Pseudo Point Nodal Superconducting Gap in Spin-Triplet UTe$_2$

Abstract

The unconventional superconductor UTe$_2$ represents a rare example of spin-triplet pairing with potentially topologically protected quantum states. However, conflicting reports on its gap structure, particularly regarding point nodes, have hindered understanding of the order parameter symmetry and topological properties. Here we report high-resolution thermal conductivity measurements on high-quality UTe$_2$ single crystals down to ~50 mK that resolve the gap anisotropy through bulk directional transport. The $b$-axis thermal conductivity $κ_b/T$ exhibits negligible residual conductivity as $T \to 0$, and its temperature dependence is consistent with a small superconducting energy gap along the $b$-axis. Under magnetic fields, the residual $κ_b/T$ shows only weak field-induced enhancement. Remarkably, a threshold field emerges at low fields for $H \parallel a$, characterized by a kink that signals a change in quasiparticle transport normal to the field. Below the threshold, $κ_b/T$ remains isotropic for all field orientations, whereas strong anisotropy between transport along and normal to the field develops above it. These signatures strongly suggest that UTe$_2$ exhibits a fully gapped state with a pseudo point-nodal structure, where gap minima approach but never reach zero. We estimate the minimal gap $Δ_{min}/Δ_0 \sim 0.1$ along the $b$-axis, where $Δ_0$ is the characteristic superconducting gap. This unusual gap structure provides crucial insights into the pairing mechanism and topology of this spin-triplet superconductor and excludes non-unitary mixing of pairing symmetries.

Figures (4)

• Figure 1: Superconducting gap structure and Doppler shift:a, b, c: Superconducting gap structure $\Delta$ (yellow) on a 2D cylindrical Fermi surface (yellow-green) for $A_u$ (a and b) and $B_{2u}$ (c) symmetries. a: Full gap opens uniformly. b: At pseudo point nodes, the gap amplitude approaches zero but remains finite. The accompanying figure (right) shows the quasiparticle dispersion in the vicinity of the gap minimum. c: The $B_{2u}$ state hosts point nodes (dark green) along the $b$-axis ($k_y$ direction). The Dirac cone emerges where the gap amplitude vanishes at a point node. d,e: Schematic representation of Doppler shift effects under magnetic field $\boldsymbol{H}$ applied perpendicular and parallel to point nodes, respectively. The supercurrent velocity $\boldsymbol{v}_s$ circulating around vortex cores is oriented perpendicular to $\boldsymbol{H}$. d: For $\boldsymbol{H} \parallel$ nodes, Doppler-shifted quasiparticle excitations are suppressed due to $\boldsymbol{v}_{\rm F}\cdot\boldsymbol{p}_{\rm s} = 0$. e: For $\boldsymbol{H} \perp$ nodes , the Doppler shift energy $\boldsymbol{v}_{\rm s}\cdot\boldsymbol{p}_{\rm s}$ exceeds the superconducting gap magnitude $\Delta$, thereby activating low-energy quasiparticle excitations in the nodal directions (red-shaded region). When thermal current $\boldsymbol{j}_Q$ is applied parallel to the nodal directions ($\boldsymbol{j}_Q \parallel$ nodes), thermal conductivity exhibits selective sensitivity to nodal excitations, while it becomes insensitive when $\boldsymbol{j}_Q \perp$ nodes.
• Figure 2: Thermal conductivity in zero magnetic field:a, The main panel shows the $b$-axis thermal conductivity divided by temperature, $\kappa_b/T$, in zero field for sample #A ($RRR \approx$410) and #B ($RRR \approx$350). Both samples exhibit a sharp discontinuity at $T_c$ (arrow). The lower inset illustrates the experimental configuration for thermal conductivity measurements with thermal current $j$$_Q$ applied along the crystallographic $b$-axis. The upper inset shows electrical resistivity along the $b$-axis in zero field plotted as a function of $T^2$ for #A and #B. $\rho_b(T)$ is well fit by $\rho_b(T)=\rho_b^0+AT^2$ (orange and cyan dashed line, respectively). b, $\kappa_b/T$ in zero field for #A and #B at very low temperatures. The inset shows $\kappa_b/T$ vs. $T^2$. The gray dashed line indicates the residual thermal conductivity expected for a superconductor with line nodes in the gap structure. c, Calculated temperature dependence of $\kappa/T$ for point nodes ($\Delta_{\rm min}=0$) and pseudo point nodes ($\Delta_{\rm min}=0.1\Delta_0$) , with heat current parallel to the nodal direction.
• Figure 3: Low temperature thermal conductivity under magnetic fields along each crystallographic axis:a,b,c, Temperature dependence of low-temperature thermal conductivity along the $b$-axis under magnetic fields applied parallel to the $a$-axis (a), $b$-axis (b), and $c$-axis (c), respectively. The dashed lines represent polynomial fitting results.
• Figure 4: Field-enhanced thermal conductivity in the $T \rightarrow 0$ limit: a--c, The zero-temperature limit of the $b$-axis thermal conductivity divided by temperature $\kappa_0^b/T$, normalized by the normal state value $(\kappa_0^b/T)_N$, plotted as a function of magnetic field applied parallel to the $a$, $b$, and $c$ axes. The magnetic fields are normalized by the corresponding upper critical field $H_{c2}^{a,b,c}$. a, $(\kappa_b^0/T)/(\kappa_b^0/T)_N$ as a function of $H/H_{c2}^{a,b,c}$ for $H$$\parallel a$, $b$ and $c$. b, $(\kappa_b^0/T)/(\kappa_b^0/T)_N$ as a function of $H/H_{c2}^{a,b}$ for $H$$\parallel a$ and $b$. For $H$$\parallel a$, a kink is observed at $H^*/H_{c2}^a \approx 0.015$ (arrow), below which $(\kappa_b^0/T)/(\kappa_b^0/T)_N$ exhibits nearly isotropic behavior, whereas pronounced anisotropy develops above this characteristic threshold field. c, The red circles show the thermal conductivity difference $\Delta (\tilde{\kappa}_b^0/T)^{\mathrm{DS}} = [\Delta (\kappa_b^0/T)/(\kappa_b^0/T)_N]_{{\boldsymbol H} \parallel a} - [\Delta (\kappa_b^0/T)/(\kappa_b^0/T)_N]_{{\boldsymbol H} \parallel b}$ for #B, which represents the difference between field configurations with and without the Doppler shift contribution. For comparison, pink open circles represent the corresponding difference calculated by subtracting the #B (${\boldsymbol H} \parallel b$) data from the #A (${\boldsymbol H} \parallel a$) data. Both datasets exhibit good agreement. The dashed line is a guide to the eye. d, Magnetic field dependence of the residual thermal conductivity term for the true point nodes ($\Delta_{\rm min}=0$) and pseudo-nodes ($\Delta_{\rm min}=0.1\Delta_0$) in a model of a Fermi surface with a square cross-section in the $a$-$b$ plane and nodes/minima along the $b$ axis, as described in the Supplementary Information. Inset: Schematic of Doppler shift effects in a model circular unit cell of the vortex lattice (radius $R$). Below the threshold field $E_{H}^\star=\Delta_{\rm min}$ the Doppler shift at distances $R>r>R^\star\sim R E_H/\Delta_{\rm min}$ is insufficient to generate unpaired quasiparticles (blue ring). For the heat current normal to the field this yields an effective thermally insulating barrier, preventing net heat conduction, see text for details.