sign checks are local explanatory deductions."},{"id":"scwiki-c6-claim-scope-2","sectionId":"scope","statement":"UPt3 is an important test case with competing descriptions of its superconducting phases. The historical E2u framework and a proposed E1u planar state organize different observations, but a candidate label is not a unique experimental determination. UTe2 has a different crystal symmetry and must not inherit UPt3's representation merely because both are uranium compounds.","evidenceType":"theory","sourceIds":["source-a329f7df31c7d3e9","source-d3edee404071ac81","source-34df31af237b76a7","source-5ff091aacae25bb0"],"conditions":"Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c6-claim-mathematical-assumptions-1","sectionId":"mathematical-assumptions","statement":"In the following local calculation, an illustrative hexagonal E2u orbital doublet is $[k_z(k_x^2-k_y^2),2k_zk_xk_y]$. With a fixed spin vector along z, its equal-amplitude chiral combination is proportional to $k_z(k_x+ik_y)^2$. It is odd under inversion. Under rotation by angle $\\alpha$ about z, its orbital factor acquires $e^{2i\\alpha}$: the continuum harmonic has $l=3$ but axial projection $m=2$, not $m=3$. Local transformation checks and finite-difference Laplacians verify these polynomial identities; they do not reproduce a cited material calculation. In a lattice, continuous angular momentum is only a comparison, not an exact conserved quantum number.","evidenceType":"theory","sourceIds":["source-8d1b859d5ffcad9f","source-d3edee404071ac81"],"conditions":"Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations."},{"id":"scwiki-c6-claim-mathematical-assumptions-2","sectionId":"mathematical-assumptions","statement":"On a unit sphere this chiral form has magnitude $|\\cos\\theta|\\sin^2\\theta$: an equatorial line zero and quadratic polar zeros. A planar E1u illustration uses two orthogonal spin vectors multiplying $k_x(5k_z^2-1)$ and $k_y(5k_z^2-1)$. Its magnitude is $\\sin\\theta|5\\cos^2\\theta-1|$, with polar point zeros and two horizontal zeros at $\\cos^2\\theta=1/5$. These are model angles, not measured node coordinates. Tsutsumi and colleagues use a related planar construction in their weak-spin-orbit scenario, with field-dependent spin orientation.","evidenceType":"theory","sourceIds":["source-34df31af237b76a7"],"conditions":"Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations."},{"id":"scwiki-c6-claim-mathematical-assumptions-3","sectionId":"mathematical-assumptions","statement":"The constant one in the spherical E1u expressions represents $k_x^2+k_y^2+k_z^2$. Restoring it makes each component a homogeneous harmonic cubic. The lower-order pair $(k_x,k_y)$ belongs to the same orbital doublet. In a local illustration with fixed orthogonal spin vectors and a unit spherical Fermi surface, add a real coefficient $a$ times this pair. The gap magnitude becomes $\\sin\\theta|5\\cos^2\\theta-1+a|$. Two distinct horizontal nodal rings occur only for $-4<a<1$, at $k_z^2=(1-a)/5$. At $a=1$ they merge into a quadratic equatorial zero; at $a=-4$ they meet the poles, giving cubic polar zeros. For real $a$ outside this interval only the polar zeros remain. This admixture is our illustration, not a fitted state reported by Tsutsumi and colleagues. For a complex coefficient, the same local model instead has squared magnitude $\\sin^2\\theta[(5\\cos^2\\theta-1+\\operatorname{Re}a)^2+(\\operatorname{Im}a)^2]$. Any nonzero imaginary part removes its horizontal zeros while retaining the poles. Odd parity and the orbital-doublet transformation rule survive, but time-reversal symmetry need not: a complex mixture cannot simply be called the same symmetry state as the real mixture. Symmetry-allowed basis mixing alone says neither which coefficients minimize a material's free energy nor where its measured nodes lie.","evidenceType":"theory","sourceIds":["source-34df31af237b76a7"],"conditions":"Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations."},{"id":"scwiki-c6-claim-mathematical-assumptions-4","sectionId":"mathematical-assumptions","statement":"A separate local geometric calculation compares the polar zeros of the unmixed models. Restricting to small polar caps, count the fraction of a unit sphere with imposed gap below $\\epsilon$. Doubling a sufficiently small cutoff approximately doubles this fraction for the quadratic E2u zero, but quadruples it for the linear E1u zero. Eight cap calculations verify these limits. This is a geometric illustration, not a density-of-states or thermal-conductivity calculation; real velocities, impurity scattering and other nodes still contribute to a measured power law.","evidenceType":"theory","sourceIds":["source-8d1b859d5ffcad9f","source-34df31af237b76a7"],"conditions":"Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations."},{"id":"scwiki-c6-claim-mathematical-assumptions-5","sectionId":"mathematical-assumptions","statement":"Our separate two-component free-energy illustration shows how split transitions can arise without determining a material representation. Set $u=|\\eta_1|^2$, $v=|\\eta_2|^2$ and relative phase $\\delta$, with dimensionless free energy $F=(t-1)u+(t-0.8)v+(u^2+v^2)/2+(0.4+0.15\\cos2\\delta)uv$. When both amplitudes are present, $\\delta=\\pm\\pi/2$ minimizes the phase term. The first onset is $t=1$; the second is $t=(0.8-0.25)/(1-0.25)\\simeq0.7333$. Five independent positive-amplitude grid comparisons agree with the analytic minima. These coefficients are chosen, not fitted to UPt3 or UTe2.","evidenceType":"theory","sourceIds":["source-8d1b859d5ffcad9f","source-a329f7df31c7d3e9"],"conditions":"Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations."},{"id":"scwiki-c6-claim-experimental-criteria-1","sectionId":"experimental-criteria","statement":"The 2002 Joynt–Taillefer review weighs UPt3's phase diagram, nodal responses and magnetic coupling. Its abstract favors E1g or E2u on balance while acknowledging that neither explains all observations. That historical conclusion is not a current universal assignment. Machida and colleagues' later angle-dependent thermal-conductivity study instead interprets the B/C-phase data using E1u. The 2011 author abstract and 2012 journal abstract differ in the stated phase coverage; this entry uses the latter and does not claim a complete version comparison.","evidenceType":"experiment","sourceIds":["source-a329f7df31c7d3e9","source-31577c7cb2ae0ac8"],"conditions":"Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c6-claim-experimental-criteria-2","sectionId":"experimental-criteria","statement":"Schemm and colleagues reported a UPt3 Kerr signal below about 0.46 K, separated from an upper susceptibility onset near 0.55 K. Their 1550 nm optical probe required an estimated local-heating allowance up to 50 mK; bulk heating inferred from mutual inductance was much smaller. Cooling in either sign of a 50 G field and then measuring with the field removed trained the signal's sign. These observations support additional time-reversal breaking in the lower-temperature phase within the experiment's controls; the optical signal alone does not select E2u. The Kerr onset is not the upper superconducting transition.","evidenceType":"experiment","sourceIds":["source-d3edee404071ac81"],"conditions":"Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c6-claim-experimental-criteria-3","sectionId":"experimental-criteria","statement":"For interpretation, record field direction, heat-current direction, temperature and response model. Vorontsov and Vekhter find that minima can become maxima along a node in a quasi-two-dimensional d-wave vortex calculation as temperature or field changes. Their treatment includes vortex scattering and finite-energy thermal weighting; it neglects Zeeman splitting and has low-field limitations. This is a methodological warning, not a calculated f-wave response for UPt3. Comparing multiple axes and independent probes remains editorial guidance: the apparent success of one chosen response model does not uniquely select a material representation.","evidenceType":"theory","sourceIds":["source-6c82ca23c0fda096"],"conditions":"Quasi-two-dimensional d-wave vortex model, rotated in-plane field and specified heat-current direction; BPT approximation, no Zeeman splitting. No f-wave or UPt3 response inferred.","verificationScope":"Selected versioned primary text and local algebra checked 2026-10-19; full material theory, final-version comparison and independent scientific review remain open.","sourceRelationship":"context-for-editorial-analysis","derivationNote":"Specific sign-reversal examples are reported theory; comparison recommendations are editorial. No new experimental result or source calculation reproduced."},{"id":"scwiki-c6-claim-experimental-criteria-4","sectionId":"experimental-criteria","statement":"As editorial guidance for a null Kerr result, record illumination, surface, residual field, cooldown history and statistical sensitivity. Independent noise and domain signals add in variance, $\\sigma_{\\rm total}^2=\\sigma_{\\rm noise}^2+\\sigma_{\\rm signal}^2$. A local constructed example, not a reanalysis of the cited measurements, verifies this identity for uncorrelated sequences. The equation requires its independence assumption; a measured absence of increased scatter is a constraint at finite sensitivity, not proof that all possible microscopic time-reversal breaking is absent.","evidenceType":"theory","sourceIds":["source-dd4e897d833ce69c"],"conditions":"Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations."},{"id":"scwiki-c6-claim-limits-and-disputes-1","sectionId":"limits-and-disputes","statement":"Hayes and colleagues reported UTe2 Kerr and calorimetric signatures supporting a multicomponent state. Their low-field warmups were compared with measurements following positive and negative training fields; the interpretation includes domains. Ajeesh and colleagues later found no spontaneous zero-field Kerr signal in four single-transition samples spanning two growth methods. Their heat-pulse procedure repeatedly crossed the transition, and trained signals varied with position and sample. The selected manuscript's optical and susceptibility sample sets are different: one must not attribute the separate double-transition susceptibility sample to the Kerr set. Retain this disagreement and its conditions rather than declaring all UTe2 samples chiral or all possible states excluded.","evidenceType":"experiment","sourceIds":["source-5ff091aacae25bb0","source-dd4e897d833ce69c"],"conditions":"Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c6-claim-limits-and-disputes-2","sectionId":"limits-and-disputes","statement":"Orthorhombic D2h has one-dimensional ordinary point-group representations, unlike hexagonal E2u. A complex combination of distinct components can still break time reversal if the energetics permit it; multidimensional point-group representations are not a universal prerequisite. Spin degeneracy, orbital structure and accidental near-degeneracy must be specified. UTe2's discovery evidence for unusual spin pairing does not independently choose an f-wave basis or a unique multicomponent state.","evidenceType":"theory","sourceIds":["source-8d1b859d5ffcad9f","source-ca6322be2694e430","source-5ff091aacae25bb0"],"conditions":"Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c6-claim-limits-and-disputes-3","sectionId":"limits-and-disputes","statement":"The local spherical polynomial examples omit the multiband and nonsymmorphic structure relevant to UPt3. Micklitz and Norman analyze its P63/mmc space group and explain how fractional-translation phases can alter odd-parity line-node constraints on a Brillouin-zone face intersected by the Fermi surface. The answer depends on spin-orbit coupling, time-reversal symmetry and band pairing; a point-group polynomial alone does not contain these ingredients. Their analysis supplies a missing theoretical context, not a unique experimental pairing assignment. Our chosen Ginzburg–Landau example does not reproduce the material phase diagram, and this entry supplies no calculated surface spectrum or bulk topological invariant.","evidenceType":"theory","sourceIds":["source-591b5b95b965e1af"],"conditions":"UPt3 space-group context and Fermi-surface intersection; spin-orbit, time-reversal and interband assumptions must be supplied. Local sphere and chosen free energy omit these inputs.","verificationScope":"Selected versioned primary text and local algebra checked 2026-10-19; full material theory, final-version comparison and independent scientific review remain open.","sourceRelationship":"context-for-editorial-analysis","derivationNote":"The primary theoretical study explains missing crystal/band ingredients; the statement about this entry is an audit of its limited local calculations."},{"id":"scwiki-c6-claim-discovery-implications-1","sectionId":"discovery-implications","statement":"The following is editorial comparison guidance: specify the crystal symmetry and basis first. Then test whether one proposed state explains phase splitting, directional nodes, spin response and symmetry-breaking observations under compatible conditions. Preserve null results, competing interpretations and source-version differences alongside positive evidence. Full scientific acceptance, complete final-version comparisons and raw-data review remain open; the model calculations here constrain language and logic, not the pairing state of a new material.","evidenceType":"theory","sourceIds":["source-8d1b859d5ffcad9f","source-d3edee404071ac81","source-34df31af237b76a7","source-dd4e897d833ce69c"],"conditions":"Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"context-for-editorial-analysis","derivationNote":"Editorial comparison guidance grounded in cited examples; not a universal measurement protocol established by these papers."}],"corrections":[{"id":"scwiki-c6-correction-1","description":"Separated historical E1g/E2u review conclusions from the later E1u proposal and explicit source-version differences.","sourceId":"source-31577c7cb2ae0ac8","scope":"Historical abstracts and selected theory passages."},{"id":"scwiki-c6-correction-2","description":"Kerr onset and upper superconducting onset retain different probes and heating uncertainty.","sourceId":"source-d3edee404071ac81","scope":"Selected methods and Fig.2 visual inspection."},{"id":"scwiki-c6-correction-3","description":"UTe2 multicomponent/TRSB narrative now includes zero-field null results and distinguishes Kerr and susceptibility sample sets.","sourceId":"source-dd4e897d833ce69c","scope":"Selected author methods/results, not universal absence of TRSB."},{"id":"scwiki-c6-correction-4","description":"Limits the horizontal-node formula to real mixing coefficients, states endpoint and complex-coefficient cases, and separates local derivations from source results.","sourceId":"source-34df31af237b76a7","scope":"Local spherical model and source-attribution clarification; no new material pairing assignment."},{"id":"scwiki-c6-correction-5","description":"Adds space-group constraints beyond point-group labels and identifies the nodal-response counterexample as d-wave theory rather than an UPt3 f-wave measurement.","sourceId":"source-591b5b95b965e1af","scope":"Selected primary theoretical passages; no measured pairing assignment or band/topology reproduction."}],"equationChecks":[{"targetSectionId":"mathematical-assumptions","tex":"k_z(k_x+ik_y)^2","check":"Four inversion/rotation/harmonic checks; axial phase winding2 differs from total l3."},{"targetSectionId":"mathematical-assumptions","tex":"\\sin\\theta|5\\cos^2\\theta-1|","check":"Five spherical evaluations, five allowed orbital admixtures and eight polar-cap fractions."},{"targetSectionId":"mathematical-assumptions","tex":"F=(t-1)u+(t-0.8)v+(u^2+v^2)/2+(0.4+0.15\\cos2\\delta)uv","check":"Five analytic minima versus independent nonnegative grids; chosen coefficients, not material fits."}],"materialCases":[{"id":"scwiki-c6-material-1","formula":"UPt3","sample":"Schemm et al. optical Kerr measurement","recordType":"experiment","tcK":null,"reportedTemperatureK":0.46,"temperatureKind":"unassigned","temperatureNote":"Approximate Kerr onset with up to50mK local-heating allowance; not upper superconducting transition","criterion":"polar-kerr-onset","pressureRegime":"unknown","pressureGPa":null,"equilibrium":"unknown","probe":"1550nm polar Kerr; field-off warmup after zero-field or±50G training; separate upper susceptibility onset","sourceId":"source-d3edee404071ac81","aliases":["heavy fermion","heavy fermions"]}],"body":"## Scope\n\nIn a rotationally invariant continuum, f-wave means orbital angular momentum $l=3$. In a crystal, specify the point group, spin or pseudospin structure and basis functions; nonsymmorphic materials also require the fractional translations of the space group. An odd cubic polynomial is not automatically a pure $l=3$ harmonic, and a node inferred from a response is not an irreducible-representation label. The discussion below assumes even-frequency pairing unless stated otherwise. Exchange symmetry must include momentum, spin, orbital indices and frequency before drawing parity conclusions in a multiorbital material. [R5](#ref-source-34df31af237b76a7) [R10](#ref-source-591b5b95b965e1af)\n\nUPt3 is an important test case with competing descriptions of its superconducting phases. The historical E2u framework and a proposed E1u planar state organize different observations, but a candidate label is not a unique experimental determination. UTe2 has a different crystal symmetry and must not inherit UPt3's representation merely because both are uranium compounds. [R2](#ref-source-a329f7df31c7d3e9) [R3](#ref-source-d3edee404071ac81) [R5](#ref-source-34df31af237b76a7) [R7](#ref-source-5ff091aacae25bb0)\n\n## Mathematical assumptions\n\nIn the following local calculation, an illustrative hexagonal E2u orbital doublet is $[k_z(k_x^2-k_y^2),2k_zk_xk_y]$. With a fixed spin vector along z, its equal-amplitude chiral combination is proportional to $k_z(k_x+ik_y)^2$. It is odd under inversion. Under rotation by angle $\\alpha$ about z, its orbital factor acquires $e^{2i\\alpha}$: the continuum harmonic has $l=3$ but axial projection $m=2$, not $m=3$. Local transformation checks and finite-difference Laplacians verify these polynomial identities; they do not reproduce a cited material calculation. In a lattice, continuous angular momentum is only a comparison, not an exact conserved quantum number. [R1](#ref-source-8d1b859d5ffcad9f) [R3](#ref-source-d3edee404071ac81)\n\nOn a unit sphere this chiral form has magnitude $|\\cos\\theta|\\sin^2\\theta$: an equatorial line zero and quadratic polar zeros. A planar E1u illustration uses two orthogonal spin vectors multiplying $k_x(5k_z^2-1)$ and $k_y(5k_z^2-1)$. Its magnitude is $\\sin\\theta|5\\cos^2\\theta-1|$, with polar point zeros and two horizontal zeros at $\\cos^2\\theta=1/5$. These are model angles, not measured node coordinates. Tsutsumi and colleagues use a related planar construction in their weak-spin-orbit scenario, with field-dependent spin orientation. [R5](#ref-source-34df31af237b76a7)\n\nThe constant one in the spherical E1u expressions represents $k_x^2+k_y^2+k_z^2$. Restoring it makes each component a homogeneous harmonic cubic. The lower-order pair $(k_x,k_y)$ belongs to the same orbital doublet. In a local illustration with fixed orthogonal spin vectors and a unit spherical Fermi surface, add a **real** coefficient $a$ times this pair. The gap magnitude becomes $\\sin\\theta|5\\cos^2\\theta-1+a|$. Two distinct horizontal nodal rings occur only for $-4<a<1$, at $k_z^2=(1-a)/5$. At $a=1$ they merge into a quadratic equatorial zero; at $a=-4$ they meet the poles, giving cubic polar zeros. For real $a$ outside this interval only the polar zeros remain. This admixture is our illustration, not a fitted state reported by Tsutsumi and colleagues. [R5](#ref-source-34df31af237b76a7)\n\nFor a **complex** coefficient, the same local model instead has squared magnitude $\\sin^2\\theta[(5\\cos^2\\theta-1+\\operatorname{Re}a)^2+(\\operatorname{Im}a)^2]$. Any nonzero imaginary part removes its horizontal zeros while retaining the poles. Odd parity and the orbital-doublet transformation rule survive, but time-reversal symmetry need not: a complex mixture cannot simply be called the same symmetry state as the real mixture. Symmetry-allowed basis mixing alone says neither which coefficients minimize a material's free energy nor where its measured nodes lie. [R5](#ref-source-34df31af237b76a7)\n\nA separate local geometric calculation compares the polar zeros of the unmixed models. Restricting to small polar caps, count the fraction of a unit sphere with imposed gap below $\\epsilon$. Doubling a sufficiently small cutoff approximately doubles this fraction for the quadratic E2u zero, but quadruples it for the linear E1u zero. Eight cap calculations verify these limits. This is a geometric illustration, not a density-of-states or thermal-conductivity calculation; real velocities, impurity scattering and other nodes still contribute to a measured power law. [R1](#ref-source-8d1b859d5ffcad9f) [R5](#ref-source-34df31af237b76a7)\n\nOur separate two-component free-energy illustration shows how split transitions can arise without determining a material representation. Set $u=|\\eta_1|^2$, $v=|\\eta_2|^2$ and relative phase $\\delta$, with dimensionless free energy $F=(t-1)u+(t-0.8)v+(u^2+v^2)/2+(0.4+0.15\\cos2\\delta)uv$. When both amplitudes are present, $\\delta=\\pm\\pi/2$ minimizes the phase term. The first onset is $t=1$; the second is $t=(0.8-0.25)/(1-0.25)\\simeq0.7333$. Five independent positive-amplitude grid comparisons agree with the analytic minima. These coefficients are chosen, not fitted to UPt3 or UTe2. [R1](#ref-source-8d1b859d5ffcad9f) [R2](#ref-source-a329f7df31c7d3e9)\n\n## Experimental criteria\n\nThe 2002 Joynt–Taillefer review weighs UPt3's phase diagram, nodal responses and magnetic coupling. Its abstract favors E1g or E2u on balance while acknowledging that neither explains all observations. That historical conclusion is not a current universal assignment. Machida and colleagues' later angle-dependent thermal-conductivity study instead interprets the B/C-phase data using E1u. The 2011 author abstract and 2012 journal abstract differ in the stated phase coverage; this entry uses the latter and does not claim a complete version comparison. [R2](#ref-source-a329f7df31c7d3e9) [R4](#ref-source-31577c7cb2ae0ac8)\n\nSchemm and colleagues reported a UPt3 Kerr signal below about 0.46 K, separated from an upper susceptibility onset near 0.55 K. Their 1550 nm optical probe required an estimated local-heating allowance up to 50 mK; bulk heating inferred from mutual inductance was much smaller. Cooling in either sign of a 50 G field and then measuring with the field removed trained the signal's sign. These observations support additional time-reversal breaking in the lower-temperature phase within the experiment's controls; the optical signal alone does not select E2u. The Kerr onset is not the upper superconducting transition. [R3](#ref-source-d3edee404071ac81)\n\nFor interpretation, record field direction, heat-current direction, temperature and response model. Vorontsov and Vekhter find that minima can become maxima along a node in a quasi-two-dimensional d-wave vortex calculation as temperature or field changes. Their treatment includes vortex scattering and finite-energy thermal weighting; it neglects Zeeman splitting and has low-field limitations. This is a methodological warning, not a calculated f-wave response for UPt3. Comparing multiple axes and independent probes remains editorial guidance: the apparent success of one chosen response model does not uniquely select a material representation. [R9](#ref-source-6c82ca23c0fda096)\n\nAs editorial guidance for a null Kerr result, record illumination, surface, residual field, cooldown history and statistical sensitivity. Independent noise and domain signals add in variance, $\\sigma_{\\rm total}^2=\\sigma_{\\rm noise}^2+\\sigma_{\\rm signal}^2$. A local constructed example, not a reanalysis of the cited measurements, verifies this identity for uncorrelated sequences. The equation requires its independence assumption; a measured absence of increased scatter is a constraint at finite sensitivity, not proof that all possible microscopic time-reversal breaking is absent. [R8](#ref-source-dd4e897d833ce69c)\n\n## Limits and disputes\n\nHayes and colleagues reported UTe2 Kerr and calorimetric signatures supporting a multicomponent state. Their low-field warmups were compared with measurements following positive and negative training fields; the interpretation includes domains. Ajeesh and colleagues later found no spontaneous zero-field Kerr signal in four single-transition samples spanning two growth methods. Their heat-pulse procedure repeatedly crossed the transition, and trained signals varied with position and sample. The selected manuscript's optical and susceptibility sample sets are different: one must not attribute the separate double-transition susceptibility sample to the Kerr set. Retain this disagreement and its conditions rather than declaring all UTe2 samples chiral or all possible states excluded. [R7](#ref-source-5ff091aacae25bb0) [R8](#ref-source-dd4e897d833ce69c)\n\nOrthorhombic D2h has one-dimensional ordinary point-group representations, unlike hexagonal E2u. A complex combination of distinct components can still break time reversal if the energetics permit it; multidimensional point-group representations are not a universal prerequisite. Spin degeneracy, orbital structure and accidental near-degeneracy must be specified. UTe2's discovery evidence for unusual spin pairing does not independently choose an f-wave basis or a unique multicomponent state. [R1](#ref-source-8d1b859d5ffcad9f) [R6](#ref-source-ca6322be2694e430) [R7](#ref-source-5ff091aacae25bb0)\n\nThe local spherical polynomial examples omit the multiband and nonsymmorphic structure relevant to UPt3. Micklitz and Norman analyze its P63/mmc space group and explain how fractional-translation phases can alter odd-parity line-node constraints on a Brillouin-zone face intersected by the Fermi surface. The answer depends on spin-orbit coupling, time-reversal symmetry and band pairing; a point-group polynomial alone does not contain these ingredients. Their analysis supplies a missing theoretical context, not a unique experimental pairing assignment. Our chosen Ginzburg–Landau example does not reproduce the material phase diagram, and this entry supplies no calculated surface spectrum or bulk topological invariant. [R10](#ref-source-591b5b95b965e1af)\n\n## Discovery implications\n\nThe following is editorial comparison guidance: specify the crystal symmetry and basis first. Then test whether one proposed state explains phase splitting, directional nodes, spin response and symmetry-breaking observations under compatible conditions. Preserve null results, competing interpretations and source-version differences alongside positive evidence. Full scientific acceptance, complete final-version comparisons and raw-data review remain open; the model calculations here constrain language and logic, not the pairing state of a new material. [R1](#ref-source-8d1b859d5ffcad9f) [R3](#ref-source-d3edee404071ac81) [R5](#ref-source-34df31af237b76a7) [R8](#ref-source-dd4e897d833ce69c)\n\n## Related topics\n\nCompare [p-wave spin and chirality](/scwiki/p-wave), [dxy directional evidence](/scwiki/dxy-pairing), [d-wave phase tests](/scwiki/d-wave), [ferromagnetic fluctuations](/scwiki/ferromagnetic-fluctuations), and [topological superconductivity](/scwiki/topological-superconductivity).","revision":"57f5d9992e70254ac7a6","sections":[{"id":"scope","title":"Scope","markdown":"## Scope\n\nIn a rotationally invariant continuum, f-wave means orbital angular momentum $l=3$. In a crystal, specify the point group, spin or pseudospin structure and basis functions; nonsymmorphic materials also require the fractional translations of the space group. An odd cubic polynomial is not automatically a pure $l=3$ harmonic, and a node inferred from a response is not an irreducible-representation label. The discussion below assumes even-frequency pairing unless stated otherwise. Exchange symmetry must include momentum, spin, orbital indices and frequency before drawing parity conclusions in a multiorbital material. [R5](#ref-source-34df31af237b76a7) [R10](#ref-source-591b5b95b965e1af)\n\nUPt3 is an important test case with competing descriptions of its superconducting phases. The historical E2u framework and a proposed E1u planar state organize different observations, but a candidate label is not a unique experimental determination. UTe2 has a different crystal symmetry and must not inherit UPt3's representation merely because both are uranium compounds. [R2](#ref-source-a329f7df31c7d3e9) [R3](#ref-source-d3edee404071ac81) [R5](#ref-source-34df31af237b76a7) [R7](#ref-source-5ff091aacae25bb0)"},{"id":"mathematical-assumptions","title":"Mathematical assumptions","markdown":"## Mathematical assumptions\n\nIn the following local calculation, an illustrative hexagonal E2u orbital doublet is $[k_z(k_x^2-k_y^2),2k_zk_xk_y]$. With a fixed spin vector along z, its equal-amplitude chiral combination is proportional to $k_z(k_x+ik_y)^2$. It is odd under inversion. Under rotation by angle $\\alpha$ about z, its orbital factor acquires $e^{2i\\alpha}$: the continuum harmonic has $l=3$ but axial projection $m=2$, not $m=3$. Local transformation checks and finite-difference Laplacians verify these polynomial identities; they do not reproduce a cited material calculation. In a lattice, continuous angular momentum is only a comparison, not an exact conserved quantum number. [R1](#ref-source-8d1b859d5ffcad9f) [R3](#ref-source-d3edee404071ac81)\n\nOn a unit sphere this chiral form has magnitude $|\\cos\\theta|\\sin^2\\theta$: an equatorial line zero and quadratic polar zeros. A planar E1u illustration uses two orthogonal spin vectors multiplying $k_x(5k_z^2-1)$ and $k_y(5k_z^2-1)$. Its magnitude is $\\sin\\theta|5\\cos^2\\theta-1|$, with polar point zeros and two horizontal zeros at $\\cos^2\\theta=1/5$. These are model angles, not measured node coordinates. Tsutsumi and colleagues use a related planar construction in their weak-spin-orbit scenario, with field-dependent spin orientation. [R5](#ref-source-34df31af237b76a7)\n\nThe constant one in the spherical E1u expressions represents $k_x^2+k_y^2+k_z^2$. Restoring it makes each component a homogeneous harmonic cubic. The lower-order pair $(k_x,k_y)$ belongs to the same orbital doublet. In a local illustration with fixed orthogonal spin vectors and a unit spherical Fermi surface, add a **real** coefficient $a$ times this pair. The gap magnitude becomes $\\sin\\theta|5\\cos^2\\theta-1+a|$. Two distinct horizontal nodal rings occur only for $-4<a<1$, at $k_z^2=(1-a)/5$. At $a=1$ they merge into a quadratic equatorial zero; at $a=-4$ they meet the poles, giving cubic polar zeros. For real $a$ outside this interval only the polar zeros remain. This admixture is our illustration, not a fitted state reported by Tsutsumi and colleagues. [R5](#ref-source-34df31af237b76a7)\n\nFor a **complex** coefficient, the same local model instead has squared magnitude $\\sin^2\\theta[(5\\cos^2\\theta-1+\\operatorname{Re}a)^2+(\\operatorname{Im}a)^2]$. Any nonzero imaginary part removes its horizontal zeros while retaining the poles. Odd parity and the orbital-doublet transformation rule survive, but time-reversal symmetry need not: a complex mixture cannot simply be called the same symmetry state as the real mixture. Symmetry-allowed basis mixing alone says neither which coefficients minimize a material's free energy nor where its measured nodes lie. [R5](#ref-source-34df31af237b76a7)\n\nA separate local geometric calculation compares the polar zeros of the unmixed models. Restricting to small polar caps, count the fraction of a unit sphere with imposed gap below $\\epsilon$. Doubling a sufficiently small cutoff approximately doubles this fraction for the quadratic E2u zero, but quadruples it for the linear E1u zero. Eight cap calculations verify these limits. This is a geometric illustration, not a density-of-states or thermal-conductivity calculation; real velocities, impurity scattering and other nodes still contribute to a measured power law. [R1](#ref-source-8d1b859d5ffcad9f) [R5](#ref-source-34df31af237b76a7)\n\nOur separate two-component free-energy illustration shows how split transitions can arise without determining a material representation. Set $u=|\\eta_1|^2$, $v=|\\eta_2|^2$ and relative phase $\\delta$, with dimensionless free energy $F=(t-1)u+(t-0.8)v+(u^2+v^2)/2+(0.4+0.15\\cos2\\delta)uv$. When both amplitudes are present, $\\delta=\\pm\\pi/2$ minimizes the phase term. The first onset is $t=1$; the second is $t=(0.8-0.25)/(1-0.25)\\simeq0.7333$. Five independent positive-amplitude grid comparisons agree with the analytic minima. These coefficients are chosen, not fitted to UPt3 or UTe2. [R1](#ref-source-8d1b859d5ffcad9f) [R2](#ref-source-a329f7df31c7d3e9)"},{"id":"experimental-criteria","title":"Experimental criteria","markdown":"## Experimental criteria\n\nThe 2002 Joynt–Taillefer review weighs UPt3's phase diagram, nodal responses and magnetic coupling. Its abstract favors E1g or E2u on balance while acknowledging that neither explains all observations. That historical conclusion is not a current universal assignment. Machida and colleagues' later angle-dependent thermal-conductivity study instead interprets the B/C-phase data using E1u. The 2011 author abstract and 2012 journal abstract differ in the stated phase coverage; this entry uses the latter and does not claim a complete version comparison. [R2](#ref-source-a329f7df31c7d3e9) [R4](#ref-source-31577c7cb2ae0ac8)\n\nSchemm and colleagues reported a UPt3 Kerr signal below about 0.46 K, separated from an upper susceptibility onset near 0.55 K. Their 1550 nm optical probe required an estimated local-heating allowance up to 50 mK; bulk heating inferred from mutual inductance was much smaller. Cooling in either sign of a 50 G field and then measuring with the field removed trained the signal's sign. These observations support additional time-reversal breaking in the lower-temperature phase within the experiment's controls; the optical signal alone does not select E2u. The Kerr onset is not the upper superconducting transition. [R3](#ref-source-d3edee404071ac81)\n\nFor interpretation, record field direction, heat-current direction, temperature and response model. Vorontsov and Vekhter find that minima can become maxima along a node in a quasi-two-dimensional d-wave vortex calculation as temperature or field changes. Their treatment includes vortex scattering and finite-energy thermal weighting; it neglects Zeeman splitting and has low-field limitations. This is a methodological warning, not a calculated f-wave response for UPt3. Comparing multiple axes and independent probes remains editorial guidance: the apparent success of one chosen response model does not uniquely select a material representation. [R9](#ref-source-6c82ca23c0fda096)\n\nAs editorial guidance for a null Kerr result, record illumination, surface, residual field, cooldown history and statistical sensitivity. Independent noise and domain signals add in variance, $\\sigma_{\\rm total}^2=\\sigma_{\\rm noise}^2+\\sigma_{\\rm signal}^2$. A local constructed example, not a reanalysis of the cited measurements, verifies this identity for uncorrelated sequences. The equation requires its independence assumption; a measured absence of increased scatter is a constraint at finite sensitivity, not proof that all possible microscopic time-reversal breaking is absent. [R8](#ref-source-dd4e897d833ce69c)"},{"id":"limits-and-disputes","title":"Limits and disputes","markdown":"## Limits and disputes\n\nHayes and colleagues reported UTe2 Kerr and calorimetric signatures supporting a multicomponent state. Their low-field warmups were compared with measurements following positive and negative training fields; the interpretation includes domains. Ajeesh and colleagues later found no spontaneous zero-field Kerr signal in four single-transition samples spanning two growth methods. Their heat-pulse procedure repeatedly crossed the transition, and trained signals varied with position and sample. The selected manuscript's optical and susceptibility sample sets are different: one must not attribute the separate double-transition susceptibility sample to the Kerr set. Retain this disagreement and its conditions rather than declaring all UTe2 samples chiral or all possible states excluded. [R7](#ref-source-5ff091aacae25bb0) [R8](#ref-source-dd4e897d833ce69c)\n\nOrthorhombic D2h has one-dimensional ordinary point-group representations, unlike hexagonal E2u. A complex combination of distinct components can still break time reversal if the energetics permit it; multidimensional point-group representations are not a universal prerequisite. Spin degeneracy, orbital structure and accidental near-degeneracy must be specified. UTe2's discovery evidence for unusual spin pairing does not independently choose an f-wave basis or a unique multicomponent state. [R1](#ref-source-8d1b859d5ffcad9f) [R6](#ref-source-ca6322be2694e430) [R7](#ref-source-5ff091aacae25bb0)\n\nThe local spherical polynomial examples omit the multiband and nonsymmorphic structure relevant to UPt3. Micklitz and Norman analyze its P63/mmc space group and explain how fractional-translation phases can alter odd-parity line-node constraints on a Brillouin-zone face intersected by the Fermi surface. The answer depends on spin-orbit coupling, time-reversal symmetry and band pairing; a point-group polynomial alone does not contain these ingredients. Their analysis supplies a missing theoretical context, not a unique experimental pairing assignment. Our chosen Ginzburg–Landau example does not reproduce the material phase diagram, and this entry supplies no calculated surface spectrum or bulk topological invariant. [R10](#ref-source-591b5b95b965e1af)"},{"id":"discovery-implications","title":"Discovery implications","markdown":"## Discovery implications\n\nThe following is editorial comparison guidance: specify the crystal symmetry and basis first. Then test whether one proposed state explains phase splitting, directional nodes, spin response and symmetry-breaking observations under compatible conditions. Preserve null results, competing interpretations and source-version differences alongside positive evidence. Full scientific acceptance, complete final-version comparisons and raw-data review remain open; the model calculations here constrain language and logic, not the pairing state of a new material. [R1](#ref-source-8d1b859d5ffcad9f) [R3](#ref-source-d3edee404071ac81) [R5](#ref-source-34df31af237b76a7) [R8](#ref-source-dd4e897d833ce69c)"},{"id":"related-topics","title":"Related topics","markdown":"## Related topics\n\nCompare [p-wave spin and chirality](/scwiki/p-wave), [dxy directional evidence](/scwiki/dxy-pairing), [d-wave phase tests](/scwiki/d-wave), [ferromagnetic fluctuations](/scwiki/ferromagnetic-fluctuations), and [topological superconductivity](/scwiki/topological-superconductivity)."},{"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| UPt3 (aliases: heavy fermion, heavy fermions); Schemm et al. optical Kerr measurement | 0.46 K; Approximate Kerr onset with up to50mK local-heating allowance; not upper superconducting transition | unassigned; polar-kerr-onset | 1550nm polar Kerr; field-off warmup after zero-field or±50G training; separate upper susceptibility onset | unknown; unknown GPa | unknown | experiment; [source](#ref-source-d3edee404071ac81) |"}],"markdown":"# f-wave pairing: crystal representations and nodal evidence\n\nCompare higher-order odd pairing, UPt3 candidate states and the limits of Kerr, nodal and multicomponent evidence.\n\n## Scope\n\nIn a rotationally invariant continuum, f-wave means orbital angular momentum $l=3$. In a crystal, specify the point group, spin or pseudospin structure and basis functions; nonsymmorphic materials also require the fractional translations of the space group. An odd cubic polynomial is not automatically a pure $l=3$ harmonic, and a node inferred from a response is not an irreducible-representation label. The discussion below assumes even-frequency pairing unless stated otherwise. Exchange symmetry must include momentum, spin, orbital indices and frequency before drawing parity conclusions in a multiorbital material. [R5](#ref-source-34df31af237b76a7) [R10](#ref-source-591b5b95b965e1af)\n\nUPt3 is an important test case with competing descriptions of its superconducting phases. The historical E2u framework and a proposed E1u planar state organize different observations, but a candidate label is not a unique experimental determination. UTe2 has a different crystal symmetry and must not inherit UPt3's representation merely because both are uranium compounds. [R2](#ref-source-a329f7df31c7d3e9) [R3](#ref-source-d3edee404071ac81) [R5](#ref-source-34df31af237b76a7) [R7](#ref-source-5ff091aacae25bb0)\n\n## Mathematical assumptions\n\nIn the following local calculation, an illustrative hexagonal E2u orbital doublet is $[k_z(k_x^2-k_y^2),2k_zk_xk_y]$. With a fixed spin vector along z, its equal-amplitude chiral combination is proportional to $k_z(k_x+ik_y)^2$. It is odd under inversion. Under rotation by angle $\\alpha$ about z, its orbital factor acquires $e^{2i\\alpha}$: the continuum harmonic has $l=3$ but axial projection $m=2$, not $m=3$. Local transformation checks and finite-difference Laplacians verify these polynomial identities; they do not reproduce a cited material calculation. In a lattice, continuous angular momentum is only a comparison, not an exact conserved quantum number. [R1](#ref-source-8d1b859d5ffcad9f) [R3](#ref-source-d3edee404071ac81)\n\nOn a unit sphere this chiral form has magnitude $|\\cos\\theta|\\sin^2\\theta$: an equatorial line zero and quadratic polar zeros. A planar E1u illustration uses two orthogonal spin vectors multiplying $k_x(5k_z^2-1)$ and $k_y(5k_z^2-1)$. Its magnitude is $\\sin\\theta|5\\cos^2\\theta-1|$, with polar point zeros and two horizontal zeros at $\\cos^2\\theta=1/5$. These are model angles, not measured node coordinates. Tsutsumi and colleagues use a related planar construction in their weak-spin-orbit scenario, with field-dependent spin orientation. [R5](#ref-source-34df31af237b76a7)\n\nThe constant one in the spherical E1u expressions represents $k_x^2+k_y^2+k_z^2$. Restoring it makes each component a homogeneous harmonic cubic. The lower-order pair $(k_x,k_y)$ belongs to the same orbital doublet. In a local illustration with fixed orthogonal spin vectors and a unit spherical Fermi surface, add a **real** coefficient $a$ times this pair. The gap magnitude becomes $\\sin\\theta|5\\cos^2\\theta-1+a|$. Two distinct horizontal nodal rings occur only for $-4<a<1$, at $k_z^2=(1-a)/5$. At $a=1$ they merge into a quadratic equatorial zero; at $a=-4$ they meet the poles, giving cubic polar zeros. For real $a$ outside this interval only the polar zeros remain. This admixture is our illustration, not a fitted state reported by Tsutsumi and colleagues. [R5](#ref-source-34df31af237b76a7)\n\nFor a **complex** coefficient, the same local model instead has squared magnitude $\\sin^2\\theta[(5\\cos^2\\theta-1+\\operatorname{Re}a)^2+(\\operatorname{Im}a)^2]$. Any nonzero imaginary part removes its horizontal zeros while retaining the poles. Odd parity and the orbital-doublet transformation rule survive, but time-reversal symmetry need not: a complex mixture cannot simply be called the same symmetry state as the real mixture. Symmetry-allowed basis mixing alone says neither which coefficients minimize a material's free energy nor where its measured nodes lie. [R5](#ref-source-34df31af237b76a7)\n\nA separate local geometric calculation compares the polar zeros of the unmixed models. Restricting to small polar caps, count the fraction of a unit sphere with imposed gap below $\\epsilon$. Doubling a sufficiently small cutoff approximately doubles this fraction for the quadratic E2u zero, but quadruples it for the linear E1u zero. Eight cap calculations verify these limits. This is a geometric illustration, not a density-of-states or thermal-conductivity calculation; real velocities, impurity scattering and other nodes still contribute to a measured power law. [R1](#ref-source-8d1b859d5ffcad9f) [R5](#ref-source-34df31af237b76a7)\n\nOur separate two-component free-energy illustration shows how split transitions can arise without determining a material representation. Set $u=|\\eta_1|^2$, $v=|\\eta_2|^2$ and relative phase $\\delta$, with dimensionless free energy $F=(t-1)u+(t-0.8)v+(u^2+v^2)/2+(0.4+0.15\\cos2\\delta)uv$. When both amplitudes are present, $\\delta=\\pm\\pi/2$ minimizes the phase term. The first onset is $t=1$; the second is $t=(0.8-0.25)/(1-0.25)\\simeq0.7333$. Five independent positive-amplitude grid comparisons agree with the analytic minima. These coefficients are chosen, not fitted to UPt3 or UTe2. [R1](#ref-source-8d1b859d5ffcad9f) [R2](#ref-source-a329f7df31c7d3e9)\n\n## Experimental criteria\n\nThe 2002 Joynt–Taillefer review weighs UPt3's phase diagram, nodal responses and magnetic coupling. Its abstract favors E1g or E2u on balance while acknowledging that neither explains all observations. That historical conclusion is not a current universal assignment. Machida and colleagues' later angle-dependent thermal-conductivity study instead interprets the B/C-phase data using E1u. The 2011 author abstract and 2012 journal abstract differ in the stated phase coverage; this entry uses the latter and does not claim a complete version comparison. [R2](#ref-source-a329f7df31c7d3e9) [R4](#ref-source-31577c7cb2ae0ac8)\n\nSchemm and colleagues reported a UPt3 Kerr signal below about 0.46 K, separated from an upper susceptibility onset near 0.55 K. Their 1550 nm optical probe required an estimated local-heating allowance up to 50 mK; bulk heating inferred from mutual inductance was much smaller. Cooling in either sign of a 50 G field and then measuring with the field removed trained the signal's sign. These observations support additional time-reversal breaking in the lower-temperature phase within the experiment's controls; the optical signal alone does not select E2u. The Kerr onset is not the upper superconducting transition. [R3](#ref-source-d3edee404071ac81)\n\nFor interpretation, record field direction, heat-current direction, temperature and response model. Vorontsov and Vekhter find that minima can become maxima along a node in a quasi-two-dimensional d-wave vortex calculation as temperature or field changes. Their treatment includes vortex scattering and finite-energy thermal weighting; it neglects Zeeman splitting and has low-field limitations. This is a methodological warning, not a calculated f-wave response for UPt3. Comparing multiple axes and independent probes remains editorial guidance: the apparent success of one chosen response model does not uniquely select a material representation. [R9](#ref-source-6c82ca23c0fda096)\n\nAs editorial guidance for a null Kerr result, record illumination, surface, residual field, cooldown history and statistical sensitivity. Independent noise and domain signals add in variance, $\\sigma_{\\rm total}^2=\\sigma_{\\rm noise}^2+\\sigma_{\\rm signal}^2$. A local constructed example, not a reanalysis of the cited measurements, verifies this identity for uncorrelated sequences. The equation requires its independence assumption; a measured absence of increased scatter is a constraint at finite sensitivity, not proof that all possible microscopic time-reversal breaking is absent. [R8](#ref-source-dd4e897d833ce69c)\n\n## Limits and disputes\n\nHayes and colleagues reported UTe2 Kerr and calorimetric signatures supporting a multicomponent state. Their low-field warmups were compared with measurements following positive and negative training fields; the interpretation includes domains. Ajeesh and colleagues later found no spontaneous zero-field Kerr signal in four single-transition samples spanning two growth methods. Their heat-pulse procedure repeatedly crossed the transition, and trained signals varied with position and sample. The selected manuscript's optical and susceptibility sample sets are different: one must not attribute the separate double-transition susceptibility sample to the Kerr set. Retain this disagreement and its conditions rather than declaring all UTe2 samples chiral or all possible states excluded. [R7](#ref-source-5ff091aacae25bb0) [R8](#ref-source-dd4e897d833ce69c)\n\nOrthorhombic D2h has one-dimensional ordinary point-group representations, unlike hexagonal E2u. A complex combination of distinct components can still break time reversal if the energetics permit it; multidimensional point-group representations are not a universal prerequisite. Spin degeneracy, orbital structure and accidental near-degeneracy must be specified. UTe2's discovery evidence for unusual spin pairing does not independently choose an f-wave basis or a unique multicomponent state. [R1](#ref-source-8d1b859d5ffcad9f) [R6](#ref-source-ca6322be2694e430) [R7](#ref-source-5ff091aacae25bb0)\n\nThe local spherical polynomial examples omit the multiband and nonsymmorphic structure relevant to UPt3. Micklitz and Norman analyze its P63/mmc space group and explain how fractional-translation phases can alter odd-parity line-node constraints on a Brillouin-zone face intersected by the Fermi surface. The answer depends on spin-orbit coupling, time-reversal symmetry and band pairing; a point-group polynomial alone does not contain these ingredients. Their analysis supplies a missing theoretical context, not a unique experimental pairing assignment. Our chosen Ginzburg–Landau example does not reproduce the material phase diagram, and this entry supplies no calculated surface spectrum or bulk topological invariant. [R10](#ref-source-591b5b95b965e1af)\n\n## Discovery implications\n\nThe following is editorial comparison guidance: specify the crystal symmetry and basis first. Then test whether one proposed state explains phase splitting, directional nodes, spin response and symmetry-breaking observations under compatible conditions. Preserve null results, competing interpretations and source-version differences alongside positive evidence. Full scientific acceptance, complete final-version comparisons and raw-data review remain open; the model calculations here constrain language and logic, not the pairing state of a new material. [R1](#ref-source-8d1b859d5ffcad9f) [R3](#ref-source-d3edee404071ac81) [R5](#ref-source-34df31af237b76a7) [R8](#ref-source-dd4e897d833ce69c)\n\n## Related topics\n\nCompare [p-wave spin and chirality](/scwiki/p-wave), [dxy directional evidence](/scwiki/dxy-pairing), [d-wave phase tests](/scwiki/d-wave), [ferromagnetic fluctuations](/scwiki/ferromagnetic-fluctuations), and [topological superconductivity](/scwiki/topological-superconductivity).\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| UPt3 (aliases: heavy fermion, heavy fermions); Schemm et al. optical Kerr measurement | 0.46 K; Approximate Kerr onset with up to50mK local-heating allowance; not upper superconducting transition | unassigned; polar-kerr-onset | 1550nm polar Kerr; field-off warmup after zero-field or±50G training; separate upper susceptibility onset | unknown; unknown GPa | unknown | experiment; [source](#ref-source-d3edee404071ac81) |\n\n## Sources\n\n### Ref source-8d1b859d5ffcad9f\n\n[R1] Phenomenological theory of unconventional superconductivity (1991). [DOI 10.1103/RevModPhys.63.239](https://doi.org/10.1103/RevModPhys.63.239). Publication: journal-article. Check: publisher-abstract-check. Checked: 2026-10-18. Identity and abstract scope checked. Full review not obtained; explicit angular illustrations are independently evaluated, not attributed as a reproduction of this review.\n### Ref source-a329f7df31c7d3e9\n\n[R2] The superconducting phases of UPt3 (2002). [DOI 10.1103/RevModPhys.74.235](https://doi.org/10.1103/RevModPhys.74.235). Publication: journal-article. Check: publisher-abstract-check. Checked: 2026-10-18. Historical E1g/E2u comparison and unresolved magnetic coupling; full review not obtained.\n### Ref source-d3edee404071ac81\n\n[R3] Observation of broken time-reversal symmetry in the heavy-fermion superconductor UPt3 (2014). [DOI 10.1126/science.1248552](https://doi.org/10.1126/science.1248552). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-18. Selected pp.1–6, Fig.2 on p.11 visually checked, and sample/apparatus passages on p.13. Published July2014; author upload October2014. Heating and training limits retained; full final-version and raw-data comparison open.\n### Ref source-31577c7cb2ae0ac8\n\n[R4] Twofold Spontaneous Symmetry Breaking in the Heavy-Fermion Superconductor UPt3 (2012). [DOI 10.1103/PhysRevLett.108.157002](https://doi.org/10.1103/PhysRevLett.108.157002). Publication: journal-article. Check: publisher-abstract-check. Checked: 2026-10-18. Journal abstract B/C coverage differs from arxiv1107.3082v1 A/B/C wording. E1u is the authors interpretation; full version/data comparison open.\n### Ref source-34df31af237b76a7\n\n[R5] A Spin Triplet Superconductor UPt3 (2012). [DOI 10.1143/JPSJ.81.074717](https://doi.org/10.1143/JPSJ.81.074717). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-18. Selected pp.1–3, Fig.1 and basis table; p.2 visually checked. Version2 July2012 marked published version by authors. Weak-SOC planar construction and competing scenarios retained; Eilenberger results not reproduced.\n### Ref source-ca6322be2694e430\n\n[R6] 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-5ff091aacae25bb0\n\n[R7] Multicomponent superconducting order parameter in UTe2 (2021). [DOI 10.1126/science.abb0272](https://doi.org/10.1126/science.abb0272). Publication: journal-article. Check: selected-first-release-text-check. Checked: 2026-10-18. Selected first-release15July2021 pp.1–2: reported Kerr,training and calorimetric interpretation. Final pagination/version and full supplement not checked.\n### Ref source-dd4e897d833ce69c\n\n[R8] Fate of Time-Reversal Symmetry Breaking in UTe2 (2023). [DOI 10.1103/PhysRevX.13.041019](https://doi.org/10.1103/PhysRevX.13.041019). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-18. Selected pp.2–3 and methods pp.5–6; Fig.1 p.3 visually checked. Kerr sample set distinguished from susceptibility set. Final publisher abstract checked; full version comparison open.\n### Ref source-6c82ca23c0fda096\n\n[R9] Nodal Structure of Quasi-Two-Dimensional Superconductors Probed by a Magnetic Field (2006). [DOI 10.1103/PhysRevLett.96.237001](https://doi.org/10.1103/PhysRevLett.96.237001). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-18. Selected pp. 1–3 and Fig. 2 visually checked: finite-energy vortex scattering, anisotropy reversal and BPT/low-field/Zeeman limitations. Publisher identity checked. Full journal comparison and vortex calculation not reproduced.\n### Ref source-591b5b95b965e1af\n\n[R10] Nodal lines and nodal loops in nonsymmorphic odd-parity superconductors (2017). [DOI 10.1103/PhysRevB.95.024508](https://doi.org/10.1103/PhysRevB.95.024508). Publication: journal-article. Check: selected-accepted-manuscript-check. Checked: 2026-10-19. Accepted manuscript dated23Dec2016, publisher identity17Jan2017; selected printedpp1-3 (PDFpp2-4), introduction andII.A-B. Space-group/fractional-phase and spin-orbit/time-reversal limits checked in text; full band/topology derivations and final-version comparison not reproduced.\n\n## Claim evidence and conditions\n\n### scwiki-c6-claim-scope-1\n\nIn a rotationally invariant continuum, f-wave means orbital angular momentum $l=3$. In a crystal, specify the point group, spin or pseudospin structure and basis functions; nonsymmorphic materials also require the fractional translations of the space group. An odd cubic polynomial is not automatically a pure $l=3$ harmonic, and a node inferred from a response is not an irreducible-representation label. The discussion below assumes even-frequency pairing unless stated otherwise. Exchange symmetry must include momentum, spin, orbital indices and frequency before drawing parity conclusions in a multiorbital material.\n\nEvidence: theory. Sources: source-34df31af237b76a7, source-591b5b95b965e1af.\n\nSource relationship: mixed-source-and-local-derivation. Tsutsumi supplies the harmonic/classification context; Micklitz-Norman supplies space-group and band-pairing context. Harmonic decomposition and separable exchange-sign checks are local explanatory deductions.\n\nConditions: Continuum l assumes rotational invariance; crystal representations include spin/pseudospin and, when relevant, nonsymmorphic fractional translations. Even-frequency default; full index exchange required.\n\nVerification scope: Selected versioned primary text and local algebra checked 2026-10-19; full material theory, final-version comparison and independent scientific review remain open.\n\n### scwiki-c6-claim-scope-2\n\nUPt3 is an important test case with competing descriptions of its superconducting phases. The historical E2u framework and a proposed E1u planar state organize different observations, but a candidate label is not a unique experimental determination. UTe2 has a different crystal symmetry and must not inherit UPt3's representation merely because both are uranium compounds.\n\nEvidence: theory. Sources: source-a329f7df31c7d3e9, source-d3edee404071ac81, source-34df31af237b76a7, source-5ff091aacae25bb0.\n\nConditions: Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-mathematical-assumptions-1\n\nIn the following local calculation, an illustrative hexagonal E2u orbital doublet is $[k_z(k_x^2-k_y^2),2k_zk_xk_y]$. With a fixed spin vector along z, its equal-amplitude chiral combination is proportional to $k_z(k_x+ik_y)^2$. It is odd under inversion. Under rotation by angle $\\alpha$ about z, its orbital factor acquires $e^{2i\\alpha}$: the continuum harmonic has $l=3$ but axial projection $m=2$, not $m=3$. Local transformation checks and finite-difference Laplacians verify these polynomial identities; they do not reproduce a cited material calculation. In a lattice, continuous angular momentum is only a comparison, not an exact conserved quantum number.\n\nEvidence: theory. Sources: source-8d1b859d5ffcad9f, source-d3edee404071ac81.\n\nSource relationship: context-for-local-derivation. Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations.\n\nConditions: Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-mathematical-assumptions-2\n\nOn a unit sphere this chiral form has magnitude $|\\cos\\theta|\\sin^2\\theta$: an equatorial line zero and quadratic polar zeros. A planar E1u illustration uses two orthogonal spin vectors multiplying $k_x(5k_z^2-1)$ and $k_y(5k_z^2-1)$. Its magnitude is $\\sin\\theta|5\\cos^2\\theta-1|$, with polar point zeros and two horizontal zeros at $\\cos^2\\theta=1/5$. These are model angles, not measured node coordinates. Tsutsumi and colleagues use a related planar construction in their weak-spin-orbit scenario, with field-dependent spin orientation.\n\nEvidence: theory. Sources: source-34df31af237b76a7.\n\nSource relationship: mixed-source-and-local-derivation. Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations.\n\nConditions: Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-mathematical-assumptions-3\n\nThe constant one in the spherical E1u expressions represents $k_x^2+k_y^2+k_z^2$. Restoring it makes each component a homogeneous harmonic cubic. The lower-order pair $(k_x,k_y)$ belongs to the same orbital doublet. In a local illustration with fixed orthogonal spin vectors and a unit spherical Fermi surface, add a real coefficient $a$ times this pair. The gap magnitude becomes $\\sin\\theta|5\\cos^2\\theta-1+a|$. Two distinct horizontal nodal rings occur only for $-4<a<1$, at $k_z^2=(1-a)/5$. At $a=1$ they merge into a quadratic equatorial zero; at $a=-4$ they meet the poles, giving cubic polar zeros. For real $a$ outside this interval only the polar zeros remain. This admixture is our illustration, not a fitted state reported by Tsutsumi and colleagues. For a complex coefficient, the same local model instead has squared magnitude $\\sin^2\\theta[(5\\cos^2\\theta-1+\\operatorname{Re}a)^2+(\\operatorname{Im}a)^2]$. Any nonzero imaginary part removes its horizontal zeros while retaining the poles. Odd parity and the orbital-doublet transformation rule survive, but time-reversal symmetry need not: a complex mixture cannot simply be called the same symmetry state as the real mixture. Symmetry-allowed basis mixing alone says neither which coefficients minimize a material's free energy nor where its measured nodes lie.\n\nEvidence: theory. Sources: source-34df31af237b76a7.\n\nSource relationship: context-for-local-derivation. Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations.\n\nConditions: Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-mathematical-assumptions-4\n\nA separate local geometric calculation compares the polar zeros of the unmixed models. Restricting to small polar caps, count the fraction of a unit sphere with imposed gap below $\\epsilon$. Doubling a sufficiently small cutoff approximately doubles this fraction for the quadratic E2u zero, but quadruples it for the linear E1u zero. Eight cap calculations verify these limits. This is a geometric illustration, not a density-of-states or thermal-conductivity calculation; real velocities, impurity scattering and other nodes still contribute to a measured power law.\n\nEvidence: theory. Sources: source-8d1b859d5ffcad9f, source-34df31af237b76a7.\n\nSource relationship: context-for-local-derivation. Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations.\n\nConditions: Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-mathematical-assumptions-5\n\nOur separate two-component free-energy illustration shows how split transitions can arise without determining a material representation. Set $u=|\\eta_1|^2$, $v=|\\eta_2|^2$ and relative phase $\\delta$, with dimensionless free energy $F=(t-1)u+(t-0.8)v+(u^2+v^2)/2+(0.4+0.15\\cos2\\delta)uv$. When both amplitudes are present, $\\delta=\\pm\\pi/2$ minimizes the phase term. The first onset is $t=1$; the second is $t=(0.8-0.25)/(1-0.25)\\simeq0.7333$. Five independent positive-amplitude grid comparisons agree with the analytic minima. These coefficients are chosen, not fitted to UPt3 or UTe2.\n\nEvidence: theory. Sources: source-8d1b859d5ffcad9f, source-a329f7df31c7d3e9.\n\nSource relationship: context-for-local-derivation. Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations.\n\nConditions: Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-experimental-criteria-1\n\nThe 2002 Joynt–Taillefer review weighs UPt3's phase diagram, nodal responses and magnetic coupling. Its abstract favors E1g or E2u on balance while acknowledging that neither explains all observations. That historical conclusion is not a current universal assignment. Machida and colleagues' later angle-dependent thermal-conductivity study instead interprets the B/C-phase data using E1u. The 2011 author abstract and 2012 journal abstract differ in the stated phase coverage; this entry uses the latter and does not claim a complete version comparison.\n\nEvidence: experiment. Sources: source-a329f7df31c7d3e9, source-31577c7cb2ae0ac8.\n\nConditions: Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-experimental-criteria-2\n\nSchemm and colleagues reported a UPt3 Kerr signal below about 0.46 K, separated from an upper susceptibility onset near 0.55 K. Their 1550 nm optical probe required an estimated local-heating allowance up to 50 mK; bulk heating inferred from mutual inductance was much smaller. Cooling in either sign of a 50 G field and then measuring with the field removed trained the signal's sign. These observations support additional time-reversal breaking in the lower-temperature phase within the experiment's controls; the optical signal alone does not select E2u. The Kerr onset is not the upper superconducting transition.\n\nEvidence: experiment. Sources: source-d3edee404071ac81.\n\nConditions: Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-experimental-criteria-3\n\nFor interpretation, record field direction, heat-current direction, temperature and response model. Vorontsov and Vekhter find that minima can become maxima along a node in a quasi-two-dimensional d-wave vortex calculation as temperature or field changes. Their treatment includes vortex scattering and finite-energy thermal weighting; it neglects Zeeman splitting and has low-field limitations. This is a methodological warning, not a calculated f-wave response for UPt3. Comparing multiple axes and independent probes remains editorial guidance: the apparent success of one chosen response model does not uniquely select a material representation.\n\nEvidence: theory. Sources: source-6c82ca23c0fda096.\n\nSource relationship: context-for-editorial-analysis. Specific sign-reversal examples are reported theory; comparison recommendations are editorial. No new experimental result or source calculation reproduced.\n\nConditions: Quasi-two-dimensional d-wave vortex model, rotated in-plane field and specified heat-current direction; BPT approximation, no Zeeman splitting. No f-wave or UPt3 response inferred.\n\nVerification scope: Selected versioned primary text and local algebra checked 2026-10-19; full material theory, final-version comparison and independent scientific review remain open.\n\n### scwiki-c6-claim-experimental-criteria-4\n\nAs editorial guidance for a null Kerr result, record illumination, surface, residual field, cooldown history and statistical sensitivity. Independent noise and domain signals add in variance, $\\sigma_{\\rm total}^2=\\sigma_{\\rm noise}^2+\\sigma_{\\rm signal}^2$. A local constructed example, not a reanalysis of the cited measurements, verifies this identity for uncorrelated sequences. The equation requires its independence assumption; a measured absence of increased scatter is a constraint at finite sensitivity, not proof that all possible microscopic time-reversal breaking is absent.\n\nEvidence: theory. Sources: source-dd4e897d833ce69c.\n\nSource relationship: context-for-local-derivation. Cited sources supply pairing-symmetry or measurement context. Polynomial transformations, coefficient admixtures, polar-cap counts, chosen free-energy coefficients and variance examples are local illustrations, not experimental fits or reproduced source calculations.\n\nConditions: Fixed spin vectors and unit sphere for gap examples; real a for two horizontal rings (-4<a<1), complex a treated separately; chosen dimensionless GL coefficients; uncorrelated noise for variance addition.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-limits-and-disputes-1\n\nHayes and colleagues reported UTe2 Kerr and calorimetric signatures supporting a multicomponent state. Their low-field warmups were compared with measurements following positive and negative training fields; the interpretation includes domains. Ajeesh and colleagues later found no spontaneous zero-field Kerr signal in four single-transition samples spanning two growth methods. Their heat-pulse procedure repeatedly crossed the transition, and trained signals varied with position and sample. The selected manuscript's optical and susceptibility sample sets are different: one must not attribute the separate double-transition susceptibility sample to the Kerr set. Retain this disagreement and its conditions rather than declaring all UTe2 samples chiral or all possible states excluded.\n\nEvidence: experiment. Sources: source-5ff091aacae25bb0, source-dd4e897d833ce69c.\n\nConditions: Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-limits-and-disputes-2\n\nOrthorhombic D2h has one-dimensional ordinary point-group representations, unlike hexagonal E2u. A complex combination of distinct components can still break time reversal if the energetics permit it; multidimensional point-group representations are not a universal prerequisite. Spin degeneracy, orbital structure and accidental near-degeneracy must be specified. UTe2's discovery evidence for unusual spin pairing does not independently choose an f-wave basis or a unique multicomponent state.\n\nEvidence: theory. Sources: source-8d1b859d5ffcad9f, source-ca6322be2694e430, source-5ff091aacae25bb0.\n\nConditions: Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c6-claim-limits-and-disputes-3\n\nThe local spherical polynomial examples omit the multiband and nonsymmorphic structure relevant to UPt3. Micklitz and Norman analyze its P63/mmc space group and explain how fractional-translation phases can alter odd-parity line-node constraints on a Brillouin-zone face intersected by the Fermi surface. The answer depends on spin-orbit coupling, time-reversal symmetry and band pairing; a point-group polynomial alone does not contain these ingredients. Their analysis supplies a missing theoretical context, not a unique experimental pairing assignment. Our chosen Ginzburg–Landau example does not reproduce the material phase diagram, and this entry supplies no calculated surface spectrum or bulk topological invariant.\n\nEvidence: theory. Sources: source-591b5b95b965e1af.\n\nSource relationship: context-for-editorial-analysis. The primary theoretical study explains missing crystal/band ingredients; the statement about this entry is an audit of its limited local calculations.\n\nConditions: UPt3 space-group context and Fermi-surface intersection; spin-orbit, time-reversal and interband assumptions must be supplied. Local sphere and chosen free energy omit these inputs.\n\nVerification scope: Selected versioned primary text and local algebra checked 2026-10-19; full material theory, final-version comparison and independent scientific review remain open.\n\n### scwiki-c6-claim-discovery-implications-1\n\nThe following is editorial comparison guidance: specify the crystal symmetry and basis first. Then test whether one proposed state explains phase splitting, directional nodes, spin response and symmetry-breaking observations under compatible conditions. Preserve null results, competing interpretations and source-version differences alongside positive evidence. Full scientific acceptance, complete final-version comparisons and raw-data review remain open; the model calculations here constrain language and logic, not the pairing state of a new material.\n\nEvidence: theory. Sources: source-8d1b859d5ffcad9f, source-d3edee404071ac81, source-34df31af237b76a7, source-dd4e897d833ce69c.\n\nSource relationship: context-for-editorial-analysis. Editorial comparison guidance grounded in cited examples; not a universal measurement protocol established by these papers.\n\nConditions: Declared point group and spin/orbital conventions; spherical harmonics,GL coefficients and variance examples are not material fits. Experimental sample/field/heating limits retained.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n## Revision and review\n\nArticle ID: scwiki-c6. Revision: 57f5d9992e70254ac7a6. Updated: 2026-10-19.\n\nReview scope: Selected UPt3 Kerr and planar-state manuscripts, historical transport/review abstracts, UTe2 contrasting Kerr reports, and independent harmonic/nodal/free-energy checks. Follow-up: version-pinned Tsutsumi II.A/Table I text, explicit local derivation attribution, real/complex admixture and endpoint checks. Selected Vorontsov-Vekhter model/Results and Micklitz-Norman accepted manuscript I/II.A-B; three frozen claims reassessed with explicit model and space-group limits.\n\n- Full scientific acceptance, complete final-version comparisons and raw-data review remain open.\n- Spherical harmonics and chosen free-energy coefficients are not a multiband material calculation.\n- Null and trained Kerr signals are scoped to sample,protocol and sensitivity; no universal exclusion claimed.\n- No definite f-wave assignment or bulk topological invariant follows from these local checks.\n\nSource: [C_pairing_symmetry/C6_f_wave.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/C_pairing_symmetry/C6_f_wave.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 788c392c4ae92a9c8c716666621564e27ca9690a53c336507be4f70358d766c4.\n","wordCount":1366}}