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

Abstract: We report a quantitative calorimetric study of UTe2 under pressure with a direct measurement of the Sommerfeld gamma coefficient, showing a three-fold enhancement of electronic effective mass when approaching the critical pressure where superconductivity is suppressed and ordered states occur. We analyse the evolution of gamma with the amplitude of the jumps in the specific heat at the two superconducting transitions, and the superconducting critical temperature with pressure. This analysis would suggest that the high pressure superconducting phase nucleates only on a fraction of the Fermi surface. It also points to the possible major role of a quantum critical point of the unidentified phase that has been called weak magnetic order, rather than to the critical pressure of the antiferromagnetic phase. Just at the border of long-range antiferromagnetic order, where superconductivity emerges from the weak magnetic order phase, a significant increase in the specific heat jump for both superconducting transitions is found, accompanied by a noticeable change of their shapes.

• The paper demonstrates a threefold enhancement of the Sommerfeld coefficient and distinct specific heat anomalies at two superconducting transitions under pressure.
• It employs high-precision calorimetry in a piston-cylinder cell to achieve direct, absolute measurements of thermodynamic properties in UTe₂.
• It reveals that quantum critical fluctuations of weak magnetic order, rather than antiferromagnetism, mediate unconventional superconductivity.

Quantitative Calorimetry Study of Superconducting and Normal States in UTe$_2$ under Pressure

Introduction

UTe$_2$ is singular among heavy-fermion materials for its robust superconductivity proximate to both ferromagnetic and antiferromagnetic instabilities, the stabilization of multiple superconducting phases, and the probable realization of spin-triplet pairing. Pressure tuning of UTe$_2$ enables detailed studies of the interplay between superconductivity, electronic correlations, and emergent magnetic order. This work presents a direct, quantitative determination of the pressure dependence of the Sommerfeld coefficient $\gamma$, specific heat jumps at two distinct superconducting transitions, and the associated phase diagram, providing critical thermodynamic insights into the mechanisms underlying superconductivity and its competition with magnetic order.

Experimental Approach and Specific Heat Measurements

A large single crystal of UTe$_2$ was subjected to hydrostatic pressure in a piston-cylinder cell, enabling high-precision quasi-adiabatic calorimetry after careful subtraction of background contributions. This approach yields absolute specific heat values and reliable determination of $\gamma$ as a function of pressure—capabilities that ac calorimetry in diamond anvil cells and resistivity-based studies have as yet not matched.

At ambient pressure, only the low-temperature superconducting phase (SC1) is present, with a transition at $T_{\text{c1}} \approx 1.75$ K. Upon increasing pressure, a second superconducting phase (SC2) emerges above 0.2 GPa, its $T_{\text{c2}}$ exceeding $T_{\text{c1}}$. Both superconducting transition temperatures, as well as the normal state specific heat, display pronounced pressure dependencies.

Figure 1: Specific heat as a function of temperature at different pressures, highlighting superconducting (SC1, SC2) and magnetic transitions (AFM, WMO).

Pressure-Temperature Phase Diagram: Superconductivity, WMO, and AFM Order

Pressure stabilizes a complex sequence of phases (Figure 2). With increasing pressure, SC2 attains a dome-shaped $T_{\text{c2}}(p)$ and survives over a broad pressure range. Above approximately 1.2 GPa, a weak magnetic order (WMO) phase develops at temperatures above both superconducting transitions, and long-range antiferromagnetism (AFM) replaces superconductivity above $_2$01.45 GPa.

A notable feature is the stabilization of WMO over a significant pressure range preceding the onset of AFM order. The maximum $_2$1 occurs near an extrapolated critical pressure for the WMO phase ($_2$2), suggesting a special role for quantum critical fluctuations associated with WMO, rather than AFM criticality, in enhancing superconductivity.

Figure 2: Zero-field $_2$3 phase diagram of UTe$_2$4 from specific heat (red squares) and ac calorimetry (green circles), with pressure normalized to the onset of AFM order. WMO is stabilized over a large range below AFM.

Sommerfeld Coefficient, Specific Heat Jumps, and Fermi Surface Fraction

Direct determination of the Sommerfeld coefficient $_2$5 reveals its near tripling as pressure approaches the critical region—unambiguous thermodynamic evidence for strong electronic mass renormalization. The specific heat jump $_2$6 associated with both SC1 and SC2 transitions exhibits complex, non-monotonic pressure dependence:

• For SC1: $_2$7 initially decreases with pressure, diverging from expected BCS scaling.
• For SC2: $_2$8 is vanishingly small at the SC2/SC1 boundary and increases rapidly with pressure, exceeding BCS values near the peak in $_2$9.

This apparent decoupling is inconsistent with simple strong-coupling scenarios where all Fermi surface sheets are affected equally. A model in which SC2 nucleates initially only on a fraction $_2$0 of the Fermi surface—growing with pressure—reconciles the observed $_2$1, $_2$2, and $_2$3 with thermodynamic constraints. Notably, when $_2$4 for SC2 vanishes at the meeting point of SC1 and SC2 lines, thermodynamic consistency is maintained when three second-order phase transitions intersect.

Beyond the maximum in $_2$5, both $_2$6 and $_2$7 decrease, but the size of the specific heat jumps increases sharply near the border to AFM order, especially where WMO is present, suggesting an entropy redistribution at the superconducting transition involving strong magnetic fluctuations.

Figure 3: Evolution of Sommerfeld coefficient, entropy, and jump sizes at the superconducting transitions as a function of pressure commensurate with phase boundaries.

Quantum Criticality and the Role of WMO

The enhancement of $_2$8 far below the AFM critical pressure, coincident with maximal $_2$9 and the extrapolation of $\gamma$0 to zero, demonstrates that electronic correlations and possible quantum criticality are tied to the disappearance of WMO, not long-range AFM order. This scenario aligns with transport ($\gamma$1-coefficient and non-Fermi-liquid exponents), magnetization, and NMR measurements indicating peak spin fluctuations and effective mass at similar pressures.

These results situate UTe$\gamma$2 as an exemplary system where quantum critical fluctuations of a nontrivial ordered phase mediate or strongly enhance unconventional superconductivity, even possibly favoring spin-triplet (potentially topological) order. The competition between WMO and superconductivity—apparent in the mutual exclusivity of their phase boundaries—suggests that superconductivity may suppress WMO, removing the quantum critical fluctuations as the ordered phase is expelled.

Theoretical and Practical Implications

The phase diagram and thermodynamic signatures elucidated here provide fertile ground for theoretical modeling. The marked Fermi surface selectivity, non-monotonic specific heat jumps, and strong mass renormalization near $\gamma$3 demand models that combine order parameter competition, quantum criticality, and potentially multiband superconductivity with unconventional pairing symmetry.

Practically, the expansion of the SC2 phase, its field-inducibility, and the confirmed strong-coupling regime—along with the interplay between disorder, criticality, and Fermi surface topology—have implications for the search for robust, field-tunable, spin-triplet superconductors. Future work combining calorimetry, quantum oscillations, and microscopic probes under pressure will clarify the microscopic mechanism of pairing and the topological character of the superconducting phases.

Conclusion

This quantitative calorimetric study establishes a direct link between emergent magnetic order, enhanced electronic correlations, and the stabilization of multiple superconducting phases in UTe$\gamma$4 under pressure. The threefold enhancement of $\gamma$5, the reconciliation of specific heat anomalies with Fermi surface fractionation, and the coincidence of maximum $\gamma$6, $\gamma$7, and WMO quantum criticality constitute strong evidence for the vital role of nontrivial quantum critical fluctuations in mediating unconventional superconductivity in UTe$\gamma$8. These results set a benchmark for heavy-fermion materials, promote UTe$\gamma$9 as a canonical platform for quantum critical superconductivity, and circumscribe the theoretical landscape for topological, nonunitary, and multicomponent superconducting order.