Quantitative thermodynamic study of superconducting and normal states in UTe2 under pressure

This quantitative calorimetric study of UTe2 under pressure reveals a three-fold enhancement of electronic effective mass near the critical pressure and suggests that the high-pressure superconducting phase nucleates on only a fraction of the Fermi surface, likely driven by a quantum critical point associated with weak magnetic order rather than antiferromagnetism.

The Gist

Imagine UTe₂ (Uranium Ditelluride) as a tiny, chaotic city made of electrons. Under normal conditions, this city is a bit messy, but when you cool it down, the electrons decide to dance together in perfect unison, creating a state called superconductivity (where electricity flows with zero resistance).

This paper is like a detective story where scientists squeeze this city with a giant hydraulic press (applying pressure) to see how the dance changes. They wanted to understand why this material has two different ways of dancing (two superconducting phases) and what happens right before the dance stops completely.

Here is the breakdown of their findings using simple analogies:

1. The Two Dances (SC1 and SC2)

At normal pressure, the electrons do one dance (SC1). But as the scientists squeeze the city harder, a second, more energetic dance (SC2) appears.

• The Analogy: Imagine a ballroom. At first, everyone does a slow waltz (SC1). As the room gets smaller (pressure increases), a fast, energetic salsa (SC2) starts happening in a corner. Eventually, the salsa gets so popular it takes over the whole room, becoming the main dance.

2. The "Heavy" Electrons

One of the most important things the scientists measured was how "heavy" the electrons feel as they move. In physics, this is called effective mass.

• The Discovery: As they squeezed the city, the electrons didn't just get slightly heavier; they became three times heavier than usual right before the superconductivity vanished.
• The Analogy: Imagine the electrons are people running through a crowd. As the pressure increases, the crowd gets so dense and reactive that the runners feel like they are wading through molasses. They are moving slower and feeling much heavier. This "heaviness" is a sign that the electrons are interacting very strongly with each other.

3. The Mystery of the "Weak Magnetic Order" (WMO)

Before the city turns into a frozen, magnetic block (Antiferromagnetism), there is a weird, fuzzy phase called Weak Magnetic Order (WMO).

• The Analogy: Think of the city as a crowd of people.
  • Normal State: Everyone is chatting randomly.
  • WMO: Everyone starts whispering in small, secret groups, but no one is shouting yet. It's a "fuzzy" state.
  • Antiferromagnetism: Everyone suddenly freezes in a rigid, alternating pattern (like a checkerboard).
• The Finding: The scientists found that the "heaviest" electrons (the peak of the molasses effect) didn't happen when the city froze completely. It happened right when the "whispering groups" (WMO) were strongest. This suggests that the superconducting dance (SC2) is actually fueled by these whispers, not the final frozen state.

4. The "Partial" Dance Floor

The paper suggests something very strange about the second dance (SC2).

• The Discovery: The "heaviness" of the electrons kept increasing even after the salsa dance (SC2) started to get slower.
• The Analogy: Imagine the salsa dance (SC2) only happens on a small stage in the middle of the ballroom, while the rest of the room is still doing the waltz. As the pressure increases, the dancers on that small stage get heavier and heavier, but the dance itself gets a bit slower.
• Why it matters: This explains why the "jump" in energy (a measure of how much the dance changes) gets huge right before the dance stops. The scientists realized that SC2 might only be "nucleating" (starting) on a tiny fraction of the electron population, while the rest of the electrons are still doing something else.

5. The Quantum Critical Point

The scientists found a "sweet spot" in pressure (around 1.2 GPa) where everything aligns:

1. The electrons are at their heaviest (maximum interaction).
2. The "whispering" (WMO) phase is at its peak.
3. The second dance (SC2) is at its strongest.

• The Analogy: It's like a tightrope walker. The most exciting moment isn't when they are standing still or when they fall; it's the exact moment they are balancing on the edge of a cliff. The paper suggests that the "Weak Magnetic Order" is that cliff edge. The superconductivity is thriving because it is dancing right on the edge of this magnetic instability.

The Big Picture Conclusion

This study is a "quantitative" (precise number-crunching) confirmation of what was previously just a guess.

• Before: Scientists knew pressure changed UTe₂, but they didn't have the exact numbers.
• Now: They proved that the electrons become three times heavier due to magnetic fluctuations.
• The Twist: The superconductivity isn't fighting against the magnetic order; it's actually born from it. The "Weak Magnetic Order" acts like a catalyst, helping the electrons pair up and dance, even though that same magnetic order eventually tries to crush the dance if you squeeze too hard.

In short: UTe₂ is a material where electricity flows perfectly because the electrons are dancing on the edge of a magnetic cliff, and squeezing the material makes that dance more intense, heavier, and more complex.

Technical Summary

1. Problem Statement

UTe$_2$ is a prominent candidate for unconventional, spin-triplet superconductivity, exhibiting multiple distinct superconducting phases (SC1 and SC2) that can be tuned by magnetic field or pressure. While previous studies using AC calorimetry and transport measurements under pressure have qualitatively identified the phase diagram—including the emergence of a high-pressure superconducting phase (SC2), a "Weak Magnetic Order" (WMO) phase, and a long-range antiferromagnetic (AFM) phase—reliable quantitative thermodynamic data has been lacking.

Specifically, there was no direct, quantitative measurement of the Sommerfeld coefficient ($\gamma$) under pressure to determine the evolution of the electronic effective mass ($m^*$) and density of states. Furthermore, the relationship between the enhancement of electronic correlations, the emergence of the WMO phase, and the stabilization of the SC2 phase near a potential quantum critical point (QCP) remained unclear. The study aims to resolve these ambiguities by providing precise calorimetric data to map the thermodynamic evolution of UTe$_2$ up to the critical pressure where superconductivity is suppressed.

2. Methodology

The authors performed a quantitative specific heat (calorimetry) study on a high-quality UTe$_2$ single crystal (mass $m = 23.73$ mg, grown by Chemical Vapor Transport) under hydrostatic pressure.

• Experimental Setup: Measurements were conducted in a piston-cylinder pressure cell using a quasi-adiabatic technique. This method allows for the direct measurement of total heat capacity, from which the specific heat of the sample is extracted by subtracting the contributions of the cell, pressure-transmitting medium (Fluorinert FC-72), and Teflon cap.
• Pressure Range: Measurements were taken from ambient pressure up to 1.48 GPa, covering the region where SC2 emerges, the WMO phase appears, and the long-range AFM order is established.
• Data Analysis:
  • The electronic specific heat coefficient ($\gamma$) was extracted from the $C/T$ values in the paramagnetic Fermi-liquid regime above ordered phases, after subtracting a pressure-independent phonon contribution.
  • Entropy ($S$) was calculated by integrating the electronic contribution of $C/T$.
  • The magnitude of the specific heat jumps ($\Delta C/T_c$) at superconducting transitions was compared against the BCS weak-coupling limit ($1.43\gamma$) and strong-coupling models.
  • A comparative phase diagram was constructed using data from this study and previous AC calorimetry data from diamond anvil cell (DAC) experiments.

3. Key Results

A. Phase Diagram Evolution

• SC1 and SC2: At ambient pressure, only SC1 is observed ($T_{c1} \approx 1.75$ K). As pressure increases, a second transition (SC2) appears. $T_{c2}$ initially rises, peaking near 1 GPa, before decreasing.
• Magnetic Orders:
  • WMO (Weak Magnetic Order): A broad anomaly appears at higher temperatures (around 4.5 K at 1.43 GPa) before the AFM transition. This phase is stabilized in a significant pressure range below the critical pressure for AFM.
  • AFM (Antiferromagnetic): Long-range AFM order emerges at $T_{AFM} \approx 3.3$ K at the highest pressure (1.48 GPa), where superconductivity is completely suppressed.
• Critical Pressure: The critical pressure ($p_c$) for the suppression of superconductivity and the onset of AFM is identified at 1.45 GPa (piston-cylinder) and 1.62 GPa (DAC).

B. Enhancement of Electronic Correlations ($\gamma$)

• The Sommerfeld coefficient $\gamma$ increases monotonically with pressure, reaching a three-fold enhancement at 1.29 GPa ($\gamma \approx 360$ mJ/mol·K$^2$).
• This maximum in $\gamma$ occurs at a pressure significantly lower than the critical pressure for AFM order ($p_c$).
• Above 1.29 GPa, as the WMO and AFM phases emerge, $\gamma$ decreases.
• This three-fold increase in $\gamma$ corresponds to a similar enhancement in the effective mass ($m^*$), confirming strong electronic correlations driven by magnetic fluctuations.

C. Specific Heat Jumps and SC2 Nucleation

• SC1: The specific heat jump ($\Delta C/T_c$) for SC1 decreases with pressure initially but shows a sudden, sharp increase at 1.43 GPa (near $p_c$), accompanied by a sharpening of the transition.
• SC2: The jump size for SC2 grows rapidly with pressure, peaking at 1.3 GPa. Crucially, the jump size extrapolates to zero at the pressure where the SC1 and SC2 transition lines meet.
• Entropy Balance: At ambient pressure, the entropy balance is consistent with $\gamma$. Under pressure, up to 1.29 GPa, the entropy rise tracks $\gamma$. However, near $p_c$, an excess entropy is observed, attributed to strong magnetic fluctuations in the WMO phase.

D. Theoretical Interpretation

The authors propose a model where the SC2 phase nucleates only on a fraction ($\alpha$) of the Fermi surface, rather than the entire surface.

• This model explains why $\gamma$ continues to increase even as $T_{c2}$ decreases (the pairing strength increases on the specific fraction of the Fermi surface).
• It accounts for the vanishing specific heat jump of SC2 at the meeting point of the SC1 and SC2 lines, satisfying thermodynamic consistency for a triple point of second-order transitions.
• The large specific heat jumps observed at 1.43 GPa are attributed to the suppression of magnetic entropy from the WMO phase upon entering the superconducting state.

4. Key Contributions

1. First Quantitative $\gamma$ Measurement: Provided the first direct, quantitative determination of the pressure-induced enhancement of the electronic effective mass in UTe$_2$, revealing a 3x increase near 1.3 GPa.
2. Clarification of the Phase Diagram: Established that the WMO phase exists in a wide pressure range below the AFM critical pressure and that the maximum of $\gamma$ and $T_{c2}$ coincides with the extrapolated critical pressure of the WMO phase ($p_{WMO}$), rather than the AFM critical pressure.
3. Fermi Surface Fraction Model: Demonstrated that the SC2 phase likely involves pairing on only a fraction of the Fermi surface, resolving discrepancies between the vanishing SC2 jump at low pressures and the strong coupling constants required to explain the data.
4. Quantum Criticality: Provided strong evidence that the WMO phase, rather than the AFM phase, hosts a quantum critical point that drives the enhancement of electronic correlations and stabilizes the SC2 phase.

5. Significance

This study fundamentally advances the understanding of unconventional superconductivity in UTe$_2$ by linking thermodynamic properties directly to the underlying electronic structure.

• It challenges the assumption that the AFM quantum critical point is the primary driver of SC2; instead, it highlights the WMO phase as the likely source of the critical fluctuations mediating pairing.
• The finding that the maximum effective mass occurs before the suppression of superconductivity suggests a complex interplay where superconductivity competes with, yet is enhanced by, the fluctuations of an intermediate magnetic order.
• The quantitative data serves as a critical benchmark for theoretical models of spin-triplet superconductivity and multi-band pairing mechanisms in heavy fermion systems.