Comparison of two superconducting phases induced by a magnetic field in UTe2

Superconductivity induced by a magnetic field near metamagnetism is a striking manifestation of magnetically-mediated superconducting pairing. After being observed in itinerant ferromagnets, this phenomenon was recently reported in the orthorhombic paramagnet UTe2. Here we explore the phase diagram of UTe2 under two magnetic-field directions: the hard magnetization axis b, and a direction titled by ≃25-30° from b in the (b,c) plane. Zero-resistivity measurements confirm that superconductivity is established beyond the metamagnetic field Hm in the tilted-field direction. While superconductivity is locked exactly at fields either smaller (for H | | b), or larger (for H tilted by ≃27° from b to c), than Hm, the variations of the Fermi-liquid coefficient in the electrical resistivity and of the residual resistivity are similar for the two field directions. The resemblance of the normal states for the two field directions puts constraints for theoretical models of superconductivity and implies that some subtle ingredients must be in play. In a magnetic field, superconductivity can be induced or reinforced near a metamagnetic transition, where ferromagnetic fluctuations are suspected to mediate the pairing strength of the Cooper pairs. Here, the authors investigate the superconductor UTe2 and report on the variation in the superconducting properties as the magnetic field is applied along two particular crystallographic axes and their relation to metamagnetism.

U nconventional superconductivity is observed in an evergrowing number of correlated-electron systems 1 , ranging from heavy-fermion 2,3 , high-temperature cuprate 4 , ironbased pnictide and chalcogenide 5 , to the newly discovered nickelate 6 and graphene-superlattice 7 families. New unusual superconducting phases continue to be discovered, such as those reported during the last two decades in the ferromagnets UGe 2 , URhGe, and UCoGe 8-10 with orthorhombic crystal structures. Instead of antiferromagnetic fluctuations, which are suspected to be the glue for superconductivity and to lead to a singlet order parameter in most heavy-fermion superconductors 3 , ferromagnetic fluctuations were proposed to drive the pairing mechanism of these ferromagnets, where the strong exchange field suggests that a spin-triplet superconducting order parameter with equalspin pairing may be realized 11 . Nuclear-magnetic-resonance (NMR) experiments brought microscopic support for such triplet state and they further highlighted the role of magnetic fluctuations 12,13 . In these three systems, a magnetic field also leads to a re-entrance or reinforcement of superconductivity, and magnetic-field-induced ferromagnetic fluctuations are suspected to directly control the pairing strength, which can be qualitatively understood as the enhancement of a strong-coupling superconducting parameter λ with field 11 . A S-shape in the temperature dependence of the superconducting critical field H c2 was observed in UGe 2 under pressure with a magnetic field along the easy magnetization axis a 14 . Re-entrance or reinforcement of superconductivity occurs in the isostructural ferromagnets URhGe and UCoGe under a magnetic field applied along their hard-magnetic axis b 15,16 . In URhGe, field-induced superconductivity coincides with a metamagnetic transition at µ 0 H m = 12 T, where enhanced magnetic fluctuations 17,18 accompany a sudden rotation of the magnetic moments (from the initial easy direction c to the direction b) 15 . In this system, a Fermi-surface instability is observed at H m , beyond which a polarized paramagnetic (PPM) regime is established [19][20][21] . In UCoGe, the situation is more subtle, since a metamagnetic transition occurs at a field µ 0 H m ≃ 50 T much higher than that of ≃15 T at which the reinforcement of superconductivity is observed 22 .
Recently, superconductivity was found to develop in the paramagnetic heavy-fermion material UTe 2 at temperatures below T sc = 1.6 K 23,24 . This system crystallizes in an orthorhombic crystal structure with space group Immm (#71, D 25 2h ) and is characterized by an anisotropic magnetic susceptibility [see Fig. 1(a)]. No sign of long-range magnetic order has been found down to the lowest temperatures (25 mK) 25 . For a magnetic field applied along the easy magnetic axis a, a large low-temperature magnetic susceptibility and a scaling plot of magnetization data were interpreted as the indication for a nearby ferromagnetic instability 23 . Following the observation of a large anisotropic upper critical field, which exceeds the normal paramagnetic limitation for all field directions 23,24 and of a tiny change in the NMR Knight shift through T sc 26 , a spin-triplet nature of superconductivity has been proposed 23 . The possibility of chiral spin-triplet superconductivity was suggested from scanning tunneling microscopy 27 and Kerreffect experiments 28 . However, while magnetic fluctuations were observed by NMR 29 and muon-spin relaxation measurements 25 , evidence supporting their ferromagnetic nature is still lacking. Furthermore, the presence of antiferromagnetic fluctuations has been reported by inelastic neutron scattering 30 .
In UTe 2 at temperatures T < T CEP ≃ 7 K, a magnetic field applied along the hard-magnetic axis b induces a first-order metamagnetic transition at µ 0 H m ≈ 35 T, which separates a lowfield correlated paramagnetic (CPM) regime from a high-field PPM regime [31][32][33] . It is accompanied by sudden jumps ΔM ≈ 0.3-0.6 μ B /U in the magnetization 31,33 and Δρ ≈ 100 μΩ cm in the residual resistivity 32 , and by a large enhancement of the effective mass at H m 31,32,34 . An empirical and almost universal relation 1 T ↔ 1 K between H m and the temperature T χ max ≈ 35 K at the maximum in the magnetic susceptibility 35 is observed, as for a large number of heavy-fermion paramagnets 36 . It indicates that the CPM regime delimited by H m and T χ max is, within a first approximation, controlled by a single energy scale. For H || b, superconductivity is reinforced above 15 T and it abruptly disappears in the PPM regime above H m 33,37 . Calorimetric studies showed the appearance of a second superconducting phase, labeled by SC2, under pressure in zero magnetic field 38 . A boundary between SC2 and the ambient-pressure and low-field superconducting phase SC1 was observed from tunnel-diodeoscillator measurements under pressure and magnetic field H || b 39 . The extrapolation of this boundary for p → 0 suggests that the superconducting phase SC2 induced under pressure and the superconducting region induced by a magnetic field H || b could be the same phase. In the following, we will label SC1 and SC2 the respective low-field and high-field superconducting regions for H || b. However, to date there is no definitive experimental evidence of a magnetic-field-induced transition between SC1 and Fig. 1 Magnetic susceptibility and phase diagram of UTe 2 . a Temperature-dependence of the inversed magnetic susceptibility 1/χ of UTe 2 in magnetic fields H applied along the three main crystallographic directions a, b, and c. Inset: Temperature-dependence of the magnetic susceptibility χ for H||a, b, and c, in a log-log scale. b Low-temperature magnetic-field versus angle phase diagram of UTe 2 , in fields applied along variable directions from b to a (angle ϕ) and from b to c (angle θ). Two low-temperature paramagnetic regimes and identified: correlated paramagnetism (CPM) and polarized paramagnetism (PPM). SC1 is the low-field superconducting phase, and SC2 and SC-PPM are the superconducting phases induced by magnetic fields H||b and H tilted by 27 ± 5°from b in the (b, c) plane, respectively. H c,2 is the critical superconducting field and H m is the metamagnetic field. Data from by Ran et al. 33 and Knebel et al. 37 were plotted in this Figure. SC2 at ambient pressure. An alternative picture without phase transition between SC1 and SC2 cannot be excluded, since the upturn of H c2 could result from a tight balance between the orbital limitation and the increase of the coupling λ with field 37 . We further note that, at ambient pressure and zero magnetic field, a single superconducting transition was identified in 23,24,40 but two separated superconducting transitions were reported in 28,41 . Figure 1(b) presents a combination of low-temperature magnetic-field versus field-angle phase diagrams of UTe 2 obtained by Ran et al. 33 and Knebel et al. 37 It summarizes the effect of magnetic fields applied in the (a, b) and (b, c) planes. A key property is that H m is minimum for H || b. It strongly increases when the field is tilted from b towards the easy magnetic axis a, and exceeds the maximum applied field (60 T) for ϕ = (b, a) > 20°. The increase of H m is softer when the field is tilted from b towards c, where it could be followed up to angles θ = (b, c) ≈ 50°. At small angles ϕ and θ, the field-reinforcement of superconductivity rapidly disappears. For the three field-directions a, b, and c, the low-temperature critical fields µ 0 H c2,a ≈ 6 T, µ 0 H c2,c ≈ 10 T, and µ 0 H c2,b ≈ 15-20 T (i.e., the extrapolated value of µ 0 H c2,b ignoring the field-reinforcement below 300 mK) delimiting the low-field superconducting phase SC1 are inversely-correlated with the lowtemperature magnetic susceptibilities χ a > χ c > χ b (see Fig. 1 and refs. 23,31,35 ). A similar inverse relation between the magnetic anisotropy and the anisotropy of H c2 was observed in many other heavy-fermion superconductors, such as URu 2 Si 2 42,43 , CeCoIn 5 44,45 , UCoGe and URhGe 11,46 . Spectacularly, a second field-induced superconducting phase was reported in UTe 2 for a field tilted from b towards c by an angle 20 < θ < 40°3 3 . This phase, labeled here as SC-PPM, was observed only in the PPM regime, in fields higher than µ 0 H m ≃ 40-45 T and up to more than 60 T 33 .
In the present work, we focus on a study by electrical resistivity of the superconducting phases SC2 and SC-PPM induced in UTe 2 at ambient pressure, under a magnetic field applied either along b, or tilted by an angle θ ≃ 27 ± 5°from b towards c. In the initial report of the SC-PPM phase, the electrical resistivity was not exactly zero, likely due to a phase issue in the pulsed-field measurement and to deviations from isothermal measurements resulting from the use of fastly varying pulsed magnetic fields 33 . The almost isothermal conditions of our experiments using longduration (rise = 70 ms, fall = 300 ms) magnetic-field pulses allow studying temperature-dependent effects in a high magnetic field. Our results show zero resistance in the SC-PPM phase, confirming its superconducting nature. We extract the full magneticfield-temperature phase diagrams of UTe 2 for H || b and H tilted by θ ≃ 27°from b to c. From a Fermi-liquid analysis we also determine the field-dependence of the residual resistivity ρ 0 and estimate the variation of the effective mass m *47 . These quantities show striking similarities for the two field-directions in contrast with the very different superconducting phase diagrams. In the discussion, elements resulting from experiments are summarized and confronted to the theoretical challenge to understand the nature of the field-induced superconducting phases in UTe 2 .

Results
Low-temperature and high-magnetic-field electrical resistivity. shows almost no heating of the samples by eddy currents in the low-temperature data, which were obtained in long-duration pulsed magnetic fields. At temperatures from T = 2.2 K to T CEP ≈ 5-6 K, at which a critical end-point is observed in the data, and under magnetic fields H || b ( Fig. 2(a)) and H tilted by θ = 27 ± 5°( Fig. 2(c)), similar and sharp first-order step-like increases of ρ are observed at µ 0 H m , which equals 34 and 45 T for the two field directions, respectively. For both directions, when the temperature is increased above T CEP , the sharp anomaly at H m changes into a broad maximum, at a field also labeled H m , which vanishes at temperatures higher than 30 K. Figure 2(b) shows that, for H || b, field-induced superconductivity develops just below H m , with an onset at a maximal temperature of 1.2 K and a zero-resistivity reached below the maximal superconducting temperature T SC ≃ 1 K. In spite of a non-zero-resistivity due to small out-of-phase contamination of the signal, this new set of data confirms, in magnetic fields extended up to 60 T, the two recent reports of field-reinforcement of superconductivity in UTe 2 for H || b 33,37 . For H tilted by θ = 27 ± 5°from b in the (b, c) plane, Fig. 2(d) shows a zero-resistivity regime in fields higher than H m . These data support the presence of a field-induced superconducting phase SC-PPM above H m 33 .
After an onset at a maximal temperature of 2 K, zero-resistivity is reached below the maximal superconducting temperature T SC ≃ 1.5 K, which is higher than the superconducting temperature reported for the field-induced phase for H || b. The magnetic field at which the zero-resistivity superconducting phase SC-PPM develops is locked to the value µ 0 H m ≃ 45 T observed for T > T SC . Inside the CPM regime, the onset of the phase SC-PPM at ≃43 T precedes the zero-resistivity-state reached beyond H m . We also confirm that the low-field superconducting phase SC1 is wellseparated from the field-induced phase SC-PPM. At the lowest temperature, the phase SC1 vanishes at a moderate critical field of ≃10 T (see Fig. 1b).
Temperature-magnetic field phase diagrams and quantum critical fluctuations. Figure 3(a) presents the magnetic-fieldtemperature phase diagram extracted here for UTe 2 in a field H || b, in agreement with ref. 37 . Although the field-induced transition between SC1 and SC2 was not observed so far at ambient pressure, the phase diagram suggests that two different superconducting regimes exist with a transition or crossover at ≃15 T. The transition temperature T SC of SC2 is maximal at a magnetic field just below µ 0 H m = 34 T. SC2 is presumably driven by the magnetic fluctuations induced on approaching the metamagnetic transition. These fluctuations also control the enhancement of the Sommerfeld coefficient γ in the heat capacity 36 and of the coefficient A of the Fermi-liquid T 2 term of the electrical resistivity 32 . We confirm here that SC2 is strictly bounded by H m , at which the magnetization was found to suddenly increase and above which a PPM regime is reached 31,33 . Figure 3(b) presents the magnetic field -temperature phase diagram extracted here for UTe 2 in a field H tilted by θ = 27 ± 5°f rom b in the (b, c) plane. While the low-field superconducting phase SC1 vanishes at a critical field µ 0 H c2 ≃ 10 T, µ 0 H m reaches 45 T at low temperature for this field direction. When the temperature is increased, the behavior is similar to that reported for H||b: H m loses its first-order character at the temperature T CEP ≈ 5-6 K. It transforms into a crossover at higher temperatures and finally disappears above 20-30 K. In agreement with the previously-published data 33 , the superconducting phase SC-PPM is only observed in fields higher than H m , and up to a superconducting critical field higher than 60 T at low temperature. A maximal field-induced superconducting temperature T SC ≈ 1.5 K appears at a field close to H m , emphasizing a direct link with the metamagnetic transition.
In many heavy-fermion magnets, a maximum of the effective mass is observed in the vicinity of a magnetic instability. It is commonly understood as resulting from the critical quantum magnetic fluctuations, coupled or not with a Fermi-surface instability 48 . Within a Fermi-liquid description, the electrical resistivity can be fitted by ρ(Τ) = ρ 0 + AT 2 , and the A coefficient varies as the square of the effective mass m * . A Fermi-liquid picture is generally valid within first approximation, and deviations from the empirical law A ∝ m *2 can result from additional electronic effects, such as changes in carrier scattering, Fermi-surface and band structure, field-induced cyclotron motion of the carriers, etc. (see for instance this work 49 ). In heavy-fermion systems, m * is mainly controlled by magnetic fluctuations related with the proximity of quantum magnetic instabilities. In several Fig. 3 Magnetic phase diagrams of UTe 2 . a Magnetic-field-temperature phase diagram of UTe 2 in a magnetic field H||b. b Magnetic-field-temperature phase diagram of UTe 2 in a magnetic field H tilted by 27 ± 5°from b in the (b, c) plane. Two low-temperature paramagnetic regimes and identified: correlated paramagnetism (CPM) and polarized paramagnetism (PPM). SC1 is the low-field superconducting phase, and SC2 and SC-PPM are the superconducting phases induced by magnetic fields H||b and H tilted by 27 ± 5°from b in the (b, c) plane, respectively. T SC is the critical superconducting temperature, T χ max is the temperature at the maximum of the magnetic susceptibility, and H m is the metamagnetic field. For the superconducting phases, colored points indicate the temperature at which zero-resistivity is reached and gray points indicate the temperature at the onset of the downwards deviation of the resistivity. CEP indicates the critical end-point of the first-order metamagnetic transition. compounds, a non-Fermi-liquid deviation from this law is observed near quantum magnetic instabilities 50 . In other compounds, as CeRu 2 Si 2 51 , CeRh 2 Si 2 52 , and URhGe 20 , a T 2 law in ρ(T) was observed down the lowest accessible temperatures at the pressure and/or magnetic-field instabilities. Recently, Fermi-liquid behaviors, including T 2 laws in the electrical resistivity, were reported at the quantum instabilities of UTe 2 under pressure 38 and magnetic field H || b 31,32,34 . In continuity with these studies, fits to the electrical-resistivity data of UTe 2 were done here for all fields investigated in the temperature windows 1.5 ≤ T ≤ 4.2 K for H || b, and 2.2 ≤ T ≤ 4.2 K for H tilted by θ = 27 ± 5°from b to c (see Supplementary Note 1 and Supplementary Fig. 7). As shown in Fig. 4(a-b), we find almost similar field-variations of A and ρ 0 at H m for the two field directions: while A increases by a factor ≃6 and passes through a maximum, ρ 0 undergoes a sharp step-like enhancement, jumping from 15 to 80 µΩ.cm. The field-variation of A reported here for H || b, in good agreement with a previous report 32 , indicates a sharp and strong enhancement of the magnetic fluctuations at H m . For H || b, a qualitatively similar enhancement of m* at H m was found by applying a Maxwell relation to magnetization data 31 and by direct heat-capacity measurements 34 .
Small differences between the two field-directions are visible from plots of A and ρ 0 versus H/H m [Fig. 4(c) and 4(d)]. While the variation of A through H m is almost symmetric for H || b, it is slightly asymmetric for H tilted by θ = 27 ± 5°from b. For the tilted-field direction, A(H) is steeper for H < H m and more gradual for H > H m . As well, the decrease of ρ 0 beyond H m is more marked for H tilted by θ = 27 ± 5°from b. Beyond these small differences, the main result here is the robust finding that the variations of A and ρ 0 are similar for the two field directions. New high-field experiments on a unique sample, using a rotation probe, are now needed for a complete angular study of the Fermi-liquid behavior.

Discussion
The ultimate goal would be to provide a full microscopic description of the different superconducting phases and their pairing mechanisms in UTe 2 . We are still far from this objective, but the experimental data presented here, in complement to those from ref. 33 , offer a broad set of constraints for theories. The role of magnetic fluctuations for superconductivity is indicated by the maximum critical temperature of the reentrant phases observed very near to H m for both field directions. A striking feature of the phase diagrams presented in Fig. 3(a-b) is that the superconducting phases SC2 for H || b and SC-PPM in a field H tilted by θ = 27 ± 5°from b towards c are bounded by the metamagnetic field H m , with a substantial difference that the phase SC2 is pinned inside the CPM regime and it does not survive in the PPM regime while, inversely, the phase SC-PPM is pinned inside the PPM regime and does not develop in the CPM regime. A natural explanation would be that the pairing mechanism changes drastically on crossing the first-order line H m , at which one would expect a difference in the nature of the critical magnetic fluctuations in the CPM and PPM regimes. This difference would change substantially for the two field-directions H || b and H tilted by 27°from b.
A rough estimation of the field-dependence of the pairing strength can be obtained from the Fermi-liquid analysis done above. A maximum of the quadratic coefficient A at H m indicates an increase of the effective mass m*, presumably controlled by  57 . In UTe 2 , the fact that the enhancement of A is almost symmetric around H m is puzzling with respect to the abrupt suppression of superconductivity for H || b, and its abrupt appearance for H tilted by θ = 27 ± 5°from b towards c. The abrupt disappearance/appearance of superconductivity at H m could also result from a sudden change of the Fermi-surface. A Fermi-surface reconstruction is compatible with the large and sudden variation of the residual resistivity ρ 0 at H m for the two field directions, but also with the sign changes in the thermo-electric power and Hall coefficient at H m for H || b 58 . However, our results raise a serious hurdle to both these pictures since the field-driven enhancement of A is very similar for H || b and H tilted by θ = 27 ± 5°from b to c. If it is an intrinsic property, the asymmetry in the field-variation of A for H tilted by 27°could suggest that the magnetic fluctuations are slightly more intense above H m for this field direction. However, this effect would be too small to explain the differences between the phases SC2 and SC-PPM. The magnetization jumps at H m are also very similar for H || b and H tilted by 27°3 3 . Extra ingredients are, thus, needed to describe the field and angle domains of stability of these two field-induced superconducting phases. Figure 5(a-c) presents views of the crystal structure of UTe 2 where the magnetic uranium ions can be seen to form a ladder structure 59 . We highlight the family of reticular (and cleaving) planes of Miller indices (0 1 1), which contain sets of ladders having the smallest inter-ladder U-U distance (d 3 = 4.89 Å). Interestingly, the direction n normal to these planes coincides, within the experimental uncertainty, with the field-direction along which the phase SC-PPM develops 33 . It lies in the (b, c) plane and has an angle θ = 23.7°with b. Figure 5(d) presents a view of the Brillouin zone. It emphasizes that the direction n in real space is equivalent to the direction k = (0 1 1) in reciprocal space. Although the connection with the pairing mechanism remains unclear, this coincidence may not be accidental and may constitute a possible line of approach for future theories. Indeed, the field-induced superconducting phases SC2 and SC-PPM may be sensitive to fine details of the Fermi-surface topology, in relation with high-symmetry directions. Further experimental studies, with a more accurate positioning of the samples (within misorientations Δθ, Δϕ < 1°), are now needed to test the robustness of the coincidence observed here.
In relation with the ladder structure, magnetic frustration has been invoked as a possible origin of the paramagnetic ground state in UTe 2 at zero-field and ambient pressure, and a competition between ferromagnetic and antiferromagnetic configurations has been discussed 59,60 . Electronic-structure calculations pointed out that the ground state is sensitive to the Coulomb repulsion, and that the ferromagnetic and antiferromagnetic configurations are energetically-close 59 . The respective roles of ferromagnetic and antiferromagnetic fluctuations in UTe 2 may, thus, be important for the superconducting phases. While UTe 2 was first proposed to be nearly ferromagnetic 23 , the nature of the pressure-induced magnetic phase, initially reported in 38 , was not determined so far. Several studies suggested that UTe 2 is not a simple nearly-ferromagnet and may be close to an antiferromagnetic instability 41,61 , which is supported by the observation of antiferromagnetic fluctuations 30 . At ambient pressure, the absence of metamagnetism in a magnetic field up to 55 T applied along the easy magnetic axis a 31,32 indicates that UTe 2 is at least not a conventional Ising paramagnet close to a ferromagnetic instability, unlike UGe 2 under pressure 62 and UCoAl at ambient pressure 63 . The negative Curie-Weiss temperatures extracted from the high-temperature magnetic susceptibility, for the three directions H || a, b, and c [see Fig. 1(a)], indicate antiferromagnetic exchange interactions (see also ref. 35 ). A broad maximum at the temperature T χ max = 35 K in the magnetic susceptibility for H || b is also compatible with the onset of antiferromagnetic fluctuations, as observed in several heavyfermion paramagnets 36 . Low-temperature downward deviations of the magnetic susceptibility for H || a,c (in comparison with its high-temperature behavior) are observed in the log-log plot shown in Inset of Fig. 1(a). These deviations confirm the formation of a heavy-fermion state below 50 K, which may coincide with the onset of antiferromagnetic fluctuations, possibly those observed by inelastic neutron scattering 30 . Interestingly, the hightemperature magnetic susceptibility for H || a varies as 1/T 0.75 over more than one decade, from 20 to 300 K. However, further investigations are needed to understand this power-law behavior. Magnetic anisotropy, which drives the preferential direction of the magnetic fluctuations, is also suspected to play a significant role for superconductivity. The inverse relationship between the low-field magnetic anisotropy and the critical fields of the phase SC1 was emphasized in the introduction. The evolution of the magnetic anisotropy in a high magnetic field may also play a role for the stabilization of the field-induced superconducting phases.
The different superconducting regimes may correspond to different order parameters, with different sensitivities to a magnetic field. It has been generally assumed that all the superconducting phases in UTe 2 have a triplet order parameter, mainly because of high values of the superconducting upper critical field, a small decrease of the NMR Knight shift below T sc 26 and a supposed proximity to ferromagnetism 23,24,59,64 . However, this still needs confirmation especially if, as pointed out above, antiferromagnetic fluctuations may play a much larger role than initially thought. The disappearance of superconducting phase SC2 as the PPM regime is entered for H || b could be related to the loss of magnetic fluctuations characteristic of the CPM regime. Thereafter, for H tilted by θ = 27 ± 5°from b to c, the phase SC-PPM could be a natural candidate for triplet superconductivity with no paramagnetic limitation. However, two questions remain: why this phase appears only for such a specific angular range, possibly in relation with the previous symmetry considerations, and especially why this phase does not develop in fields smaller than H m ? Interestingly, other superconducting phases develop in UTe 2 under pressure combined with a magnetic field applied along the easy axis a 61 , and the resulting isobar magnetic-field-temperature phase diagrams have similar features than that reported for another compound with multiple superconducting phases, UPt 3 at ambient pressure 65 .
A full understanding of the magnetic fluctuations and their feedback on the superconducting pairing undoubtedly requires the knowledge of the Fermi-surface and electronic structure of UTe 2 . As mentioned above, calculated Fermi surfaces strongly depend on the Coulomb repulsion U: for large values of U, twodimensional Fermi surfaces along c similar to that of ThTe 2 and corresponding to a localized f-electrons limit have been expected 59,60,66 . For quasi one-dimensional 67 or quasi twodimensional 68,69 Fermi surfaces, Ginzburg-Landau theories, which neglect the role of magnetic fluctuations, also predict that the orbital limit could be suppressed for particular field directions. However, while angle-resolved-photo-emission-spectroscopy revealed a light low-dimensional band, they also showed the presence of a heavy three-dimensional band centered around the point Z of reciprocal space 70 . The observation of lowdimensional features in the bulk properties (for instance stronglyanisotropic electrical resistivity) are now needed to support a lowdimensional Fermi-surface model of superconductivity for UTe 2 .
Rich-phase diagrams were obtained for UTe 2 under different field directions and pressures. Although the measurements presented here and in other works 33,[37][38][39]41,61,71 start to bring a clear picture of the complex phase diagram of UTe 2 , which includes multiple superconducting and magnetic phases, we are still far from a deep understanding of its electronic properties. A target is now to perform microscopic studies to identify the nature of the magnetic fluctuations and their change through H m . In relation with these magnetic fluctuations, a challenge will be to identify the nearby long-range-ordered magnetic phases. The objective to characterize the Fermi-surface in the different phases is also emphasized. Beyond the need for solid experimental findings, theoretical developments are needed to describe the superconducting pairing mechanism(s) and order parameter(s). This is a stiff challenge, but the rare flurry of stunning phenomena observed in UTe 2 fully justifies such forthcoming efforts.

Methods
Samples. Single crystals of UTe 2 were prepared by the chemical vapor transport method with iodine as transport agent. Their structure and orientation was checked by single-crystal X-ray diffraction. A sharp bulk transition at T sc = 1.6 K was indicated from specific heat measurements, while zero-resistivity at temperatures below T sc was confirmed by zero-field AC resistivity measurements. Samples #5, #6, and #7, whose electrical-resistivity data are presented here, have similar residual-resistivity ratios ρ(300 K)/ρ(2 K) ≃ 25 to those of samples #1, #2, and #3 studied previously 32,37 , indicating similar sample qualities.
Pulsed-field experiments. Electrical-resistivity measurements were performed at the Laboratoire National des Champs Magnétiques Intenses (LNCMI) in Toulouse under long-duration pulsed magnetic fields, either up to 68 T (30 ms raise and 100 ms fall) and combined with an 4 He cryostat offering temperatures down to 1.4 K, or up to 58 T (55 ms rise and 300 ms fall) and combined by a homedeveloped dilution fridge made of a non-metallic mixing chamber offering temperatures down to 100 mK. A standard four-probe method with currents I || a, at a frequency of 20-70 kHz, and a digital lock-in detection were used. Resistivity data were normalized so that the maximal value, at a temperature of ≈65 K and at zero-field, reaches 450 µΩ.cm (a different normalization lead to a maximum of 650 µΩ.cm in a previous work 32 ). Normalization was made following absolute resistivity measurements on samples whose geometrical shape was known. The measurements in different field directions were done on different samples, and we cannot exclude that the small differences, as those seen in the variations of A and ρ 0 extracted from a T 2 law, have an extrinsic origin (they could result from a limit of reproducibility in our measurements). Concerning the tilted-field direction, the choice for an angle θ ≃ 27°was made following the initial study made by Ran et al. 33 , where electrical-resistivity measurements indicated that the phase SC-PPM is centered at a tilt angle θ ≃ 23.7°, while tunnel-diode-oscillator measurements showed that it is centered around θ ≃ 33°.

Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.