Spin-lattice relaxation for point-node-like s-wave superconductivity in f-electron systems

Using an f-d-p model for UTe2, this study demonstrates that while a point-node-like s-wave pairing state exhibits a reduced but still robust Hebel-Slichter peak, it remains inconsistent with experimental nuclear magnetic resonance measurements.

The Gist

Imagine a material called UTe2 as a bustling dance floor where electrons are the dancers. Scientists have been trying to figure out exactly how these electrons pair up to create a special state called superconductivity (where electricity flows with zero resistance).

Here is a simple breakdown of what this paper investigated and found, using some everyday analogies.

The Mystery of the Dance Floor

Scientists know UTe2 is a superconductor, but they are arguing about the "rules of the dance."

• The Clue 1 (The Heat): When they measure how much heat the material holds, it behaves like a dance floor with a few empty spots (called "nodes") where dancers can move freely. This suggests the pairing isn't a perfect, uniform circle.
• The Clue 2 (The Spin): Recent measurements of the dancers' "spin" (their orientation) suggest they are pairing up in a specific way that usually implies a perfect, smooth dance floor with no empty spots.

The New Theory: The "Point-Node" Dance

The authors of this paper proposed a theory to solve this puzzle. They used a complex mathematical model (the f-d-p model) to simulate the electrons.

• The Result: Their math suggested the electrons form an s-wave pair (a standard, stable type of pairing) but with a twist: it has accidental "point nodes."
• The Analogy: Imagine a perfectly round trampoline (the standard s-wave state). Now, imagine someone poked two tiny holes right at the corners. The trampoline is still mostly round, but those tiny holes allow for the specific "heat behavior" scientists observed. This is the "point-node-like" state.

The Test: The "Hebel-Slichter Peak"

To see if this theory is true, the scientists looked at a specific signal called the spin-lattice relaxation rate (measured by a technique called NMR).

• The Expectation: In a standard, perfect superconductor, when the temperature drops just below the freezing point of the superconducting state, the NMR signal usually spikes dramatically. This is called the Hebel-Slichter peak.
• The Analogy: Think of this peak like a sudden, loud cheer from the crowd right when the music starts. In a perfect, smooth dance floor, the crowd goes wild immediately.
• The Reality in UTe2: Real experiments on UTe2 show no loud cheer. The signal is flat. There is no peak.

The Experiment: Does the "Hole" Theory Explain the Silence?

The authors asked: "If our theory is right (that there are tiny holes in the dance floor), would that explain why the crowd doesn't cheer?"

• The Logic: They thought maybe the "holes" (the nodes) would smooth out the crowd's reaction, making the loud cheer quieter or broader, so it wouldn't be noticed.
• The Calculation: They ran computer simulations to see what happens to the "cheer" (the peak) when you have these tiny holes versus a perfect floor.

The Verdict: The Theory Doesn't Fit

The results were surprising:

1. The Cheer is Still There: Even with the "tiny holes" (the point-node-like state), the loud cheer (the Hebel-Slichter peak) remained very strong. It was slightly smaller than in a perfect floor, but still very obvious.
2. The "Disorder" Factor: They also checked if "messiness" in the material (like dirt on the dance floor) could kill the cheer. They found that while messiness does kill the cheer, it kills it in both the perfect floor and the "holey" floor equally. So, the "holes" alone aren't the reason the cheer is missing in real life.

The Conclusion

The paper concludes that while their "point-node-like" theory explains the heat measurements perfectly, it fails to explain the NMR measurements.

• Simple Summary: The theory predicts a loud cheer that should be heard, but in the real world, the crowd is silent. Therefore, this specific "point-node-like" dance style is likely not what is happening in UTe2, even though it looks good on paper for other reasons.

The scientists are left with a puzzle: They need to find a new explanation for why the electrons pair up in UTe2 in a way that creates the "holes" (for the heat) but also silences the "cheer" (for the NMR).

Technical Summary

Technical Summary: Spin-lattice relaxation for point-node–like s-wave superconductivity in f-electron systems

Problem Statement
The superconducting mechanism and gap structure of UTe$_2$ remain subjects of intense debate. While the material exhibits an upper critical field exceeding the Pauli limit—suggesting spin-triplet pairing—recent nuclear magnetic resonance (NMR) Knight shift measurements indicate a spin-singlet state or a specific spin-triplet $A_u$ state. Furthermore, specific heat and penetration depth measurements suggest the existence of point nodes, which contradicts the fully gapped nature typically associated with standard spin-triplet states.

Previous theoretical work by the authors, utilizing an $f$-$d$-$p$ effective model and third-order perturbation theory (TOPT), proposed that the most probable pairing state in UTe$_2$ is a highly anisotropic, point-node–like $s$-wave state. This state is unconventional, mediated by on-site Coulomb repulsion rather than magnetic fluctuations, and successfully reproduces the observed $T^3$ specific heat behavior. However, a critical discrepancy remains: NMR measurements in UTe$_2$ report the absence of a Hebel-Slichter peak in the spin-lattice relaxation rate ($1/T_1$) just below the critical temperature ($T_c$). In conventional isotropic $s$-wave superconductors, a large coherence peak in the density of states (DOS) typically generates a prominent Hebel-Slichter peak. The authors investigate whether the gap anisotropy and accidental nodes of their proposed point-node–like $s$-wave state can explain the suppression of this peak, thereby reconciling their theoretical model with experimental NMR data.

Methodology
The study employs the $f$-$d$-$p$ model, an effective six-orbital tight-binding Hamiltonian for UTe$_2$ that reproduces quasi-two-dimensional Fermi surfaces ($\alpha$ and $\beta$ bands) and antiferromagnetic fluctuations at $\mathbf{Q} \approx (0, \pm\pi, 0)$.

1. Gap Function Determination: The linearized Eliashberg equation was solved within TOPT to obtain the momentum-dependent gap function, $\Delta_a(\mathbf{k})$. This yielded a point-node–like $s$-wave pairing state with accidental nodes at the corners of the $\alpha$-surface in the $k_z=0$ plane.
2. Spin-Lattice Relaxation Calculation: The temperature dependence of the spin-lattice relaxation rate, $1/T_1$, was calculated using the derived gap structure. The calculation involved:
   • Deriving the retarded spin susceptibility $\chi^R_{+-}(\mathbf{Q}, \Omega)$ via analytic continuation of the thermal spin susceptibility.
   • Expressing $1/T_1$ in terms of partial density of states (PDOS) and coherence factors, specifically focusing on the $p$ and $p'$ orbitals to mimic light-element signals used in actual NMR experiments.
   • Separating the gap into a momentum-dependent part, $g_a(\mathbf{k})$, and a temperature-dependent magnitude, $\Delta(T)$, estimated by solving the BCS gap equation numerically.
   • Introducing a phenomenological damping parameter $\gamma$ to account for quasi-particle lifetime and disorder effects.
3. Comparative Analysis: The authors compared the calculated $1/T_1$ and PDOS for the point-node–like $s$-wave state against a hypothetical isotropic $s$-wave state (constructed by momentum-averaging the gap function) across varying values of $\gamma$.

Key Results

• Hebel-Slichter Peak Robustness: In the isotropic $s$-wave case, a large Hebel-Slichter peak is observed for low damping ($\gamma = 0.0001, 0.001$). While the peak diminishes as $\gamma$ increases to $0.005$, it remains observable.
• Point-Node Suppression: In the point-node–like $s$-wave case, the Hebel-Slichter peak is indeed smaller than in the isotropic case due to the broadening of the DOS coherence peaks caused by gap anisotropy. However, the peak remains "robust" and significant for low $\gamma$.
• Role of Damping: When $\gamma$ is increased to $0.005$, the Hebel-Slichter peak in the point-node–like state becomes negligibly small. Crucially, the peak in the isotropic $s$-wave state is suppressed to the same order of magnitude under the same conditions.
• PDOS Analysis: The calculated PDOS for the point-node–like state shows a single, broadened coherence peak (dominated by the $\beta$-band), whereas the isotropic state exhibits two sharp coherence peaks. The broadening in the anisotropic case suppresses low-energy excitations just below $T_c$, reducing the peak height, but does not eliminate it unless disorder ($\gamma$) is sufficiently large.

Conclusions and Significance
The paper concludes that while the point-node–like $s$-wave pairing state successfully explains the $T^3$ specific heat behavior and the reduction of the NMR Knight shift, it fails to account for the complete absence of the Hebel-Slichter peak observed in UTe$_2$ NMR measurements.

The authors demonstrate that the gap anisotropy inherent to the point-node–like state is insufficient to suppress the Hebel-Slichter peak to the level of non-observability without simultaneously suppressing the peak in a standard isotropic $s$-wave state to the same degree. Therefore, the scenario that DOS broadening due to gap anisotropy is the primary reason for the missing peak in UTe$_2$ is deemed unrealistic. Consequently, the point-node–like $s$-wave pairing state proposed in their previous work is inconsistent with the spin-lattice relaxation results, suggesting that the superconducting gap structure of UTe$_2$ remains an open question that cannot be resolved solely by this specific unconventional $s$-wave model.