# Interplay between magnetism and superconductivity in UTe2

Di S. Wei  Geballe Laboratory for Advanced Materials, Stanford University,
Stanford, CA 94305  Department of Applied Physics, Stanford University,
Stanford, CA 94305  Stanford Institute for Materials and Energy Sciences, SLAC
National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025
David Saykin  Geballe Laboratory for Advanced Materials, Stanford University,
Stanford, CA 94305  Department of Physics, Stanford University, Stanford, CA
94305  Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025  Oliver Y.
Miller  Geballe Laboratory for Advanced Materials, Stanford University,
Stanford, CA 94305  Department of Physics, Stanford University, Stanford, CA
94305  Sheng Ran  Department of Physics, Quantum Materials Center, University
of Maryland, College Park, MD 20742, USA.  NIST Center for Neutron Research,
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA.
Department of Physics, Washington University in St. Louis, St. Louis, MO
63130, USA  Shanta R. Saha  Department of Physics, Quantum Materials Center,
University of Maryland, College Park, MD 20742, USA.  Daniel F. Agterberg
Department of Physics, University of Wisconsin, Milwaukee, Milwaukee,
Wisconsin 53201, USA.  Jörg Schmalian  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, Karlsruhe 76021, Germany
Nicholas P. Butch  Department of Physics, Quantum Materials Center, University
of Maryland, College Park, MD 20742, USA.  NIST Center for Neutron Research,
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA.
Johnpierre Paglione  Department of Physics, Quantum Materials Center,
University of Maryland, College Park, MD 20742, USA.  NIST Center for Neutron
Research, National Institute of Standards and Technology, Gaithersburg, MD
20899, USA.  The Canadian Institute for Advanced Research, Toronto, Ontario,
Canada.  Aharon Kapitulnik  Geballe Laboratory for Advanced Materials,
Stanford University, Stanford, CA 94305  Department of Applied Physics,
Stanford University, Stanford, CA 94305  Stanford Institute for Materials and
Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road,
Menlo Park, CA 94025  Department of Physics, Stanford University, Stanford, CA
94305

###### Abstract

Time-reversal symmetry breaking (TRSB) in UTe2 was inferred from observations
of a spontaneous Kerr response in the superconducting state after cooling in
zero magnetic field, while a finite $c$-axis magnetic field training was
further used to determine the nature of the non-unitary composite order-
parameter of this material. Here we present an extensive study of the
magnetic-field-trained Kerr effect, which unveils a unique critical state of
pinned ferromagnetic vortices. We show that a remanent Kerr signal that
appears following the removal of a training magnetic field, which reflects the
response of the TRSB order parameter and the external magnetic field through
the paramagnetic susceptibility. This unambiguously demonstrate the importance
of the ferromagnetic fluctuations and their intimate relation to the composite
order parameter. Focusing the measurement to the center of the sample, we are
able to accurately determine the maximum field that is screened by the
critical state and the respective critical current. Measurements in the
presence of magnetic field show the tendency of the superconductor to produce
shielding currents that oppose the increase in vortex-induced magnetization
due to the diverging paramagnetic susceptibility.

##  I Introduction

UTe2 is an example of superconductivity in the presence of ferromagnetic
fluctuations that persist to temperatures approaching zero, suggesting a
magnetic quantum critical phenomenon. The observations of two successive
superconducting transitions in specific heat and spontaneous polar Kerr effect
(PKE) [1] along the crystallographic $c$-axis led to the conclusion that
superconductivity in UTe2 is characterized by a two-component order-parameter
that breaks time reversal symmetry [1, 2, 3]. While several possibilities were
considered for the symmetry representation, the one that seemed to agree with
the data was $(\psi_{1}\in B_{3u},\psi_{2}\in B_{2u})$ [1, 3], implying a non-
unitary order-parameter, which we previously argued can be stabilized by
magnetic fluctuations [1].

However, the nature of these magnetic fluctuations is still not fully
understood. Muon spin relaxation indicate coexistence of ferromagnetic
fluctuations and superconductivity, where the temperature dependence of the
dynamic relaxation rate down to 0.4 K agrees with the self-consistent
renormalization theory of spin fluctuations for a three-dimensional weak
itinerant ferromagnetic metal [4]. Magnetization measurements can be collapsed
onto a single curve using a theory of metallic ferromagnetic quantum
criticality [5], strongly suggesting a zero-temperature ferromagnetic
transition. However, recent neutron scattering results show that magnetic
fluctuations in UTe2 are dominated by incommensurate spin fluctuations near an
antiferromagnetic ordering wave vector and extends to at least 2.6 meV,
suggesting that these fluctuations play an important role in the development
of the superconducting state [6]. Furthermore, recent transport and
thermodynamics studies under hydrostatic pressure suggest that the
superconducting transition temperature is maximized near a putative
antiferromagnetic quantum critical point occurring at a modest pressure of
$\sim 1.3$ GPa [7]. While two superconducting states have been confirmed, the
nature of the magnetic interaction associated with each of the states could be
different [7, 8]. Nevertheless, extrapolation from other similar uranium-based
superconductors including URhGe, UCoGe and UGe2 strongly support a spin
triplet state mediated by ferromagnetic fluctuations [1, 2, 3]. Focusing on
the superconducting state at ambient pressure, one way to resolve the magnetic
nature of the superconducting state of UTe2 is to test what magnetic state the
superconducting order-parameter nucleates upon interaction with the magnetic
fluctuations in both, the Meissner and vortex states.

While the time-reversal symmetry breaking (TRSB) property could be determined
unambiguously from the observation of spontaneous Kerr response in the
superconducting state after cooling in zero magnetic field, understanding the
field-cool data require more attention. First, the strong ferromagnetic
fluctuations at low temperatures may counter the onset of Meissner currents.
In addition, indications are that UTe2 exhibits very strong pinning effects
[9]. For example, removing the magnetic field at low temperatures after a
field-cool may result in a finite remanent magnetization due to trapped flux
[10], which interacts with the fluctuating magnetic moments as well as with
the TRSB superconductor [1]. Such a behavior would be enhanced when the
material is cooled above the lower critical field, $H_{c1}$, and a critical
state concept is expected to be applicable [11, 12]. In this case Kerr effect
will be an ideal probe of the local magnetization, particularly aiming at the
center of the sample’s surface perpendicular to the applied magnetic field.

In this paper we report on high resolution polar Kerr effect measurements of
single crystal UTe2 samples under various conditions involving combinations of
field cool (FC), zero field cool (ZFC), field warmup (FW) and zero field
warmup (ZFW). The assemblage of our results suggest strong correlations
between the time-reversal symmetry breaking order-parameter, the ferromagnetic
susceptibility, and the external magnetic field - when applied. First, as
previously reported [1], the observation of a finite Kerr response under ZFW
conditions that followed ZFC, unambiguously prove the time reversal symmetry
breaking of the superconducting order-parameter. The varying magnitude of the
effect (with a maximum amplitude of $\sim 0.4\mu$rad) and its random sign
observed under those conditions is expected from a spontaneous symmetry
breaking effect that exhibit two-fold domains. Furthermore, ZFW measurements
that followed a finite FC schedule show a low-temperature, field-independent,
Kerr response of $\sim 0.4\mu$rad when the cooling field is $\lesssim 30$G,
and an increasing low temperature Kerr response for larger cooling magnetic
fields. This remanent Kerr response, which originates from a combination of
remanent magnetization associated with trapped vortices, as well as the
superconducting order-parameter extrapolates to saturate at $H^{*}\sim 920$G.
The above behavior, as well as the temperature dependence of the remanent Kerr
effect, particularly the way it vanishes near $T_{c}$ suggest an unusual
interplay between magnetism and the TRSB order-parameter in a non-unitary
superconductor.

Our observation that the TRSB in UTe2 can be trained by a magnetic field along
the crystallographic $c$-axis (coinciding with the $z$-direction in our
notations) requires the presence of a term $\sim
H_{z}i(\psi_{1}\psi_{2}^{*}-\psi_{1}^{*}\psi_{2})\equiv H\tilde{\psi}$. Hence,
$\tilde{\psi}$ is a TRSB composite order-parameter. A resulting $z$-axis
magnetization will yield the following contribution to the free energy

$f_{m}=\alpha m^{2}+\gamma m\tilde{\psi}-mH$ (1)

Since the Curie temperature is suppressed to $T=0$, and to match with the
normal state, $2\alpha=\chi_{n}(T)^{-1}$ is the inverse normal-state magnetic
susceptibility. At zero magnetic field the magnetization associated with the
order-parameter can be shown to stabilize the nonunitary state [1]. However,
in the presence of magnetic field, and particularly in the vortex state, the
resulting magnetization should reflect a competition between the normal state
component and the superconducting tendency to screen the nucleation of a
finite magnetization. Here we present a simple analysis of the electromagnetic
response of a strongly paramagnetic superconductor, which seems to account for
the experimental results that demonstrate the effect of screening in strongly
reduce the magnetization as the temperature tends to zero.

##  II Experiment

To study the interplay between time reversal symmetry breaking (TRSB) in the
superconducting state of UTe2, and the vortex state, we performed high
resolution polar Kerr effects (PKE) using a Zero-Area Sagnac Interferometer
(ZASI) that probes the sample at a wavelength of 1550 nm, in a He-3 cryostat
with base temperature $\lesssim 0.3$ mK. The UTe2 single crystal used in this
study and a basic schematic of the low-temperature setup are shown in Fig. 1,
while the full apparatus is described in the supplementary Information (SI)
[13].

In general, the Kerr effect is defined through an asymmetry of reflection
amplitudes of right- and left- circularly polarized light from a given
material, yielding a Kerr rotation angle $\theta_{K}$, and is observed only if
reciprocity is broken. Thus, in the absence of any time-dependence or
irreversible effects, a finite Kerr effect unambiguously points to TRSB in
that material system. From its nature, Kerr effect is described within the
general theory of scattering, and thus it is ideal to probe TRSB in
superconductors since it is not subjected to shielding (Meissner) effects that
counter the magnetization. However, restrictions on the possible observation
of a finite Kerr effect suggest that for pure crystals, measurements at
frequencies where interband pairing effects are important yield a maximum
signal, although still very small [14, 15, 16, 17].

In the Kerr measurements reported here we use a collimated beam of diameter
$\sim 10.6~{}\mu$m, which emerges from a lens and quarter wave plate assembly
that were designed to minimize sensitivity to changes in both angle and
distance relative to the sample as to prevent temperature-dependent changes in
optical alignment [18]. For all the measurements the beam is aimed at the
center of the sample of average diameter 1.2 mm. However, probing the system
at near-IR frequencies ($\omega$), which is much larger than the
superconducting gap energy ($\Delta$) will reduce a typical ferromagnetic-like
response of order $\sim 1$ rad, by a factor of $(\Delta/\hbar\omega)^{2}\sim
10^{-7}$, yielding a theoretically predicted signal of about 0.1-1 $\mu$rad.
Thus, a sensitive device is needed to detect such small signals, and the ZASI
with its high degree of common-mode rejection for any reciprocal effects (e.g.
linear birefringence, optical activity, etc.) is probably the most suitable
one for this study.

In a typical FC Kerr experiment of a weak-pinning superconductor, the sample
is first cooled in a field (FC) lower than $H_{c1}(0)$. The Kerr effect is
then measured at zero-field while warming up (ZFW). In this case the ZFW
measurement yields results similar to the largest Kerr value obtained in a
zero-field cool (ZFC) followed by ZFW experiment, indicating a single domain
Kerr response. If the sample is cooled in a field larger than $H_{c1}(0)$,
some flux will be trapped after removing the magnetic field at low
temperatures, which, depending on the magnetic susceptibility of the material,
may result in a finite Kerr effect. Determining the magnetic contribution
above $T_{c}$, we expect that upon cooling in a field $H_{c1}\lesssim H\ll
H_{c2}$, a normal state contribution will be smaller by at least a factor
$H/H_{c2}$. Thus, without explicit magnetism (that is, magnetic response
beyond Pauli paramagnetism or Landau Diamagnetism), such effects are
undetectable. Indeed, in previous Kerr measurements of Sr2RuO4 [19], where
Kerr effect of $\sim 0.1\mu$rad was detected, or the heavy fermion Uranium-
based superconductors UPt3 [20] and URu2Si2 [21], and the filled-skutterdite
PrOs4Sb12 [22] (giving larger signal of $\sim 0.4$ to 0.7 $\mu$rad, which is
expected due to their strong spin-orbit interaction), the same magnitude of
Kerr effect was observed irrespective of the magnitude off the orienting FC.
Testing the apparatus with reciprocal reflecting media such as simple BCS
superconductors (or just mirrors), as well as the spin-singlet d-wave heavy-
fermion compound CeCoIn5 yielded a null result as expected [23].

Figure 1: (a) Crystal used for Kerr measurements. (b) End part of the
apparatus (for full description see SI [13]). (c) Five different Zero-field-
cool Kerr data for a superconducting UTe2 with $T_{c}\approx 1.55$K

##  III Results

Zero field data: Polar Kerr effect measurements were performed at low
temperatures on a single crystal of UTe2, after first cooling it to a base
temperature of 0.3 K in ambient field of $\sim 0.2$ G (which we will denote as
zero field) and measuring while warming the sample to above the
superconducting transition of $T_{c}\approx 1.55$ K. The cumulative results of
five cooldowns are shown in Fig. 1c, and are consistent with measurements on a
second crystal. We note that different cooldown result in a random size and
random sign of the signal, bound by a maximum signal of $|\theta_{K}|\approx
0.4\mu$rad at $0.3$ K. While these zero-field cool data may point to domain
formation, it is typically assumed that with a two-state order-parameter, as
we anticipate in UTe2, domains may be costly, thus often measurements will
result in the full signal.

Low field data: Assuming the zero-field data arises from the TRSB order-
parameter, we use field cool data, with the field oriented along the
$c$-direction of the crystal, to test for the ability to couple to that order-
parameter, and at the same time orient the sample to a single domain. This is
a crucial part of the correct determination of the possible components of the
proposed nonunitary order-parameter [1].

Figure 2: Very Low Field Cool - Measurements at zero-field Warmup. Note that
the saturated Kerr angle is the same for this field range. Here dashed-dotted
line (red) is a fit to $[1-(T/T_{c})^{2}]$ with $T_{c}=1.7$ K, while dashed
line (black) is a fit to a $[1-(T/T_{c})]^{1/2}$ with $T_{c}=1.55$ K, which
should dominate near the actual TRSB temperature.

First we note that for FC measurements of $H\lesssim 15$G we see no difference
from ZFC experiments, that is, values can fluctuate in each cooldown similar
to Fig. 1. Figure 2 shows a set of 6 different experiments where the sample
was cooled in magnetic fields of $\pm 20$, $\pm 25$, and $\pm 30$G, and the
Kerr angle was measured in a ZFW configuration. The low-temperature Kerr angle
of this collection of data seems to saturate at the same value, which also
coincides with the maximum value obtained in the ZFC experiments. We therefore
conclude that this value of $\theta_{K}\approx 0.4~{}\mu$rad is the intrinsic
contribution due to the TRSB order-parameter. Attempting to fit the
temperature behavior to the standard form of an order parameter
$\propto[1-(T/T_{c})^{2}]$ yields a reasonable, but with a slightly higher
$T_{c}$ than where the Kerr effect seems to vanish. A better fit
$\propto[1-(T/T_{c})]^{1/2}$ with the actual $T_{c}$ seems to hold near where
the Kerr effect vanishes, which should dominate near the actual TRSB
temperature, thus indicating the existence of a composite order parameter.
However, below $\sim T_{c}/2$, $\theta_{K}(T)$ develops an anomalous behavior,
exhibiting more flat temperature dependence. Such a behavior may be a
precursor to a more pronounced flat temperature behavior that we observe upon
cooling in higher magnetic fields, and is typical for remanent magnetization
behavior of a “hard superconductor,” which exhibits large vortex critical
current [10, 24]. Its occurrence in UTe2 is a central theme of this
manuscript.

Figure 3: Low Field Cool. dashed line box is the low field regime shown in
Fig. 3.

Increasing the training magnetic field above $\sim 30$G, Fig. 3 shows that the
low-temperature Kerr angle saturation value now increases with field, and
continues to show a rather flat behavior in approaching $T_{c}$, thus further
support our hypothesis that the Kerr response through $\sigma_{xy}(\omega)$
originates from the intrinsic magnetism. However, the increase in remanent
Kerr value indicates that the vortex state induced by the external cooling
field now dominates, which is observed through the interaction with the
induced ferromagnetism.

Wider field range: When further increasing the magnetic field, a closer
resemblance to the behavior of the magnetization of a strong-pinning
superconductor in the critical state is revealed. For example, inspection of
remanent magnetization in field cooled measurements on URu2Si2 and UPt3 [25]
show an evolution very similar to our Kerr measurements in Fig. 3 and the
subsequent increased field regime shown in Fig. 4. At the same time, we notice
that in our previous measurements of Kerr effect in this materials the TRSB
Kerr signal was independent of the training field [20, 21], thus corroborating
the claim that the Kerr effect in these materials detects directly the TRSB
order-parameter.

Figure 4: Temperature dependence remanent Kerr effect after Field Cool. Dashed
line denotes the “envelope” behavior as the cooling field is increased. It
extrapolates to $\Delta\theta_{K}\approx 8.0\mu$rad (dashed line), which would
correspond to an approximate cooling field of $\sim 920$G using extrapolation
of the fit shown in Fig. 6

Measurements in finite magnetic field: The conclusion in [1] that the order-
parameter is non-unitary with $(\psi_{1}\in B_{3u},\psi_{2}\in B_{2u})$, which
is stabilized by magnetic fluctuations, suggests the possibility that in the
presence of magnetic field the composite order-parameter may appear at a
higher temperature, either at the higher $T_{c}$, or as fluctuation induced
effect [26]. Assuming a standard correspondence between the Kerr and
magnetization measurements (see e.g. [27]), Fig. 5 shows a comparison of Kerr
effect measurement below 16K and SQUID magnetometry of a similar sample from
the same batch. Measurements were done at 240G along the $c$-axis (below $\sim
1000$G the magnetic susceptibility is roughly field independent, possibly
exhibiting a weak minimum at $\sim 240$ G. See SI [13]). Also marked in Fig. 5
is the $T_{c}$ as determined from Fig. 3. While there is a good correspondence
between the Kerr response and the magnetization, an anomaly in the Kerr effect
is detected in the 2 to $\sim$ 6 K, which could be traced to the interplay
between the magnetization along different axes in that regime (see SI [13]).

Figure 5:  Susceptibility and Kerr effect measurements of UTe2 in magnetic
fields below $H^{*}$. For Kerr effect he sample was cooled in magnetic field
of 240G and then measured when warmed up in 240G (data points represent bin-
average of 0.3 K). Susceptibility data was measured in a SQUID magnetometer
down to 1.8 K, and was fitted to the Kerr data in the temperature range of 7
to 8 K assuming $\theta_{K}\propto\chi=M/H$. The vertical line is the zero-
field warmup Tc.

##  IV Analysis and Discussion

Magneto-optical effects are described within quantum theory as the interaction
of photons with electron spins through spin-orbit interaction (see e.g. [28]).
In a ferromagnetic material the Kerr angle is found to scale with the
magnetization [29], while inherently antiferromagnetic systems may still
exhibit a finite Kerr effect depending on their local symmetry (see e.g.
[30]).

Focusing on the measurements on UTe2, we first note that for all FC
measurements of $H\lesssim 15$ G, including ZFC, the sample exhibits a random
in sign and in magnitude Kerr signal, which is bound by
$|\theta_{K}(0)|\approx 0.4~{}\mu$rad. This clearly establishes an intrinsic
TRSB associated with the superconducting order-parameter rather than a mere
magnetic response. However, as will be discussed below, UTe2 is a strong-
pinning superconductor, clearly exhibiting critical state features, which will
require further considerations in analyzing the low field data.

The low-field data: The fact that no field dependence is observed for ZFW
measurements that follows either ZFC or FC of $H\lesssim 30$ G, suggests that
this very low field effect is a result of either the TRSB order-parameter, or
a constant remanent magnetization, which is induced by an internal ordering
field rather than the external cooling field. This may occur if the cooling
field $H$ aligns the order-parameter $\tilde{\psi}$, which in turn induces a
finite magnetization through the bilinear coupling term as in Eqn. 1. Indeed,
Paulsen et al. [9] reported that expulsion of flux in FC experiments was
negligible in fields greater than just a few G. Such bulk magnetization
measurements may reflect a critical state that persists even below $H_{c1}$
[31, 24]. By contrast, in our Kerr measurements the $\sim 10.6~{}\mu$m
diameter beam points at the center of the sample of average diameter 1.2 mm
(see Fig. 1b). Thus, in such measurement along the applied magnetic fields, we
expect the center of the sample to practically be vortex free (In the pure
sense of the critical state model it is completely vortex free).

Flux pinning, the critical state and high field data: UTe2 is a strongly
paramagnetic material, with large magnetic susceptibility above the
superconducting $T_{c}$. Further observations of strong magnetic hysteresis,
weak Meissner effect, and large critical current [9] imply the presence of a
critical state [10], where cooling in magnetic field may result in large
remanent magnetization. While this may impact field-cool Kerr measurements,
the configuration of the experiment may also provide new insight into the
critical state phenomenon in this material. Moreover, it was shown by Clem and
Hao [24] that with strong pinning, even below $H_{c1}(0)$, cooling in a field
results in remanent magnetization that is only weakly temperature dependent
when warming up, and disappears only above the irreversibility temperature
$T_{irr}<T_{c}$ [25]. Increasing further the magnetic field will result in an
increase in the remanent magnetization, until a field $H^{*}(T)$ is reached,
where for a given temperature $T$, full flux penetration into the center of
the sample occurs. Assuming that in this field regime the Kerr angle is
proportional to the remanent magnetization in the material, we first
extrapolate the envelope of the temperature dependence of the $\theta_{K}(T)$
to $T=0$, obtaining an extrapolated value of $8.0\pm 0.3\mu$rad. Next we plot
the saturated Kerr angle from Fig. 4 as a function of cooling field as shown
in Fig. 6. Combining these two measurements we find that the Kerr angle at
$T=0$ will be maximum in a field-cool of $H_{max}\approx 920$G.

Figure 6: saturated Kerr vs. field UTe2

Since we assume that the vortex state, which induces the ferromagnetic
magnetization, is in the critical state, the vortex distribution in space is
determined by the critical current which is assumed constant [11]. In this
model the vortex-density profile (i.e. the profile of the magnetic induction
$B(r)$) is a straight line, matching the external field at the sample surface.
If the external field is taken to zero before warming up, this profile
determines the variation of the remanent magnetization in space. Obviously,
with the above approximation the maximum remanent magnetization is obtained
for a magnetic field $H^{*}$ for which a first vortex is introduced to the
center of the sample. Similarly, we will need to take the external magnetic
field to $-H^{*}$ to subsequently have the center of the sample completely
screened. For a sample of averaged diameter $D$, the above arguments imply:

$H^{*}=\frac{2\pi J_{c}D}{c},\ \ \ \ \ {\rm or}\ \ \ \ \ \ J_{c}=\frac{c}{2\pi
D}H^{*}$ (2)

Taking for our data $H^{*}\approx H_{max}=920G$, we obtain $J_{c}\approx
1.05\times 10^{4}$A/cm2, which is very close to the critical currents recently
measured for UTe2 of $J_{c}^{a}\approx 0.8\times 10^{4}$A/cm2 and
$J_{c}^{b}\approx 1.6\times 10^{4}$A/cm2 [9]. This agreement between the Kerr
data and the direct measurements of the critical current further support our
interpretation of the origin of the Kerr signal above $H_{c1}$.

Figure 7:  Expanded vertical scale of field cool, field warmup of Fig. 5(b).
Zero Kerr angle was offsetted to the plateau around 4K. The SQUID
susceptibility data (red full circles) is the same as in Fig. 5 for this
temperature range. The dotted line is the extrapolation of the SQUID-measured
susceptibility shown in Fig. 5 fitted below 4K to the lowest susceptibility
temperature measurement of 1.8K (represented with a faint vertical line). The
vertical red line is the zero-field warmup $T_{c}=1.55$K, which meets the
local averaged Kerr data at the $\times$ symbol.

Further support of the critical state scenario is discussed in the SI: [13]),
where we analyze a hysteresis loop at a field much smaller than $H^{*}$,
demonstrating how the evolution of the superconducting order counter-acts the
increase in magnetization below $T_{c}$. While cooling in a field nucleates
remanent magnetization, cooling at zero field and increasing the field to
$H_{c1}\ll H<H^{*}$ results in a weak Kerr effect at the center of the sample,
which is further reduced almost completely as the field is reduced to zero.

It is important to note the difference between UTe2 and other uranium-based
superconductors that exhibit a critical state. Focusing on UPt3 and URu2Si2,
Kozioł et al., [25] show remanent magnetization curves for these two materials
for field below their respective $H^{*}$. At the same time, our own Kerr
measurements on these materials [20, 21] show that the Kerr effect in FC
measurements is independent of cooling field, and coincide with the envelope
of the zero-field cool experiments. The reason is, as explained above, that
with no explicit magnetism, the trapped vortices only carry a Kerr effect due
to Pauli paramagnetism or Landau diamagnetism, which is reduced by those
vortices’ relative density of $H/H_{c2}$. Such an effect is undetectable with
our nanoradians resolution. At the same time, the Kerr measurements on UTe2 do
resemble the bulk magnetization measurements of UPt3 and URu2Si2 [25] (see SI:
[13]). This clearly indicate that the core of the vortices in UTe2 are
ferromagnetic, with the core magnetization aligned with the direction of the
cooling field. This then confirms the tendency of the material towards
ferromagnetism and further suggests that ferromagnetic fluctuations affect the
superconducting order-parameter.

TRSB order-parameter in magnetic field: Inspection of Fig. 5, the normal state
(FW) Kerr data shows a standard increase consistent with a paramagnetic
susceptibility of the material, before experiencing a sharp increase below
$\sim 6$K which seems to return to the original susceptibility curve below
$\sim 2$K. Indeed, a change in course of the magnetic susceptibility was
previously noted in this temperature range by several groups [32, 33],
typically attributed to the $c$-direction being perpendicular to the easy
axis-$a$ (See also SI: [13]). To better understand the observed behavior, we
show in Fig. 7 the expanded data below 4K (data points represent bin-average
of 100 mK), which includes the superconducting transition region. Since
susceptibility data is only available down to 1.8K, our analysis needs to rely
on the extrapolation of this normal state data to lower temperatures, however,
inspection of the trend of the data already suggests that the extrapolated
susceptibility will go above the average of Kerr angle at $T_{c}$, marked with
the cross ($\times$) symbol at the $T_{c}$ of the ZFW measurements (e.g. Fig.
4). To illustrate this claim, we fit the susceptibility data below 4K
(4${}^{t}h$ order polynomial give the best fit), and extrapolate it lower than
the measured 1.8 K. The extrapolated susceptibility, which represents the
normal-state magnetic response, clearly overshoots above the actual Kerr data.
This behavior indicate the emergence of superconducting fluctuations.

Since our experiment does not detect the Meissner currents that shield the
whole crystal, the data must indicate the competition between the paramagnetic
susceptibility and the superconducting order-parameter that develops in the
interior of the sample. This in turn suggests that the proximity to the
superconducting state is felt already $\sim 0.3$ K above the Kerr-
detected-$T_{c}$. However, as $T_{c}$ is approached, the magnetic response
weakens and the Kerr angle at the center of the sample decreases towards its
value at $T_{c}$. Such an effect cannot be described by Eqn. 1, which
considers a uniform magnetic field and ignores screening effects. Instead, to
understand this effect we need to include the full elecrodynamics of the
superconductor and its interaction with the magnetization. Since the Kerr
effect is expected to follow the magnetization, including the TRSB effect of
the superconducting order-parameter, we revise Eqn. 1, writing

$f_{m}=\alpha m^{2}+c|\nabla m|^{2}+\gamma m\tilde{\psi}-mh$ (3)

where we added a term that allows for non-uniform magnetization, which is
expected in the vortex state and the external magnetic field is replaced with
the local magnetic field, $\vec{h}(\vec{r})$, that we will take as pointing in
the $c$-direction (thus omit the vector sign.) This local field arises from
the solution of the Ginzburg-Landau equations and thus its profile through the
vortices already include the field penetrating through the core and the
shielding currents that extend a penetration-depth distance $\lambda$ beyond
the vortex core. Moreover, we recall that
$\langle\vec{h}(\vec{r})\rangle=\vec{B}$, the magnetic induction of the
superconductor. A more detailed approach, aimed at analyzing the magnetic
structure in a ferromagnetic superconductor was recently presented by
Devizorova et al.[34]. However, here we were only interested in the magnitude
of the average magnetization. Thus, to obtain the equilibrium contribution of
the paramagnetism to the magnetization in the superconductor, we minimize the
magnetic free energy with respect to $m$, obtaining:

$\frac{1}{\chi_{n}}\left[1-a^{2}\nabla^{2}\right]m=h-\gamma\tilde{\psi}$ (4)

where, as in Eqn. 1 we identify $2\alpha=\chi_{n}(T)^{-1}$ as the inverse of
the normal-state magnetic susceptibility, and $a^{2}=c/\alpha$ is the magnetic
stiffness parameter, which represents a microscopic length of order lattice
constant. Now, $(h-\gamma\tilde{\psi})$ acts as an effective local magnetic
field that follow the vortex lattice periodicity. As we solve for the
magnetization and substitute back in Eqn. 3, we clearly observe that the large
magnetic susceptibility caused by ferromagnetic fluctuations further enhances
the effective field $\sim\chi h$ that couples to the composite order
parameter. Taking the Fourier transform of Eqn. 4, we arrive at the
$q$-component of the transform for the magnetization:

$m_{q}=\chi_{n}\frac{h_{q}-\gamma\tilde{\psi}_{q}}{1+a^{2}q^{2}}$ (5)

However, since $h_{q}$ arises from the London equation, it should decay with
the penetration length $\lambda$ away from the vortex center, thus
$h_{q}=h(0)/(1+\lambda^{2}q^{2})$. We thus expect

$m_{q}=\frac{\chi_{n}}{1+a^{2}q^{2}}\frac{h(0)-\gamma\langle\tilde{\psi}\rangle}{1+\lambda^{2}q^{2}}$
(6)

The total magnetization density is then

$\langle
m\rangle=\sum_{q}\frac{\chi_{n}}{1+a^{2}q^{2}}\frac{B-\gamma\langle\tilde{\psi}\rangle}{1+\lambda^{2}q^{2}}$
(7)

Near $T_{c}$ the sum is dominated by $q\to 0$, yielding the same result we
would expect from Eqn. 1, that is:

$\langle m\rangle_{T\approx
T_{c}}=\chi_{n}\left(B-\gamma\langle\tilde{\psi}\rangle\right)$ (8)

However, as the temperature tends towards zero, we cannot ignore the shorter
wavelengths, which will be dominated by the vortex lattice wave-vector,
$q\approx\sqrt{\Phi_{0}/B}$. Since in that regime $a^{2}B/\Phi_{0}\ll 1$,
while $\lambda^{2}B/\Phi_{0}\approx B/H_{c1}\gg 1$, we can change the sum to
integral, and (up to order of ${\rm ln}\kappa$) we obtain:

$\langle m\rangle_{T\ll
T_{c}}\approx\chi_{n}\frac{H_{c1}}{B}\left(B-\gamma\langle\tilde{\psi}\rangle\right)$
(9)

which demonstrates the effect of screening. Here $\chi_{n}(T)\to\infty$ as
$T\to 0$, before saturating to its full value at $T=0$. Thus without the large
reduction of $H_{c1}/B$, the magnetization would monotonically increase
towards zero temperature. Our experimental data for Kerr effect at 240 G,
which is much larger than $H_{c1}$ indeed agree with the trend calculated
above. This unique behavior is clearly demonstrated in Fig. 7.

##  V Conclusions

UTe2 exhibits strong paramagnetic fluctuations, with no apparent finite-
temperature magnetic order. Using magneto-optic polar Kerr effect to study the
magnetic behavior of UTe2 crystals, while focusing our detection beams to the
center of the sample, away from any surface Meissner currents, we are able to
study the pure superconducting wavefunction response at very low field, and
the evolution of the vortex critical state at higher field. This approach
allows for a clear demonstration of the properties of a critical state that
appears in this material in the presence of an applied magnetic field,
including the determination of the critical current density for vortex motion
when the field is applied in the $c$-direction. When vortices are present, a
ferromagnetic magnetization is detected, confirming the proximity to a
ferromagnetic critical point, possibly a quantum-critical one. Subtracting out
the magnetic effect, the order-parameter shows an orbital TRSB effect of size
similar to the B-phase of UPt3 [20] and the low temperature behavior of the
superconducting state of URu2Si2 [21]. Finally, measurements in the presence
of magnetic field clearly show the tendency of the superconductor to produce
shielding currents that oppose the increase in vortex-induced magnetization
due to the diverging paramagnetic susceptibility.

Acknowledgments: We acknowledge discussions with Steve Kivelson and Ronny
Thomale. Work at Stanford University was supported by the Department of
Energy, Office of Basic Energy Sciences, under contract no. DE-AC02-76SF00515.
Work at University of Wisconsin was supported by the Department of Energy,
Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering under Award DE-SC0021971. Research at the University of Maryland
was supported by the Air Force Office of Scientific Research Award No.
FA9550-14-1-0332 (support of T.M.), the Department of Energy Award No. DE-
SC-0019154 (specific heat experiments), the Gordon and Betty Moore
Foundation’s EPiQS Initiative through Grant No. GBMF9071 (materials
synthesis), NIST, and the Maryland Quantum Materials Center. Sample synthesis
and preliminary characterization were supported by NIST. Work at Karlsruhe was
supported by the Deutsche Forschungsgemeinschaft (German Research Foundation)
Project ER 463/14-1.

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SUPPLEMENTARY INFORMATION

Interplay between magnetism and superconductivity in UTe2

D.S. Wei,1,2,3, D. Saykin,1,3,4 O. Y. Miller,1,4 D. F. Agterberg,5 N. P.
Butch,6,7 J. Paglione,6,7,8 and Aharon Kapitulnik,1,2,3,4 1Geballe Laboratory
for Advanced Materials, Stanford University, Stanford, CA 94305
2Department of Applied Physics, Stanford University, Stanford, CA 94305
3Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory,
2575 Sand Hill Road, Menlo Park, CA 94025
4Department of Physics, Stanford University, Stanford, CA 94305
5Department of Physics, University of Wisconsin, Milwaukee, Milwaukee,
Wisconsin 53201, USA.
6Department of Physics, Quantum Materials Center, University of Maryland,
College Park, MD 20742, USA
7NIST Center for Neutron Research, National Institute of Standards and
Technology, Gaithersburg, MD 20899, USA.
8The Canadian Institute for Advanced Research, Toronto, Ontario, Canada.

##  MATERIALS, METHODS AND ADDITIONAL INFORMATION

### Single Crystals Growth

The UTe2 single crystals were grown by a chemical vapor transport method.
Powder x-ray diffraction (XRD), neutron scattering and Laue XRD measurements
indicate that the single crystals are of high quality. The details of the
sample growth and characterization are given in Refs [32, 35].

### Measurement Apparatus

High-resolution measurements of Kerr rotation were performed using a fiber-
based zero-area loop Sagnac interferometer as first described in Ref. [19].
Following the main components shown in Fig. S1, two linearly polarized
counterpropagating beams comprising the interferometer are isolated along the
fast and slow axes of $\sim$10 m of polarization-maintaining fiber and are
modulated in phase with a frequency that matches the transit time of light
through the interferometer. The linearly polarized light traveling along each
of these paths passes through a quarter-waveplate and is converted into
circularly polarized light just above the sample. In the absence of magnetic
field, the apparatus is completely reciprocal by symmetry except for the
sample. Thus, upon reflection, one branch of the interferometer acquires a
phase shift of $+\theta_{K}$, while its orthogonally polarized counterpart
acquires an opposite phase shift of $-\theta_{K}$. The two phase shifts are
added at the detector and extracted from the ratio of the first and second
harmonics of the modulated signal.

Figure S1:  Schematic of the zero-area loop Sagnac interferometer used in this
study. On the right is the end-part showing one beam that enters the quarter-
waveplate from the fast axis as right circularly polarized, and reflected back
into the slow axis as left circularly polarized after experiencing a Kerr
angle shift. A second beam propagates in the direction of the sample in the
slow axis and is reflected into the fast axis.

### Field-Cool Measurements

To compare the field cool observed in UTe2 to other uranium based, but non-
paramagnetic systems, we show in Fig. S2 remanent magnetization and Kerr
effect data for UPt3. Focusing first on the UPt3 results, we observe that the
remanent magnetization in Fig. S2(a) “ignores” the second transition where
TRSB is onset for this material, and is fully described a a critical state
model with relatively strong pinning [25]. By contrast, the Kerr effect in
Fig. S2(b) is independent of whether it was obtained in a field-cool or zero-
field-cool measurement, and it starts at the known lower $T_{c}$ of the
material [20]. In (c) we show a set of remanent Kerr effect as a function of
magnetic field for UTe2, which clearly resemble the behavior of the
magnetization of UPt3, but not the Kerr effect response.

Figure S2:  Field-cool Remanent Magnetization (a) after cooling in magnetic
fields of 182, 25 and 5 G (from Ref. [25]) vs. field-cool Kerr measurements
(b) (from Ref. [20, 18]) for UPt3 crystals. The $-50$ G FC data was flipped
from a negative to positive $\theta_{K}$. Here solid line is a fit to
$\theta_{K}\propto\sqrt{[1-(T/T_{c-})^{2}][1-(T/T_{c+})^{2}]}$, where
$T_{c+}=0.55$ K and $T_{c+}=0.47$ K. Vertical dashed lines mark the two
superconducting transitions, with the lower one associated with the TRSB
transition as observed by Kerr effect.

### Kerr Effect Measurements at Zero-Field

Kerr angle data was acquired at a rate of 1 sample per second. Due to the slow
time constant of the RF lock-in amplifier used to measure the signal, each
data point is correlated with its 8 closest neighbors. In order to accurately
compute error bars, we eliminated these correlations by dividing the data into
50-point chunks and averaging the points in each chunk, leaving us with a set
of averages $\\{x_{i}\\}$. These averages are then almost completely
uncorrelated and thus represent independent samples with 50 sec. of averaging
time. Before plotting the data vs temperature, the chunk averages are then
binned into temperature bins, with all the xi in each bin averaged together.
The points plotted in all $\theta_{K}(T)$ graphs are these bin averages. The
error bars given are the 1-$\sigma$ standard error of the mean, computed as
the standard deviation of the $x_{i}$ in each bin divided by the square root
of the number of chunks.

### Kerr Effect Measurements at finite magnetic field

UTe2 is paramagnetic with a finite positive susceptibility, which works
against the Meissner currents in the superconducting state, and against the
diamagnetism of the optics (here we note that there are several components of
the optics such as a few inches of fiber strand, lenses and quarter waveplate,
all transparent insulators and thus exhibit diamagnetism in magnetic field.)
However, since at these low fields both contributions are linear in magnetic
field, we can write for the Kerr response in the presence of small magnetic
field:

$\Delta\theta_{K}=-aH+\chi_{K}(T)H$ (S.10)

where $a$ is the optics diamagnetic contribution and $\chi_{K}(T)$ is the
temperature-dependent Kerr effect due to the magnetic susceptibility of the
material in the normal state.

Figure S3: (a): Susceptibility of UTe2 in magnetic fields below $H^{*}$. (b):
Kerr effect measurements. The sample was cooled in magnetic field of 240G and
then measured when warmed up in 240G (A) and in zero-field (B). The vertical
red line is the zero-field warmup Tc.

Following our analysis in the previous section, we determined that at e.g.
240G the remanent magnetization is as large as the field applied, and
therefore the diamagnetic contribution due to the optics can be extracted as
the difference between the in-field and zero-field Kerr angle at base
temperature, thus $aH\approx 0.5\mu$rad as is evidence from Fig. S3.

Establishing the effect of the optics, we can subtract this effect from any
finite field measurement. In particular we show in Fig. S4 a hysteresis loop
performed at 0.3 K in the magnetic field range of $\pm 500$ G. The sample was
first cooled to base temperature in zero magnetic field. As explained in the
main text, in such ZFC the observed Kerr effect will typically lie in the
range of $\pm 0.4~{}\mu$rad, where smaller Kerr angles correspond to domain
averaging effect. The Kerr effect was then measured while increasing the
magnetic field to $+500$ G, then decreasing it to $-500$ G, increasing it to
$+500$ G, and finally decreasing it to zero. While the scatter in the data is
rather large, presumably due to domain effects and vortex motion, it is clear
that unlike the field-cool data shown above, after cooling in zero field the
Kerr response is weak, indicating the good shielding of the sample in
preventing vortices to reach the center of the sample (note: the Kerr effect
is measured with a beam of size $\sim 10~{}\mu$m at the center of the $\sim
1.2$ mm size sample.) At the same time we clearly observe the sharp increase
in Kerr angle as we approach zero magnetic field, indicating the critical
current associated with the presence of trapped flux, which needs to change
direction as the field is reversed. Compared to the field cool measurements at
500 G (Fig. 4 in the main text), we see that only 15% of the flux is trapped
at the center of the sample where Kerr effect is measured. Below $\sim 250$ G
similar loops yield a zero-field intersect of $<0.4~{}\mu$rad, indicating a
more intrinsic effect, which we argue to be the effect of the order-parameter
itself.

Figure S4:  Hysteresis of Kerr angle between -500 and +500 G. Arrows mark the
direction of the field. The multi-looped experiment starts with a ZFC to a
value consistent with Fig. (1)c. Then in circled section (1) (green markers)
the field increased to +500 G, decreased to -500 G in section (2) (blue
markers) , increased back to +500 G in section (3) (red markers) and then back
to zero-field in section (4) (purple markers). Note: The optics contribution
described in Eqn. S.10 was subtracted from the data. The dashed line connect
the mean of the data points, and is bolder near zero field to emphasize the
increase in Kerr effect.

### Magnetic Susceptibility

Figure S5 below shows the low-temperature part of the magnetic susceptibility
of a UTe2 crystal measured in a SQUID magnetometer down to 1.8 K. Focusing on
the $c$-axis, Fig. S5 indicates that the low field susceptibility is very
weakly field dependent, with a shallow minimum at $\sim 240$ G. This plot is
therefore important in comparing the susceptibility to the Kerr effect that we
analyze in the main text (See Figs. 5 and 7.) Fig. S5 is the magnetic
susceptibility along the three principal axes of the crystal, showing a weaker
than Curie-law for either direction.

Figure S5: Magnetic susceptibillity of UTe2 single crystal above $T_{c}$. (a)
Measurements along the $c$-axis at 10 G, 240 G and 1000 G, showing weak
magnetic field dependence and a shallow minimum in the field range of
$\sim$240 G. (b) Susceptibility at 1000 G. Note that $a$-axis (easy axis)
response is largest, while along the $b$-axis a minimum around 3 K is barely
visible.

### The Lower Critical Field of UTe2

A first task is therefore to determine the lower critical field $H_{c1}$,
which for UTe2 is not simple. Low field magnetization isotherms were reported
by Paulsen et al. [9] along the $a$\- and $b$\- axes, indicating a reasonable
fit to the “standard” form $H_{c1}(T)=H_{c1}(0)[1-(T/T_{c})^{2}]$, with
$H^{a}_{c1}(0)=23$G and $H^{b}_{c1}(0)=42$G, and the good fits could be an
indication of the reliability of the measurement. However, these measurements
do not seem to agree with complementary measurements of the thermodynamic and
upper critical fields, expecting:

$H_{c1}H_{c2}=H_{c}^{2}[{\rm ln}\kappa+c],$ (S.11)

where $\kappa$ is the Ginzburg-Landau (GL) parameter, and $c\approx 0.5$ is a
constant. In fact, a test of the $H_{c1}(0)$ values against the separately
measured upper-critical fields $H_{c2}(0)$, and thermodynamic critical field
$H_{c}(0)$ indicate that only the $a$-axis direction agrees with the expected
relation. Furthermore, the anisotropy of $H_{c1}$ along $a$\- and $b$\- axes
is found to be opposite to that expected from the anisotropy of the respective
$H_{c2}(T)$, even close to $T_{c}$ [9]. These results might indicate that
estimating $H^{c}_{c1}(T)$ from standard data may be complicated. For example,
using the thermodynamic critical field, $H_{c}(0)=490$G [9], and
$H^{c}_{c2}(0)=7.5$ T (as extrapolated from low-field in [32]), and a GL-
parameter of $\kappa\approx 110$, estimated separately from
$H_{c2}(0)=\sqrt{2}\kappa H_{c}(0)$, we obtain an estimate of
$H^{c}_{c1}(0)\approx 17$G, while in the same paper [9], using remanent
magnetization arguments including demagnetization factors, a value of $\sim
13$ G was deduced. At the same time, Bae et al. [36] estimated the effective
penetration depth $\lambda^{ab}_{eff}$, when a magnetic field is applied along
the $c$-axis to be in the range of 1126 nm to 947 nm. Taking the shorter value
gives a maximum estimate of $H^{c}_{c1}(0)\approx 8.5$G. Turning to our own
data of the value of the low-temperature saturated Kerr effect taken from
Figs. 3 and 4 and plotted in Fig. 6, we also find that $10{\rm G}\lesssim
H^{c}_{c1}(0)\lesssim 25{\rm G}$. All these results suggest that it is safe to
assume $H_{c1}(0)\lesssim 20$G, which will be important for the discussion
below.