# Magnetic reshuffling and feedback on superconductivity in UTe2 under
pressure

M. Vališka  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, 38000,
Grenoble, France  Charles University, Faculty of Mathematics and Physics,
Department of Condensed Matter Physics, Ke Karlovu 5, Prague 2, 121 16, Czech
Republic  W. Knafo  Laboratoire National des Champs Magnétiques Intenses -
EMFL, CNRS, Univ. Grenoble Alpes, INSA-T, Univ. Toulouse 3, 31400 Toulouse,
France  G. Knebel  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS,
38000, Grenoble, France  G. Lapertot  Univ. Grenoble Alpes, CEA, Grenoble INP,
IRIG, PHELIQS, 38000, Grenoble, France  D. Aoki  Institute for Materials
Research, Tohoku University, Ibaraki 311-1313, Japan  D. Braithwaite  Univ.
Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, 38000, Grenoble, France

###### Abstract

The discovery of superconductivity in the heavy-fermion paramagnet UTe2 has
attracted a lot of attention, particularly due to the reinforcement of
superconductivity near pressure- and magnetic-field-induced magnetic quantum
phase transitions. A challenge is now to characterize the effects of combined
pressure and magnetic fields applied along variable directions in this
strongly anisotropic paramagnet. Here, we present an investigation of the
electrical resistivity of UTe2 under pressure up to 3 GPa and pulsed magnetic
fields up to 58 T along the hard magnetic crystallographic directions
$\mathbf{b}$ and $\mathbf{c}$. We construct three-dimensional phase diagrams
and show that, near the critical pressure, a field-enhancement of
superconductivity coincides with a boost of the effective mass related to the
collapse of metamagnetic and critical fields at the boundaries of the
correlated paramagnetic regime and magnetically-ordered phase, respectively.
Beyond the critical pressure, field-induced transitions precede the
destruction of the magnetically-ordered phase, suggesting an antiferromagnetic
nature. By bringing new elements about the interplay between magnetism and
superconductivity, our work appeals for microscopic theories describing the
anisotropic properties of UTe2 under pressure and magnetic field.

##  I Introduction

The recent discovery of superconductivity in the strongly correlated system
UTe2 has sparked enormous interest Ran  _et al._ (2019a); Aoki  _et al._
(2019). This orthorhombic compound is a paramagnet with anisotropic magnetic
properties Ikeda  _et al._ (2006); Ran  _et al._ (2019a); Aoki  _et al._
(2019); Miyake  _et al._ (2019, ): the magnetic susceptibility along the
$a$-axis increases strongly at low temperature, leading to the initial
suggestion that the system is very close to ferromagnetic order Ran  _et al._
(2019a), whereas the other directions are ”hard” magnetization axes, with
$\mathbf{b}$ being the hardest at low temperature. But the properties of the
superconducting state are the most striking aspect, and in particular the
strong enhancement of superconductivity when a magnetic field $\mathbf{H}$ is
applied along the $b$-axis Ran  _et al._ (2019b); Knebel  _et al._ (2019). In
this case superconductivity persists in magnetic fields up to
$\mu_{0}H_{m}=35$ T, where a first order metamagnetic transition occurs with a
large jump of the magnetization Miyake  _et al._ (2019), a similarly large
jump in the residual electrical resistivity of the normal state Knafo  _et
al._ (2019), and the destruction of superconductivity Knebel  _et al._ (2019);
Ran  _et al._ (2019b); Niu  _et al._ (2020). Even more remarkably when the
field is tilted by about $30~{}^{\circ}$ from the $b$-axis in the hard
$\mathbf{b}-\mathbf{c}$ plane, superconductivity re-emerges above
$\mu_{0}H_{m}\simeq 40$ T for this angle Ran  _et al._ (2019b); Knafo  _et
al._ (2021). The extremely high values of the upper critical field, $H_{c2}$,
compared to the initial superconducting critical temperature ($T_{sc}=1.6$ K)
suggest a probable spin-triplet order parameter, at least in some parts of the
phase diagram. This enhancement of superconductivity is very reminiscent of
the phenomenon found in the ferromagnetic superconductors URhGe Lévy  _et al._
(2005) and UCoGe Aoki  _et al._ (2009). However, in these cases the
reinforcement of superconductivity, when a field is applied along a hard
magnetic axis, is understood as a consequence of the collapse of
ferromagnetism, since an enhancement of the ferromagnetic fluctuations have
been shown to be responsible for the superconducting pairing Wu  _et al._
(2017); Tokunaga  _et al._ (2015). This explanation can obviously not be
directly transposed to UTe2 where no sign of magnetic ordering has been found
down to very low temperatures Sundar  _et al._ (2019); Paulsen  _et al._
(2021). Low-dimensional antiferromagnetic fluctuations were reported,
suggesting that UTe2, whose U ions form a magnetic ladder structure, is
subject to antiferromagnetic exchange leading to antiferromagnetic
correlations Duan  _et al._ (2020); Knafo  _et al._ . The opening of a gap
associated with these antiferromagnetic fluctuations was also observed in the
superconducting phase Duan  _et al._ ; Raymond  _et al._ . The magnetic
properties of UTe2 are thus associated to its unusual superconducting
properties. A full description of the relationship between the two is
essential to understand superconductivity in UTe2, and may well advance our
understanding of magnetically-mediated superconductivity in general.

Applying pressure is the tool of choice to tune magnetism in strongly
correlated systems. Often pressure ($p$) can drive a system towards and
through a magnetic instability, giving a direct probe of the relationship
between magnetism and superconductivity. For UTe2 it has already been shown
that hydrostatic pressure induces an enhancement of $T_{sc}$ by a factor 2,
reaching about 3 K Braithwaite  _et al._ (2019); Knebel  _et al._ (2020); Ran
_et al._ (2020); Thomas  _et al._ (2020). Pressure has also revealed further
complexities of this system’s superconducting state, with multiple
superconducting order parameters appearing Braithwaite  _et al._ (2019);
Thomas  _et al._ (2020); Aoki  _et al._ (2020); Lin  _et al._ (2020). It was
also shown that above a critical pressure $p_{c}\simeq 1.5-1.7$ GPa, magnetic
order occurs with the concomitant disappearance of superconductivity
Braithwaite  _et al._ (2019); Knebel  _et al._ (2020); Ran  _et al._ (2020);
Thomas  _et al._ (2020); Aoki  _et al._ (2020); Lin  _et al._ (2020).
Furthermore the metamagnetic field decreases strongly with pressure for
$\mathbf{H}\parallel\mathbf{b}$ Knebel  _et al._ (2020), and when field is
applied along the $a$-axis (the easy axis at ambient pressure) the multiple
superconducting states have quite different behaviours Aoki  _et al._ (2020).
A large enhancement of $H_{c2}$ for $\mathbf{H}\parallel\mathbf{c}$ was also
found with an inversion of the anisotropy of $H_{c2}$ in the
$\mathbf{b}-\mathbf{c}$ plane for pressures close to $p_{c}$ Knebel  _et al._
(2020). A recent study showed that the magnetic anisotropy is also
significantly changed with pressure, with $\mathbf{b}$ becoming the easy axis
above $p_{c}$ Li  _et al._ (2021). In addition, re-entrant superconductivity
occurs for $\mathbf{H}\parallel\mathbf{c}$ just above the critical pressure
Aoki  _et al._ (2021). The above studies were mainly performed in static
magnetic fields, but the highest static fields available, even in dedicated
facilities, are insufficient to reveal all the physics in UTe2. Here we report
on an experiment combining high pressure and pulsed magnetic fields up to 60
T. These measurements were also performed over a wide temperature range giving
new insight into the evolution of the magnetic properties and their feedback
on superconductivity in three-dimensional (3D) $(H,p,T)$ phase diagrams of
UTe2.

Figure 1: (a) Zero-field temperature-dependence of the electrical resistivity
$\rho$ of UTe2 under pressure. The grey squares show the curve at high
pressure and high field that we take as a background. (b) Temperature
dependence of resistivity $\Delta\rho$ determined after subtraction of the
background.

##  II Methods

We used a previously described pressure cell Braithwaite  _et al._ (2016)
allowing magnetoresistivity measurements in pulsed magnetic fields up to 60 T
and temperatures down to 1.4 K. Single crystals of UTe2 were grown by the
chemical vapor transport technique as described elsewhere Aoki  _et al._
(2019). The crystals were oriented by X-ray Laue diffraction and cut to bar
shaped samples of about $0.8\times 0.2\times 0.1$ mm3. Two successive
experiments were performed in a pressure cell, offering the simultaneous
measurement of the electrical resistivity of two samples in a magnetic field
along the $\mathbf{b}$ and $\mathbf{c}$ directions. First experiment was
performed at pressures above $p_{c}$ on samples $\sharp$A with
$\mathbf{H}\parallel\mathbf{b}$ and $\sharp$B with
$\mathbf{H}\parallel\mathbf{c}$ (see Supplementary Materials SM ). Second
experiment was performed under a large set of pressures below and above
$p_{c}$, from 0.3 to 3.1 GPa, on samples $\sharp$C with
$\mathbf{H}\parallel\mathbf{b}$ and $\sharp$D with
$\mathbf{H}\parallel\mathbf{c}$, and corresponds to the data presented in this
manuscript (see also Supplementary Materials SM ). Both experiments gave
similar results, although the samples set up for
$\mathbf{H}\parallel\mathbf{b}$ displayed an unintentional misalignment (see
Section III), probably having moved on pressurization. A piece of lead was
also mounted in the cell to determine the pressure. High-pressure
magnetoresistivity measurements were performed at the Laboratoire National des
Champs Magnétiques Intenses (LNCMI) in Toulouse under long-duration (50 ms
rise and 300 ms fall) pulsed magnetic fields up to 58 T and temperatures down
to 1.4 K. A standard four-probe method with a current
$\mathbf{I}\parallel\mathbf{a}$ of 0.5 mA, at a frequency of 15-70 kHz and
digital lock-in detection was used. The temperature dependence of the
resistivity was also measured directly in zero field.

##  III Results

###  III.1 Zero-field high-pressure properties

Figure 2: Phase diagram of UTe2 from these and previous Aoki  _et al._ (2021)
measurements. CPM and CPM/WMO denote the correlated paramagnetic regimes
stabilized at pressures below and above $p_{c}$, respectively, SC the
superconducting phase, and MO the magnetically-ordered phase.

Zero-field electrical resistivity $\rho(T)$ curves measured at different
pressures are shown in Fig. 1(a-b). Below $p_{c}$, the onset of
superconductivity is visible at low temperatures, leading to zero resistivity.
Above $p_{c}$ a different anomaly appears in the resistivity curve (labeled
$T_{M}$) that is almost certainly the signature of long range magnetic order.
In the 1.8 GPa curve a further anomaly is apparent at higher temperature
($T_{\rho}^{kink}$). A similar anomaly has been seen in other resistivity
studies at pressures just above $p_{c}$ Thomas  _et al._ (2020) and a broad
anomaly is also visible in the magnetization Li  _et al._ (2021). Another
feature of the $\rho(T)$ curves is a broad maximum, which occurs at a
temperature $T_{\rho}^{max}\simeq 60$ K at ambient pressure. $T_{\rho}^{max}$
decreases with pressure down to about 20 K at $p_{c}$ and then slightly shifts
to higher temperatures again as pressure is increased above $p_{c}$. This
maximum is a general feature in heavy fermion systems and is an indication of
the energy scale below which the cross-over between a high temperature
paramagnetic state (PM) to a coherent heavy fermion state or correlated
paramagnetic state (CPM) occurs.

In Fig. 2 we show the zero-field pressure-temperature phase diagram
constructed from our measurements together with data from previous studies
Aoki  _et al._ (2021). The superconducting critical temperature $T_{sc}$
increases from an initial value of 1.7 K up to a maximum value of about 3 K at
a pressure of about 1.3 GPa. $T_{sc}$ then decreases abruptly and disappears
at a critical pressure $p_{c}\simeq 1.5$ GPa. Reported values of $p_{c}$ vary,
probably due to different pressure conditions, ranging from about 1.4 to 1.7
GPa Braithwaite  _et al._ (2019); Knebel  _et al._ (2020); Ran  _et al._
(2020); Thomas  _et al._ (2020); Aoki  _et al._ (2020); Lin  _et al._ (2020);
Aoki  _et al._ (2021). An interesting feature is the maximum in the
susceptibility $T_{\chi(\mathbf{H}\parallel\mathbf{b})}^{max}$ observed for
$\mathbf{H}\parallel\mathbf{b}$ below $p_{c}$ and for
$\mathbf{H}\parallel\mathbf{a},\mathbf{c}$ above $p_{c}$ Ikeda  _et al._
(2006); Ran  _et al._ (2019a); Knebel  _et al._ (2020); Aoki  _et al._ (2021).
$T_{\rho}^{max}$ and $T_{\chi(\mathbf{H}\parallel\mathbf{b})}^{max}$ show
similar behavior with pressure up to $p_{c}$, although their values are quite
different. Of course, neither feature is a precise indication of this energy
scale which is anyway a cross-over, but the large temperature difference here
is due to the effect of other contributions to the resistivity, including (but
not limited to) phonon scattering, which should be subtracted to get the
magnetic scattering. The temperature shift induced by this effect can be quite
significant in the case of UTe2, as the maximum is rather broad and weak. To
check, we subtracted the background shown in Fig. 1(a), corresponding to a
$\rho(T)$ curve obtained under combined high pressure and magnetic field. This
background corresponds to a high-field regime where the ground state is
magnetically polarized, and for which a significant part of the magnetic
correlations has been suppressed. Empirically we see that, once the background
is subtracted, the maximum is much more pronounced and occurs at a temperature
$T_{\Delta\rho}^{max}$ quite close to
$T_{\chi(\mathbf{H}\parallel\mathbf{b})}^{max}$, which delimitates a CPM
regime [see Fig. 1(b)]. Beyond the critical pressure, a switch of the magnetic
properties is observed, in relation with the onset of long-range magnetic
ordering (MO), and possibly higher-temperature correlated paramagnetism or
short-range weak magnetic ordering (noted CPM/WMO, see later): a maximum in
the magnetic susceptibility is observed for
$\mathbf{H}\parallel\mathbf{a},\mathbf{c}$ at the temperatures
$T_{\chi(\mathbf{H}\parallel\mathbf{a})}^{max}$ and
$T_{\chi(\mathbf{H}\parallel\mathbf{c})}^{max}$, respectively, but no maximum
of the magnetic susceptibility is observed for $\mathbf{H}\parallel\mathbf{b}$
Li  _et al._ (2021). Knowing that it is also possible to follow the maximum in
$\Delta\rho$ to high fields as will be shown later, we therefore conclude that
$T_{\Delta\rho}^{max}$ is a good criterion to follow the cross-over to the CPM
regime as a function of pressure, field and temperature.

###  III.2 High-field and high-pressure electrical resistivity

In Fig. 3(a-b) we show the magnetoresistivity curves at the lowest temperature
(1.4 K) for different pressures and the two magnetic-field orientations. Most
of the main results of this study are already apparent here. For the sample
set up with $\mathbf{H}\parallel\mathbf{b}$, at low pressure the first-order
metamagnetic transition appears as a huge and sharp increase of the
resistivity, similar to what is seen at ambient pressure. However, here at 0.3
GPa this occurs at $\mu_{0}H_{m}\simeq 43$ T, a field significantly higher
than at ambient pressure (about 35 T) whereas it has previously been shown
that $H_{m}$ decreases with pressure Knebel  _et al._ (2020); Li  _et al._
(2021). The most likely explanation for this discrepancy is that the sample
was somewhat misaligned in respect to the field, probably having moved inside
the pressure cell on pressurization. Indeed, it has been shown that $H_{m}$
increases when the field is rotated away from the $b$-axis in both the
$\mathbf{b}-\mathbf{c}$ and $\mathbf{b}-\mathbf{a}$ planes Ran  _et al._
(2019b). The value of $H_{m}$ found here would imply quite a large
misalignment, between 15 and 30 ∘. However, as the effect of a magnetic field
up to 35 T applied along the $b$-axis has already been well studied Knebel
_et al._ (2020), we will see that this tilted configuration allows us to
capture the essential physics for $\mathbf{H}\parallel\mathbf{b}$, as well as
revealing interesting results for a field applied with some misalignment from
the $b$-axis. In the following we will refer to this field configuration as
$\mathbf{H}\approx\parallel\mathbf{b}$. On increasing pressure, the
metamagnetic transition remains clear up to $p_{c}$ where $H_{m}$ decreases to
about 12 T. At higher pressure the aspect of the curve changes and shows
several features that we will explicit further on. For the sample with
$\mathbf{H}\parallel\mathbf{c}$ the resistivity curve is basically featureless
at 0.3 GPa, similar to the ambient pressure results Knafo  _et al._ (2019). In
the high pressure curves (1.8 and 2.4 GPa), the zero field resistivity at 1.4
K has increased considerably, and shows a decrease with field in two steps
with the corresponding field values marked here as $H_{c}$ and $H_{m}^{*}$.
Concerning the superconductivity, the lowest temperature reached in this study
(1.4 K) is only slightly below the ambient pressure superconducting critical
temperature, so at low pressures we see almost no trace of superconductivity.
However, $T_{sc}$ increases significantly with pressure, and our measurements
give a good indication of the superconducting phase diagram. Two important
effects are visible here. First for $\mathbf{H}\approx\parallel\mathbf{b}$, we
see that at 1.3 GPa, superconductivity extends up to $\mu_{0}H_{m}\simeq 20$
T. At ambient pressure, it has been shown that for a field perfectly aligned
along the $b$-axis, superconductivity is reinforced with field and exists up
to $H_{m}$, but that this effect disappears with a misalignment of just a few
degrees, and $H_{c2}$ is considerably reduced. With the misalignment necessary
to explain the large value of $H_{m}$ in our case we would certainly not
expect superconductivity to extend up to $H_{m}$ at ambient pressure, so this
implies that pressure strongly changes the phase diagram when the field is
rotated away from the $b$-axis. The second remarkable effect is seen for
$\mathbf{H}\parallel\mathbf{c}$, where at 1.55 GPa the resistivity is not zero
at low fields, but a re-entrant superconducting state with zero resistivity is
found at high field, between approximately 8 and 18 T, similar to the previous
reports Ran  _et al._ (2020); Aoki  _et al._ (2021).

Figure 3: Magnetoresistivity $\rho(H)$ curves at different pressures and at
the lowest temperature (1.4 K) for the configurations (a)
$\mathbf{H}\approx\parallel\mathbf{b}$ and (b)
$\mathbf{H}\parallel\mathbf{c}$. Reconstructed temperature dependence of the
resistivity $\Delta\rho$ determined after subtraction of the background under
magnetic field (c) $\mathbf{H}\approx\parallel\mathbf{b}$ at 1.3 GPa and (d)
$\mathbf{H}\parallel\mathbf{c}$ at 1.8 GPa (2b). Figure 4: Left-hand graphs:
magnetoresistivity curves for the configuration
$\mathbf{H}\approx\parallel\mathbf{b}$ at different temperatures for the
pressures (a) $p=1$ GPa, (b) $p=1.55$ GPa, and (c) $p=1.8$ GPa. Right-hand
graphs: obtained magnetic-field-temperature phase diagrams of the
superconducting and magnetically ordered phases, and of the CPM regime
delimited by $H_{m}$ and $T_{\Delta\rho}^{max}$, for the pressures (d) $p=1$
GPa, (e) $p=1.55$ GPa, and (f) $p=1.8$ GPa. CPM denotes the correlated
paramagnetic regimes , PPM the polarized paramagnetic regime, SC the
superconducting phase, and MO the magnetically-ordered phase. Figure 5: Left-
hand graphs: magnetoresistivity curves for the configuration
$\mathbf{H}\parallel\mathbf{c}$ at different temperatures for the pressures
(a) $p=1.3$ GPa, (b) $p=1.55$ GPa, and (c) $p=1.8$ GPa. Right-hand graphs:
obtained magnetic-field-temperature phase diagrams of the superconducting and
magnetically ordered phases, and of the CPM regime delimited by $H_{m}$ and
$T_{\Delta\rho}^{max}$, for the pressures (d) $p=1.3$ GPa, (e) $p=1.55$ GPa,
and (f) $p=1.8$ GPa. CPM and CPM/WMO denote the correlated paramagnetic
regimes stabilized at pressures below and above $p_{c}$, PPM the polarized
paramagnetic regime, SC the superconducting phase, and MO the magnetically-
ordered phase. Figure 6: 3D magnetic-field-pressure-temperature phase diagrams
for the configurations (a) $\mathbf{H}\approx\parallel\mathbf{b}$ and (b)
$\mathbf{H}\parallel\mathbf{c}$. CPM and CPM/WMO denote the correlated
paramagnetic regimes stabilized at pressures below and above $p_{c}$, PPM the
polarized paramagnetic regime, SC the superconducting phase, and MO the
magnetically-ordered phase. Figure 7: Top graphs: Field dependence of the
quadratic temperature dependence coefficient $A$ at different pressures for
the configurations (a) $\mathbf{H}\approx\parallel\mathbf{b}$ and (b)
$\mathbf{H}\parallel\mathbf{c}$. Lower graphs (c-d): colour plot of $A$ with
superimposed pressure-magnetic-field phase diagrams showing how, for both
configurations, the enhancement of superconductivity coincides with an
enhancement of $A$. The coefficient $A$ was not determined within the grey
regions.

A fuller understanding of the above effects and the complex phase diagram can
be obtained by looking in detail at the temperature dependence of the
magnetoresistivity. First, we examine the case for
$\mathbf{H}\approx\parallel\mathbf{b}$. Fig. 4 (left-hand graphs) shows the
resistivity curves for different temperatures and three pressures. For $p=1$
and 1.55 GPa, i.e. $p<p_{c}$, the curves are qualitatively similar to the
ambient pressure results Knafo  _et al._ (2019). At low temperature, the
first-order metamagnetic transition to the polarized paramagnetic (PPM) regime
appears as a sharp and large increase of the resistivity. As temperature is
increased this anomaly transforms into a broad maximum indicating a crossover
delimiting the CPM regime. $H_{m}$ decreases with pressure as mentioned
previously, and apart from the larger value of $H_{m}$ due to the misalignment
of the field our results are similar to the previous study under pressure for
$\mathbf{H}\parallel\mathbf{b}$ Knebel  _et al._ (2020). The phase diagrams
drawn from these anomalies are shown in the right-hand panels of Fig. 4. The
field where the maximum of the resistivity occurs decreases as pressure is
increased, similarly to the ambient pressure results Knafo  _et al._ (2019),
and connects with the temperature
$T_{\chi(\mathbf{H}\parallel\mathbf{a})}^{max}$ that also decreases with
pressure Li  _et al._ (2021). We can see that the temperature
$T_{\Delta\rho}^{max}$ at the maximum of the zero-field resistivity after
subtraction of the background described above, which corresponds approximately
to $T_{\chi}^{max}$ at zero field, also corresponds to $H_{m}$ in the range
where both features can be seen. This implies that $T_{\Delta\rho}^{max}$ is
indeed a good criterion to determine the boundary of the CPM regime. The
superconducting phase diagrams are also plotted for $p=1$ and 1.55 GPa. For
both pressures $H_{c2}$ shows an S-shape and, as already mentioned, at 1.55
GPa superconductivity extends up to $H_{m}$ despite the probable misalignment
of field. These results will be discussed in more detail further on. The lower
panels of Fig. 4 show the results for $\mathbf{H}\approx\parallel\mathbf{b}$
at 1.8 GPa. The high temperature magnetoresistivity curves retain the
characteristic broad maximum seen at pressures below $p_{c}$, indicating a
cross-over into a polarized state. This maximum disappears at temperatures
above 10-15 K, corresponding to the value of $T_{\Delta\rho}^{max}$, and
becomes a broad decrease of the resistivity with field. However, the
magnetoresistivity curves at low temperature are quite different. Now
$p>p_{c}$ and the ground state is almost certainly some kind of long range
magnetic order below $T_{M}\simeq 3$ K. For $T<T_{M}$ a pronounced kink can be
seen at $\mu_{0}H_{c}\simeq 13$ T corresponding to the transition from the
long range magnetic order to the polarized paramagnetic state. Several other
features are visible in the magnetoresistivity at the fields
$\mu_{0}H_{r,1}=2.3$ T, $\mu_{0}H_{r,2}=4.3$ T, and $\mu_{0}H_{r,3}=9.3$ T
$<\mu_{0}H_{c}$ at $T=1.4$ K. These transitions are presumably related to
magnetic moment reorientations within the magnetically ordered phase (see
Supplementary Materials SM ).

We now look at the case $\mathbf{H}\parallel\mathbf{c}$ (Fig. 5). For this
configuration no metamagnetic transition has been observed at ambient pressure
at least up to 70 T Ran  _et al._ (2019a). For the pressures 1.3 and 1.55 GPa,
i.e., $p<p_{c}$, at high temperatures no particular feature is apparent in the
magnetoresistivity. The most interesting result here concerns the
superconductivity. At ambient pressure the $H_{c2}(T)$ curve for
$\mathbf{H}\parallel\mathbf{c}$ shows no indication of enhancement of
superconductivity with field. However several studies have already shown that
under pressure the situation changes, with the slope of $H_{c2}$ becoming
extremely steep Braithwaite  _et al._ (2019); Knebel  _et al._ (2020),
possibly indicating the appearance of a field-enhancement of superconductivity
for $\mathbf{H}\parallel\mathbf{c}$, and re-entrant superconductivity
appearing close to the critical pressure. These effects are confirmed here:
the upper and middle panels of Fig. 5 show that $\mu_{0}H_{c2}$ reaches about
20 T at 1.3 GPa and that re-entrant superconductivity develops at 1.55 GPa,
where $\mu_{0}H_{c2}$ exceeds 20 T, respectively. The lower panels of Fig. 5
show that at 1.8 GPa, i.e., for $p>p_{c}$, the critical field $H_{c}$ where
the low-temperature long-range magnetic order is destroyed can be seen as a
well-defined kink in the curves for temperatures below $T_{M}\simeq 3$ K. At
the lowest temperature the critical field reaches $\mu_{0}H_{c}\simeq 20$ T
and it shifts slightly to lower field as the temperature increases. Similarly
to the $\mathbf{H}\approx\parallel\mathbf{b}$ configuration, three anomalies
in the electrical resistivity can be defined at the critical fields
$\mu_{0}H_{r,1}=2.8$ T, $\mu_{0}H_{r,2}=8.1$ T, and $\mu_{0}H_{r,3}=12.3$ T
$<\mu_{0}H_{c}$ at $T=1.4$ K, within the magnetically ordered state (see
Supplementary Materials SM ). As for the configuration
$\mathbf{H}\approx\parallel\mathbf{b}$, all these features disappear when the
temperature is raised above $T_{M}$, confirming their link to the low-
temperature magnetic order. However, for $p>p_{c}$, in contrast to the
$\mathbf{H}\approx\parallel\mathbf{b}$ configuration a pronounced kink remains
at higher field suggesting a transition or a well-defined cross-over into the
polarized state. We denote this field $\mu_{0}H_{m}^{*}\simeq 30$ T and
speculate on its pseudo-metamagnetic nature, in analogy with the metamagnetic
field where the polarized state occurs at low pressure for
$\mathbf{H}\approx\parallel\mathbf{b}$, although here it does not show a sharp
first-order transition. The nature of the regime between $H_{c}$ and
$H_{m}^{*}$ is not clear. Anomalies have previously been seen in the
resistivity, the specific heat and the magnetization Thomas  _et al._ (2020);
Li  _et al._ (2021) at a temperature higher than $T_{M}$ for pressures close
above $p_{c}$. Indeed, in the present study a clear kink can be seen in the
zero-field resistivity at 1.8 GPa (Fig. 3). It has been suggested that this
phase could correspond to static Thomas  _et al._ (2020) or short-range Li
_et al._ (2021) weak magnetic order (WMO). The field $H_{m}^{*}$ probably
corresponds to a transition or crossover between this phase and the polarized
paramagnetic regime. Interestingly, a maximum in the magnetic susceptibility
was observed at a temperature $T_{\chi}^{max}=11$ K for a pressure $p=1.8$ GPa
$>p_{c}$ and a magnetic field $\mathbf{H}\parallel\mathbf{c}$ Li  _et al._
(2021). Similarly to the low-pressure CPM regime delimited in a magnetic field
$\mathbf{H}\parallel\mathbf{b}$ by
$T_{\chi(\mathbf{H}\parallel\mathbf{b})}^{max}$ and $H_{m}$ Knafo  _et al._
(2019); Miyake  _et al._ (2019) (see also other heavy-fermion systems Aoki
_et al._ (2013); Knafo (3458)), the WMO regime may also correspond to a second
CPM regime delimited in a magnetic field $\mathbf{H}\parallel\mathbf{c}$ by
$T_{\chi(\mathbf{H}\parallel\mathbf{c})}^{max}$ and $H_{m}^{*}$. In the
following, we will label this regime as CPM/WMO.

###  III.3 Magnetic quantum criticality and superconductivity

The full 3D phase diagrams obtained for both configurations of magnetic field
$\mathbf{H}\parallel\approx\mathbf{b}$ and $\mathbf{H}\parallel\mathbf{c}$ are
represented in Fig. 6. We see that the phase diagram for the magnetic order is
quite similar for both configurations, with the ordering temperature being
suppressed with field, but with a well-defined transition even at high field.
A similar behavior has also been seen for $\mathbf{H}$ applied along the easy
magnetic axis $\mathbf{a}$, where the critical field is even smaller Aoki  _et
al._ (2020). The succession of field-induced transitions at fields
$H_{r,i}<H_{c}$, with $i=1-3$, (presented in Fig. 4 and 5, but not in Fig. 6
for clarity) indicates the stabilization of different magnetic structures.
Such behaviour would not be expected for a ferromagnetically-ordered phase, at
least for a field applied along the easy axis, strongly suggesting that the
magnetic order is of an antiferromagnetic (or spin-density-wave) type, as
already inferred from previous results Braithwaite  _et al._ (2019); Knebel
_et al._ (2020); Thomas  _et al._ (2020); Aoki  _et al._ (2020). A swich of
polarization processes occurs at $p_{c}$ and leads to quite different 3D phase
diagrams for the two configurations $\mathbf{H}\parallel\approx\mathbf{b}$ and
$\mathbf{H}\parallel\mathbf{c}$:

  * •

At pressures below $p_{c}$, for $\mathbf{H}\approx\parallel\mathbf{b}$ the CPM
regime appears as a well-defined 3D bubble, delimited by the first-order
transition at $H_{m}$ at low temperature and the temperature
$T_{\Delta\rho}^{max}$ (or $T_{\chi}^{max}$) at low field. However, for
$\mathbf{H}\parallel\mathbf{c}$, while at zero field the cross-over between
the low-temperature CPM and high-temperature PM regimes is obviously the same,
there is no signature of transition to the PPM regime with field, and it is
likely that a change develops smoothly as a continuous rotation of the
moments. This is consistent with the absence of a maximum in the magnetic
susceptibility for $\mathbf{H}\parallel\mathbf{c}$, which is almost Curie-
Weiss-like down to the lowest temperatures Ikeda  _et al._ (2006); Knafo  _et
al._ (2021), and with the fact that $T_{\Delta\rho}^{max}$ increases with
applied field for $\mathbf{H}\parallel\mathbf{c}$.

  * •

For pressures above $p_{c}$ the situation is quite different. Now a 3D bubble
formed by the CPM/WMO regime occurs for $\mathbf{H}\parallel\mathbf{c}$, with
a quite well-defined transition into the PPM regime, whereas for
$\mathbf{H}\approx\parallel\mathbf{b}$ this probably occurs as a broad cross-
over. This is also consistent with the appearance of a maximum of magnetic
susceptibility for $\mathbf{H}\parallel\mathbf{c}$ and the disappearance of
such maximum for $\mathbf{H}\parallel\mathbf{b}$ under pressures beyond the
critical pressure Li  _et al._ (2021).

This reshuffling of the magnetic properties at the critical pressure $p_{c}$
is related to a switch of the anisotropy of the magnetic susceptibility, with
the hard magnetic axis $\mathbf{b}$ at ambient pressure becoming the easy
magnetic axis at high-pressure Li  _et al._ (2021).

In heavy-fermion materials, magnetic quantum criticality, generally
accompanied by a Lifshitz Fermi-surface instability, is reflected in the
enhancement of the effective mass $m^{*}$ as the field is increased towards
$H_{m}$ seen by a direct measurement of the specific heat Imajo  _et al._
(2019), as well as from magnetization Miyake  _et al._ (2019) and resistivity
Knafo  _et al._ (2019) measurements. Magnetic fluctuations are often
considered as the origin of the large effective mass $m^{*}$ observed in these
materials Knafo (3458). The quadratic temperature coefficient $A$, obtained by
a fit of the resistivity to a Fermi-liquid behavior $\rho=\rho_{0}+AT^{2}$,
varies as $m^{*2}$ within first approximation and shows a pronounced maximum
at $H_{m}$ for $\mathbf{H}\parallel\mathbf{b}$, as well for a field tilted by
30 ∘ in the $\mathbf{b}-\mathbf{c}$ plane Knafo  _et al._ (2021). In Fig. 7 we
show the field dependence of the $A$ coefficient for different pressures,
extracted from the reconstructed temperature dependences of the resistivity
for both configurations. For the configuration
$\mathbf{H}\approx\parallel\mathbf{b}$, at the lowest pressures the
metamagnetic transition appears as a very sharp peak at $H_{m}$, starting at
about 43 T at 0.3 GPa. As pressure is increased the peak position moves to
lower fields, the value of $A$ increases, and above 1.3 GPa the peak starts to
broaden noticeably. For the highest pressures ($p>p_{c}$) the low field
($H<H_{c}$) points are omitted as the onset of magnetic order occurring close
to the lowest temperature measured here made the analysis meaningless in this
case. For the configuration $\mathbf{H}\parallel\mathbf{c}$, at low pressure
no feature is visible in $A(H)$. However, above 1 GPa a broad maximum becomes
apparent that shifts to higher fields and becomes more pronounced as pressure
is increased, remaining visible even for $p>p_{c}$.

The considerable changes of the magnetic properties with pressure have strong
consequences on the superconductivity. In the 3D phase diagrams of Fig. 6, an
enhancement of superconductivity can be seen close to $p_{c}$ for both
configurations of magnetic field $\mathbf{H}\approx\parallel\mathbf{b}$ and
$\mathbf{H}\parallel\mathbf{c}$. Previous studies have shown that field re-
entrant superconductivity develops at ambient pressure and low temperature for
a sample perfectly aligned with $\mathbf{H}\parallel\mathbf{b}$ and that at a
temperature of 1.4 K the sample should be superconducting at all fields up to
$H_{m}$ once a small pressure is applied Knebel  _et al._ (2019, 2020). Here,
due to misalignment, no re-entrant superconductivity is seen for
$\mathbf{H}\approx\parallel\mathbf{b}$ at low pressure and $H_{c2}$ is quite
low at $T=1.4$ K. However, we find that superconductivity extends up to
$H_{m}$ as pressure is increased. A similar result was also found recently for
a sample oriented at an angle of 30 ∘ from the $b$-axis in the
$\mathbf{b}-\mathbf{c}$ plane Ran  _et al._ . Concerning the configuration
$\mathbf{H}\parallel\mathbf{c}$, we have evidenced the presence of field-
induced superconductivity under pressures $p\lesssim p_{c}$. The lower panels
(c-d) of Fig. 7 show the electronic $(p,H)$ phase diagrams at our base
temperature, $T=1.4$ K, with the evolution of $A$ as a color plot. They
emphasize the relationship between the enhancement of $A$ and the high-field
stabilization of superconductivity. Field-reinforced or field-induced
superconductivity is observed close to the critical pressure, where the
collapse of the field scales $H_{m}$ and $H_{m}^{*}$ and enhanced $A$
coefficients are observed. For $\mathbf{H}\approx\parallel\mathbf{b}$
superconductivity survives up to $H_{m}$ under pressures near to $p_{c}$,
where $A$ reaches its maximum value at $H_{m}$. For
$\mathbf{H}\parallel\mathbf{c}$ and $p=1.55$ GPa, although no field-induced
magnetic transition is observed and temperatures larger than $T_{sc}$,
superconductivity may result from the proximity of critical magnetic
fluctuations, as indicated by the enhancement of $A$ in a nearby region of the
phase diagram. $A$ is maximum near $H_{m}^{*}$ for $p\gtrsim p_{c}$, and the
field-induced superconducting phase which develops for $p\lesssim p_{c}$
appears as a prolongation of the $H_{m}^{*}$ line.

##  IV Discussion

Quantum criticality, either purely magnetic or accompanied by a Fermi-surface
instability, is suspected to be a driving force for superconductivity in many
heavy-fermion systems. In UTe2 this is evidenced by the enhancement of
superconductivity on approaching the magnetic phase transition under pressure,
and on approaching metamagnetic transitions with field. For the latter case, a
quantitative analysis has shown that the re-entrant superconducting behaviour
for $\mathbf{H}\parallel\mathbf{b}$ can be explained by a monotonic increase
of the pairing strength related to the increase of the effective mass $m^{*}$
Knebel  _et al._ (2019); Brison . Furthermore, the rapid disappearance of the
re-entrant behaviour as soon as the field is rotated away from the $b$-axis is
a natural consequence, mainly due to the increase of $H_{m}$. Indeed, as the
enhancement of the pairing strength occurs at higher fields, it is no longer
sufficient to overcome the orbital and possibly paramagnetic pair-breaking
effects at lower fields. A small anisotropy of $m^{*}$, that is maximum for
$\mathbf{H}\parallel\mathbf{b}$ can further amplify this phenomenon. From this
picture, it is easy to understand our result. As can be seen from the $A(H)$
curves, as pressure is increased the enhancement of $m^{*}$ and consequently
of the pairing strength will simultaneously occur at lower fields and become
stronger, allowing the field enhancement of superconductivity to be effective
over a much wider angular range. We expect that, with a measurement made at
lower temperatures, we would have found a superconducting state extending up
to $H_{m}$ at even lower pressure. An open question is whether on approaching
$p_{c}$ the field enhancement of superconductivity is still effective or not.
In a previous study with $\mathbf{H}\parallel\mathbf{b}$, the characteristic
S-shape of $H_{c2}$ was lost for pressures of 1 GPa and above Knebel  _et al._
(2020). The initial slope remained large but $H_{c2}$ showed a pronounced
curvature that could be an indication of increasingly effective Pauli
limitation with pressure. In the present study, for
$\mathbf{H}\approx\parallel\mathbf{b}$ we see the S-shape of $H_{c2}$ at 1.55
GPa implying that the field reinforcement effect might still be active. This
would be consistent with the pressure and field dependence of $A$, which still
shows a pronounced maximum at $H_{m}$, and reaches higher values than at
ambient pressure. What is clear from the study with configuration
$\mathbf{H}\approx\parallel\mathbf{b}$ is that the metamagnetic transition at
$H_{m}$ still acts as a very effective cut-off for superconductivity which
does not survive in the polarized state. This is similar to the ambient
pressure behavior for $\mathbf{H}\parallel\mathbf{b}$ but now concerns a wider
angular range. It contrasts with the surprising re-entrant superconducting
phase only seen in the PPM regime stabilized above $H_{m}$ in the
configuration with a magnetic field $H$ tilted by 30 ∘ from $\mathbf{b}$
toward $\mathbf{c}$ at ambient pressure Ran  _et al._ (2019b); Knafo  _et al._
(2021). Recently a study performed in this configuration under pressure showed
that superconductivity extends continuously below and above $H_{m}$ Ran  _et
al._ . It seems that here we are not in this configuration, implying that we
probably have also a component of field along the $a$-axis, and so the
extension of superconductivity up to $H_{m}$ may be found also for
misalignement of the field in the $\mathbf{a}-\mathbf{b}$ plane under
pressure.

For $\mathbf{H}\parallel\mathbf{c}$, the extremely steep slope found for
$H_{c2}$ on approaching $p_{c}$, as well as the very high values ($25-30$ T)
found for $H_{c2}$ here and in a previous study Aoki  _et al._ (2021),
strongly suggest that pressure causes an enhancement of the superconducting
pairing strength to come into play. Fig. 7 shows clearly how this enhancement
seems to be linked to the maximum of $A(H)$ that appears under pressure, and
thus to the new field scale $H_{m}^{*}$, corresponding to the crossover to the
polarized state from the CPM/WMO state. Approaching $p_{c}$ we find a re-
entrant behaviour of superconductivity as has been reported previously Aoki
_et al._ (2021). While the field enhancement of the pairing strength is
certainly favorable for this re-entrant behavior, the main ingredient is
probably the competition between the magnetically ordered and superconducting
phases. Indeed previous studies (in a magnetic field along $\mathbf{c}$ Aoki
_et al._ (2021) or along an undetermined direction Ran  _et al._ (2020))
showed that superconductivity only appears above the field necessary to
suppress the magnetic order. This would suggest that in the present work the
measurement at 1.55 GPa is actually at a pressure slightly above $p_{c}$. For
$\mathbf{H}\parallel\mathbf{c}$ and close to $p_{c}$, a question is whether
superconductivity can develop in the correlated or polarized paramagnetic
regimes, or in both.

Interestingly, the electrical resistivity $\rho$ measured here with a current
$\mathbf{I}\parallel\mathbf{a}$ captures the physics driving the maxima in the
magnetic susceptibility for different directions of magnetic field. Fig. 2
shows the similar values of $T_{\Delta\rho}^{max}$ and
$T_{\chi(\mathbf{H}\parallel\mathbf{b})}^{max}$ for $p<p_{c}$, and of
$T_{\Delta\rho}^{max}$ and
$T_{\chi(\mathbf{H}\parallel\mathbf{a})}^{max}\simeq
T_{\chi(\mathbf{H}\parallel\mathbf{c})}^{max}$ for $p\gtrsim p_{c}$ Li  _et
al._ (2021). A transverse relationship between the electrical resistivity and
the magnetic susceptibility anisotropies may be the consequence of an
anisotropic Kondo hybridization between conduction and localized $f$ electrons
in UTe2. Rich information about the electronic interactions responsible for
the magnetic fluctuations in UTe2 in its normal non-superconducting phases may
be accessed via a careful investigation of the anisotropy of the electrical
resistivity, with different electrical-current directions (see [Eo  _et al._ ,
]), under different magnetic-field directions possibly combined with pressure.

We have seen that the application of pressure and high magnetic field on UTe2
leads to an extremely complex phase diagram with a complete reshuffling of the
magnetic anisotropy and strong associated effects on superconductivity. This
helps us understand the intricate relationship between superconductivity and
magnetism in this system. The clear signature of the destruction of magnetic
order with field, as well as the signature of several field induced
transitions inside the magnetically ordered phase, show that this order may be
of antiferromagnetic or SDW type. Different domains of stability and exclusion
of superconductivity are found under pressure and magnetic fields. Critical
magnetic fluctuations, possibly of ferromagnetic kind, associated with $H_{m}$
and $H_{m}^{*}$, may induce the large values of $A$ observed here, in relation
with the enhancement of the superconducting pairing mechanism. However,
superconductivity does not necessarily occur in all parts of the phase diagram
where $A$ is enhanced. Quite simple pictures can explain some parts of the
phase diagram, but understanding why superconductivity is destroyed below or
beyond a metamagnetic field, depending on the field direction, and in the
magnetically-ordered phase stabilized under pressure remains a theoretical
challenge. Experimentally, a full knowledge of the angle dependence of
superconductivity under pressure and very high magnetic field would help gain
a full understanding of superconductivity in UTe2. Our study is a step in this
direction.

## Acknowledgements

We thank J. Flouquet and J.-P. Brison for useful discussions. We acknowledge
financial support from the Cross-Disciplinary Program on Instrumentation and
Detection of CEA, the French Alternative Energies and Atomic Energy
Commission, KAKENHI (JP15H05882, JP15H05884, JP15K21732, JP16H04006,
JP15H05745, JP19H00646,JP19K03736) and GIMRT (19H0416, 19H0414), and the
French national research agency Programme Investissements d’Avenir under
Program No. ANR-11-IDEX-0002-02, Reference No. ANR-10-LABX-0037-NEXT, and the
collaborative research project FRESCO.

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