Bulk signatures of re-entrant superconductivity in UTe$_2$ from ultrasound measurements

We report bulk ultrasound measurements up to 80 T and down to 0.5 K of the field re-entrant superconducting phase of the unconventional superconductor UTe$2$. Clear bulk signatures of superconductivity are observed in the longitudinal elastic mode $c{11}$ for fields applied at a tilt angle of $θ_{b-c} =30^\circ$ from $b$-axis. We confirm an upper critical field of $H_{\rm c2}\approx65$ T at 0.5 K and bulk superconductivity which survives up to $T\approx 2$ K for fields above the metamagnetic transition. The $c_{11}$ mode has propagation and displacement vectors along the $a$-axis, and for fields applied at a tilt angle of $θ_{b-c} =30^\circ$, this mode is sensitive to the elasticity of the vortex lattice. The anomalies observed in $c_{11}$ are in part reminiscent of superconducting vortices pinned to lattice defects. Nonetheless, an excess attenuation, with respect to the normal state, is observed throughout the entire superconducting phase, suggesting unusual vortex dynamics and pinning in the field re-entrant superconducting phase of UTe$_2$.

💡 Research Summary

This paper presents a comprehensive bulk investigation of the field‑re‑entrant superconducting phase (denoted SC3) in the heavy‑fermion compound UTe₂ using pulsed‑field ultrasound spectroscopy. Measurements were performed down to 0.5 K and up to 80 T, with the magnetic field tilted by θ₍b‑c₎ = 30° away from the crystallographic b‑axis (i.e., in the b‑c plane). The authors focused on the longitudinal acoustic mode c₁₁, whose propagation vector k and polarization u are both aligned along the a‑axis, making it sensitive to the elastic response of a possible vortex lattice.

Key experimental observations are:

1. Metamagnetic transition (Hm) – For fields below ~41 T (the metamagnetic transition), the relative sound velocity Δv/v decreases smoothly with increasing field while the attenuation Δα rises, reproducing behavior known from previous studies with H∥b. At H ≈ Hm a sharp step‑like anomaly appears in both Δv/v and Δα, confirming the correct tilt angle.

2. Re‑entrant superconductivity (SC3) and hysteresis – Below the superconducting transition temperature (T ≈ 2 K) and for H > Hm, a new hysteretic feature emerges. On the down‑sweep of the magnetic‑field pulse, both Δv/v and Δα are larger than in the normal (field‑polarized) state, whereas on the up‑sweep the anomalies are much weaker. The hysteresis opens at a field HSC that tracks the upper critical field Hc₂(T) (≈65 T at 0.5 K) and persists up to Hm. Its magnitude grows with decreasing temperature and shows a clear dependence on the ultrasound frequency (70–388 MHz). The authors rule out trivial heating effects (eddy‑current heating, magnetocaloric heating) through detailed supplemental analyses.

3. Mode selectivity – The transverse mode c₆₆ (k∥a, u∥b) shows no hysteresis under identical conditions, indicating that the observed anomalies in c₁₁ are not caused by simple magnetostriction (ΔLₐ) but rather by a mechanism that couples specifically to the longitudinal strain along a.

To interpret these findings, two theoretical frameworks are examined:

• Landau‑Khalatnikov (LK) order‑parameter relaxation predicts a reduction of both sound velocity and attenuation below Tc, opposite to the experimental trend, and cannot reproduce the observed frequency dependence or hysteresis without invoking unrealistically long relaxation times (τ ≈ 10⁻¹⁰ s ≪ pulse duration). Hence, LK relaxation is insufficient.

• Vortex‑lattice pinning provides a natural explanation. In the mixed state, pinned vortices contribute an additional elastic term Δcᵥᵢⱼ to the overall elastic modulus. For the c₁₁ geometry (k∥u⊥H) the measurement probes the vortex‑lattice compression modulus cᵥL, which can increase the sound velocity when vortices are pinned (critical‑state model predicts c₁₁ ≈ B²/4π). The excess attenuation is then attributed to vortex‑motion dissipation (flux‑flow) that remains sizable even in the pinned regime, especially at high frequencies. The absence of a signal in c₆₆ is consistent with the hierarchy cᵥ₆₆ ≪ cᵥ₁₁ for a strongly type‑II superconductor (κ ≫ 1). A phenomenological thermally assisted flux flow (TAFF) model reproduces the temperature dependence of the hysteresis and the evolution of HSC(T).

Overall, the study establishes that the SC3 phase in UTe₂ is a genuine bulk superconducting state, not a surface or filamentary effect. The ultrasound signatures—enhanced stiffness on the down‑sweep and persistent excess attenuation—point to unconventional vortex dynamics, possibly involving strong pinning to intrinsic lattice defects or magnetic textures. This bulk evidence supports scenarios where the re‑entrant superconductivity is stabilized by proximity to a metamagnetic quantum critical point or by a Jaccarino‑Peter–type exchange field compensation. Moreover, the work demonstrates the power of high‑field ultrasound as a bulk probe of exotic superconductivity in strongly correlated systems.