Expansion of the high field-boosted superconductivity in UTe2 under pressure

Magnetic field induced superconductivity is a fascinating quantum phenomenon, whose origin is yet to be fully understood. The recently discovered spin triplet superconductor, UTe2, exhibits two such superconducting phases, with the second one reentering in the magnetic field of 45 T and persisting up to 65 T. More surprisingly, in order to induce this superconducting phase, the magnetic field has to be applied in a special angle range, not along any high symmetry crystalline direction. Here we investigated the evolution of this high-field induced superconducting phase under pressure. Two superconducting phases merges together under pressure, and the zero resistance persists up to 45 T, the field limit of the current study. We also reveal that the high field-induced superconducting phase is completely decoupled from the first order field polarized phase transition, different from previously known example of field induced superconductivity in URhGe, indicating a superconductivity boosted by a different paring mechanism.


INTRODUCTION
The recent evidence for spin triplet superconductivity in UTe 2 has opened an avenue for the study of topological superconductivity 1 . The superconducting state of UTe 2 closely resembles that of ferromagnetic superconductors, but the normal state is paramagnetic and shows no indication of magnetic ordering 2-4 . Spin triplet pairing is strongly indicated by the extremely large, anisotropic upper critical field H c2 1,5 , nodes on the superconducting gap 6,7 , and the temperature independent NMR Knight shift in the superconducting state 8,9 .
The superconducting order parameter comprises two components and breaks time reversal symmetry 10,11 . A nontrivial topology is suggested by the observation of chiral in-gap bound states by scanning tunneling spectroscopy 12 .
Even more striking, UTe 2 hosts two independent field-induced superconducting phases [13][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 measurements [17][18][19] , as well as the lack of a ferromagnetic ground state, has led to suggestions that the field-induced superconductivity in UTe 2 is due to reduced dimensionality instead of magnetic fluctuations 20,21 : a magnetic field applied parallel to quasi 2D conducting layers will stabilize superconductivity when the magnetic energy reaches the hopping amplitude between the conducting layers 20,22 .
In this work we investigate the evolution of the magnetic field-induced superconducting phases in UTe 2 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.

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 ∆f , which is sensitive to the change of both electrical resistance and magnetic susceptibility. Two samples were studied for which the magnetic field was applied approximately 25 degrees and 30 degrees, respectively, away from the b axis towards c.
At ambient pressure, three distinct phases were observed as shown in Fig. 1  Absent in this field configuration is another field induced superconducting phase that is observed on the low-field side of the metamagnetic field H FP when 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 P x , from SC PM1 to SC PM2 , in zero magnetic field at approximately The pressure-magnetic field phase diagram for the magnetic field in this angle range is summarized in Fig. 1b. All three phases manifest clear evolution under pressure, as seen in R and ∆f in Fig. 2. The metamagnetic field is monotonically suppressed by the applied pressure, similar to the behavior for field along b 24 , although it starts at a higher value. The metamagnetic field vanishes at a critical pressure P c 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 ∆f change discontinuously on passing through the metamagnetic field in the normal state, implying that it maintains the first-order metamagnetic transition observed at ambient pressure 15,23,29 .
Upon initial increase of the pressure, the stability of the both superconducting phases is enhanced: the upper critical field of SC PM , H c2 , increases and the critical onset field of SC FP , H l , which coincides with the metamagnetic field, decreases. In an intermediate crossover pressure range, the phase boundary between SC PM and SC FP 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. 3b). As the pressure further increases, the upper critical field of SC PM is limited by the metamagnetic field and decreases, but the critical onset field of SC FP starts to increase, and the two superconducting phases are no longer connected. When the metamagnetic field vanishes, SC PM is suppressed completely.
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 SC PM has a smaller value and critical onset field of SC FP has a larger value at ambient pressure. The SC PM and SC FP 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 P c .
The discontinuous nature of the metamagnetic transition at H FP is conspicuous in the resistance data. At low pressures (Fig. 3c), a sharp upward jump marking the FP phase boundary is replaced at low temperatures by a sharp downward jump marking the SC FP phase boundary. An additional hallmark of first-order transitions, namely field-hysteresis is also readily apparent (inset of Fig. 3c). In the intermediate pressure range, H FP limits the lower-field superconducting phase SC PM (Fig. 3d). Here, the transitions associated with H FP are again hysteretic (inset of Fig. 3d). Interestingly, the decoupled SC FP 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 (Fig. 3a, b).
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 H FP , upon which Fermi surface reconstruction has been suggested based on thermoelectric power and Hall effect measurements 23,30,31 . In addition, in the low pressure range, SC FP only appears on the high-field side of the metamagnetic field ( Fig. 2a, b), while SC PM 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 SC PM . This behavior is apparent in both panels of Fig. 2c, where the high-temperature part of the upper critical field of SC PM 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 SC PM coincides with the metamagnetic field, the behavior suddenly changes and the upper critical field of SC PM becomes almost temperature independent down to low temperatures. Taken together, these facts imply that the SC PM and SC FP phases separated by the metamagnetic transition might have different superconducting pairing 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 field were they not limited by the metamagnetic field.
In the high pressure range, the critical onset field of SC FP and the metamagnetic field 6 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 SC FP phases. These anomalies, denoted A FP , are pronounced in the ∆f data, and noticeable in R data (Fig. 4) is not accompanied by a zero resistance state (Fig. 3a), but a plausible explanation for this is that A FP is actually partially superconducting due to the effects of energy-level broadening are stronger at lower field. Based on the inverse-field periodicity of A FP , the next Landau level will be centered at approximately 100 T, achieving zero resistance at magnetic fields as low as 90 T, which is practically achievable using the strongest available non-destructive pulsed magnet systems.

Crystal synthesis
Single crystals of UTe 2 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/cm 3 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 xray powder diffraction using a Rigaku x-ray diffractometer with Cu-K α 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 an uncertainty of one standard deviation.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.

COMPETING INTERESTS
The authors declare no competing interests.  the high-field reentrant superconductivity SC F P stabilizes in the FP phase. Applied pressure enhances SC P M 1 , which meets a decreasing H F P at a crossover region P x . Here, SC P M 1 transitions to SC P M 2 , which survives up to P c , where it is replaced by magnetic order M. Below P x , H F P is a lower bound on SC F P , but above P x , the two are decoupled and SC F P survives beyond P c . An anomaly A F P , suggestive of Landau-level superconductivity, emerges above P c . The phase diagram is based on the resistance data shown in Fig. 2    pressures and emergence of high-pressure phase A F P . b) The resistance at pressures 0.69, 0.85, and 1.12 GPa is zero between 11 T and 45 T at 0.4 K. Zero resistance persists to zero magnetic field at these pressure values. c) For fields lower than H F P , the superconducting transitions H C2 are broad, but H F P is sharp. As shown in inset, H F P is hysteretic, reflecting the 1st order transition, both at high temperatures in the FP phase and low temperature in the SC F P phase. d) Much sharper superconducting phase boundaries occur once H F P limits SC F P . As shown in inset, low-field phase boundaries are hysteretic and first order, as is the onset of SC F P .