Introduction

The recent evidence for spin-triplet superconductivity in UTe2 has opened an avenue for the study of topological superconductivity1. The superconducting state of UTe2 closely resembles that of ferromagnetic superconductors, but the normal state is paramagnetic and shows no indication of magnetic ordering2,3,4. Spin-triplet pairing is strongly indicated by the extremely large, anisotropic upper critical field Hc21,5, nodes on the superconducting gap6,7, and the temperature-independent NMR Knight shift in the superconducting state8,9. The superconducting order parameter comprises two components and breaks time-reversal symmetry10,11. A nontrivial topology is suggested by the observation of chiral in-gap bound states by scanning tunneling spectroscopy12.

Even more striking, UTe2 hosts two independent field-induced superconducting phases13,14,15,16, with the higher-field phase reentering at a high magnetic field of 45 T and persisting up to 65 T when the magnetic field is aligned over a limited angular range about the normal direction of the (011) plane. The quasi-two-dimensional Fermi surface revealed by band structure calculations and photoemission measurements17,18,19, as well as the lack of a ferromagnetic ground state, has led to suggestions that the field-induced superconductivity in UTe2 is due to reduced dimensionality instead of magnetic fluctuations20,21: a magnetic field applied parallel to quasi 2D conducting layers will stabilize superconductivity when the magnetic energy reaches the hopping amplitude between the conducting layers20,22.

In this work, we investigate the evolution of the magnetic field-induced superconducting phases in UTe2 as pressure is applied to samples oriented specifically along the off-axis angle which stabilizes the high-field phase. Over a range of pressures near 1 GPa, the two different superconducting phases merge together, and the electrical resistance remains zero up to at least 45 T, a remarkably large value for a superconductor with a 3 K critical temperature. The high-field-induced superconducting phase is completely decoupled from the first-order transition to a field-polarized state, suggesting that magnetic fluctuations may not be crucial to this reentrant superconductivity. At pressures exceeding the critical pressure at which metamagnetic phase transition extrapolates to zero magnetic field, we observe features with the same temperature dependence as the high-field-induced superconducting phase, further investigation of which might shed light on the mechanism reentrant superconductivity.

Results and discussion

Pressure-magnetic field phase diagram

To characterize the evolution of the high-field-induced superconducting phase under pressure, we performed complementary measurements of electrical resistance R and tunnel diode oscillator (TDO) frequency Δf, which is sensitive to the change of both electrical resistance and magnetic susceptibility. Two samples were studied for which the magnetic field was applied ~25° and 30°, respectively, away from the b axis toward c. At ambient pressure, three distinct phases were observed as shown in Fig. 1: a field-polarized state FP with greatly enhanced magnetization and resistance13,15,16,23; the low-field superconducting phase, SCPM, coexisting with the paramagnetic state; and the high-field-induced superconducting phase, SCFP, existing inside the field-polarized state. The criteria used to infer critical magnetic field for each phase are explained in the Supplementary Information (see Supplementary Figs. 4–7).

Fig. 1: The stability of superconducting and magnetic phases of UTe2.
figure 1

a The zero-field superconducting phase SCPM1 exhibits a strongly direction-dependent upper critical field HC2. Along the crystallographic b axis, the reentrant superconducting phase SCPM2 is stabilized up to a first-order magnetic transition at HFP, above which exists a magnetically polarized phase FP. Applied pressure uniformly suppresses all field scales and HFP bounds superconductivity up to a critical pressure Pc. b For a magnetic field oriented between 25 and 30° between the b- and c axes, the high-field reentrant superconductivity SCFP stabilizes in the FP phase. Applied pressure enhances SCPM1, which meets a decreasing HFP at a crossover region Px. Here, SCPM1 transitions to SCPM2, which survives up to Pc, where it is replaced by magnetic order M. Below Px, HFP is a lower bound on SCFP, but above Px, the two are decoupled and SCFP survives beyond Pc. An anomaly AFP, suggestive of Landau-level superconductivity, emerges above Pc. The phase diagram is based on the resistance data shown in Fig. 2.

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Absent in this field configuration is another field-induced superconducting phase that is observed on the low-field side of the metamagnetic field HFP when the magnetic field is applied along the b axis. For magnetic field along the b axis, applied pressure suppresses the metamagnetic field to zero at 1.5 GPa and forces a phase transition inside the paramagnetic state at a crossover pressure Px, from SCPM1 to SCPM2, in zero magnetic field at ~1 GPa24. Recent symmetry analysis indicates that the SCPM1 phase has a two-component order parameter while the order parameter of SCPM2 phase only has one component11. Our measurements in this study do not exhibit any field-induced features24, so we infer that the SCPM1-SCPM2 boundary is nearly vertical in the H−P plane. At pressures exceeding 1.5 GPa, long-range magnetic order is stabilized25,26,27,28, whose features have been interpreted in terms of both ferromagnetism24,25 and antiferromagnetism27. Without an unambiguous experimental proof of the nature of this phase, in this study we label it as a magnetically ordered phase, M.

The pressure-magnetic field phase diagram for the magnetic field in this angle range is summarized in Fig. 1. All three phases manifest clear evolution under pressure, as seen in R and Δf in Fig. 2. The metamagnetic field is monotonically suppressed by the applied pressure, similar to the behavior for field along b24, although it starts at a higher value. The metamagnetic field vanishes at a critical pressure Pc between 1.47 and 1.54 GPa, giving rise to a spontaneous polarized state in zero magnetic field beyond this pressure. Over the entire pressure range, both R and Δf change discontinuously on passing through the metamagnetic field in the normal state, implying that it maintains the first-order metamagnetic transition observed at ambient pressure15,23,29.

Fig. 2: High-field TDO frequency and resistance at different pressures.
figure 2

TDO frequency is sensitive to the change of both electrical resistance and magnetic susceptibility of the sample. Magnetic field is applied at 30° from b axis toward c axis for TDO measurement, and 25° from b axis toward c axis for resistance measurement. a In the low-pressure region, HFP serves as a lower bound to SCFP, whose dome is cut off discontinuously. b At crossover pressures Px, HFP falls approximately between SCFP and SCPM, whose ranges of stability would otherwise overlap. c Above Px, HFP serves as the upper field limit for SCPM. d Above Pc, long-range ordered magnetism M sets in while SCFP survives and the anomalous feature AFP emerges as a local maximum in Δf. Solid dots are from experimental data, dashed lines are guide for the eye. Note that TDO and resistance measurements are performed for slightly different magnetic field direction, which leads to the slightly different phase diagrams for each pressure.

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Upon initial increase of the pressure, the stability of both superconducting phases is enhanced: the upper critical field of SCPM, Hc2, increases, and the critical onset field of SCFP, Hl, which coincides with the metamagnetic field, decreases. In an intermediate crossover pressure range, the phase boundary between SCPM and SCFP is no longer visible in the electrical resistance; this remains zero up to 45 T at base temperature, which is noteworthy as it is the largest DC magnetic field currently available to experiment (Fig. 3). As the pressure further increases, the upper critical field of SCPM is limited by the metamagnetic field and decreases, but the critical onset field of SCFP starts to increase, and the two superconducting phases are no longer connected. When the metamagnetic field vanishes, SCPM is suppressed completely.

Fig. 3: High-field magnetoresistance, showing phase boundaries at different pressures.
figure 3

Magnetic field is applied at 25° from b axis toward c axis. a Pressure dependence of 0.4 K magnetoresistance, showing the sharpening of phase boundaries at higher pressures and emergence of high-pressure phase AFP. b The resistance at pressures 0.69, 0.85, and 1.12 GPa is zero between 11 and 45 T at 0.4 K. Zero resistance persists to zero magnetic field at these pressure values. c For fields lower than HFP, the superconducting transitions HC2 are broad, but HFP is sharp. As shown in inset, HFP is hysteretic, reflecting the 1st order transition, both at high temperatures in the FP phase and low temperature in the SCFP phase. d Much sharper superconducting phase boundaries occur once HFP limits SCFP. As shown in inset, low-field phase boundaries are hysteretic and first order, as is the onset of SCFP.

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Subtle differences between the electrical resistance and TDO samples highlight the effects of their slight angular offset (Fig. 2). As the TDO sample sits at a slightly larger angle away from the b axis, the upper critical field of SCPM has a smaller value and the critical onset field of SCFP has a larger value at ambient pressure. The SCPM and SCFP phases remain connected over a much smaller pressure range. Similarly, the field-polarized state starts above a higher magnetic field because the metamagnetic field is larger, but it is suppressed faster upon increasing pressure and vanishes also at Pc.

The discontinuous nature of the metamagnetic transition at HFP is conspicuous in the resistance data (Fig. 3). At low pressures, a sharp upward jump marking the FP phase boundary is replaced at low temperatures by a sharp downward jump marking the SCFP phase boundary. An additional hallmark of first-order transitions, namely field-hysteresis is also readily apparent. In the intermediate pressure range, HFP limits the lower-field superconducting phase SCPM. Here, the transitions associated with HFP are again hysteretic. Interestingly, the decoupled SCFP phase appears to also exhibit some small hysteresis, the origin of which may be associated with details of the field reentrance. These features are consistent at all measured pressures.

These details reveal important points about the relationships between the many electronic phases. The low- and high-field superconducting phases always exist on opposite sides of the metamagnetic transition HFP, upon which Fermi surface reconstruction has been suggested based on thermoelectric power and Hall effect measurements23,30,31. In addition, in the low-pressure range, SCFP only appears on the high-field side of the metamagnetic field (Fig. 2a, b), while SCPM only exists at fields lower than the metamagnetic field. The role of the metamagnetic field switches in the intermediate pressure range, where the metamagnetic field truncates the low-field phase SCPM. This behavior is apparent in both panels of Fig. 2c, where the high-temperature part of the upper critical field of SCPM curve rises rapidly as temperature decreases, extrapolating to field values far higher than the metamagnetic field, but then, as soon as the upper critical field of SCPM coincides with the metamagnetic field, the behavior suddenly changes and the upper critical field of SCPM becomes almost temperature-independent down to low temperatures. Taken together, these facts imply that the SCPM and SCFP phases separated by the metamagnetic transition might have different superconducting pairings that are unique to PM and FP, respectively. It is particularly striking that both low- and high-field superconducting phases look like they would cover much larger ranges of the field were they not limited by the metamagnetic field.

In the high-pressure range, the critical onset field of SCFP and the metamagnetic field are well separated, by more than 20 T. This is crucial for the understanding of the pairing mechanism of the SCFP phase. In the case of URhGe, the reentrance of superconductivity can be explained in terms of ferromagnetic fluctuations parallel to the direction of the magnetic field. In that case, reentrant SC occurs in the vicinity of the magnetic critical field. In UTe2 at low pressure, the SCFP phase resembles somewhat the reentrant phase in URhGe, leading to the speculation that magnetic fluctuations are also responsible for reentrant superconductivity in UTe2. However, here we show clearly that the SCFP phases can exist in the region far away from the field-polarized phase line, indicating a possibility scenario that magnetic fluctuations are not responsible for the pairing mechanism. Future experiments to investigate magnetic fluctuations in the vicinity of the SCFP phase at pressure above 1.3 GPa will potentially shed light on the pairing mechanism.

Anomaly in the high-pressure region

A striking characteristic of the high-pressure FP phase is the emergence of additional features in the field range between the M and SCFP phases. These anomalies, denoted AFP, are pronounced in the Δf data, and noticeable in R data (Fig. 4). It is not clear whether these anomalies correspond to a thermodynamically distinct phase. In order to trace the evolution of these anomalies, we introduced criteria to define the boundaries as shown in the Supplementary Materials. These anomalies exhibit a clear temperature dependence below 1.2 K. This notable similarity to the temperature dependence of SCFP suggests that AFP is a closely related phenomenon, with a similar energy scale to that of the superconductivity. This distinguishes AFP from the zero-field magnetically ordered phase M, which has a higher-temperature ordering temperature relative to superconductivity, that continues to increase with applied pressure.

Fig. 4: Additional anomalies emerge in the high-pressure region.
figure 4

AFP phase appears at 1.54 GPa in both TDO and resistance data, which have the same temperature dependence as SCFP phase, indicating AFP phase might be the precursor of superconductivity. Magnetic field is applied at 30° from b axis toward c axis for TDO measurement, and 25° from b axis toward c axis for resistance measurement.

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An exciting possibility is that AFP is a precursor to superconductivity. Previous theoretical studies have shown that in extreme magnetic field Landau levels will have a dramatic influence on the low-temperature behavior of the upper critical field32,33,34. Indeed, a more recent theoretical study indicates that SCFP might be a form of superconductivity that is enhanced by high-field Landau quantization of the conduction electrons35. Therefore, AFP and SCFP may actually be the same superconducting phase occurring at different Landau levels, analogous to Shubnikov-de Haas oscillations. A challenge to this interpretation is that AFP is not accompanied by a zero-resistance state (Fig. 3a), but a plausible explanation for this is that AFP is actually partially superconducting due to the effects of energy-level broadening are stronger at the lower field. Based on the inverse-field periodicity of AFP, the next Landau level will be centered at ~100 T, achieving zero resistance at magnetic fields as low as 90 T, which is practically achievable using the strongest available nondestructive pulsed magnet systems.

Methods

Crystal synthesis

Single crystals of UTe2 were synthesized by the chemical vapor transport method using iodine as the transport agent. Elements of U and Te with atomic ratio 2:3 were sealed in an evacuated quartz tube, together with 3 mg/cm3 iodine. The ampoule was gradually heated up and hold in the temperature gradient of 1060/1000 °C for 7 days, after which it was furnace cooled to room temperature. The crystal structure was determined by X-ray powder diffraction using a Rigaku X-ray diffractometer with Cu–Kα radiation. Crystal orientation was determined by Laue X-ray diffraction performed with a Photonic Science X-ray measurement system.

Measurement

Magnetoresistance and tunnel diode oscillator (TDO) measurements were performed at the National High Magnetic Field Laboratory, Tallahassee, using the 45-T hybrid magnet. A non-magnetic piston-cylinder pressure cell was used for measurements under pressure up to 1.57 GPa, with Daphne oil 7575 as the pressure medium. Pressure was calibrated at low temperatures by measuring the fluorescence wavelength of ruby, which has a known temperature and pressure dependence. The TDO technique uses an LC oscillator circuit biased by a tunnel diode whose resonant frequency is determined by the values of LC components, with the inductance L given by a coil that contains the sample under study; the change of its electrical resistance and magnetic properties results in a change in resonant frequency. Identification of commercial equipment does not imply recommendation or endorsement by NIST. Error bars correspond to uncertainty of one standard deviation.