Vanishing Phase Stiffness and Fluctuation-Dominated Superconductivity: Evidence for Inter-Band Pairing in UTe$_2$

TL;DR

This work shows that three-dimensional superconductivity in UTe2 hosts an unusually broad fluctuation regime under pressure, extending far above and below the transition temperature. Through ultrasonic probes of the elastic modulus $c_{33}$ and attenuation $\alpha_{33}$, the authors document a crossover from mean-field behavior at ambient pressure to a fluctuation-dominated SC2 phase with a dramatically reduced phase stiffness. A multiband, inter-band pairing scenario driven by ferromagnetic fluctuations naturally yields the observed low phase stiffness and short coherence length, challenging conventional single-band GL descriptions. The findings imply a distinct pairing channel in SC2 with substantial implications for high kinetic-inductance devices and for understanding multiband, fluctuation-rich superconductivity in correlated materials.

Abstract

Superconductivity in three dimensions is almost universally governed by Ginzburg-Landau mean field theory, with critical fluctuations typically confined to within a few percent of the transition temperature ($T_{\rm c}$). We report that the heavy-Fermion superconductor UTe$_2$ exhibits a fluctuation regime that extends over a temperature range as wide as $T_{\rm c}$ itself -- the largest observed for any three-dimensional superconductor. Through ultrasound measurements of the elastic moduli and sound attenuation, we find that UTe$_2$ transitions from a mean-field-like state at ambient pressure to a fluctuation dominated state at higher pressures. This regime is marked by elastic softening and an increase in sound attenuation that onsets well above $T_{\rm c}$, with the attenuation remaining anomalously high deep in the superconducting state. Our analysis shows that these features stem from an extremely low superfluid phase stiffness. This results in a kinetic inductance as high as that of granular aluminum, but achieved in the clean limit. We propose that this exotic state is driven by dominant inter-band pairing mediated by ferromagnetic fluctuations, leading to "local" cooper pairs with a coherence length of only a few lattice constants.

Figures (7)

• Figure 1: a) The temperature-pressure phase diagram of UTe2 in zero magnetic field. Phase boundaries (black lines) are guides to the eye adapted from Braithwaite et. al. braithwaiteMultipleSuperconductingPhases2019. The two superconducting phases, SC1 and SC2, are shaded red and blue, respectively. Colored dots indicate transitions measured in the present work, and the pressure above which we observe two transitions is denoted by $P^{\star}$. b) Specific heat of UTe2 at ambient pressure as a function of temperature from ranNearlyFerromagneticSpintriplet2019. A sharp upwards jump in $C/T$ is observed at $T_{\rm c1}$ upon entering the SC1 state. c) Ambient pressure $c_{33}$ elastic modulus as a function of temperature. A sharp downward jump $\Delta c_{33}\xspace/c_{33}\xspace$ is seen at $T_{\rm c1}$. d) The longitudinal sound attenuation $\alpha_{33}$ at ambient pressure as a function of temperature. An order-parameter relaxation peak $\Delta \alpha_\text{LK}$ is seen just below $T_{\rm c}$. As the quasiparticle density of states is gapped out below $T_{\rm c}$, we see the conventional decrease $\Delta \alpha_\text{gap}$ in the attenuation. Fits to different models of gap attenuation are shown.
• Figure 2: Longitudinal sound attenuation $\alpha_{33}$ and fractional change in elastic modulus $\Delta c_{33}\xspace/c_{33}\xspace$ as a function of temperature at different pressures. $T_{\rm c1}$ and $T_{\rm c2}$ are indicated with gray lines. $\alpha_{33}$ exhibits a peak upon entering the SC1 state at the lowest three pressures ($P <P^{\star}\xspace$). At the higher three pressures ($P >P^{\star}\xspace$), the peak increases in magnitude and occurs upon entering the SC2 phase. The transition from the SC2 phase to the SC1 phase is clearly seen in the form of a kink as the attenuation decreases in the SC2 phase. Both features remain sharp at all pressures, indicating minimal pressure inhomogeniety. $c_{33}$ exhibits a sharp drop upon entering the SC1 state at the lowest two pressures, as expected for a superconducting transition. We see fluctuations onset at pressures greater than 0.19 GPa, in the form of a decrease in $c_{33}$ and increase in $\alpha_{33}$, at temperatures greater than $T_{\rm c}$. The size of the jump in $c_{33}$ increases as the pressure is increased above $P^{\star}$. The insets show a small jump that corresponds to the SC2-SC1 transition, which is seen at the highest two pressures. At the highest pressure of 0.77 GPa, the attenuation deep inside the SC2 superconducting state is larger than it is in the normal state, which is highly unusual for a superconductor.
• Figure 3: a) The elastic modulus $c_{33}$ across the superconducting transition as a function of temperature, with the background subtracted out. We also show the elastic response of YBa$_2$Cu$_3$O$_7$---a superconductor with a relatively large fluctuation region. b) Theoretical predictions for $c_{33}$ using the Gaussian fluctuations model from the analysis section. The Ginzburg number $Gi$ quantifies the strength of the fluctuations and increases at higher pressures.
• Figure 4: A cartoon depicting intra- and inter-- band pairing for a simple 2-band parabolic dispersion. Blue dots represent intra-band pairs at momenta $k$ and $-k$, and red dots represent inter-band pairs at momenta $k'$ and $-k'$. The intra-band pairs sit at the Fermi energy $E_F$. The gray shaded area represents the pairing energy shell. For an inter-band pairing state, this pairing energy ($\varepsilon_c$) must be greater than the band splitting $\varepsilon_B$samokhinGinzburgLandauEnergyMultiband2024, which is denoted by the red shaded area.
• Figure 5: Sound attenuation and the shear elastic modulus $c_{55}$. There is a sharp drop in the attenuation at $T_{\rm c1}$ at all pressures, following the conventional expectation of BCS theory. Sharp kinks are seen in the elastic modulus at both $T_{\rm c1}$ and $T_{\rm c2}$. Insets show a zoomed-in elastic modulus, clearly showing two $T_{\rm c}$'s for $P>P^{\star}\xspace$.