Observation of a pronounced Hebel-Slichter peak in the spin-lattice relaxation rate and implications for gap and pairing symmetry in LaNiGa$_2$

The observation of a pronounced Hebel-Slichter peak in the NQR spin-lattice relaxation rate of LaNiGa$_2$ strongly supports a two-band singlet BCS-like pairing model with two distinct gaps, thereby challenging previous proposals of non-unitary triplet pairing with time-reversal symmetry breaking.

The Gist

Imagine a superconductor as a busy dance floor where electrons usually move chaotically. When the temperature drops low enough, these electrons pair up and start dancing in perfect unison, allowing electricity to flow without any resistance.

For a long time, scientists thought the material LaNiGa2 was a very special, rare dancer. They believed it was doing a "triplet" dance, where the partners spin in the same direction (like two people spinning clockwise together). This was a big deal because it suggested the material was breaking a fundamental rule of physics called "time-reversal symmetry," essentially meaning the dance looked different if you played the video backward.

However, this new paper is like a detective stepping in to re-examine the evidence. The researchers looked at how the atomic nuclei in the material "relax" (calm down) after being disturbed, a process that acts like a high-speed camera capturing the details of the electron dance.

Here is what they found, using some simple analogies:

The "Hebel-Slichter" Peak: A Crowd Surge

In a standard, "normal" superconductor dance, right as the music starts (when the material becomes superconducting), there is a sudden, massive surge of activity. Scientists call this the Hebel-Slichter peak. It's like when a crowd at a concert suddenly jumps up and cheers in perfect unison the moment the beat drops.

The researchers found a very loud, clear cheer (a pronounced peak) in LaNiGa2.

The Problem with the "Triplet" Theory

Previously, scientists thought LaNiGa2 was a "non-unitary triplet" dancer. Imagine a triplet dance where the partners are spinning in the same direction, but one partner is slightly faster than the other.

• The Theory: If the partners had different speeds (different energy gaps), the paper argues that the "cheer" (the peak) would be muffled or disappear entirely. It would be like a crowd trying to cheer in unison but half of them are clapping slowly and the other half quickly; the result is a messy, weak sound.
• The Reality: The data showed a loud, clear cheer.
• The Conclusion: For the "triplet" theory to work with this loud cheer, the partners would have to be spinning at the exact same speed (identical gaps). But if they are spinning at the exact same speed, the dance is no longer "non-unitary," and it stops breaking the time-reversal symmetry rule.

The "Singlet" Alternative: A Classic Waltz

The researchers then tested a different theory: that LaNiGa2 is actually a "singlet" dancer. In this dance, partners spin in opposite directions (one clockwise, one counter-clockwise), which is the standard move for most superconductors.

• The Fit: When they modeled the data as a "two-band singlet" dance (where there are two slightly different groups of dancers, but they all follow the standard rules), the model perfectly matched the loud, clear cheer observed in the experiment.

The Verdict

The paper concludes that the evidence points away from the exotic "triplet" dance and toward a more conventional "singlet" dance.

• The Old Idea: LaNiGa2 is a rare, exotic dancer breaking time-reversal symmetry.
• The New Finding: The loud "cheer" in the data suggests it's actually a standard dancer, just with two slightly different groups of partners.

The authors admit that other experiments (using muons, which are like tiny magnetic probes) previously suggested the exotic "triplet" dance was happening. However, those measurements were very close to the limit of what the equipment could detect. This new study suggests that if LaNiGa2 is doing the exotic dance, it's doing it in a way that doesn't produce the loud "cheer" we see. Since the cheer is there, the exotic dance is unlikely.

In short: The paper argues that LaNiGa2 is likely not the exotic, time-reversal-breaking material we thought it was, but rather a more conventional superconductor that just happens to have two different energy gaps. To be sure, they suggest we need to test single crystals (perfectly formed dancers) rather than the powdered samples used so far, and use other tools to double-check the "time-reversal" claim.

Technical Summary

Technical Summary: Observation of a Pronounced Hebel-Slichter Peak in LaNiGa2 and Implications for Pairing Symmetry

Problem Statement
The superconductor LaNiGa2 ($T_c \approx 2.1$ K) has been a subject of intense theoretical and experimental interest due to its non-symmorphic orthorhombic crystal structure ($Cmcm$), which supports non-trivial band topology and band degeneracies at the Fermi level. Previous measurements, including specific heat, penetration depth, and muon spin resonance ($\mu$SR), suggested the existence of two distinct superconducting gaps and spontaneous magnetic fields. These observations led to a proposal that LaNiGa2 hosts a fully gapped, internally antisymmetric, non-unitary triplet (INT) pairing state. In this INT scenario, the condensate involves equal-spin pairing ($S=1$) with two distinct gap magnitudes, resulting in finite internal magnetization and time-reversal symmetry breaking (TRSB). However, the spin structure of the superconducting condensate remains unproven by these thermodynamic and magnetic probes, which are primarily sensitive to the density of states rather than the spin symmetry of the Cooper pairs.

Methodology
To probe the spin structure directly, the authors performed zero-field nuclear quadrupolar resonance (NQR) measurements on the $^{139}$La nucleus ($I=7/2$) in a finely powdered LaNiGa2 sample ($T_c = 1.83$ K). The study focused on the temperature dependence of the nuclear spin-lattice relaxation rate ($T_1^{-1}$) down to 240 mK. The relaxation rate is driven by spin-flip scattering of nuclear spins with Bogoliubov quasiparticles, making it highly sensitive to the coherence factors and gap symmetry of the superconducting state.

The experimental data were compared against two theoretical models:

1. The INT Model: A two-band model with equal-spin triplet pairing, allowing for non-unitarity (TRSB) and two distinct gap magnitudes ($\Delta_1 \neq \Delta_2$). The authors calculated $T_1^{-1}$ for various orientations of the triplet $\mathbf{d}$-vector relative to the nuclear quantization axis.
2. The Two-Band Singlet Model: A conventional two-gap BCS-like model with spin-singlet pairing and inter-band coupling, featuring two distinct gap magnitudes but no TRSB.

The calculations utilized the Bogoliubov-de Gennes (BdG) formalism, incorporating the specific hyperfine interaction between La nuclear spins and electron spins in the two bands. The authors accounted for the nuclear resonance frequency ($\omega_N$) and the energy conservation constraints in the scattering processes.

Key Results
The NQR data reveal a pronounced Hebel-Slichter (HS) coherence peak in $T_1^{-1}$ just below $T_c$, where the relaxation rate increases by a factor of approximately 2.5 relative to the normal state. At lower temperatures, the relaxation rate exhibits activated (exponential) behavior, consistent with a fully gapped superconductor. The slope change in the Arrhenius plot suggests the presence of two gaps with values estimated at $\approx 2k_B T_c$ and $< 1.0 k_B T_c$.

• Analysis of the INT Model: The authors find that the INT model fails to reproduce the robust HS peak observed in the data unless the two superconducting gaps are exactly equal in magnitude ($\Delta_1 = \Delta_2$). If the gaps are unequal (a requirement for non-unitarity and TRSB in this model), the coherence peak is rapidly suppressed or eliminated due to energy conservation constraints in the spin-flip scattering processes. Even when the equal-spin pairing axis is misaligned with the nuclear quantization axis, a weak peak arises only under artificial conditions (e.g., anomalously small nuclear frequencies) that are not physically credible. Furthermore, the unitary case ($\Delta_1 = \Delta_2$) implies no TRSB, contradicting the premise of the non-unitary INT state.
• Analysis of the Singlet Model: In contrast, the two-band singlet BCS model with distinct gap magnitudes provides an excellent fit to the experimental data, including the magnitude and temperature dependence of the HS peak. The fitted gap values ($\Delta_1/k_B T_c \approx 1.41$, $\Delta_2/k_B T_c \approx 1.95$) are consistent with previous specific heat and magnetic susceptibility measurements.
• Comparative Analysis: The magnitude of the HS peak in LaNiGa2 is significantly larger than in many other unconventional superconductors and is comparable to conventional s-wave superconductors like MgB2 and CaPd2As2, further supporting a conventional pairing mechanism.

Significance and Claims
The paper concludes that the observation of a robust Hebel-Slichter coherence peak in LaNiGa2 is inconsistent with the previously proposed non-unitary triplet (INT) pairing state, which requires unequal gaps to exhibit TRSB. The data are instead well-described by a two-band singlet pairing model with distinct gaps.

The authors state that these findings "raise doubts on the identification of non-unitary triplet-pairing with time-reversal symmetry breaking in this material." They acknowledge that their results are not fully consistent with previous $\mu$SR measurements that reported TRSB, noting that those measurements were conducted on polycrystalline samples near the detection limit. The paper does not definitively rule out TRSB but suggests that the pairing symmetry is likely singlet rather than triplet. The authors propose that future measurements on single crystals and complementary techniques, such as optical Kerr rotation and ultrasonic attenuation, are necessary to definitively confirm or refute the presence of TRSB and clarify the pairing symmetry in LaNiGa2.