Thermodynamic Discovery of Tetracriticality and Emergent Multicomponent Superconductivity in UTe$_2$

Abstract

The candidate topological superconductor UTe$_2$ exhibits a complex phase diagram with multiple superconducting states, yet the nature of their coexistence has remained a central mystery. In particular, the apparent intersection of two second-order phase boundaries at a ``triple point'' in the pressure-temperature phase diagram is thermodynamically forbidden, suggesting hidden phase transitions or a fundamental misunderstanding of the superconductivity in UTe$_2$. Here, we use pulse-echo ultrasound to resolve this puzzle by discovering a new phase boundary that is characterized by a unique ``upward jump" in the sound velocity -- direct thermodynamic evidence for a re-entrant phase transition. Our results establish $\left(P^{\star},T^{\star}\right)$ as a tetracrtical point, beyond which the ambient and pressure-induced superconducting order parameters form a multi-component state. We use the measured phase diagram to construct a Ginzburg-Landau theory that shows that strong competition between the two superconducting order parameters drives the re-entrance and leads to phase locking that suppress superconducting fluctuations. These findings provide the definitive magnetic field-temperature-pressure phase diagram and establish a thermodynamic foundation for multi-component -- and potentially topological -- superconductivity in UTe$_2$.

Figures (5)

• Figure 1: Proposed temperature-pressure phase diagrams for UTe2. The elusive multicritical point is indicated by a black star. The well-established SC1 and SC2 phase boundaries are shown as solid red and blue lines respectively. a) A proposed phase diagram where the SC2 phase boundary extends past the critical point, with a region of co-existence of the SC1 and SC2 phases (shaded purple). Such a scenario implies the existence of a phase boundary indicated with a blue dashed line---this boundary has not yet been observed. b) In a second proposal, the SC2 phase boundary terminates upon meeting the SC1 phase boundary. Given the weak heat capacity anomalies, this requires a third-order superconducting transition, where second derivatives of the free energy (such as heat capacity) remain continuous yipThermodynamicConsiderationsPhase1991braithwaiteMultipleSuperconductingPhases2019.
• Figure 2: The phase diagram of UTe2 near the tetracritical point. a) The zero-field phase diagram of UTe2 constructed in this work. $T_{\rm c1}$, $T_{\rm c2}$ and $T_{\rm c2}^*$ are represented by solid triangles, hollow triangles, and a star respectively. The solid red and blue lines are guides to the eye that represent the SC1 and SC2 phase boundaries, respectively. Regions in phase space where the SC1 and SC2 order parameters are non-zero are shaded red and blue, respectively. Inset: The UTe2 phase diagram over a larger pressure range, with $T_{\rm c1}$ and $T_{\rm c2}$ lines adapted from Braithwaite et al. braithwaiteMultipleSuperconductingPhases2019 b) The $c_{55}$ elastic modulus versus temperature in UTe2 at different pressures. For $P<0.2$ GPa, we find a single, sharp kink at $T_{\rm c1}$. At $P>0.2$ GPa, we find sharp kinks at both $T_{\rm c2}$ and $T_{\rm c1}$. c) The $c_{33}$ elastic modulus, measured simultaneously with the $c_{55}$ elastic modulus. For $P<0.2$ GPa, we find a downward jump at $T_{\rm c1}$ when UTe2 enters the SC1 state. This jump grows for $P>0.2$ GPa, and occurs at $T_{\rm c2}$. At $P=0.21$ GPa, we find an upward jump at $T_{\rm c2}^*$ as the sample exits the SC2 superconducting state upon cooling. Inset: The jump at $T_{\rm c1}$ at 0.77 GPa. The data shown are from a different sample optimized for a $c_{33}$ measurement. We show quantitative similarity between both samples in the SI.
• Figure 3: Field-temperature phase diagrams of UTe2 at fixed pressure. Corresponding $c_{55}$ and $c_{33}$ data are shown directly below. $T_{\rm c1}$, $T_{\rm c2}$ and $T_{\rm c2}^*$ measured in this work are represented by solid triangles, hollow triangles, and stars respectively. a-d) Phase diagrams. Regions in phase space where the SC1, SC2, and SC1+SC2 order parameters are non-zero are shaded in red, blue, and purple, respectively. e-h) $c_{55}$ elastic modulus data. $T_{\rm c1}$ and $T_{\rm c2}$ are seen as sharp kinks in the data. j-m) $c_{33}$ elastic modulus data. The downward jumps upon cooling correspond to $T_{\rm c1}$ or $T_{\rm c2}$, while the upward jumps correspond to $T_{\rm c2}^*$.
• Figure 4: Field-Temperature-Pressure phase diagram of UTe2. SC1 and SC2 phase boundaries are represented by red and blue sheets respectively. The solid red and blue lines on the $B=0$ plane represent the zero-field phase boundary from this work (\ref{['fig:zerofield']}a), while the solid black lines represent the constant-pressure phase boundaries shown in \ref{['fig:infield']}a-d. Dashed black lines are adapted from Vasina et. al. vasinaConnectingHighFieldHighPressure2025. The zero temperature ground states of the system are shown on the $T=0$ plane. The tetracritical point in the $P-T$ plane---$\left(P^{\star}\xspace,T^{\star}\xspace\right)$, indicated with a light-blue star---becomes a line of tetracritical points in $B_b$-$P$-$T$ space, indicated by the light-blue line. This line terminates at 12 tesla and 1 kelvin at ambient pressure. Between 12 and 18 tesla, the SC1+SC2 state exists at ambient pressure.
• Figure 5: Order parameter competition in UTe2. a-c) The order parameters magnitudes for the SC1 ($|\psi_1|$) and SC2 ($|\psi_2|$) states versus temperature at different pressures. $T_{\rm c1}$ is indicated by a solid gray line, while $T_{\rm c2}$ and $T_{\rm c2}^*$ are indicated by dashed gray lines. Insets show where each pressure lies on the phase diagram of \ref{['fig:zerofield']}a. d-f) Ultrasonic attenuation $\alpha_{33}$ versus temperature at different pressures. We find an parameter relaxation peak as the sample enters the SC1 state at 0 GPa, and as the sample enters and exits the SC2 state at 0.21 GPa. At 0.77 GPa, anomalously large phase fluctuations kamatVanishingPhaseStiffness2026 cause the ultrasonic attenuation inside the SC2 state to be larger than the normal state value. $\alpha_{33}$ decreases as the SC1 order parameter grows and suppresses phase fluctuations below $T_{\rm c1}$.