{"schemaVersion":"1.0","revision":"fa961ab24277a91737c3","article":{"id":"scwiki-c7","topicId":"scwiki-topic-c7","code":"C7","slug":"chiral-pairing","title":"Chiral pairing: time reversal, topology and boundary evidence","summary":"Separate orbital handedness, broken time reversal, band topology and the signals used to test them.","aliases":["chiral pairing","chirality","Chern number","time reversal","TRSB","edge current","kagome","roton PDW","s+is"],"evidenceTypes":["theory","experiment"],"relatedCodes":["C5","C6","C3","B11","B6"],"editorialStatus":"scoped-check","contentState":"pilot","modifiedAt":"2026-10-19T14:45:44.059096+00:00","review":{"checkedAt":"2026-10-19","scope":"Selected topology, boundary-current and optical manuscripts; kagome muon/STM passages; inherited UPt3 and Sr2RuO4 scoped evidence; independent lattice and angular checks. Follow-up: distinguish KV3Sb5 zero-field and transverse-field responses; check a local three-dimensional slice example.","limitations":["Full scientific acceptance, complete final-version comparisons and raw-data review remain open.","Fixed-gap lattice topology and sharp-edge continuum currents are not realistic material calculations.","Charge-order, superconducting and surface temperature scales are not interchangeable.","No current crystal candidate is declared conclusively chiral by these local checks."]},"provenance":{"repository":"JackZH26/jzis-superconductivity","commit":"0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf","path":"C_pairing_symmetry/C7_chiral_pairing.md","sha256":"83702d492b9ec3f0db41d54a045ebecc8ce06f0cd72755d36a041efcf3746344","sourceId":"file-19025829180d940a"},"sourceMap":[{"startLine":1,"endLine":15,"targetSections":["scope"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; symmetry, topology and probe boundaries expanded with primary checks."},{"startLine":16,"endLine":28,"targetSections":["scope","mathematical-assumptions"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; symmetry, topology and probe boundaries expanded with primary checks."},{"startLine":29,"endLine":38,"targetSections":["mathematical-assumptions","experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; symmetry, topology and probe boundaries expanded with primary checks."},{"startLine":39,"endLine":46,"targetSections":["experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; symmetry, topology and probe boundaries expanded with primary checks."},{"startLine":47,"endLine":58,"targetSections":["experimental-criteria","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; symmetry, topology and probe boundaries expanded with primary checks."},{"startLine":59,"endLine":78,"targetSections":["discovery-implications","related-topics"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; symmetry, topology and probe boundaries expanded with primary checks."}],"references":[{"id":"source-a33c639b7db4f05a","doi":"10.1103/PhysRevB.61.10267","title":"Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect","year":2000,"url":"https://doi.org/10.1103/PhysRevB.61.10267","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/cond-mat/9906453v3","note":"Selected pp. 4–7: continuum assumptions, weak/strong topology and charge Hall distinction. Version 3 January 2000; final-version comparison open. Local lattice example is independent."}},{"id":"source-69d33a869fbbb0c6","doi":"10.1088/0034-4885/79/5/054502","title":"Chiral superconductors","year":2016,"url":"https://doi.org/10.1088/0034-4885/79/5/054502","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-19","evidenceUrl":"https://arxiv.org/pdf/1512.01151v2","note":"Kallin and Berlinsky, version 2 April 2016. Selected pp. 5–7 and 15–17: multiband formalism, half-quantum vortices and optical response. Historical candidate assignments are not current confirmations. Follow-up: III.2, Eqs.16-21, edge modes versus nonquantized charge current and screening. Local stacked-slice example is not attributed to this review."}},{"id":"source-25d6a80b561d5ad2","doi":"10.1103/PhysRevB.90.224519","title":"Vanishing edge currents in non-p-wave topological chiral superconductors","year":2014,"url":"https://doi.org/10.1103/PhysRevB.90.224519","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1410.0377v2","note":"Selected pp. 2–3; p. 3 visually checked. Version 2 marked published version by authors. Independent angular quadrature checks Eqs. 6–11; full lattice/self-consistent calculations not reproduced."}},{"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-c7f8b99a795aba41","doi":"10.1038/s41586-019-1596-2","title":"Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance","year":2019,"url":"https://doi.org/10.1038/s41586-019-1596-2","publicationStatus":"journal-article","verification":{"scope":"author-manuscript-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1904.00047v1","note":"Pustogow et al. v1 selected pp.4–7; p.7 RF-energy and heating passage visually checked. Zero-strain0.7107T versus strained1.9980T conditions distinguished. Published identity checked; full final-version equivalence and subsequent literature survey not completed."}},{"id":"source-484da7abcc165cd3","doi":"10.1038/s41586-021-04327-z","title":"Time-reversal symmetry-breaking charge order in a kagome superconductor","year":2022,"url":"https://doi.org/10.1038/s41586-021-04327-z","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-19","evidenceUrl":"https://arxiv.org/pdf/2106.13443v2","note":"Selected pp. 2–3 and 6 plus experimental-methods passages; Fig. 1 on p. 2 visually checked. December 2021 author version 2, published February 2022. Nuclear/electronic and charge/superconducting distinctions retained; full final-version/data review open. Follow-up: v2 text, Magnetic Response Across Charge Order, Unconventional Superconductivity, Eq.1, Fig.1/3 captions and Methods. Gamma versus transverse Gaussian damping distinguished. Publisher page was inaccessible; no new final-version or figure-pixel review."}},{"id":"source-057c43c86174f961","doi":"10.1038/s41586-021-03983-5","title":"Roton pair density wave in a strong-coupling kagome superconductor","year":2021,"url":"https://doi.org/10.1038/s41586-021-03983-5","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/2103.09188v2","note":"Selected pp. 3–7 and methods pp. 17–18; Fig. 3 on p. 14 visually checked. Author version 2 September 2021 has a different title and mistyped journal volume 559; publisher confirms Nature 599. Final-version/raw-data comparison open."}},{"id":"source-3c7aa9b4c8da3abf","doi":"10.1103/PhysRevB.81.134522","title":"Three-band superconductivity and the order parameter that breaks time-reversal symmetry","year":2010,"url":"https://doi.org/10.1103/PhysRevB.81.134522","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.134522","note":"Published 23 April 2010. Abstract supports frustrated three-band TRSB possibility; full phase diagram not reviewed. Local fixed-phase example is illustrative only."}},{"id":"source-c4434644191d0174","doi":"10.1103/sgcz-9rc7","title":"Muon Knight Shift as a Precise Probe of the Superconducting Symmetry of Sr2RuO4","year":2026,"url":"https://doi.org/10.1103/sgcz-9rc7","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/2512.14107v1","note":"Selected pp.1–5 and Fig.3 visually checked. Published9February2026; manuscript16December2025. Same-crystal field-cooled subtraction and0.5K lower magnetization limit retained. Supplement, final-version comparison and raw-data reproduction open."}}],"claims":[{"id":"scwiki-c7-claim-scope-1","sectionId":"scope","statement":"Chiral pairing describes a superconducting order with a handed orbital structure, such as a complex combination of two directional basis functions. Three questions must remain separate: whether time reversal is broken, whether the pairing is chiral, and whether a specified gapped Hamiltonian has a nonzero topological invariant. A Kerr signal, a complex coefficient or a named lattice does not answer all three. Kallin and Berlinsky's 2016 review is a historical guide; its candidate-material assignments are not current confirmations.","evidenceType":"theory","sourceIds":["source-69d33a869fbbb0c6"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c7-claim-scope-2","sectionId":"scope","statement":"Broken time reversal need not mean orbital chirality. For example, frustrated interband pair scattering can favor complex relative phases among otherwise s-wave gaps. Stanev and Tešanović studied this possibility in a three-band model. An illustrative vector of gaps on three inequivalent labelled bands, $(1,e^{2\\pi i/3},e^{-2\\pi i/3})$, cannot be made equal to its conjugate by one common phase, although each component has no angular winding. This statement assumes only a common U(1) freedom; it is not a stability calculation or a material assignment.","evidenceType":"theory","sourceIds":["source-3c7aa9b4c8da3abf"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"The three-band model motivates a local fixed-phase vector; the common-phase test is an author illustration, not the source phase diagram."},{"id":"scwiki-c7-claim-mathematical-assumptions-1","sectionId":"mathematical-assumptions","statement":"Read and Green distinguish weak- and strong-pairing phases of a two-dimensional spinless complex p-wave model. Their continuum chemical potential is measured relative to the band bottom: its zero marks a gap-closing transition in that model. The broken symmetry alone does not distinguish the two phases. This continuum criterion must not be copied unchanged into a lattice or multiband calculation.","evidenceType":"theory","sourceIds":["source-a33c639b7db4f05a"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c7-claim-mathematical-assumptions-2","sectionId":"mathematical-assumptions","statement":"For an independent lattice illustration, set hopping and pairing scales to one and use $H(\\mathbf{k})=\\mathbf{h}(\\mathbf{k})\\cdot\\boldsymbol{\\tau}$, with $\\mathbf{h}=(\\sin k_x,\\chi\\sin k_y,-2\\cos k_x-2\\cos k_y-\\mu)$ and $\\chi=\\pm1$. In this Nambu convention the off-diagonal gap is $\\sin k_x-i\\chi\\sin k_y$. Define the occupied-band Berry connection as $\\mathbf{A}=i\\langle u_-|\\nabla_{\\mathbf{k}}u_-\\rangle$. With the orientation $(k_x,k_y)$, the lower-band Chern number for $\\chi=1$ is $0,-1,+1,0$ in the intervals separated by $\\mu=-4,0,4$. Reversing $\\chi$ reverses its sign.","evidenceType":"theory","sourceIds":["source-a33c639b7db4f05a","source-69d33a869fbbb0c6"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"The references supply topology context; the stated lattice convention and Chern calculation are local, not reproduced material results."},{"id":"scwiki-c7-claim-mathematical-assumptions-3","sectionId":"mathematical-assumptions","statement":"Sixteen calculations compare two mesh sizes, four chemical potentials and both handedness choices. Berry-link phases from explicit occupied eigenvectors agree with independently summed spherical solid angles. Four analytic gap-closing checks identify the excluded critical points; no gapped-band invariant is assigned there. The same complex gap can thus accompany a zero or nonzero Chern number. The calculation fixes the gap by hand and omits self-consistency, realistic bands and disorder.","evidenceType":"theory","sourceIds":["source-a33c639b7db4f05a"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Local mesh and analytic checks of the declared lattice Hamiltonian, not a primary-paper numerical replication."},{"id":"scwiki-c7-claim-mathematical-assumptions-4","sectionId":"mathematical-assumptions","statement":"A boundary calculation illustrates a different distinction. In Huang, Taylor and Kallin's ideal sharp-edge continuum treatment, occupied bound states and continuum states both contribute to the integrated current. For axial harmonic index $m=1$, their respective values are $1/2$ and $-1/4$ in units $\\hbar k_F^2/(4\\pi m^*)$ per spin component. For integer $m>1$, each integrated contribution vanishes in this limit despite the chiral edge branches. Independent angular quadrature reproduces the analytic sums for six values of $m$.","evidenceType":"theory","sourceIds":["source-25d6a80b561d5ad2"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"The ideal sharp-edge equations and assumptions are reported theory; their numerical angular quadrature is a local check."},{"id":"scwiki-c7-claim-mathematical-assumptions-5","sectionId":"mathematical-assumptions","statement":"That cancellation is not a universal absence of current in higher-order chiral pairing. The same paper shows that making the density vary slowly over distances much longer than the coherence length restores a current; lattice effects also matter. A sharp specular boundary, a smooth trap and a disordered multiband surface are different problems. The calculation above does not predict a magnetic-field amplitude for a real sample.","evidenceType":"theory","sourceIds":["source-25d6a80b561d5ad2"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c7-claim-experimental-criteria-1","sectionId":"experimental-criteria","statement":"Kerr rotation probes an optical response. In the clean Galilean-invariant limit, a spatially uniform electric field couples to center-of-mass motion and the charge Hall response vanishes even for complex pairing. At the single-particle level, the clean one-band current-vertex structure also gives a cancellation; multiband and impurity processes require separate treatment. Optical frequency, refractive response and geometry matter. Neither a null Kerr measurement nor one nonzero angle can be converted directly into a Chern number.","evidenceType":"theory","sourceIds":["source-a33c639b7db4f05a","source-69d33a869fbbb0c6"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c7-claim-experimental-criteria-2","sectionId":"experimental-criteria","statement":"For UPt3, retain Schemm and colleagues' distinction between the Kerr onset near 0.46 K and the upper susceptibility onset near 0.55 K, including optical heating and field-training controls. For Sr2RuO4, Pustogow and colleagues' oxygen NMR response constrains a simple fixed-d-vector interpretation. The later 2026 muon Knight-shift study also has field-dependent extraction and low-temperature magnetization limits. These are complementary constraints, not a license to restore a historically favored chiral p-wave label as settled fact.","evidenceType":"experiment","sourceIds":["source-d3edee404071ac81","source-c7f8b99a795aba41","source-c4434644191d0174"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c7-claim-experimental-criteria-3","sectionId":"experimental-criteria","statement":"Mielke and colleagues studied KV3Sb5, not the whole AV3Sb5 family interchangeably. The additional zero-field exponential relaxation rate $\\Gamma$ rises below charge order near 80 K and stays approximately constant across the superconducting transition near 1.1 K. This is not a statement about every muon signal: field-cooled transverse-field measurements at 5 and 10 mT show increased Gaussian damping below that superconducting transition, attributed to the flux-line lattice. Zero-field data alone do not establish time-reversal breaking: nuclear and electric-field-gradient effects can contribute. The authors combine zero-field and high-field evidence for their charge-order interpretation without resolving the orbital-current pattern. Residual-field compensation better than 30 mG and longitudinal decoupling at 50 G belong to the zero-field controls, not the transverse-field vortex measurement. The 80 K feature is not a superconducting chirality onset.","evidenceType":"experiment","sourceIds":["source-484da7abcc165cd3"],"conditions":"KV3Sb5 crystal measurements, not an AV3Sb5-family assignment. Separate zero-field exponential Gamma from field-cooled transverse-field Gaussian damping at 5 and 10 mT; charge order near 80 K is not Tc near 1.1 K.","verificationScope":"Selected primary text and local slice calculation checked 2026-10-19; no independent review, raw-data reanalysis or final-version equivalence claimed.","sourceRelationship":"direct-support","derivationNote":"Selected versioned experimental text separates zero-field exponential relaxation, transverse-field vortex damping and charge-order interpretation; no raw-data refit."},{"id":"scwiki-c7-claim-experimental-criteria-4","sectionId":"experimental-criteria","statement":"Chen and colleagues studied CsV3Sb5 with tunneling and Josephson spectroscopy. Their selected manuscript distinguishes bulk signatures near 2.8 K from a surface transition near 2.3 K, and uses a calibrated 300 mK electron temperature. Spatial gap and conductance modulations motivate their roton-PDW interpretation. The authors explicitly state that roton excitations were not directly observed. A proposed complex vortex–antivortex pattern is therefore an interpretation to test, not a measured uniform chiral d-wave order parameter.","evidenceType":"experiment","sourceIds":["source-057c43c86174f961"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c7-claim-limits-and-disputes-1","sectionId":"limits-and-disputes","statement":"Local magnitude alone does not resolve handedness: $|\\Delta|^2=|\\Delta^*|^2$. Four complex spatial examples verify this identity without making a tunneling-response model. A Josephson signal can support superconducting coherence, while determining orbital phase structure requires suitable reference geometry and controls. Fourier peaks, gap-depth maps and zero-bias peaks must retain their individual evidential roles.","evidenceType":"theory","sourceIds":["source-057c43c86174f961"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"Conjugation invariance is local algebra; the cited spectroscopy motivates the evidential distinction, not a derivation of the identity."},{"id":"scwiki-c7-claim-limits-and-disputes-2","sectionId":"limits-and-disputes","statement":"Do not require every time-reversal-breaking state to belong to a two-dimensional crystal representation. Complex combinations of distinct representations or interband phases can behave differently. Likewise, half-quantum vortices can occur in equal-spin-pairing states without establishing orbital chirality. A strain response or vortex signature must be interpreted in the actual symmetry and spin-orbit setting.","evidenceType":"theory","sourceIds":["source-69d33a869fbbb0c6","source-3c7aa9b4c8da3abf"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."},{"id":"scwiki-c7-claim-limits-and-disputes-3","sectionId":"limits-and-disputes","statement":"The preceding Chern numbers apply only to gapped two-dimensional Hamiltonians. A local three-dimensional extension illustrates the restriction: add $-2\\cos k_z$ to $h_z$ and fix $\\mu=-4$. Each fixed-$k_z$ slice has $\\mu_{\\rm eff}=-4+2\\cos k_z$. In the same occupied-band convention, $C(k_z)=-\\chi$ for $|k_z|<\\pi/2$ and zero for $\\pi/2<|k_z|\\leq\\pi$; it is undefined on the nodal slices $k_z=\\pm\\pi/2$. Solving all three components of $\\mathbf{h}=0$ gives only $(0,0,\\pm\\pi/2)$. Sixteen numerical comparisons and this analytic zero classification check the chosen model, not a material or a calculation reproduced from the references. Thus a single gapped two-dimensional integer cannot be assigned unchanged to this gapless three-dimensional example. Edge spectra, thermal transport, optical response and screened equilibrium charge currents still require their own calculations. Complete scientific acceptance, final-version comparisons, raw-data review and material-specific modelling remain open.","evidenceType":"theory","sourceIds":["source-69d33a869fbbb0c6"],"conditions":"Local spinless fixed-gap lattice: t=t_z=Delta=1, mu=-4, chi=+/-1, kz in [-pi,pi], occupied-band A=i<u-|grad u-> and oriented kx,ky. Chern numbers exclude gapless slices.","verificationScope":"Selected primary text and local slice calculation checked 2026-10-19; no independent review, raw-data reanalysis or final-version equivalence claimed.","sourceRelationship":"context-for-local-derivation","derivationNote":"The stacked lattice and slice calculation are new author illustrations. References supply topology and response context; no actual material invariant or measured current is inferred."},{"id":"scwiki-c7-claim-discovery-implications-1","sectionId":"discovery-implications","statement":"First identify the transition and its probe. Then specify the order-parameter basis, domain controls, bulk gap and Hamiltonian used for any topology claim. Compare positive and null evidence under compatible conditions, retaining charge-order and superconducting temperature scales separately. A useful discovery target is a state that satisfies these linked tests, not a material selected merely for a complex gap notation or a kagome lattice.","evidenceType":"theory","sourceIds":["source-a33c639b7db4f05a","source-d3edee404071ac81","source-484da7abcc165cd3","source-057c43c86174f961"],"conditions":"Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.","verificationScope":"Named primary passages and independent examples only; complete scientific acceptance remains open."}],"corrections":[{"id":"scwiki-c7-correction-1","description":"Restored both review authors, Kallin and Berlinsky, and used C. Mielke III attribution from the original record.","sourceId":"source-69d33a869fbbb0c6","scope":"Bibliographic identity; historical assignments remain bounded."},{"id":"scwiki-c7-correction-2","description":"Separated sharp-boundary current cancellation from soft-boundary behavior and from the bulk Chern invariant.","sourceId":"source-25d6a80b561d5ad2","scope":"Selected equations and independent angular quadrature."},{"id":"scwiki-c7-correction-3","description":"Kagome charge-order response, surface superconductivity and proposed roton-PDW interpretation retain distinct evidential roles.","sourceId":"source-057c43c86174f961","scope":"Selected methods/results; no universal chirality verdict."},{"id":"scwiki-c7-correction-4","description":"Specified zero-field exponential Gamma rather than all muon relaxation; retained the separate transverse-field superconducting damping increase.","sourceId":"source-484da7abcc165cd3","scope":"Selected author manuscript v2 text and captions, not raw-data refitting or complete final-version comparison."}],"equationChecks":[{"targetSectionId":"mathematical-assumptions","tex":"H(\\mathbf{k})=\\mathbf{h}(\\mathbf{k})\\cdot\\boldsymbol{\\tau}","check":"Sixteen occupied-eigenvector Berry-link and spherical-area comparisons, plus four analytic gap-closing checks; conventions explicit."},{"targetSectionId":"mathematical-assumptions","tex":"\\hbar k_F^2/(4\\pi m^*)","check":"Six sharp-edge angular integrals versus analytic bound-state and continuum sums."},{"targetSectionId":"limits-and-disputes","tex":"|\\Delta|^2=|\\Delta^*|^2","check":"Four complex spatial examples; three separate global-phase tests, without a microscopic stability claim."},{"targetSectionId":"limits-and-disputes","tex":"C(k_z)=-\\chi\\quad (|k_z|<\\pi/2)","check":"Local stacked lattice, 16 Berry-link versus sphere-area comparisons on gapped slices, analytic classification of all four in-plane sine-zero candidates, and two gap-closing points; no Chern number on nodal slices."}],"materialCases":[{"id":"scwiki-c7-material-1","formula":"KV3Sb5","sample":"Mielke et al. aligned crystal mosaic, muon charge-order response","recordType":"experiment","tcK":null,"reportedTemperatureK":80,"temperatureKind":"unassigned","temperatureNote":"Approximate charge-order scale, not superconducting Tc. The additional zero-field exponential Gamma remains approximately constant across Tc near 1.1 K; transverse-field vortex damping increases below Tc and is a different observable.","criterion":"charge-order-associated-relaxation","pressureRegime":"unknown","pressureGPa":null,"equilibrium":"unknown","probe":"Zero-field muon relaxation with residual-field compensation and longitudinal/high-field controls","sourceId":"source-484da7abcc165cd3"},{"id":"scwiki-c7-material-2","formula":"CsV3Sb5","sample":"Chen et al. tunneling study on prepared surfaces","recordType":"experiment","tcK":null,"reportedTemperatureK":0.3,"temperatureKind":"measurement","temperatureNote":"Calibrated electron temperature for spectroscopy, not a transition temperature. Surface and bulk transitions reported separately.","criterion":"spectroscopy-temperature","pressureRegime":"unknown","pressureGPa":null,"equilibrium":"unknown","probe":"STM/STS and Nb-tip Josephson spectroscopy; surface preparation and drift correction specified","sourceId":"source-057c43c86174f961"}],"body":"## Scope\n\nChiral pairing describes a superconducting order with a handed orbital structure, such as a complex combination of two directional basis functions. Three questions must remain separate: whether time reversal is broken, whether the pairing is chiral, and whether a specified gapped Hamiltonian has a nonzero topological invariant. A Kerr signal, a complex coefficient or a named lattice does not answer all three. Kallin and Berlinsky's 2016 review is a historical guide; its candidate-material assignments are not current confirmations. [R2](#ref-source-69d33a869fbbb0c6)\n\nBroken time reversal need not mean orbital chirality. For example, frustrated interband pair scattering can favor complex relative phases among otherwise s-wave gaps. Stanev and Tešanović studied this possibility in a three-band model. An illustrative vector of gaps on three inequivalent labelled bands, $(1,e^{2\\pi i/3},e^{-2\\pi i/3})$, cannot be made equal to its conjugate by one common phase, although each component has no angular winding. This statement assumes only a common U(1) freedom; it is not a stability calculation or a material assignment. [R8](#ref-source-3c7aa9b4c8da3abf)\n\n## Mathematical assumptions\n\nRead and Green distinguish weak- and strong-pairing phases of a two-dimensional spinless complex p-wave model. Their continuum chemical potential is measured relative to the band bottom: its zero marks a gap-closing transition in that model. The broken symmetry alone does not distinguish the two phases. This continuum criterion must not be copied unchanged into a lattice or multiband calculation. [R1](#ref-source-a33c639b7db4f05a)\n\nFor an independent lattice illustration, set hopping and pairing scales to one and use $H(\\mathbf{k})=\\mathbf{h}(\\mathbf{k})\\cdot\\boldsymbol{\\tau}$, with $\\mathbf{h}=(\\sin k_x,\\chi\\sin k_y,-2\\cos k_x-2\\cos k_y-\\mu)$ and $\\chi=\\pm1$. In this Nambu convention the off-diagonal gap is $\\sin k_x-i\\chi\\sin k_y$. Define the occupied-band Berry connection as $\\mathbf{A}=i\\langle u_-|\\nabla_{\\mathbf{k}}u_-\\rangle$. With the orientation $(k_x,k_y)$, the lower-band Chern number for $\\chi=1$ is $0,-1,+1,0$ in the intervals separated by $\\mu=-4,0,4$. Reversing $\\chi$ reverses its sign. [R1](#ref-source-a33c639b7db4f05a) [R2](#ref-source-69d33a869fbbb0c6)\n\nSixteen calculations compare two mesh sizes, four chemical potentials and both handedness choices. Berry-link phases from explicit occupied eigenvectors agree with independently summed spherical solid angles. Four analytic gap-closing checks identify the excluded critical points; no gapped-band invariant is assigned there. The same complex gap can thus accompany a zero or nonzero Chern number. The calculation fixes the gap by hand and omits self-consistency, realistic bands and disorder. [R1](#ref-source-a33c639b7db4f05a)\n\nA boundary calculation illustrates a different distinction. In Huang, Taylor and Kallin's ideal sharp-edge continuum treatment, occupied bound states and continuum states both contribute to the integrated current. For axial harmonic index $m=1$, their respective values are $1/2$ and $-1/4$ in units $\\hbar k_F^2/(4\\pi m^*)$ per spin component. For integer $m>1$, each integrated contribution vanishes in this limit despite the chiral edge branches. Independent angular quadrature reproduces the analytic sums for six values of $m$. [R3](#ref-source-25d6a80b561d5ad2)\n\nThat cancellation is not a universal absence of current in higher-order chiral pairing. The same paper shows that making the density vary slowly over distances much longer than the coherence length restores a current; lattice effects also matter. A sharp specular boundary, a smooth trap and a disordered multiband surface are different problems. The calculation above does not predict a magnetic-field amplitude for a real sample. [R3](#ref-source-25d6a80b561d5ad2)\n\n## Experimental criteria\n\nKerr rotation probes an optical response. In the clean Galilean-invariant limit, a spatially uniform electric field couples to center-of-mass motion and the charge Hall response vanishes even for complex pairing. At the single-particle level, the clean one-band current-vertex structure also gives a cancellation; multiband and impurity processes require separate treatment. Optical frequency, refractive response and geometry matter. Neither a null Kerr measurement nor one nonzero angle can be converted directly into a Chern number. [R1](#ref-source-a33c639b7db4f05a) [R2](#ref-source-69d33a869fbbb0c6)\n\nFor UPt3, retain Schemm and colleagues' distinction between the Kerr onset near 0.46 K and the upper susceptibility onset near 0.55 K, including optical heating and field-training controls. For Sr2RuO4, Pustogow and colleagues' oxygen NMR response constrains a simple fixed-d-vector interpretation. The later 2026 muon Knight-shift study also has field-dependent extraction and low-temperature magnetization limits. These are complementary constraints, not a license to restore a historically favored chiral p-wave label as settled fact. [R4](#ref-source-d3edee404071ac81) [R5](#ref-source-c7f8b99a795aba41) [R9](#ref-source-c4434644191d0174)\n\nMielke and colleagues studied KV3Sb5, not the whole AV3Sb5 family interchangeably. The additional zero-field exponential relaxation rate $\\Gamma$ rises below charge order near 80 K and stays approximately constant across the superconducting transition near 1.1 K. This is not a statement about every muon signal: field-cooled transverse-field measurements at 5 and 10 mT show increased Gaussian damping below that superconducting transition, attributed to the flux-line lattice. Zero-field data alone do not establish time-reversal breaking: nuclear and electric-field-gradient effects can contribute. The authors combine zero-field and high-field evidence for their charge-order interpretation without resolving the orbital-current pattern. Residual-field compensation better than 30 mG and longitudinal decoupling at 50 G belong to the zero-field controls, not the transverse-field vortex measurement. The 80 K feature is not a superconducting chirality onset. [R6](#ref-source-484da7abcc165cd3)\n\nChen and colleagues studied CsV3Sb5 with tunneling and Josephson spectroscopy. Their selected manuscript distinguishes bulk signatures near 2.8 K from a surface transition near 2.3 K, and uses a calibrated 300 mK electron temperature. Spatial gap and conductance modulations motivate their roton-PDW interpretation. The authors explicitly state that roton excitations were not directly observed. A proposed complex vortex–antivortex pattern is therefore an interpretation to test, not a measured uniform chiral d-wave order parameter. [R7](#ref-source-057c43c86174f961)\n\n## Limits and disputes\n\nLocal magnitude alone does not resolve handedness: $|\\Delta|^2=|\\Delta^*|^2$. Four complex spatial examples verify this identity without making a tunneling-response model. A Josephson signal can support superconducting coherence, while determining orbital phase structure requires suitable reference geometry and controls. Fourier peaks, gap-depth maps and zero-bias peaks must retain their individual evidential roles. [R7](#ref-source-057c43c86174f961)\n\nDo not require every time-reversal-breaking state to belong to a two-dimensional crystal representation. Complex combinations of distinct representations or interband phases can behave differently. Likewise, half-quantum vortices can occur in equal-spin-pairing states without establishing orbital chirality. A strain response or vortex signature must be interpreted in the actual symmetry and spin-orbit setting. [R2](#ref-source-69d33a869fbbb0c6) [R8](#ref-source-3c7aa9b4c8da3abf)\n\nThe preceding Chern numbers apply only to gapped two-dimensional Hamiltonians. A local three-dimensional extension illustrates the restriction: add $-2\\cos k_z$ to $h_z$ and fix $\\mu=-4$. Each fixed-$k_z$ slice has $\\mu_{\\rm eff}=-4+2\\cos k_z$. In the same occupied-band convention, $C(k_z)=-\\chi$ for $|k_z|<\\pi/2$ and zero for $\\pi/2<|k_z|\\leq\\pi$; it is undefined on the nodal slices $k_z=\\pm\\pi/2$. Solving all three components of $\\mathbf{h}=0$ gives only $(0,0,\\pm\\pi/2)$. Sixteen numerical comparisons and this analytic zero classification check the chosen model, not a material or a calculation reproduced from the references. Thus a single gapped two-dimensional integer cannot be assigned unchanged to this gapless three-dimensional example. Edge spectra, thermal transport, optical response and screened equilibrium charge currents still require their own calculations. Complete scientific acceptance, final-version comparisons, raw-data review and material-specific modelling remain open. [R2](#ref-source-69d33a869fbbb0c6)\n\n## Discovery implications\n\nFirst identify the transition and its probe. Then specify the order-parameter basis, domain controls, bulk gap and Hamiltonian used for any topology claim. Compare positive and null evidence under compatible conditions, retaining charge-order and superconducting temperature scales separately. A useful discovery target is a state that satisfies these linked tests, not a material selected merely for a complex gap notation or a kagome lattice. [R1](#ref-source-a33c639b7db4f05a) [R4](#ref-source-d3edee404071ac81) [R6](#ref-source-484da7abcc165cd3) [R7](#ref-source-057c43c86174f961)\n\n## Related topics\n\nContinue with [p-wave spin structure](/scwiki/p-wave), [f-wave candidate states](/scwiki/f-wave), [d-wave phase tests](/scwiki/d-wave), [topological superconductivity](/scwiki/topological-superconductivity), and [charge and density-wave pairing](/scwiki/charge-fluctuation-pairing).","revision":"359431ef5b4c1b075aaf","sections":[{"id":"scope","title":"Scope","markdown":"## Scope\n\nChiral pairing describes a superconducting order with a handed orbital structure, such as a complex combination of two directional basis functions. Three questions must remain separate: whether time reversal is broken, whether the pairing is chiral, and whether a specified gapped Hamiltonian has a nonzero topological invariant. A Kerr signal, a complex coefficient or a named lattice does not answer all three. Kallin and Berlinsky's 2016 review is a historical guide; its candidate-material assignments are not current confirmations. [R2](#ref-source-69d33a869fbbb0c6)\n\nBroken time reversal need not mean orbital chirality. For example, frustrated interband pair scattering can favor complex relative phases among otherwise s-wave gaps. Stanev and Tešanović studied this possibility in a three-band model. An illustrative vector of gaps on three inequivalent labelled bands, $(1,e^{2\\pi i/3},e^{-2\\pi i/3})$, cannot be made equal to its conjugate by one common phase, although each component has no angular winding. This statement assumes only a common U(1) freedom; it is not a stability calculation or a material assignment. [R8](#ref-source-3c7aa9b4c8da3abf)"},{"id":"mathematical-assumptions","title":"Mathematical assumptions","markdown":"## Mathematical assumptions\n\nRead and Green distinguish weak- and strong-pairing phases of a two-dimensional spinless complex p-wave model. Their continuum chemical potential is measured relative to the band bottom: its zero marks a gap-closing transition in that model. The broken symmetry alone does not distinguish the two phases. This continuum criterion must not be copied unchanged into a lattice or multiband calculation. [R1](#ref-source-a33c639b7db4f05a)\n\nFor an independent lattice illustration, set hopping and pairing scales to one and use $H(\\mathbf{k})=\\mathbf{h}(\\mathbf{k})\\cdot\\boldsymbol{\\tau}$, with $\\mathbf{h}=(\\sin k_x,\\chi\\sin k_y,-2\\cos k_x-2\\cos k_y-\\mu)$ and $\\chi=\\pm1$. In this Nambu convention the off-diagonal gap is $\\sin k_x-i\\chi\\sin k_y$. Define the occupied-band Berry connection as $\\mathbf{A}=i\\langle u_-|\\nabla_{\\mathbf{k}}u_-\\rangle$. With the orientation $(k_x,k_y)$, the lower-band Chern number for $\\chi=1$ is $0,-1,+1,0$ in the intervals separated by $\\mu=-4,0,4$. Reversing $\\chi$ reverses its sign. [R1](#ref-source-a33c639b7db4f05a) [R2](#ref-source-69d33a869fbbb0c6)\n\nSixteen calculations compare two mesh sizes, four chemical potentials and both handedness choices. Berry-link phases from explicit occupied eigenvectors agree with independently summed spherical solid angles. Four analytic gap-closing checks identify the excluded critical points; no gapped-band invariant is assigned there. The same complex gap can thus accompany a zero or nonzero Chern number. The calculation fixes the gap by hand and omits self-consistency, realistic bands and disorder. [R1](#ref-source-a33c639b7db4f05a)\n\nA boundary calculation illustrates a different distinction. In Huang, Taylor and Kallin's ideal sharp-edge continuum treatment, occupied bound states and continuum states both contribute to the integrated current. For axial harmonic index $m=1$, their respective values are $1/2$ and $-1/4$ in units $\\hbar k_F^2/(4\\pi m^*)$ per spin component. For integer $m>1$, each integrated contribution vanishes in this limit despite the chiral edge branches. Independent angular quadrature reproduces the analytic sums for six values of $m$. [R3](#ref-source-25d6a80b561d5ad2)\n\nThat cancellation is not a universal absence of current in higher-order chiral pairing. The same paper shows that making the density vary slowly over distances much longer than the coherence length restores a current; lattice effects also matter. A sharp specular boundary, a smooth trap and a disordered multiband surface are different problems. The calculation above does not predict a magnetic-field amplitude for a real sample. [R3](#ref-source-25d6a80b561d5ad2)"},{"id":"experimental-criteria","title":"Experimental criteria","markdown":"## Experimental criteria\n\nKerr rotation probes an optical response. In the clean Galilean-invariant limit, a spatially uniform electric field couples to center-of-mass motion and the charge Hall response vanishes even for complex pairing. At the single-particle level, the clean one-band current-vertex structure also gives a cancellation; multiband and impurity processes require separate treatment. Optical frequency, refractive response and geometry matter. Neither a null Kerr measurement nor one nonzero angle can be converted directly into a Chern number. [R1](#ref-source-a33c639b7db4f05a) [R2](#ref-source-69d33a869fbbb0c6)\n\nFor UPt3, retain Schemm and colleagues' distinction between the Kerr onset near 0.46 K and the upper susceptibility onset near 0.55 K, including optical heating and field-training controls. For Sr2RuO4, Pustogow and colleagues' oxygen NMR response constrains a simple fixed-d-vector interpretation. The later 2026 muon Knight-shift study also has field-dependent extraction and low-temperature magnetization limits. These are complementary constraints, not a license to restore a historically favored chiral p-wave label as settled fact. [R4](#ref-source-d3edee404071ac81) [R5](#ref-source-c7f8b99a795aba41) [R9](#ref-source-c4434644191d0174)\n\nMielke and colleagues studied KV3Sb5, not the whole AV3Sb5 family interchangeably. The additional zero-field exponential relaxation rate $\\Gamma$ rises below charge order near 80 K and stays approximately constant across the superconducting transition near 1.1 K. This is not a statement about every muon signal: field-cooled transverse-field measurements at 5 and 10 mT show increased Gaussian damping below that superconducting transition, attributed to the flux-line lattice. Zero-field data alone do not establish time-reversal breaking: nuclear and electric-field-gradient effects can contribute. The authors combine zero-field and high-field evidence for their charge-order interpretation without resolving the orbital-current pattern. Residual-field compensation better than 30 mG and longitudinal decoupling at 50 G belong to the zero-field controls, not the transverse-field vortex measurement. The 80 K feature is not a superconducting chirality onset. [R6](#ref-source-484da7abcc165cd3)\n\nChen and colleagues studied CsV3Sb5 with tunneling and Josephson spectroscopy. Their selected manuscript distinguishes bulk signatures near 2.8 K from a surface transition near 2.3 K, and uses a calibrated 300 mK electron temperature. Spatial gap and conductance modulations motivate their roton-PDW interpretation. The authors explicitly state that roton excitations were not directly observed. A proposed complex vortex–antivortex pattern is therefore an interpretation to test, not a measured uniform chiral d-wave order parameter. [R7](#ref-source-057c43c86174f961)"},{"id":"limits-and-disputes","title":"Limits and disputes","markdown":"## Limits and disputes\n\nLocal magnitude alone does not resolve handedness: $|\\Delta|^2=|\\Delta^*|^2$. Four complex spatial examples verify this identity without making a tunneling-response model. A Josephson signal can support superconducting coherence, while determining orbital phase structure requires suitable reference geometry and controls. Fourier peaks, gap-depth maps and zero-bias peaks must retain their individual evidential roles. [R7](#ref-source-057c43c86174f961)\n\nDo not require every time-reversal-breaking state to belong to a two-dimensional crystal representation. Complex combinations of distinct representations or interband phases can behave differently. Likewise, half-quantum vortices can occur in equal-spin-pairing states without establishing orbital chirality. A strain response or vortex signature must be interpreted in the actual symmetry and spin-orbit setting. [R2](#ref-source-69d33a869fbbb0c6) [R8](#ref-source-3c7aa9b4c8da3abf)\n\nThe preceding Chern numbers apply only to gapped two-dimensional Hamiltonians. A local three-dimensional extension illustrates the restriction: add $-2\\cos k_z$ to $h_z$ and fix $\\mu=-4$. Each fixed-$k_z$ slice has $\\mu_{\\rm eff}=-4+2\\cos k_z$. In the same occupied-band convention, $C(k_z)=-\\chi$ for $|k_z|<\\pi/2$ and zero for $\\pi/2<|k_z|\\leq\\pi$; it is undefined on the nodal slices $k_z=\\pm\\pi/2$. Solving all three components of $\\mathbf{h}=0$ gives only $(0,0,\\pm\\pi/2)$. Sixteen numerical comparisons and this analytic zero classification check the chosen model, not a material or a calculation reproduced from the references. Thus a single gapped two-dimensional integer cannot be assigned unchanged to this gapless three-dimensional example. Edge spectra, thermal transport, optical response and screened equilibrium charge currents still require their own calculations. Complete scientific acceptance, final-version comparisons, raw-data review and material-specific modelling remain open. [R2](#ref-source-69d33a869fbbb0c6)"},{"id":"discovery-implications","title":"Discovery implications","markdown":"## Discovery implications\n\nFirst identify the transition and its probe. Then specify the order-parameter basis, domain controls, bulk gap and Hamiltonian used for any topology claim. Compare positive and null evidence under compatible conditions, retaining charge-order and superconducting temperature scales separately. A useful discovery target is a state that satisfies these linked tests, not a material selected merely for a complex gap notation or a kagome lattice. [R1](#ref-source-a33c639b7db4f05a) [R4](#ref-source-d3edee404071ac81) [R6](#ref-source-484da7abcc165cd3) [R7](#ref-source-057c43c86174f961)"},{"id":"related-topics","title":"Related topics","markdown":"## Related topics\n\nContinue with [p-wave spin structure](/scwiki/p-wave), [f-wave candidate states](/scwiki/f-wave), [d-wave phase tests](/scwiki/d-wave), [topological superconductivity](/scwiki/topological-superconductivity), and [charge and density-wave pairing](/scwiki/charge-fluctuation-pairing)."},{"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| KV3Sb5; Mielke et al. aligned crystal mosaic, muon charge-order response | 80 K; Approximate charge-order scale, not superconducting Tc. The additional zero-field exponential Gamma remains approximately constant across Tc near 1.1 K; transverse-field vortex damping increases below Tc and is a different observable. | unassigned; charge-order-associated-relaxation | Zero-field muon relaxation with residual-field compensation and longitudinal/high-field controls | unknown; unknown GPa | unknown | experiment; [source](#ref-source-484da7abcc165cd3) |\n| CsV3Sb5; Chen et al. tunneling study on prepared surfaces | 0.3 K; Calibrated electron temperature for spectroscopy, not a transition temperature. Surface and bulk transitions reported separately. | measurement; spectroscopy-temperature | STM/STS and Nb-tip Josephson spectroscopy; surface preparation and drift correction specified | unknown; unknown GPa | unknown | experiment; [source](#ref-source-057c43c86174f961) |"}],"markdown":"# Chiral pairing: time reversal, topology and boundary evidence\n\nSeparate orbital handedness, broken time reversal, band topology and the signals used to test them.\n\n## Scope\n\nChiral pairing describes a superconducting order with a handed orbital structure, such as a complex combination of two directional basis functions. Three questions must remain separate: whether time reversal is broken, whether the pairing is chiral, and whether a specified gapped Hamiltonian has a nonzero topological invariant. A Kerr signal, a complex coefficient or a named lattice does not answer all three. Kallin and Berlinsky's 2016 review is a historical guide; its candidate-material assignments are not current confirmations. [R2](#ref-source-69d33a869fbbb0c6)\n\nBroken time reversal need not mean orbital chirality. For example, frustrated interband pair scattering can favor complex relative phases among otherwise s-wave gaps. Stanev and Tešanović studied this possibility in a three-band model. An illustrative vector of gaps on three inequivalent labelled bands, $(1,e^{2\\pi i/3},e^{-2\\pi i/3})$, cannot be made equal to its conjugate by one common phase, although each component has no angular winding. This statement assumes only a common U(1) freedom; it is not a stability calculation or a material assignment. [R8](#ref-source-3c7aa9b4c8da3abf)\n\n## Mathematical assumptions\n\nRead and Green distinguish weak- and strong-pairing phases of a two-dimensional spinless complex p-wave model. Their continuum chemical potential is measured relative to the band bottom: its zero marks a gap-closing transition in that model. The broken symmetry alone does not distinguish the two phases. This continuum criterion must not be copied unchanged into a lattice or multiband calculation. [R1](#ref-source-a33c639b7db4f05a)\n\nFor an independent lattice illustration, set hopping and pairing scales to one and use $H(\\mathbf{k})=\\mathbf{h}(\\mathbf{k})\\cdot\\boldsymbol{\\tau}$, with $\\mathbf{h}=(\\sin k_x,\\chi\\sin k_y,-2\\cos k_x-2\\cos k_y-\\mu)$ and $\\chi=\\pm1$. In this Nambu convention the off-diagonal gap is $\\sin k_x-i\\chi\\sin k_y$. Define the occupied-band Berry connection as $\\mathbf{A}=i\\langle u_-|\\nabla_{\\mathbf{k}}u_-\\rangle$. With the orientation $(k_x,k_y)$, the lower-band Chern number for $\\chi=1$ is $0,-1,+1,0$ in the intervals separated by $\\mu=-4,0,4$. Reversing $\\chi$ reverses its sign. [R1](#ref-source-a33c639b7db4f05a) [R2](#ref-source-69d33a869fbbb0c6)\n\nSixteen calculations compare two mesh sizes, four chemical potentials and both handedness choices. Berry-link phases from explicit occupied eigenvectors agree with independently summed spherical solid angles. Four analytic gap-closing checks identify the excluded critical points; no gapped-band invariant is assigned there. The same complex gap can thus accompany a zero or nonzero Chern number. The calculation fixes the gap by hand and omits self-consistency, realistic bands and disorder. [R1](#ref-source-a33c639b7db4f05a)\n\nA boundary calculation illustrates a different distinction. In Huang, Taylor and Kallin's ideal sharp-edge continuum treatment, occupied bound states and continuum states both contribute to the integrated current. For axial harmonic index $m=1$, their respective values are $1/2$ and $-1/4$ in units $\\hbar k_F^2/(4\\pi m^*)$ per spin component. For integer $m>1$, each integrated contribution vanishes in this limit despite the chiral edge branches. Independent angular quadrature reproduces the analytic sums for six values of $m$. [R3](#ref-source-25d6a80b561d5ad2)\n\nThat cancellation is not a universal absence of current in higher-order chiral pairing. The same paper shows that making the density vary slowly over distances much longer than the coherence length restores a current; lattice effects also matter. A sharp specular boundary, a smooth trap and a disordered multiband surface are different problems. The calculation above does not predict a magnetic-field amplitude for a real sample. [R3](#ref-source-25d6a80b561d5ad2)\n\n## Experimental criteria\n\nKerr rotation probes an optical response. In the clean Galilean-invariant limit, a spatially uniform electric field couples to center-of-mass motion and the charge Hall response vanishes even for complex pairing. At the single-particle level, the clean one-band current-vertex structure also gives a cancellation; multiband and impurity processes require separate treatment. Optical frequency, refractive response and geometry matter. Neither a null Kerr measurement nor one nonzero angle can be converted directly into a Chern number. [R1](#ref-source-a33c639b7db4f05a) [R2](#ref-source-69d33a869fbbb0c6)\n\nFor UPt3, retain Schemm and colleagues' distinction between the Kerr onset near 0.46 K and the upper susceptibility onset near 0.55 K, including optical heating and field-training controls. For Sr2RuO4, Pustogow and colleagues' oxygen NMR response constrains a simple fixed-d-vector interpretation. The later 2026 muon Knight-shift study also has field-dependent extraction and low-temperature magnetization limits. These are complementary constraints, not a license to restore a historically favored chiral p-wave label as settled fact. [R4](#ref-source-d3edee404071ac81) [R5](#ref-source-c7f8b99a795aba41) [R9](#ref-source-c4434644191d0174)\n\nMielke and colleagues studied KV3Sb5, not the whole AV3Sb5 family interchangeably. The additional zero-field exponential relaxation rate $\\Gamma$ rises below charge order near 80 K and stays approximately constant across the superconducting transition near 1.1 K. This is not a statement about every muon signal: field-cooled transverse-field measurements at 5 and 10 mT show increased Gaussian damping below that superconducting transition, attributed to the flux-line lattice. Zero-field data alone do not establish time-reversal breaking: nuclear and electric-field-gradient effects can contribute. The authors combine zero-field and high-field evidence for their charge-order interpretation without resolving the orbital-current pattern. Residual-field compensation better than 30 mG and longitudinal decoupling at 50 G belong to the zero-field controls, not the transverse-field vortex measurement. The 80 K feature is not a superconducting chirality onset. [R6](#ref-source-484da7abcc165cd3)\n\nChen and colleagues studied CsV3Sb5 with tunneling and Josephson spectroscopy. Their selected manuscript distinguishes bulk signatures near 2.8 K from a surface transition near 2.3 K, and uses a calibrated 300 mK electron temperature. Spatial gap and conductance modulations motivate their roton-PDW interpretation. The authors explicitly state that roton excitations were not directly observed. A proposed complex vortex–antivortex pattern is therefore an interpretation to test, not a measured uniform chiral d-wave order parameter. [R7](#ref-source-057c43c86174f961)\n\n## Limits and disputes\n\nLocal magnitude alone does not resolve handedness: $|\\Delta|^2=|\\Delta^*|^2$. Four complex spatial examples verify this identity without making a tunneling-response model. A Josephson signal can support superconducting coherence, while determining orbital phase structure requires suitable reference geometry and controls. Fourier peaks, gap-depth maps and zero-bias peaks must retain their individual evidential roles. [R7](#ref-source-057c43c86174f961)\n\nDo not require every time-reversal-breaking state to belong to a two-dimensional crystal representation. Complex combinations of distinct representations or interband phases can behave differently. Likewise, half-quantum vortices can occur in equal-spin-pairing states without establishing orbital chirality. A strain response or vortex signature must be interpreted in the actual symmetry and spin-orbit setting. [R2](#ref-source-69d33a869fbbb0c6) [R8](#ref-source-3c7aa9b4c8da3abf)\n\nThe preceding Chern numbers apply only to gapped two-dimensional Hamiltonians. A local three-dimensional extension illustrates the restriction: add $-2\\cos k_z$ to $h_z$ and fix $\\mu=-4$. Each fixed-$k_z$ slice has $\\mu_{\\rm eff}=-4+2\\cos k_z$. In the same occupied-band convention, $C(k_z)=-\\chi$ for $|k_z|<\\pi/2$ and zero for $\\pi/2<|k_z|\\leq\\pi$; it is undefined on the nodal slices $k_z=\\pm\\pi/2$. Solving all three components of $\\mathbf{h}=0$ gives only $(0,0,\\pm\\pi/2)$. Sixteen numerical comparisons and this analytic zero classification check the chosen model, not a material or a calculation reproduced from the references. Thus a single gapped two-dimensional integer cannot be assigned unchanged to this gapless three-dimensional example. Edge spectra, thermal transport, optical response and screened equilibrium charge currents still require their own calculations. Complete scientific acceptance, final-version comparisons, raw-data review and material-specific modelling remain open. [R2](#ref-source-69d33a869fbbb0c6)\n\n## Discovery implications\n\nFirst identify the transition and its probe. Then specify the order-parameter basis, domain controls, bulk gap and Hamiltonian used for any topology claim. Compare positive and null evidence under compatible conditions, retaining charge-order and superconducting temperature scales separately. A useful discovery target is a state that satisfies these linked tests, not a material selected merely for a complex gap notation or a kagome lattice. [R1](#ref-source-a33c639b7db4f05a) [R4](#ref-source-d3edee404071ac81) [R6](#ref-source-484da7abcc165cd3) [R7](#ref-source-057c43c86174f961)\n\n## Related topics\n\nContinue with [p-wave spin structure](/scwiki/p-wave), [f-wave candidate states](/scwiki/f-wave), [d-wave phase tests](/scwiki/d-wave), [topological superconductivity](/scwiki/topological-superconductivity), and [charge and density-wave pairing](/scwiki/charge-fluctuation-pairing).\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| KV3Sb5; Mielke et al. aligned crystal mosaic, muon charge-order response | 80 K; Approximate charge-order scale, not superconducting Tc. The additional zero-field exponential Gamma remains approximately constant across Tc near 1.1 K; transverse-field vortex damping increases below Tc and is a different observable. | unassigned; charge-order-associated-relaxation | Zero-field muon relaxation with residual-field compensation and longitudinal/high-field controls | unknown; unknown GPa | unknown | experiment; [source](#ref-source-484da7abcc165cd3) |\n| CsV3Sb5; Chen et al. tunneling study on prepared surfaces | 0.3 K; Calibrated electron temperature for spectroscopy, not a transition temperature. Surface and bulk transitions reported separately. | measurement; spectroscopy-temperature | STM/STS and Nb-tip Josephson spectroscopy; surface preparation and drift correction specified | unknown; unknown GPa | unknown | experiment; [source](#ref-source-057c43c86174f961) |\n\n## Sources\n\n### Ref source-a33c639b7db4f05a\n\n[R1] Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect (2000). [DOI 10.1103/PhysRevB.61.10267](https://doi.org/10.1103/PhysRevB.61.10267). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-18. Selected pp. 4–7: continuum assumptions, weak/strong topology and charge Hall distinction. Version 3 January 2000; final-version comparison open. Local lattice example is independent.\n### Ref source-69d33a869fbbb0c6\n\n[R2] Chiral superconductors (2016). [DOI 10.1088/0034-4885/79/5/054502](https://doi.org/10.1088/0034-4885/79/5/054502). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-19. Kallin and Berlinsky, version 2 April 2016. Selected pp. 5–7 and 15–17: multiband formalism, half-quantum vortices and optical response. Historical candidate assignments are not current confirmations. Follow-up: III.2, Eqs.16-21, edge modes versus nonquantized charge current and screening. Local stacked-slice example is not attributed to this review.\n### Ref source-25d6a80b561d5ad2\n\n[R3] Vanishing edge currents in non-p-wave topological chiral superconductors (2014). [DOI 10.1103/PhysRevB.90.224519](https://doi.org/10.1103/PhysRevB.90.224519). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-18. Selected pp. 2–3; p. 3 visually checked. Version 2 marked published version by authors. Independent angular quadrature checks Eqs. 6–11; full lattice/self-consistent calculations not reproduced.\n### Ref source-d3edee404071ac81\n\n[R4] 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-c7f8b99a795aba41\n\n[R5] Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance (2019). [DOI 10.1038/s41586-019-1596-2](https://doi.org/10.1038/s41586-019-1596-2). Publication: journal-article. Check: author-manuscript-scoped-check. Checked: 2026-10-18. Pustogow et al. v1 selected pp.4–7; p.7 RF-energy and heating passage visually checked. Zero-strain0.7107T versus strained1.9980T conditions distinguished. Published identity checked; full final-version equivalence and subsequent literature survey not completed.\n### Ref source-484da7abcc165cd3\n\n[R6] Time-reversal symmetry-breaking charge order in a kagome superconductor (2022). [DOI 10.1038/s41586-021-04327-z](https://doi.org/10.1038/s41586-021-04327-z). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-19. Selected pp. 2–3 and 6 plus experimental-methods passages; Fig. 1 on p. 2 visually checked. December 2021 author version 2, published February 2022. Nuclear/electronic and charge/superconducting distinctions retained; full final-version/data review open. Follow-up: v2 text, Magnetic Response Across Charge Order, Unconventional Superconductivity, Eq.1, Fig.1/3 captions and Methods. Gamma versus transverse Gaussian damping distinguished. Publisher page was inaccessible; no new final-version or figure-pixel review.\n### Ref source-057c43c86174f961\n\n[R7] Roton pair density wave in a strong-coupling kagome superconductor (2021). [DOI 10.1038/s41586-021-03983-5](https://doi.org/10.1038/s41586-021-03983-5). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-18. Selected pp. 3–7 and methods pp. 17–18; Fig. 3 on p. 14 visually checked. Author version 2 September 2021 has a different title and mistyped journal volume 559; publisher confirms Nature 599. Final-version/raw-data comparison open.\n### Ref source-3c7aa9b4c8da3abf\n\n[R8] Three-band superconductivity and the order parameter that breaks time-reversal symmetry (2010). [DOI 10.1103/PhysRevB.81.134522](https://doi.org/10.1103/PhysRevB.81.134522). Publication: journal-article. Check: publisher-abstract-check. Checked: 2026-10-18. Published 23 April 2010. Abstract supports frustrated three-band TRSB possibility; full phase diagram not reviewed. Local fixed-phase example is illustrative only.\n### Ref source-c4434644191d0174\n\n[R9] Muon Knight Shift as a Precise Probe of the Superconducting Symmetry of Sr2RuO4 (2026). [DOI 10.1103/sgcz-9rc7](https://doi.org/10.1103/sgcz-9rc7). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-18. Selected pp.1–5 and Fig.3 visually checked. Published9February2026; manuscript16December2025. Same-crystal field-cooled subtraction and0.5K lower magnetization limit retained. Supplement, final-version comparison and raw-data reproduction open.\n\n## Claim evidence and conditions\n\n### scwiki-c7-claim-scope-1\n\nChiral pairing describes a superconducting order with a handed orbital structure, such as a complex combination of two directional basis functions. Three questions must remain separate: whether time reversal is broken, whether the pairing is chiral, and whether a specified gapped Hamiltonian has a nonzero topological invariant. A Kerr signal, a complex coefficient or a named lattice does not answer all three. Kallin and Berlinsky's 2016 review is a historical guide; its candidate-material assignments are not current confirmations.\n\nEvidence: theory. Sources: source-69d33a869fbbb0c6.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-scope-2\n\nBroken time reversal need not mean orbital chirality. For example, frustrated interband pair scattering can favor complex relative phases among otherwise s-wave gaps. Stanev and Tešanović studied this possibility in a three-band model. An illustrative vector of gaps on three inequivalent labelled bands, $(1,e^{2\\pi i/3},e^{-2\\pi i/3})$, cannot be made equal to its conjugate by one common phase, although each component has no angular winding. This statement assumes only a common U(1) freedom; it is not a stability calculation or a material assignment.\n\nEvidence: theory. Sources: source-3c7aa9b4c8da3abf.\n\nSource relationship: mixed-source-and-local-derivation. The three-band model motivates a local fixed-phase vector; the common-phase test is an author illustration, not the source phase diagram.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-mathematical-assumptions-1\n\nRead and Green distinguish weak- and strong-pairing phases of a two-dimensional spinless complex p-wave model. Their continuum chemical potential is measured relative to the band bottom: its zero marks a gap-closing transition in that model. The broken symmetry alone does not distinguish the two phases. This continuum criterion must not be copied unchanged into a lattice or multiband calculation.\n\nEvidence: theory. Sources: source-a33c639b7db4f05a.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-mathematical-assumptions-2\n\nFor an independent lattice illustration, set hopping and pairing scales to one and use $H(\\mathbf{k})=\\mathbf{h}(\\mathbf{k})\\cdot\\boldsymbol{\\tau}$, with $\\mathbf{h}=(\\sin k_x,\\chi\\sin k_y,-2\\cos k_x-2\\cos k_y-\\mu)$ and $\\chi=\\pm1$. In this Nambu convention the off-diagonal gap is $\\sin k_x-i\\chi\\sin k_y$. Define the occupied-band Berry connection as $\\mathbf{A}=i\\langle u_-|\\nabla_{\\mathbf{k}}u_-\\rangle$. With the orientation $(k_x,k_y)$, the lower-band Chern number for $\\chi=1$ is $0,-1,+1,0$ in the intervals separated by $\\mu=-4,0,4$. Reversing $\\chi$ reverses its sign.\n\nEvidence: theory. Sources: source-a33c639b7db4f05a, source-69d33a869fbbb0c6.\n\nSource relationship: context-for-local-derivation. The references supply topology context; the stated lattice convention and Chern calculation are local, not reproduced material results.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-mathematical-assumptions-3\n\nSixteen calculations compare two mesh sizes, four chemical potentials and both handedness choices. Berry-link phases from explicit occupied eigenvectors agree with independently summed spherical solid angles. Four analytic gap-closing checks identify the excluded critical points; no gapped-band invariant is assigned there. The same complex gap can thus accompany a zero or nonzero Chern number. The calculation fixes the gap by hand and omits self-consistency, realistic bands and disorder.\n\nEvidence: theory. Sources: source-a33c639b7db4f05a.\n\nSource relationship: context-for-local-derivation. Local mesh and analytic checks of the declared lattice Hamiltonian, not a primary-paper numerical replication.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-mathematical-assumptions-4\n\nA boundary calculation illustrates a different distinction. In Huang, Taylor and Kallin's ideal sharp-edge continuum treatment, occupied bound states and continuum states both contribute to the integrated current. For axial harmonic index $m=1$, their respective values are $1/2$ and $-1/4$ in units $\\hbar k_F^2/(4\\pi m^*)$ per spin component. For integer $m>1$, each integrated contribution vanishes in this limit despite the chiral edge branches. Independent angular quadrature reproduces the analytic sums for six values of $m$.\n\nEvidence: theory. Sources: source-25d6a80b561d5ad2.\n\nSource relationship: mixed-source-and-local-derivation. The ideal sharp-edge equations and assumptions are reported theory; their numerical angular quadrature is a local check.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-mathematical-assumptions-5\n\nThat cancellation is not a universal absence of current in higher-order chiral pairing. The same paper shows that making the density vary slowly over distances much longer than the coherence length restores a current; lattice effects also matter. A sharp specular boundary, a smooth trap and a disordered multiband surface are different problems. The calculation above does not predict a magnetic-field amplitude for a real sample.\n\nEvidence: theory. Sources: source-25d6a80b561d5ad2.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-experimental-criteria-1\n\nKerr rotation probes an optical response. In the clean Galilean-invariant limit, a spatially uniform electric field couples to center-of-mass motion and the charge Hall response vanishes even for complex pairing. At the single-particle level, the clean one-band current-vertex structure also gives a cancellation; multiband and impurity processes require separate treatment. Optical frequency, refractive response and geometry matter. Neither a null Kerr measurement nor one nonzero angle can be converted directly into a Chern number.\n\nEvidence: theory. Sources: source-a33c639b7db4f05a, source-69d33a869fbbb0c6.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-experimental-criteria-2\n\nFor UPt3, retain Schemm and colleagues' distinction between the Kerr onset near 0.46 K and the upper susceptibility onset near 0.55 K, including optical heating and field-training controls. For Sr2RuO4, Pustogow and colleagues' oxygen NMR response constrains a simple fixed-d-vector interpretation. The later 2026 muon Knight-shift study also has field-dependent extraction and low-temperature magnetization limits. These are complementary constraints, not a license to restore a historically favored chiral p-wave label as settled fact.\n\nEvidence: experiment. Sources: source-d3edee404071ac81, source-c7f8b99a795aba41, source-c4434644191d0174.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-experimental-criteria-3\n\nMielke and colleagues studied KV3Sb5, not the whole AV3Sb5 family interchangeably. The additional zero-field exponential relaxation rate $\\Gamma$ rises below charge order near 80 K and stays approximately constant across the superconducting transition near 1.1 K. This is not a statement about every muon signal: field-cooled transverse-field measurements at 5 and 10 mT show increased Gaussian damping below that superconducting transition, attributed to the flux-line lattice. Zero-field data alone do not establish time-reversal breaking: nuclear and electric-field-gradient effects can contribute. The authors combine zero-field and high-field evidence for their charge-order interpretation without resolving the orbital-current pattern. Residual-field compensation better than 30 mG and longitudinal decoupling at 50 G belong to the zero-field controls, not the transverse-field vortex measurement. The 80 K feature is not a superconducting chirality onset.\n\nEvidence: experiment. Sources: source-484da7abcc165cd3.\n\nSource relationship: direct-support. Selected versioned experimental text separates zero-field exponential relaxation, transverse-field vortex damping and charge-order interpretation; no raw-data refit.\n\nConditions: KV3Sb5 crystal measurements, not an AV3Sb5-family assignment. Separate zero-field exponential Gamma from field-cooled transverse-field Gaussian damping at 5 and 10 mT; charge order near 80 K is not Tc near 1.1 K.\n\nVerification scope: Selected primary text and local slice calculation checked 2026-10-19; no independent review, raw-data reanalysis or final-version equivalence claimed.\n\n### scwiki-c7-claim-experimental-criteria-4\n\nChen and colleagues studied CsV3Sb5 with tunneling and Josephson spectroscopy. Their selected manuscript distinguishes bulk signatures near 2.8 K from a surface transition near 2.3 K, and uses a calibrated 300 mK electron temperature. Spatial gap and conductance modulations motivate their roton-PDW interpretation. The authors explicitly state that roton excitations were not directly observed. A proposed complex vortex–antivortex pattern is therefore an interpretation to test, not a measured uniform chiral d-wave order parameter.\n\nEvidence: experiment. Sources: source-057c43c86174f961.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-limits-and-disputes-1\n\nLocal magnitude alone does not resolve handedness: $|\\Delta|^2=|\\Delta^*|^2$. Four complex spatial examples verify this identity without making a tunneling-response model. A Josephson signal can support superconducting coherence, while determining orbital phase structure requires suitable reference geometry and controls. Fourier peaks, gap-depth maps and zero-bias peaks must retain their individual evidential roles.\n\nEvidence: theory. Sources: source-057c43c86174f961.\n\nSource relationship: context-for-local-derivation. Conjugation invariance is local algebra; the cited spectroscopy motivates the evidential distinction, not a derivation of the identity.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-limits-and-disputes-2\n\nDo not require every time-reversal-breaking state to belong to a two-dimensional crystal representation. Complex combinations of distinct representations or interband phases can behave differently. Likewise, half-quantum vortices can occur in equal-spin-pairing states without establishing orbital chirality. A strain response or vortex signature must be interpreted in the actual symmetry and spin-orbit setting.\n\nEvidence: theory. Sources: source-69d33a869fbbb0c6, source-3c7aa9b4c8da3abf.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n### scwiki-c7-claim-limits-and-disputes-3\n\nThe preceding Chern numbers apply only to gapped two-dimensional Hamiltonians. A local three-dimensional extension illustrates the restriction: add $-2\\cos k_z$ to $h_z$ and fix $\\mu=-4$. Each fixed-$k_z$ slice has $\\mu_{\\rm eff}=-4+2\\cos k_z$. In the same occupied-band convention, $C(k_z)=-\\chi$ for $|k_z|<\\pi/2$ and zero for $\\pi/2<|k_z|\\leq\\pi$; it is undefined on the nodal slices $k_z=\\pm\\pi/2$. Solving all three components of $\\mathbf{h}=0$ gives only $(0,0,\\pm\\pi/2)$. Sixteen numerical comparisons and this analytic zero classification check the chosen model, not a material or a calculation reproduced from the references. Thus a single gapped two-dimensional integer cannot be assigned unchanged to this gapless three-dimensional example. Edge spectra, thermal transport, optical response and screened equilibrium charge currents still require their own calculations. Complete scientific acceptance, final-version comparisons, raw-data review and material-specific modelling remain open.\n\nEvidence: theory. Sources: source-69d33a869fbbb0c6.\n\nSource relationship: context-for-local-derivation. The stacked lattice and slice calculation are new author illustrations. References supply topology and response context; no actual material invariant or measured current is inferred.\n\nConditions: Local spinless fixed-gap lattice: t=t_z=Delta=1, mu=-4, chi=+/-1, kz in [-pi,pi], occupied-band A=i<u-|grad u-> and oriented kx,ky. Chern numbers exclude gapless slices.\n\nVerification scope: Selected primary text and local slice calculation checked 2026-10-19; no independent review, raw-data reanalysis or final-version equivalence claimed.\n\n### scwiki-c7-claim-discovery-implications-1\n\nFirst identify the transition and its probe. Then specify the order-parameter basis, domain controls, bulk gap and Hamiltonian used for any topology claim. Compare positive and null evidence under compatible conditions, retaining charge-order and superconducting temperature scales separately. A useful discovery target is a state that satisfies these linked tests, not a material selected merely for a complex gap notation or a kagome lattice.\n\nEvidence: theory. Sources: source-a33c639b7db4f05a, source-d3edee404071ac81, source-484da7abcc165cd3, source-057c43c86174f961.\n\nConditions: Explicit band, Berry and boundary conventions; fixed-gap models are not material fits. Experimental temperatures retain sample, probe and phase distinctions.\n\nVerification scope: Named primary passages and independent examples only; complete scientific acceptance remains open.\n\n## Revision and review\n\nArticle ID: scwiki-c7. Revision: 359431ef5b4c1b075aaf. Updated: 2026-10-19T14:45:44.059096+00:00.\n\nReview scope: Selected topology, boundary-current and optical manuscripts; kagome muon/STM passages; inherited UPt3 and Sr2RuO4 scoped evidence; independent lattice and angular checks. Follow-up: distinguish KV3Sb5 zero-field and transverse-field responses; check a local three-dimensional slice example.\n\n- Full scientific acceptance, complete final-version comparisons and raw-data review remain open.\n- Fixed-gap lattice topology and sharp-edge continuum currents are not realistic material calculations.\n- Charge-order, superconducting and surface temperature scales are not interchangeable.\n- No current crystal candidate is declared conclusively chiral by these local checks.\n\nSource: [C_pairing_symmetry/C7_chiral_pairing.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/C_pairing_symmetry/C7_chiral_pairing.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 83702d492b9ec3f0db41d54a045ebecc8ce06f0cd72755d36a041efcf3746344.\n","wordCount":1193}}