careful studies have revealed a correlation between sample quality, $T_{\rm
sc}$ value and amplitude of the residual term (see Fig. 38, see also Refs.
[137, 102]). As already emphasized, determining the origin of $\gamma_{\rm
res}$ is today a fully open problem. Empirically, it has been proposed to
subtract a “divergent quantum critical contribution” to the density of states
with a $T^{-1/3}$ temperature dependence [48]. This could reconcile the
specific heat temperature dependence with the presence of gap nodes, and with
the observed low temperature behaviour of the thermal conductivity. However,
there is presently no theoretical explanation for such a contribution.

Finally, let us note that with a strong coupling regime, and the possible
existence of small gaps as in other heavy-fermion superconductors, the
interpretation of power laws in terms of gap nodes requires probably more
caution than met in the presently published works.

####  5.5.5 Hunt for ferromagnetic fluctuations

A last ”proof” of triplet superconductivity, is the proximity to a
ferromagnetic instability which should favour ferromagnetic fluctuations, and
a pairing mechanism through the exchange of ferromagnetic fluctuations. As
explained in section 4.4, this proximity to a ferromagnetic instability is
nowadays not universally accepted, and experimentally, only incommensurate
fluctuations near an antiferromagnetic wave vector have been observed (see
section 4.4).

So let us make a few points clear. First, even if ferromagnetic fluctuations
would certainly favour spin triplet pairing, spin triplet superconductivity
can also arise without ferromagnetic fluctuations; this possibility has even
been predicted long ago [138], and continues to stimulate theoretical work
(see [139] for a quite recent example, referring also to UTe2).
Experimentally, a good example is UPt3, where ferromagnetic excitations have
never been observed [140]. Nevertheless, the most popular symmetry for the
superconducting order parameter is the odd-parity ($f$-wave) $E_{2u}$ model
[141]. Furthermore, in another field, twisted bylayer graphene is strongly
suspected to present a triplet superconducting phase, surviving also in high
fields much above the paramagnetic limit [142], without any sign of
ferromagnetic fluctuations.

As discussed in section 5.4.1, the reinforcement of superconductivity for
fields along the $b$-axis implying a field induced increase of the pairing
strength resembles the situation of URhGe or UCoGe. Nevertheless, this does
not prove that ferromagnetic fluctuations are at the origin of the pairing. In
UTe2, the microscopic mechanism responsible for the field-induced boost of the
pairing strength is unknown, and seems connected to the meta-magnetic
transition, which bares more resemblance to that found in antiferromagnetic
systems than in ferromagnets (see section 4.2).

Maybe a stronger case for the role of ferromagnetic fluctuations in the
pairing mechanism of UTe2 would be the observation of a decrease of the
pairing strength along the ”easy” a-axis. Such a strong decrease has been
identified in UCoGe as responsible for the very strong anisotropy of the upper
critical field and the acute angular dependence of this upper critical field,
but also of $1/T_{1}T$ [83, 79]. It was predicted theoretically for a pairing
mechanism arising from ferromagnetic fluctuations, and is much less model-
dependent than the behaviour for fields along the intermediate of hard axis
[143]. Such a decrease is suspected in UTe2 as it could explain the anomalous
anisotropy of the lower critical field $H_{\rm c1}$, opposite to expectations
from that of $H_{\rm c2}$ even in the Ginzburg-Landau regime close to $T_{\rm
sc}$ [110]. It is also supported by the anomalous curvature of $H_{\rm c2}$
for $H\parallel a$-axis as measured by specific heat [109]. More quantitative
work is required to confirm this point, which appears nevertheless as a
promising way to demonstrate that the putative ferromagnetic fluctuations
would play a strong role in the pairing mechanism of UTe2 at ambient pressure.
Besides, several predictions were made, based on realistic theoretical models
of the electronic band structure of UTe2 and its associated Fermi surface, for
the possible and dominant magnetic fluctuations (see section 3.1).

Figure 39:  Comparison of energy scans below (0.25K) and above (2K) $Tsc$ in
UTe2, including Bose factor corrections, at the antiferromagnetic wave-vector
$\mathbf{Q}=(0,0.59,0)$. Figure from [144].

Recently, inelastic neutron scattering measurements [144] have revealed a
resonance appearing below $T_{\rm sc}$ (see Fig. 39) at the wave-vector where
antiferromagnetic fluctuations are strongest (section 4.4), on 27 aligned
single crystals. This result has been confirmed even more recently by triple
axis neutron experiments on a much smaller sample but with a 10 times smaller
wave-vector integration and using a unique single crystal [145].

This is again a difference with the ferromagnetic superconductors URhGe and
UCoGe, where such a resonance has not been observed. Quantitatively, it is
centered at a large energy $E_{r}\approx 7.9k_{B}T_{\rm sc}$, leading to a
ratio $E_{r}/2\Delta\approx 2$, taking for the value of the gap $\Delta$ the
estimation from the scanning tunneling microscopy (STM) measurements
($\Delta\approx 0.25$ meV [146]). This is far above the value
$E_{r}/2\Delta\approx 0.64$ reported in the unconventional spin singlet
high-$T_{\rm c}$ cuprates, or in the iron pnictide superconductors [147]. In
the other heavy fermions where such a resonance has been observed [145],
$E_{r}/2\Delta$ remains close or lower than $1$, except in UBe13 with
$E_{r}/2\Delta\approx 1.9$. Note that UBe13 is the only one, with UTe2,
strongly suspected to be $p$-wave [148]. Below $E_{r}$, a spin-gap opens in
UTe2 like in other spin singlet superconductors. There is yet no definite
interpretation of the significance of this resonance. Following the analogy
with the cuprates or iron pnictide superconductors, it might support
antiferromagnetic fluctuations as the dominant pairing mechanism [144].
However, the differences with these well-studied unconventional singlet
superconductors, and notably the location of this (broad) resonance in the
overdamped region above $2\Delta$ might indicate a different origin; for
example, the spectral weight thus redistributed in the superconducting phase
could arise from itinerant modes, connected to some Fermi surface nesting
features not directly involved in the pairing, or even from more local modes
[145].

###  5.6 Topological superconductivity

Potential topological properties of the superconducting excitations in UTe2
where put forward already in the very first publication UTe2 [1], initially
purely on general grounds due to the spin-triplet pairing. However, this
question became rapidly more focused after a first STM study claiming to have
detected a signature of chiral surface excitations [146]. This was followed by
Kerr-effect measurements [149, 96], as well as London penetration depth
measurements [131, 97] all claiming the detection of anomalous effects
connected to a form of chiral state.

####  5.6.1 Basics of Topological superconductivity

Realization of topological superconductivity (TSC) and Majorana fermions is
one of the current central issues in modern condensed matter physics.
Stimulated by the proposal to use Majorana fermions as qubits for quantum
computation [150], about two decades of intensive study have revealed the
conditions for the emergence of TSC [151, 152, 153]. Even apart from
application to quantum computation, TSC in itself is an intriguing topological
phase of matter, namely, a new type of unconventional superconductivity.
Classical definition of unconventional superconductivity relies on the
symmetry; non-$s$-wave superconductors are unconventional. Topology provides
another classification; topology of the wave function can be trivial or
nontrivial. From this respect, several platforms of TSC have been identified
(see the review [153]).

Figure 40: Platform of topological superconductivity.

Although there is no one-to-one correspondence between symmetry and topology,
they are closely linked to each other. Figure 40 is a rough sketch of the
platform of TSC based on the symmetry of superconductivity. We have a dilemma
for the materials design. The $s$-wave superconductors are ubiquitous in
nature. But, they are mostly trivial even from the viewpoint of topology. Fine
tuning of parameters may enable TSC [154, 155, 156, 157, 158, 159], and indeed
tremendous efforts have been devoted for the realization of topological
$s$-wave superconductivity in artificial nanowires and heterostructures.
However, it is considered that “evidence” for nanowires needs further
scientific verification [160].

On the other hand, spin-triplet superconductors are strong candidates for
topological superconductivity, as they are topologically nontrivial in most
cases [161, 162, 163, 164]. But, there is a disadvantage; spin-triplet
superconductors are rare in nature. Heavy fermion superconductors UPt3, UGe2,
URhGe, UCoGe, and UTe2 are strong candidates, as discussed in this article.

The $d$-wave superconductors fall into an intermediate category. They are less
rare than the spin-triplet superconductors, and fine tuning of parameters is
not necessary for realizing TSC [165, 166, 167]. However, heterostructure
[165, 167] or intense laser light [166] is required in the present proposals.

####  5.6.2 Possible topological superconducting states in UTe2

Topological phases protected by crystalline symmetry, named topological
crystalline superconductors or higher-order topological superconductors, are
not considered here. Indeed, existing proposals for UTe2 are for the TSC in
the canonical sense, which may be protected by the local symmetries, namely,
the time-reversal symmetry, particle-hole symmetry, and chiral symmetry. Such
topological phases are classified based on the topological periodic table, so-
called ten-fold way. A relevant part of the table for superconductors is shown
in Table 5.

Table 5:  A part of the topological periodic table relevant for superconductors. Topological classification is determined by the Altland-Zirnbauer symmetry class (AS class) and the spatial dimension $d$. Time-reversal symmetric superconductors belong to the DIII class, while time-reversal symmetry breaking superconductors belong to the D class. AZ class | $d=1$ | $d=2$ | $d=3$
---|---|---|---
D | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | 0
DIII | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$

The topological periodic table tells us that three-dimensional (3D)
superconductors are characterized by an integer topological invariant
($\mathbb{Z}$), which is known as the 3D winding number $\omega$, when the
time-reversal symmetry is preserved (DIII class). In other words, 3D
superconductors can be topological superconductors specified by the winding
number. On the other hand, all the 3D superconductors are topologically
trivial when the time-reversal symmetry is broken (“$0$” in the D class).
However, even when the 3D invariant is trivial or absent, superconducting
states can be distinguished from completely trivial states by the low-
dimensional topological invariants. For instance, the $k_{z}=0$ plane of the
Brillouin zone can be regarded as an effective two-dimensional (2D) system,
which can be specified by the 2D topological invariant, namely, the
$\mathbb{Z}_{2}$ invariant $\nu_{i}$ ($i=1,2,3$) in the DIII class or the
Chern number in the D class. If the 2D invariant is nontrivial, Majorana
fermion may appear on surfaces preserving the quantum number $k_{z}$.

Up to now, the possibility of TSC in UTe2 has been discussed mainly in
relation with the issue of time-reversal symmetry breaking. If the time-
reversal symmetry is broken, the 3D odd-parity superconductors are likely to
be Weyl superconductors [168], which are gapless superconductors with Weyl
nodes protected by topology. The topological protection is ensured by the
Chern number. Conversely, we have a chance to see TSC in the strong sense,
when the odd-parity superconductors preserve the time-reversal symmetry. The
2D and 3D topological invariants in the DIII class have been investigated
based on the band structure calculations for UTe2 [16]. The results in Table 6
predict the TSC when the Fermi surfaces are as Fig. 4(b) or Fig. 4(c). All the
odd-parity superconducting states, namely, the $A_{u}$, $B_{1u}$, $B_{2u}$,
and $B_{3u}$ states, are specified by at least one $\mathbb{Z}_{2}$ invariant.
The $A_{u}$ state is also a 3D topological superconductor with an odd winding
number, $\omega\in 2\mathbb{Z}+1$.

Table 6: Gap structures, nontrivial topological indices, and surfaces hosting stable Majorana states for the Fermi surfaces in Figs. 4(b) and (c).  FSs (i) or (ii) |  |  |
---|---|---|---
IR | Gap structure | Topological index | Surfaces
$A_{u}$ | Full gap | $(\omega,\nu_{1},\nu_{2},\nu_{3})$ | (100), (010), (001)
$B_{1u}$ | Point node ($\Lambda$) | $\nu_{3}$ | (100), (010)
$B_{2u}$ | Point node ($\Delta$) | $\nu_{2}$ | (100), (001)
$B_{3u}$ | Point node ($\Sigma,F)$ | $\nu_{1}$ | (010), (001)
Table 7: Gap structures, nontrivial topological indices, and surfaces hosting stable Majorana states for the Fermi surfaces in Fig. 4(d).  FSs (iii) |  |  |
---|---|---|---
IR | Gap structure | Topological index | Surfaces
$A_{u}$ | Full gap | $\omega\in 2\mathbb{Z}$ | Unpredicted
$B_{1u}$ | Full gap | Trivial | None
$B_{2u}$ | Point node ($\Delta,U$) | Trivial | None
$B_{3u}$ | Point node ($\Sigma,F$) | Trivial | None

If the Fermi surfaces are as Fig. 4(d), all the 2D $\mathbb{Z}_{2}$ invariants
are trivial, as shown in Table 7. Thus, only the $A_{u}$ state can be a
topological superconductor because the winding number must be zero for mirror-
even superconducting states [169], namely, the $B_{1u}$, $B_{2u}$, and
$B_{3u}$ states. We can also state that the winding number has to be an even
integer, $\omega\in 2\mathbb{Z}$, in the $A_{u}$ state. Estimation of $\omega$
requires detailed information of the pair potential and Fermi surfaces in this
case.

In the orthorhombic systems with $D_{2h}$ point group, double superconducting
transitions are needed to break the time-reversal symmetry (see section 5.2).
Thus, the time-reversal symmetric superconducting phase must exist at least
near the transition temperature, even if the time-reversal symmetry is broken
at zero temperature. In this phase, TSC can be predicted based on the topology
of Fermi surfaces, as we demonstrated in Tables 6 and 7. Experimental
verification of Fermi surfaces in UTe2 is awaited also for development in the
field of TSC.

####  5.6.3 Double superconducting transition.

(a)

(b)

Figure 41: (a) Double transition as seen on the specific heat of several UTe2
samples (taken from Ref. [149]). (b) Sample with a double transition from
Grenoble. After cutting in two parts, only the higher one, quite sharp,
appears in one of the two pieces. See also Ref. [170].

A main experimental debate on the superconducting state of UTe2 is whether or
not the double transition observed at ambient pressure (see Fig 41-(a)) is an
intrinsic feature or a simple problem of sample inhomogeneities. The
importance of this issue arises from the requirement of such an intrinsic
double transition in order to break time-reversal symmetry in a system having
only 1D IR: see section above and 5.2, as well as the discussion on [149].
Such a double transition implies that the two (mixed) irreducible
representations of the mow temperature phase are nearly degenerate, something
considered as rather unlikely [149].

Of course, another possibility would be that the superconducting transition is
first order, as this removes the necessity for the two irreducible
representations of being completely degenerate. The mechanism leading to a
first order superconducting transition might imply for example a simultaneous
ferromagnetic order (with a jump of magnetisation), or a volume change.
However, as already mentioned, no sign of a ferromagnetic order as been
observed below $T_{\rm sc}$ [110] and thermal expansion measurements only
detected a jump of the thermal dilatation, not a discontinuity of the volume
[37]. Moreover, no hysteresis at $T_{\rm sc}$ has been detected. Weakly first
order transitions can be difficult to detect, but the main experimental
efforts are directed these days to confirm or disclaim the intrinsic nature of
the double transition rather than explore the possibility of a (hidden) first
order transition.

To test whether the double transition is intrinsic or not requires either to
be able to show that the transition emerge from different parts of the
crystal, or at the opposite to show that the two transitions behave
differently under an external tuning parameter like magnetic field or
pressure. At first sight, the mere fact that sharp single transitions are
observed on the vast majority of UTe2 single crystals favours an extrinsic
origin; for some crystals $\frac{\Delta T_{\rm sc}}{T_{\rm sc}}$ can be as
small as $0.01$ ($1\%$), far sharper than usually observed in even very high
quality crystals of the reference heavy-fermion superconductor CeCoIn5 with
$\frac{\Delta T_{\rm sc}}{T_{\rm sc}}\approx 0.03$. Another unfavourable point
for an intrinsic origin is that, when the double transition is observed, the
peak heights and temperature splitting are not reproducible, at the opposite
of UPt3 for example; in UPt3, either broad transitions, or reproducible double
transitions (with improved sample quality) were observed [171]. By contrast in
UTe2, a single crystal grown in Grenoble and showing two distinct
superconducting transitions at $T_{\rm sc1}$=1.4 K and $T_{\rm sc2}$=1.6 K was
clearly inhomogeneous; after being cut in two pieces, one piece shows still a
double transition, while the second piece had only one sharp transition at
$T_{\rm sc2}$ (see Fig. 41(b)). The situation of the double transition in UTe2
seems closer to the case of PrOs4Sb12 [172] or URu2Si2 [173], where single
transitions could be isolated in parts of crystals showing initially double
transitions.

Conversely, in the first paper claiming an intrinsic double transition [149],
it was mentioned that the two transitions behaved differently under field
$H\parallel c$-axis. However, the large broadening of the transitions in this
field direction, which does not correspond to an extrema of the angular
dependence of $H_{\rm c2}$, makes it difficult to be fully convinced that the
difference is real. Remarkably, the difference between the lower and higher
transition temperature is field independent for all directions in the measured
field and temperature range [149].

Another attempt was made by studying the evolution under pressure of both
transitions [27]. However, the weakness of the specific heat jump at the upper
transition above a few kbars renders the identification of the second
transition quite uncertain toward zero pressure, and the proposed evolution of
the two transitions will not convince a sceptic. The most recent work done by
the same group [37] explores the possibility of determining the pressure
evolution of both transitions from the thermal dilatation measurement at zero
pressure, using Ehrenfest relations:

$\frac{dT_{\rm sc}}{dp}=\frac{\Delta\alpha V}{\Delta C_{\rm p}/T}$

Experimentally, the difficulty is to measure precisely the volume thermal
expansion jump $\Delta\alpha$ at $T_{\rm sc}$ (requiring a measurement in the
three crystallographic directions, with possible opposite signs, and sample
dimensions smaller along the $c$-axis) and the specific heat jump, on samples
where a double transition is usually associated with a rather strong
broadening of the transitions. It was claimed in [37] that an opposite sign
was detected for the pressure evolution of both transitions, which would be
reliable, even if the absolute values of the slopes is not so reliable (owing
to the experimental difficulties just mentioned above). However, like for the
direct pressure of the field measurements, a serious improvement on the sample
quality is required to confirm this claim, and have convincing support for the
intrinsic nature of this double transition. In the course of writing this
review, a new version of this work has been put on arXiv [170], with more
systematic measurements on samples from different batches, and is now claiming
that the double transition at ambient pressure in their samples is arising
purely from sample inhomogeneities.

Essentially, the experimental difficulty are the broad transitions on samples
with double transitions, and/or the weak amplitude of one of the two
transitions. And certainly, there is room for sample improvement; as explained
in [37, 102], the differences in the samples is expected to arise from a
difference in sample stoichiometry, which is indeed most difficult to control
reproducibility in chemical vapor transport synthesis of systems which are not
line compounds (UPt3 has a congruent melting). This is certainly a most urgent
problem to be solved, together with the availability of high quality crystals
with large enough dimensions along the $c$-axis.

####  5.6.4 Chiral superconductivity

Early attempts to detect a tiny magnetic signature coming from a potential
time reversal symmetry breaking in the superconducting state by muon
spectroscopy failed, due to the very strong relaxation induced by local low
energy magnetic fluctuations [44]. So the polar Kerr effect is up to now the
only measurement pointing directly to a time reversal symmetry breaking of the
superconducting state (see Fig. 42) [149]. Indeed, when cooling the sample in
zero field through $T_{\rm sc}$, a sizeable polar Kerr angle of maximum
amplitude 500 nrad is detected in some runs. This Kerr effect can be “trained”
by cooling in a very small field ($\pm$ 2.5 mT, $H\parallel c$-axis), which is
considered as resulting from a field-control of the formation of chiral
domains. Moreover, the field training along $c$-axis imposes a symmetry of
type $B_{3u}\pm iB_{2u}$ or $B_{1u}\pm iA_{u}$ for odd-parity order parameters
[149].

Figure 42:  Measurement of the Kerr angle in zero field, when cooling below
the superconducting transtition (at $T_{\rm sc}=1.6K$). On the same UTe2
sample, some runs (like run 1) show a growth of the Kerr angle below $T_{\rm
sc}$, while other do not (like run2). It has been argued that this difference
is due to the formation of domains in the zero field experiment. Figure taken
from Ref. [149].  Figure 43: Measurement of the temperature dependence of the
change of the London penetration depth in UTe2, along the $a$ (red), $b$
(blue) and $c$ (green) axes. The anomalous power law along the $c$ axis,
smaller than 2, is analyzed as arising form pairs of Weyl nodes of the gap,
supporting a chiral superconducting state in UTe2. From [97].

For such a mixture of different irreducible representations, no symmetry-
imposed nodes of the gap can survive. However, all the chiral states proposed
for UTe2 happen to be also non unitary states, with an order parameter of the
form $\bm{d_{1}}$ \+ i $\bm{d_{2}}$, where $\bm{d_{1}}$ and $\bm{d_{2}}$ are
two (real) ${\bm{d}}$-vectors belonging to two different IR. Hence, the
resulting non unitary states have two branches in their excitation spectrum,
with a gap proportional to:

$\Delta_{\rm\pm}\propto\sqrt{|\bm{d_{1}}|^{2}+|\bm{d_{2}}|^{2}\pm
2|\bm{d_{1}}\land\bm{d_{2}}|}$ (7)

Then, if for some wave-vector of the Fermi surface the conditions
$|\bm{d_{1}}|=|\bm{d_{2}}|$ and $\bm{d_{1}}\perp\bm{d_{2}}$ are met, there is
a gap node in the branch $\Delta_{\rm-}$ at this point [149], called a Weyl
node due to the chirality of the excitations. For both the $B_{3u}\pm iB_{2u}$
or $B_{1u}\pm iA_{u}$, four such nodes (at least) should exist in the mirror
planes $k_{x}=0$ and $k_{y}=0$, provided the Fermi surface exist at these
locations [149].

A quite complete picture of the possible nodes and their location according to
the relative weight of the 2 irreducible representations is given in the
supplemental part of [97]. Regarding comparison with experiments, the
constraint on the mixed irreducible representations from the Kerr effect
arises from symmetry considerations on the training effect. It remains to be
confirmed that the signal becomes non-zero only below the second transition,
which is not so clear from the raw data, and if the change in chiral domains
is the correct explanation for the absence or presence of a non-zero Kerr
angle depending on the runs.

The proposals from the London penetration depth measurements come from
anomalous responses. All measurements [48, 131, 97] find a $T^{2}$ dependence
for the low temperature variation of $\lambda_{L}$ when the field is along the
$c$-axis, probing “$a-b$” plane electrodynamics. However, in [131] an
anomalous response is found on the dissipative part, attributed to a surface
“normal fluid” of chiral excitations. Quantitatively, this phenomenological
explanation is however puzzling as it assigns to these excitations a coherence
length with a Fermi velocity of very light bands, around 100 times larger than
those deduced from the upper critical field measurements [110].

The work in [97] is the first which has measured the response in all three
directions and endeavours to extract the temperature dependence of the
superfluid density in the three crystallographic directions. It requires
careful analysis of the sample geometries and the possible values for the
London penetration depth at zero temperature, and high precision in the
measurements and corrections to extract the values along the c-axis. If a
$T^{2}$ dependence of $\Delta\lambda_{L}$ is found indeed at low temperature
for $\lambda_{L}$ along the $a$ and $b$ axis, a weaker power law ($\approx
T^{1.4}$) is found along the c-axis (see Fig. 43). This exponent can be
reproduced by calculations with a $B_{3u}+iA_{u}$ order parameter, having
possibly Weyl nodes in the neighbourhood of the $c$ and $b$ axes. From the
results presented in [97]-Supplement, it remains however difficult to
reconcile the power laws observed in the other directions with those obtained
from the model calculation. The very anisotropic Fermi surface of UTe2 might
be an explanation for this discrepancy, but clearly, more work is required to
settle the existence of these Weyl nodes.

Last but not least, the STM studies [146] which were first to propose a chiral
order parameter in UTe2, also detect two anomalous behaviours in their
spectrum. First, they find a very large residual density of states at zero
energy in the superconducting state (around 60 $\%$ of the normal state value
[146]), which could arise from the zero energy modes associated to Weyl nodes.
Second, they find a reproducible asymmetry in the spectroscopy of the
superconducting excitations near step edges on the surface (see Fig. 44),
which they attribute in a phenomenological way to some chirality of the
excitations. According to the surface orientation where these excitations are
observed, the chiral axis ($\bm{d_{1}}\land\bm{d_{2}}$) is required to lie
along the $a$-axis, Hence they suggest a $B_{1u}\pm iB_{2u}$ state, stating
also that more work is required to check if this interpretation is correct.

Figure 44: Asymmetric spectra (bottom) measured along step edges, with colors
(light to dark blue or red) corresponding to that of the dots on the top
figure, showing the position where they were recorded. Measurements performed
at 0.3K deep in the superconducting state. The asymmetry of the spectra is
interpreted as arising from tunneling processes at non-zero in plane momentum
near the step edges, probing the chiral dispertion of in-gap surface states.
From [146].

To conclude on this issue of topological chiral superconductivity, there are
presently several puzzling results and experiments, but still no converging or
even coherent proposals for the mixed IR order parameter that would explain
these results. The most urgent task is certainly to clarify the origin of the
observed double transition in the specific heat, as it put strong constraints
which will determine the framework for the possible explanations of the
results. And for quantitative comparison with predictions, a much more
complete knowledge of the Fermi surface is clearly mandatory.

##  6 High pressure phase diagram

###  6.1 Zero field measurements

A new surprise arises from pressure studies of UTe2. Surprisingly, two
superconducting phases $SC1$ and $SC2$ have been discovered for pressures
above 0.3 GPa in zero magnetic field. These superconducting transitions are
clearly observed in specific heat measurements [174, 27]. Initially $T_{\rm
sc}$ decreases with pressure, and above 0.3 GPa the lower transition
extrapolates linearly to zero for a pressure around 1.6 GPa. The high
temperature superconducting phase increases in temperature up to 3 K at $p\sim
1.2$ GPa, and is rapidly suppressed for higher pressures, likely by a first
order transition: above the critical pressure $p_{\rm c}\approx 1.6$ GPa a new
magnetically ordered phase emerges. This high pressure phase diagram,
presented in Fig. 45, has been confirmed by experiments performed in Grenoble
[174], Maryland [39] Los Alamos [27] and Oarai [175] (the critical pressure
varies slightly due to different pressure conditions in the different
experiments). A rather intriguing point is that the enhancement of $T_{\rm
sc}$ under pressure by roughly a factor of two compared to the zero pressure
value, which is similar to the field-enhancement at zero pressure of the
pairing constant $\lambda$ from 1 to 2 (and the associated $T_{\rm sc}$
increase from $\approx 1.5$ K to $\approx 3$ K), from $H=0$ up to the
metamagnetic transition at $H_{\rm m}$ (see section 5.5.1 and Fig. 29). This
suggests that increasing pressure favors a similar pairing reinforcement than
magnetic field along the $b$-axis.

Figure 45: Pressure temperature diagrams of UTe2 from different groups. Left
panel: From AC calorimetry in a diamond anvil cell with argon as pressure
medium, At zero pressure only a single transition is observed and above 0.3
GPa two distinct superconducting transitions appear. Superconductivity is
suppressed above 1.7 GPa, and a new magnetic phase is observed. Inset: From
resistivity measurements in a modified Bridgman cell with Fluorinert as
pressure medium (from Ref. [174]). Middle panel: From resistivity measurements
in a piston cylinder pressure cell with Daphne oil 7373 as pressure medium.
Superconductivity is suppressed above 1.5 GPa, and two magnetic transitions
are observed above (from Ref. [39]). Right panel: From resistivity (squares)
and AC calorimetry (circles, diamonds) performed in the same piston-clamp
pressure cell using Daphne oil 7373 as pressure medium. The shaded area marks
the region where resistivity and specific heat show different $T_{\rm sc}$
values, indicating sample inhomogeneity due to the first order nature of the
transition. In the magnetic ordered state, two different magnetic transition
are reported (from Ref. [27]).

All present studies of the high pressure phase diagram converge to the point
that the transition from the superconducting state to the magnetically ordered
state is first order, as indicated in Fig. 45, and there may be a small
pressure range where both superconductivity and magnetic order coexist. Direct
microscopic measurements to determine the magnetic order are not performed
yet, but recent experiments seem to conclude that the ground state above
$p_{c}$ is antiferromagnetic [27, 175, 40, 176].

The characteristic crossover temperature $T^{\ast}\approx 12$ K at $p=0$ (see
section 4.1), which marks the low temperature regime in UTe2, decreases
linearly with pressure down to $T=2$ K close to $p_{\rm c}$ (see Fig. 45) [39]
in good agreement with the thermal expansion data [36]. In Ref. [27], evidence
was given for a pressure-induced antiferromagnetic quantum critical point in
intermediate valence UTe2; a slight change of the valence from a dominant U3+
to a U4+ configuration was observed in X-ray absorption spectroscopy [27]; it
was claimed to be at the origin of the change of the magnetic anisotropy under
pressure on both sides of the critical pressure $p_{\rm c}$ [40].

###  6.2 Field measurements

Careful studies by resistivity and ac calorimetry under pressure demonstrate
the occurrence of multiple phase transitions induced by a magnetic field,
identified as superconducting transitions [59, 175]. Presently, these field-
induced transitions between superconducting phases have been explored only for
$H\parallel a$. In this field direction, a pronounced enhancement of $H_{\rm
c2}^{a}$ has been observed by resistivity at low temperatures at 0.5 GPa [59].
This was rapidly confirmed by ac calorimetry, uncovering moreover the
occurence of multiple superconducting phases below $H_{\rm c2}$, as shown in
Fig. 46.

For $H\parallel b$ under pressure, resistivity measurements show that $H_{\rm
c2}^{b}$ remains limited by $H_{\rm m}$, and that the pressure dependence of
$H_{\rm m}$ follows that of $T_{\rm max}$ (see Fig. 47) [59]. It has been also
observed that the singular field enhancement of superconductivity vanishes
above 1 GPa. By tunnel diode oscillator experiments a transition from a low
field superconducting phase to a high field phase (the reinforced
superconductivity) has been evidenced and it has been stressed that the low
field phase disappears near 1 GPa, and for higher pressures up to $p_{\rm c}$
only the (at $p=0$) high - field phase is observed.[177] However, this demands
still confirmation by thermodynamic measurements under high pressure.

Figure 46: Magnetic field $H$ versus temperature $T$ phase diagrams of UTe2
for field applied along the $a$ axis at different pressure from ac calorimetry
measurements. Line are guides to the eyes. (taken from Ref.[175].

For $H\parallel c$, the upper critical field $H_{\rm c2}^{c}$ has been
determined by magnetoresistivity and also by ac calorimetry. On approaching
the critical pressure $p_{\rm c}$, a strong increase and an almost vertical
slope of $\partial H_{\rm c2}/\partial T$ is observed [174, 59]. Close to the
critical pressure at $p_{\rm c}=1.45$ GPa, long range antiferromagnetic order
(MO) seems to restrict the superconducting domain to a small region at high
magnetic fields and low temperatures. At $p=1.55$ GPa no superconductivity is
observed in zero field, however field-reentrant superconductivity seems to
survive just above the critical field $H_{\rm MO}$ of the magnetic order.
Finally, at 1.61 GPa, no trace of superconductivity is observed anymore [178].

Figure 47: Magnetic field versus temperature phase diagram for different
pressures. Full symbols indicate the upper critical field, open squares to the
metamagnetic field $H_{\rm m}$ and open circles to a crossover above the
critical end point, which is marked by the arrows $H_{\rm m}$. Lines are
guides to the eyes. (taken from Ref. [59].

The phase diagrams for different pressures on both sides of $p_{\rm c}$ is
shown in Fig. 48. From a maximum of the resistivity data for field along the
three axis, another line can be drawn at higher fields or temperatures than
the MO phase, delimiting a “weak magnetic ordered phase” (WMO). This WMO phase
is found by the different groups and different techniques used to probe UTe2
under pressure. Notably the specific heat experiment of Ref. [27], have well
demonstrated that if the MO phase is characterized by a sharp specific heat
jump, as expected for true long range magnetic ordered phases, only a rather
large specific heat anomaly occurs at the WMO line (called AFM1 in this work).
The WMO phase could be a phase with only short-ranged magnetic correlations.

As regards the origin of the different superconducting phases, there are still
competing explanations, and the most prominent are discussed in the
theoretical section 7.2. However, there are also experimental challenges. A
first one arises from the zero field phase diagram (Fig. 45), and the
appearance above $0.3$ GPa of a higher $T_{\rm sc}$ phase. The specific heat
anomaly at the highest $T_{\rm sc}$ transition is very small, and grows with
pressure, keeping a rough balance with the lower one: the proportion is
opposite when the lowest $T_{\rm sc}$ transition is suppressed. A first
natural explanation is a spatially inhomogeneous transition, with a sample
volume split between the two superconducting phases, and a repartition
evolving with pressure. This would explain why the lower transition keeps the
same slope $\frac{dT_{\rm sc}}{dp}$ below and above 0.3GPa. Explaining why
such a “phase separation” would happen in UTe2 is not easy, notably because
the mechanism should be “intrinsic”, owing to the reproducibility of the
experiments by the different groups. A more appealing alternative is that
these two phases are coexisting homogeneously. Thermodynamically, the crossing
of the two lines in the phase diagram of Fig. 45 without a change of slope of
$\frac{dT_{\rm sc}}{dp}$ is possible in an homogeneous sample if the specific
heat jump of the upper transition is vanishing at the crossing [174], which is
indeed the case experimentally within error bars. This coincidence of $T_{\rm
sc}$ and crossing of the lines, imposed by thermodynamics for homogeneous
transitions, has a priori no reason to exist in the inhomogeneous scenario.

Regarding the field dependence of these two phases as measured along the
a-axis (see Fig. 48), the change of behaviour of $H_{\rm c2}$ at low
temperatures is quite remarkable. In the higher $T_{\rm sc}$ phase, the very
strong negative curvature of $H_{\rm c2}$ suggests a dominant paramagnetic
limit for $H\parallel a$, close to the value expected for weak coupling
singlet superconductors. By contrast, at low temperatures, the larger $H_{\rm
c2}$ of the low $T_{\rm sc}$ phase indicates a suppression of the paramagnetic
limit, a feature probably very difficult to understand if the low $T_{\rm sc}$
phase is not spin triplet. Hence, even though many questions remain for a
detailed explanation of these pressure-field-temperature superconducting phase
diagrams, these $H_{\rm c2}$ measurements under pressure for $H\parallel a$
are probably one of the strongest support for a spin triplet superconducting
state in UTe2.

During the writing this review, a pressure study of the reentrant high-field
superconducting phase for field angles between 25∘ and 45∘ in the $b-c$ plane
has appeared [179]. Depending on the angle, between 0.8 and 1 GPa, the field
reinforced superconducting phase reaches $H_{\rm m}$. But for higher pressures
(1.3 GPa), remarkably, the superconducting phase above $H_{\rm m}$ appears
clearly detached from the metamagnetic field, suggesting that a different
mechanism is required for both phases below and above $H_{\rm m}$, as that
above $H_{\rm m}$ clearly requires high magnetic fields (and not only a strong
magnetic polarization). Additionally, one may wonder if the high field phases
measured close to $p_{\rm c}$ along the $a$ or $c$ axes are related to this
high field phase above $H_{\rm m}$. Following the angular dependence of the
reentrant superconducting phase above $H_{\rm m}$ under pressure is certainly
a good experimental challenge.

Figure 48: Phase diagrams for different pressures close to the critical
pressure for a magnetic field applied along the $c$ axis. (taken from Ref.
[178].

The combination of pressure and magnetic fields has lead to discover an
extremely rich multiphase-superconducting phase diagram, which is related to
competing interactions occurring already in the normal phase, as previously
discussed. This is also related to another unusual property of UTe2. Indeed,
as opposed to most other heavy-fermion superconductors, in UTe2, the
electronic Grüneisen parameter of the normal phase $\Gamma_{\rm e}\approx-30$
has the same sign as the Grüneisen parameter $\Gamma_{\rm
sc}\approx-\frac{\partial\log T_{\rm sc}}{\partial\log V}$ of the
superconducting phase, and with quite similar values [37]. In the other
U-based superconductors like UPt3, URu2Si2, and UBe13 [180] or ferromagnetic
superconductors like URhGe [181, 77] or UCoGe [78], $\Gamma_{\rm e}$ and
$\Gamma_{\rm sc}$ have opposite signs. This is natural as an increase of the
effective mass is associated with a decrease of $T^{\ast}$, and with an
increase of $T_{\rm sc}$ if it originates from the pressure dependence of the
pairing mechanism. The singularity points out that superconductivity in UTe2
is the result of different competing interactions.

The rather weak value of $\Gamma_{\rm e}\approx-30$ may appear quite
antagonistic to the location of UTe2 near a ferromagnetic or antiferromagnetic
second order magnetic instability. This value is close to those found in
intermediate valence compounds [180]. However, in the case of a ferromagnetic
instability, experiments as well as theory, generally points out that the
instability should be first order. Large Grüneisen parameters can be expected
only if the first order transition is weak. As reported below, recent
experiments performed in order to follow the pressure dependence of the
magnetization confirm the key role of valence fluctuations under pressure
[40].

Figure 49: (Upper panel) $M/H$ measured a a magnetic field of 1 T applied
along the $a$ axis for different pressures. (Middle panel) $M/H$ for a field
of 1 T along the $b$ axis. (lower panel) Pressure dependence of the maximum of
the $b$ axis susceptibility and the characteristic temperature $T^{\ast}$ of
the $a$ axis susceptibility. (taken from Reference [40]

The compressibility of UTe2 is rather high with a bulk modulus $G=57$ GPa
[38]. Applying a pressure of $p\sim p_{\rm c}\sim 1.5$ GPa leads to a volume
shrinking of 3%, i. e. roughly the same as measured on the melting curve of
3He going from the liquid to the solid phase [40]. For such volume shrinking,
a drastic change can occur in the electronic properties. In the specific case
of UTe2, the major phenomenon seems to be a switch from a low pressure phase,
where the magnetic properties are initially driven by the dominant U3+
configuration to a high pressure phase at $p>p_{\rm c}$, where the starting
dominant configuration will be that of U4+. Tiny pressure increase of the
volume can modify the magnetic anisotropy (see Ref. [182] for Tm
chalcogenide). The local consequence is a drastic change in the anisotropy of
the measured susceptibility: below $p_{\rm c}$ at low temperature $\chi_{a}$
remains higher than $\chi_{b}$, while above $p_{\rm c}$ the anisotropy is
reversed. In addition, similar to Kondo lattice insulators such as SmS and
SmB6 a complete change of the Fermi surface can be expected [40]. According to
LDA calculations under pressure, the energy gap at the Fermi energy closes
around $2.8\,{\rm GPa}$; the position of the 5$f$ level shifts upward under
pressure [183]. Figure 49 shows for $H\parallel a$ and $b$ the temperature
variation of $M/H$ measured at 1 T for different pressures. At 2 K the
magnetic hierarchy for a field of 5 T is $M_{a}\sim 3.5M_{c}\sim 6M_{b}$,
while above $p_{\rm c}$ at $p=1.7$ GPa it becomes $M_{b}\sim 1.2M_{a}\sim
3M_{c}$. For $p=1.7$ GPa, on cooling $\chi_{a}$ shows a maximum at $T_{\rm
WMO}$ but decreases strongly at the magnetic ordering temperature as observed
for antiferromagnetic materials along their axis where the magnetic moments
will be aligned. The location of UTe2 at ambient pressure close to its U3+
$5f^{3}$ configuration is in excellent agreement with core level spectroscopy
[24] but differs from angular dependent photoemission spectroscopy (ARPES)
analysis, where a valence close to $5f^{2}$ configuration has been claimed
[17] (see section 3.2). The Ising anisotropy observed at ambient pressure is
in good agreement with the crystal field splitting of the 5$f$ level with $\pm
J_{z}$ doublet ground state. Switching to the $5f^{2}$ configuration, a
drastic change in the crystal field scheme will appear; the $5f^{2}$
configuration is quite analogous to Pr3+ situation, with various crystal field
singlets.

Independent from the precise definition of the crossover temperature
$T^{\ast}$ at zero pressure (maxima of $C/T$, $\rho_{c}$ or
$\partial\rho_{a}/\partial T$, minima of $\alpha$ or $\partial M_{a}/\partial
T$), $T^{\ast}$ collapses just above $p_{\rm c}$ (see Fig. 49). The finite
value of $T^{\ast}$ at $p_{\rm c}$ suggests that the change from the U3+ to
U4+ configuration could be first order. The verification of the first order
nature of this valence change needs experimental detection, like a volume
discontinuity at $p_{\rm c}$. The situation is complex because pressure
modifies the relative strength of the correlations and at the same time
increases of the width of the electronic bands. Moreover, the instability at
$p_{\rm c}$ can be driven by a Fermi surface Lifshitz transition. An
interesting theoretical objective would be to precise quantitatively how the
anisotropy of the interactions is suppressed under pressure and what is its
role in the hierarchy of the different pairing channels.

##  7 Theoretical perspectives

###  7.1 Magnetism

Despite the claim of nearly ferromagnetic magnetic correlations in the first
paper [1], it is now expected that the magnetic correlations in UTe2 are not
limited to a single spin fluctuation component. The magnetic interactions and
fluctuations in UTe2 have been theoretically investigated based on the
localized and itinerant picture of $f$-electrons. The magnetic interactions
for localized $f$-electrons were calculated from first-principles [15], and
various magnetic structures in Fig. 50 including ferromagnetic and
antiferromagnetic states were predicted. The magnetic fluctuations due to the
itinerant $f$-electrons were calculated based on the model reproducing the
band structure for an intermediate Coulomb interaction (Fig. 4(c)). In this
calculation [184], a 24-band periodic Anderson model for the Te$5p$-, U$6d$-,
and U$5f$-electrons has been adopted, and an effective parameter $p$
multiplying the $f$-$f$ and $c$-$f$ hybridization is taken into account.
Because the hybridization in the heavy fermion states is enhanced by the
applied pressure, the parameter $p$ is considered as an indication of the
pressure effect. Fig. 51 shows the momentum dependence of the magnetic
susceptibility deduced from these calculations.

Figure 50: Magnetic structures proposed in Ref. [15].  Figure 51: Magnetic
susceptibility calculated based on the 24-band periodic Anderson model [184].
(a-f) $\chi_{a}(\bm{q},0)$, (g-l) $\chi_{b}(\bm{q},0)$, and (m-r)
$\chi_{c}(\bm{q},0)$. Momentum dependence on the $q_{x}$-$q_{y}$ plane at
$q_{z}=0$ is drawn for various parameters $p$ corresponding to the applied
pressure. Figures taken from Ref. [184].

Both Figs. 50 and 51 show an instability between ferromagnetic and
antiferromagnetic order on increasing respectively the Hubbard $U$ or the
hybridization. Thus, competing or coexisting ferromagnetic and
antiferromagnetic fluctuations are expected in UTe2. The dominant magnetic
fluctuations may change as a function of the magnetic field and applied
pressure. This is at least roughly consistent with the experimental
indications of antiferromagnetic correlation [42, 43, 27, 175] as well as
ferromagnetic one [1, 2, 41, 44]. From Fig. 51 it is predicted that the
dominantly ferromagnetic correlations change to antiferromagnetic correlations
on increasing the pressure. This appears consistent with experimental
observations [1, 2, 41, 44, 27, 175].

For the parameter at ambient pressure $p=1.0$, the ferromagnetic fluctuations
show a moderately Ising anisotropy, $\chi_{a}>\chi_{b}\simeq\chi_{c}$ and
$\chi_{a}/\chi_{b}\sim 4$. In a large pressure region $p>2$, the
antiferromagnetic fluctuations are dominant, and a weak anisotropy stabilizes
the antiferromagnetic order with magnetic moment along the a-axis. These
calculated results on magnetic anisotropy are surprisingly consistent with the
experiments although the degeneracy of the $j=5/2$ multiplet is not taken into
account in the 24-band periodic Anderson model. In this model, the magnetic
anisotropy arises from the local Rashba spin-orbit coupling which may appear
in locally noncentrosymmetric crystals [185, 186]. Indeed, UTe2 is a case,
where the global point group is centrosymmetric $D_{2h}$, but the local
symmetry of U sites is noncentrosymmetric $C_{2v}$ [184, 108]. Note that the
magnetic anisotropy plays an essential role in the following results on
superconductivity. It should also be noticed that the predicted magnetic wave-
vector of antiferromagnetic fluctuations is not equivalent to the experiment
at ambient pressure [43]. It is ${\bm{Q}}\parallel$[100] in Fig. 51, while
${\bm{Q}}\parallel$[010] in Ref. [43]. This discrepancy may be resolved by
taking into account the $j=5/2$ multiplet and/or tuning the hybridization
parameters.

###  7.2 Theories for superconductivity in UTe2

To clarify the superconducting phase of UTe2, microscopic theoretical studies
have been conducted based on the 24-band periodic Anderson model [184] and on
the model with exchange interactions [108]. In Ref. [184], the magnetic-
fluctuation-mediated superconductivity has been studied with employing the
same model as for Fig. 51, and the pressure dependence of the superconducting
instability has been predicted. In this model, at ambient pressure, the
ferromagnetic fluctuations stabilize odd-parity spin-triplet
superconductivity, and the moderately Ising-type magnetic anisotropy favors a
${\bm{d}}$-vector perpendicular to the a-axis, consistent with an intuitive
expectation. On the other hand, in the high pressure region, antiferromagnetic
fluctuations dominate and rather stabilize an even-parity spin-singlet
superconducting state.

Figure 52: (a) Eigenvalues $\lambda$ of the Eliashberg equation for various
IRs of $D_{2h}$ point group. The parameter $p\geq 1$ indicates applied
pressure. The Coulomb interaction is set so that the Stoner factor is
$\alpha_{\rm sf}=0.98$. (b) Transition temperatures of the $A_{u}$, $B_{3u}$,
and $A_{g}$ superconducting states for a fixed Coulomb interaction $U=1.9$.
Figures taken from Ref. [184].

Such a pressure dependence of superconductivity is theoretically captured by
the eigenvalues of the linearized Eliashberg equation [187]. The maximum
eigenvalue determines the stable superconducting state. Figure 52 shows the
$p$ dependence of the maximum eigenvalues for several IRs of the $D_{2h}$
point group, revealing stable odd-parity superconductivity at low pressures
$p\leq 2$ as well as stable even-parity superconductivity at high pressures
$p>2$. For the odd-parity superconductivity, the $B_{3u}$ state and $A_{u}$
state are almost degenerate. This is probably a consequence of the Ising-type
ferromagnetic fluctuations with $\chi_{a}>\chi_{b}\sim\chi_{c}$. Because the
dominant component of the pair potential is ${\bm{d}}({\bm{k}})=k_{b}\hat{c}$
in the $B_{3u}$ state and ${\bm{d}}({\bm{k}})=k_{b}\hat{b}$ in the $A_{u}$
state, these states are approximately related by a rotation of
${\bm{d}}$-vector in the $b$-$c$ plane. When the magnetic fluctuations are
nearly isotropic in this plane, the $B_{3u}$ state and $A_{u}$ state are
nearly degenerate. The other odd-parity superconducting states of different
IRs, namely, the $B_{1u}$ and $B_{2u}$ states, do not contain a pair potential
with orbital component $k_{b}$ and spin component perpendicular to the
$a$-axis, and therefore, these states are not stable. The even-parity
superconducting states are similarly classified into the $A_{g}$, $B_{1g}$,
$B_{2g}$, and $B_{3g}$ states (Table 1). Figure 52 shows that the $A_{g}$
state is stable in the high pressure region. Although this symmetry is
equivalent to the conventional $s$-wave superconductors, the order parameter
changes sign in the Brillouin zone like in the $s_{\pm}$-wave superconducting
state of iron-based superconductors. Thus, the $A_{g}$ state is gapless (line
node) [184]. By contrast, the $A_{u}$ state is nearly point-nodal but fully
gapped, while in the $B_{3u}$ state the quasiparticle spectrum has point
nodes. As we discussed in section 5.5.4, existing experimental results
indicate point nodes in the superconducting gap at ambient pressure. This is
consistent with the odd-parity $A_{u}$ and $B_{3u}$ states rather than with
the even-parity $A_{g}$ state.

After recent theoretical work [188], it has been recognized that the naive
expectation for the superconducting gap structure based on the point group
symmetry and basis functions [91] (see Table 1 for example) breaks down in
some cases. However, an exhaustive classification theory remains consistent
with the classical theory [189] for the orthorhombic and symmorphic UTe2
system [16]. We show the nodal structure of UTe2 predicted based on the band
structure calculation and symmetry analysis in Tables 6 and 7. When time-
reversal symmetry is broken, odd-parity superconducting states are likely to
possess Weyl nodes [168], which are topologically-protected point nodes.
Prediction of Weyl superconducting states is compatible with several
experiments claiming to have revealed point nodes of the gap (see e.g. [97]).
However, for more accurate comparison with predictions, we need detailed
information on the Fermi surface and pair potential, particularly for Weyl
nodes. Indeed, the Weyl nodes are not constrained by symmetries, so they can
appear at any point of the Fermi surface, depending on the amplitude of the
different components of the order parameter (see Ref. [149] and section
5.6.4). This is indeed an intriguing property of the Weyl nodes: Weyl points
in superconductors are not protected by crystalline symmetry in most cases,
although they are stable due to the topological protection.

Figure 53:  A theoretical interpretation of phase diagrams (a) in the $P$-$T$
plane and (b) in the $T$-$H_{a}$ plane. Figures taken from Ref. [184].

The analysis of the periodic Anderson model (Fig. 52), has allowed to propose
a first microscopic interpretation of the experimental superconducting phase
diagrams in the pressure-temperature ($P$-$T$) plane and the temperature-
magnetic-field ($T$-$H_{a}$) plane. This interpretation is presented on Fig.
53. A superconducting phase transition from the odd-parity $B_{3u}$ or $A_{u}$
state to the even-parity $A_{g}$ state occurs under the applied pressure,
coinciding with the crossover in magnetic fluctuations from ferromagnetic to
antiferromagnetic. The coexisting mixed-parity superconducting state would be
stabilized in the intermediate pressure region (Fig. 53(a)). In the mixed-
parity superconducting state, either the $B_{3u}$($A_{u}$) + $A_{g}$ state or
the $B_{3u}$($A_{u}$) + $iA_{g}$ state are expected [190, 191, 192], and space
inversion symmetry is spontaneously broken. The former is similar to states
from noncentrosymmetric superconductors, where the crystalline inversion
symmetry is broken [98, 99]. The latter is more exotic and breaks both
inversion and time-reversal symmetries. These states are distinguished by the
preserved symmetry; the time-reversal symmetry is preserved in the
$B_{3u}$($A_{u}$) + $A_{g}$ state, while the combination of space inversion
and time-reversal operations, called $PT$ symmetry, is preserved in the
$B_{3u}$($A_{u}$) + $iA_{g}$ state. From the viewpoint of symmetry, the former
corresponds to the odd-parity electric multipole state, while the latter
corresponds to the odd-parity magnetic multipole state [94]. When the
conventional BCS theory for intraband pairing applies, the latter is stable
[191], and thus, the $B_{3u}$($A_{u}$) + $iA_{g}$ state is the most likely in
UTe2.

Such a spontaneously parity-mixed pairing state was referred to in a review
article for 3He published forty years ago with the comment “there seems at
present no experimental evidence which require this hypothesis” [45]. Even
now, such experimental evidence has not been reported. Hence, it is desirable
to examine whether UTe2 may be the first material where such a state would
exist. Even apart from issues of UTe2, smoking gun experiments for the
inversion symmetry breaking in superconductors are awaited. At present, no
experiments have detected an inversion symmetry breaking in the bulk, unlike
for the time-reversal symmetry breaking and rotation symmetry breaking. The
polar Kerr effect and $\mu$SR measurements have been widely used as probes of
broken time-reversal symmetry in superconductors [193]. The rotation symmetry
breaking in superconductors, named nematic superconductivity, can be studied
by various bulk measurements [194]. On the other hand, theoretical exploration
of the spontaneously mixed-parity superconducting state is now in progress
[192, 195, 196].

Corresponding to Fig. 46, the $T$-$H_{a}$ phase diagram under pressure is
drawn in Fig. 53(b). A purely odd-parity superconducting state will be stable
in the high field region because the paramagnetic depairing effect is partly
avoided. This phase is denoted as $A_{u}+B_{3u}$ because the $B_{3u}$ and
$A_{u}$ states can not be distinguished by symmetry under a magnetic field
$H\parallel a$ (see Table 2). Compared with Fig. 53(b) we see rich phases in
the experimental phase diagram (Fig. 46). This may be due to the degeneracy of
the $B_{3u}$ and $A_{u}$ states. For instance, the $B_{3u}$ dominant state and
the $A_{u}$ dominant state may appear in the phase diagram as
thermodynamically distinct phases.

In this section, we have discussed a scenario based on the ferromagnetic-
antiferromagnetic crossover. However, the symmetry of superconductivity and
the interpretation of the multiple phases in UTe2 are far from settled. Even
at ambient pressure, experimental results have been interpreted based on
various scenarios, as shown in Table 4. Accordingly, theories proposed various
superconducting states. Most of them assume purely odd-parity spin-triplet
superconductivity, such as the $B_{3u}+iB_{2u}$ state [108] and the
$B_{3u}+iB_{1u}$ state [19]. Assuming that long-lived ferromagnetic
fluctuation plays the same role as the ferromagnetic long-range order and
ignoring the spin-orbit coupling, a spin-triplet pairing state with
${\bm{d}}({\bm{k}})=(k_{b}+ik_{c})(\hat{b}+i\hat{c})$ was also proposed [109,
106, 197]. This state is classified into a $B_{3u}+iA_{u}$ state in the
terminology of IR. Firm theoretical grounds for the assumptions have not been
obtained up to now. To settle these issues, a close comparison between
theories and experiments, which is lacking at present, is highly desired. In
particular, a smoking gun experiment revealing the symmetry breaking is
awaited.

As detailed discussions have already exposed (sections 4.3, 4.4), the nature
of magnetic correlations in UTe2 and its pressure and magnetic field
dependence are still open issues. In particular, only the antiferromagnetic
fluctuation have been observed in the neutron scattering [43, 42], despite
various indications for the ferromagnetic correlation and spin-triplet
superconductivity. This issue is essential for the mechanism of
superconductivity. Comparing the theoretical results in Fig. 51 and Fig. 52,
we notice that the spin-triplet superconductivity is stable in the
ferromagnetic-antiferromagnetic crossover region around $p=2$. Thus, strictly
ferromagnetic correlations are not necessary for spin-triplet
superconductivity.

##  8 Conclusions

UTe2 appears to be an exciting new paramagnetic heavy fermion superconductor.
Regarding the superconducting phase, the number of experimental and
theoretical investigations triggered by the first publication [1] three years
ago is certainly a record in the field of heavy-fermion superconductors. This
enthusiasm has been supported by the heady prospects of topological spin
triplet superconductivity, the stunning discoveries of high-field reinforced
and induced superconducting states, as well as pressure-induced exotic
superconducting phase diagrams. Naturally, for such a “young” system, most
results and hypothesis require confirmation.

Progress has already been done on the crystals, displaying now superconducting
transitions larger than $1.8$ K, smaller residual specific heat terms, but
still not good enough purity which would allow for the observation of quantum
oscillations. Hence, the low temperature bulk Fermi surface, and the
experimental determination of the electronic structure are still
controversial.

The difficult subject of topological superconductivity and possibly Weyl nodes
requires first of all a clarification of the origin of the double
superconducting transition, observed in some samples, a target probably
accessible with improvements of the crystals, as well as the pressure and
thermal dilatation measurements already performed.

Determining the role of the competing magnetic fluctuations in the pairing
mechanism for the different regions of temperature, pressure and magnetic
field is probably the greatest challenge, like in all unconventional
superconductors. It requires a precise experimental knowledge and theoretical
description of the normal phase of UTe2. Large progress has already been
achieved by neutron scattering and magnetic resonance experiments, but the
role of short range ferromagnetic fluctuation has still to be clarified. The
rapid progress of theoretical approaches, starting from realistic band
calculations of these complex materials to propose predictions on the magnetic
and superconducting properties brings a new hope. UTe2 is certainly an
excellent case study for these new methods, owing to the richness of its phase
diagrams, and stimulated by the amazing possibility of mixed singlet-triplet
states due to spontaneous space inversion symmetry breaking in the
superconducting state.

## Acknowledgments

We thank our colleagues D.Braithwaite, W. Knafo, G. Lapertot, A. Miyake, Q.
Niu, C. Paulsen, A. Pourret, A. Rosuel, S. Raymond, G. Seyfahrt, I. Sheikin,
M. Vališka, S. Rousseau, C. Marcenat, T. Helm, M. Kimata, F. Honda, D. Li, Y.
Homma, Y. J. Sato, Y. Shimizu, A. Nakamura, G. Nakamine, K. Kinjo, S.
Kitagawa, H. Sakai, and S. Kambe for collaborations in the various
experiments. We also thank J. Ishizuka, S. Sumita, and A. Daido for
collaborations. We acknowledge very helpful critical reading of the manuscript
by M. Houzet, J.-P. Sanchez, and H. Suderow. We further thank for fruitful
discussions K. Asayama, H. Harima, K. Miyake, K. Machida, M. Houzet, M.
Zhitomirsky, V. Mineev, S. Fujimoto, S. Fujimori, K. Izawa, S. Kittaka, T.
Sakakibara, S. Imajo, Y. Kohama, K. Willa, F. Hardy, C. Meingast, and D. F.
Agterberg.

We have got financial support from the Cross-Disciplinary Program on
Instrumentation of CEA, the French Alternative Energies and the Atomic Energy
Commission, the French National Agency for Research ANR within the project
FRESCO No. ANR-20-CE30-0020 and FETTOM ANR-19-CE30-0037, the CEA Exploratory
program TOPOHALL, JSPS KAKENHI (JP18H05227, JP18H01178, JP20H05159,
JP19H00646, JP20K20889, JP20H00130, JP20KK0061), SPIRITS 2020 of Kyoto
University, and ICC-IMR. We acknowledge support of the LNCMI-CNRS, member the
European Magnetic Field Laboratory (EMFL).

##  9 References

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