#  $\mathbf{c}$ axis electrical transport at the metamagnetic transition in
the heavy-fermion superconductor UTe2 under pressure

G. Knebel  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, F-38000
Grenoble, France  A. Pourret  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG,
PHELIQS, F-38000 Grenoble, France  S. Rousseau  Univ. Grenoble Alpes, CEA,
Grenoble INP, IRIG, PHELIQS, F-38000 Grenoble, France  N. Marquardt  Univ.
Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, F-38000 Grenoble, France  D.
Braithwaite  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, F-38000
Grenoble, France  F. Honda  Institute for Materials Research, Tohoku
University, Oarai, Ibaraki, 311-1313, Japan  Central Institute of Radioisotope
Science and Safety, Kyushu University, Fukuoka 819-0395, Japan  D. Aoki  Univ.
Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, F-38000 Grenoble, France
Institute for Materials Research, Tohoku University, Oarai, Ibaraki, 311-1313,
Japan  G. Lapertot  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS,
F-38000 Grenoble, France  W. Knafo  Laboratoire National des Champs
Magnétiques Intenses - EMFL, CNRS, Univ. Grenoble Alpes, INSA-T, Univ.
Toulouse 3, F-31400 Toulouse, France  G. Seyfarth  Laboratoire National des
Champs Magnétiques Intenses (LNCMI), CNRS, Univ. Grenoble Alpes, F-38042
Grenoble, France  J-P. Brison  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG,
PHELIQS, F-38000 Grenoble, France  J. Flouquet  Univ. Grenoble Alpes, CEA,
Grenoble INP, IRIG, PHELIQS, F-38000 Grenoble, France<EMAIL_ADDRESS>
###### Abstract

The electrical resistivity of the unconventional superconductor UTe2 shows
very anisotropic behavior in the normal state depending on the current
direction. In the present paper we show that the maximum in the resistivity
$\rho_{c}$ for current applied along the $c$ axis at $T^{\rm
max}_{\rho_{c}}\approx 14.75$ K follows the minimum in the thermal expansion
$T_{\alpha}^{\star}$ along $b$ axis. Under a magnetic field applied along the
$b$ axis, $T^{\rm max}_{\rho_{c}}$ can be tracked up to the critical point of
the first order metamagnetic transition, which is located near 6 K and 34.5 T.
Surprisingly, at the metamagnetic field $H_{m}$ the resistivity $\rho_{c}$
shows a steplike decrease while the resistivities $\rho_{a}$ and $\rho_{b}$,
for current along the $a$ and $b$ axis, respectively, show a steplike
increase. Under hydrostatic pressure $T^{\rm max}_{\rho_{c}}$ and $H_{m}$
decrease significantly up to the critical pressure $p_{c}$ at which
superconductivity is suppressed and a long range antiferromagnetic order
appears. We show that the phase diagram at different pressures can be scaled
by $T^{\rm max}_{\rho_{c}}$ in field and temperature suggesting that this
temperature scale is governing the main interactions in the normal state.

###### pacs:

71.18.+y, 71.27.+a, 72.15.Jf, 74.70.Tx

##  I Introduction

The discovery of superconductivity in UTe2 has raised an intense research
effort due to the possible spin-triplet pairing [1, 2, 3]. Indication for this
rare pairing state is the strong enhancement of the superconducting upper
critical field $H_{c2}$, which exceeds by far the Pauli limit of
superconductivity for all three crystallographic axes. Another one is the
change of this paramgnetic limit in the different superconducting phases
observed under pressure for field along the easy a-axis [4]. Knight shift
measurements in magnetic resonance experiments (NMR) suggest a possible spin-
triplet superconducting state too [5, 6, 7]. Moreover, a quite singular
property of superconductivity in UTe2 is that the pairing strength itself is
sensitive to the magnetic field for all crystal axes [8]. For a field applied
along the $b$ axis, superconductivity is enhanced above $H^{\star}\approx 17$
T [9, 10], and a phase transition between two susperconducting phases, the
low-field superconducting phase (lfSC) and a high-field phase (hfSC), have
been evidenced by specific heat [8], and independently by combined NMR and ac
susceptibility measurements [11]. This shows that the field-induced hfSC is a
bulk superconducting phase. It is limited by a metamagnetic transition at
$\mu_{0}H_{m}\approx 35$ T [9, 10]. The first order metamagnetic transition is
characterized by a large jump of the magnetization [12], a volume
discontinuity [13], and a jump in the magnetoresistance with the current
applied along the $a$ axis [14]. It is accompanied by a strong increase of the
Sommerfeld coefficient $\gamma$ of the specific heat [15, 16, 8] and of the
$A$ coefficent of the Fermi-liquid resistivity [14, 17]. Recent NMR
measurements show that critical longitudinal magnetic fluctuations starts to
develop for $H>H^{\star}$ and are diverging in the vicinity of $H_{m}$ [18].
An abrupt Fermi-surface change at $H_{m}$ has been discussed from Hall effect
measurements [19], but constitutes still an open question [20].
Superconductivity in UTe2 was claimed to be topologically non-trivial
following the detection of chiral-edge states in scanning-tunneling microscopy
(STM) [21], and of broken time-reversal symmetry in Kerr effect studies[22].
However, these results are still discussed controversally and may depend on
the sample quality [23, 24].

Under hydrostatic pressure, two superconducting phases occur above 0.3 GPa,
indicating multiphase superconductivity at zero field [25, 26, 27]. Different
superconducting phases have been confirmed by measurements under magnetic
field applied along the $a$ and $b$ axes at constant pressure [28, 27]. The
superconducting phases are suppressed at a critical pressure $p_{c}\approx
1.5$–1.7 GPa (depending on the pressure medium) and a magnetically ordered
state occurs. Initially a ferromagnetic state has been expected [29], but
recent neutron diffraction experiments under confirm an incommensurate
antiferromagnetic order above $p_{c}$ [30]. The magnetic anisotropy is
reversed at $p_{c}$ [31, 32, 33, 34]. While the $a$ axis is the easy
magnetization axis at ambient pressure, it switches to the $b$ axis in the
pressure induced magnetic phase. The change in the magnetic anisotropy has
also a feedback on the superconducting state under pressure. Close to $p_{c}$,
the upper critical field $H_{c2}$ is the highest for the $c$ axis, and even
reentrant superconductivity occurs in this direction [4, 29, 35, 33].

To understand the occurrence of these various superconducting phases in UTe2
the knowledge of the electronic structure and of the magnetic fluctuations is
necessary. Various electronic band structure calculations indicate cylindrical
Fermi surfaces along the $c$ axis [36, 37, 38, 39, 40, 41]. Experimentally the
Fermi surface of UTe2 is not fully determined, but angle resolved
photoemission spectroscopy (ARPES) experiments [39] and quantum oscillations
have confirmed the quasi-two-dimensional Fermi-surface sheets [42, 43, 44,
45].

Resistivity measurements with different current directions suggest that UTe2
is a three dimensional metal [46] indicating that either the cylinders are
strongly corrugated or that a three-dimensional Fermi surface is missing in
quantum oscillation experiments. For currents applied along the $a$ and $b$
axes, the temperature dependence of the resistivity shows a broad maximum near
60 K and drops for lower temperatures, indicating the formation of coherent
quasiparticle bands characteristic of a heavy-fermion state. Once a non-
magnetic background of the scattering is subtracted, this maximum is shifted
down to the temperature scale $T^{\rm max}_{\chi_{b}}\approx 35$ K, for which
the susceptibility and the Hall effect for $H\parallel b$ are maximum [33, 17,
19]. For a current applied along the $c$ axis a distinct maximum of the
resistivity appears only at $T^{\rm max}_{\rho_{c}}\approx 14.5$ K [46, 17,
47, 48]. Near this characteristic temperature several other quantities show
anomalies: the electronic specific heat has a broad maximum [49], the thermal
expansion coefficients along the $c$ and $b$ axes have a minimum [49, 26], the
thermoelectric power has a minimum [50], and the $a$ axis susceptibility
$\chi_{a}$ as well as the Knight shift becomes constant at lower temperatures
[51]. The anomaly at $T^{\star}$ corresponds to the crossover to a coherent
low temperature regime [46, 52]. Furthermore, inelastic neutron scattering
experiments show the development of a magnetic excitation below $T\approx 60$
K at the incommensurate wavevector $\mathbf{k_{1}}=(0,0.57,0)$ which is
maximal at the energy transfer 3-4 meV [53, 54, 52]. They also reveal that
these fluctuations are low dimensional and antiferromagnetic, saturating below
$T^{\star}\approx 15$ K [53, 54]. They become gapped in the superconducting
state [55, 56].

In the present paper, we study the field and pressure dependence of the
different fluctuations by electrical resistivity measurements. The paper is
divided in different sections. In Section II we will present the experimental
details and give a summary of all samples studied in this work. Section III
presents ambient pressure results: we will show the field dependence
($H\parallel b$) of the $T^{\star}$ anomaly in the resistivity with current
applied along the $c$ axis ($\rho_{c}$) and longitudinal thermal expansion
measurements ($\alpha_{b}$) up to the metamagnetic field. We will demonstrate
that the maximum $\rho_{c}$ at $T^{\star}$ can be followed up to the
metamagnetic transition. Further we will discuss the anisotropy of the
magnetoresistivity for currents applied along the $a$, $b$, and $c$ axes.
Finally, we will compare in the superconducting phase diagram in a field
$H\parallel b$ for different samples with different $T_{sc}$. We find that the
lfSC phase scales with $T_{sc}$, while the hfSC is strongly affected by sample
quality.

In the following Section IV we will concentrate on the resistivity
measurements under high pressure. In Section IV.1 the temperature dependence
of the resitivity under pressure for a current applied along the $c$ axis are
presented and compared to those with a current along the $a$ axis. We present
the pressure-temperature phase diagram of UTe2 at zero magnetic fields. In
Section IV.2 we present the temperature and field dependence of $\rho_{c}$
under pressure. In Section IV.3 we discuss the obtained magnetic field –
temperature phase diagrams for the different pressures. We show that the phase
diagrams for the different pressures scale with the temperature $T^{\star}$
indicating the importance of the fluctuations connected with this energy
scale. Magnetoresistivity gives evidence that the order, which occurs above
the critical pressure, is antiferromagnetic in agreement with recent neutron
diffraction results [30]. Finally in Section V we give a short conclusion.
Additional data supporting the analyses are presented in the Appendix A.

##  II Experimental methods

In this study single crystals of UTe2 grown by chemical vapor transport (CVT)
using iodine as transport agent [57] and by the molten salt flux (MSF) method
[58] have been investigated. An overview of all samples is given in Tab. 1. At
ambient pressure we studied the dependence of the current direction on the
electrical resistivity and on the $A$ coefficient of the Fermi-liquid
resistivity. For $J\parallel b$ (sample S4) and $J\parallel c$ (S1a, S5)
single crystals grown by CVT have been investigated with superconducting
transitions $T_{sc}\approx 1.85$ K at zero field (throughout the paper, the
superconducting transition temperature $T_{sc}$ in the resistivity is defined
by the criteria $\rho=0$, see Fig. 18 in the Appendix). CVT grown samples have
been cut from large crystals with a spark cutter after alignment using a Laue
photograph. Typical dimensions of these CVT crystals are $1\times 0.3\times
0.15$ mm3. Two different crystals have been used for measurements with current
$J\parallel c$. For sample S1a, the alignment of the contacts has probably not
been perfectly done due to its irregular, non bare-like shape, and the
absolute value of the resistivity of this sample is smaller than expected from
literature [46, 47]. We studied in addition a second sample S5 ($0.8\times
0.15\times 0.1$ mm3) which has a similarly high resistance; unfortunately,
here the field orientation is not perfect as the measured metamagnetic
transition in this experiment is higher than expected (data are shown in the
Appendix Fig. 19). For resistivity measurements with current $J\parallel a$
axis ($\rho_{a}$) we used a MSF sample with a $T_{sc}\approx 2$ K. The MSF
sample had a naturally needlelike shape along the $a$ axis with dimensions of
$2\times 0.2\times 0.2$ mm3. The high-pressure resistivity experiments with
current alonc $c$ have been performed on a bareshape-like sample (S1b) which
is cut from the same single crystal as S1a with current applied along the $c$
axis. For the measurements with current along $a$ we refer to our previous
results reported partly in Ref. [4].

Table 1: Summary of single crystals used in this study and the corresponding measurements. | crystal rowth |  $T_{sc}$ (K) | measurement
---|---|---|---
S1a | CVT | 1.84 |  $\rho_{c}(T,H)$, $J\parallel\sim c$, $p=0$
|  |  | (current not perfectly aligned)
S1b | CVT | 1.84 |  $\rho_{c}(T,H)$, $J\parallel c$, $p\neq 0$
S2 | CVT | 1.45 |  $\alpha_{b}$, same crystal as in [8]
S3 | MSF | 2.0 |  $\rho_{a}(H)$, $J\parallel a$, $p=0$
S4 | CVT | 1.82 |  $\rho_{b}(H)$, $J\parallel b$, $p=0$
S5 | CVT | 1.85 |  $\rho_{c}(H)$, $J\parallel c$, $p=0$
|  |  | (not perfectly aligned in field)

Electrical resistivity measurements have been performed with a standard four
point lock-in technique with a maximal applied current of 0.5 mA for the
measurements at ambient pressure and 1 mA in the high pressure cell. For all
samples, electrical contacts to the samples have been performed by spot
welding 15 $\mu$m diameter Au wires on a freshly polished surface of the
crystal as it is known that the surface of UTe2 is very sensitive to air. To
strengthen the electrical contacts mechanically, tiny drops of conducting
silver paste have been added on top of the welded contacts. The contact
resistance is typically of the order of several m$\Omega$, but this has not
been further studied.

The high-field transport measurements have been performed at the high magnetic
field laboratory LNCMI in Grenoble using the resistive 36 T magnet M9 using a
3He cryostat with a base temperature near 400 mK. We used a MP35N piston
cylinder pressure cell to apply pressures up to 1.61 GPa with Daphne 7373 oil
as pressure medium for the high-pressure resistivity experiment. The outer
diameter of the pressure cell is only 15 mm which fits very close to the inner
diameter of the vaccuum can of the cryostat (16 mm). The high-pressure
experiments have been performed up to a maximal field of 35 T. The temperature
has been measured with a calibrated RuO2 chip which is glued on the pressure
cell. The lowest temperature in the pressure cell was $T_{\rm min}\approx 1.2$
K.

Thermal expansion measurements at zero pressure have been performed using a
high resolution capacitive dilatometer [59] in the LNCMI Grenoble on the
magnet M10 with a maximal field of 30 T in the temperature range from 2 to 25
K. The CVT grown sample (S2) with $T_{sc}\approx 1.5$ K was already studied in
Ref. [8] by magnetostriction experiments.

##  III Ambient pressure results

###  III.1 c axis transport and field dependence of $\mathbf{T^{\star}}$

Figure 1: Temperature dependence of the electrical resistivity $\rho_{c}(T)$
of UTe2 with current $J\parallel c$ at different magnetic fields $H\parallel
b$ (measured on sample S1a). The maximum of the resistivity $T^{\rm
max}_{\rho_{c}}$ shifts with field from $T^{\rm max}_{\rho_{c}}=14.75$ K at
$H=0$ to 5.5 K at 34 T.

Figure 1 displays the temperature dependence of the $c$ axis resistivity
$\rho_{c}$ for different magnetic fields applied along the $b$ axis below 20
K. A maximum of the resistivity at $T^{\rm max}_{\rho_{c}}\approx 14.75$ K
occurs at $H=0$, similar to $\rho_{c}$ measurements [46, 17]. However, as
already mentioned above, the absolute value of the resistivity $\rho_{c}$ at
low temperature is smaller than in previous reports [46, 47, 17], which might
be due to a non-perfect alignment of the current with respect to the $c$ axis.
Under magnetic field $T^{\rm max}_{\rho_{c}}$ shifts to a lower temperature
and the maximum gets more pronounced. At the highest field of 34 T, just below
the metamagnetic transition, $T^{\rm max}_{\rho_{c}}\approx 5.5$ K. It is
close to the critical point of the first order metamagnetic transition which
is located near 6 K and 34.5 T. Superconductivity, defined by $\rho=0$, occurs
in zero field below $T_{sc}=1.89$ K. The minimum of $T_{sc}$ ($T_{sc}=0.88$ K)
is found at $H^{\star}\approx 18$ T. For higher fields the temperature of zero
resistivity increases again and at 34 T, $T_{sc}=1.35$ K. This behavior of
$H_{c2}(T)$ is similar to that of previous reports [9, 10, 60]. The highest
$T_{sc}$ of the hfSC phase is found just below $H_{m}$.

In the normal state above $T_{sc}$ the resistivity $\rho_{c}$ follows a Fermi-
liquid temperature dependence $\rho(T)=\rho_{0}+AT^{2}$, where $\rho_{0}$ is
the residual resistivity and $A$ the coefficient of the electron-electron
scattering term. This is shown in Fig. 20 in the Appendix, which shows
$\rho_{c}$ as a function of $T^{2}$ for several magnetic fields [61]. While at
10 T the fitting range is up to about 6 K, for the highest field this range is
strongly reduced and the maximum temperature of the fit is about 4 K, i.e. the
fitting range of the $T^{2}$ dependence is small so that the following
analysis of the $A$ coefficient is rather qualitative. Anyway, in heavy-
fermion systems the relation $A\propto\gamma^{2}$ is often obeyed indicating
that local fluctuations dominate [62]. In a simplified picture, the $A$
coefficient is an indirect measure the square of the effective mass $m^{*}$ of
the charge carriers. One has to keep in mind that resistivity in difference to
specific heat is a directional probe and so in addition to the scattering time
it depends also on the topology of the Fermi surface and not only on the
density of states. The field dependence of $A$ and the residual resistivity
$\rho_{0}$ will be discussed below.

Figure 2: Longitudinal thermal expansion $\alpha_{b}$ as a function of
temperature for different magnetic fields applied along $b$ axis up to 29 T.
$\alpha_{b}(T)$ shows a pronounced minimum, which we refer as
$T^{\star}_{\alpha}$ in the text. Measurements are performed on sample S2.

Figure 2 shows the longitudinal thermal expansion coefficient along the $b$
axis $\alpha_{b}$ as a function of temperature at different magnetic fields.
$\alpha_{b}(T)$ is negative and has a very broad minimum at
$T^{\star}_{\alpha}\approx 14.5$ K in zero field. A magnetic field applied
along the $b$ axis shifts $T^{\star}_{\alpha}$ to lower temperatures and the
minimum sharpens significantly. We can follow the minimum up to the highest
field of 29 T and $T^{\star}_{\alpha}(H)$ determined from thermal expansion is
in excellent agreement with the maximum $T^{\rm max}_{\rho_{c}}$ in the $c$
axis transport. Fig. 3 shows the phase diagram of UTe2 at ambient pressure
determined from the $c$ axis transport and the thermal expansion along $b$
axis. This shows that the thermodynamic anomaly detected at
$T^{\star}_{\alpha}$ connects to the metamagnetic transition at $H_{m}$. The
$T^{\star}_{\alpha}\approx T^{\rm max}_{\rho_{c}}$ is a signature of a
crossover to the low temperature state. In heavy fermion compounds this
crossover arises from the interplay of the onsite and intersite magnetic
interactions, which result in the formation of the coherent heavy-fermion
state on cooling. In UTe2 this crossover is also visible in the evolution of
the magnetic fluctuations at finite wave vectors [54, 53, 52, 51, 46]. In
addition to the magnetic fluctuations, crystal field effects may also play a
role in the microscopic origin of the $T^{\star}$ anomaly [63].

Figure 3: Phase diagram of UTe2 for field along the $b$-axis from the $c$ axis
transport and the thermal expansion measurements. $T_{sc}$ is the
superconducting transition temperature, $T^{\star}_{\alpha}$ the minimum in
the thermal expansion coefficient $\alpha_{b}$, $T^{\rm max}_{\rho_{c}}$ the
maximum in $\rho_{c}(T)$, and $H_{m}$ the field of the metamagnetic
transition. Clearly, the maximum $T^{\rm max}_{\rho_{c}}$ coincides with the
temperature of the minimum in the thermal expansion. $T_{sc}$ is determined by
the criterium $\rho=0$ and error bars are smaller than the symbol size.

###  III.2 Anisotropic transport at $H_{m}$

Figure 4: Magnetoresistivity for $H\parallel b$ at different temperature with
the current applied along $a$, $b$ and $c$ axes. The inset in every panel
gives the field range of the metamagnetic transition in an enlarged scale.
[MSF-grown sample S3 for $J||a$, CVT grown samples for $J||b$ (S4) and $J||c$
(S1a).]

Previous magnetoresistivity measurements with a current $J\parallel a$ axis
showed that $\rho_{a}(H)$ in the normal state just above $T_{sc}$ increases
abruptly at the metamagnetic transition $H_{m}$ by a factor of 4 [14, 64, 19].
A hysteresis between field up and down sweeps of about 0.2 T indicates the
first order nature of the transition. The transition is sharp up to a critical
point of $\approx 6$ K where the the hysteresis vanishes. At higher
temperatures a broad crossover from a coherent correlated paramagnetic to an
incoherent paramagnetic state occurs. A similar abrupt change in the Hall
resistivity as a function of field at $H_{m}$ together with a change of the
main charge carriers observed in thermoelectric power experiments [19] suggest
that an abrupt change of the Fermi surface occurs at the metamagnetic
transition $H_{m}$. Figure 4 displays the magnetoresistivity with a current
injection along the $a$, $b$, and $c$ axes as a function of the magnetic field
applied along the $b$ axis. For the $J\parallel a$ axis, we used a high
quality MSF grown sample with a $T_{sc}=2$ K. $\rho_{a}(H)$ of the MSF sample
with the higher $T_{sc}$ shows a similar field dependence in the normal state
as reported previously [14, 64] The samples used for the different current
directions show re-entrant superconductivity at the lowest temperature up to
$H_{m}$.

For a current $J\parallel b$ axis, $\rho_{b}$ shows an extremely large
positive jump at $H_{m}$, by a factor 8.5 at 2.25 K in the normal state, see
Fig. 4(b). This jump is much stronger than that being observed for a current
applied along the $a$ axis.

Astonishingly, for a current $J\parallel c$ axis, the magnetoresistivity
$\rho_{c}(H)$ drops at the metamagnetic transition as shown in the lower panel
of Fig. 4(c); at 3 K the magnetoresistivity decreases by a factor of 1.45, i.
e., the change of the magnetoresistivity at the metamagnetic transition is
much smaller and opposite to that observed along the $a$ and $b$ directions,
and for $T$ near $T_{sc}$ no strong anomaly occurs near $H_{m}$. With
increasing temperatures above $T_{sc}$ the drop of the resistivity along $c$
becomes stronger and changes into a crossover above the critical point of the
metamagnetic transition line, like for the other current directions (see also
Ref. [17]). These distinct differences between the magnetoresistivity measured
for the different current directions might be due to the anisotropic quasi-2D
Fermi surface of UTe2 [42, 43, 44], but a detailed microscopic picture of the
anisotropic scattering is still missing. In Ref. [46], within a simple two
band model, it has been proposed that the conduction along the $c$ axis is
dominated by a heavy $Z$ pocket of the Fermi surface detected in ARPES
measurements, while the conduction along the $a$ and $b$-axis directions is
dominated mainly by the cylindrical Fermi surfaces. Recent dHvA quantum
oscillations experiments did not detect such a small 3D Fermi-surface pocket
[42, 43, 45] in difference to Ref. [44] where a 3D Fermi surface pocket may be
observed using a tunnel diode oscillator circuit (TDO); future experimental
investigations have to clarify this point.

An anisotropy of the transport properties at a metamagnetic transition has
been also observed in other heavy-fermion compounds. In the paramagnetic and
nearly antiferromagnetic CeRu2Si2, where the metamagnetic transition occurs
for a field applied along the easy $c$ axis, the magnetoresitivity shows a
positive jump in the transversal configutation $J\parallel a$, while a peak
occurs for $J\parallel c$ [65]. In the paramagnetic and nearly ferromagnetic
UCoAl the magnetoresistance has a positive jump at $\mu_{0}H_{m}\approx 0.6$ T
for $J\parallel H\parallel c$, which is the easy magnetization axis, but a
negative jump occurs in the transversale configuration [66]. And in
antiferromagnetic UPd2Al3, a sharp peak occurs at $H_{m}$ in the longitudinal
configuration while a strong decrease of almost 50 % occurs in the transverse
magnetoresistance [67]. In all of these examples, the metamagnetic transition
is accompanied with a Fermi surface reconstruction [68, 69, 70], as proposed
for UTe2.

Figure 5: Normalized $A$ coefficient determined from a Fermi liquid
temperature dependence $\rho(T)=\rho_{0}+AT^{2}$ of the resistivity for
current applied along the $a$, $b$, and $c$ directions as a function of
$H/H_{m}$. Full symbols are determined from the measured field sweeps at
different temperatures. Fits have been performed for $T<3.5$ K $J||a$ (sample
S3), $T<4.5$ K for $J||b$ (S4),, and $T<4$ K for $J||c$ (S5, data shown in Fig
19). Open symbols for $J\parallel c$ are from fits of data shown in Fig. 1
(sample S1a). The lower panel (b) shows the residual resistivity $\rho_{0}$
obtained from the Fermi liquid fit for the three current directions.

The field dependences of the normalized $A$ coefficient and of the residual
resistivity $\rho_{0}$ extracted from Fermi-liquid fits are shown in Fig. 5 as
a function of $H/H_{m}$. $A$ and $\rho_{0}$ were determined from the
temperature dependent measurements (open circles for $J\parallel c$) and from
the field sweeps at constant temperature (full symbols), see also Figs. 19,
20, 21 in the Appendix. We compare $A(H)$ normalized to its zero field value
for the different current directions in Fig. 5 (see also Fig. 22 in the
Appendix which shows the absolute values). The anisotropy of the $A$
coefficient at zero field is similar to that in Ref. [46]. The field
dependence of $A$ for $J\parallel a$ determined on the new MCF sample is in
excellent agreement with those published in Ref. [14]. However, there are
distinct differences in the field dependence of $A(H)$ for the different
directions.

For $J\parallel a$ axis the absolute value of $A$ is the smallest and it
increases from about $A=0.6\,\mu\Omega\,\rm{cm}\,K^{-2}$ at zero field to 4
$\mu\Omega\,\rm{cm}\,K^{-2}$ at $H_{m}$. Astonishingly, the field enhancement
of $A$ is symmetric around $H_{m}$, which was not expected owing to the first
order character of the metamagnetic transition. For current along the $b$
axis, the absolute value of $A$ equals $5\,\mu\Omega\,\rm{cm}\,K^{-2}$ at zero
field. $A(H)$ increases very strongly by a factor of 10, when the metamagnetic
transition is approached. (The absolute value of $A(H)$ for the $b$-axis
direction may be over estimated as it is only determined from the measured
field dependences at different temperatures, and no temperature dependent
measurements at fixed field have been performed (see Fig. 21).)

In zero field, the value of $A$ is the highest for a current applied along the
$c$ axis, with $A=8.3\,\mu\Omega\,\rm{cm}\,K^{-2}$. On approaching the
metamagnetic transition under magnetic field, $A$ increases only by a factor
of 3.4 at $H_{m}$ for $J\parallel c$ and decreases abruply just above the
metamagnetic transition. If we compare the value at zero field with Sommerfeld
coefficient at zero field of 120 mJ mol-1K-2 we find that $A/\gamma^{2}$ is
much higher than the standard value of the Kadowaki Woods ratio of $1\times
10^{-5}$ $\mu\Omega$ cm(K mol/mJ)2, commonly expected in a Kondo lattice. The
main reason for this strong difference is the quasi-2D Fermi surface in UTe2,
and thus, leading to a strongly anisotropic $A$ coefficient. By contrast, if
we compare the field dependence of the specific heat divided by temperature
$C/T$ at 1.8 K and that of the $A$ coefficient determined from $\rho_{c}$ (see
Fig. 23 in the Appendix), we find an excellent agreement of both properties.
Especially, the strong drop of the $C/T|_{T=1.8\,K}$ does not occur in the
$A(H)$ for the other directions.

Figure 5(b) shows the field dependence of the residual resistivity $\rho_{0}$
normalized to its value in zero field. We clearly see that $\rho_{0}$ for the
$a$ and $b$ axes increases by a factor of at least 15 ($a$) or 20 ($b$), while
the change of $\rho_{0}$ along $c$ is very small. The strong increase of the
residual resistivity for $J\parallel a$ and $b$ cannot be only due to a change
in the charge carrier density, as this should influence the change of the
magnetoresistance for $J\parallel c$ at $H_{m}$ in rather similar way. In a
simple picture it may also be an indication for a Fermi surface change at
$H_{m}$ as has been previously suggested from the thermoelectric response
through $H_{m}$ [19]. Surely, the change of the Fermi surface above $H_{m}$ is
still an open question.

###  III.3 Comparison of the superconducting phase diagrams

In Fig. 6(a) we compare the superconducting upper critical field $H_{c2}$ with
$H\parallel b$ for samples with different values of $T_{sc}$ varying from 1.45
K to 2 K. All samples show field-reentrant superconductivity in fields above
$H^{\star}\approx 16-18$ T. The thermodynamic phase diagram has been only
determined in one of the samples ($T_{sc}=1.84$ K by specific heat
measurements [8]). The exact field of the reinforcement superconducting
transition is difficult to determine from the transport experiments as it does
not coincide exactly with the bulk transition [8, 60]. As shown in Ref. [8],
the anomaly in specific heat corresponding to the bulk transition to the hfSC
phase appears as a hump and is extremely broad [71]. However, $H^{\star}$ is
roughly independent of the sample quality [60]. We also see in Fig. 6(a) that
the value of the measured metamagnetic field $H_{m}$ varies from 33.8–34.75 T
for the different samples. The lowest metamagnetic field has been found for
the sample with the highest $T_{sc}$, while in samples with a lower $T_{sc}$ a
slightly higher $H_{m}$ has been observed. The robustness of $H_{m}$ for
different sample qualities has also been reported in Ref. [72]

Figure 6: (a) Comparison of the superconducting phase diagram of samples with
different $T_{sc}$ for field $H\parallel b$. (Red circles are taken from Ref.
[9], yellow squares from specific heat measurements of Ref. [8], green
triangles up determined from data in Fig. 1, violet triangles from Fig. 3(a).)
The field of the metamagnetic transition is marked by open symbols,
respectively. (b) Upper critical field normalized by $T_{sc}$ versus
$T/T_{sc}$ for the different samples.

In Fig. 6(b) we plot $H_{c2}$ normalized by $T_{sc}$ as a function of
$T/T_{sc}$. In this representation $H_{c2}(T/T_{sc})$ in the lfSC phase scales
on a single curve for samples with different $T_{sc}$, while in the high-field
phase hfSC the critical field strong variations occurs depending on the
samples. In general, in the clean limit, the orbital superconducting upper
critical field at zero temperature depends on the Fermi velocities $v_{\rm
F_{\perp}}$ perpendicular to the applied magnetic field $H$ and $T_{sc}$ as
$H_{c2}^{\rm orb}\propto(\frac{T_{sc}}{v_{\rm F_{\perp}}})^{2}$. However, in
Fig. 6(b) we plot $\frac{\mu_{0}H_{c2}}{T_{sc}}$ vs $\frac{T}{T_{sc}}$. This
normalization is much better than that with a purely orbital limit
$\frac{H_{c2}}{T_{sc}^{2}}$ vs $\frac{T}{T_{sc}}$ (see Fig. 24 in the
Appendix). As discussed in Ref. [8] the upper critical field for $H\parallel
b$ is not described by a simple orbital limit. Surprisingly, as shown in Fig.
6, the low lfSC and hfSC phases show very different behavior for the different
samples. While the bulk nature of the hfSC phase has been clearly shown by
specific heat, thermal expansion and ac susceptibility [8, 11, 60], the bulk
transition itself is intrinsically very broad. The thermal expansion
experiments clearly indicate that the vortex dynamics are different in the
lfSC and hfSC phases [8]. This is also supported by measurements of the
critical current [73], indicating that pinning in this phase depends strongly
on sample quality and impurities. Here, we observe that $H^{\star}$ (Fig 6a),
like $H_{m}$, appears weakly dependent on the sample purity, [26, 60] even
though $T_{sc}$ changes very significantly between samples.

##  IV $\mathbf{c}$ axis electrical transport under pressure

###  IV.1 Comparison of $\mathbf{c}$ axis and $\mathbf{a}$ axis resistivity in
zero field

Figure 7: Temperature dependence of the resistivity for (a) $\rho_{a}$
(current $J||a$, data from the pressure experiment reported in Ref. [4]) in
UTe2 at different pressures on a logarithmic scale. The black and grey arrows
indicate the ordering temperature $T_{\rm MO}$ and the crossover to a
correlated regime $T_{\rm WMO}$. (b) Temperature dependence of $\rho_{c}$
(current $J||c$). The arrows indicate $T_{\rm MO}$, $T_{\rm WMO}$, and $T^{\rm
max}_{\rho_{c}}$.

Figure 7 displays the temperature dependence of the resistivity with a current
along the $a$ axis ($\rho_{a}$, upper panel) and along the $c$ axis
($\rho_{c}$, lower panel) for different pressures at zero magnetic field. For
both current directions, the general behavior at high temperature is similar
to that at ambient pressure. $\rho_{a}(T)$ increases on cooling down to a
temperature $T^{\rm max}_{\rho_{a}}$ and decreases to lower temperatures. The
temperature of the broad maximum in $\rho_{a}$, $T^{\rm max}_{\rho_{a}}$,
decreases with increasing pressure, and is minimum near the critical pressure
$p_{c}\approx 1.5$ GPa. At $p_{c}$ superconductivity is suppressed and the
magnetically ordered ground state is formed (see Fig. 8). $T^{\rm
max}_{\rho_{a}}$ increases slightly above $p_{c}$. The exact position of
$p_{c}$ depends on the pressure conditions; $p_{c}\sim 1.7$ GPa has been
reported in Refs. [25] and [33] using anvil pressure cells.

The temperature dependence of $\rho_{a}$ shows a kink for $p=1.48$ GPa near
the temperature $T_{\rm WMO}\approx 6$ K and a second anomaly near $T_{\rm
MO}\approx 3.5$ K in agreement with the previous report of two magnetic
anomalies above $p_{c}$ [32, 35, 31]. While at $T_{\rm MO}$ long range
magnetic order appears, $T_{\rm WMO}$ has been identified in magnetization
measurements under pressure as a crossover to a weakly magnetically ordered
state [31]. With increasing pressure, $T_{\rm WMO}$ shifts to higher
temperature, gets less pronounced and cannot be resolved anymore above 2 GPa
as it gets close to $T^{\rm max}_{\rho_{a}}$. The lower anomaly at $T_{\rm
MO}$ is almost pressure independent up to 2.51 GPa, the highest pressure in
this experiment (see Fig. 8). Very close to $p_{c}$ we observe at 1.48 GPa not
only the magnetic anomaly, but also still the onset of a very broad
superconducting anomaly. Similar coexistence of magnetism and
superconductivity has been observed in Ref. [32] by ac calorimetry. At the
border of an antiferromagnetic instability, it is difficult to observe a
magnetic transition inside the superconducting state ($T_{\rm MO}<T_{sc}$),
whereas superconductivity occurs often inhomogeneously inside the magnetically
ordered state when $T_{sc}<T_{\rm MO}$. A very well studied example for this
competition of magnetic order and superconductivity is given by CeRhIn5 [74,
75] with the inhomogeneous appearance of superconductivity below its critical
pressure ($T_{sc}<T_{N}$ ), followed by the rapid suppression of magnetic
order above $p_{c}$ when $T_{sc}>T_{N}$.

As shown in Fig. 7(b), $\rho_{c}$ decreases with decreasing temperature for
all pressures above 50 K. Similarly to the zero pressure data, at low
temperatures a pronounced maximum $T^{\rm max}_{\rho_{c}}$ occurs. This
maximum shifts to lower temperatures with pressure up to 1.32 GPa. At this
pressure, just below $p_{c}$, it almost coincides with the onset of
superconductivity. The maximum of $T_{sc}\approx 3$ K is observed at 1.2 GPa
for both samples. The normal-state resistivity changes drastically above the
critical pressure $p_{c}$. At $p=1.61$ GPa, $\rho_{c}$ increases strongly
below 10 K by a factor of two, shows a small plateau below 7 K and increases
again below 3.45 K with a maximum at 2.4 K. The strong increase of the
resistivity is attributed to a short range magnetic order at $T_{\rm
WMO}\approx 7.5$ K and antiferromagnetic order below 3.5 K. A common feature
between the $a$ and $c$ axes transport is the increase of the resistivity at
the lower magnetic transition $T_{\rm MO}$, which may indicate the opening of
an electronic gap when entering in the magnetic state. The residual
resistivity $\rho_{0}$ increases for both current directions strongly through
$p_{c}$, and the anisotropy of the residual resistivity $\rho_{c0}/\rho_{a0}$
is still of the order of three in the magnetically ordered state.

Figure 8: Pressure-temperature phase diagram of UTe2 at zero magnetic field
indicating the pressure dependence of $T^{\rm max}_{\rho_{a}}$, $T^{\rm
max}_{\chi_{b}}$ (taken from Ref. [4]), $T^{\rm max}_{\rho_{c}}$,
$T^{*}_{\chi_{a}}$ (taken from Ref. [31]), the superconducting transition
temperature $T_{sc}$ (red symbols, triangles taken from Ref. [4]), and the
magnetic transition temperatures $T_{\rm MO}$ and $T_{\rm WMO}$. (Triangles-up
and circles are from the $a$ and $c$ axis resistivity, respectively.)

In Fig. 8 we summarize the pressure-temperature phase diagram of UTe2 from the
present resistivity measurements combined with previous studies [4, 31]. The
characteristic temperatures determined from the temperature dependence of the
resistivity $T^{\rm max}_{\rho_{a}}$, $T^{\rm max}_{\rho_{c}}$, the maximum of
the magnetic susceptibility measured along the $b$ axis, $T^{\rm
max}_{\chi_{b}}$, and $T^{*}_{\chi_{a}}$, which marks a broad shoulder in the
susceptibility measured along the $a$ axis [31], decrease up to the critical
pressure $p_{c}\approx 1.5$ GPa. $T^{*}_{\chi_{a}}$ follows roughly $T^{\rm
max}_{\rho_{c}}$. Recently we have shown that $T^{\rm max}_{\rho_{a}}$ scales
with the maximum of the magnetic susceptibility $T^{\rm max}_{\chi_{b}}$ [33,
17] as a function of magnetic field and pressure, when a background
contribution to the resistivity is subtracted. The background corresponds to
the resistivity in the high-field regime above $H_{m}$, where the system is in
a polarized state and magnetic fluctuations are strongly suppressed. The
magnetic interactions change drastically at the critical pressure and for
$p>p_{c}$, the crystallographic $b$ axis becomes the axis of easy
magnetization in the magnetic ordered state, while the $a$ axis is an
intermediate axis [31, 34].

###  IV.2 $\mathbf{c}$ axis transport under pressure and in magnetic field
$\mathbf{H\parallel b}$

Next we focus on the $c$ axis resistivity $\rho_{c}$ for field applied along
the $b$ axis at different pressures.

Figure 9: (a) Temperature dependence of the resistivity at 0.35 GPa for
different magnetic fields. Solid (dashed) lines are for fields below (above)
the metamagnetic transition. (b) Field dependence of the resistivity for
different temperatures. (c) Zoom on the magnetoresistivity around the
metamagnetic transition.

Figure 9(a) shows the temperature dependence of $\rho_{c}$ for different
fields $H\parallel b$ at 0.35 GPa. Near $T^{\rm max}_{\rho_{c}}\approx 12$ K
the resistivity shows a maximum which is slightly lower than at ambient
pressure. Under magnetic field, like at $p=0$ (see Fig. 1), the maximum shifts
to lower temperatures and, at 27.5 T, we find $T^{\rm max}_{\rho_{c}}\approx
5.5$ K. Above $H_{m}=29.5$ T, at 30 T and for higher fields, the temperature
dependence of $\rho_{c}$ changes significantly and instead of a maximum, a
sharp drop of the resistivity is observed below 5 K. By further increasing the
field, this anomaly is shifted to slightly higher temperatures. At low
temperatures $\rho_{c}(T)$ has a $T^{2}$ temperature dependence in zero
magnetic field up to 5 K, but with increasing field the $T^{2}$ range
decreases significantly due to the low value of $T^{\rm max}_{\rho_{c}}\approx
5$ K. Even above $H_{m}$ no clear $T^{2}$ is observed in the measured
temperature range down to 1.5 K. The superconducting transition temperature
decreases from $T_{sc}=2.28$ K at zero field down to $T_{sc}=1.65$ K at 8 T
and is almost field-independent in higher fields up to 27.5 T. Thus the field
enhancement of superconductivity is less pronounced compared to ambient
pressure, but a phase line between the lfSC and hfSC phases still exists [27].

In Fig. 9(b) we show the field dependence $\rho_{c}(H)$ up to 35 T for
different temperatures at $p=0.35$ GPa. At the lowest temperature,
superconductivity survives up to $H_{m}=29.5$ T. A large jump to the normal
state resistivity occurs at $H_{m}$. $\rho_{c}(H)$ shows a hysteresis between
the up and down sweeps of the field [see Fig. 9(c)] indicating the first order
nature of the transition at $H_{m}$. At 1.65 K, the signature of the first
order metamagnetic transition is very weak. At higher temperatures, we observe
a marked hysteresis at the metamagnetic transition and the magnetoresistance
$\rho_{c}(H)$ decreases strongly. At 6 K no hysteresis is observed anymore,
indicating that the critical point of the first order transition is located
between 4.5–6 K, thus lower than at ambient pressure.

Figure 10: (a) Temperature dependence of the resistivity at 1 GPa for
different magnetic fields. Solid (dashed) lines are for fields below (above)
the metamagnetic transition. (b) Field dependence of the resisitivity for
different temperatures. (c) Zoom on the magnetoresistivity around the
metamagnetic transition.

Figures 10 and 11 show the resistivity for higher pressures (1 GPa and 1.32
GPa respectively) approaching the critical pressure. From the
magnetoresistance at 1 GPa we conclude that the metamagnetic transition field
is further reduced down to $\mu_{0}H_{m}=18.5$ T [see Fig.10 (b)]. The
temperature dependence of the resistivity has changed slightly compared to
lower pressures, and only the characteristic temperatures are lower. The
maximum in $\rho(T)$ decreases from $T^{\rm max}_{\rho_{c}}=7.5$ K at $H=0$ to
3.24 K at $\mu_{0}H=17.5$ T. For fields above $H_{m}$ the absolute value of
the resistivity has decreased and a shallow maximum in the temperature
dependence appears at 4.8 K at 20 T. This maximum shifts to higher
temperatures with increasing magnetic field. The superconducting transition is
monotonously suppressed from 3.05 K at $H=0$ to 2.07 K at 17.5 T. Due to the
high superconducting transition temperature and to the low $T^{\rm
max}_{\rho_{c}}$, no $T^{2}$ temperature dependence is observed in the normal
state above the superconducting transition. A Fermi-liquid $T^{2}$ dependence
is only recovered above the metamagnetic transition in the polarized phase for
fields above $H_{m}$, but the maximum temperature is about 3.5 K, thus the
$T^{2}$ regime is very narrow. We observe zero resistivity up to $H_{m}$ at
1.3 and 2 K [see Fig. 10(b,c)]. We do not find any signature of a
reinforcement of superconductivity, but the upper critical field is
monotonously suppressed.

In the normal state a maximum develops in the magnetoresistance at
temperatures higher than the critical point, which is near 3.5 K and 17.5 T at
1 GPa. The maximum shifts to higher temperatures with increasing magnetic
fields.

Figure 11: (a) Temperature dependence of the resistivity at 1.32 GPa for
different magnetic fields. Solid (dashed) lines are for fields below (above)
the metamagnetic transition. (b) Field dependence of the resistivity for
different temperatures. (c) Zoom on the magnetoresistivity around the
metamagnetic transition.

Figure 11 presents the temperature and field dependence of $\rho_{c}$ close to
the critical pressure at 1.32 GPa. At zero magnetic field the resistivity
increases with decreasing temperature, superconductivity sets in near 3 K and
zero resistivity is attained at 2.65 K. The shallow maximum near 3.05 K at
zero field may already be a signature of the onset of superconductivity. The
characteristic temperature $T^{\rm max}_{\rho_{c}}$ seems to be lower or at
most of the same order than $T_{sc}$. Under magnetic field up to 9 T a shallow
maximum occurs just above the onset of superconductivity. Above $H\approx 9$
T, close to $H_{m}$, a broad maximum in $\rho_{c}(T)$ occurs and is shifted to
higher temperatures for higher magnetic fields. At low temperatures a Fermi-
liquid regime occurs.

Figure 12: (a) Temperature dependence of the resistivity at 1.61 GPa for
different magnetic fields. $T_{\rm MO}$ indicates the magnetic ordering
temperature and $T_{\rm WMO}$ the crossover to a short-range correlated regime
at zero field, respectively. (b) Field dependence of the resistivity for
different temperatures measured at the LNCMI. Blue arrows indicate the
direction of the field sweep. $H_{c}$ and $H_{k}$ mark the critical field of
the disappearance of the hysteresis and a well defined kink in the
magnetoresistivity at 1.5 K. The inset in (b) shows the difference
$\Delta\rho_{c}$ between field up and field down sweeps at 1.5 K, with the
opening of the hysteresis at $H_{c}=10.3$ T.  Figure 13: Field dependence of
the $c$ axis resistivity at 1.61 GPa for different temperatures above (a) and
below $T_{M}$ (b). A strong hysteresis occurs in the magnetic state. $H_{c}$
indicates the critical field of the magnetic order. $H_{k}$ indicates a kink
in the resistivity (see also Fig. 12 above).  Figure 14: Magnetic and
superconducting phase diagram of UTe2 at 0.35, 1.1, 1.32, and 1.61 GPa for
field applied along the $b$ axis defined from the resistivity measurements
with $J\parallel c$. Full green circles (squares) indicate the maximum of the
resistivity $T^{\rm max}_{\rho_{c}}$ as function of temperature below (above)
the metamagnetic transition. The superconducting transition at $T_{sc}(H)$
(full red circles) is defined by zero resistivity. The metamagnetic transition
at $H_{m}$ is marked by blue triangles for pressures below $p_{c}\approx 1.45$
GPa. SC marks the superconducting phase, PPM the polarized paramagnetic phase
above $H_{m}$. At 1.61 GPa long range magnetic order (MO) occurs below $T_{\rm
MO}$ up to $H_{c}$. $T_{\rm WMO}$ (open circles) is defined by the inflection
point of the increase of the $c$ axis resistivity and shows a similar field
dependence as the maximum of the resistivity $T^{\rm max}_{\rho_{c}}$. No
metamagnetism occurs. Open blue circles Open blue circles (determined from
extrema of the magnetoresistivity) correspond probably moment reorientations
in the MO state. $H_{c}$ indicates the critical field for the magnetically
ordered state. $H_{k}$ indicates the kink in the magnetoresistivity in the
polarized high field regime.

The field dependence $\rho_{c}(H)$ is shown in Fig. 11(b) for different
temperatures. Panel (c) displays the field range around the metamagnetic
transition in an enlarged scale. We first discuss $\rho(H)$ at the lowest
temperature $T=1.3$ K [see Fig. 11(b)]. Zero resistivity is observed up to
$\approx 9.6$ T in the field sweep with increasing field. For higher fields
the resistivity increases sharply and has a change of slope as a function of
field near 10.15 T. A clear signature of the metamagnetic transition is
missing. By further increasing the field, $\rho_{c}(H)$ at 1.3 K has a maximum
at 13.3 T, which marks the onset of superconductivity near $H_{c2}$ in the
field sweep. A well defined kink in $\rho_{c}(H)$ occurs at $H_{k}=17.1$ T in
the normal state. For even higher fields the magnetoresistivity has a minimum
around 26 T. With increasing temperature these features get broadened and the
kink cannot be followed above 2 K. Figure 11(c) displays the anomaly of the
metamagnetic transition on an enlarged scale. While at 2.3 K almost no anomaly
at the metamagnetic transition occurs due to the closeness to the onset of
superconductivity (at 8.7 T), at 2.5 and 2.8 K only a tiny anomaly indicates
the metamagnetic transition; no jump or marked decrease in $\rho_{c}(H)$
occurs anymore. Therefore the critical point of the first order transition can
no more be resolved due to superconductivity. For temperatures above 2.8 K no
signature of $H_{m}$ is detected. At 4 K the magnetoresistivity shows a
minimum at 9 T and a maximum at 11.4 T. This maximum shifts to higher fields
with increasing temperature, similarly to the temperature $T^{\rm
max}_{\rho_{c}}$ of the maximum in $\rho_{c}(T)$ at fixed field.

Finally, in Figs. 12 and 13 we show the temperature and field dependencies of
$c$ axis resistivity at $p=1.61$ GPa. On cooling the resistivity shows a
smooth increase at zero field below 10 K with an inflection point at $T_{\rm
WMO}\approx 7.5$ K, and a sharp increase at $T_{\rm MO}=3.45$ K. Under field,
$T_{\rm WMO}$ increases slightly in temperature up to 10 T, and much faster
for higher fields when it gets less pronounced, as the absolute value of the
resistivity decreases (see Fig. 25 in the Appendix which shows $d\rho/dT$ for
different magnetic fields). On the contrary $T_{\rm MO}$ decreases with
increasing field, and its signature stays sharp under field up to 10 K.

A strongly negative magnetoresistance up to fields of 30 T is visible in Fig.
12(b). It decreases by more than a factor 10 from zero field to 20 T
indicating the suppression of the magnetic scattering related to the magnetic
correlations established below the characteristic temperature $T_{\rm WMO}$.
This strong magnetoresistance at low temperatures contrasts with the rather
low variation of the magnetoresistance by only 2% at 15 K. For temperatures
above $T_{\rm MO}$ the magnetoresistance does not show any hysteresis and
decreases monotonously below 20 T. On the contrary, below $T_{\rm MO}$ the
magnetoresistance shows a significant hysteresis for field up and field down
sweeps. We define the field $H_{c}$, where the hysteresis disappears as the
critical field of the magnetically ordered state, see inset in Fig. 12.
$H_{c}$ determined here from the field sweeps coincides with the temperature
$T_{\rm MO}$ determined from the temperature sweeps. Inside the magnetically
ordered state several additional anomalies occur. In Figs. 12(b) and 13(b)
maxima of the magnetoresistance are found in fields $H<H_{c}$. These findings
are similar to those reported in Refs. [27, 33]. The observation of clear
anomalies in the magnetic ordered state is presumably an indication for re-
orientations of the magnetic moments inside the magnetically ordered state. Of
course, thermodynamic measurements are needed to determine the phase lines
inside the ordered state. Very recently, an incommensurate antiferromagnetic
order has been observed by neutron diffraction experiments [30].

For fields $H>H_{c}$ we observe at low temperatures an additional kink at
$H_{k}$ in the magnetoresistance. A similar feature had been already observed
at lower pressures in the polarized state at $p=1.32$ GPa. The signature of
$H_{k}$ disappears for temperatures above 3 K. A comparable kink in the
magnetoresistance has been observed for an angle of 25 deg from the $b$ toward
the $c$ axis at about 21 T [76]. The origin of this anomaly is still an open
question.

###  IV.3 Discussion of the pressure dependence

Figure 15: Superconducting upper critical field $H_{c2}$ for different
pressures. Dashed lines indicate the metamagnetic field for each pressure,
which is the upper field limit of superconductivity.  Figure 16: Magnetic
phase diagram of UTe2 for field along the $b$ axis normalized to the
temperature $T^{\rm max}_{\rho_{c}}(H=0)$ at zero magnetic field as a function
of the normalized temperature. Circles (squares) give $T^{\rm max}_{\rho_{c}}$
below (above) the metamagnetic transition, stars indicate $H_{m}(T)$ for the
different pressures.

Figure 14 summarizes the $H-T$ phase diagrams at different pressures
determined from the resistivity measurements with current $J\parallel c$ and
field $H\parallel b$. For all pressures below $p_{c}$, $T^{\rm
max}_{\rho_{c}}$ is connected to the metamagnetic transition.
Superconductivity is observed up to $H_{m}$, which decreases with pressure.
For fields higher than $H_{m}$ a crossover to a polarized regime occurs as a
function of temperature. Close to the critical pressure an additional kink
occurs in the polarized state at low temperatures in $\rho_{c}(H)$ at $H_{k}$,
which persists also above $p_{c}$. Above the critical pressure, at 1.61 GPa,
two magnetic anomalies occur: the lower anomaly $T_{\rm MO}$ is a transition
to a long range magnetically ordered state. Its antiferromagnetic nature was
suggested by the different phase lines inside the ordered state and has been
recently shown by neutron diffraction under high pressure [30]. On the
contrary, the presence of a true phase transition at $T_{\rm WMO}$ is less
clear, and not only broad features occurs in the resistivity, but also in
magnetization [31] or specific heat measurements [32]. The field dependence of
$T_{\rm WMO}(H)$ resembles that of a ferromagnet under field applied along the
easy magnetization axis, as the transition at $T_{\rm WMO}$ smears out under a
magnetic field and shifts to higher temperatures in a similar way as the
maximum $T^{\rm max}_{\rho_{c}}$. However, the microscopic origin of $T_{\rm
WMO}$ has not been identified today, crystal field effects may also be not
negligible. We recall that the magnetic anisotropy changes at $p_{c}$ with $b$
being the easy and $c$ the hard magnetization axis [31]. The coexistence and
competition of antiferromagnetic and ferromagnetic fluctuations has been
suggested from NMR measurements [77].

The pressure evolution of the superconducting phase diagram is similar to
those previously discussed in Ref. [4] and shown in more detail in Fig. 15.
The superconducting transition temperature increases with applied pressure and
superconductivity is stable under magnetic field up to the metamagnetic
transition, but the temperature dependence of the upper critical field
$H_{c2}(T)$ changes significantly under pressure. At low pressure (see at 0.35
GPa) a reinforcement of superconductivity under magnetic field can still be
observed. The critical field $H^{\star}$ of the upturn of $H_{c2}(T)$ is
already strongly reduced from about $17$ T at zero pressure to around $8$ T at
0.35 GPa. The pressure 0.35 GPa is very close to the pressure, where ac
calorimetry at zero field indicates two different superconducting phases [25,
28, 32]. Previous reports of the upper critical field for $H\parallel b$ [4,
27] in that pressure range show at low field a concave curvature indicating
the increase of pairing strength under magnetic field. As initially $T_{sc}$
decreases with increasing pressure, the data at 0.35 GPa suggest that for this
pressure already two superconducting phases should exist, since $T_{sc}$ is
higher than at zero pressure. Thus, on cooling at zero magnetic field we
should enter first in a high-temperature superconducting phase. Of course,
with resistivity measurements, it is not possible to discern different
superconducting phases and detect phase lines inside a superconducting phase.
The critical temperature $T_{sc}$ determined by resistivity measurements is
always that of the phase with the higher $T_{sc}$. In Refs. [27, 78] it has
been proposed that the lfSC is suppressed with pressure and embedded in the
high-temperature superconducting phase. The high-temperature superconducting
phase is proposed to be the pressure continuation the hfSC. More detailed
studies by thermodynamic probes of the phase line between the lfSC and hfSC
phases for $H\parallel b$ are needed in future. They should be performed in
the vicinity of the pressure where the high-temperature superconducting phase
appears and under magnetic field applied along $b$.

Obviously, the temperature dependence of $H_{c2}$ changes on approaching
$p_{c}$. At $p=1$ GPa we do not observe any signature of a field enhancement
of superconductivity up to $H_{m}$ in agreement with previous studies [4, 27].
The strong curvature of $H_{c2}(T)$ might indicate the increase of the
paramagnetic limitation near $p_{c}$ for $H\parallel b$ [4]. The question of
the pairing symmetry of the superconducting state near $p_{c}$ is still open.
On one side, the NMR Knight shift stays constant on cooling through $T_{sc}$
favoring an odd parity pairing. On the other side, if the high-temperature
superconducting phase corresponds to the pressure evolution of the hfSC state,
it could also be a singlet state. In Ref. [8] it has been proposed that the
hfSC may be spin singlet and driven by critical fluctuations on approaching
the metamagnetic transition. Under pressure, the hfSC state may survives up to
$p_{c}$. It has even been predicted [79] that antiferromagnetic fluctuations
would be reinforced under pressure favoring a spin-singlet state for the high-
temperature superconducting phase above 0.3 GPa. As the magnetically ordered
state MO is antiferromagnetic, antiferromagnetic fluctuations may drive the
superconductivity close to the critical pressure. However, even at ambient
pressure and low field the pairing mechanism in UTe2 is still not settled.

Next we will concentrate on the normal state properties. As shown in Fig. 14
the temperature $T^{\rm max}_{\rho_{c}}$ of the maximum in the $c$ axis
resistivity is decreasing with applied pressure. For all pressures below
$p_{c}$ the field dependence of $T^{\rm max}_{\rho_{c}}$ connects to the
metamagnetic transition $H_{m}$, where $T^{\rm max}_{\rho_{c}}$ has its lowest
value. For fields above $H_{m}$, $T^{\rm max}_{\rho_{c}}$ marks the crossover
to a polarized paramagnetic regime. If we identify the minimum value of
$T^{\rm max}_{\rho_{c}}$ as the critical point of the first order metamagnetic
transition, we determine that this critical point decreases almost linearly
with pressure from about (5.6 K, 34 T) at zero pressure to (2.55 K, 8.85 T) at
1.32 GPa. At 1.32 GPa the signature of the metamagnetic transition is almost
lost and only a tiny anomaly is visible.

Figure 17: (a) $A$ coefficient as a function of magnetic field. Solid arrows
indicate the position of $H_{m}$ for the different pressures, the dashed arrow
the critical field of the magnetic order. Due to the closeness of $T^{\rm
max}_{\rho_{c}}$ and $T_{sc}$, $A$ can be only determined at 1 and 1.32 GPa
for $H>H_{m}$, where superconductivity is suppressed. Above the critical
pressure at 1.61 GPa a Fermi liquid is only observed for fields above $H_{c}$.
Dashed lines are guides to the eye. (b) $A$ as a function of the normalized
field, $H_{m}$ for $p<p_{c}$ and $H_{c}$ for $p>p_{c}$. Lines are guides to
the eye.

In Fig. 16 we plot the phase diagram of the normal state scaled in field and
temperature by the temperature $T^{\rm max}_{\rho_{c}}(H=0)$. Importantly, the
phase diagrams of the different pressures below $p_{c}$ scale almost
perfectly. Only for $p=1.32$ GPa the scaling is less good, but this may be due
to the difficulty to determine $T^{\rm max}_{\rho_{c}}$ at low field correctly
due to the high superconducting temperature, $T_{sc}\approx T^{\rm
max}_{\rho_{c}}$. Our thermal expansion measurements at zero pressure clearly
show the correspondence $T^{*}\approx T^{\star}_{\alpha}\approx T^{\rm
max}_{\rho_{c}}\approx 14.5$ K. This characteristic temperature scale connects
to the metamagnetic transition at $H_{m}$. The microscopic origin of
$T^{\star}$ is clearly related to the interplay of magnetic fluctuations and
the formation of a coherent heavy-fermion state. This has been shown by
different NMR and also inelastic neutron scattering experiments at ambient
pressure. Inelastic neutron scattering evidenced the development of
antiferromagnetic fluctuations at wave vector $\mathbf{k_{1}}=(0,0.57,0)$
below 60 K which saturate below 15 K [53, 54, 52]. The magnetic fluctuations
in UTe2 depend strongly on the particular ladder like structure of the U atoms
along the $a$ axis with the rung along $c$ axis. Two dimensional
antiferromagnetic fluctuations originating from magnetic ladders coupled along
$b$ were captured by inelastic neutron scattering and have a characteristic
energy scale of 3-4 meV. The temperature dependence of these fluctuations is
compatible with that from the NMR relaxation rate $1/T_{1}T$ [80, 51]. Under
pressure, $1/T_{1}T$ measured on the different Te-sites, also show a constant
behavior up to $T^{\star}$ [77, 34] and $T^{\star}$ from the NMR scales with
$T^{\rm max}_{\rho_{c}}$ determined from the resistivity. This seems to be the
dominant energy scale which determines the pressure and field dependence of
the phase diagram.

We emphasize that electronic correlations increase on approaching the critical
pressure. The field dependence of the $A$ coefficient from our measurements
with a current along the $c$ axis is shown in Fig. 17(a) as a function of the
field and in (b) as a function of the normalized field $H/H_{m}$ ($p<p_{c}$)
or $H/H_{c}$ ($p>p_{c}$). (Figs. 26 and 27 in the Appendix show the
experimental data and the corresponding fits.) At 0.35 GPa $A(H)$ has still a
similar field dependence as at ambient pressure and $A$ shows a steplike
decrease just above the metamagnetic transition. For higher pressures, due to
the lower $T^{\rm max}_{\rho_{c}}$, the onset of superconductivity prevents
the determination of $A$ below the metamagnetic transition in $\rho_{c}$. Thus
$A(H)$ has been only determined in the field range above $H_{c}$. It shows a
strong monotonous decrease. The enhancement of the $A$ coefficient at $H_{m}$
is the largest close to the critical pressure and comparable to $A$ on
approaching $H_{c}$ in the magnetic state. This indicates that quantum
criticality in UTe2 is important for the formation of the antiferromagnetic
state. This leads to the enhancement of superconductivity under magnetic field
close to the critical pressure and to the reentrant behavior of
superconductivity along the $c$ axis [35, 33]. However, the transition from
the superconducting ground state to the magnetically ordered may be first
order and connected to a strong change in the electronic structure. Indication
for this is the abrupt increase in the residual resistivity through $p_{c}$
[35, 81]. Direct microscopic evidence for this is given by the change by 7
percent of the 5$f$ count through $p_{c}$ towards the U4+ configuration [81].
As already mentioned, this change in the electronic structure goes along with
a drastic change in the magnetic anisotropy.

##  V Conclusions

We have shown that the resistivity in UTe2 depends strongly on the electrical
current direction. The measurements with $J\parallel c$ clearly reveal the
important energy scales. In particular, at the metamagnetic transtion at
$\mu_{0}H_{m}\approx 34.5$ T the resistivity $\rho_{c}$ strongly decreases for
current applied along the $c$ axis, while $\rho_{a}$ and $\rho_{b}$ strongly
increase (current along $a$ or $b$ axis respectively). The field dependence of
the $A$ coefficient for $J\parallel c$ seems to replicate $\gamma(H)$ better
than for the other current directions. The maximum $T^{\rm max}_{\rho_{c}}$ in
the temperature dependence of $\rho_{c}$ decrease on approaching the critical
point of the first order metamagnetic transition line. $T^{\rm
max}_{\rho_{c}}$ coincides with the temperature of the minimum observed in the
thermal expansion $\alpha_{b}$ along the $b$ axis. Under hydrostatic pressure
$T^{\rm max}_{\rho_{c}}$ and $H_{m}$ decreases up to the critical pressure
$p_{c}$. The phase diagram in the normal state below $p_{c}$ scales with
$T^{\rm max}_{\rho_{c}}$, indicating the importance of this low energy scale:
the coherence along the $c$ axis governs the low temperature behavior.
Superconductivity is observed for all pressures up to $H_{m}$, which collapses
at $p_{c}$. At zero pressure the low field superconducting phase lfSC scales
with $T_{sc}$ independently of the sample quality, while the high field
superconducting phase hfSC is strongly sample dependent. This indicated that
these two phases may emerge from different pairing mechanisms [8]. Accurate
determination of the phase diagrams under pressure by thermodynamic
measurements under magnetic field along $b$ are strongly required, in
particular close to the pressure where the high-temperature superconducting
phase occurs. This will elucidate whether the high temperature superconducting
phase is the pressure evolution of the field-induced hfSC phase, or not. A
challenge in future will be to precise the field and pressure dependence of
the valence and spin correlations and their feedback on superconductivity
under pressure.

##  VI Acknowledgments

We thank K. Ishida, Y. Yanase and M.E. Zhitomirsky for fruitful discussions.
We received financial support from the French Research Agency ANR within the
projects FRESCO No. ANR-20-CE30-0020 and FETTOM No. ANR-19-CE30-0037. Further
finacial support has been provided by the JPSJ programs KAKENHI P22H04933,
JP20K20889, JP20H00130, JP20KK0061. We acknowledge support of the LNCMI-CNRS,
member of the European Magnetic Field Laboratory (EMFL) and from the
Laboratorie d’excellence LANEF (ANR-10-LABEX-51-01).

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*

##  Appendix A Supplementary data

In the Appendix we show additional figures to which we refer in the main text.

Figure 18: Example for the definition of the superconducting transition
temparation $T_{sc}$ and the maximum $T^{\star}$ from the resistivity data at
$p=0$ and $h=0$. In the phase diagrams in the following the error bars for
$T_{sc}$ is in all cases smaller than the symbol size and thus we do not plot
error bars. We note that the discussion of the transition width from
resistivity data alone is difficult as it is not a bulk probe. The upper inset
shows a zoom on the anomaly at $T^{\star}$, the horizontal arrow shows the
maximal error of the determination.  Figure 19: (a) Magnetoresistance
$\rho_{c}$ for field $H\parallel\sim b$ and current along $c$ measured on
sample S5 for different temperatures $T$ between 1 and 6 K. The metamagnetic
transition appears at $H_{m}=35.4$ T in the field up sweeps. The hysteresis of
the metamagnetic transition is about 0.4 T. While for $T=1.1$ K the jump at
$H_{m}$ is positive, it is negative and very tiny at 2.1 K in the when the
sample is in the normal state for all fields.The lower panel (b) shows the
temperature dependence extracted from the field dependent measurements. Fermi
liquid fits (dashed lines) are performed in the temperature range $T<4$ K.
Figure 20: (a) Temperature dependence of the resistivity $\rho_{c}$ as a
function of $T^{2}$ for different magnetic fields up to 34 T. The dashed-lines
present a fit with $\rho_{c}=\rho_{0}+AT^{2}$. The range of the $T^{2}$ regime
decreases with increasing magnetic field. Close to the metamagnetic field a
fit with an free exponent $n$ (Figure 19(a)) would result in an exponent
$n=3$, but we in the analysis of the Fermi liquid $A$ coefficient, we fixed
the exponent to be $n=2$.  Figure 21: (a) Temperature dependence of the
resistivity for different magnetic fields reconstructed from the
magnetoresistance measurements show for field $H\parallel\sim b$ measured on
sample S3 for $J\parallel a$ (upper panel) and on sample S4 for $J\parallel b$
(lower panel). $T^{2}$ fits are indicated by the dotted lines, due to the
limited temperatures error bars for the $A$ coefficient shown in Fig. 5 are
rather large. Figure 22: (a) Fermi-liquid coefficient $A$ as a function of the
magnetic field $H\parallel b$ with current $J$ applied along $a$, $b$, and $c$
axis. (b) Field dependence of the residual resistivity $rho_{0}$ for the
different current directions.  Figure 23: Field dependence of the $A$
coefficient determined from the resistivity measurements with $J\parallel c$
and $(C/T)^{2}$ measured at 1.8 K as a function of magnetic field normalized
to the metamagnetic transition field $H_{m}$. Both quantities scale
surprisingly well. Especially the very sharp drop at the metamagnetic
transition is only observed in the measurement of the $A$ coefficient with
current applied along the $c$ axis, while for current in the other directions,
the enhancement of the Fermi-liquid $A$ coefficient occurs to be more
symmetric around $H_{m}$.  Figure 24: Upper critical field $H_{c2}$ normalized
by $T_{sc}^{2}$ versus $T/T_{sc}$ for the different samples. This scaling
would correspond to an purely orbital limited upper critical field. However
the scaling of the critical field only by $T_{sc}$ shown in Fig. 6 seems to be
better.  Figure 25: Temperature derivative $d\rho_{c}/dT$ at $p=1.61$ GPa for
different magnetic fields. The arros indicate the magnetic transition
temperature $T_{\rm MO}$ and the temperature of the crossover $T_{\rm WMO}$.
Figure 26: Temperature dependence of the resistivity reconstructed from the
measured magnetoresistance for different magnetic field for different
pressures, (a) 0.35 GPa, (b) 1 GPa, and (c) 1.32 GPa.  Figure 27: Temperature
dependence of the resistiviy $\rho_{c}$ at 1.61 GPa as a function of
temperature up to 5 K for different magnetic fields in the polarized magnetic
regime above the critical field $H_{c}$ of the antiferromagnetic state. The
dashed lines indicate fits with a Fermi liquid temperature dependence
$\rho_{c}=\rho_{0}+AT^{2}$. The temperature range of the Fermi-liquid range
decreases strongly on appoaching $H_{c}$. The inset shows the data at 25 T and
30 T in an expanded view.