# Unconventional Superconductivity in UTe2

D. Aoki1, J.-P. Brison2, J. Flouquet2, K. Ishida3, G. Knebel2, Y. Tokunaga4,
Y. Yanase5,6 1IMR, Tohoku University, Oarai, Ibaraki, 311-1313, Japan  2Univ.
Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, F-38000 Grenoble, France
3Department of Physics, Kyoto University, Kyoto 606-8502, Japan  4ASRC, Japan
Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan  5Department of Physics,
Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
6Institute for Molecular Science, Okazaki 444-8585, Japan
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###### Abstract

The novel spin-triplet superconductor candidate UTe2 was discovered only
recently at the end of 2018 and attracted enormous attention. We review key
experimental and theoretical progress which has been achieved in different
laboratories. UTe2 is a heavy-fermion paramagnet, but right after the
discovery of superconductivity it has been expected to be close to a
ferromagnetic instability showing many similarities to the U-based
ferromagnetic superconductors, URhGe and UCoGe. The competition between
different types of magnetic interactions and the duality between the local and
itinerant character of the $5f$ Uranium electrons, as well as the shift of the
U valence appear as key parameters in the rich phase diagrams discovered
recently under extreme conditions like low temperature, high magnetic field,
and pressure. We discuss macroscopic and microscopic experiments at low
temperature to clarify the normal phase properties at ambient pressure for
field applied along the three axis of this orthorhombic structure. Special
attention will be given to the occurrence of a metamagnetic transition at
$H_{\rm m}=35$ T for a magnetic field applied along the hard magnetic axis
$b$. Adding external pressure leads to strong changes in the magnetic and
electronic properties with a direct feedback on superconductivity. Attention
will be given on the possible evolution of the Fermi surface as a function of
magnetic field and pressure.

Superconductivity in UTe2 is extremely rich exhibiting various unconventional
behaviors which will be highlighted. It shows an exceptionally huge
superconducting upper critical field with a re-entrant behavior under magnetic
field and the occurrence of multiple superconducting phases in the temperature
field, pressure phase diagram. There is evidence for spin-triplet pairing.
Experimental indications exist for chiral superconductivity and spontaneous
time reversal symmetry breaking in the superconducting state. The different
theoretical approaches will be described. Notably we discuss that UTe2 is a
possible example for the realization of a fascinating topological
superconductor. The recent discovery of superconductivity in UTe2 reemphasizes
that U-based heavy fermion compounds give unique examples to study and
understand the strong interplay between the normal and superconducting
properties in strongly correlated electron systems.

##  1 Introduction

Superconductivity in UTe2 has been discovered only in late 2018 and rapidly
confirmed [1, 2]. While UTe2 is paramagnetic already in the first report [1]
the possibility of spin-triplet superconductivity and the importance of
ferromagnetic fluctuations for the superconducting pairing has been
emphasized. It has been proposed that UTe2 belongs as paramagnetic end member
to the family of the ferromagnetic superconductors like URhGe and UCoGe.
However, rapidly it turned out that key differences in the crystal structure
and by consequence on the magnetic interactions and the electronic structure
make that the studies on UTe2 open new unique phenomena.

The electronic structure of UTe2 is very particular. In band structure
calculations within the local density approximation (LDA) a small gap of
$\Delta=130$ K occurs at the Fermi level. Thus, neglecting the electronic
correlations, UTe2 would be a small gap insulator [2]. Remarkably, the value
of $\Delta$ is comparable to the antiferromagnetic interactions (as determined
from the value of the Curie Weiss temperature above 100 K). As a consequence,
on cooling the convergence of these two mechanisms lead to a very different
scenario with a metallic ground state, the possibility of supplementary
ferromagnetic interactions at low temperatures interfering the high
temperature antiferromagnetic ones. From thermoelectric power, Hall effect and
from the link between the $\gamma$ coefficient of the specific heat and the
upper critical field it has been shown that UTe2 appears as a good metal with
roughly one charge carrier/U atom [3]. Modifying the electronic band structure
through a magnetic field induced Zeeman shift or damping the magnetic
interactions by external pressure will lead to a metamagnetic transition with
a change in the balance between ferromagnetic and antiferromagnetic
interactions, crossing through valence transition and Lifshitz transition [4].
All these ingredients will have a direct impact on the Cooper pairing with the
observation of the multiple superconducting phases.

The aim of this review is to present the status (August 2021) of research on
unconventional superconductivity in UTe2 from our point of view. It is a
rapidly emerging field, and we apologize if interesting experiments or
developments are incompletely covered. In sections 2 and 3, we present
respectively what is known on the crystalline and electronic structure of
UTe2. In section 4 we present the normal state properties of UTe2 with focus
on electronic transport and thermodynamics, as well as magnetic fluctuations
detected by nuclear spin resonance (NMR) and inelastic neutron scattering
experiments. In section 5 we review the superconducting properties of UTe2 and
discuss the possible symmetry and topology of unconventional superconducting
phases. Section 6 introduces the properties of UTe2 under high pressure and we
discuss the superconducting phase diagrams under pressure and the magnetic
properties. Finally in section 7 we present perspectives on the
superconductivity in UTe2 from a theoretical point of view.

##  2 Crystal structure of UTe2

UTe2 crystallizes in a body centered orthorhombic structure (see Fig. 1) with
the symmorphic space group $Immm$ (No. 71, $D_{2h}^{25}$). The lattice
parameters at ambient temperature are $a=4.159$ Å, $b=6.124$ Å, and $c=13.945$
Å, respectively, and no change in the crystal structure is found down to low
temperatures [5]. There are four formula units per unit cell, and the 4 U
atoms occupy $4i$ site. The 8 Te atoms are located at two different sites in
the unit cell; $4j$ and $4h$ sites with point symmetries $mm2$ and $m2m$.
Their respective sites are denoted Te(1) and Te(2). The U atoms are surrounded
by two Te(1) and four Te(2) atoms in form of a bi-trigonal prism. The Te(2)
atoms form chains along the $b$ axis with a distance of 3.05 Å, which is the
shortest distance between all atoms in the structure. The U atoms form a two-
leg ladder structure along the $a$ axis with the shortest distance between the
U atoms of about 3.78 Å along the rung in $c$ direction. It is significantly
smaller than distance of the leg of 4.16 Å along the $a$ axis. The distance
between two such ladder chains is 4.91 Å and and next neighboring chains are
displaced by $c/2$. In Fig. 1 the nearest neighbors of the U atoms are shown,
the arrows indicate the relative changes in the inter-atomic distances on
cooling [5, 6]. Remarkably, the distance between the two U atoms on the rung
shrinks by 60% more that the lattice parameter along the $c$ axis with
decreasing temperature [5]. Finally we note that the shortest U - U distance
is larger than the so-called Hill limit [7] $(\approx 3.5$Å). We would expect
that the U moments are localized and that at low temperature magnetic order
occur. In the simple picture, unconventional superconductivity is not expected
[8]. Although the lattice constant of $c$-axis is rather long compared to
those of $a$ and $b$-axes, the Brillouin zone is not flattened very much along
$c$-axis, because of the “body-centered” orthorhombic structure, suggesting
the formation of 3D Fermi surfaces rather than 2D Fermi surfaces. The shape of
Brillouin zone looks similar to that of the well-known ThCr2Si2-type
structure, but it is elongated along $a$-axis. We further want to emphasise
that the $[011]$ direction, corresponds to the direction tilted by 23.7 deg
from $b$ to $c$-axis, at which the field-reentrant superconductivity is
observed above the metamagnetic transition field $H_{\rm m}$ (see section
5.5.1 below) [9, 10]. It should be stressed that the inversion center of the
structure is not located at the U site, but exists at the middle of the U rung
in spite of the body-centered orthorhombic structure with the global inversion
symmetry. The situation is similar to the case for the recently discovered
heavy fermion superconductor CeRh2As2 with CaBe2Ge2-type structure (space
group: $P4/nmm$) [11], where the inversion center is not located at the Ce
atom with global inversion symmetry, leading to the broken inversion symmetry
locally.

Figure 1: (Left)Body-centered orthorhombic crystal structure of UTe2. (Right)
Zoom on the nearest neighbors of the U atoms, the arrows indicate the relative
changes in inter-atomic distances when the temperature is lowered. The main
effect of temperature on the structure is the reduction of the U - U distance
on the ladder rungs as indicated by the red arrows in the right figure [5, 6].

The substitution of Te by Se in UTe2 leads to the drastic change of the
crystal structure. At least up to U(Te0.8Se0.2)2, the space group $Immm$ is
retained, but further substituting with Se, the space group changes to $Pnma$,
where ferromagnetic order occurs at relatively high temperatures with the
semi-metallic electronic state. For example, U(Te0.36Se0.64)2 orders
ferromagnetically below $T_{\rm Curie}\sim 69\,{\rm K}$ [12]. $\beta$-USe2 is
a ferromagnet with $T_{\rm Curie}=14\,{\rm K}$, showing the semiconducting
behavior. We note that $Pnma$ is a subgroup of $Pmmn$ (No. 59, $D^{13}_{2h}$),
which is a subgroup of $Immm$.

##  3 Electronic structure of UTe2

###  3.1 Theoretical predictions

As we mentioned in the introduction, a naive LDA band calculation for UTe2
predicted an insulating state with a small gap of 13 meV [2]. This is clearly
incompatible with experiments showing the metallic conduction and
superconducting instability. The discrepancy indicates that the electron
correlation effects are crucially important in UTe2. A way to take into
account the electron correlations from first-principles is the GGA+U method,
where GGA means “generalized gradient approximation” and the Hubbard $U$ is
the on-site Coulomb repulsion, responsible for the development of strong
electronic correlations. The GGA+U calculation for the paramagnetic state
indeed revealed an insulator-metal transition [13]. As shown in Fig. 2, the
insulating gap appearing in the LDA calculations closes and conducting
carriers appear for $U>1$ eV. For a large $U$, the density of electron
carriers reaches $n_{{\rm e}\uparrow}\simeq 0.2$ per spin. This leads to
electron and hole carrier densities $n_{\rm e}=n_{\rm h}\simeq 0.4$ since UTe2
is a compensated metal. The total density $n_{\rm e}+n_{\rm h}\simeq 0.8$ is
close to unity and consistent with experiments [3]. These results have been
obtained by the relativistic full-potential linearized augmented plane wave +
local orbitals method within the GGA approximation. The effect of Hubbard
interaction $U$ is taken into account at the level of a static mean-field
approximation. Another DFT+U calculation, however within the LSDA
approximation and in a hypothetical ferromagnetic ground state [14], also
predicts the metallic state for similar values of $U$. A long-range
ferromagnetic order has not been observed in UTe2, however it is argued [14]
that locally, the electronic structure could be considered as ferromagnetic
due to slow long-range ferromagnetic fluctuations. Finally, the metal-
insulator transition due to strong electronic correlations has also been
confirmed by the density functional theory combined with the dynamical mean
field theory (DFT+DMFT) [15], however, results are presented in this last case
for much larger values of $U\simeq 7$ eV.

Figure 2: The GGA+U calculations for the Coulomb interaction dependence of (a)
the band gap at the Fermi level and (b) the electron carrier density per spin
[13]. Insulator-metal transition occurs around $U=1$ eV. Metallic states with
different topology of FSs are labeled by (i)-(iii). Figures taken from Ref.
[16].

Discussion of the mechanism of insulator-metal transition by the Coulomb
interaction, can be done with the results from the GGA approximation, giving
the density of states for various values of $U$ (Fig. 3). The density of
states at $U=0$ eV reveals an hybridization gap at the Fermi energy. Because
the density of states near the Fermi level mainly comes from the heavy
$f$-electron bands, the result of the LDA calculation is regarded as a Kondo
insulating state. Branching the Coulomb interaction shifts the hybridization
gap to the high energy side. Such an energy shift of the hybridization gap
should be prohibited in the single-orbital periodic Anderson model so as to
preserve the charge neutrality. However, all the $J=5/2$ multiplets contribute
to the low-energy electronic states in UTe2. Hence the multi-orbital nature
allows the energy shift, and the resulting insulator-metal transition.

Figure 3: The GGA+U calculations for the partial density of states for U-5f
orbitals of $j=5/2$, $j_{z}=\pm 1/2$, $\pm 3/2$, and $\pm 5/2$ near the Fermi
level [13]. Dashed line represents the total density of states. Figures taken
from Ref. [16].

Fermi surfaces appear above the insulator-metal transition. Figure 4 shows the
Fermi surfaces for various values of $U$. We see Lifshitz transitions; the
topology of Fermi surfaces changes with increasing $U$. We denote the regions
(i), (ii), and (iii) in Fig. 2 based on the different topology of the Fermi
surfaces in Figs. 4(b), 4(c), and 4(d).

Figure 4: (a) First Brillouin zone and symmetry points of body-centered
orthorhombic crystals. (b)-(d) Fermi surfaces of UTe2 by GGA+U calculations
for (b) $U=1.0$ eV [region (i)], (c) $U=1.1$ eV [region (ii)], and (d) $U=2.0$
eV [region (iii)] [13]. The electron sheet (cyan and red) and the hole sheet
(blue and yellow) are highlighted by colors. Figures taken from Ref. [16].

The Fermi surfaces in the large Coulomb interaction region [Fig. 4(d)]
resemble the results of the DFT+DMFT calculation which assumes a large $U$
[15, 17]. Thus, the rectangular Fermi surfaces are consistently obtained by
the DFT+$U$ and DFT+DMFT calculations, and they are similar to the Fermi
surfaces of ThTe2 without $f$-electrons [17, 18, 19]. Combining these results,
we suppose that the $f$-electrons are almost localized in the large Coulomb
interaction region. Indeed, the rectangular Fermi surfaces are formed by the
conduction electrons mainly consisting of U-6$d$ electrons and Te(2)-5$p$
electrons. Interestingly, they have a quasi-one-dimensional character, and the
conducting directions are orthogonal to each other. Thus, the perpendicular
quasi-one-dimensional Fermi surfaces of 6$d$ and 5$p$ electrons hybridize and
form rectangular Fermi surfaces in Fig. 4(d). The electronic state in the
large Coulomb interaction region is well understood in this way. However, such
electronic structure is not compatible with expected properties of UTe2. For
instance, the large specific heat [1, 2] indicates the presence of an
itinerant heavy $f$-electron band. A recent observation of moderate
anisotropic electrical conductivity [20] requires three-dimensional Fermi
surfaces. On the other hand, only light quasi-one-dimensional Fermi surfaces
appear in the large $U$ region.

The expected properties, namely, the heavy, three-dimensional, and itinerant
$f$-electron bands, are fulfilled in the intermediate Coulomb interaction
region. Low-energy electronic states appear around the X point in the
Brillouin zone, when we decrease the Coulomb interaction $U$. This heavy
electron band hybridizes with the conduction electron band, and then, the
ring-shaped Fermi surface of Fig. 4(c) is formed. This band structure is more
realistic for UTe2, although further verification is needed as other
calculations predict qualitatively different band structure [21] which is in
reasonable agreement with the soft X-ray ARPES experiment [22]. Comparison
with experiments is highly desirable. In this regard, an ARPES experiment [17]
reported low-energy electronic states consistent with Fig. 4(c). In addition
to the light rectangular Fermi surfaces, a signature of the heavy band around
the X point has been observed. Note that the X point in the conventional
notation is equivalent to the Z point in Ref. [17] (Fig. 6). Further ARPES
studies clarifying the sensitivity to the bulk and surface states are desired.
Quantum oscillation measurements are also awaited as the Fermi surfaces of
heavy-fermion systems, such as UPt3 [23], were precisely determined by them.

The strength of Coulomb interaction also affects the number of $f$-electrons
per U ion. Generally speaking, Coulomb interaction decreases the $f$-electron
number: $n_{f}=2$ for the localized $5f^{2}$ state, while $n_{f}=3$ for the
itinerant $5f^{3}$ state. Theoretical values are reported as $n_{f}\simeq 2.3$
[17], $n_{f}\simeq 2.5$ [16], $n_{f}\simeq 2.73$ [21]. As mentioned in the
next section, core level photo-emission experiments support an intermediate
valence state [24].

###  3.2 Experiment

The electronic structure of UTe2 has been studied by angle resolved
photoemission spectroscopy (ARPES) using different photon energies.
Experiments in Japan have been performed on the soft X-ray beam line BL23SU at
Spring 8 using photon energies between $h\nu=560-800$ eV [22, 24], and
experiments in the US using synchrotron radiation at the Advanced Light Source
(Berkeley) and an helium lamp light source (New York University) using photon
energies $h\nu=10-150$ eV [17]. The results obtained in these experiments are
very different and present, at first glance, contradicting views of the
electronic structure. This may be related to the different inelastic mean free
path of the photoelectrons which is a function of the kinetic energy of the
photon [25] and in general, low energy photoemission can be very surface
sensitive instead of probing the bulk. From the universal curve [25, 26] one
can expect that the soft X-ray experiments probe 15Å depth, while for the low
energy spectroscopy near 100 eV the probing depth is two times lower and thus
more surface sensitive.

Figure 5 shows the valence band spectrum obtained by the angle-integrated
photoelectron spectroscopy (AIPES) experiments performed at an energy of
$h\nu=800$ eV [22]. The sharp peak below $E_{\rm F}$ is due to the U 5$f$
state with an itinerant character. The long tail is indicative of an anomalous
admixture of the U 5$f$ and Te 5$p$ bands observed at a higher binding energy.
A similar peak is found for the partial U 5$f$ density of states obtained by
resonant photoelectron spectroscopy at the U $4d-5f$ absorption edge
($h\nu=736$ eV) which suggests that the U $5f$ states in UTe2 are itinerant in
nature, but with some hybridization with the Te 5$p$ states. The lower panel
in Fig. 5 indicates the theoretical U $5f$ and Te $5p$ partial density of
states. This calculation has been performed treating all the 5$f$ electrons as
itinerant (see Ref. [22]).

The core-level spectroscopy reveals that a mixed-valence state is realized and
the dominant contribution is the itinerant 5$f^{3}$ configuration, by
comparison to UB2 with an itinerant 5$f^{3}$ configuration, and to UPd3 with a
localized 5$f^{2}$ configuration [24]. This has been also concluded from the
Uranium L3 X-ray absorption near-edge spectroscopy [27]. These itinerant 5$f$
states are in agreement with the electronic structure calculations presented
in Refs. [22, 21].

Figure 5: AIPES spectrum measured at $h\nu=800\,{\rm eV}$ and the theoretical
partial DOS of U 5$f$ and Te 5$p$. Figure taken from Ref. [22].

Contrary to the core-level spectroscopy, the spectrum at the U $5d-5f$
resonance ($h\nu=98$ eV) shows no peak structure close to the Fermi energy,
but clearly enhanced intensities at $-0.7$ eV binding energy [17]. This
results is in good agreement with the first principle-based dynamical mean
field theory (DFT + DMFT) [17]. The spectra are interpreted to originate from
light Te bands and the U-$5f$ states occurs to be mainly localized, i.e. do
not contribute to the Fermi surface. The conclusion of the experiment is that
the the dominant configuration is U-5$f^{2}$.

The overall band structure obtained from the different ARPES experiments is
again very different. The soft X-ray experiments (at photon energies
$h\nu\approx 600$ eV) are well explained by the band structure calculation
based on the 5$f$-itinerant model. However, the fine structure near the Fermi
level was unresolved experimentally in the soft X-ray experiments, owing to
insufficient resolution compared to the bandwidth of the flat bands.

In difference, the high resolution ARPES experiments using a helium light
source [17] revealed two light quasi-one-dimensional bands at the Fermi level,
attributed to the U and Te chains. Figures 6(a)-(c) show two light Fermi
surfaces observed at $\Gamma$–$X$ and $\Gamma$–$Y$. These Fermi surfaces
coincide with the calculated Fermi surfaces for ThTe2. Another heavy pocket is
also detected at Z point, as shown in Fig. 6(d). This Fermi surface seems to
be inconsistent with the results from the soft X-ray ARPES experiments [21],
which are more bulk sensitive, than these ARPES measurements using a helium
light source ($hv=30-150$ eV). Hence, no clear conclusion on the electronic
structure on UTe2 emerges from the photoemission experiments performed up to
now.

Figure 6: Quasi-one-dimensional Fermi surfaces obtained by high resolution
ARPES experiments [17]. (a)ARPES measurements at the $(001)$ surface. (b)(c)
the band structure near the Fermi level along $k_{x}$ and $k_{y}$ directions.
(d) Three dimensional dispersion and a heavy Z-point electron pocket. Figure
taken from Ref. [17].

No quantum oscillations have been detected, neither by de Haas-van Alphen
(dHvA) nor by Shubnikov-de Haas (SdH) effect. Possible reasons why dHvA or SdH
oscillations are not detected are: 1) the sample quality is insufficient, 2)
the effective mass is large by comparison to the Fermi surface volume, 3) the
curvature factor of the Fermi surfaces are unfavorable. The residual
resistivity ratio ($\mbox{RRR}\sim 40$) of the best available crystals of UTe2
should be enough for detecting at least a part of the Fermi surfaces by dHvA
or SdH experiments. Compared to the very large initial slope of the upper
critical field, the moderately enhanced $\gamma$-value may point to the
existence of rather small pockets (moderate $\gamma$) with large effective
masses (large $H_{\rm c2}$). Hence, if the carrier number was small, and only
these small Fermi surfaces with heavy quasi-particles existed, they would be
very difficult to detect by quantum oscillations, as it is predicted in UBe13.
However, in UTe2, the carrier number indicated by Hall experiments [3] seems
not so small, revealing some volume of Fermi surfaces. In addition, the 2D
Fermi surfaces suggested by the low energy ARPES experiments and by the band
calculations should be easily detected by dHvA or SdH experiments, because of
the light effective mass of the associated carriers and the favourable
curvature of the Fermi surface.

Naturally, the large $H_{\rm c2}$ also impedes to detect the dHvA or SdH
signals in the conventional measurements using a superconducting magnet up to
$15\,{\rm T}$. Nevertheless, the higher field experiments using a resistive
magnet or a pulse field magnet also show no quantum oscillations up to now.
This failure of a detection of any quantum oscillations in UTe2 is one of the
deep puzzles of UTe2, calling for still improved sample quality. Progress in
the knowledge of the Fermi surface topology is a clear key challenge; a next
generation of experiments is highly demanded, both for ARPES and quantum
oscillations measurements.

##  4 Normal state properties

###  4.1 Basic properties of UTe2 and behavior for field $H\parallel a$, the
easy magnetization axis at $p=0$

The magnetic susceptibility $\chi$ of UTe2 is shown in Fig. 7 for the three
crystallographic axes $a$, $b$, $c$. For all directions the susceptibility
follows a Curie Weiss behavior $\chi=C/(T-\Theta)$ for temperatures above 150
K with respective Curie-Weiss temperatures $\Theta=-60$, -110, and -135 K
along the $a$, $b$, and $c$ axis [28, 10]. The negative value of $\Theta$ may
be, at first glance, indicative for a dominant antiferromagnetic exchange at
high temperatures. The corresponding effective magnetic moments obtained from
the fits above 150 K are $\mu_{\rm eff}=3.44\,\mu_{\rm B}$, $3.5\,\mu_{\rm
B}$, and $3.33\,\mu_{\rm B}$ for the $a$, $b$, and $c$ axis, respectively.
These values are very close to the free ion values 3.58 $\mu_{\rm B}$ for
$5f^{2}$ (U4+) or 3.62 $\mu_{\rm B}$ for $5f^{3}$ (U3+) states. However, the
crystal field splitting and the Kondo effect may have also influence on the
temperature dependence of the susceptibility. On cooling, $\chi_{a}$ growths
stronger than the Curie Weiss behavior below 100 K, starts to saturate around
20 K, but finally increases strongly in small fields below
$T_{a}^{\ast}\approx 12$ K down to the onset of superconductivity. We note
that on approaching the superconducting transition at $T_{\rm sc}\approx 1.6$
K, at low field no Fermi-liquid behavior $\chi_{a}=\chi_{0}+aT^{2}$ is
detected. A quantum critical scaling of $M_{a}/T$ versus $H/T^{1.5}$ at low
temperature (the subtraction of a linear term in $M_{a}(H)$) has been proposed
to indicate the proximity to a ferromagnetic quantum critical point [1].
However, this scaling rises two open questions namely the origin for the
subtraction of the additional term and the scaling exponents apply for
disordered metals near ferromagnetic quantum criticality [29]. Thus, in
principle it should not be applicable for UTe2, which shows unconventional
superconductivity and is in the clean limit. The increase of the $a$ axis
susceptibility below 10 K remains to be clarified.

Figure 7: Magnetization divided by magnetic field $m/H$ measured at 0.1 T for
the three crystallographic axes. The black dashed line indicates the Curie-
Weiss behavior for the $a$ axis. The gray dashed line is a fit to a power law
temperature dependence below 10 K. The inset shows the field dependence of the
magnetization measured at 1.8 K. (taken from Ref. [1])

In difference to the $a$ axis, the susceptibility $\chi_{b}$ follows the Curie
Weiss behavior down to 50 K, and it shows a broad maximum at $T_{\rm max}\sim
35$ K. Such a maximum is often observed in heavy-fermion paramagnetic systems
along the easy magnetization axis, see e.g. studies on the CeRu2Si2 family
[30, 31], as a signature of the development of antiferromagnetic correlations
and the formation of the coherent heavy-fermion state. As we will see below,
the energy scale of $T_{\rm max}$ corresponds to that of the field induced
metamagnetic transition for a field applied along the $b$ axis [32, 33]. At
300 K the $c$ axis susceptibility $\chi_{c}$ is smallest. On cooling, only a
smooth increase of susceptibility $\chi_{c}$ is observed which is slightly
stronger than the Curie-Weiss behavior below 150 K, and only for $T<25$ K the
$b$ axis is the magnetically hard axis with $\chi_{b}<\chi_{c}$. At 2 K, for
$H=7$ T, the magnetization $M_{a}$ along the $a$ axis reaches 0.5 $\mu_{\rm
B}$/U, while $M_{b}$ and $M_{c}$ get 0.09 and 0.15 $\mu_{\rm B}$/U (see inset
in Fig. 7). The $a$ axis, which corresponds to the ladder direction, is the
initially easy magnetization axis, and the $b$ axis, which is perpendicular to
the ladder rungs, is the hard magnetic axis. The anisotropy $M_{a}/M_{b}\sim
5.5$, is not very strongly Ising-type in UTe2. It can result from crystal
field effects and also by the interaction themselves governed by local
properties and the Fermi surface topology. High field magnetization
measurements along the $a$ axis indicate that the magnetization increases up
to 20 T and tends to saturate for $H>40$ T to a value of $M=1.1\mu_{\rm
B}/{\rm f.u.}$ in a fully polarized state (see inset of Fig. 9). It is
worthwhile to note that just above $T_{sc}$ the susceptibility
$\chi_{a}=\partial M_{a}/\partial H$ becomes lower than that of
$\chi_{b}=\partial M_{b}/\partial H$ at $H_{a-b}\approx 16$ T, i.e. roughly
the field when the upper critical field $H_{\rm c2}$ along $b$ shows a clear
field reinforcement.

Figure 8: Temperature dependence of the electrical resistivity of of UTe2 for
a current applied along different crystallographic axes. (taken from Ref.
[20])

The electrical resistivity in UTe2 is anisotropic and it depends strongly on
the current direction [20] as shown in Fig. 8. While for a current injected
along $a$ or $b$ the resistivity shows a shallow maximum around 60 K and
starts to decrease strongly below 50 K due to the formation of the coherent
Kondo-lattice, the resistivity with current direction along the $c$ axis
decreases from 300 K with a distinct minimum near 40 K and shows a pronounced
maximum near the characteristic temperature $T^{\ast}\approx 14$ K. The
derivative of the resistivity $d\rho/dT$ shows a maximum for the three current
directions near $T_{\rho}^{*}\approx 15$ K for $a$ and $b$ axis current and
near 7.5 K for $c$ axis current. Thus, the different location of
$T^{\ast}_{\rho}$ from the derivative of the resistivity marks clearly the
crossover behavior. The microscopic origin of this distinct different behavior
on current directions is not determined right now, but it may be related to
the characteristic Fermi surface topology [20]. The residual resistivity
$\rho_{0}$ varies at least by a factor five between the current direction
$j\parallel a$ axis and $j\parallel c$ axis. In addition the $A$ coefficient
of the Fermi liquid resistivity $\rho(T)=\rho_{0}+AT^{2}$, observed below 5 K,
depends also from the current direction. For current along $a$, $A<1$
$\mu\Omega$ cm K-2 in all reports, for current along $b$ $A\approx 1.5$
$\mu\Omega$ cm K-2, while it is strongly enhanced for current along $c$ axis
and $A\approx 7$ $\mu\Omega$ cm K-2 is observed. As roughly $A/\rho_{0}$
appears current invariant, it suggests that the $(a,b)$ plane is the easy
conducting plane.

Figure 9: Field dependence of the $A$ coefficient of the resistivity for
$H\parallel a$ axis, taken from Refs. [2, 32]. The inset shows the field
dependence of the magnetization at $T=1.4$ K.  Figure 10: Temperature
dependence of $dM/dT$ for different fields along the $a$ axis up to 7 T.

In a crude single band scheme, the field dependence of the $A$ coefficient
give some hints to the field dependence of the effective mass $m^{\ast}$. In a
first approach the Kadowaki-Woods ratio is fulfilled in UTe2 (The ratio
$A/\gamma^{2}\approx 10^{-5}\,\mu\Omega\,{\rm cm}({\rm mJ}\,{\rm
mol^{-1}K}^{-2})^{-2}$ has an universal value in many different heavy fermion
materials) [34, 35]. Of course, more detailed thermodynamic experiments are
needed in future to obtain more profound knowledge of the field dependence of
the effective mass $m^{\ast}$ (see Fig. 9). In first experiments, the $A$
coefficient with the current along the $a$ axis and a magnetic field
$H\parallel a$ show a rapid decrease of $A$ for $H$ up to 6 T followed by an
almost constant value for $H>6$ T [2]. Measurements using pulsed magnetic
fields show that $A$ is continuously decreasing up to the highest field of 60
T [32, 10]. At a field of $H_{a}\sim 6$ T a field-induced Lifshitz transition
has been clearly observed in thermoelectric power experiments at low
temperature [3]. At the field of the Lifshitz transition a small anomaly
occurs in the field-derivative $dM(T)/dH$ [33]. An important point is that at
this field the field-induced magnetization reaches a critical value of 0.4
$\mu_{\rm B}$ [3], which is very close to the value of the magnetization where
a first order metamagnetic transition along $b$ with a large jump of the
magnetization occurs, as discussed below in section 4.2. The thermoelectric
power experiments show further anomalies near 10.5 and 21 T, which may hint to
further Fermi surface changes, but up to now the details of the Fermi surface
topology are not determined.

We note that for $H\parallel a$, a clear change of regime occurs near
$T^{\ast}\approx 12-15$ K, below which the susceptibility starts increasing
strongly to lower temperatures at low magnetic fields. Similar to the
resistivity, the characteristic temperature $T^{\ast}$ shows up very clearly
as a pronounced minimum in $\frac{dM_{a}}{dT}$ for $H\to 0$ near 14 K;
increasing the magnetic field leads to an increase of $T^{*}$ which can be
followed by a clear minimum of $\frac{\partial M_{a}}{\partial T}$ as a
function of $H$, $T^{\ast}$ reaches 20 K at $H=14$ T (see Fig. 10 and Ref.
[36]). In difference to the $a$ axis susceptibility, the temperature
dependence of the electronic specific heat $(C_{\rm el}\propto\gamma T)$ as
well as the that of the electrical resistivity follows a Fermi-liquid behavior
($\rho\propto T^{2}$) above the superconducting transition. Figure 11(b) shows
the electronic specific heat divided by temperature $C_{\rm el}/T$ as a
function of temperature. The superconducting anomaly at $T_{\rm sc}=1.6$ K is
very sharp. The Sommerfeld coefficient $\gamma\approx 0.12$ J$\cdot$mol-1K-2
just above $T_{\rm sc}$ indicates the moderate heavy fermion behavior. In the
electronic contribution of the specific heat $C_{\rm el}/T$ a broad maximum
occurs at $T^{\ast}\approx T^{\ast}_{a}\approx 12$ K. Under magnetic field
along the $a$ axis this maximum increases with field and can be located at 15
K at 9 T [36].

Figure 11: Electronic part of the thermal expansion along the $c$ axis (upper
panel), the electronic contribution of the specific heat $C_{\rm el}/T$
(middle panel), and the derivatived $d\rho_{a}/dT$ as function of temperature.
The upper inset in (a) shows the temperature dependence of the electronic
Grüneisen parameter $\Gamma_{\rm el}$. The lower inset indicates the
electronic entropy as function of temperature (after Ref. [36]).  Figure 12:
Linear thermal expansion coefficient as function of temperature for the three
orthorhombic axes and the volume. (taken from Ref. [37]) Figure 13:
Temperature dependence of the longitudinal thermoelectric power $S$ at $H=0$
and 9 T along the $a$ axis. (taken from Reference [3])

A clear signature of $T^{\ast}$ is also found in the temperature dependence of
the thermal expansion
$\alpha_{i}=\frac{1}{\ell_{i}}\frac{\partial(\Delta\ell_{i}/\ell_{i})}{\partial
T}$ by a minimum for $\alpha_{b}$ and $\alpha_{c}$ and a change of sign for
$\alpha_{a}$, as shown in Fig. 12 [37, 36]. As the values for $\alpha_{b}$ and
$\alpha_{c}$ are much larger than that for $\alpha_{a}$, the volume thermal
expansion $\alpha_{\rm V}$ is negative below 30 K and shows a minimum at
$T^{\ast}$. Knowing the value of the bulk modulus $B=57$ GPa [38], the
corresponding Grüneisen parameter $\Gamma_{el}=B\frac{\partial
T^{\ast}}{\partial p}=B\frac{\alpha_{\rm el}}{C_{\rm e}l}$ extrapolates to
$\Gamma_{\rm el}(T=0)=-30$ while at $T^{\ast}\approx 12$ K, $\Gamma_{\rm
e}\approx-17$ has only half of the extrapolation to $T\to 0$. Just below
$T^{\ast}$, the free energy cannot be reduced to an unique variation of
$T/T^{\ast}$ which would have lead to a constant Grüneisen parameter:
correlations play a major role. Taking $\Gamma_{\rm el}(T=0)$, the decrease of
$T^{\ast}$ with pressure is estimated to be $-6\,{\rm K/GPa}$, which is close
to the value detected by pressure experiments, $-7.8\,{\rm K/GPa}$ [39],
$-5.8\,{\rm K/GPa}$ [40]. Other signatures of $T^{\ast}$ are also a maximum of
the electrical resistivity with current along the $c$ axis [20] and in
thermoelectric power measurements with the thermal gradient along the $a$ axis
which shows a minimum (see Fig. 13) [3]. Other indications of the
characteristic temperature $T^{\ast}$ from microscopic measurements as
discussed later (see section 4.3) are the strong enhancement of the spin-spin
relaxation rate $1/T_{2}$ in the NMR experiment [41] and the saturation of the
antiferromagnetic fluctuations detected by the inelastic neutron scattering
experiment [42]. In agreement with the field-increase with $H\parallel a$ of
$T^{\ast}$ in $\frac{\partial M_{a}}{\partial T}$, a similar field dependence
is observed for the extreme in $C_{\rm e}/T$ and in $\alpha_{c}$.
Interestingly, the integrated entropy reaches $R\ln 2$ at $T\sim 50$ K
indicating a doublet ground state; with the hypothesis of a major U3+
configuration (i.e. $5f^{3}$), a doublet can be expected in agreement with the
measured anisotropy of the magnetization $M_{a}/M_{b}$ measured at 2 K [21].

The increase of the susceptibility $\chi_{a}$ on cooling below 50 K and the
increase of the crossover temperature $T^{\ast}$ for field applied along the
$a$ axis favor the picture that in addition to the antiferromagnetic
fluctuations detected by inelastic neutron scattering [43, 42] low energy
ferromagnetic fluctuations develop at low temperatures (see discussion on NMR
and $\mu$SR data in section 4.3) [41, 44]. At first glance an attempt may be
to try to derive a Fermi-liquid parameter $F_{0}^{a}$ for $H\parallel a$, as
it was done for liquid 3He [45]. However, a first difficulty is that
$\chi_{a}$ at lowest temperatures does not follow a Fermi-liquid behavior
above $T_{\rm sc}$ as discussed in section 4.1; a second particularity is
that, contrary to 3He, where the moment of the quasiparticle is known, the
corresponding quasiparticle may carry a moment very different than the one
Bohr magneton of the free electronic carrier. Thus the value of the Wilson
Ratio $R_{\rm W}\propto\chi/\gamma$ as well as the Korringa ratio ${\cal
K}\propto 1/(K^{2}T_{1}T)$, where $K$ is the Knight shift of the NMR resonance
and $1/T_{1}$ the spin-lattice relaxation rate, which would indicate a nearly
ferromagnetic state in UTe2 (see Ref. [36]), can be misleading in the
derivation of $F_{0}^{a}$ taking into account the dual localized and itinerant
character of the $5f$ electrons and the complexity of the multi band structure
[46, 47].

An open question is the origin of the strong increase of the susceptibility
$\chi_{a}$ below $T^{\ast}$. Using simple arguments on an extra contribution
of $\chi_{a}$ from a Fermi liquid estimate (which may be reached above 6 T, or
on the necessity to respect the entropy balance at $T_{\rm sc}$, a simple
amount of free paramagnetic Ising centers near 1% is sufficient to give a bath
leading on cooling to an apparent residual $\gamma_{\rm res}$ value in the
superconducting state below $T_{\rm sc}$. This extra contribution has been
neglected for the discussion of the superconducting properties [48]. It is
regarded as totally separated from the superconducting condensate. The
proximity to valence, magnetic and electronic instabilities will favor the
creation of defects strongly coupled to the lattice properties.

Figure 14: Field dependence of the characteristic temperature $T^{\ast}$
normalized to its zero field value for $H||a$. $T^{\ast}$ is obtained from the
maximum of $d\rho_{a}/dT$, the maximum of the electronic specific heat $C_{\rm
el}/T$, from the minimum of $dM_{a}/dT$. We also show the field dependence of
$[A(H)/A(0)]^{-0.5}$, which is roughly proportional to the field dependence of
$1/m^{\ast}$ (see also Ref. [36]).

Figure 14 shows the relative field dependence of the characteristic
temperature $T^{\ast}$ for a field along the easy magnetization axis $a$
defined either by the temperature derivative of the $a$ axis resistivity (data
from Ref. [32]), the maximum of the electronic specific heat $C_{\rm e}/T$,
the minimum of the temperature derivative of the magnetic susceptibility
$dM_{a}/dT$, or the inverse of the square root of the normalized $A$
coefficient of the resistivity. At least $T^{\ast}$ derived from $dM_{a}/dT$
indicates a change of the regime for $H\sim 6$ T; it is less clear from
$T^{\ast}\propto 1/\sqrt{A}$ scaling. We note that within the accuracy of the
determination of the $A$ coefficient from the pulsed field experiments, no
clear field saturation of $A$ occurs, despite the saturation of the
magnetization above 30 T. The driving force which may control $T^{\ast}$ may
be the result of the entrance in a coherent motion between the itinerant
quasiparticles and the magnetic fluctuations originating from the local $5f$
uranium centers and not a characteristic energy of a given magnetic
fluctuation. $T^{\ast}$ emerges clearly in the temperature dependence of the
detected incommensurate antiferromagnetic fluctuations [42].

Figure 15: Temperature derivative of the magnetization versus temperature
measured at 1000 Oe for different angles from the $a$ axis to the $b$ axis in
steps of 10 deg.

As we will see from the concomitant analysis of the upper critical field
$H_{\rm c2}$ and the lower critical field $H_{c1}$ one may suspect that low
energy ferromagnetic fluctuations play a major role in the superconducting
pairing at low pressure with an initial strong field decrease for $H\parallel
a$. In a simple classical case, a good way to detect the field dependence of
the pairing is to go back to the temperature derivative of the magnetization,
as it is discussed in URhGe [49]. Following the Maxwell relations, $(\partial
M/\partial T)$ at constant field is linked to the field derivative of the
entropy as $(\frac{\partial M}{\partial T})_{H}=(\frac{\partial S}{\partial
H})_{T}$. If the magnetization $M$ has a Fermi liquid $T^{2}$ dependence, we
can derive from the temperature dependence of the magnetization the field
derivative of the $\gamma$ value of the specific heat, as $\frac{\partial
M}{\partial T}=T\frac{\partial\gamma}{\partial H}$. Figure 15 shows the
temperature dependence of $\partial M/\partial T$ measured at 1000 Oe for
different angles from the $a$ axis to the $b$ axis. For $H\parallel a$
$\partial M/\partial T$ is negative and it shows a pronounced minimum near
$T^{\ast}\approx 15$ K. The position of the minimum is unchanged with
increasing angle towards the $b$ axis. It gets lower in intensity and near 70
degree from $a$, it has vanished. For lower temperatures a negative maximum
occurs on cooling for $H\parallel a$ and the absolute value of $\partial
M/\partial T$ is strongly increasing and is a signature of the complexity of
the magnetic response. For $H\parallel b$, $\partial M/\partial T$ changes
sign from negative to positive near $T_{\chi}^{\rm max}\approx 35$ K and a
positive maximum occurs near 15 K. The sign change at lowest temperature may
indicate some paramagnetic impurities.

Thus, the magnetization measurements give already a first hint on the field
dependence of the superconducting pairing interaction, as in a simple picture
$\gamma(H)\propto m^{\ast}(H)\propto\lambda(H)$. It is obvious that the field-
decrease of $\partial S/\partial H$ occurs for field along the $a$ axis and a
field-increase of $\partial S/\partial H$ for $H\parallel b$ for $H\parallel
a$ the magnetic field will lower the pairing while for $H\parallel b$ an
increase of the pairing is expected. Despite the complexity of the $\partial
M/\partial T$ response for $H\parallel a$, a rough estimation of
$\partial\gamma/H$ is in agreement with the possible strong collapse of the
pairing, as will be discussed later.

###  4.2 $H\parallel b$: Metamagnetism and $T_{\rm max}$

The remarkable feature in the normal state for a magnetic field applied along
the $b$ axis is the occurrence of a metamagnetic transition at $H_{\rm
m}\approx 35$ T. Its link with a ferromagnetic instability is not obvious and
it remains an intriguing phenomena. Usually, one would expect that
metamagnetism in a nearly ferromagnetic system will occur along the easy
magnetic axis, as e.g. observed in UCoAl [50] or in paramagnetic state above
the critical pressure in UGe2 [51, 52]. In a crude model scheme of density of
states the pronounced maximum in $C_{\rm el}/T$ at $T^{\ast}$ is a mark of the
minima of the density of states at $E_{\rm F}$, which seems in agreement with
electronic structure calculations [17] and also with the observation of a
maximum in the temperature dependence of the susceptibility $\chi_{b}$ along
the $b$ axis at $T_{\rm max}$.

Figure 16: Magnetization versus magnetic field applied along the $b$ direction
at different temperatures. (taken from Reference [33])

Below 8 K field dependent magnetization measurements clearly show a first
order metamagnetic transition at $H_{\rm m}\approx 35$ T at
$M_{b}^{\ast}\approx 0.4\mu_{\rm B}$ with a huge jump of $\Delta
M_{b}^{\ast}\approx 0.5\mu_{\rm B}$ and a hysteresis between field up and down
sweeps, which increases to low temperatures [33, 9] (see Fig. 16).
Interestingly, the slope $\frac{\partial M}{\partial H}$ is almost the same
below and above $H_{\rm m}$ (see Fig. 16) as if the metamagnetic instability
corresponds mainly to a jump of the local $5f$ magnetization. Above a critical
end point $T_{\rm CEP}\approx 6-8$ K, where the hysteresis of the transition
vanishes, the first order transition changes into a crossover and the
magnetization $M_{b}$ shows a marked inflection point as function of field
along the $b$ axis. Magnetoresistivity measurements for $H\parallel b$ show
that this jump of the magnetization is accompanied by a huge jump of the
residual resistivity $\rho_{0}$ at $H_{\rm m}$ by a factor of 4, as shown in
Fig. 17. Increasing temperature above 7 K the first order nature gets lost and
the maximum in the magnetoresistivity and the Hall effect indicates a
crossover which seems to be connected to the maximum in the magnetic
susceptibility $\chi_{b}$ [32, 53].

Figure 17: Magnetoresistivity of UTe2 for magnetic field applied along the $b$
axis. (taken from Reference [32]) Figure 18: Magnetic field dependence of
$\gamma_{\rm N}$ normalized to $\gamma_{\rm N}(H=0)$ as function of field
normalized to $H_{\rm m}$. The red squares are from specific heat measurements
in pulsed magnetic field [54], blue circles from magnetization measurements
using the Maxwell relation [33], and green triangles from the $A$ coefficient
of the resistivity [32]. The inset gives a zoom on the field range near
$H_{\rm m}$. (Figure taken from Reference [54])

In addition, the $A$ coefficient of the resistivity increases strongly on
approaching $H_{\rm m}$ and has a strong maximum at $H_{\rm m}$ [32]. A rather
similar enhancement is observed in $\gamma(H)$ derived from the temperature
dependence of the magnetization $M_{b}(T)$ by using the Maxwell relation [33],
and also by direct measurements of the specific heat at high magnetic fields
[54] as shown in Fig. 18. From this last measurements, just below $H_{\rm m}$
we find $\frac{\gamma(H_{\rm m}-\varepsilon)}{\gamma_{0}}\sim 1.8$ while just
above $H_{\rm m}$ the ratio $\frac{\gamma(H_{\rm
m}+\varepsilon)}{\gamma_{0}}\sim 1.6$. Thus there may be a discontinuity of
$\gamma_{\rm n}(H)$ at the first order metamagnetic transition. Recently,
taking into account the magnetocaloric effect and the Clausius-Clapeyron
equation, a discontinuity of the Sommerfeld coefficient $\gamma$ through
$H_{\rm m}$ has been established [55] (see also Fig. 27 in section 4.5). It is
interesting to note that an almost similar field dependence of the $A$
coefficient has been reported for an angle of 30 deg from the $b$ to the $c$
axis, where the field enhanced superconductivity occurs only above the
metamagnetic transition [9, 10]. Again, bthe recent magnetocaloric
measurements find a jump of $\gamma$ on crossing $H_{\rm m}$ [56]. A remaining
open question is to analyse the weight of nonuniform and time dependent
magnetic fluctuations during the pulse which can be induced on crossing
$H_{\rm m}$ in pulse magnetic fields [57].

Figure 19: Field variation for $H||b$ of the characteristic temperature
$T^{\ast}$ normalized to its zero field value. $T^{\ast}$ is obtained from the
maximum of $d\rho_{a}/dT$. We also show the field dependence of
$[A(H)/A(0)]^{-0.5}$ [32], and the normalized $\gamma$ value
$[\gamma(h)/\gamma(0)]^{-1}$ [54] which is roughly proportional to the field
dependence of $1/m^{\ast}$. The inset compares the field dependence of
$T^{\ast}$ and $T_{\rm max}$ taken from Ref. [53].

Following the previous considerations for $H\parallel a$, as the derivative of
the resistivity resembles the specific heat anomaly around a magnetic phase
transition [58] the relative field dependence of $T^{\ast}_{\rho}$ for
$H\parallel b$ is reported on Fig. 19 to track the field variation of the
crossover regime. We also added the relative field dependence of
$T^{\ast}_{\gamma}\sim 1/\gamma$, $T^{\ast}_{A}\sim 1/\sqrt{A}$ normalized to
their zero field values. The inset in Fig. 19 compares the field dependence of
the crossover temperatures $T^{\ast}$ and $T_{\max}$. Clearly,
$T^{\ast}_{\rho}$ and $T_{\max}$ end at the critical end point of the first
order transition $T_{\rm CEP}\approx 7$ K at 34.5 T, while $T^{\ast}_{\gamma}$
and $T^{\ast}_{A}$ show deep minima. A basic question is why wsuch an
enhancement of the effective mass (reflected in the increase of
$\gamma\propto\sqrt{A}$) on approaching the metamagnetic field can occur at a
first order phase transition. In the framework of ferromagnetic fluctuations
an enhancement of $\gamma$ on both sides of $H_{\rm m}$ [56] has been
proposed. However, the microscopic origin of the metamagnetism remains an open
question.

Interestingly, we calculate the Grüneisen parameter of $H_{\rm m}$,
$\Gamma_{H_{\rm m}}=-\frac{\partial\ln H_{\rm m}}{\partial\ln V}=-21$ taking
the observed pressure decrease of $H_{\rm m}$, which is about - 15 T/GPa [59].
It is quite near the values of $\Gamma_{\rm el}(T^{\ast})$ and $\Gamma_{\rm
el}(T=0)$. Thus the collapse of $T^{\ast}$ and $H_{\rm m}$ will occur in the
same pressure range, as shown experimentally [39, 59] (see section 6 below).
Furthermore, the comparable values of the different Grüneisen parameters
$\Gamma$ stress that the driving phenomena must be the same up to $H_{\rm m}$
despite possible changes in the magnetic coupling. Knowing the pressure
variation of $H_{\rm m}$ and the jump of the magnetization at $H_{\rm m}$, a
relative volume shrinking near $10^{-3}$ is associated to the first order
transition at $H_{\rm m}$. This shrinking is 30 times lower than the volume
contraction going from $p=0$ to the critical pressure $p_{\rm c}$ [38].

It is obvious that under a magnetic field the Zeeman splitting of the heavy
bands drives an eletronic instability for a critical value of the
magnetization $M^{\ast}_{b}\sim 0.4\mu_{\rm B}$ at $H_{\rm m}$. At the
opposite to the behavior of $\gamma$ and the $A$ coefficient on crossing the
metamagnetic transition, the thermoelectric power changes steplike at $H_{\rm
m}$ and a sign change from negative to positive occurs at low temperature
[53]. This is a clear indication that the majority heat carriers change at the
metamagnetic transition and gives a strong hint to a Fermi surface
instability. Additional support for this is given by Hall effect measurements
which also indicate a decrease of the charge carrier number by a factor of
nine, which presents a clear hint to a Fermi surface instability occurring at
$H_{\rm m}$ [53].

###  4.3 Spin dynamics probed by nuclear magnetic resonance

The nature of the spin fluctuations in UTe2 has been investigated by means of
125Te-NMR [41]. 125Te nuclei ($I=1/2$) carries no quadrupole moment and has a
gyromagnetic ratio $\gamma_{\rm N}/2\pi=13.45$ MHz/T, so that the nuclear
relaxations occur purely in magnetic origin. Figure 20 shows the temperature
dependence of $(1/T_{1}T)_{\alpha}$ measured in a single crystal for fields
applied along all three crystal axis directions ($\alpha$= $a$, $b$ and $c$).
$(1/T_{1}T)_{b,c}$ exhibit a strong temperature dependence in contrast with
flat behavior for $(1/T_{1}T)_{a}$. With decreasing temperature,
$(1/T_{1}T)_{b,c}$ increase rapidly and tend to saturate below 10 K. There is
no obvious field dependence below $5$ T in low fields.

In general, $1/T_{1}T$ measured in a field along the $\alpha$ direction is
associated with the imaginary part of the dynamic susceptibility
$\chi^{{}^{\prime\prime}}_{\beta,\gamma}({\bm{q}},\omega_{\rm n})$ along the
$\beta$ and $\gamma$ directions perpendicular to $\alpha$ [60, 61]. Hence the
directional dynamic susceptibility components for each orthorhombic crystal
axis,

$R_{\alpha}=\frac{\gamma_{\rm n}^{2}k_{\rm
B}}{2}\sum_{{\bm{q}}}|A^{\alpha}_{\rm
hf}|^{2}\frac{\chi^{{}^{\prime\prime}}_{\alpha}({\bm{q}},\omega_{\rm
n})}{\omega_{\rm n}}$ (1)

can be evaluated using the relations $(1/T_{1}T)_{a}=R_{b}+R_{c}$,
$(1/T_{1}T)_{b}=R_{a}+R_{c}$ and $(1/T_{1}T)_{c}=R_{a}+R_{b}$, where
$\omega_{n}$ is the NMR resonance frequency and $A_{\rm hf}^{\alpha}$ the
hyperfine coupling constant. The results are shown in Fig. 21. Since there is
no $(1/T_{1}T)_{a}$ data below 20 K (due to the divergence of $1/T_{2}$ as
discussed later), $R_{\alpha}$ was not estimated in that temperature region.
Figure 21 demonstrates that UTe2 exhibits a strong anisotropy for the
dynamical spin susceptibilities, i.e., $R_{a}\gg R_{c}>R_{b}$ above 20 K,
which is, however, moderate, compared to the case for UCoGe [61]. Note that
the static ($\chi_{\alpha}$) and dynamical spin susceptibilities
($R_{\alpha}$) often possess contrasting anisotropies in $f$-electron systems
[62, 63, 64, 65]. In the present case, however, nearly the same anisotropy has
been found for them.

Figure 20: The temperature dependence of $(1/T_{1}T)_{\alpha}$ for fields
applied along the three crystalline axes [41]. Figure 21: Orientation-resolved
dynamic susceptibility $R_{\alpha}$ (see text) along the three crystalline
axis directions in UTe2. The inset shows the temperature dependence of
$1/T_{2}$ for fields applied along the the $a$\- axis [41].

The inset of Fig. 21 shows the temperature dependence of the spin-spin
relaxation rate $1/T_{2}$ at 1.87 and 3.26 T. $1/T_{2}$ starts to increase
upon cooling below about 40 K, and diverges near 20 K. The $1/T_{2}$ observed
here reflects the magnitude of the longitudinal component (parallel to the
applied field) of slow spin fluctuations near zero frequency ($\omega\sim 0$).
That is, $1/T_{2}\propto G_{\|}(0)$, where
$G_{\alpha}(\omega)=\int_{-\infty}^{\infty}\left<h_{\alpha}(t)h_{\alpha}(0)\right>\exp(i\omega
t)dt$ is the spectral density of the fluctuating hyperfine field [66, 67]. In
general, the development of such a low frequency mode of fluctuations implies
the onset of static order along the $a$-axis. However, neither specific heat
nor other bulk measurements have detected any signature of a phase transition
around 10 - 20 K [1, 2, 28], but a broad anomaly in specific heat at
$T^{\ast}\approx 12$ K. Clearly, there is a crossover temperature for magnetic
fluctuations near 20 K for $H\parallel a$, below which the time scale of the
longitudinal fluctuations along the $a$ axis become extremely slow, of the
order of NMR frequencies ($\sim$MHz), i.e. very slow by comparison to the
antiferromagnetic correlations detected in neutron scattering [42] (see
section 4.4).

Slow spin fluctuations has been also detected by $\mu$SR at zero external
field. However, These spin fluctuations develop only below 5 K [44]. The
discrepancy in temperature between NMR (performed in fields $H>1.8$ T) and
$\mu$SR ($H=0$ T) suggests that there exists strong field-dependence of the
slow fluctuations at low temperatures. The $\mu$SR experiment also
demonstrates the coexistence of the slow fluctuations with superconductivity
below $T_{\rm SC}$.

So far it has not been concluded from the NMR data whether the observed slow
fluctuations are ferromagnetic in character. The $\mu$SR experiment suggests
that the temperature dependence of the muon relaxations rate follows to the
power-law equation $\propto T^{-4/3}$ in an intermediate temperature range of
$0.4<T<4.9$ K [44]; the behavior is expected from the self-consistent
renormalization (SCR) theory near a ferromagnetic quantum critical point in a
three-dimensional metal [68]. However, the deviation from this power law at
the lowest temperatures has still to be clarified; the occurrence of
ferromagnetic fluctuations in UTe2 is still puzzling.

###  4.4 Spin dynamics probed by neutron scattering

As inelastic neutron scattering is wave vector resolved it allows to
characterize fully the magnetic excitations. A first inelastic neutron
scattering experiment has been performed at the Cold Neutron Chopper
Spectrometer at Oak Ridge National Laboratory on a large assembly of 61
oriented single crystals to obtain a total mass of 700 mg, but with the
constraint of a mosaicity of about 15 degrees of the full assembly [43].
Surprisingly, it led to the discovery of antiferromagnetic fluctuations at the
incommensurate wavevectors $\mathbf{k_{1}}=(0,0.57,0)$ and
$\mathbf{k_{2}}=(0,0.43,0)$ (in reciprocal lattice units) which extends at
least up to 2.6 meV as shown in Fig. 22. The obtained incommensurate wave
vectors are very close to the commensurate antiferromagnetic vector
$(0,0.5,0)$. The emergence of these antiferromagnetic fluctuations is
explained by electronic structure calculations and it has been concluded that
the RKKY interaction between the 5$f$ moments drive the antiferromagnetic
fluctuations [43].

Figure 22: (a) Energy dependence of the magnetic excitations measured along
the $[0,K,0]$ direction revealing two excitations at
$\mathbf{k_{1}}=(0,0.57,0)$ and $\mathbf{k_{2}}=(0,1.57,0)$ integrated between
$-0.5<L<0.5$. Panels (b) and (c) show 1D cuts along the $K$ direction at
different energies $E$ throug the vectors $\mathbf{k_{1}}$ and
$\mathbf{k_{2}}$. (Figure taken from Ref. [43].)  Figure 23: Energy scans
measured at the temperature $T=2.1$ K and the momentum transfers
$\mathbf{Q_{1}}=(0,1.43,0)$ characteristic of antiferromagnetic fuctuations
(red squares). The other $\mathbf{Q}$ vectors are characteristic for
ferromagnetic fluctuations, but no remarkable signal compared to the
background (taken at $\mathbf{Q_{0}}=(0,0.16,0.19$) or the signal taken at
$T=60$ K has been observed. For $\mathbf{Q_{1}}$ the line is a fit to the data
by a quasi-elastic Lorentzian shape. (Figure taken from Ref. [42].)

Recently, neutron scattering experiments using a one single crystal confirmed
the incommensurate magnetic fluctuations. This experiment has been performed
on the triple axis spectrometer Thales at the Institute Laue Langevin with
energy transfers in the range $0.6<E<7.5$ meV using a crystal with a total
mass of 241 mg with a mosaicity lower than 2 degrees [42]. Figure 23
represents energy scans at $T=2.1$ K at momentum transfers
$\mathbf{Q_{1}}=(0,1.43,0)$ which corresponds to the incommensurate wave
vector $\mathbf{k_{1}}$ ($\mathbf{Q_{1}}=\tau-\mathbf{k_{1}}$, where
$\tau=(0,2,0)$ is a nuclear Bragg peak) and $\mathbf{Q}=(0,2.1,0)$,
$\mathbf{Q}=(0,1.07,1.07)$, and $\mathbf{Q}=(0,1.07,3.07)$ which are
characteristic for ferromagnetic fluctuations. Obviously, within the
experimental resolution, the intensities at the ferromagnetic wave vectors are
very close to the background level in this energy window.

Figure 24: Temperature dependence, in a double logarithmic presentation, of
the relaxation rate $\Gamma(\mathbf{k_{1}})$ (blue) and of the real part of
the static susceptibility at $\mathbf{k_{1}}$ (red). (Figure taken from Ref.
[42].)

The energy scans at $\mathbf{Q_{1}}=(0,1.43,0)$ have been performed up to 60
K. Figure 24 shows the temperature dependence of the relaxation rate of the
incommensurate fluctuations at $\mathbf{k_{1}}$ and of the static
susceptibility $\chi^{\prime}$ at $\mathbf{k_{1}}$. Both quantities point out
again a characteristic temperature of 15 K, close to the value of $T^{\ast}$
detected in thermodynamic, electric transport and NMR measurements. With
increasing temperature $\Gamma(\mathbf{k_{1}})$ increases strongly; an open
question is whether antiferromagnetic correlations persists above 100 K as
suggested by the Curie-Weiss temperature.

Figure 25: $(0,1.43,Q_{L})$ scan at the energy transfer $E=3$ meV and the
temperature $T=2.1$ K. (Figure taken from Ref. [42].)

Fig. 25 displays a $Q_{L}$ scan at the momentum transfer
$\mathbf{Q}=(0,1.43,Q_{L})$ at the energy transfer 3 meV. It shows a sine-wave
evolution of the signal as discussed in Ref. [42], best fit to the data is
made by assuming fluctuating magnetic moments $\mu\parallel a$ and that the
magnetic moments on the ladder rung are ferromagnetically coupled [42]. The
sine-wave modulation indicates that the magnetic correlations along the $c$
axis, i.e. the inter-ladder coupling in this directions can be neglected.

In difference to the initially expected ferromagnetic correlations, neutron
scattering experiments detect only incommensurate magnetic fluctuations close
to an antiferromagnetic wave vector. These antiferromagnetic correlations
involve ferromagnetically ordered moments $\mu\parallel a$ on the ladder rung,
with antiferromagnetic inter-ladder coupling along the $b$ axis, and a quasi
magnetic decoupling along the $c$. Future neutron experiments in the normal
state above $T_{\rm sc}$ have to address the field dependence of these
fluctuations in a magnetic field applied along the $a$ and $b$ axes.

###  4.5 Comparison to the normal state properties of URhGe and UCoGe

After the discovery of superconductivity, the paramagnetic UTe2 has been
considered to show very similar properties to the ferromagnetic
superconductors URhGe and UCoGe [1]. However, there are very basic differences
between them. Firstly, the URhGe and UCoGe crystallize in the TiNiSi-type
orthorhombic structure with the space group $Pnma$ (#62, $D^{16}_{2h}$), which
belongs to the non-symmorphic space group. A particularity of this structure
is that the U atoms form a zig-zag chain along the $a$ direction and thus, the
local inversion symmetry at the U site is broken. The shortest distances
between U atoms are 3.497 and 3.481Å for URhGe and UCoGe, respectively. It is
much smaller than in UTe2 and thus much closer to the Hill limit. Secondly,
both, URhGe and UCoGe order ferromagnetic with respective Curie temperatures
of $T_{\rm Curie}=9.5$ and 2.5 K. Bulk superconductivity occurs below $T_{\rm
sc}\approx 0.25$ K in URhGe [69] and below $T_{\rm sc}\approx 0.6$ K in UCoGe
[70, 71, 72, 73]. Most spectacularly, re-entrance of superconductivity in
URhGe and a strong enhancement of the superconducting upper critical field in
UCoGe along the $b$ axis near the field where the ferromagnetic order
collapses in the transverse field [74, 75].

In URhGe the re-entrance of superconductivity is accompanied by a
reorientation of the magnetic moment from the easy $c$ axis under magnetic
field to the $b$ axis, which results to a strong increase of the magnetization
for $H\parallel b$ at $H_{\rm m}\sim 12\,{\rm T}$ [74]. The origin of the
metamagnetic transition along the hard magnetization $b$ axis is well
understood (see also [76]). The transverse magnetic field along the $b$ axis
leads to a field decrease of its Curie temperature [74, 49]. At zero field,
the ferromagnetic moment is 0.4 $\mu_{B}$/U atom. In the temperature
dependence of the inverse of the static susceptibility $\chi$ (see Fig. 26),
it is obvious that $1/\chi$ along the three crystallographic axes has never a
linear $T$ dependence over a significant temperature range from which
definitely a Curie Weiss term can be extracted. Near $T\sim 300$ K the
respective Curie Weiss temperatures would be $\Theta=-496$ K, -137 K and
$-122$ K for the $a$,$b$ and $c$ axes, respectively. On cooling, for both $c$
and $b$ axes a strong decrease of $|\Theta|$ can be observed and finally for
the $c$ axis a dominant ferromagnetic interactions towards $T_{\rm
Curie}\approx 9.5$ K. A striking point is that in low fields due to the
formation of a ferromagnetic order with a magnetic moment of $0.4\mu_{\rm B}$
aligned along the $c$ axis, the susceptibility $\chi_{c}$ is small, but the
susceptibility $\chi_{b}$ along the $b$ axis is three times larger than
$\chi_{c}$. Thus, the comparison of the susceptibilities suggests that in high
enough field the $b$ axis will become the easy magnetization axis and even a
favorable axis for the development of a ferromagnetic component along $b$
[74].

Figure 26: Temperature dependence of the inverse susceptibility $1/\chi(T)$ of
URhGe measured at a magnetic field of $H=10$ kOe for the $a$, $b$ and $c$
axis.  Figure 27: Comparison of the magnetization along the $b$ axis of UTe2
and URhGe normalized to the respective metamagnetic field. The inset shows the
field dependence of the normalized specific heat coefficient
$\gamma(H)/\gamma(0)$ as function of the normalized field $H/H_{\rm m}$ along
the $b$ axis. (Data for URhGe are taken from Ref. [49], data for UTe2 from
Ref. [55].)

As URhGe orders ferromagnetically, on cooling no characteristic crossover
temperature will occur above the ferromagnetic Curie temperature $T_{\rm
Curie}$ in thermodynamic properties such as $C/T$ or the thermal expansion
$\alpha_{\rm V}$ at least from 30 K to $T_{\rm Curie}$. At 30 K the electronic
Grüneisen parameter $\Gamma_{\rm el}(30{\rm K})\approx+2$ and it rises up to
$+15$ below $T_{\rm Curie}$ [77, 78]. As pointed out, in URhGe $\Gamma_{\rm
e}$ and $\Gamma_{\rm T_{\rm sc}}$ are quite opposite in excellent agreement
that there is only one dominant mechanism in the effective mass enhancement
and in the pairing mechanism (see below) [79].

Similarly to UTe2, metamagnetism is observed in URhGe for a critical value of
the magnetization $M_{b}^{\ast}\approx 0.4\mu_{\rm B}$, however the jump of
the $\Delta M_{b}^{\ast}$ is five to six times smaller than in UTe2 (see Fig.
27. This weak $\Delta M_{b}^{\ast}$ value may explain the rather tiny Fermi
surface change [80] and thus the persistence of field reentrant
superconductivity on a narrow field range above $H_{m}$. Qualitatively the
relative enhancement of the Sommerfeld coefficient
$\frac{\gamma(H)}{\gamma(H_{m})}$ as a function of $H/H_{m}$ are rather
similar, however with a larger ratio for UTe2 [81].

In URhGe, spin fluctuations are strongly enhanced around $H_{\rm m}$ along
both the $b$-axis (longitudinal mode, increase of $1/T_{2}$ ) and the $c$-axis
(transverse mode, increase of $1/T_{1}$). As a consequence of the strong
enhancement of the magnetic fluctuations, the reentrant superconductivity has
a higher $T_{\rm sc}\approx 0.45$ K than at zero field ($T_{\rm sc}\approx
0.25$ K).

In UCoGe, the spin fluctuations develop at low temperatures as a feature of a
system close to the ferromagnetic instability [61, 82, 83]. As in URhGe only
ferromagnetic interactions are involved. In the normal state the temperature
dependence of $1/T_{1}$ measured by 59Co-NQR is nearly constant above a
characteristic temperature of 40 K, which is regarded cross-over temperature
to a heavy fermion state at low temperatures [82]. Below 10 K $1/T_{1}$
increases strongly and shows a large peak at the ferromagnetic transition at
$T_{\rm Curie}\approx 2.5$ K. From the temperature dependence of the 59Co
nuclear quadrupole resonance spectrum however, it has been concluded that the
ferromagnetic transition in UCoGe is already at ambient pressure of weakly
first order. $1/T_{1}$ and Knight shift measured by NMR confirm the strong
Ising type character of the magnetic fluctuations. The observed fluctuations
are in longitudinal mode ($H\parallel c$) and they are drastically suppressed
by external fields applied along the same direction (i.e., parallel to the
easy-magnetization axis) [83]. The anisotropy of the superconducting upper
critical field can be explained by the observed magnetic anisotropy of the
fluctuations [83, 84]. Such a strong characteristic feature has not been
observed for URhGe.

A key difference of UTe2 compared to the ferromagnetic superconductors URhGe
and UCoGe is the absence of long-range magnetic order in the ground state. As
a consequence the characteristic energy of spin fluctuations is expected to be
much higher at low temperatures around $T_{\rm sc}=1.6$ K, and indeed, in UTe2
there is no signature of continuous softening of fluctuations in $1/T_{1}T$.
Instead, $1/T_{1}T$ shows a nearly $T$-independent behavior below 10 K in UTe2
[41]; the behavior expected for an ordinary heavy electron metal.
Interestingly, however, $1/T_{2}$ for $H\parallel a$ detects slow fluctuations
in longitudinal mode along the easy-magnetization axis in the temperatures
region where superconductivity appears. To understand the field-enhancement of
superconductivity it will be crucial to explore the spin dynamics in UTe2 for
a magnetic field above 16 T for $H\parallel b$.

At least UCoGe is at ambient pressure ($p=0$) at the verge of a
ferromagnetic–paramagnetic instability which will occur at $p_{\rm c}\approx
1$ GPa [85, 86, 87]. Already at $p=0$ the ordered moment of 0.06 $\mu_{\rm B}$
is weak. The susceptibility $\chi_{c}$ along the $c$ axis is under high
magnetic field up to 50 T higher than for the other directions [88]. In
difference to URhGe, for a transverse field applied along the $b$ axis, which
suppresses $T_{\rm Curie}$ near 13 T no metamagnetism has been observed and
thus the enhancement of superconductivity in UCoGe near 15 T is not connected
to a metamagnetic transition. Under pressure the ferromagnetic order strongly
increases in URhGe and a magnetic quantum critical regime can not been
attained by pressure at leat up to 14 GPa [89]. It is worthwhile to notice
that around $p_{c}$ the resistivity in UCoGe does not show a Fermi liquid
$T^{2}$ term, but a linear $T$ dependence on approaching $T_{\rm sc}$; the
recovery of a $T^{2}$ temperature dependence is observed only above 4 GPa far
above $p_{\rm c}$ [87].

##  5 Superconductivity

After the initial excitement triggered by the remarkable properties of UTe2
under high magnetic field, strongly suggesting a spin triplet superconducting
state, experimental and theoretical works have focused on three main issues:

  * •

evidences for a spin triplet superconducting state, covering not only the
remarkable high-field behaviour of $H_{\rm c2}$ for $H\parallel b$, but also
the detection of the spin state by NMR, as well as that of gap nodes and the
search for ferromagnetic fluctuations,

  * •

topological properties, which quickly focused on the possibility of a chiral
(time reversal symmetry breaking) state,

  * •

mechanisms for the high field-reinforced or induced superconducting phases.

###  5.1 Basics of unconventional superconductivity

Before discussing the experimental and theoretical investigations on these
issues in UTe2, let us introduce some basics of unconventional
superconductivity. Superconducting phases are characterized by the pair
potential, which may also be called gap function or superconducting order
parameter. The pair potential appears in the off-diagonal part of the
Bogoliubov-de Gennes Hamiltonian. For spin $1/2$ fermions [90, 11], it is
described by the $2\times 2$ matrix [45],

$\displaystyle\hat{\Delta}({\bm{k}})=\left(\psi({\bm{k}})+{\bm{d}}({\bm{k}})\cdot{\bm{\sigma}}\right)i\sigma_{y},$
(2)

with the Pauli matrix, ${\bm{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})$. The
vector component ${\bm{d}}({\bm{k}})$, called ${\bm{d}}$-vector, represents
the spin-triplet Cooper pairs, while the scalar component $\psi({\bm{k}})$
represents the spin-singlet Cooper pairs. To fulfill the fermions anti-
commutation relation, the ${\bm{d}}$-vector must be an odd function with
respect to the momentum ${\bm{k}}$. Furthermore, the ${\bm{d}}$-vector is an
axial vector as the angular momentum is. Therefore, the ${\bm{d}}$-vector,
namely, the order parameter of spin-triplet superconductivity has odd parity
under spatial inversion. By contrast, the scalar order parameter
$\psi({\bm{k}})$ for spin-singlet superconductivity has even parity. Thus,
there is a close link between the parity and spin of Cooper pairs; spin-
singlet superconductivity is even-parity superconductivity, while spin-triplet
is odd-parity. This correspondence ensures that the spin-triplet
superconductivity can be distinguished from the spin-singlet superconductivity
when space inversion symmetry is preserved. Thus, the spin-triplet
superconductivity is well-defined even when the spin is not a good quantum
number due to spin-orbit coupling or magnetic order.

In superconductors, various crystalline symmetries other than space inversion
may be preserved. Indeed, the superconducting phases are classified by the
crystalline point group symmetry [91]; according to the Landau theory for
phase transitions, the order parameter at the second order phase transition
must belong to an irreducible representation (IR) of the total symmetry group.
In other words, transition temperatures of the superconducting order in
different IRs are nonequivalent. For example, the IRs of $D_{4h}$ and $D_{2h}$
point groups are listed in Table 1. Basis functions reveal typical order
parameters of the corresponding IRs. The $D_{4h}$ group corresponds to
tetragonal crystals such as CeCoIn5 [92] and CeRh2As2 [11], while the $D_{2h}$
group is the symmetry of the spin-triplet superconductor candidates UGe2,
URhGe, UCoGe [93], and (the main topic of this article) UTe2.

Table 1:  Classification of superconducting phases in the (a) $D_{4h}$ and (b)
$D_{2h}$ point group symmetry. Spin-orbit coupling is taken into account.
Irreducible representations (IR), basis functions, and parity for spatial
inversion are listed. A superconducting order parameter of the corresponding
IR is a linear combination of the basis functions as Eqs. (3) and (4), with
coefficients determined by microscopic properties.

(a) $D_{4h}$
---
IR | Basis functions | Parity
$A_{1u}$ |  $k_{a}\hat{a}+k_{b}\hat{b}$, $k_{c}\hat{c}$ | -
$A_{2u}$ | $k_{b}\hat{a}-k_{a}\hat{b}$ | -
$B_{1u}$ | $k_{a}\hat{a}-k_{b}\hat{b}$ | -
$B_{2u}$ | $k_{b}\hat{a}+k_{a}\hat{b}$ | -
$E_{u}$ |  $\\{k_{a}\hat{c},k_{b}\hat{c}\\}$, $\\{k_{c}\hat{a},k_{c}\hat{b}\\}$ | -
$A_{1g}$ | $1$ | +
$A_{2g}$ | $k_{a}k_{b}(k_{a}^{2}-k_{b}^{2})$ | +
$B_{1g}$ | $k_{a}^{2}-k_{b}^{2}$ | +
$B_{2g}$ | $k_{a}k_{b}$ | +
$E_{g}$ | $\\{k_{a}k_{c},k_{b}k_{c}\\}$ | +

(b) $D_{2h}$
---
IR | Basis functions | Parity
$A_{u}$ |  $k_{a}\hat{a}$, $k_{b}\hat{b}$, $k_{c}\hat{c}$ | -
$B_{1u}$ |  $k_{b}\hat{a}$, $k_{a}\hat{b}$, $k_{a}k_{b}k_{c}\hat{c}$ | -
$B_{2u}$ |  $k_{a}\hat{c}$, $k_{c}\hat{a}$, $k_{a}k_{b}k_{c}\hat{b}$ | -
$B_{3u}$ |  $k_{c}\hat{b}$, $k_{b}\hat{c}$, $k_{a}k_{b}k_{c}\hat{a}$ | -
$A_{g}$ | 1 | +
$B_{1g}$ | $k_{x}k_{y}$ | +
$B_{2g}$ | $k_{x}k_{z}$ | +
$B_{3g}$ | $k_{y}k_{z}$ | +

###  5.2 Spontaneous symmetry breaking in unconventional superconductors

In superconductors, the relationship between the symmetry of order parameter
and the symmetry breaking is different from other phase transitions. This is
because the hidden $U(1)$ gauge degree of freedom becomes apparent in
superconductors. For example, odd-parity superconductivity preserves space
inversion symmetry in combination with a $U(1)$ gauge operation, although the
space inversion symmetry in the naive sense is broken. Since the $U(1)$ gauge
degree of freedom is inactive in most properties, it is regarded that the
space inversion symmetry is preserved in odd-parity superconductors. This
should be contrasted to the diagonal long-range order, such as parity-
violating antiferromagnets and odd-parity multipole order [94, 95], where the
odd-parity order parameter breaks the space inversion symmetry. In the same
sense, all the superconducting transitions of one-dimensional IR break neither
the crystalline symmetry nor the time-reversal symmetry. Spontaneous symmetry
breaking occurs (1), when the order parameter belongs to a multi-dimensional
IR, or (2), when transition temperatures of nonequivalent IRs are accidentally
coinciding (degenerate IRs) and order parameters distinguished by symmetry
coexist.

An example of case (1) is seen in the $D_{4h}$ group; it contains the two-
dimensional $E_{g}$ and $E_{u}$ representations. In the case of the $E_{u}$
representation, the time-reversal symmetry is broken when the
${\bm{d}}$-vector is ${\bm{d}}({\bm{k}})=(k_{a}\pm ik_{b})\hat{c}$ (chiral
superconductivity). On the other hand, the fourfold rotation symmetry is
broken when ${\bm{d}}({\bm{k}})=k_{a}\hat{c}$ (nematic superconductivity). The
spontaneous symmetry breaking of this kind has been investigated for a long
time [91]. On the other hand, in the $D_{2h}$ group, all the IRs are one-
dimensional. Thus, only the case (2) can occur, and the two-stage (double)
superconducting transition or first-order one is required for the symmetry
breaking. Positive and negative evidence for a two-stage transition in UTe2
will be discussed later (section 5.6.3), and the reliability of the positive
evidence is one of the actual central issues. The double superconducting
transitions at least under the pressure reached a consensus, although those at
ambient pressure are under debate. If the double transition indeed occurs, the
combination of the IRs determines which symmetries are broken. For example,
the space inversion parity is different between the odd-parity representation
(spin-triplet superconductivity) and the even-parity (spin-singlet
superconductivity). Therefore, the coexistent mixed-parity state does not have
a definite parity, indicating a broken inversion symmetry. Such a possibility
will be discussed below for UTe2. Similarly, a $A_{u}+B_{3u}$ state breaks the
mirror and rotation symmetries, while maintaining the inversion symmetry. The
time-reversal symmetry is broken when the relative phase of the order
parameters is neither $0$ nor $\pi$. All the odd-parity superconducting states
with broken time-reversal symmetry are chiral superconducting states, allowing
a non-zero orbital magnetic moment. The chiral axis depends on the
combination: For example, the $B_{3u}+iB_{2u}$ or $B_{1u}+iA_{u}$ state has
the chiral axis along the $c$-axis. The polar Kerr rotation measurement would
imply such an orientation of the chiral axis [96]. On the other hand, in the
$B_{3u}+iA_{u}$ state supported by the London penetration depth measurements
[97], the chiral axis is along the $a$-axis. The chiral axis corresponds to
the direction of spontaneous magnetic (orbital) moment.

The above discussion also applies to the mixing of order parameters by
external perturbation. For example, the space inversion symmetry may be broken
by the crystalline structure, and an odd-parity superconducting order
parameter can be admixed with an even-parity one, as has been studied in the
literature [98, 99]. In the same way, the magnetic field may break the
crystalline symmetry. In the case of UTe2, the $D_{2h}$ group is reduced to
$C_{2h}$ when the magnetic field is applied along a symmetry axis. Therefore,
the superconducting state may be described by the combination of IRs, such as
$A_{u}+iB_{3u}$ for ${\bm{H}}\parallel\hat{a}$. The relative phase should be
$\pm\frac{\pi}{2}$ because the time-reversal symmetry is broken in the
magnetic field. Mixed superconducting order parameters for magnetic fields
along the symmetry axes are listed in Table 2. The field direction is
equivalent to the chiral axis of the superconducting state. Intuitively
speaking, the spontaneous magnetic moment along the chiral axis is induced by
the external magnetic field.

Table 2:  Classification of odd-parity superconducting phases in UTe2 under the magnetic field. Typical order parameters belonging to each IR are listed in Table 1. IR of $C_{2h}$ | $A_{u}$ | $B_{u}$
---|---|---
${\bm{H}}\parallel\hat{a}$ | $A_{u}+iB_{3u}$ | $B_{1u}+iB_{2u}$
${\bm{H}}\parallel\hat{b}$ | $A_{u}+iB_{2u}$ | $B_{1u}+iB_{3u}$
${\bm{H}}\parallel\hat{c}$ | $A_{u}+iB_{1u}$ | $B_{2u}+iB_{3u}$

###  5.3 Spin susceptibility of spin triplet superconductors

As opposed to spin singlet superconductors, spin-triplet superconductors
possess spin degrees of freedom, well represented by their superconducting
order parameter. This order parameter can have different representations, the
most natural one being to write it as:

$\displaystyle\ket{\Psi(\mathbf{\hat{k}})}$
$\displaystyle=\Delta^{\uparrow}(\mathbf{\hat{k}})\ket{\uparrow\uparrow}+\Delta^{\downarrow}(\mathbf{\hat{k}})\ket{\downarrow\downarrow}+\Delta^{0}(\mathbf{\hat{k}})(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})$

where $\Delta^{\uparrow}$, $\Delta^{\downarrow}$, and $\Delta^{0}$ are the
spin-up, spin-down and spin-zero amplitudes of the superconducting order
parameter for the chosen quantization axis. However, the ${\bm{d}}$-vector
representations used here is most commonly used for spin triplet
superconductors, as it is ”independent” from the chosen quantization axis.
(see Ref.[45] for its definition and use in the seminal example of superfluid
3He, or [100] for a detailed connection between both representations). Indeed,
for a quantization axis in a direction characterized by a unit vector
$\bm{u}$, the amplitude of the $S_{z}=0$ component of the spin triplet state
for this quantization axis is simply proportional to ${\bm{d}}\cdot\bm{u}$.

Hence, the spin susceptibility in the superconducting state is closely related
to the direction of the ${\bm{d}}$-vector; the spin susceptibility decreases
when the ${\bm{d}}$-vector is not perpendicular to the magnetic field $\bm{B}$
($|{\bm{d}}\cdot\bm{B}|$ finite). Experimentally, the NMR Knight shift
measurement reflects the electronic spin susceptibility, giving access to the
${\bm{d}}$-vector direction in the spin-triplet superconducting state.
According to the group-theoretical classification in Table 1, all the odd-
parity superconducting states possess ${\bm{d}}$-vector components along the
$a$, $b$, and $c$ axis. For instance, the ${\bm{d}}$-vectors in the $A_{u}$
and $B_{3u}$ states are written as

$\displaystyle{\bm{d}}^{A_{u}}({\bm{k}})=$
$\displaystyle\alpha\,k_{a}\,\hat{a}+\beta\,k_{b}\,\hat{b}+\gamma\,k_{c}\,\hat{c}$
$\displaystyle+{\rm(higher\,\,order\,\,terms)},$ (3)
$\displaystyle{\bm{d}}^{B_{3u}}({\bm{k}})=$
$\displaystyle\alpha\,k_{a}k_{b}k_{c}\,\hat{a}+\beta\,k_{c}\,\hat{b}+\gamma\,k_{b}\,\hat{c}$
$\displaystyle+{\rm(higher\,\,order\,\,terms)}.$ (4)

Thus, the spin susceptibility for all the symmetry axes should decrease, when
we take into account not only the $p$-wave (first order in $k$) and also the
$f$-wave (third order in $k$) components. This is also true for the other
$B_{1u}$ and $B_{2u}$ states.

Generally speaking, the ${\bm{d}}$-vector component parallel to the magnetic
field is suppressed as

$\displaystyle\frac{d|\alpha|}{dH_{a}^{2}}<0,\hskip
14.22636pt\frac{d|\beta|}{dH_{b}^{2}}<0,\hskip
14.22636pt\frac{d|\gamma|}{dH_{c}^{2}}<0,$ (5)

with the coefficients in eqs. (3) and (4). When the above ${\bm{d}}$-vector
component vanishes, the decrease in the Knight shift should disappear. The
robustness against the magnetic field is related to the mechanism of
superconductivity. For instance, the $\beta$-term is robust when the
superconductivity is mainly stabilized by the pairing interaction leading to
the $\beta$-term, and vice versa.

###  5.4 Evidences for spin triplet superconductivity

Experimentally, the two most direct probes of the Cooper pairs spin state are
the measurement of the Knight shift by NMR discussed theoretically above, and
the determination of the upper critical field, $H_{\rm c2}$ revealing (or not)
paramagnetic limitation. Let us start with the upper critical field, which is
one of the most salient and puzzling property of UTe2.

####  5.4.1 Upper critical field

A striking feature of the upper critical field in UTe2 [see Fig. 28(a)],
mentioned immediately in the first papers reporting its superconducting phase
[1, 2], is that the paramagnetic limit is violated in the three principal
crystallographic directions, which is taken as a strong support for spin
triplet superconductivity. Indeed, for a free electron value of the
gyromagnetic factor ($g=2$), and a the weak-coupling (BCS) limit for the
superconducting state, the Pauli limit is related to the critical temperature
by the relation $H_{\rm P}[T]=1.84T_{\rm sc}[K]$. Hence, this limit should be
of order 3 T in UTe2, whereas experimental values of $H_{\rm c2}$ are above,
6, 11 and 15 T along the $a$, $c$ and $b$-axis respectively. The violation is
strongest in the hard axis $b$ direction, and even for field above 15 T a
strong reinforcement of superconductivity occurs up to the metamagnetic
transition at $H_{\rm m}$.

(a)

(b)

Figure 28: (a)-$H_{\rm c2}$ measured in the three crystallographic directions
by electrical transport measurement. Note the strong reinforcement of $H_{\rm
c2}$ for $H\parallel b$ up to the metamagnetic transition (open square
symbols) at $H_{\rm m}\approx 34.5$ T. Data from [101, 59]. (b)-Angular
dependence of $H_{\rm c2}$ from the $b$ up to 8 deg turned to the $a$ axis. In
addition we added the $H_{\rm c2}$ from another sample, and those taken from
Ref. [1]. Figure taken from [101]

Despite this large possible influence of the paramagnetic limitation on the
upper critical field of UTe2, deriving the spin state of the Cooper pairs from
the measured upper critical field is not straightforward. First of all, owing
to the large specific heat jump (Fig. 29(a)) observed in all samples (see e.g.
Ref. [102]), it is clear that UTe2 is a strong coupling superconductor, which
pushes the paramagnetic limit to larger field values. For example, a strong
coupling constant $\lambda=1$ already doubles the weak-coupling Pauli limit
[103]. Hence, the data reported in [2] for $H_{\rm c2}^{a}$ (we use the
notation $H_{\rm c2}^{i}$ for the upper critical field along the
$i$-direction) can also be fitted for a moderate strong coupling constant
$\lambda=0.75$ with a finite paramagnetic limit, if the $g$ factor is kept
below $g<0.5$. This emphasises that the interpretation of the paramagnetic
limit of the upper critical field requires a knowledge of the effective
$g$-factor of the charge carriers, as well as of its anisotropy. A well-known
difficulty with uranium-based superconductors is the estimation of the spin-
orbit interaction. For spin-triplet superconductors, it acts potentially to
lock (strong spin-orbit limit) the ${\bm{d}}$-vector direction to the
crystallographic axes, and to yield anisotropic $g$-factors. For example, in
the singlet superconductor of URu2Si2, $g$, measured by quantum oscillations,
is of order 2 for field along c and vanishes for fields in the basal plane
[104, 105]. Experimentally, not much can be said yet on the value and on the
effects of this spin-orbit interaction in UTe2. Theoretically both the weak
and strong spin-orbit limit are considered regarding the orientation of the
${\bm{d}}$-vector, and some models [106, 107] explicitly predict a strongly
reduced paramagnetic limit in the transverse ($b$ and $c$) directions due to a
very anisotropic spin-orbit interaction. Indeed, it is quite puzzling that
$H_{\rm c2}$ so strongly violates the paramagnetic limit for field along $c$
or $b$, as most models of the order parameter assume, at least in zero field,
that the ${\bm{d}}$-vector has no component along the $a$ axis but non-zero
components along both the $c$ and $b$ axes. To reconcile these order
parameters with the apparent absence of paramagnetic limitation of $H_{\rm
c2}$, a very anisotropic $g$-factor [107], and/or a rotation of the
${\bm{d}}$-vector under field is proposed [16, 108, 109]. However, another
factor could play a key role in UTe2: the field dependence of the pairing
strength.

This effect, and notably its reinforcement under fields perpendicular to the
easy axis have been documented in ferromagnetic superconductors [79, 93], and
proposed again for UTe2 due to the strong reinforcement observed for fields
along the $b$-axis [1, 101]. In the first report [1] a strong increase of
$H_{\rm c2}^{b}$ is found around 1.2 K. Since then, it has been shown that the
increase of $H_{\rm c2}^{b}$, is sample dependent (see Fig. 28(b) and Refs.
[101, 53]). To the best of our knowledge, no bulk measurement has confirmed
this field reinforcement of superconductivity for $H\parallel b$-axis. In Ref.
[101], at $T=0.4$ K for $H>16$ T $H_{\rm c2}^{b}$ along the $b$ axis reaches a
turning point and for a perfect alignment along the $b$ axis $H_{\rm c2}^{b}$
increases up to $H_{\rm m}$. Tilting the magnetic field by 8 deg from the $b$
to the $a$ axis leads to a rapid suppression of the field reinforced
superconductivity (see Fig. 28(b)). The field- reinforcement of
superconductivity for $H\parallel b$ is clearly correlated with the strong
increase of $\gamma$ on approaching $H_{\rm m}$ [33, 54, 101]. The strong
coupling parameter $\lambda(H)$ deduced from calculations of the upper
critical field is shown in Fig. 29 and compared to that observed in the
ferromagnetic superconductors URhGe and UCoGe. For a perfect alignment along
the $b$ axis $\lambda$ varies from 1 at $H=0$ to 2 at $H_{\rm m}$.

(a)

(b)

Figure 29: (a)Specific heat of UTe2, showing a large and sharp specific heat
jump despite a sizeable residual term, classifying UTe2 as a strong coupling
superconductor. Data from [102] (b) Field dependence of the pairing strength
in UTe2 in the three directions (green), as deduced from strong coupling fits
of $H_{\rm c2}$. Note that $\lambda$ is almost doubled between zero field and
$H_{\rm m}$. Also reported is our calculations for the ferromagnetic
superconductors UCoGe (red) and URhGe (blue). Diamonds: $a$-axis, circles:
$b$-axis, squares: $c$-axis. In UTe2, for $H\parallel a$, the field dependence
is deduced from an estimation of the slope of $H_{\rm c2}$ compatible with the
measurements of $H_{\rm c1}$ [110]. A very similar behaviour would be obtained
from the fit of $H_{\rm c2}\parallel a$ as measured by specific heat (see Fig.
30).

Note also that the reinforcement of $H_{\rm c2}$ along the $b$-axis remains to
be confirmed by bulk thermodynamic measurements; up to now, it has only been
detected by electrical transport measurements [1, 101, 39, 42] or by the
Seebeck effect [3].

Figure 30:  (a) $H_{\rm c2}$ determined by specific heat measurements along
the three crystallographic directions. (b) On the zoom close to $T_{\rm sc}$,
note the very strong slope of $H_{\rm c2}\parallel a$-axis, comparable to that
along the $b$-axis, as well as the anomalous very strong curvature of $H_{\rm
c2}\parallel a$ at very low fields [109].

This is an important issue, as thermodynamic measurements seem also to yield a
quantitatively different behaviour of $H_{\rm c2}$ from that of transport
measurements at low fields. For example, the determination of $H_{\rm c2}$
from specific heat measurements in [109] suggests a slope of $H_{\rm c2}$ at
$T_{\rm sc}$ along the $a$-axis much larger than along the $c$-axis, and
rather close to the value along the $b$-axis (see Fig. 30). So, contrary to
the determination of $H_{\rm c2}$ from transport measurements, which are
easily polluted close to $T_{\rm sc}$ by inhomogeneities and the presence of a
small amount of higher-$T_{\rm sc}$ phases, bulk specific heat measurements of
the initial slope of $H_{\rm c2}$ are in better agreement with band
calculations predicting a “2D-like” anisotropy, with lower Fermi velocity
along the $c$ axis than along $a$ or $b$. This anisotropy of the Fermi
velocities is also supported by the anisotropy of the resistivity (see section
4.1). This remains true as long as the effects of a potential variation of the
pairing strength with field are ignored.

Nevertheless, $H_{\rm c2}$ for $H\parallel a$ as determined in [109] cannot be
fitted in the whole temperature range by any combination of orbital and
paramagnetic limit, owing to the strong curvature close to $T_{\rm sc}$ which
does not match the “large value” of $H_{\rm c2}(0)$. So it is likely that the
very strong curvature observed for $H\parallel a$ results from a strong
suppression of the pairing strength along the easy axis, as also suggested by
lower critical field measurements (see Fig. 31 and their comparison to $H_{\rm
c2}$ data [110]). This does not help to identify the spin state of the Cooper
pairs, however, a strong suppression of the pairing strength along the easy
axis is a support to ferromagnetic fluctuations as playing a key role in the
pairing mechanism, and hence to a spin-triplet state with vanishing component
of the ${\bm{d}}$-vector along the $a$-axis.

Figure 31:  $H_{\rm c1}$ determined by analysis of magnetization measurements:
red, $H\parallel a$, blue, $H\parallel b$, green, $H\parallel c$, lines are
guides to the eye. Note that $H_{\rm c1}\parallel a$ and $H_{\rm c1}\parallel
b$ have the same anisotropy than $H_{\rm c2}$, whereas it should be opposite.
And $H_{\rm c1}\parallel c$ is larger than in the two other directions, which
can be reconciled by thermodynamic measurements of $H_{\rm c2}$ (see Fig. 30)
but not by transport (see Fig. 28-(a)).

###  5.5 Mechanisms for the high field-reinforced or induced superconducting
phases

####  5.5.1 Field-reinforced superconductivity.

Regarding the possible mechanism behind the field reinforcement of $H_{\rm
c2}$, we want first to stress that this question is possibly related, however
different from that of a possible change of symmetry of the superconducting
order parameter under field. Indeed several theoretical works have proposed a
field-induced transition between different irreducible representations for
field applied along the $b$-axis [16, 108], notably because most proposed
${\bm{d}}$-vector for the ground state of UTe2 have a non-zero component along
the $b$-axis (and no component along the easy $a$-axis), which imply a finite
paramagnetic limitation for $H$ along $b$. As experiment shows that the
paramagnetic limit is strongly violated for this axis (by more than a factor
10), it strongly suggests that the reinforced phase above 15T requires a
change of symmetry of the superconducting order parameter. For example, as
already mentionned, the saturation of $H_{\rm c2}$ imposed by the paramagnetic
limit could be lifted by a rotation of the ${\bm{d}}$-vectors, such that the
upper phase has no component of ${\bm{d}}$ along the $b$-axis (e.g. switching
from $B_{3u}$ to $B_{2u}$ in high fields [108, 16]). Nevertheless, such a
transition between different symmetry states alone will not explain the upturn
of $H_{\rm c2}$: the enhancement of $T_{\rm sc}$ for fields above $15$ T
requires in addition a field-reinforcement of the dominant pairing mechanism
above that field.

This issue did not trigger as many theoretical works as the putative
topological properties of the superconducting ground state of UTe2. Yet, these
striking experimental behaviours call for explanations. An original proposal
for the field reinforcement of $H_{\rm c2}$ along the $b$-axis, different from
that of a field-induced increase of the paring strength, is the Lebed
mechanism [111, 112] . This mechanism could lead to a complete suppression of
the orbital limitation in quasi 1D [113] or quasi 2D [112] superconductors.
Hence, in quasi 1D [113] or quasi 2D triplet superconductors with no
paramagnetic limitation, $T_{\rm sc}$ in high fields could be the same as in
zero field. As well explained by Lebed himself [112], the physical origin of
this suppression of the orbital limitation is that in the quasi-classical
picture, electron trajectories oscillate in the directions perpendicular to
the applied field, with an amplitude inversely proportional to the field. So
in a quasi-2D superconductor with field in the conducting plane, under high
field, electrons become confined in the plane, a geometry for which the
orbital effect is suppressed. This physical picture shows that this mechanism
can apply if well defined conducting planes separated by “insulating regions”
exist, like in the cuprates or in some organic materials. This is required for
the confinement of the charge carriers within these planes at high fields to
make sense. However, UTe2 is a 3D system, notably as regards the flat f-bands
which are dominant for superconductivity, even though some Fermi sheets are
predicted to have a 2D or even 1D character [114, 16]. The “failure” of the
Lebed mechanism applied to UTe2 is best seen through an estimation of the
characteristic field $B^{\\*}$ where field induced reinforcement should
appear. The quasi 2D behaviour of UTe2, if any, would appear within the
($a,b$) planes with limited coupling along the $c$-axis. For fields along $b$,
if $d$ is the distance between ($a,b$) planes, $\xi_{0}^{b}$ the coherence
length controlling $H_{\rm c2}\parallel b$, and $H^{{}^{\prime}}_{c2}$ the
slope of $H_{\rm c2}$ at $T_{\rm sc}$, $B^{*}$ should be of order:

$\displaystyle B^{*}$ $\displaystyle=\frac{\hbar
v_{c}}{v_{a}qd^{2}}=\frac{\Phi_{0}}{d^{2}}\frac{v_{c}}{\pi v_{a}}$ (6)
$\displaystyle=2H_{c2}(\parallel
b)\left(\frac{\xi_{0}^{b}}{d}\right)^{2}\frac{H^{{}^{\prime}}_{c2}(\parallel
c)}{H^{{}^{\prime}}_{c2}(\parallel a)}$

Putting numbers, with the largest distance between uranium ions along c of
order $0.38$ nm, $B^{*}$ is above $1000$ T. UTe2 is not 2D enough for this
mechanism to help understanding the re-entrant behaviour of $H_{\rm
c2}\parallel b$.

Microscopically, the only work we are aware of trying to understand how the
metamagnetic transition could trigger the reinforcement of $H_{\rm c2}$ is
that of Ref. [56], where it is predicted that ferromagnetic longitudinal
fluctuations might appear along the $b$-axis close to $H_{\rm m}$, leading
also to the observed increase of the specific heat Sommerfeld coefficient on
approaching $H_{\rm m}$ [33, 54, 32, 55]. It gives some justifications for the
estimates of the evolution of the strong coupling parameter made with crude
models from the behaviour of $H_{\rm c2}$ ([1, 101, 59]) as discussed in
section 5.4.1 (and see Fig. 29(b)). Nevertheless, many arbitrary choices made
for these estimations, still require to be checked or invalidated by future
experiments. Notably:

  * •

$\frac{\partial\lambda}{\partial H}$ is assumed finite and positive for
$H\parallel b$, in the initial paper [1], and to reconcile $H_{\rm c2}$ and
$H_{\rm c1}$ anisotropies [110]. This seems also coherent with the evolution
of the Sommerfeld $\gamma$ coefficient [33, 54], however, precise low field
measurements of $\gamma(H)$ in all three principal crystallographic directions
are still missing. If $\gamma(H)$ in UTe2 is also governed by the field
dependence of the interaction responsible for the pairing, which might be less
obvious than in the ferromagnetic superconductors due to the presence of
competing interactions (see section 4), then this measurement is a good test
of the hypothesis on $\frac{\partial\lambda}{\partial H}$ (at least
qualitatively).

  * •

when considering the relationship between $\lambda$ and $\gamma$, an important
parameter is the zero field value of $\lambda$. Indeed, in the weak coupling
limit, $\lambda$ could be field dependent and have a strong influence on the
field dependence of $H_{\rm c2}$, with almost no noticeable effects on
$\gamma$: in the weak coupling limit ($\lambda\rightarrow 0$), $T_{\rm sc}$
depends exponentially on $\lambda$ through a BCS-like relation $T_{\rm
sc}\propto\exp\left(-\frac{1}{\lambda-\mu^{*}}\right)$, whereas
$\gamma\propto(1+\lambda)$. Moreover, for paring mediated by magnetic
fluctuations, the relation between $\gamma$ and $\lambda$ is not as simple as
for the phonon case notably in the strong coupling regime [115].

  * •

With $\lambda$ increasing strongly with field along the $b$-axis, it is also
possible to add a paramagnetic limitation, which will be enhanced by the
strong coupling regime being further reinforced under field. For example, the
reported field dependence of $\lambda$ for UTe2 in Fig. 29 assumes a
paramagnetic limitation corresponding, for s-wave superconductivity, to a
value of the gyromagnetic factor $g\approx 0.9$ for $H\parallel b$ (and $g=0$
for $H\parallel a$ and $H\parallel c$). The resulting smooth behaviour of
$\lambda(H)$ is even qualitatively more consistent with the reported field
dependence of the Sommerfeld coefficient [33, 54] than $\lambda(H)$ deduced
without paramagnetic limitation (see [101]).

Hence, the combined strong coupling regime with increased coupling strength
along the $b$-axis, as well as the spin-orbit effects strongly weakens the
argument that the clear violation of the paramagnetic limit along the three
axis is a smoking gun for $p$-wave superconductivity in UTe2.

####  5.5.2 Superconductivity above $H_{\rm m}$

The disappearance of superconductivity above $H_{\rm m}$ for $b$-axis seems
associated with a strong increase of the residual resistivity linked to the
Fermi surface reconstruction at $H_{\rm m}$ (see section 4.2). It may also
point out a change of the pairing mechanism on crossing $H_{\rm m}$ associated
to a change in the nature of the interactions. Surprisingly, when tilting the
magnetic field from the $b$-axis toward the $c$-axis, pulsed field experiment
at the National High Magnetic Field Laboratory in Los Alamos revealed a new
superconducting pocket above $H_{\rm m}$ [9], for angles $\theta$ between 25∘
and 45∘ hence around the to $[011]$ direction, with a maximum $H_{\rm c2}$
above 60 T close to 35∘, see Fig. 32. This result was confirmed by further
experiments made in the High Magnetic Field Laboratory in Toulouse [42].
Magnetization measurements, realized in the different groups show little
dependence of the magnetization jump with angle: experiments in the US report
$\Delta M\sim 0.3\mu_{\rm B}$ for $H\parallel b$ or $H$ at $\theta\approx
35^{\circ}$ [9], whereas experiments in Japan report a larger jump $\Delta
M\sim 0.6\mu_{\rm B}$ for $H\parallel b$ [33] or at $\theta\approx 28^{\circ}$
[55] (see Fig. 33).

Figure 32: Magnetic field - angle phase diagram of the upper critical field
$H_{\rm c2}$ indicating different superconducting phases. Remarkably, in the
($b-c$) plane superconductivity occurs above $H_{\rm m}$ in the angular range
from 23 to 45 degrees. Figure taken from Ref. [9]). Figure 33: (upper panel)
Magnetization curves extrapolated to 0 K of UTe2 for $H\parallel b$ and for
$\theta\approx 28$ deg from the $b$ to the $c$ axis. (lower panel) Magnetic
field dependence of $\Delta\gamma$ derived from the magnetization data.
(Figure taken from Ref. [55]

As discussed in section 4.2, some normal state properties like the $A$
coefficient of resistivity for current along the $a$-axis, or the specific
heat Sommerfeld coefficient deduced thanks to Maxwell relations from $M(T,H)$
seem, at first glance, rather “symmetric” with respect to $H_{\rm m}$, and
change little with angle; not only $\Delta M$ [9, 55], but also the $A$
coefficient of the resistivity [42]. This would suggest that no drastic
changes occur as regards the density of states either across $H_{\rm m}$ or
when rotating the field from $b$ toward $[011]$, which contrasts with Hall
effect or Seebeck measurement detecting a Fermi surface instability and
carrier number change at $H_{\rm m}$. More detailed direct specific heat
measurement however seem to indicate the presence of a sizeable jump at
$H_{\rm m}$, of order 20% [54], and new magnetization measurements confirm
this negative jump of the $\gamma$ term above $H_{\rm m}$ for $H\parallel b$
and predict a positive jump of the $\gamma$ term of the same order for $H$
applied at $\theta\approx 28^{\circ}$ (Fig. 33). Clearly, more work is
required to clarify the normal state behaviour at $H_{\rm m}$.

Up to now, the origin of this new superconducting phase was considered as a
real puzzle, notably when contrasted to the disappearance of superconductivity
above $H_{\rm m}$ for $H\parallel b$. The Lebed mechanism has been proposed as
a possible route [39], however, the same arguments as given above for the
field reinforced phase ($H\parallel b$) apply as well for H at an angle
$\theta\approx 35^{\circ}$ from the $b$-axis. Another possibility, evoked but
refuted in Ref. [9], is a Jaccarino-Peter mechanism. This mechanism, very
efficient and well documented in some Chevrel phases (see e.g. [116, 117]), is
a compensation of the Zeeman effect of the conduction electrons in the Cooper
pairs by an internal exchange field arising from the field-induced
polarization of local magnetic moments. It could fit with the expected local
nature of the uranium moments in UTe2 (see section 2), and the negative
exchange field expected to have led to the heavy (Kondo) bands. It is possible
to set the requirements for such a mechanism to be effective at a finite angle
but not for $H\parallel b$ [118]. If this mechanism is at work, it would imply
that $H_{\rm c2}$ along the $b$-axis is also influenced by the paramagnetic
effect (see discussion above) either due to a non-zero component of the
${\bm{d}}$-vector along $b$ or to spin-singlet pairing.

Naturally, if it is confirmed that contrary to the initial indications from
transport measurements, the density of states has an opposite jump at $H_{\rm
m}$ for $H\parallel b$ or $H$ at 30∘, it would be easier to understand this
new high-field induced phase, without a Jaccarino-Peter mechanism. A new
pairing mechanism would also be expected, either dominated by
antiferromagnetic fluctuations if they are responsible for the metamagnetic
transition, or by ferromagnetic fluctuations along the $b$-axis if a mechanism
such as proposed in [56] is at work. In both cases, it calls for a microscopic
investigations on the magnetic properties above $H_{\rm m}$, either to explore
the validity of the Jaccarino-Peter hypothesis or to understand the interplay
between Fermi surface anomalies, metamagnetic transition and the pairing
mechanism.

####  5.5.3 Knight-shift measurements in the superconducting state

Knight shift measures the effective field produced by the electrons, and is
proportional to the local spin susceptibility at the nucleus. As opposed to
the upper critical field, which is sensitive to the spin state only for large-
enough fields, when the paramagnetic effect can be dominant compared to the
orbital effect, Knight shift measurements by NMR have larger accuracy in the
low or intermediate field range, and moreover are performed at constant field,
which avoids the caveats connected to the possible field-dependence of the
pairing strength.

In an even parity spin-singlet superconductor the spin susceptibility is
isotropically suppressed in the superconducting state. For the spin-triplet
case, the spin part of the Knight shift decreases in the superconducting state
when the magnetic field $H$ is parallel to the ${\bm{d}}$ vector
($\mbox{\boldmath$H$}\parallel{\bm{d}}$); however, it remains unchanged in the
case of $\mbox{\boldmath$H$}\perp{\bm{d}}$. Hence, the ${\bm{d}}$-vector
component can be derived from the measurements of the temperature dependence
of the Knight shift along each crystalline axis.

Already in the first report on superconductivity in UTe2, the NMR Knight shift
measured on a powdered sample has been presented and no change in the NMR
frequency through $T_{\rm sc}$ has been observed [1]. However, the very tiny
Knight-shift value in the normal state ($K_{n}\sim 0.01$ %) is unexpected for
a U-based compound near the magnetic instability. Moreover, to identify the
spin state thoroughly, NMR measurements on a single crystalline sample in
field applied along each axis are required. A very good sensitivity on high
quality single crystals was obtained in the next studies [119, 120, 121],
using in addition 125Te-enriched samples (99.9% of 125Te): the signal
intensity of the enriched sample was roughly one order of magnitude larger
than that of the Te-natural sample.

Up to now, the NMR spectrum when $H\parallel a$ could not be observed below 20
K due to the quasi-divergence of the nuclear spin-spin relaxation rate
$1/T_{2}$ [41], as discussed in section 4.3. Along the $b$ and $c$ axes, the
NMR measurements support the spin-triplet pairing and reveal that the
${\bm{d}}$-vector has $\hat{b}$ and $\hat{c}$ components that show an
anisotropic response against $H$ in the $bc$ plane.

Figure 34:  Temperature dependence of $\Delta K_{b}$ (A) and $\Delta K_{c}$
(B) measured at 1, 3, and 5.5 T (see in the text)[120]. Dashed lines are added
as a guide to the eye. Temperature dependence of the AC susceptibility $\chi$
at the same magnetic fields along the $b$ (C) and $c$ (D) axes. (Figure from
Ref. [120])

Figures 34 (A) and (B) present the temperature variation of $\Delta K_{b}(T)$
(A) and $\Delta K_{c}(T)$ (B) at $\mu_{0}H$ = 1, 3, and 5.5 T, which are
compared with the temperature dependence of the AC susceptibility. Here,
$\Delta K_{i}(T)$ ($i$ = $b$ and $c$) is defined as $\Delta K_{i}(T)\equiv
K_{i}(T)-K_{n,i}$ with the normal-state Knight shift denoted as $K_{n,i}$. The
decrease in $\Delta K_{b}(T)$ observed at 1 T is $\sim$ 0.1% [119], and almost
the same for $\Delta K_{c}$. The decrease in $\Delta K_{b}$ in the SC state is
almost field-independent above 3 T. By contrast, when the field increases,
$\Delta K_{c}$ vanishes even at the lowest temperatures, as shown in Fig. 34
(B).

Figure 35:  The magnetic field dependence of the decrease in $\Delta K_{b}$
(diamonds) and $\Delta K_{c}$ (circles)[120]. The solid and dashed lines
represent the calculated SC diamagnetic shielding effects $\Delta K_{{\rm
dia},b}$ and $\Delta K_{{\rm dia},c}$, respectively. The inset shows the $H$
dependence of the decrease in $K_{{\rm spin},i}$ ($\Delta K_{{\rm spin},i}$)
estimated by subtracting $\Delta K_{{\rm dia},i}$ from $\Delta K_{i}$.

The field-dependence of $\Delta K_{i}$ at the lowest temperature ($\sim 0.1$
K) is plotted in Fig. 35. In general, $\Delta K_{i}$ is attributed to two
contributions: a decrease in the spin part of $K_{i}$ ($\Delta K_{{\rm
spin,}i}$) in the SC state and a change of the internal field due to the SC
diamagnetic shielding effect $\Delta K_{{\rm dia},i}$ [122].

The field dependence of $\Delta K_{{\rm dia},i}$ ($i$ = $b$ and $c$) is
estimated from the experimental results, and depicted with the solid and
dotted curves in Fig. 35. The details are given in Ref. [120]. As highlighted
by Paulsen et al., the $H_{\rm c1}$ value along the $b$ axis is unexpectedly
large and is two times the estimated value from $H_{c}$[110]. Even if such an
unexpectedly large $H_{c1}$ is adopted for $H\parallel b$, the observed field
dependence of the $K_{b}$ decrease cannot be accounted for solely by the field
dependence of $\Delta K_{{\rm dia},b}$; hence, it indicates that $K_{\rm
spin}$ along the $b$ and $c$ axes decreases in the SC state, and the decrease
in $K_{\rm spin}$ shows an anisotropic response against applied field, as
shown in the inset of Fig. 35. This suggests that the decrease in the spin
susceptibility is maintained at least up to 6.5 T when $H\parallel b$. By
contrast, it is suppressed gradually when $H\parallel c$, and the spin
susceptibility does not change anymore with temperature across $T_{\rm sc}$ at
5.5 T.

A crucial point to interpret the amplitude of the observed decrease of $\Delta
K$ is to estimate the spin contribution of the itinerant quasiparticles to the
normal-state $K$. In NMR studies of transition metals, the spin-part of the
Knight shift has been evaluated from the $K-\chi$ plot, where it was assumed
that the orbital shift is temperature-independent and $K_{\rm spin}$ is
temperature-dependent[123]. However, it has been understood that such a simple
estimation is not valid in heavy-fermion superconductors due to the strong
spin-orbit coupling, particularly for the estimation of $K_{\rm spin}$ related
to superconductivity. In such a case, $\Delta K_{\rm spin}$ can also be
roughly estimated from the change of the electronic term $\gamma_{\rm el}$ in
the specific heat below $T_{\rm sc}$[124]

$\Delta~{}K_{\rm\gamma}=\frac{A_{\rm
hf}}{N_{A}\mu_{B}}\Delta\chi_{\rm\gamma}=\frac{A_{\rm
hf}}{N_{A}\mu_{B}}\frac{\Delta\gamma_{\rm el}(\mu_{\rm
eff})^{2}}{\pi^{2}k_{B}^{2}}R.$

Here, $N_{A}$ is Avogadro’s number, $\mu_{B}$ is Bohr magneton,
$\Delta\chi_{\gamma}$ is the change of the quasi-particle spin susceptibility
below $T_{\rm sc}$ evaluated from the decrease of $\gamma_{\rm el}$, $\mu_{\rm
eff}$ is the effective moment, ($k_{B}$ is Boltzmann’s constant,) and $R$ is
the Wilson ratio. In UTe2, by adopting $\Delta\gamma_{\rm el}\sim$ 60
mJ/mol$\cdot$K2[93] and $A_{\rm hf}$ determined from a $K-\chi$ plot[41],
$\Delta K_{\gamma}$ along the $b$ and $c$ axes are estimated and are shown in
Table 3, where we used $(\mu_{\rm
eff})^{2}=g^{2}\,j(j+1)\mu_{B}^{2}=3\,\mu_{B}^{2}$ for the free-electron value
($g$ = 2, $j$ = 1/2) and $R=1$. In the Table 3, the experimental reduction of
$K$ in the SC state $\Delta K_{\rm spin}$ and these values in UPd2Al3 [125,
126] and URu2Si2 [127] are also shown. It is noted that $\Delta K_{\gamma}$
determined by this method is in good agreement with $\Delta K_{\rm spin}$ in
UPd2Al3 [124] and URu2Si2 [127], which are identified as a spin-singlet
superconductors. In UTe2, $K_{\rm spin}$ along both axes is approximately one
order of magnitude smaller than $\Delta K_{\gamma}$, suggestive of a large
residual spin component at $T$ = 0. The presence of such a large spin
component seems incompatible with a spin-singlet state, and favors the spin-
triplet superconductivity scenario. We point out that such a tiny decrease of
the Knight shift in the superconducting state was also reported in the multi-
phase superconductor UPt3, where the decrease in the superconducting state
measured in low fields ($\sim 0.2$ T) is only $0.5\sim 1.4$% of $\Delta
K_{\gamma}$ [128].

However in UTe2, even for a spin-triplet scenario, the observed decrease in
directions perpendicular to the easy axis is far too small, and it shows the
limitations of the estimate from $\Delta\gamma_{\rm el}$ and the Wilson ratio;
taking proper account of the spin orbit effect on the spin susceptibility
anisotropy and the band splitting is required. This is well illustrated by a
recent theoretical calculation based on the spin-triplet superconducting
scenario[107], and anisotropic spin-orbit coupling, which does reproduce such
a tiny $\Delta K_{\rm spin}$. The main ingredient of the model is precisely
the band splitting induced by a strongly anisotropic spin-orbit coupling,
which lifts the degeneracy between the different values of $|j_{z}|$, with
quantization axis along the easy $a$-axis. For transverse field directions,
the normal state susceptibility is governed by interband terms (indexed by
different values of $|j_{z}|$), so that the decrease of the spin
susceptibility in the superconducting state, only affected by the intra-band
term with $|j_{z}|=1/2$, remains very small. Hence in these strongly
correlated systems, a precise microscopic modeling of the multi-orbital
$f$-electron bands is required to properly interpret the anisotropy and
magnitude of the decrease of the Knight-shift in the superconducting state.

Table 3: Physical quantities on U-based superconductors. Observed reduction of
the electronic term in the specific-heat measurement ($\Delta\gamma_{\rm
el}$), spin susceptibility ( $\Delta\chi_{\gamma}$) estimated from the Wilson
ration with $\Delta\gamma_{\rm el}$, Hyperfine coupling constant ($A_{\rm
hf}$), spin-part Knight shift ($\Delta K_{\gamma}$) estimated from the
$\Delta\gamma_{\rm el}$, observed Knight-shift reduction in the SC state
($\Delta K_{\rm spin}$) are shown.

| $T_{\rm sc}$ | $\Delta\gamma_{\rm el}$ | $\Delta\chi_{\gamma}$ | $H$ | $A_{\rm hf}$ | $\Delta K_{\gamma}$ | $\Delta K_{\rm spin}$
---|---|---|---|---|---|---|---
| (K) | (mJ/mol K2) | ($10^{-3}$ emu/mol) |  | (T / $\mu_{\rm B}$) | (%) | (%)
UTe2 | 1.6 | 60 | 0.82 | $\parallel b$ | 5.18 | 0.76 | 0.058
|  |  |  | $\parallel c$ | 3.9 | 0.57 | 0.08
UPd2Al3 | 2 | 150 | 2.0 | $\perp c$ | 0.35 | 0.13 | 0.11
|  |  |  | $\parallel c$ | 0.35 | 0.13 | 0.08
URu2Si2 | 1.2 | 65.5 | 0.90 | $\perp c$ | 0.36 | 0.058 |  $\sim 0$ *
|  |  |  | $\parallel c$ | 0.36 | 0.058 | 0.05
UPt3 | 0.55/0.5 | 420 | 6.0 | $\perp c$ | -7.1 | -8.8 | -0.05
|  |  |  | $\parallel c$ | -8.4 | -7.3 | -0.10

*Absence of $\Delta K_{\rm spin}$ in URu2Si2 $H\perp c$ is interpreted as the anisotropy of the $g$-factor [104, 105].

Figure 36:  Magnetic field dependence of the change in the spin component of
the Knight shift in the SC state, $\Delta K_{\rm spin}$[121]. The dotted line
indicates $\Delta K_{\rm spin}$ = 0. The dashed lines are a guide for the eye.
(Inset) 125Te-NMR spectrum at $\sim 0.1$ K (SC state) measured at 8, 10.5, and
12.5 T. For comparison, the normal-state NMR spectrum is also shown.

Considering that the spin susceptibility decreases with a similar magnitude
for $H\parallel b$ and $H\parallel c$ at low fields, a ${\bm{d}}$-vector
containing non-zero $\hat{b}$ and $\hat{c}$ components at low $H$ is very
likely. In addition, with the strong anisotropy along the $a$ axis in the
normal-state spin susceptibility, the ${\bm{d}}$ vector could well be
perpendicular to the $a$ axis. Hence, considering the possible SC order
parameters for $D_{2h}$ point group symmetry, presented in section 5.1, Table
1, the $B_{3u}$ state is a promising candidate: see also discussion below
(section 5.5.4) on the possible gap nodes. However with the present
experiments, an $A_{u}$ state with non-zero $k$-dependent components of
${\bm{d}}$ along the $a$, $b$ and $c$-axes cannot be excluded. To distinguish
the two possibilities, Knight-shift measurements along the $a$ axis, probing
the $a$ component of the ${\bm{d}}$ vector, would provide valuable
information.

Another major open issue is the behaviour of the ${\bm{d}}$-vector under high
field for $H\parallel b$ or $c$-axes. It is expected that the
${\bm{d}}$-vector orientation changes under field, and that the observed
suppression of $\Delta K$ in both directions under high field could result
from this re-orientation. How strongly the ${\bm{d}}$-vector is pinned in its
low-field orientation, and whether the re-orientation occurs as a crossover or
a phase transitions, are some of the main points to be investigated. This
should lead to additional constraints on the possible symmetries for the order
parameter.

The anisotropic response of the superconducting spin susceptibility shown
above suggests that the $c$ component in the ${\bm{d}}$-vector is continuously
suppressed when $H\parallel c$ and vanishes at 5.5 T. For $H\parallel b$, the
field dependence of $\Delta K_{\rm spin}$ in Fig. 36, suggests that the $b$
component is constant up to 7 T, and that a ${\bm{d}}$-vector rotation occurs
above 7 T, leading to a vanishing $b$ component of the ${\bm{d}}$-vector at
12.5 T. We point out that 12.5 T is not so far from the critical value of
$H_{\rm{c2}}\parallel b$ at which $T_{\rm sc}$ is minimum, until it increases
with $H$ (see Fig. 28). Hence as already stated, the field-reinforced
superconductivity might coincide with a different SC state where the
${\bm{d}}$-vector can now rotate away from the $b$ axis. A further
experimental investigation by other measurements is required to probe the
presence of an anomaly between these two SC states.

####  5.5.4 Gap nodes

Unconventional superconductivity is very often characterised by the presence
of gap nodes imposed by the broken crystal symmetries, and the most common
types of nodes are line nodes. However, in the case of triplet
superconductors, Blount’s theorem [129] states that the most general order
parameter can only have point nodes (or no nodes), not line nodes. It has been
shown that line nodes can nevertheless exist for odd-parity superconductors in
non-symmorphic crystals (those having symmetries combining point group
elements with non primitive translations), like UPt3, URhGe, UCoGe [130].
UTe2, which is symmorphic, should be constrained by Blount’s theorem. Hence in
UTe2, identifying the types of nodes of the gap, if any, can be another route
to probe spin-triplet superconductivity.

Several measurements have claimed to identify point nodes in the gap of UTe2:
thermal conductivity ($\kappa$) [48], London penetration depth ($\lambda_{L}$)
[48, 131, 97], specific heat ($C_{p}$) [48, 109], NMR $1/T_{1}T$ relaxation
rate [119]. Table 4 summarises the different order parameter symmetries and
node types and positions proposed for the various experiments.

Table 4:  Proposals for the gap nodes and corresponding spin-triplet order
parameters, deduced from various measurements.

Experiment | type of nodes | position | order parameter | reference
---|---|---|---|---
$\kappa$ | points |  $k_{a}$ & $k_{b}$ | ? | [48]
$\lambda_{L}$ | points | in (a,b) plane |  $B_{2u}$ or $B_{3u}$ | [48]
$\lambda_{L}$ | points | close to (a,b) plane | chiral | [131]
$\lambda_{L}$ | points | close to $k_{b}$ & $k_{c}$ | $B_{3u}+iA_{u}$ | [97]
$C_{p}$ | points | ? | ? | [48]
$C_{p}$ (angular H-dependance) | points | $k_{a}$ |  $k_{b}+ik_{c}$ or $k_{b}\bm{c}+k_{c}\bm{b}$ ($B_{3u}$) | [109]
$1/T_{1}T$ |  points or fully gapped | ? |  multigap with $\Delta_{2}\approx\Delta_{1}/10$ | [119]

As can be seen from Tab. 4, if all measurements essentially converge to the
presence of point nodes of gap, the position of these nodes and the proposals
for the corresponding superconducting order parameters largely vary. It is
quite remarkable that so many experiments could already probe the nodal
structure of this system so soon after its discovery; this is in strong
contrast to the case of URhGe and UCoGe, where the existence of line or point
nodes is still today a completely open question. The much higher $T_{\rm sc}$
of UTe2 certainly helps. But most remarkably, the extrapolation of the thermal
conductivity down to the lowest temperatures reveals a negligible residual
term (Fig. 37) [48].

Figure 37: Thermal conductivity at low temperatures in the superconducting
state of UTe2. It shows only a very small residual term at zero temperature in
zero field, as opposed to the behaviour of specific heat (see Fig. 29), or
Fig. 3c of Ref. [1]. Under field, the residual term increases rapidly, which
is also taken as supporting point nodes. Figure taken from Ref. [48].

This contrasts with other systems like UCoGe, where the residual terms, even
for samples with residual resistivity ratios (RRR) above 150, prevent the
observation of the intrinsic power laws and their potential anisotropy [72].
The reason for such small residual terms despite a RRR barely larger than 40
in UTe2 (the samples studied had RRR $\approx 22$) remains unclear, like the
sharpness of the specific heat transition (except for some samples displaying
double or multiple transitions, see Fig. 41). A path for progress is to probe
gap nodes along the $c$-axis; UTe2 crystals have usually a much smaller
dimension along the $c$-axis, so that transport measurements are difficult in
this direction.

Regarding the London penetration depth $\lambda_{L}$, the situation is similar
because the largest surfaces parallel to the field have the largest
contribution: extracting $\lambda_{L}$ for the $c$-axis requires high
precision to subtract two large quantities [97]. Today, the London penetration
depth measurements presented in Ref. [97] are the only one probing the nodes
in all directions, and surprisingly, the observed power laws suggest nodes
close to the three axis directions (see Fig 43 in section 5.6.4 below), which
is not easily enforced by symmetry or topological arguments .

Beyond the question of the nodes, that of a possible multigap structure is
also very acute as it can strongly influence the low temperature density of
excitations, independently of the issue of spin triplet or singlet
superconductivity. Indeed, it has been realized that in most heavy-fermion
superconductors, multigap superconductivity sets-in, with large gaps on the
heavy $f$-bands and smaller gaps on the light-band mass (see e.g. UNi2Al3
[132], PrOs4Sb12 [133], CeCoIn5 [134], URu2Si2 [135] and others). UTe2 should
be such a case, with its Fermi sheets having strong $f$ character while $p$ or
$d$ bands from Te should yield 1D or 2D like Fermi sheets with light masses
(see Fig. 4). Presently, the only measurement which strongly supports the
existence of a very small gap, preventing to distinguish between nodes or
fully gapped excitations, is the NMR relaxation rate. All other measurements
claimed the absence of low energy gaps, mainly from the field dependence of
$\kappa$ [48] or $C_{p}$ [109] at very low temperatures. Nevertheless, due to
the absence of a quantitative comparison to theoretical predictions, it is not
clear if the contradiction with $1/T_{1}T$ is a real deep problem; further
studies should certainly clarify this important point, which could otherwise
change all conclusions drawn from the observed power laws.

Figure 38:  Correlations between transition temperature with sample quality as
measured by the amplitude of the residual $C/T$ term, $\gamma_{0}$ scaled by
$\gamma_{\rm N}$ in the normal state obtained from 35 different batches. The
inset is the zoom at low residual $C/T$.

Another factor which should be taken into account is the question of the
strong coupling regime. The same NMR measurement of $1/T_{1}T$ showed the
absence of a Hebel-Slichter peak, consistent with an unconventional pairing
state, and a fast decrease of the relaxation rate close to $T_{\rm sc}$,
evidencing the presence of a gap at least twice larger than the BCS weak-
coupling value [119]. This strong coupling regime is also supported by the
specific heat measurements of the superconducting transition. Indeed, for the
lowest $T_{\rm sc}$ samples, the specific heat jump at $T_{\rm sc}$ is
$\frac{\Delta C}{\gamma T_{\rm sc}}\sim 1.5$ instead of 1.43 predicted for a
weak-coupling BCS singlet state [1]. And this value raises up to above 1.8 for
samples with $T_{\rm sc}$ at 1.77 K [102]. This is all the more remarkable
that UTe2 displays a low temperature anomaly, apparently independent of the
superconducting state [109], which may contribute to the normal state specific
heat at $T_{\rm sc}$ [48], leading to an over-estimation of the Sommerfeld
coefficient $\gamma$ and an under-estimation of $\frac{\Delta C}{\gamma T_{\rm
sc}}$. Moreover, if gap nodes are indeed present, the predicted weak coupling
specific-heat jump should also be lower than 1.43 [136], as can be easily
understood from the entropy balance. So the observed large values of
$\frac{\Delta C}{\gamma T_{\rm sc}}$ strongly push UTe2 toward the strong
coupling regime.

As regards the discussion of gap nodes from the specific heat measurement,
they are complicated by the presence of this low temperature anomaly. In an
intermediate temperature range, between 0.1 and 0.5 K, $C/T$ seems to
extrapolate to a residual value $\gamma_{\rm res}\approx$ 60
mJ$\cdot$mol-1K-2, which is roughly half of the value of the normal state
value $\gamma_{\rm N}$. This residual term measured on different crystals and
in different laboratories seems at first site quite reproducible, almost
independent on sample quality.[1, 2, 137, 48, 102, 109]. However, systematic