# Singlet and triplet pairing: spin, parity and spin–orbit coupling

Classify pairing with orbital and frequency exchange, and distinguish measured correlations from a proposed order parameter.

## Scope

Spin singlet and spin triplet describe antisymmetric and symmetric two-electron spin states. They do not, by themselves, identify a crystal representation, a pairing mechanism, chirality or topology. The familiar even-parity singlet / odd-parity triplet rule assumes even-frequency pairing and a symmetric orbital or band exchange sector. State these assumptions before applying it to a material. [R1](#ref-source-8d1b859d5ffcad9f) [R4](#ref-source-0342c1624c6eb6d0)

Spin–orbit coupling (SOC) makes physical spin different from a band label. In a time-reversal-invariant centrosymmetric normal state, inversion combined with time reversal produces a degenerate pair at each momentum. Such a doublet need not transform under crystal rotations as an ordinary spin-1/2 pseudospin: Samokhin identifies exceptional twofold bands where the usual axial-vector transformation of the triplet component fails. His classification retains even/odd intraband components in its static convention; the exception concerns crystal transformation rules, not the Pauli principle. [R5](#ref-source-f2133d67ac862ff4)

## Mathematical assumptions

For a homogeneous pair correlation with combined spin and orbital indices $a,b$, fermion exchange requires $F_{ab}(\mathbf k,i\omega_n)=-F_{ba}(-\mathbf k,-i\omega_n)$. If spin, momentum, orbital and frequency exchange can each be diagonalized, their signs obey $s_Ss_Ps_Os_\omega=-1$. Here frequency exchange is not the physical time-reversal operation. For $s_O=s_\omega=+1$, spin singlet implies even momentum parity and triplet implies odd momentum parity. Reversing either the orbital or frequency sign reverses that implication. An independent matrix check tests all 16 sign combinations, accepting eight and rejecting eight. These are allowed symmetry sectors, not evidence that all are stable phases. [R4](#ref-source-0342c1624c6eb6d0)

In a chosen spin-1/2 basis, write the static gap matrix $\Delta=(\psi+\mathbf d\cdot\boldsymbol\sigma)i\sigma_y$. The scalar is the spin-antisymmetric component; the vector multiplies symmetric spin matrices. An analogous decomposition in a band doublet must retain that basis's actual symmetry matrices. In higher-angular-momentum low-energy bands, coupled angular-momentum sectors replace a simple singlet/triplet classification; microscopic fermion antisymmetry still applies. Leggett's helium framework and crystal symmetry classifications are useful references, with their respective physical settings kept separate. The 1976 Leggett correction withdraws a historical factor-of-two discrepancy claim. [R1](#ref-source-8d1b859d5ffcad9f) [R2](#ref-source-b4553aaaeacb3c3d) [R3](#ref-source-24556d6dba4d8cc1) [R4](#ref-source-0342c1624c6eb6d0) [R5](#ref-source-f2133d67ac862ff4)

For the illustrative noncentrosymmetric Hamiltonian $h=\xi I+\alpha\mathbf g\cdot\boldsymbol\sigma$, take real odd $\mathbf g(-\mathbf k)=-\mathbf g(\mathbf k)$ and an imposed even singlet gap $\psi i\sigma_y$. The anomalous Green function can contain a triplet component even though the imposed triplet gap is zero. Twelve independent matrix inversions reproduce the helicity-projector expression, in the specified Nambu convention, $F=-\sum_{\lambda=\pm}\psi P_\lambda i\sigma_y/[\omega_n^2+(\xi+\lambda\alpha|\mathbf g|)^2+|\psi|^2]$, where $P_\lambda=(I+\lambda\hat{\mathbf g}\cdot\boldsymbol\sigma)/2$. Different denominators generate a triplet correlation; they coincide at the particle-hole-symmetric test point. This distinguishes a pair correlation from a self-consistent triplet order parameter. The calculation is an imposed mean-field example, not a prediction of a material's mixing ratio. [R6](#ref-source-6c944305286c9cad)

For an imposed aligned triplet component $\mathbf d=d\hat{\mathbf g}$ with nonzero $|\mathbf g|$, the gap-matrix singular values are $|\psi+d|$ and $|\psi-d|$. In the declared zero-field model, take even $\xi$, even complex scalars $\psi,d$, real odd $\mathbf g$, and no additional interband pairing. Direct diagonalization gives $E_\lambda^2=(\xi+\lambda\alpha|\mathbf g|)^2+|\psi+\lambda d|^2$, with $\lambda=\pm1$. A zero-energy node requires both $\xi+\lambda\alpha|\mathbf g|=0$ and $\psi+\lambda d=0$ at the same momentum on the same helicity Fermi sheet. Cancellation away from that sheet leaves a finite excitation energy. Equal magnitudes alone do not suffice: $d=i\psi$ gives both gap magnitudes $\sqrt{2}|\psi|$ for nonzero $\psi$. We checked 120 aligned spin-basis BdG matrices, plus controls for on-sheet cancellation, off-sheet cancellation, relative phase, transverse triplet pairing and zero SOC. A transverse triplet component mixes helicities, so this scalar formula must not be used unchanged. At $\mathbf g=0$, use the original spin matrix instead of the undefined unit vector. These are local imposed-model checks; the cited papers supply model context and its correction, not a self-consistent mixed-state solution. Interactions, bands and the gap equation must determine the actual amplitudes and any material nodes. [R6](#ref-source-6c944305286c9cad) [R7](#ref-source-46e49166d702226d)

Frigeri and colleagues' weak-coupling calculation neglects small singlet–triplet coupling terms associated with particle-hole asymmetry and analyzes the channels separately. Its leading triplet suppression contains $|\mathbf d|^2-|\hat{\mathbf g}\cdot\mathbf d|^2$: alignment avoids this particular suppression. The erratum adds an omitted linear term to the MnSi SOC vector, $\mathbf g=\alpha_1\mathbf k+\alpha_2[k_x(k_y^2-k_z^2),k_y(k_z^2-k_x^2),k_z(k_x^2-k_y^2)]$. Thus a conclusion about suppression of a p-wave component depends on the relative coefficients; alignment alone does not choose the stable state. Three independent angular examples check that dependence without calculating a transition temperature. The erratum also rescales the original Fig. 2 SOC parameter by dividing it by $\sqrt{3/2}$. [R6](#ref-source-6c944305286c9cad) [R7](#ref-source-46e49166d702226d)

## Experimental criteria

A Knight shift includes spin and orbital contributions and depends on field direction, SOC, hyperfine coupling and the superconducting measurement conditions. Pustogow and colleagues' 2019 Sr2RuO4 study used reduced radio-frequency excitation to address heating and reported a spin-susceptibility reduction. The selected author-version passages checked here do not establish every possible representation. The 2026 muon result adds a low-field constraint, with a finite temperature floor and extrapolation assumptions; it also does not constitute a complete order-parameter identification. Compare the actual susceptibility tensor and conditions with competing states. [R9](#ref-source-c7f8b99a795aba41) [R10](#ref-source-c4434644191d0174)

An upper critical field above a simple weak-coupling Pauli estimate is not a stand-alone triplet test. Orbital depairing, SOC, strong coupling, multiband effects and field direction must be addressed. In particular, Frigeri's field calculation neglects orbital depairing and considers a continuous instability: its formally large paramagnetic limit is not an unlimited physical upper critical field. In UTe2, the discovery paper's measured field range and higher extrapolated scale must remain distinct. [R6](#ref-source-6c944305286c9cad) [R14](#ref-source-ca6322be2694e430)

UGe2, URhGe and UCoGe motivate triplet and equal-spin pairing through superconductivity associated with ferromagnetism, but discovery evidence does not determine a complete odd-parity representation. Check microscopic coexistence and sample volume rather than infer them solely from two transition anomalies. UTe2 must not be placed in the same category of established long-range ferromagnetic superconductors; its historical interpretation is not a substitute for a current material-specific review. The uranium discovery abstracts and selected UTe2 text have different review depths here. [R11](#ref-source-db2917242488a101) [R12](#ref-source-15aec45a868e9fb9) [R13](#ref-source-a4695ba5bfa789f3) [R14](#ref-source-ca6322be2694e430)

Phase-sensitive junctions, impurity response and spin-sensitive probes can provide complementary constraints, but interfaces can select or generate correlations absent from the bulk gap. Tanaka and Golubov's proximity theory explicitly permits odd-frequency, even-parity triplet correlations in a diffusive normal metal. That is not by itself an intrinsic odd-frequency bulk superconducting phase. The publisher abstract has been checked; the full junction derivation remains open. [R8](#ref-source-778c4d1f166c8630)

## Limits and disputes

Do not infer a singlet–triplet mixing fraction from crystal inversion breaking. Symmetry permits compatible components; microscopic interactions decide their magnitude. Conversely, an orbital-antisymmetric spin-triplet sector can have even momentum parity at even frequency. Physical-spin pairing and the intraband pseudospin gap can therefore carry different descriptions after projection. Clepkens and colleagues' interorbital “shadowed-triplet” construction illustrates this distinction as a model proposed for Sr2RuO4, not an experimentally established assignment. Only its author abstract and publication identity are checked here. [R4](#ref-source-0342c1624c6eb6d0) [R6](#ref-source-6c944305286c9cad) [R15](#ref-source-2928c6abb9344456)

Neither an unchanged Knight shift nor a measured decrease alone fixes all SOC-coupled possibilities. A ferromagnetic background, a large critical field, nodes or broken time reversal similarly fails to identify a unique representation. Keep spin structure, momentum parity, orbital exchange, frequency dependence, crystal representation and topology as separate questions. The finite tests on this page validate specified algebra; they do not establish energetic stability or reproduce susceptibility data. [R1](#ref-source-8d1b859d5ffcad9f) [R4](#ref-source-0342c1624c6eb6d0) [R5](#ref-source-f2133d67ac862ff4) [R9](#ref-source-c7f8b99a795aba41)

## Discovery implications

For each candidate, record inversion and magnetic symmetries, the electronic basis, orbital or band indices, frequency convention, field direction, temperature, heating controls, measured observables and alternative explanations. Report a proposed gap only at the level supported by mutually consistent probes and a stated response model. Retain both Frigeri papers when using the corrected SOC argument. Full scientific acceptance, complete version comparisons and experimental reproduction remain open. [R5](#ref-source-f2133d67ac862ff4) [R6](#ref-source-6c944305286c9cad) [R7](#ref-source-46e49166d702226d) [R9](#ref-source-c7f8b99a795aba41)

## Related topics

Continue with [p-wave pairing](/scwiki/p-wave), [f-wave pairing](/scwiki/f-wave), [chiral pairing](/scwiki/chiral-pairing) and [ferromagnetic fluctuations](/scwiki/ferromagnetic-fluctuations).

## Material cases

No sample-specific material records are included in this article. Model results are not experimental transition temperatures.

## Sources

### Ref source-8d1b859d5ffcad9f

[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.
### Ref source-b4553aaaeacb3c3d

[R2] A theoretical description of the new phases of liquid 3He (1975). [DOI 10.1103/RevModPhys.47.331](https://doi.org/10.1103/RevModPhys.47.331). Publication: journal-article. Check: publisher-abstract-with-erratum-check. Checked: 2026-10-18. ABM/BW framework abstract checked with1976 erratum; full derivations not reviewed. Explicit local matrices are independent illustrations.
### Ref source-24556d6dba4d8cc1

[R3] Erratum: A theoretical description of the new phases of liquid 3He (1976). [DOI 10.1103/RevModPhys.48.357](https://doi.org/10.1103/RevModPhys.48.357). Publication: journal-article. Check: primary-erratum-text-scoped-check. Checked: 2026-10-18. Complete brief text checked: withdraws claimed factor-of-two disagreement with Brinkman–Smith near Tc on p.401. Direct download403; no visual scan claimed.
### Ref source-0342c1624c6eb6d0

[R4] Superconductivity in spin-3/2 systems: symmetry classification, odd-frequency pairs, and Bogoliubov Fermi surfaces (2021). [DOI 10.1103/PhysRevResearch.3.033255](https://doi.org/10.1103/PhysRevResearch.3.033255). Publication: journal-article. Check: selected-author-text-check. Checked: 2026-10-18. Author v2, 18 September 2021, marked published version; pp.1–2 read, p.2 visually checked. Selected exchange classification only; full higher-spin derivation and Bogoliubov-Fermi-surface claims not reviewed.
### Ref source-f2133d67ac862ff4

[R5] Symmetry of superconducting pairing in non-pseudospin electron bands (2019). [DOI 10.1103/PhysRevB.100.054501](https://doi.org/10.1103/PhysRevB.100.054501). Publication: journal-article. Check: selected-author-text-check. Checked: 2026-10-18. Author v1, 18 July 2019; pp.1–2 and selected pp.4–6 equations 7–20 read. Static intraband parity and non-pseudospin rotation distinction; complete exceptional nodal tables not audited.
### Ref source-6c944305286c9cad

[R6] Superconductivity without Inversion Symmetry: MnSi versus CePt3Si (2004). [DOI 10.1103/PhysRevLett.92.097001](https://doi.org/10.1103/PhysRevLett.92.097001). Publication: journal-article. Check: selected-author-text-and-erratum-check. Checked: 2026-10-19. Re-read v3 (28 July 2004) model Eqs. 2-7, helicity sheets, neglected coupling of order alpha/E_F, Eq. 9 alignment restriction and appended erratum (linear MnSi term and Fig. 2 rescaling). Provides context, not a self-consistent mixed-gap prediction. No new PDF pixel review or material data reanalysis.
### Ref source-46e49166d702226d

[R7] Erratum: Superconductivity without Inversion Symmetry: MnSi versus CePt3Si (2004). [DOI 10.1103/PhysRevLett.93.099903](https://doi.org/10.1103/PhysRevLett.93.099903). Publication: journal-article. Check: selected-author-text-and-erratum-check. Checked: 2026-10-19. Re-read v3 (28 July 2004) model Eqs. 2-7, helicity sheets, neglected coupling of order alpha/E_F, Eq. 9 alignment restriction and appended erratum (linear MnSi term and Fig. 2 rescaling). Provides context, not a self-consistent mixed-gap prediction. No new PDF pixel review or material data reanalysis.
### Ref source-778c4d1f166c8630

[R8] Theory of the Proximity Effect in Junctions with Unconventional Superconductors (2007). [DOI 10.1103/PhysRevLett.98.037003](https://doi.org/10.1103/PhysRevLett.98.037003). Publication: journal-article. Check: publisher-abstract-check. Checked: 2026-10-18. Published 16 January 2007. Diffusive normal-metal proximity correlations and frequency/spin/parity mapping checked in abstract; full derivation not reviewed.
### Ref source-c7f8b99a795aba41

[R9] 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.
### Ref source-c4434644191d0174

[R10] 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.
### Ref source-db2917242488a101

[R11] Superconductivity on the border of itinerant-electron ferromagnetism in UGe2 (2000). [DOI 10.1038/35020500](https://doi.org/10.1038/35020500). Publication: journal-article. Check: publisher-abstract-scoped-check. Checked: 2026-10-18. Saxena et al. publisher indexed abstract and Crossref identity checked. Restricted-pressure superconductivity; original curves and microscopic coexistence not audited. Replaces inherited DOI35050500 whose actual target remained unresolved.
### Ref source-15aec45a868e9fb9

[R12] Coexistence of superconductivity and ferromagnetism in URhGe (2001). [DOI 10.1038/35098048](https://doi.org/10.1038/35098048). Publication: journal-article. Check: publisher-abstract-scoped-check. Checked: 2026-10-18. Aoki et al. publisher indexed abstract: ambient-pressure thermodynamic evidence, with qualified triplet interpretation. Corrects inherited35065045, which identifies unrelated semiconductor melting research. Full data not reanalyzed.
### Ref source-a4695ba5bfa789f3

[R13] Superconductivity on the Border of Weak Itinerant Ferromagnetism in UCoGe (2007). [DOI 10.1103/PhysRevLett.99.067006](https://doi.org/10.1103/PhysRevLett.99.067006). Publication: journal-article. Check: publisher-abstract-scoped-check. Checked: 2026-10-18. Huy et al. original abstract: ambient-pressure bulk signatures. Curie3K and best-sample resistive0.8K are distinct; no sample Tc extracted here. Full curves not reviewed.
### Ref source-ca6322be2694e430

[R14] 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.
### Ref source-2928c6abb9344456

[R15] Higher angular momentum pairings in interorbital shadowed-triplet superconductors: Application to Sr2RuO4 (2021). [DOI 10.1103/PhysRevB.104.104512](https://doi.org/10.1103/PhysRevB.104.104512). Publication: journal-article. Check: author-abstract-and-publication-check. Checked: 2026-10-18. Author v2 dated 30 September 2021. Abstract and journal identity checked; model even-parity interorbital triplet and intraband pseudospin-singlet distinction, not a material assignment. Full calculation not reviewed.

## Claim evidence and conditions

### scwiki-c9-claim-scope-1

Spin singlet and spin triplet describe antisymmetric and symmetric two-electron spin states. They do not, by themselves, identify a crystal representation, a pairing mechanism, chirality or topology. The familiar even-parity singlet / odd-parity triplet rule assumes even-frequency pairing and a symmetric orbital or band exchange sector. State these assumptions before applying it to a material.

Evidence: theory. Sources: source-8d1b859d5ffcad9f, source-0342c1624c6eb6d0.

Conditions: Distinguish physical spin, band doublet and crystal representation; static intraband parity is conditional on the named basis.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-scope-2

Spin–orbit coupling (SOC) makes physical spin different from a band label. In a time-reversal-invariant centrosymmetric normal state, inversion combined with time reversal produces a degenerate pair at each momentum. Such a doublet need not transform under crystal rotations as an ordinary spin-1/2 pseudospin: Samokhin identifies exceptional twofold bands where the usual axial-vector transformation of the triplet component fails. His classification retains even/odd intraband components in its static convention; the exception concerns crystal transformation rules, not the Pauli principle.

Evidence: theory. Sources: source-f2133d67ac862ff4.

Conditions: Distinguish physical spin, band doublet and crystal representation; static intraband parity is conditional on the named basis.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-mathematical-assumptions-1

For a homogeneous pair correlation with combined spin and orbital indices $a,b$, fermion exchange requires $F_{ab}(\mathbf k,i\omega_n)=-F_{ba}(-\mathbf k,-i\omega_n)$. If spin, momentum, orbital and frequency exchange can each be diagonalized, their signs obey $s_Ss_Ps_Os_\omega=-1$. Here frequency exchange is not the physical time-reversal operation. For $s_O=s_\omega=+1$, spin singlet implies even momentum parity and triplet implies odd momentum parity. Reversing either the orbital or frequency sign reverses that implication. An independent matrix check tests all 16 sign combinations, accepting eight and rejecting eight. These are allowed symmetry sectors, not evidence that all are stable phases.

Evidence: theory. Sources: source-0342c1624c6eb6d0.

Source relationship: mixed-source-and-local-derivation. The cited exchange classification supplies context; the finite 16-sector matrix enumeration is a local algebra check, not independent scientific review.

Conditions: Exchange-product rule for resolved sectors. Imposed Nambu/helicity gap examples, no self-consistency, material stability or transition prediction. Frigeri approximation and erratum both retained.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-mathematical-assumptions-2

In a chosen spin-1/2 basis, write the static gap matrix $\Delta=(\psi+\mathbf d\cdot\boldsymbol\sigma)i\sigma_y$. The scalar is the spin-antisymmetric component; the vector multiplies symmetric spin matrices. An analogous decomposition in a band doublet must retain that basis's actual symmetry matrices. In higher-angular-momentum low-energy bands, coupled angular-momentum sectors replace a simple singlet/triplet classification; microscopic fermion antisymmetry still applies. Leggett's helium framework and crystal symmetry classifications are useful references, with their respective physical settings kept separate. The 1976 Leggett correction withdraws a historical factor-of-two discrepancy claim.

Evidence: theory. Sources: source-8d1b859d5ffcad9f, source-b4553aaaeacb3c3d, source-24556d6dba4d8cc1, source-0342c1624c6eb6d0, source-f2133d67ac862ff4.

Conditions: Exchange-product rule for resolved sectors. Imposed Nambu/helicity gap examples, no self-consistency, material stability or transition prediction. Frigeri approximation and erratum both retained.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-mathematical-assumptions-3

For the illustrative noncentrosymmetric Hamiltonian $h=\xi I+\alpha\mathbf g\cdot\boldsymbol\sigma$, take real odd $\mathbf g(-\mathbf k)=-\mathbf g(\mathbf k)$ and an imposed even singlet gap $\psi i\sigma_y$. The anomalous Green function can contain a triplet component even though the imposed triplet gap is zero. Twelve independent matrix inversions reproduce the helicity-projector expression, in the specified Nambu convention, $F=-\sum_{\lambda=\pm}\psi P_\lambda i\sigma_y/[\omega_n^2+(\xi+\lambda\alpha|\mathbf g|)^2+|\psi|^2]$, where $P_\lambda=(I+\lambda\hat{\mathbf g}\cdot\boldsymbol\sigma)/2$. Different denominators generate a triplet correlation; they coincide at the particle-hole-symmetric test point. This distinguishes a pair correlation from a self-consistent triplet order parameter. The calculation is an imposed mean-field example, not a prediction of a material's mixing ratio.

Evidence: theory. Sources: source-6c944305286c9cad.

Source relationship: mixed-source-and-local-derivation. The primary spin/SOC Hamiltonian supplies context; the imposed singlet-gap Green-function inversion is a local derivation and does not solve a triplet gap equation.

Conditions: Exchange-product rule for resolved sectors. Imposed Nambu/helicity gap examples, no self-consistency, material stability or transition prediction. Frigeri approximation and erratum both retained.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-mathematical-assumptions-4

For an imposed aligned triplet component $\mathbf d=d\hat{\mathbf g}$ with nonzero $|\mathbf g|$, the gap-matrix singular values are $|\psi+d|$ and $|\psi-d|$. In the declared zero-field model, take even $\xi$, even complex scalars $\psi,d$, real odd $\mathbf g$, and no additional interband pairing. Direct diagonalization gives $E_\lambda^2=(\xi+\lambda\alpha|\mathbf g|)^2+|\psi+\lambda d|^2$, with $\lambda=\pm1$. A zero-energy node requires both $\xi+\lambda\alpha|\mathbf g|=0$ and $\psi+\lambda d=0$ at the same momentum on the same helicity Fermi sheet. Cancellation away from that sheet leaves a finite excitation energy. Equal magnitudes alone do not suffice: $d=i\psi$ gives both gap magnitudes $\sqrt{2}|\psi|$ for nonzero $\psi$. We checked 120 aligned spin-basis BdG matrices, plus controls for on-sheet cancellation, off-sheet cancellation, relative phase, transverse triplet pairing and zero SOC. A transverse triplet component mixes helicities, so this scalar formula must not be used unchanged. At $\mathbf g=0$, use the original spin matrix instead of the undefined unit vector. These are local imposed-model checks; the cited papers supply model context and its correction, not a self-consistent mixed-state solution. Interactions, bands and the gap equation must determine the actual amplitudes and any material nodes.

Evidence: theory. Sources: source-6c944305286c9cad, source-46e49166d702226d.

Source relationship: context-for-local-derivation. Frigeri main text supplies the spin/SOC model and explicitly neglects small channel mixing; its erratum corrects the SOC vector. The mixed-gap spectrum, 120 matrix diagonalizations and five controls are local calculations, not results attributed to those papers.

Conditions: Zero-field imposed 4x4 BdG model; even xi, psi and scalar d; real odd nonzero g; dvector=d ghat; no extra interband pairing. Both the normal helicity energy and its complex gap must vanish at the same momentum. No self-consistent phase, material node or mixing-fraction assignment.

Verification scope: Selected Frigeri v3 model and appended erratum read on 2026-10-19; local full BdG spectrum and boundary controls checked; no material or independent scientific acceptance.

### scwiki-c9-claim-mathematical-assumptions-5

Frigeri and colleagues' weak-coupling calculation neglects small singlet–triplet coupling terms associated with particle-hole asymmetry and analyzes the channels separately. Its leading triplet suppression contains $|\mathbf d|^2-|\hat{\mathbf g}\cdot\mathbf d|^2$: alignment avoids this particular suppression. The erratum adds an omitted linear term to the MnSi SOC vector, $\mathbf g=\alpha_1\mathbf k+\alpha_2[k_x(k_y^2-k_z^2),k_y(k_z^2-k_x^2),k_z(k_x^2-k_y^2)]$. Thus a conclusion about suppression of a p-wave component depends on the relative coefficients; alignment alone does not choose the stable state. Three independent angular examples check that dependence without calculating a transition temperature. The erratum also rescales the original Fig. 2 SOC parameter by dividing it by $\sqrt{3/2}$.

Evidence: theory. Sources: source-6c944305286c9cad, source-46e49166d702226d.

Source relationship: mixed-source-and-local-derivation. The weak-coupling approximations and corrected SOC vector come from the main paper and erratum; three angular checks are local illustrations, not transition-temperature calculations.

Conditions: Exchange-product rule for resolved sectors. Imposed Nambu/helicity gap examples, no self-consistency, material stability or transition prediction. Frigeri approximation and erratum both retained.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-experimental-criteria-1

A Knight shift includes spin and orbital contributions and depends on field direction, SOC, hyperfine coupling and the superconducting measurement conditions. Pustogow and colleagues' 2019 Sr2RuO4 study used reduced radio-frequency excitation to address heating and reported a spin-susceptibility reduction. The selected author-version passages checked here do not establish every possible representation. The 2026 muon result adds a low-field constraint, with a finite temperature floor and extrapolation assumptions; it also does not constitute a complete order-parameter identification. Compare the actual susceptibility tensor and conditions with competing states.

Evidence: experiment. Sources: source-c7f8b99a795aba41, source-c4434644191d0174.

Conditions: Each probe and source has independent review scope; no single observable establishes a complete representation.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-experimental-criteria-2

An upper critical field above a simple weak-coupling Pauli estimate is not a stand-alone triplet test. Orbital depairing, SOC, strong coupling, multiband effects and field direction must be addressed. In particular, Frigeri's field calculation neglects orbital depairing and considers a continuous instability: its formally large paramagnetic limit is not an unlimited physical upper critical field. In UTe2, the discovery paper's measured field range and higher extrapolated scale must remain distinct.

Evidence: experiment. Sources: source-6c944305286c9cad, source-ca6322be2694e430.

Conditions: Each probe and source has independent review scope; no single observable establishes a complete representation.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-experimental-criteria-3

UGe2, URhGe and UCoGe motivate triplet and equal-spin pairing through superconductivity associated with ferromagnetism, but discovery evidence does not determine a complete odd-parity representation. Check microscopic coexistence and sample volume rather than infer them solely from two transition anomalies. UTe2 must not be placed in the same category of established long-range ferromagnetic superconductors; its historical interpretation is not a substitute for a current material-specific review. The uranium discovery abstracts and selected UTe2 text have different review depths here.

Evidence: experiment. Sources: source-db2917242488a101, source-15aec45a868e9fb9, source-a4695ba5bfa789f3, source-ca6322be2694e430.

Conditions: Each probe and source has independent review scope; no single observable establishes a complete representation.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-experimental-criteria-4

Phase-sensitive junctions, impurity response and spin-sensitive probes can provide complementary constraints, but interfaces can select or generate correlations absent from the bulk gap. Tanaka and Golubov's proximity theory explicitly permits odd-frequency, even-parity triplet correlations in a diffusive normal metal. That is not by itself an intrinsic odd-frequency bulk superconducting phase. The publisher abstract has been checked; the full junction derivation remains open.

Evidence: theory. Sources: source-778c4d1f166c8630.

Conditions: Each probe and source has independent review scope; no single observable establishes a complete representation.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-limits-and-disputes-1

Do not infer a singlet–triplet mixing fraction from crystal inversion breaking. Symmetry permits compatible components; microscopic interactions decide their magnitude. Conversely, an orbital-antisymmetric spin-triplet sector can have even momentum parity at even frequency. Physical-spin pairing and the intraband pseudospin gap can therefore carry different descriptions after projection. Clepkens and colleagues' interorbital “shadowed-triplet” construction illustrates this distinction as a model proposed for Sr2RuO4, not an experimentally established assignment. Only its author abstract and publication identity are checked here.

Evidence: theory. Sources: source-0342c1624c6eb6d0, source-6c944305286c9cad, source-2928c6abb9344456.

Conditions: Allowed correlations, model gap and measured order parameter are distinct; abstract-level sources do not receive full-text acceptance.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-limits-and-disputes-2

Neither an unchanged Knight shift nor a measured decrease alone fixes all SOC-coupled possibilities. A ferromagnetic background, a large critical field, nodes or broken time reversal similarly fails to identify a unique representation. Keep spin structure, momentum parity, orbital exchange, frequency dependence, crystal representation and topology as separate questions. The finite tests on this page validate specified algebra; they do not establish energetic stability or reproduce susceptibility data.

Evidence: theory. Sources: source-8d1b859d5ffcad9f, source-0342c1624c6eb6d0, source-f2133d67ac862ff4, source-c7f8b99a795aba41.

Conditions: Allowed correlations, model gap and measured order parameter are distinct; abstract-level sources do not receive full-text acceptance.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

### scwiki-c9-claim-discovery-implications-1

For each candidate, record inversion and magnetic symmetries, the electronic basis, orbital or band indices, frequency convention, field direction, temperature, heating controls, measured observables and alternative explanations. Report a proposed gap only at the level supported by mutually consistent probes and a stated response model. Retain both Frigeri papers when using the corrected SOC argument. Full scientific acceptance, complete version comparisons and experimental reproduction remain open.

Evidence: theory. Sources: source-f2133d67ac862ff4, source-6c944305286c9cad, source-46e49166d702226d, source-c7f8b99a795aba41.

Conditions: Full scientific acceptance open; preserve source/version and competing explanations.

Verification scope: Selected primary passages and independent finite examples; full scientific acceptance remains open.

## Revision and review

Article ID: scwiki-c9. Revision: 50b2abe31c49de268079. Updated: 2026-10-19T15:10:03.920700+00:00.

Review scope: Selected Frigeri and Samokhin model passages, complete Frigeri erratum text, selected Dutta exchange classification, inherited NMR/uranium evidence and explicit abstract-only proximity/shadowed-triplet sources. Independent exchange, Green-function and gap examples. Follow-up: Frigeri v3 model/approximations and appended erratum re-read; local aligned BdG gap versus zero-energy-node conditions tested with 120 cases and five controls. Existing reviews of other sources were not repeated.

- Full scientific acceptance and complete version comparisons remain open.
- Finite imposed models do not calculate stable phases, material mixing fractions or transition temperatures.
- Proximity and shadowed-triplet papers have abstract-level review only; most uranium discovery evidence is abstract-level.
- Susceptibility datasets, full supplements and higher-spin nodal classifications have not been reproduced.

Source: [C_pairing_symmetry/C9_singlet_vs_triplet.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/C_pairing_symmetry/C9_singlet_vs_triplet.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 bdeb21d3f5b339cb657753fc5a64c6662c8d935451392be0f84c21119d637e36.