# Robust nodal behavior in the thermal conductivity of superconducting UTe2

Ian M. Hayes  Maryland Quantum Materials Center, Department of Physics,
University of Maryland, College Park, MD 20742, USA.  Tristin E. Metz
Maryland Quantum Materials Center, Department of Physics, University of
Maryland, College Park, MD 20742, USA.  Corey E. Frank  Maryland Quantum
Materials Center, Department of Physics, University of Maryland, College Park,
MD 20742, USA.  NIST Center for Neutron Research, National Institute of
Standards and Technology, Gaithersburg, MD 20899, USA.  Shanta R. Saha
Maryland Quantum Materials Center, Department of Physics, University of
Maryland, College Park, MD 20742, USA.  NIST Center for Neutron Research,
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA.
Nicholas P. Butch  Maryland Quantum Materials Center, Department of Physics,
University of Maryland, College Park, MD 20742, USA.  NIST Center for Neutron
Research, National Institute of Standards and Technology, Gaithersburg, MD
20899, USA.  Vivek Mishra  P.J. Hirschfeld  Department of Physics, University
of Florida, Gainesville, FL, 32611-8440  Johnpierre Paglione<EMAIL_ADDRESS>Maryland Quantum Materials Center, Department of Physics, University of
Maryland, College Park, MD 20742, USA.  The Canadian Institute for Advanced
Research, Toronto, Ontario, Canada.

###### Abstract

The superconducting state of the heavy-fermion metal UTe2 has attracted
considerable interest because of evidence for spin-triplet Cooper pairing and
non-trivial topology. Progress on these questions requires identifying the
presence or absence of nodes in the superconducting gap function and their
dimension. In this article we report a comprehensive study of the influence of
disorder on the thermal transport in the superconducting state of UTe2.
Through detailed measurements of the magnetic field dependence of the thermal
conductivity in the zero-temperature limit, we obtain clear evidence for the
presence of point nodes in the superconducting gap for all samples with
transition temperatures ranging from 1.6 K to 2.1 K obtained by different
synthesis methods, including a refined self-flux method. This robustness
implies the presence of symmetry-imposed nodes throughout the range studied,
further confirmed via disorder-dependent calculations of the thermal transport
in a model with a single pair of nodes. In addition to capturing the
temperature dependence of the thermal conductivity up to $T_{c}$, this model
allows us to limit the possible locations of the nodes, suggesting a B1u or
B2u symmetry for the superconducting order parameter. Additionally, comparing
the new, ultra-high conductivity samples to older samples reveals a crossover
between a low-field and a high field regime at a single value of the magnetic
field in all samples. In the high field regime, the thermal conductivity at
different disorder levels differ from each other by a simple offset,
suggesting that some simple principle determines the physics of the mixed
state, a fact which may illuminate trends observed in other clean nodal
superconductors.

Ever since the discovery of equal-spin pairing in superfluid 3He-A, there has
been a search for an analogous equal-spin paired state in a
superconductorLeggett (1975). This search was given new impetus by the
realization that a triplet superconductor could host majorana zero modes that
would enable fault-tolerant quantum computation.Nayak  _et al._ (2008) The
discovery of superconductivity in uranium ditelluride (UTe2) was a watershed
moment for the field because the superconducting state immediately showed two
clear indications of triplet pairing. The first was that the upper critical
field $H_{c2}$ exceeds the Pauli limit for all orientations. The second was
that the nuclear magnetic resonance (NMR) Knight shift–which is a measure of
the local spin susceptibility in the superconducting state–shows almost no
change through the superconducting transition temperature $T_{c}$.Ran  _et
al._ (2019); Fujibayashi  _et al._ (2022) Both of these phenomena are hard to
obtain unless the electron spins are aligned in the superconducting state.

These observations triggered extensive efforts to characterize the orbital
part of the order parameter. Early measurements of the thermal conductivity
and penetration depth pointed to a superconducting gap function with point
nodes.Metz  _et al._ (2019) Comparison of the penetration depth in different
directions favored a multi-component gap structure with nodes away from the
high symmetry axes, an observation that made sense in the context of magneto-
optic Kerr rotation measurements that indicated that the superconducting order
parameter broke time reversal symmetry.Ishihara  _et al._ (2023); Hayes  _et
al._ (2021)

In parallel, progress on the synthesis of UTe2 yielded samples with notably
higher $T_{c}$ values (up to 2 K compared to 1.6 K) and significantly reduced
residual resistivity ($\rho(0K)$) values, indicating significantly reduced
levels of disorder.Rosa  _et al._ (2022); Sakai  _et al._ (2022) Although the
$H_{c2}$ values of these samples also exceed the Pauli limit, they differ in
several other respects from the first generation samples. In particular, the
evidence for time reversal symmetry-breaking did not reproduce Ajeesh  _et
al._ (2023) and the NMR Knight shift change through $T_{c}$ was observed to
occur in all crystallographic directions Matsumura  _et al._ (2023). These
observations have been interpreted to favor a single-component order
parameter, and in some cases a nodeless gap function Matsumura  _et al._
(2023); Suetsugu  _et al._ (2023), which in turn suggests that key features of
the superconducting state were inaccessible in the older generation of
samples.

Here we use measurements of the thermal conductivity $\kappa$ to investigate
this possible disorder-dependent evolution of the superconducting state in
UTe2. We study both the finite and zero-temperature limits in samples that
vary by a factor of $\sim$20 in residual resistivity, and $\sim$30% in
$T_{c}$. Throughout this range of sample quality we are able to confirm the
presence of point nodes. This conclusion rests on two qualitative features in
the data: the absence of a nonzero intercept in $\kappa/T$ at zero
temperature, which rules out line nodes, and an immediate increase in this
intercept with the application of magnetic field, which rules out a full gap.
Our wide range of disorder levels allows us to confirm this nodal structure by
performing calculations of $\kappa/T$ in a simple model of an axial $p$-wave
state as functions of both temperature and magnetic field, assuming scattering
of disorder and inelastic processes. The calculations are successful at
capturing the evolution of the peak below $T_{c}$ with increasing disorder.
Because thermal conductivity is direction-sensitive, these calculations also
exhibit different features when the nodes are parallel to or perpendicular to
the direction of current flow. Comparing these two scenarios with the data
supports the conclusion that the nodes are not near the crystallographic
$a$-axis. Given our best current understanding of the Fermi surface and the
current evidence in favor of a single-component order parameter, this implies
a gap structure belonging to either the B2u or B3u irreducible representation
of the crystal point group, consistent with conclusions reached in recent
ultrasound and heat capacity experiments.Theuss  _et al._ (2023); Lee  _et
al._ (2023) Finally, the field dependence of thermal transport in the zero
temperature limit, $\kappa(H,T$$\rightarrow$0)/$T$, exhibits a crossover near
1 T to a regime where it grows at a uniform rate independent of the level of
disorder, suggesting a universal behavior in the mixed state that is
reminiscent of other ultra-clean nodal superconductors.

##  I Results and Discussion

Figure 1: Variation of the $a$-axis thermal conductivity of UTe2 with
disorder. The measured thermal conductivity of three samples of UTe2 with
varying residual resistance ratios (RRRs) of 30 (S1, green), 50 (S2, blue) and
600 (S3, purple) and superconducting transition temperatures of 1.6 K (S1),
1.8 K (S2) and 2.1 K (S3) are presented as a function of temperature. The
overall shape of the curves is similar for all three samples, but with
noticeably stronger peaks below $T_{c}$ in the higher RRR samples. The
estimated electron contribution to the total thermal conductivity estimated by
the Weidemann-Franz law ($L_{0}/\rho(T)$) is shown for each sample as a thin
line. The inset shows the measured thermal conductivity data below 0.5 K, with
clear limiting behavior approaching $\kappa/T$=0 as $T\to 0$ for all samples.

Data are presented on four single-crystal samples of UTe2 with varying Tc
values and normal state scattering rates as deduced from their residual
resistivity ratios (RRR = $\rho(300~{}$K$)/\rho(0~{}$K$)$). Sample S1
($T_{c}$=1.6 K, RRR=30), grown by the chemical vapor transport (CVT) method
and whose thermal conductivity was previously reported Metz  _et al._ (2019),
is compared to sample S2 ($T_{c}$ =1.8 K,RRR=50) grown by CVT at lower
temperatures (950C and 860C), and samples S3 ($T_{c}$ =2.1 K,RRR=600) and S4
($T_{c}$ =2.1 K, RRR=100) grown by a novel tellurium flux method described in
the Materials and Methods section.

Electrical and thermal transport of samples S1, S2, and S3 are presented in
Figure 1, plotted as the total measured thermal conductivity $\kappa/T(T)$ and
the electron contribution $L_{0}/\rho(T)$ for each sample estimated from the
Wiedemann-Franz law, $\kappa/T=L_{0}/\sigma$ (all for currents directed along
the crystallographic $a-$axis). In the normal state, all three samples show
qualitatively similar behavior as a function of temperature, albeit with
larger absolute values of $\kappa/T$ in the samples with lower residual
resistivities. Below $T_{c}$, all three samples exhibit a significant increase
in $\kappa/T$ that peaks near $\sim$1 K but with a peak magnitude that varies
considerably between the samples, being the most pronounced in S3 followed by
smaller magnitudes for S2 and S1, respectively. The magnitude of this
enhancement varies inversely with the residual resistivity of each sample,
also reflected in the RRR values (RRR= 30, 50, and 600 for samples S1, S2 ans
S3 respectively). As explained below (see ”Theorretical Analysis” Section),
this behavior is qualitatively consistent with that of many unconventional
superconductors, which frequently exhibit such enhancements in $\kappa/T$
below the superconducting transition that scale inversely with the elastic
scattering rate.

Figure 2: Thermal transport of UTe2 in the superconducting state. Data are
presented for four comparative samples with varying $T_{c}$ and RRR values as
indicated. Panels (a), (b), and (c) present the magnetic field evolution of
$\kappa(T)/T$ at low temperatures for the new samples measured in this study.
Qualitatively similar behavior is observed in all samples, with zero-field
data trending clearly toward zero conductivity values as $T\to 0$ for all
samples, and rapid increases of the extrapolated value with increasing
magnetic fields. The extrapolated $T\to 0$ values for each curve are plotted
as a function of magnetic field $H$ in panels (d), (e) and (f). Panel (d)
plots the full field range up to the upper critical field $H_{c2}$, with data
normalized to its normal state value above $H_{c2}$ and plotted against
normalized field $H/H_{c2}$, showing the apparent depression of conductivity
with increasing RRR value of sample. Panel (e) presents the same normalized
data as panel (a) but zoomed in to the very low field range where the
universal rapid initial rise is evident. Panel (f) presents the same data but
plotted as a function of absolute conductivity and magnetic field values,
demonstrating the enhancement of absolute conductivity with increasing RRR and
$T_{c}$ values, but also the remarkably consistent rate of growth above 1 T
fields as discussed in the text. The inset to Panel (f) shows a field sweep of
sample S3 at 50mK, demonstrating that the peak shown in the zero temperature
extrapolation is not an artifact.

As discussed below, the enhanced thermal conductivity below $T_{c}$ can be
naturally explained using model calculations that vary the electron scattering
rate, but these models also require a mixture of electron and phonon
contributions. To start, we focus on the most reliable method of measuring the
electronic response due to the superconducting gap structure, which is to
study the evolution of the zero-temperature extrapolation of $\kappa/T$ with
magnetic field. In this limit phonons necessarily vanish and the behavior of
the extrapolated value, $\kappa(T\rightarrow 0,H)/T$, can only reflect the
properties of the superconducting state. In this regime, the change of the
thermal conductivity under a small magnetic field is dominated by a shift in
the quasiparticle spectrum that is induced by the presence of the supercurrent
associated with vortices.Shakeripour  _et al._ (2009) If the quasiparticle
spectrum has a gap, then a nonzero magnetic field is required before
$\kappa/T$ exhibits any increase. In contrast, if there are nodes in the
spectrum then at arbitrarily small fields the density of states will be
enhanced and $\kappa/T$ will increase continuously from zero field. This
effect has been used to diagnose the presence of nodes in the gap structure of
many superconductors Shakeripour  _et al._ (2009), including in the earliest
work on UTe2 Metz  _et al._ (2019).

Figure 2 presents the temperature and magnetic field dependences of the
fermionic thermal conductivity, $\kappa(T,H)/T$, for samples S2, S3, and S4,
plotted as a function of temperature for various fixed fields
($\kappa(T,H)/T$, panels a-c), as well as a function of field in the $T\to 0$
limit ($\kappa_{0}(H)/T$, panels d-f). In all of these measurements, the
magnetic field was applied in the longitudinal configuration, with field
parallel to both the heat current and the crystallographic $a-$axis. The
overall shape of $\kappa(T,H)/T$ is qualitatively similar for all samples,
with a rapid increase with temperature that is mildly dependent on magnetic
field strength at low fields, but with a residual $T\to 0$ intercept
$\kappa_{0}/T$ that exhibits a strong field dependence. As shown in Figs.
2d)-f), $\kappa_{0}/T$ undergoes an initial rapid rise as the field is
increased from zero, followed by an intermediate field regime where it grows
more slowly. Once the magnetic field is increased to nearly $H_{c2}$,
$\kappa_{0}/T$ rises rapidly to the normal state value. We study the low field
regime most fully in the highest RRR sample, S3, where data were taken down to
30 mK for six magnetic fields up to 0.5 T provides the most robust $T\to 0$
extrapolations (Fig. 2c). $\kappa(T)/T$ exhibits a clear, rigid upshift with
increasing fields starting as low as 50 mT. We emphasize that this rise at low
fields is readily visible in our raw data at the lowest temperatures and is
therefore not sensitive to the details of the $T\rightarrow 0$ extrapolation.
However, because the temperature growth in $\kappa/T$ is stronger than linear,
this extrapolation can be misleading if data is not measured below $\sim$100
mK, as has been reported recently.Suetsugu  _et al._ (2023)

The idea that this low field rise is coming from a Doppler-shifted density of
states is consistent with the variation in the magnitude of this effect in
these samples. To factor out changes coming from variation in the mean free
path of the different samples we can use the zero temperature value of
$\kappa/T$ in the normal state above $H_{c2}$. In the normal state all of
these samples should have the same carrier density, so changes in
$\kappa_{0}/T$ should scale with the mean free path. In Fig 2e) we plot
$\kappa(T\rightarrow 0,H)/\kappa(T\rightarrow 0,H_{c2})$ as a function of
normalized magnetic field, $H/H_{c2}$. All for samples show a comparable
slope, suggesting that they have a common density of states that is being
tuned by the magnetic field.

Further support for the Doppler shift interpretation comes from the fact that
$\kappa/T$ rises immediately starting at zero field. There are various
scenarios in which a fully gapped superconductor can exhibit a small rise in
$\kappa/T$ at magnetic field values well below $H_{c2}$, including the
presence of very deep gap minima and multi-gap scenarios. However, in any
scenario with a full gap there will be a finite field range within which
$\kappa/T$ is approximately zero, with the rise in $\kappa/T$ accelerating
only as the magnetic field overcomes the minimum of the gap, leading to a
concave up shape for $\kappa_{0}(H)/T$. By contrast, the extrapolated value of
$\kappa(T,H)/T$ grows most quickly near zero field for all of our samples,
allowing us to conclusively rule out a fully gapped superconducting state in
UTe2 at low pressures and magnetic fields.

Having concluded that UTe2 poses nodes, the next question is whether these are
point nodes or line nodes. There is a well-known theoretical result that in a
superconductor with line nodes the residual value of $\kappa/T$ in the zero
temperature limit is bounded from below.Graf  _et al._ (1996) This has been
found to be the case empirically in several line-node superconductors
including KFe2As2 and YBCO Reid  _et al._ (2012). This universal residual term
is given by
$\kappa_{0}/T=\frac{1}{3}\gamma_{0}v_{f}^{2}\frac{a\hbar}{2\mu\Delta_{0}}$
Graf  _et al._ (1996). Using the best estimates from the literature for
$2\Delta_{0}$=$3.5k_{B}T$, $\gamma_{0}$=125 mJ/mol K2, $\mu=2$ and
$v_{f}=10,000$ m/s, this yields a residual term of about 0.25 mW/K2cm. As
shown in the Fig. 1 inset, the zero-field, ultra-low temperature behavior of
$\kappa/T$ for samples S1, S2, and S3 clearly extrapolates to well below 0.25
mW/K2cm at $T$=0, violating the minimum value expected for a line node
superconductor. Together with the evidence for a rapid rise in $\kappa_{0}/T$
at low fields, this provides conclusive evidence for symmetry imposed point
nodes in the gap structure of UTe2. This is consistent with conclusions from
nearly all thermodynamic experiments to date – including specific heat Ran
_et al._ (2019); Rosa  _et al._ (2022); Kittaka  _et al._ (2020), penetration
depth Metz  _et al._ (2019); Ishihara  _et al._ (2023) and NMR Nakamine  _et
al._ (2021, 2019) – and, together with the theoretical analysis presented
below, point to a gap structure consistent with point nodes lying away from
the crystallographic $a$-axis.

Finally, we identify some intriguing universal behavior of $\kappa_{0}(H)/T$
in the intermediate field regime. As discussed above and shown in Fig 2 panels
e) and f), the higher RRR samples show a more rapid rise in $\kappa/T$ with
applied magnetic field. However this trend is confined to a region below 1 T.
At higher fields, the unnormalized $\kappa(T\rightarrow 0,H)/T$ grows at a
nearly uniform rate in all samples, despite the factor of $\sim 20$ difference
in their residual resistivities. As the disorder is reduced, the conductivity
is simply increased by a fixed amount independent of field. Generically, the
physics of the mixed state is expected to be quite complicated since it
involves quasiparticle dynamics in the inhomogeneous medium of a vortex
lattice, as well as the dynamics of the vortices themselves. The fact that
disorder influences this regime in an extremely simple way in UTe2 strongly
suggests that there is a simple principle or physical picture that determines
$\kappa/T$ in this regime. This potentially opens up the mixed state to more
thorough theoretical treatment than has been attempted previously.

The most important observation to be made about the data in this regime is
that this simple pattern exists in $\kappa(T\rightarrow 0,H)/T$ as a function
of disorder. However, we offer a few thoughts about what it might imply for
the physics of the mixed state. First of all, this separation – between an
overall conductivity scale that is sensitive to disorder, and a field
dependence that is not – is suggestive of parallel conductivity channels. If a
resistivity shows a simple offset between samples with different disorder
levels, it is easy to interpret that behavior as arising from the presence of
multiple scattering rates. Given that this pattern arises in the conductivity,
it is more easily explained as arising from the presence of two conductivity
channels, one of which is field independent and affected by disorder, and one
of which sensitive to magnetic field but insensitive to disorder. Second, the
fact that the boundary between these two regimes lies at the same value of the
magnetic field in all samples suggests that this is not a vortex-lattice
transition. Such a transition should depend on the density of pinning sites
and therefore is expected to be disorder-dependent. Third, a fixed field scale
translates to a specific vortex density, so it seems likely that this high
field regime is defined by some length scale in the system becoming comparable
to the inter-vortex distance.

One consequence of this disorder-independence of the rise of conductivity with
field is that the shape of $\kappa_{0}(H)/T$ is flatter in cleaner samples
when plotted as a fraction of the normal state thermal conductivity (i.e. Fig.
2d ). This trend has been noted in several very clean unconventional
superconductors such as KFe2As2 and YBCO.Reid  _et al._ (2012) To the best of
our knowledge, none of the data sets on these materials presents a wide enough
array of disorder values to thoroughly test whether there is a disorder-
independent growth rate for $\kappa/T(H)$ in any of these materials. If it is
the case that any of these other clean nodal superconductors show similar
behavior as a function of disorder, it would create a basis for understanding
the mixed state that is independent of the details of pairing. This creates a
powerful motivation for more thorough studies of the mixed state in other
clean nodal superconductors.

Finally, there is the striking observation that for sample S3, which has the
highest RRR value, $\kappa(T\rightarrow 0,H)/T$ actually reaches a maximum at
0.5 T before decreasing and settling into a smooth rise in the intermediate
field regime. While it may be tempting to interpret this local maximum as an
indication that some distinctive physics is present around 0.5 T, such as a
phase transition, the simplest interpretation is probably that this peak is
just a consequence of the crossover from the low-field regime dominated by the
Doppler shift of the quasi-particle spectrum to the intermediate field regime
which is dominated by some other physics. All of our samples exhibit a
crossover at this absolute field scale, so it is reasonable that in the
highest RRR sample $\kappa_{0}(H)/T$ grows fast enough that it overshoots the
value that it will ultimately settle into in the universal mixed regime above
1 T.

Figure 3: Thermal conductivity for an axial state on a spherical Fermi
surface. The calculated thermal conductivity is presented for a gap model with
$\Delta=\Delta_{0}\sin\theta\exp(i\phi)$, where $\theta$ is the polar angle
and $\phi$ is the azimuthal angle, for heat currents directed perpendicular
(left columns) and parallel (right columns) to the nodal positions, and for
two different disorder levels (0.98$T_{c0}$ in black, 0.85$T_{c0}$ in red).
Panels (a) and (b) present electronic thermal conductivity as functions of
temperature for weak ($\tan\theta_{s}=2$, dashed lines) and for the strong
($\tan\theta_{s}=10$, solid lines) pointlike impurity scatterers. Panels (c)
and (d) include the contribution from the inelastic scattering to the
electronic thermal conductivities in the presence of strong scatterers with
$\tan\theta_{s}=10$, and panels (e) and (f) add phonon contributions (blue
dashed lines) modeled assuming defect scattering as described in the text. The
total thermal conductivity (magenta solid lines) thus is calculated to include
impurity scattering with $\tan\theta_{s}=10$ for $T_{c}=0.98T_{c0}$, inelastic
scattering $\Gamma_{in}=0.2T_{c0}$ at $T=T_{c0}$ and phonon contributions.

##  II Theoretical analysis

Here we validate the point node picture quantitatively with a model
calculation of the temperature dependence of $\kappa/T$. The theory of thermal
conductivity in triplet superconductors is very similar to that for singlet
superconductorsAmbegaokar and Griffin (1965); Kadanoff and Martin (1963),
provided the triplet order parameter matrix is unitary. Most of the commonly
studied triplet states, including the 3He-A phase, are in this class. In that
case the familiar quasiparticle energies $E_{\bf k}=\sqrt{{\xi_{\bf
k}}^{2}+|{\bf d}({\bf k})|^{2}}$ enter the thermal current response, and the
same expressions can be used for triplet superconductorsSchmitt-Rink  _et al._
(1986, 1986); Hirschfeld  _et al._ (1986a, 1988a). In the case of nonunitary
states, additional terms involving ${\bf q}\equiv i{\bf d}({\bf k})\times{\bf
d}^{*}({\bf k})$ occur in both the quasiparticle energies and in the weights
of two-particle processesMishra  _et al._ (2024). In the nonunitary case, the
zeros of $|{\bf d}({\bf k})|^{2}$ differ from those of the zeros of $E_{{\bf
k}}$ even on the Fermi surface $\xi_{{\bf k}}=0$ (“spectral nodes”). Indeed,
it was proposed by Ishihara et al.Ishihara  _et al._ (2023) that in UTe2,
complex linear combinations of 1D irreducible representations could support
spectral nodes pointing in generic directions in the orthorhombic Brillouin
zone, explaining early experiments indicating time reversal symmetry breaking
Hayes  _et al._ (2021). By contrast, order parameters corresponding to a
single 1D irreducible representation are necessarily real, with nodes aligned
along high-symmetry axes. Previous work measured penetration depth for
currents running along different directions, and concluded from the universal
$\delta\lambda(T)\sim T^{2}$ behavior that nodes were located at generic
positions in the Brillouin zone, requiring nonunitary statesIshihara  _et al._
(2023).

Thermal conductivity is complex to analyze due to the fact that thermal
currents can be carried by both electrons and phonons, each of which can
scatter from multiple physical sources. In cuprates, it was shown that
electronic conductivity limited by disorder was dominant at low temperatures,
while a combination of inelastic electron-electron scattering and phonon-
electron scattering dominated the intermediate and high temperature
regimesHirschfeld and Putikka (1996). Intriguingly, the directional thermal
conductivity of sample S1 Metz  _et al._ (2019) is unexpectedly relatively
isotropic, at least in the context of a system with point nodes along a single
high-symmetry axis. This could reflect a distribution of nodes away from high
symmetry directions, as suggested in Ref. Ishihara  _et al._ (2023), or a
larger than expected phonon contribution. We note, for instance, that the low
temperature thermal conductivity of samples S1, S2 and S3 are not strikingly
different as shown in Figure 1 (inset), suggesting that phonon conduction is
not negligible. Theoretically, calculated values of electronic $\kappa/T$ are
too small at low $T$ and fall systematically below the data, providing further
support for this conclusion.

Figure 4: Qualitative fits to the experimental data .  Panels (a)-(c) show the
qualitative fit to the temperature dependence of $\kappa/T$ normalized to
their respective normal state values $\kappa_{\mathrm{N}}/\mathrm{T}$ at
$T_{c}$. The $s$-wave scattering phase-shift is $\tan^{-1}3$, and $\kappa/T$
along the nodal direction (blue) and perpendicular to the nodal direction
(red) include contribution from phonons and inelastic scattering. Filled
circles show the experimental data for heat current applied along the
$a$-axis. Panels (d)-(f) show the fits (lines) to the field dependence of the
zero temperature limit values of experimental data (circles) measured with
heat current parallel to applied fields, normalized to the zero temperature
limit values of $\kappa/T$ measured at $H_{c2}$. Legends are equivalent for
all panels and each column represents same values of $T_{c}$ and impurity
parameters. Note that the impurity scattering rates are chosen to match the
experimental $T_{c}$ values.

We assume the standard Bardeen-Rickayzen-Tewordt theory of phonon thermal
conductivityBardeen  _et al._ (1959); Uher (1990) with phonon mean free path
limited by defects. For simplicity, we ask initially what information can be
gleaned from the qualitative features of the temperature- and field-dependence
of the measured thermal conductivity along the $a$-axis direction only. The
enhancement of the electronic mean free path below $T_{c}$ due to the collapse
of inelastic scattering, followed by a saturation of the mean free path due to
disorder, led in the cuprates to a peak in $\kappa_{el}$ that was very
sensitive to disorder. This behavior is similar to that observed in Fig. 1 for
our UTe2 samples, so we attempt a similar analysis here, with phonon
contributions $\kappa_{ph}$ and inelastic scattering due to electronic
interactions $1/\tau_{inel}$ that by assumption do not depend on pairbreaking
disorder. The electronic part $\kappa_{el}$ is calculated assuming independent
elastic and inelastic contributions, $1/\tau=1/\tau_{el}+1/\tau_{inel}$, and
added to the phonon contribution. The model and parameters used are described
in the Appendix.

In Fig. 3, we show the theoretical thermal conductivity $\kappa(T)/T$
calculated with these assumptions in an axial state, ${\bf d}({\bf
k})=(k_{x}+ik_{y}){\hat{y}}\propto\sin\theta$, with $\theta$ the polar angle
with respect to the $z$-axis in momentum space, taken to lie along the
crystallographic $a$-axis direction. This choice of order parameter is
obviously a dramatic oversimplication compared to the states that have been
discussed previously in the orthorhombic crystal, including nonunitary ones;
nevertheless the qualitative features at low $T,H$ should depend only on the
existence of linear point nodes, and their location relative to the field and
current directions. Panels (a),(b) of Fig. 3 show some representative
electronic thermal conductivity plots assuming scattering from impurities
only, for a range of impurity parameters; the plot is based on the theory
reviewed in Appendix A. First, note that the observed $\kappa_{el}/T\sim
T^{2}$ behavior is expected over an intermediate temperature range for a state
with point nodes located in a generic position relative to the heat current
flow, and impurity scattering in the intermediate-to-unitary limit. At
asymptotically low temperatures, the $T$ dependence is strongly influenced by
self-consistency, despite the lack of leading universal constant
$\kappa_{el}/T$ as in the line node caseGraf  _et al._ (1996); Norman and
Hirschfeld (1996). Instead, the behavior of $\kappa_{el}/T$ as $T\rightarrow
0$ is very flatHirschfeld  _et al._ (1986b, 1988b). Furthermore, as shown in
Fig. 3(b), deviations from unitary or near-unitary scattering lead to poor
agreement with the data due to low-$T$ upturns.

Fig. 3(c) and (d) now illustrate the effect of inelastic scattering on the
electronic thermal conductivity. The strong peak below $T_{c}$ in $\kappa/T$
is strongly reminiscent of cuprates, where it is attributed, e.g. in
YBCOHirschfeld and Putikka (1996), to the collapse of the quasiparticle spin
fluctuation relaxation rate in the superconducting state as the gap opens
below $T_{c}$. In the current model, however, the collapse is somewhat weaker
due to the point nodes, and the contribution of the inelastic scattering falls
off even more rapidly than in the line node case as $T\rightarrow 0$ (Appendix
A). In Fig. 3 (e),(f), we show the effect of adding the phonon contribution to
the thermal conductivity. Phonons are modeled assuming defect scattering as
described in Refs. Uher (1990); Hirschfeld and Putikka (1996) and reviewed in
Appendix A. Within the current model, this leads to
$\kappa/T=\kappa_{el}/T+\kappa_{ph}/T$ peaks at somewhat lower temperatures
than experiment, but agrees qualitatively with the overall behavior with
temperature (Fig. 4).

The experimental data shown in Fig. 1 vary monotonically at low $T$, and there
is no linear-$T$ term in $\kappa$ from impurity scattering, as expected for an
axial state realized in sufficiently clean samplesGraf  _et al._ (1996);
Norman and Hirschfeld (1996). We show in Fig. 4(a) a comparison of the best
fit of the theory for a combination of impurity, inelastic electron scattering
and phonon scattering parameters, with the experimental data. Since the
variation of physical properties with sample quality is a pressing issue in
UTe2, we also present results in Fig.4(a)-(c) for concentrations corresponding
to $T_{c}$ values of 2.1 K, 1.8 K and 1.6 K in theory and experiment. We see
that the height of the inelastic peak at intermediate temperatures grows with
decreasing disorder, as in Fig. 1, reflecting the growth of the inelastic
quasiparticle mean free path to much lower temperatures before being cut off
by the elastic mean free path. While many uncertainties exist with this
simplified model as discussed below, the best fit is obtained assuming the
nodes are away from the $a$-axis, most readily observed in the large
calculated anisotropy for node-parallel and -perpendicular heat currents in
Fig. 4a). In part this conclusion arises from the sensitivity of the
$j_{q}\parallel\hat{a}$ result to impurity phase shift deviations from
unitarity, as reflected in the low-$T$ bump not seen in experiment. But it is
also consistent with recent experiments suggesting the nodal positions are
located away from the $a$-axis, in particular the recent ultrasound
experiments that find less sensitivity to strain along the $b$-axis as
evidence for a B2u order parameter Theuss  _et al._ (2023), as well as a small
anisotropy in penetration depth consistent with $b$\- and $c$-axis nodal
directions Ishihara  _et al._ (2023).

We now compare with the expected field dependence for a state with point
nodes, calculated within the Doppler shift approximation where quasiparticle
motion in the vortex state is treated semiclassically, such that each
quasiparticle has its energy shifted $E_{\bf k}\rightarrow{\bf v}_{s}\cdot{\bf
k}$, where ${\bf v}_{s}$ is the superfluid velocity. The Doppler shift
contribution from extended quasiparticle states in this case can be shown to
be $\kappa/T(H\rightarrow 0)\sim H/H_{c2}$, very different from the
exponentially activated dependence expected for a fully gapped system, and
somewhat different from the $H\log H$ behavior expected for line nodes. The
vortex lattice itself is assumed to be pinned (with other assumptions and
parameters described in Appendix A). In Fig. 4 (d)-(f), we show the calculated
field dependent $\kappa$, normalized to its $\Delta=0$ (normal state)
$T\rightarrow 0$ value, which in the current description corresponds to the
electronic impurity scattering contribution only. This is the same procedure
employed in the normalization of the experimental data in Fig. 2(d),(f). While
the theory does not reproduce the apparent small peak at low fields for the
purest sample, it is roughly consistent in magnitude and behavior with the
overall field behavior for $H\ll H_{c2}$. Thus, while the best fit for sample
S3 (Fig. 4(d)) is consistent with the conclusion drawn from the
$T$-dependence, that nodes are away from the principal $a$-axis direction, the
results for the dirtier samples, S1 and S2 in (e),(f), appear to disagree.
However, the rough suppression of the overall field dependence variation with
disorder is captured by the theory. We note that this magnitude of the field
variation includes a factor describing the geometry of the vortex lattice and
the coherence lengths; this factor is not known quantitatively but should be
of order unity and is chosen to be the same for all three disorder levels.
Thus from the field dependence alone it is difficult to determine the
positions of the nodes relative to the heat current. On the other hand, it is
clear that the field dependence is most consistent with point nodes, and
inconsistent a fully gapped state.

##  III Conclusions

In this work, we report the first systematic study of the low-temperature
thermal transport of UTe2 as a function of scattering rate and superconducting
transition temperature, utilizing crystals obtained by different growth
methods including a unique synthesis route using tellurium flux to obtain
residual resistance ratios as high as 600. Given the variations in sample
properties reported for the evolving generations of crystalline quality, the
direct measure of the density of low-energy quasiparticle excitations this
experiment provides yields conclusive information about the intrinsic nature
of the superconducting gap symmetry and structure. We have measured the
temperature- and magnetic-field dependence of the thermal conductivity of four
samples with varying $T_{c}$ values and normal state scattering rates,
revealing a robust pattern of behavior in the superconducting state of UTe2.
Our experimental data can be understood within the standard theory of a
superconductor with point nodes in the presence of varying levels of disorder,
including inelastic electron-electron scattering that leads to a peak in the
temperature dependence below $T_{c}$ for the cleaner samples. The zero-
temperature and low-field limiting behaviors of thermal transport are shown to
only be consistent with the presence of a superconducting gap with point
nodes, allowing us to rule out any full-gap scenarios. Theoretical analysis of
a simple model $p$-wave state supports a picture with nodes located away from
the crystalline $a$-axis, suggesting a B1u or B2u symmetry for the low-field,
ambient-pressure superconducting order parameter of UTe2 that is consistent
with recent experiments. The sample-independent rate of increase of thermal
conductivity observed in the zero-temperature, intermediate-field mixed state
is suggestive of a universal phenomenology for ultra-clean nodal
superconductors that deserves further attention.

Acknowledgements. The authors acknowledge useful conversations with D.
Agterberg, J.C. Davis, V. Madhavan, and S. Anlage. Research at the University
of Maryland was supported by the Department of Energy Award No. DE-SC-0019154
(transport experiments), the Gordon and Betty Moore Foundation’s EPiQS
Initiative through Grant No. GBMF9071 (materials synthesis), NIST, and the
Maryland Quantum Materials Center. PJH was supported by NSF-DMR-2231821.

##  Appendix A  Materials and Methods.

Two different growth methods were employed to produce crystals with reduced
normal state scattering rate. The first was a chemical vapor transport method
analogous to the one used in ref Metz  _et al._ (2019), but at reduced
temperatures. This was motivated by previous work showing that lower growth
temperatures can increase $T_{c}$ and reduce normal state scattering in UTe2
Rosa  _et al._ (2022). The sample measured here was grown for one week in a
temperature gradient of 950C to 860C, with a uranium to tellurium ratio of
1:1.85. Samples from this growth batch consistently exhibited superconducting
transitions at 1.8 kelvin. The residual resistance ratios of crystals in this
batch varied from 30-50.

Further increases in $T_{c}$ were obtained through a novel tellurium-flux
growth method. Previous attempts to grow UTe2 in a tellurium flux have only
yielded non-superconducting crystals with very low residual resistivity ratios
of about 2 or three. The best available analyses of these samples suggest that
they are uranium deficient at the few percent level Sakai  _et al._ (2022).
Motivated by these observations, we attempted Te-flux growths with greater
concentrations of uranium. For these growths, uranium and tellurium chunks
were combined in a ratio of nearly one to two, placed inside an alumina
crucible and sealed in a quartz ampoule with an argon atmosphere of about
100mbar. The growths were heated rapidly to 1185C, held there for five hours
and then cooled to 1000C at 4C/hour. Because the uranium to tellurium ratio
was close to the stoichiometry of the UTe2, relatively little excess tellurium
remained after the growth cycle. This allowed the ampoule to be simply cooled
to room temperature and the residual flux was removed from the crystals
mechanically.

Figure A1: Characterization of superconducting UTe2 samples grown in Tellurium
flux. Panel (a) shows the $a-$axis resistivity of four samples of Te-flux
grown UTe2. All four samples show a superconducting transition around 2 K and
RRRs above $\sim$100\. The data are normalized to the average value of
$\rho(300~{}K)$ for consistency. Panel (b) shows the superconducting
transition in heat capacity of S3 from the main text. The size of the jump at
Tc is comparable to that of high-Tc samples of UTe2 grown by other methods.

In growths with $\sim$30$\%$ uranium content, the RRRs of the samples improved
to about seven–as opposed to two or three as found in samples grown with an
initial uranium concentration of 22$\%$ or 25$\%$Sakai  _et al._ (2022). At a
32$\%$ uranium concentration, the growths produced some superconducting
crystals. These crystals make up about a third of the the overall yield of the
growths—with the remainder being nonsuperconducting UTe2 crystals—and they
exhibit superconducting transition temperatures of about 2K and elevated
residual resistivity ratios of between 100 and 600. By contrast, the non-
superconducting crystals in these growths show RRRs of about 3. These
observations suggest chemical similarity to crystals grown in a NaCl/KCl flux,
which also exhibit superconductivity at 2K and arise characteristically as a
minority phase in those growthsSakai  _et al._ (2022). Further work will be
required to determine exactly why superconductivity emerges suddenly in
growths with a 32$\%$ uranium concentration, but we can offer a few
observations here. According to published binary phase diagrams, a Te-U system
with 32$\%$ or more uranium will first form the phase U7Te12 as it cools and
then form UTe2 crystals. The fact that superconductivity is not observed in
any crystals grown at 30$\%$ or less strongly suggests that the presence of
U7Te12 in the melt is at least part of the explanation. However, no
macroscopic U7Te12 crystals could be found in the batch, and the phase did not
appear in powder x-ray patterns. On the other hand, magnetization measurements
sometimes revealed traces of U-Te binary compounds on the tellurium-rich side
of UTe2, including U7Te12. This is yet another similarity that these crystals
possess to those grown in a NaCl/KCl flux Sakai  _et al._ (2022). These
observations suggests that U7Te12 may be forming as expected but then
transforming into UTe2 at lower temperatures, creating crystals in a uranium-
rich environment that is thought to be favorable to superconductivity in this
material. In any case, the existence of a second synthesis route to high RRR
samples of UTe2 should provide valuable opportunities to study the microscopic
factors responsible for the differences in Tc in this system.

These Te-flux grown, superconducting crystals where characterized by x-ray
diffraction, which confirmed that the crystal structure was that of UTe2, with
resistivity measurements, which showed superconductivity at 2K and
substantially enhanced RRRs, and with heat capacity measurements, which
confirmed the bulk nature of the superconducting transition (see Fig A2).

Resistivity data were collected with a conventional four-terminal, AC lock-in
technique. Heat capacity was measured by a quasiadiabatic technique where the
calorimeter and sample where heated by about one percent of the bath
temperature and allowed to relax. The heat capacity was then extracted by
fitting the the rising and falling curves to an exponential function.Thermal
conductivity measurements were performed by a one heater, two thermometer, DC
method. The typical thermal gradient was between one and two percent of the
bath temperature. Contacts to the samples where made with a bismuth-tin-copper
solder that maintains good thermal conductivity throughout the temperature and
field range studied.

To study the $T\rightarrow 0$ behavior of the thermal conductivity it is
necessary to extrapolate our $\kappa(T)/T$ curves to zero kelvin. The fits
shown in Fig 2 panels a-c were done by finding the best fit to the data of a
single power law, $\kappa/t\sim T^{n}$ where both $n$ and the intercept were
allowed to vary. For some curves, a good fit to a single power law could only
be obtained over a limited temperature range. This was especially true for the
highest RRR sample at low fields. Because of this, the fittings were generally
done for data below 300mK, and were restricted to even lower temperatures if
no visually acceptable fit could be found. Finally, although the zero
temperature extrapolations are useful for modeling and conveying the data, we
would like to emphasize that all of the trends in the data that support the
conclusion of point nodes are visually obvious in the raw data and so the
conclusion that point nodes are present in the gap does not depend on the
details of these extrapolations.

Figure A2: High field $a-$axis thermal conductivity of sample S3. Compared to
the lower RRR samples, S3 shows dramatically higher thermal conductivity in
the normal state at low temperatures, consistent with its higher extrapolated
electrical conductivity. Magnetic field is still aligned with the crystal
$a-$axis.

##  Appendix B Theoretical details.

### Thermal conductivity for a superconductor with point nodes

The simple model order parameter for a unitary superconductor with point nodes
is given by an axial state. On a spherical Fermi surface, the axial state is
given by

$\Delta_{\theta}=\Delta_{0}\sin\theta e^{i\phi},$ (1)

where $\theta$ is the polar angle, $\phi$ is the azimuthal angle and
$\Delta_{0}$ is the superconducting gap maximum. This state has point nodes
along the $\hat{z}$ direction at the north and south poles of the Fermi
surface. The diagonal thermal conductivity for the axial state Hirschfeld  _et
al._ (1988b) is given by

$\displaystyle\frac{\kappa_{ii}}{T}$ $\displaystyle=$ $\displaystyle\hbar
k_{B}^{2}\int_{-\infty}^{\infty}d\omega\frac{\omega^{2}}{T^{2}}\left(-\frac{dn_{F}(\omega)}{d\omega}\right)$
(2) $\displaystyle\times$ $\displaystyle\left\langle
N_{0}v_{Fi}^{2}\frac{1}{2\mathrm{Re}\sqrt{\Delta_{\theta}^{2}-\tilde{\omega}^{2}}}\left[1+\frac{|\tilde{\omega}|^{2}-|\Delta_{\theta}|^{2}}{\left|\Delta_{\theta}^{2}-\tilde{\omega}^{2}\right|}\right]\right\rangle_{FS},$

where $N_{0}$ is the single-particle density of states per spin at the Fermi
level in the normal state, $v_{Fi}$ is the Fermi velocity along the $i^{th}$
direction, $T$ is the temperature, and $n_{F}(\omega)$ is the Fermi function.
$\langle\dots\rangle_{FS}$ denotes an average over the Fermi surface. The
effect of elastic impurity scattering is embedded in $\tilde{\omega}$, which
is obtained by solving the following self-consistent equation,

$\tilde{\omega}=\omega+i0^{+}+\frac{n_{imp}}{\pi
N_{0}}\frac{g_{0}}{\cot^{2}\theta_{s}-g_{0}^{2}}.$ (3)

Here $n_{imp}$ is the impurity concentration, $\theta_{s}\equiv\tan^{-1}(\pi
N_{0}V_{imp})$ is the $s$-wave scattering phase shift which is related to the
strength of the impurity potential, and $g_{0}$ is

$g_{0}(\tilde{\omega})=\left\langle\frac{\tilde{\omega}}{\sqrt{\Delta_{\theta}^{2}-\tilde{\omega}^{2}}}\right\rangle_{FS}.$
(4)

In the zero temperature limit,

$\displaystyle\frac{\kappa_{ii}}{T}\Big{|}_{T\rightarrow 0}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}\hbar k_{B}^{2}}{3}\left\langle
N_{0}v_{Fi}^{2}\frac{1}{2\mathrm{Re}\sqrt{\Delta_{\theta}^{2}-\tilde{\omega}^{2}_{0}}}\right.$
(5)
$\displaystyle\left.\left[1+\frac{|\tilde{\omega}_{0}|^{2}-|\Delta_{\theta}|^{2}}{\left|\Delta_{\theta}^{2}-\tilde{\omega}_{0}^{2}\right|}\right]\right\rangle_{FS},$

Here $\tilde{\omega}_{0}$ is $\tilde{\omega}(\omega=0)$. In case of point
nodes, $\Delta_{\theta}$ is zero on finite number of points on the Fermi
surface, hence, in the absence of any scattering
($\tilde{\omega}_{0}\rightarrow 0$), $\kappa/T$ vanishes. Unlike states with
line nodes, for weak disorder an axial state has no residual density of
states, nor a limiting low-$T$ term in the thermal conductivityHirschfeld  _et
al._ (1986a). Such terms arise only above a critical disorder level depending
on the phase shift. For the purpose of fitting the thermal conductivity data,
we have assumed that the highest $T_{c}$ sample corresponds to $2\%$ reduction
in $T_{c}$, and the impurity concentration is chosen to give this suppression
with $\theta_{s}=\tan^{-1}(3)$. For $T_{c}=1.8$K and $T_{c}=1.65K$ samples,
the impurity concentrations are chosen to give $15\%$ and $25\%$ $T_{c}$
suppression, respectively. The $T_{c}$ is determined by solving,

$\displaystyle\ln\left(\frac{T_{c}}{T_{c0}}\right)$ $\displaystyle=$
$\displaystyle\Psi\left[\frac{1}{2}\right]-\Psi\left[\frac{1}{2}+\frac{\Gamma_{N}}{2\pi
T_{c}}\right].$ (6)

Here $\Psi$ denotes the digamma function and the normal state scattering rate
$\Gamma_{N}$ is $({n_{imp}}/{\pi N_{0}})/{(\cot^{2}\theta_{s}+1)}$.

####  B.0.1 Inelastic scattering

As discussed in the main text, the effect of inelastic scattering and
contribution from phonons need to be included in order to describe the
experimental data. In general, the treatment of simultaneous elastic and
inelastic scattering is a complex problem, however here we simply assume that
source of scattering is independent, and combine the individual contributions
when evaluating thermal conductivity. This can be done by adding the inelastic
scattering rate to $\tilde{\omega}$ in Eq. (2),

$\displaystyle\tilde{\omega}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{\omega}+\frac{i}{2\tau_{inel}}=\mathrm{Re}\left[\tilde{\omega}\right]+\frac{i}{2\tau_{el}}+\frac{i}{2\tau_{inel}}$
(7)

The imaginary part of $\tilde{\omega}$ is the elastic scattering rate. We
further assume that the origin of the inelastic scattering is a boson that
could be phonons or spin-fluctuations, and the bosonic dispersion and the
electron-fermion coupling do not have any significant momentum and spin
dependence. The low energy bosonic density of states is,

$\mathcal{D}(\Omega)=\frac{\Omega}{\Omega^{2}+\Omega_{0}^{2}},$ (8)

where $\Omega$ is the energy of the boson and $\Omega_{0}$ is the
characteristic energy scale associated with the bosonic mode and it is assumed
to be largest energy scale compare to $\Delta_{0}$, T or $\hbar/2\tau_{el}$.
Under these assumptions, the inelastic scattering rate is,

$\displaystyle\frac{1}{2\tau_{inel}}$ $\displaystyle=$
$\displaystyle\frac{g_{bf}^{2}}{2}\int_{-\infty}^{\infty}d\Omega
N(\omega-\Omega)\mathcal{D}(\Omega)$ (9)
$\displaystyle\times\left[\coth\left(\frac{\Omega}{2T}\right)+\coth\left(\frac{\omega-\Omega}{2T}\right)\right],$

where $g_{bf}$ is the effective electron-boson coupling constant and
$N(\omega)$ is the quasi-particle density of states. In the zero temperature
limit, the inelastic scattering rate reduces to

$\frac{\hbar}{2\tau_{inel}}={g_{bf}^{2}}\int_{0}^{\omega}d\Omega
N(\omega-\Omega)\mathcal{D}(\Omega).$ (10)

At low energies $\omega\leq\Delta_{0}$, the density of states can be
approximated as power law $N(\omega)\sim N_{0}(\omega/\Delta_{0})^{\alpha}$,
and the bosonic density of states can be approximated as
$\Omega/\Omega_{0}^{2}$ and the inelastic lifetime becomes,

$\frac{\hbar}{2\tau_{inel}}\approx\frac{N_{0}g_{bf}^{2}}{\Omega_{0}^{2}\Delta_{0}^{\alpha}(\alpha^{2}+3\alpha+2)}\omega^{\alpha+2}.$
(11)

For line nodes, $\alpha=1$ leads to $\frac{1}{2\tau_{inel}}\propto\omega^{3}$
and for point nodes $\alpha=2$ that gives
$\frac{1}{2\tau_{inel}}\propto\omega^{4}$. The integrand in Eq. (2) is sharply
peaked at $\omega=0$, and observing the weak $\omega$ dependence of inelastic
scattering rate, we only retain the temperature dependence on
$\tau_{inel}^{-1}$ at $\omega=0$. At $\omega=0$, the inelastic scattering rate
becomes,

$\frac{\hbar}{2\tau_{inel}}={g_{bf}^{2}}\int_{0}^{\infty}d\Omega\frac{N(-\Omega)\mathcal{D}(\Omega)}{\sinh(\Omega/T)}.$
(12)

This integral is also peaked at $\Omega=0$, therefore, it can be approximated
as,

$\frac{\hbar}{2\tau_{inel}}\approx\frac{{g_{bf}^{2}}}{\Omega_{0}^{2}}\int_{0}^{\infty}d\Omega\frac{N(\Omega)\Omega}{\sinh(\Omega/T)},$
(13)

which yields $T^{4}$ temperature dependence for point nodes and $T^{3}$ for
line nodes. The prefactor ${{g_{bf}^{2}}}/{\Omega_{0}^{2}}$ can be fixed by
the value of the inelastic scattering rate at $T_{c}$ and it is used as a
fitting parameter. We find that the inelastic scattering rate
${\hbar}/{2\tau_{inel}}=0.2T_{c0}$ at $T_{c0}$ gives a reasonable fit to the
experimental data. Apart from the inelastic scattering rate, the electron-
boson scattering also affects the effective mass. However, the correction to
the effective mass is weak, noting that the effective mass is quite high in
UTe2.

####  B.0.2 Phonon thermal conductivity

The general expression for the phonon thermal conductivity isUher (1990);
Bardeen  _et al._ (1959),

$\displaystyle\kappa_{ph}$ $\displaystyle=$
$\displaystyle\frac{k_{B}^{4}T^{3}}{2\pi^{2}\hbar^{3}v_{s}^{4}}\int_{0}^{\infty}dx\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}\ell(x,T),$
(14)

where $v_{s}$ is the sound velocity, $x$ is $\hbar\omega/k_{B}T$, $\omega$ is
the phonon energy and $\ell(x,T)$ is the effective mean free path given by,

$\displaystyle\frac{1}{\ell(x,T)}$ $\displaystyle=$
$\displaystyle\frac{1}{\ell_{B}}+\frac{1}{\ell_{d}}+\frac{1}{\ell_{e-ph}}+\frac{1}{\ell_{U}},$
(15)

where $\ell_{B}$, $\ell_{d}$, $\ell_{e-ph}$ and $\ell_{U}$ are the mean free
path due to boundary, defect, electron-phonon, Umklapp scattering processes,
respectively.

In order to determine the relevant scattering mechanism for phonons below
T$\lesssim$2K, we need to estimate the sound velocity and Debye temperature.
Within the Debye model the low temperature specific heat due to phonons is
given by $C_{ph}(T)=A_{ph}T^{3}$, where $A_{ph}$ is,

$A_{ph}=\frac{12\pi^{4}}{5}N_{atoms}k_{B}\frac{1}{\Theta_{D}^{3}}.$ (16)

Here $N_{atoms}$ is the number of atoms per mole, and $\Theta_{D}$ is the
Debye temperature. Using the experimentally estimated value
$A_{ph}=2.84$mJK-4mole-1Metz  _et al._ (2019) and
$N_{atoms}=$1.8$\times$1024/mole, we find $\Theta_{D}=$127K. The sound
velocity $v_{s}\equiv\omega_{D}\sqrt[3]{V_{u.c.}/6\pi^{2}N_{p.u.c}}$ is
1.3km/s using the unit cell volume $V_{u.c.}=$355$\mathrm{\AA}^{3}$, and
number of atoms per unit cell $N_{p.u.c}=$12Aoki  _et al._ (2022). We are
interested in phonon thermal conductivity below T${}_{c}\ll\Theta_{D}$,
therefore, Umklapp scattering can be ignored. There is no evidence for strong
electron-phonon couplingYang and Lee (2023), and in the superconducting state
$\ell_{e-ph}$ is expected to be very large due to reduced quasi-particle
density of statesTewordt and Wölkhausen (1989). Hence, the boundary scattering
$\ell_{B}$ and scattering by the point-defects $\ell_{d}$ are the two main
mechanisms that determine the phonon mean free path. The boundary scattering
is determined by the sample size and the amount of diffusive scattering by the
boundaries. On the other hand, the scattering by the point defects is an
energy dependent process. The mean free path due to point defects is given
byKlemens (1955),

$\frac{1}{\ell_{d}(x,T)}=\left[\frac{81\pi^{4}}{4V_{u.c.}}N_{p.u.c.}^{4}\right]^{\frac{1}{3}}\left(\frac{T}{\Theta_{D}}\right)^{4}x^{4}\sum_{i}N_{di}\left(\frac{\delta
M_{i}}{M_{i}}\right)^{2}.$ (17)

Here $N_{di}$ is the number of $i^{th}$ kind of defect, $M_{i}$ is the mass of
$i^{th}$ kind of ion and $\delta M_{i}$ is the mass difference of kind of
defect from $M_{i}$. For a vacancy $\delta M_{i}=M_{i}$ and for a different
type of isotope $\delta M_{i}=M_{i}-M_{isotope}$, where $M_{isotope}$ is the
isotope mass. We rewrite the $\kappa_{ph}$,

$\frac{\kappa_{ph}}{T}=\alpha_{0}T^{2}\int_{0}^{\infty}dx\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}\frac{1}{1+\alpha_{1}T^{4}x^{4}},$
(18)

where $\alpha_{0}$ and $\alpha_{1}$ are,

$\displaystyle\alpha_{0}$ $\displaystyle=$
$\displaystyle\frac{k_{B}^{4}\ell_{B}}{2\pi^{2}\hbar^{3}v_{s}^{4}},$ (19)
$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\left[\frac{81\pi^{4}}{4V_{u.c.}}N_{p.u.c.}^{4}\right]^{\frac{1}{3}}\frac{\ell_{B}}{\Theta_{D}^{4}}\sum_{i}N_{di}\left(\frac{\delta
M_{i}}{M_{i}}\right)^{2}.$ (20)

Since it is not possible to determine the effective boundary scattering length
in the presence of realistic boundaries causing diffusive scattering, or the
number of defects, we use $\alpha_{0}$ and $\alpha_{1}$ as independent two
fitting parameters to capture the effect of these two types of phonon mean
free paths. Table 1 shows the parameters $\alpha_{0}$ and $\alpha_{1}$ used to
fit the data.

$T_{c}$ (K) |  $\alpha_{0}$ (mW/K4/cm) |  $\alpha_{1}$ (K-4) | $\kappa_{ph}/\kappa_{el}|_{T=T_{c}}$
---|---|---|---
2.1 | 1.92 | 0.113 | 0.89
1.8 | 1.57 | 0.124 | 1.16
1.65 | 0.84 | 0.151 | 0.74
Table 1: Parameters used to calculate the phonon thermal conductivity for each
sample.

####  B.0.3 Magnetic field dependent thermal conductivity

Within the semi-classical formalismVolovik (1993); Kübert and Hirschfeld
(1998); Matsuda  _et al._ (2006), the effect of the magnetic field can be
described by shifting the quasiparticle energy by a Doppler shift, which is,

$\delta\omega=\frac{1}{2}m^{\ast}\mathbf{v}_{F}\cdot\mathbf{v}_{s},$ (21)

where $\mathbf{v}_{s}$ is the superfluid velocity, $\mathbf{v}_{F}$ is the
Fermi velocity, and $m^{\ast}$ is the effective mass. The superfluid velocity
is,

$\displaystyle\mathbf{v}_{s}=\frac{\hbar}{2m^{\ast}r}\hat{\Psi},$ (22)

where $r$ is the distance from the vortex core in the real space and $\Psi$ is
the winding angle. We rewrite the Doppler shift energy as,

$\displaystyle\delta\omega$ $\displaystyle=$
$\displaystyle\Delta_{0}\frac{1}{\rho}\mathcal{B}\sqrt{\frac{\pi\eta}{16}}\sqrt{\frac{H}{H_{c2}}}f(\phi,\theta,\Psi),$
(23)

where $\rho\equiv r/R_{H}$ is the distance from the vortex core in the units
of magnetic length scale $R_{H}\equiv\sqrt{\Phi_{0}/\eta H}$. Here $\Phi_{0}$
is the magnetic flux quanta, and $\eta$ is a dimensionless parameter of the
order of unity. The value of $\eta$ depends on the vortex lattice structure
and we use $\eta=\sqrt{3}/4$ which correspond to a triangular vortex lattice,
another dimensionless parameter
$\mathcal{B}=\hbar|v_{F}|/\pi\Delta_{0}\xi_{0}$ is used as a free parameter in
our calculations and we have set $\mathcal{B}=2.5$. The upper critical field
$H_{c2}\equiv\Phi_{0}/\pi\xi_{0}^{2}$. We have assumed that the upper critical
fields are isotropic along all directions and the Fermi surface is a sphere.
The true anisotropy of the system can in principle be accounted for by
assuming anisotropic coherence lengths, Fermi velocities, and including the
effects of disorder on these quantities, which we have not done, in part
because we were analyzing data for $\bf J_{Q},H\parallel\hat{a}$ only. In Eq.
(23), function $f(\phi,\theta,\Psi)$ contains the information about the
magnetic field direction and the Fermi surface topology. For a magnetic field
along the nodes $\hat{z}$ axis and perpendicular to the nodes $\hat{x}$, $f$
are,

$\displaystyle f(\phi,\theta,\Psi)_{H\parallel\hat{z}}$ $\displaystyle=$
$\displaystyle\sin\theta\sin\left(\phi-\Psi\right)$ (24) $\displaystyle
f(\phi,\theta,\Psi)_{H\parallel\hat{x}}$ $\displaystyle=$
$\displaystyle\left[\cos\Psi\sin\theta\sin\phi+\sin\Psi\cos\theta\right].$
(25)

In order to calculate the effect magnetic field on thermal conductivity, we
first solve Eq. (3) with Doppler shifted quasiparticle energy
$\omega\rightarrow\omega-\delta\omega$. This makes impurity renormalized
quasi-particle energy a function of $\rho$, $\Psi$. Now, using
$\tilde{\omega}(\rho,\Psi)$, we calculate $\kappa(\rho,\Psi)/T$, and finally,
we average $\kappa$ over the vortex lattice unit cell. The averaging over the
vortex lattice unit cell is defined as,

$\kappa=\frac{1}{\mathbf{A}}\int_{0}^{1}d\rho\rho\int_{0}^{2\pi}d\Psi\kappa(\rho,\Psi),$
(26)

where $\mathbf{A}$ is the vortex lattice unit cell area.

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