Intimate relationship between spin configuration in the triplet pair and superconductivity in UTe$_2$

Spin-triplet superconductivity is an intriguing quantum coherent state with both spin and orbital degrees of freedom, which holds significant potential for future applications in quantum technology. However, how the spin of the triplet pairs responds to an external magnetic field remains poorly understood. This is mainly due to the absence of suitable spin-triplet superconductors. Here, we report results of Knight-shift and ac-susceptibility measurements on UTe$2$. We demonstrate that the spin susceptibility, which slightly decreases compared to the normal-state value below the superconducting (SC) transition temperature $T{\rm c}$, is rapidly restored and nearly recovers to the normal-state values around 5 T, well below the SC upper critical field $H_{c2}$ when the magnetic field is applied along the $c$ axis ($H \parallel c$). In addition, we found that $H_{\rm c2}$ of superconductivity becomes larger when the SC spin aligns with the magnetic field. By considering the results on $H \parallel b$, our results suggest the presence of a close relationship between the spin configuration of the triplet pair and $H_{\rm c2}$, as well as the anisotropic pinning interaction acting on the triplet pairs. These phenomena, which have never been observed in spin-singlet superconductors, represent characteristic features unique to spin-triplet superconductors. We discuss the similarities between superconductivity in UTe$_2$ and superfluid $^3$He, focusing on their spin-triplet pairing states.

💡 Research Summary

This paper presents a comprehensive investigation of the spin‑triplet superconductivity in the uranium‑based compound UTe₂, focusing on how the spin configuration of the Cooper pairs responds to external magnetic fields and how this response influences superconducting properties such as the upper critical field (Hc₂). Using high‑quality single crystals (Tc ≈ 2.1 K) enriched with ^125Te, the authors performed ^125Te nuclear magnetic resonance (NMR) Knight‑shift measurements together with ac‑susceptibility (χ_ac) experiments for magnetic fields applied along the crystallographic b‑ and c‑axes, covering a field range up to 24 T and temperatures down to ~50 mK.

Key experimental observations are: (i) for H∥c, the Knight shift associated with the Te(II) site (Kc,II) shows a modest reduction (~0.2 %) below Tc, indicating a small decrease of the spin susceptibility χ_spin in the superconducting state. This reduction is rapidly suppressed as the field is increased, and the Knight shift recovers to its normal‑state value already at ≈5 T, well below the measured Hc₂. The recovery is much faster than the field‑induced recovery of the residual Sommerfeld coefficient γ₀(H), demonstrating that the spin part of the susceptibility, not just the diamagnetic shielding, is being restored.

(ii) For H∥b, the decrease of the Knight shift is more pronounced, and the recovery with field occurs at similar fields, but a striking increase of Hc₂ is observed when the spin of the triplet pair aligns with the external field. In the high‑field superconducting (HFSC) regime (μ₀H > 16 T) the upper critical field exceeds 30 T, indicating that Pauli pair‑breaking is essentially absent and only orbital limiting remains.

The authors interpret these results within the framework of odd‑parity superconductivity in an orthorhombic D₂h crystal. The possible order‑parameter irreducible representations are Au, B₁u, B₂u, and B₃u. Earlier Knight‑shift work suggested that the d‑vector (the direction of zero spin projection) has components along all three crystallographic axes, consistent with an Au state—analogous to the A‑phase of superfluid ^3He. When a magnetic field is applied, the crystal symmetry is reduced to C₂h, allowing mixing between Au and the Bu representations. This mixing makes the d‑vector flexible: it can rotate to align with the field, producing the observed rapid recovery of the spin susceptibility.

A central insight is the existence of anisotropic “pinning” of the triplet pairs. The field‑induced alignment of the d‑vector reduces the Pauli depairing effect, thereby enhancing Hc₂. This pinning is highly direction‑dependent: for H∥c the Knight shift recovers quickly, while for H∥b the enhancement of Hc₂ is dramatic. Such behavior is unique to spin‑triplet superconductors; spin‑singlet systems never display a field‑controlled increase of Hc₂ through spin alignment.

The paper also draws parallels with superfluid ^3He. The Au state with a three‑axis d‑vector resembles the ^3He A‑phase, while the high‑field superconducting phase of UTe₂ may be viewed as analogous to the B‑phase, both featuring multiple superconducting phases tuned by field or pressure. However, UTe₂ differs in that strong electronic correlations, pronounced Ising anisotropy, and metamagnetic transitions introduce additional complexity absent in the nearly isotropic ^3He system.

Methodologically, the authors took care to eliminate artifacts: they calibrated the magnetic field using ^65Cu NMR, distinguished the two Te sites by their distinct Knight shifts, and subtracted the bulk diamagnetic contribution by comparing the two sites. RF heating was minimized by reducing pulse power, and the ac‑susceptibility was extracted from shifts in the NMR tank circuit resonance frequency.

Limitations include the difficulty of accessing the full high‑field regime for H∥b with NMR (the highest field used was 24 T, while Hc₂ may exceed 30 T), and the need for even higher fields to map the complete evolution of the d‑vector in the HFSC phase. Future work could combine NMR with torque magnetometry or neutron scattering to directly probe the d‑vector orientation.

In summary, the study provides compelling evidence that in UTe₂ the spin configuration of the triplet Cooper pairs is not rigid but can be reoriented by modest magnetic fields, leading to a rapid restoration of spin susceptibility and a substantial enhancement of the upper critical field. This demonstrates a close, previously unobserved relationship between spin orientation, anisotropic pinning, and superconducting robustness in a spin‑triplet superconductor, and it establishes UTe₂ as a solid‑state platform that mirrors many of the exotic features of superfluid ^3He while offering new avenues for spin‑controlled quantum technologies.