Pairing symmetry

Pairing symmetry is the core debate. Early papers argued for spin-triplet, odd-parity, possibly nonunitary superconductivity. Cleaner crystals forced revisions on residual density of states, gap nodes, and time-reversal symmetry. What follows is evidence-tiered, not a single accepted order parameter.

Pauli-limit violation and NMR

Discovery-era ¹²⁵Te NMR on powdered CVT material reported a temperature-independent Knight shift through T_c, a classic spin-triplet fingerprint in that geometry[1]. Combined with H_c2≫H_P, the Science paper concluded parallel-spin pairing with large residual γ interpreted as a half-gapped nonunitary condensate[1]. That residual-γ reading is sample-quality sensitive: high-T_c crystals show much smaller residual γ, so “half the electrons remain ungapped” is not universal[6][7].

On ultraclean MSF crystals with T_c=2.1 K, Matsumura et al. find spin susceptibility decreases along b and c as well as a, constraining the d-vector to have components along all three axes[3]. Still compatible with odd-parity triplet pairing; not with a single fixed d̂∥a picture.

Odd parity: Josephson selection rules

Li et al. report In/UTe₂ Josephson selection rules: finite coupling on (001), none on (1−10), argued as odd-parity B_1u (Γ₂⁻) at zero/low field, with Andreev surface bound states on (1−10)[8]. This evidence is independent of Knight-shift ambiguities. Interface quality and facet assignment remain caveats[8].

Full gap versus nodes

Metz et al. (2019) argued for point nodes from penetration-depth / microwave data on early crystals[2]. Suetsugu et al. (2024), using thermal conductivity on ultraclean MSF crystals, report vanishing residual κ/T and field dependence like a full-gap superconductor, favoring A_u over B_3u with a-axis nodes[4]. Parallel thermal-transport work still argues for nodal behavior (feed: arXiv:2402.19353). Unresolved. Sample quality and residual normal fluid are the usual suspects[4][7].

Multicomponent order?

Hayes et al. combined polar Kerr TRSB with split specific-heat transitions to argue for multicomponent order (B_3u+iB_2u or B_1u+iA_u)[5]. Ajeesh et al. find no spontaneous Kerr on single-T_c crystals and conclude intrinsic TRSB is unlikely—removing the strongest push toward chiral multicomponent order at ambient pressure[6]. Multicomponent physics remains plausible under pressure and in high field even if ambient zero-field order is single-component[7].

Bottom line

Odd-parity / triplet pairing remains the leading working hypothesis[1][3][8]. The precise irreducible representation, whether the gap is fully open, and whether time-reversal is broken are not settled[4][6][7].

Order-parameter vocabulary (minimal)

Spin singlet vs triplet: Singlet pairs have total spin 0 and even orbital parity in centrosymmetric crystals; triplet pairs have spin 1 and odd orbital parity. Knight shift and Pauli-limit behavior are the workhorse spin diagnostics.

d-vector: The triplet gap is written with a vector d(k) in spin space. Nodes and TRSB depend on whether d is real (unitary) or complex (nonunitary).

Unitary vs nonunitary: Nonunitary states have d×d* ≠ 0, produce unequal spin gaps, and spontaneously magnetize. Discovery papers leaned nonunitary; clean-crystal residual γ no longer forces that extreme limit.

A_u vs B_iu: A_u is fully symmetric odd-parity (analogous to ³He-B in simplified language). B_1u/B_2u/B_3u transform like k_z, k_y, k_x-type form factors and typically support point nodes along high-symmetry axes unless accidental gaps open.

How to weigh a new pairing paper

1. Does it use MSF crystals with residual γ near single digits (mJ/mol·K²)?
2. Is the probe bulk (specific heat, thermal conductivity, NMR) or surface (STM, Kerr)?
3. Does the claim require TRSB? If yes, cite the Ajeesh/Azari tension explicitly.
4. Are nodes required by symmetry of the proposed irrep, or accidental?
5. Is high-field data being used to infer zero-field pairing (risky)?

How the consensus shifted

2019–2020: nonunitary triplet with point nodes and large residual γ was the default story from discovery NMR, specific heat, and penetration depth. 2021: multicomponent TRSB (Hayes) temporarily dominated outlook sections. 2022–2023: MSF crystals and single-transition thermodynamics undercut residual-γ and double-T_c pillars. 2023–2024: Kerr/μSR nulls on clean crystals; thermal conductivity full-gap reports; Josephson odd-parity selection rules. 2025: axis-resolved Knight shifts on T_c=2.1 K crystals. The field did not “prove singlet”; it narrowed which triplet scenarios remain viable.

What the discovery NMR actually showed

Ran et al. measured ¹²⁵Te NMR on powdered crystals at 15.1 MHz. The Knight shift did not drop through T_c, while the tank-circuit resonance frequency confirmed diamagnetic shielding. In singlet superconductors the spin susceptibility falls; the absence of that fall is the triplet argument. Powder averages axes, so the measurement cannot locate the d-vector. It also cannot distinguish a truly constant spin susceptibility from a partial cancellation among axes. That is exactly why Matsumura et al.’s later axis-resolved work on MSF crystals is a necessary sequel rather than a minor extension.

The same discovery paper’s residual γ≈55 mJ mol⁻¹ K⁻² was nearly sample-independent among their CVT pieces and was therefore called intrinsic nonunitary half-gapping. Later MSF crystals with residual γ of only a few mJ mol⁻¹ K⁻² falsify that universality without falsifying triplet pairing itself. Lesson: residual DOS is not a safe order-parameter fingerprint in this material.

Josephson parity tests in practice

Odd-parity selection rules compare Josephson coupling on crystal faces that transform differently under inversion-odd form factors. A finite supercurrent on (001) with none on (1−10), as reported by Li et al., matches B_1u-type expectations more naturally than even-parity singlets. Practical failure modes include faceting errors, damaged interfaces, and proximity-induced singlet pairing in the indium counterelectrode region. Still, as an independent parity channel, Josephson data deserve weight alongside NMR.

Specific-heat power laws

Ran et al. reported C(T) below T_c following a power law with n≈3.2 after subtracting residual γ, inconsistent with a fully isotropic BCS exponential and taken as evidence for point nodes of a nonunitary state. Suetsugu et al.’s later vanishing residual κ/T on MSF crystals pulls the other way. Power-law specific heat can also arise from impurity bands. The safe statement is: early CVT thermodynamics looked nodal; clean-crystal thermal transport often looks fully gapped; both cannot be naively averaged into “UTe₂ has nodes.”

Uniaxial stress tests

Girod et al. (Phys. Rev. B 106, L121101, 2022) examined thermodynamics and transport under uniaxial stress. Ajeesh et al. emphasize that shear stress expected to split a B_3u+iB_2u multicomponent state did not produce the anticipated T_c splitting on single-transition samples. That negative result is as important as any positive Kerr report when weighing ambient multicomponent scenarios.

Claim → evidence table

Update: PNAS DOI for Josephson odd-parity

Li et al. is now verified as Proc. Natl. Acad. Sci. U.S.A. 122, e2419734122 (2025), DOI 10.1073/pnas.2419734122 (arXiv:2309.08668). Their Table of irreps shows why (001) coupling with (1−10) null selects B_1u among allowed D_2h channels. Combined with Matsumura’s three-axis Knight shifts on MSF crystals and Pauli-limit violation, odd-parity triplet remains the leading ambient low-field hypothesis—while gap nodes and TRSB stay separately contested.

What to realistically expect

Expect: More Josephson replications; more axis-resolved NMR on shared MSF batches; slow arbitration of nodes vs full gap via thermal conductivity.

Do not expect: Immediate consensus on a single irrep label in every textbook—high-field phases may not share the zero-field irrep (Li et al. note this explicitly).

References

1. Ran, S., Eckberg, C., Ding, Q.-P., Furukawa, Y., Metz, T., Saha, S. R., Liu, I.-L., Zic, M., Kim, H., Paglione, J., & Butch, N. P. “Nearly ferromagnetic spin-triplet superconductivity.” Science 365, 684–687 (2019). DOI 10.1126/science.aav8645.
2. Metz, T., Bae, S., Ran, S., Liu, I.-L., Eo, Y. S., Fuhrman, W. T., Agterberg, D. F., Anlage, S. M., Butch, N. P., & Paglione, J. “Point-node gap structure of the spin-triplet superconductor UTe₂.” Physical Review B 100, 220504 (2019). DOI 10.1103/PhysRevB.100.220504.
3. Matsumura, H., Takahashi, Y., Kinjo, K., Kitagawa, S., Ishida, K., Tokunaga, Y., et al. “b-axis and c-axis Knight shift measurements in the superconducting state on ultraclean UTe₂ with T_c=2.1 K.” Physical Review B 111, 174526 (2025). DOI 10.1103/PhysRevB.111.174526.
4. Suetsugu, S., Shimomura, M., Kamimura, M., et al. “Fully gapped pairing state in spin-triplet superconductor UTe₂.” Science Advances 10, eadk3772 (2024). DOI 10.1126/sciadv.adk3772.
5. Hayes, I. M., Wei, D. S., Metz, T., Zhang, J., Eo, Y. S., Ran, S., Saha, S. R., Collini, J., Butch, N. P., Agterberg, D. F., Kapitulnik, A., & Paglione, J. “Multicomponent superconducting order parameter in UTe₂.” Science 373, 797–801 (2021). DOI 10.1126/science.abb0272.
6. Ajeesh, M. O., Bordelon, M., Girod, C., Mishra, S., Ronning, F., Bauer, E. D., Maiorov, B., Thompson, J. D., Rosa, P. F. S., & Thomas, S. M. “Fate of Time-Reversal Symmetry Breaking in UTe₂.” Physical Review X 13, 041019 (2023). DOI 10.1103/PhysRevX.13.041019.
7. Aoki, D., Brison, J.-P., Flouquet, J., Ishida, K., Knebel, G., Tokunaga, Y., & Yanase, Y. “Unconventional superconductivity in UTe₂.” Journal of Physics: Condensed Matter 34, 243002 (2022). DOI 10.1088/1361-648X/ac5863.
8. Li, Z., et al. “Observation of odd-parity superconductivity in UTe₂.” Li, Z., Moir, C., McKee, N., Lee-Wong, E., Baumbach, R., Maple, M. B., & Liu, Y. “Observation of odd-parity superconductivity in UTe₂.” Proc. Natl. Acad. Sci. U.S.A. 122, e2419734122 (2025). DOI 10.1073/pnas.2419734122. arXiv:2309.08668.