# Absence of a bulk charge density wave signature in x-ray measurements of
UTe2

Caitlin S. Kengle  Department of Physics, University of Illinois, Urbana,
Illinois 61801, USA  Materials Research Laboratory, University of Illinois,
Urbana, Illinois 61801, USA  Dipanjan Chaudhuri  Department of Physics,
University of Illinois, Urbana, Illinois 61801, USA  Materials Research
Laboratory, University of Illinois, Urbana, Illinois 61801, USA  Xuefei Guo
Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801,
USA  Thomas A. Johnson  Department of Physics, University of Illinois, Urbana,
Illinois 61801, USA  Materials Research Laboratory, University of Illinois,
Urbana, Illinois 61801, USA  Simon Bettler  Department of Physics, University
of Illinois, Urbana, Illinois 61801, USA  Materials Research Laboratory,
University of Illinois, Urbana, Illinois 61801, USA  Wolfgang Simeth
Laboratory for Neutron and Muon Instrumentation, Paul Scherrer Institute,
Villigen PSI, Switzerland  Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  Matthew J. Krogstad  Materials Science
Division, Argonne National Laboratory, 9700 South Cass Avenue, Lemont,
Illinois 60439, USA  Zahir Islam  Advanced Photon Source, Argonne National
Laboratory, 9700 South Cass Avenue, Lemont, Illinois 60439, USA  Sheng Ran
Maryland Quantum Materials Center, Department of Physics, University of
Maryland, College Park, MD, USA  NIST Center for Neutron Research, National
Institute of Standards and Technology, Gaithersburg, MD, USA  Department of
Physics, Washington University in St. Louis, St Louis, MO, USA  Shanta R. Saha
Maryland Quantum Materials Center, Department of Physics, University of
Maryland, College Park, MD, USA  Johnpierre Paglione  Maryland Quantum
Materials Center, Department of Physics, University of Maryland, College Park,
MD, USA  Canadian Institute for Advanced Research, Toronto, Ontario, Canada
Nicholas P. Butch  Maryland Quantum Materials Center, Department of Physics,
University of Maryland, College Park, MD, USA  NIST Center for Neutron
Research, National Institute of Standards and Technology, Gaithersburg, MD,
USA  Eduardo Fradkin  Department of Physics, University of Illinois, Urbana,
Illinois 61801, USA  Anthony J Leggett Institute for Condensed Matter Theory,
University of Illinois, Urbana, Illinois 61801, USA  Vidya Madhavan
Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801,
USA  Peter Abbamonte<EMAIL_ADDRESS>Department of Physics, University
of Illinois, Urbana, Illinois 61801, USA  Materials Research Laboratory,
University of Illinois, Urbana, Illinois 61801, USA

## Abstract

The long-sought pair density wave (PDW) is an exotic phase of matter in which
charge density wave (CDW) order is intertwined with the amplitude or phase of
coexisting, superconducting order [1, 2]. Originally predicted to exist in
copper-oxides, circumstantial evidence for PDW order now exists in a variety
of materials. Recently, scanning tunneling microscopy (STM) studies have
reported evidence for a three-component charge density wave (CDW) at the
surface of the heavy-fermion superconductor, UTe2, persisting below its
superconducting transition temperature [3, 4, 5]. Here, we use hard x-ray
diffraction measurements on crystals of UTe2 at $T=1.9$ K and $12$ K to search
for a bulk signature of this CDW. Using STM measurements as a constraint, we
calculate the expected locations of CDW superlattice peaks, and sweep a large
volume of reciprocal space in search of a signature. We failed to find any
evidence for a CDW near any of the expected superlattice positions in many
Brillouin zones. We estimate an upper bound on the CDW lattice distortion of
$u_{max}\lesssim 4\times 10^{-3}\mathrm{\AA}$. Our results suggest that the
CDW observed in STM is either purely electronic, somehow lacking a signature
in the structural lattice, or is restricted to the material surface.

##  Introduction

Strongly correlated electronic systems often exhibit complex phase diagrams
characterized by multiple ordered phases in close proximity. These broken-
symmetry phases often compete or cooperate with one another, and exotic ground
states, including unconventional superconductivity, can emerge as a result [6,
7, 8]. When the phases cooperate to form a single ground state, the orders are
said to be intertwined. A pair density wave (PDW) is a canonical example of
such intertwined order in which superconductivity intimately interacts with
charge density wave (CDW) order, resulting in a spatial modulation of the
superconducting order parameter [9, 7]. Evidence for a PDW in real materials
is mixed, the most compelling case currently being the cuprates [10, 11]. It
is therefore of great interest to search for evidence of PDW phases in other
materials.

Recently, superconductivity was discovered in the actinide heavy fermion
compound, UTe2 with a transition temperature, $T_{c}$, as high as $2$ K, and
possible spin-triplet pairing [12, 13, 14]. Evidence for an exotic pairing
mechanism comes from its extremely high, strongly anisotropic upper critical
field [12]. Additionally, multiple re-entrant superconducting phases have been
observed at high magnetic fields, termed “Lazarus superconductivity” [15, 16],
and under hydrostatic pressure [17, 18, 19]. The presence of a PDW may provide
insight to the nature of the superconducting order parameter, a source of
active debate in UTe2 [20, 14, 13, 21, 15, 12, 22, 23, 24, 25, 26].

Scanning tunneling microscopy (STM) is an important method for detecting CDWs
and PDWs on the surfaces of materials [4, 27, 28, 29]. This method was notably
used to detect a Cooper pair CDW in Bi2Sr2CaCu2O8+δ [27]. Similar observations
were recently made in UTe2. Using both normal and superconducting STM tips,
charge order peaks were identified in the tunneling spectra, indicating the
possible existence of a pair density wave phase below $T\approx 4.5$ K [3, 4,
5]. The charge density wave and pair density wave have coincident wave
vectors, though the PDW is shifted in phase by $\pi$ [4]. While these
observations are compelling, it is important to corroborate them with a
different technique.

X-ray scattering is the quintessential method for detecting and characterizing
CDWs [30, 31, 32]. X-ray diffraction measures the energy-integrated (i.e.,
“equal time”) density-density correlation function of a material, giving a
direct measurement of the charge density of the bulk, complimentary to STM,
which measures the wavefunction overlap of a tip and a surface in the presence
of a local electric field. Many charge density waves have been discovered
using X-ray diffraction in 1D systems, [33, 34], layered chalcogenides [35,
36], and copper-oxides [10, 37, 38, 39], among many others. Here, we apply
this method to investigate CDW behavior in UTe2.

## STM Summary

UTe2 has a proclivity to cleave along the (011) surface normal. It has an
orthorhombic crystal structure (space group $Immm$) with lattice parameters
$a=4.12\,\mathrm{\AA}$, $b=6.08\,\mathrm{\AA}$, and $c=13.96\,\mathrm{\AA}$.
The Brillouin zone of UTe2, with the (011) direction indicated, is shown in
Fig. 1 (a). A representative STM image of the (011) surface is shown in Fig. 1
(b). The figure, reproduced from [3], shows the Fourier transform of the local
density of states taken at the Fermi energy, $E_{F}$, at $T=300$ mK. Three CDW
modulations are visible in the data at incommensurate wavevectors,
$q_{1}^{CDW}$, $q_{3}^{CDW}$, and $q_{3}^{CDW}$. These modulations, described
in detail in Refs. [3, 4, 5], appear below $T\approx 4.5$ K and possibly
persist up to temperatures as high as $T\approx 10$ K. In the surface
coordinates defined in Ref. [3], which we summarize in Fig. 1(c) , the CDW
wave vectors have coordinates $q_{1}^{CDW}=(0.43q_{x},q_{y})$,
$q_{2}^{CDW}=(0.43q_{x},-q_{y})$, and $q_{3}^{CDW}=(0.57q_{x},0)$, where
$q_{x}$ and $q_{y}$ are the vector components of a nearby Bragg peak. Here we
seek to see if these CDW modulations are present in the bulk of UTe2 through
low-temperature X-ray diffraction measurements.

Figure 1:  Relationship between STM measurements and 3D reciprocal space for
UTe2. (a) Body centered orthorhombic Brillouin zone. The green arrows
indicated by $b_{1,2,3}$ represent the primitive reciprocal lattice vectors.
The black arrows represent the vectors discussed in Eqs. 3a-3b. Surface
projected Brillouin zone. (b) Zero bias STM measurement within the CDW phase
(reproduced from [3]). (c) 3D Bragg peaks projected to (011) surface with the
corresponding 2D surface projected Brillouin zone, confirming the consistency
with the peaks found in STM. Vector components between the $q=0$ point and the
relevant Bragg peak and CDW positions according to Ref. [3].

## Results

A major advantage of high energy X-rays is that they give access to a large
volume of momentum space. Additionally, high energy X-rays have proven
effective in detecting purely electronic CDWs, which occur without a
significant periodic lattice distortion [37]. Representative 2D cuts through
3D reciprocal space are shown in Fig. 2 (a) with $l=5$ and (b) with $k=2$.
Sharp Bragg peaks at integer Miller indices, $(h,k,l)$, illustrate the
crystalline quality of the samples.

Figure 2 (c) shows a scan of the $kl$ plane taken at non-integer $h=-0.43$
r.l.u. at $T=1.9$ K, which shows the primary background features: scattering
from the beryllium dome and radiation shield and streaking from Bragg peaks
caused by the finite bandwidth of the incident X-ray beam. These features were
found to be temperature-independent from our base temperature of $T=1.9$ K to
$T=12$ K, which is above the CDW phase transition observed in Refs. [3, 4, 5]
[Fig. 3]. No sign of a CDW was observed in this particular cut of three-
dimensional reciprocal space.

The wide reciprocal space cut shown in Figure 2 (c), done at fixed $h=-0.57$,
should intersect the $q^{CDW}_{1}$ and $q^{CDW}_{2}$ positions from the $h=-1$
zone as well as the $q^{CDW}_{3}$ position from the $h=0$ zone. However, no
CDW signal is observed above the experimental background at any of the three
expected locations.

[t]

Figure 2: $T=1.9$ K measurement of reciprocal space in UTe2 crystal. (a) $hk$
cut along $l=5$ r.l.u. and (b) $hl$ cut along $k=2$ r.l.u. showing Bragg peak
reflections obeying $h+k+l=2n$. (c) $kl$ cut along non-integer $h=-0.43$
r.l.u. The weak signal comes from scattering from the beryllium domes and from
the tails of Bragg peaks caused by the finite bandwidth of the X-ray source.
All panels have the same intensity scale.

With hopes of uncovering a more subtle effect, we performed temperature
difference measurements, comparing the same regions of reciprocal space at
$T=1.9$ K and $T=12$ K, the latter being above the putative CDW ordering
temperature [3, 4, 5]. We focused on three Brillouin zones near $(-1,1,5)$,
$(-1,3,5)$, and $(-1,2,5)$, shown in Fig. 3 (these data sets were integrated
over a narrow range of $h$ indicated in square brackets in each panel). The
red lines indicate, in each region, the possible CDW locations constrained
from Eqs. 6 and Eqs. 7. No changes with temperature were observed in any of
these three Brillouin zones, or any of the other Brillouin zones mapped in
Fig. 2.

Figure 3: Diffraction images taken below and above the CDW transition
temperatures in the $kl$ plane at the appropriate $h$ values. The red line
represents the calculated CDW positions in this zone for (a) $q^{CDW}_{1}$,
(b) $q^{CDW}_{2}$, and (c) $q^{CDW}_{3}$ at $T=1.9$ K. To compare, the
calculated position of (d) $q^{CDW}_{1}$, (e) $q^{CDW}_{2}$, and (f)
$q^{CDW}_{3}$ at $T=12$ K, above the CDW transition temperature, are also
shown. Streaks in (a) and (d) come from the finite bandwidth of the incident
X-ray beam. Rings in (b, c, e, f) come from scattering from amorphous
beryllium of the sample enclosure. All images have the same intensity scale.

Lacking a clear signature of a CDW, we can nevertheless use our data to place
an upper bound on the CDW amplitude in UTe2. The background level of our
measurements places an upper limit on the CDW peak intensity. The majority of
the background comes from the beryllium rings (clearly visible in the $hk$
plane and line cuts), whose average photon counting intensity is $\sim 10$ Hz
in this reciprocal space region. Assuming the photon noise is Poisson
distributed, this places an upper bound on the maximum CDW intensity of
$I_{CDW}^{max}\leq\sqrt{10}\approx 3$ Hz.

How this intensity translates into a bound on the CDW amplitude depends on the
CDW correlation length. In the STM data, the CDW peaks are of similar width to
the crystalline Bragg peaks (Fig. 1(b)) whose width, in turn, is limited by
the field of view of the STM image. The CDW width may also be field-of-view
limited, or it may be inherently that broad. We therefore consider two cases:
(1) The intrinsic width of the CDW is as observed in the STM measurements, and
(2) the CDW has the same width as the structural Bragg peaks observed in our
X-ray measurements. The latter case is summarized in Fig. 4.

The amplitude of the CDW is proportional to the integrated intensity of the
CDW reflection, which can be estimated,

$I_{CDW}^{Integrated}\lesssim\frac{1}{2}(I_{CDW}^{max})(f_{CDW})^{3}$ (1)

where $f_{CDW}^{3}$ is the three-dimensional width of the CDW in reciprocal
space. In the case where the CDW width is taken to be the same as in STM
measurements, $f_{CDW}^{3}=(\xi^{-1}_{STM})^{3}$, where $\xi^{-1}_{STM}$ is
the full-width at half-maximum of the CDW peak in STM, measured along the
$q_{x}$ direction in Fig. 1(b), which we have assumed to be isotropic. In the
other case, $f_{CDW}^{3}=(\delta_{h}\delta_{k}\delta_{l})$ where
$\delta_{h,k,l}$ represent the widths of the nearest Bragg peak in the three
orthogonal reciprocal space directions.

Figure 4 (a-b) shows cuts through the $(-1,2,5)$ Bragg peak. Panels (c-e) show
line cuts through the maxima in the $h$, $k$, and $l$, respectively. The peak
has a maximum intensity of $1.74\times 10^{6}$ Hz and widths in the $h$, $k$,
and $l$ directions of $0.043\mathrm{\AA}^{-1}$, $0.032\mathrm{\AA}^{-1}$, and
$0.050\mathrm{\AA}^{-1}$, respectively. The integrated intensity of this Bragg
peak can then be approximated in the same way as the CDW peak shown in Eq. 1,
$I_{Bragg}^{Integrated}=\frac{1}{2}(I_{Bragg}^{max})(\delta_{h}\delta_{k}\delta_{l})$,
where $\delta_{h,k,l}$ are the measured widths along the crystallographic
directions. Using this number, we can place an upper limit on the charge
amplitude of the CDW,

$\frac{\rho_{CDW}^{2}}{\rho_{(-1,2,5)}^{2}}\lesssim\frac{I_{CDW}^{Integrated}}{I_{Bragg}^{Integrated}},$
(2)

where $\rho_{(-1,2,5)}$ is the calculated structure factor of the $(-1,2,5)$
Bragg peak and $\rho_{CDW}$ is the CDW amplitude. The structure factor of the
$(-1,2,5)$ Bragg peak is calculated to be
$\rho_{(-1,2,5)}=169.38\>\mathrm{e}^{-}/v_{\mathrm{cell}}$, where
$v_{\mathrm{cell}}$ is the volume of the standard orthorhombic unit cell.
Comparing this to the CDW charge density gives $\rho_{CDW}\lesssim
2.13\>e^{-}/v_{\mathrm{cell}}$ in the case that the CDWs have the same width
as in STM measurements or $\rho_{CDW}\lesssim 0.70\>e^{-}/v_{\mathrm{cell}}$
in the case that the CDWs have the same width as the (-1,2,5) Bragg peak.

## Conclusions

Our study indicates that UTe2 does not exhibit a bulk CDW of the sort that
occurs in many other transition metal dichalcogenides, such as NbSe2 or TaS2
[40], which exhibit pronounced CDW signatures in both STM and X-ray
measurements [35, 41]. Note that, while we have used STM measurements as a
guide for where to look in reciprocal space, we performed a full, 3D momentum
sweep and would have observed the CDW, even if its wave vector deviated from
what was observed in STM.

How to interpret the upper bound on the amplitude, $\rho_{CDW}$, depends on
whether one assumes the CDW is electronic, the scattering arising purely from
a valence modulation, or structural, the scattering arising from a periodic
distortion to the crystalline lattice.

Assuming the CDW is purely structural, the magnitude of the lattice
displacement should be of order $u_{max}/a\sim\rho_{CDW}/\rho_{tot}$, where
$\rho_{tot}$ represents the total charge density [40]. Assuming the order is
long-ranged (see Results), the amplitude bound $\rho_{CDW}\lesssim
0.70\>e^{-}/v_{\mathrm{cell}}$ translates into an atomic displacement
$u_{max}/a\lesssim 8.93\times 10^{-4}$, where $a$ is the lattice parameter of
UTe2. Alternatively, if we now consider the case that the CDW has a short
correlation length, the atomic displacement would be $u_{max}/a\lesssim
2.72\times 10^{-3}$. Both of these numbers are markedly lower than the atomic
displacements in other dichalcogenide CDWs, which have distortions $u_{max}/a$
ranging from 1-7% [40]. We conclude that whatever CDW is observed in STM
experiments in UTe2 has a completely different character from CDWs in other
dichalcogenides.

We can also consider the case in which the CDW is assumed to be purely
electronic, meaning the atoms do not shift from their equilibrium positions,
and the CDW is dominated by a modulation in the valence electron density [37,
42, 43]. In this case, our X-ray measurements allow for an amplitude of up to
$\sim 2\>e^{-}$ per unit cell. This would represent a significant fraction of
the valence band [19, 44, 45]. Such an amplitude might be comparable to the
effect seen in STM, though would be difficult to reconcile with the absence of
a resistive transition in bulk transport measurements in this range of
temperatures [14, 46, 47]. We cannot, however, completely rule out this
possibility.

Figure 4: Scan through (-1,2,5) Bragg peak at $T=1.9$ K in the (a) $hk$ plane
and (b) $hl$ plane. Both plots are integrated $\pm 0.05$ r.l.u. in the out-of-
plane direction. Line cut through Bragg peak along the (c) $h$, (d), $k$, and
(e) $l$ directions. Signal from amorphous beryllium appear in (a), (c) and
(d).

A final explanation is that the CDW observed in STM measurements is restricted
to the surface of UTe2. This would explain why its presence is so robust in
multiple STM measurements [3, 4, 5] and absent in transport measurements and
the bulk-sensitive X-ray measurements presented here. This scenario would be
reminiscent of the CDW seen in STM measurements of NaxCa2-xCuO2Cl2 [48]. Such
a CDW transition can be explained through the framework of extraordinary phase
transitions, in which the critical behavior is different at the surface
compared to in the bulk of material [49, 50]. As a result the transition
temperature of the surface can be higher than that of the bulk. This allows
for the possibility that the CDW in UTe2 could be of either electronic or
structural origin, the absence of a bulk signature in X-ray and transport
measurements being a consequence of the small volume fraction of the effect.
In this scenario, one might expect different surface terminations to lead to
different surface ordering effects.

##  Methods

### Low-temperature X-ray diffraction measurements

Experiments were done at beamline 6ID-D at the Advanced Photon Source at
Argonne National Laboratory. The sample was grown under the same conditions as
those in [3]. The incident X-ray energy was $E_{i}=100$ keV (wavelength
$\lambda=0.124\mathrm{\AA}$), at which the penetration depth in UTe2 is $\sim
0.6$ mm, making this a bulk measurement.

The sample was loaded into a vacuum chamber with transparent beryllium dome
and radiation shield held at a base pressure $\sim 10^{-5}$ Torr. The sample
was encapsulated in GE varnish to reduce time exposed to air during sample
transport. Sample temperature was controlled by an ARS DE302 three-stage
cryocooler in which pre-cooled helium gas was fed into an open circuit Joule-
Thomson stage to reach a base temperature of $T=1.9$ K.

Data were collected using a Pilatus 3 x CdTe 2M detector at $T=1.9$ and $12$
K. Full, three-dimensional (3D) reciprocal space maps were performed by
rotating the sample in tbe beam by 145∘ using a Huber four circle
diffractometer. The lattice parameters varied from
$a=4.099(20)\>\mathrm{\AA}$, $b=6.137(28)\>\mathrm{\AA}$, and
$c=14.145(64)\>\mathrm{\AA}$ at $T=1.9$ K to $a=4.101(20)\>\mathrm{\AA}$,
$b=6.135(28)\>\mathrm{\AA}$, and $c=14.147(65)\>\mathrm{\AA}$ at $T=12$ K.
Discrepancies between these values and those reported in Ref. [51] are likely
a due to varying sample mosaicity.

### Calculation of CDW Wave Vectors from STM Data

STM measurements have found the locations of CDW peaks on the (011) surface
[3, 4, 5]. However, since STM is a surface measurement the periodicity of the
CDW in the direction perpendicular to the surface is unconstrained. Using STM
results we place partial constraints on the Miller indices, and identify what
regions of momentum space the CDW is expected to be visible to bulk X-ray
diffraction measurements, making the assumption that the wave vectors at the
surface are the same as in the bulk.

As noted before, all measurements of the CDW in STM are on the cleaved $(011)$
surface. In sample coordinates, the unit vector normal to this surface
$\hat{\textbf{z}}=\nicefrac{{\overrightarrow{\textbf{z}}}}{{|\textbf{z}|}}=\nicefrac{{\left(0,c,b\right)}}{{\sqrt{b^{2}+c^{2}}}}$.
The two vectors orthonormal to $\hat{\textbf{z}}$, which define the coordinate
system of the surface, are

$\displaystyle\hat{\textbf{x}}$ $\displaystyle=\left(1,0,0\right),$ (3a)
$\displaystyle\hat{\textbf{y}}$
$\displaystyle=\hat{\textbf{z}}\times\hat{\textbf{x}},$ (3b)

where $\hat{\textbf{x}}$ is defined to coincide with the $(h,k,l)=(1,0,0)$
crystallographic direction.

The peaks observed in an STM measurement [e.g., Fig. 1(b)] are projections of
the bulk 3D Bragg peaks, which are indexed by Miller indices $(h,k,l)$, onto
to the $(x,y)$ plane defined by Eqs. 3a and 3b. The projection is given by the
two-component vector

$\displaystyle{q}_{2D}$
$\displaystyle=\biggl{(}\hat{\textbf{x}}\cdot\textbf{{q}}_{3D},\hat{\textbf{y}}\cdot\textbf{{q}}_{3D}\biggr{)}$
(4)

where
$\textbf{q}_{3D}=\biggl{(}h\dfrac{2\pi}{a},k\dfrac{2\pi}{b},l\dfrac{2\pi}{c}\biggr{)}$.
The locations of these projected points, and the associated 2D Brillouin zone
of the surface, are shown in Fig. 1(c) and are consistent with STM data in
Fig. 1(b). (Note that we have included only peaks with $h+k+l=2n$, where $n$
is an integer, the others being forbidden by crystal symmetry). The 2D
surface-projected Brillouin zone shown in light blue in panels (b) and (c) was
created by drawing perpendicular bisectors between the surface-projected Bragg
peaks.

In terms of the surface basis vectors Eq. 3, the CDW modulations observed in
Ref. [3], are
${q}^{\mathrm{CDW}}_{1}=\biggl{(}0.43\dfrac{2\pi}{a},\dfrac{2\pi}{\sqrt{b^{2}+c^{2}}}\biggr{)}$,
${q}^{\mathrm{CDW}}_{2}=\biggl{(}0.43\dfrac{2\pi}{a},\dfrac{-2\pi}{\sqrt{b^{2}+c^{2}}}\biggr{)}$,
and ${q}^{\mathrm{CDW}}_{3}=\biggl{(}0.57\dfrac{2\pi}{a},0\biggr{)}$. The 3D
positions of each of the three CDW modulations, $q_{i}^{CDW}$, such as can be
constrained by a 2D STM measurement, are determined by equating,

$q_{2D}=q_{i}^{CDW}.$ (5)

Doing so, we find that the STM experiment fully constrains the $h$ components
of each of the three CDWs, i.e.,

$\displaystyle h^{CDW}_{1}$ $\displaystyle=0.43\>\mathrm{r.l.u.}$ (6a)
$\displaystyle h^{CDW}_{2}$ $\displaystyle=0.43\>\mathrm{r.l.u.}$ (6b)
$\displaystyle h^{CDW}_{3}$ $\displaystyle=0.57\>\mathrm{r.l.u.}.$ (6c)

The $k$ and $l$ components cannot be determined independently, but Eq. 5
establishes a relationship between them:

$\displaystyle k^{CDW}_{1}$ $\displaystyle=l^{CDW}_{1}-1\>\mathrm{r.l.u.}$
(7a) $\displaystyle k^{CDW}_{2}$
$\displaystyle=l^{CDW}_{2}+1\>\mathrm{r.l.u.}$ (7b) $\displaystyle
k^{CDW}_{3}$ $\displaystyle=l^{CDW}_{3}$ (7c)

Note that, unlike Eq. 6, Eq. 7 contain only natural numbers. This suggests,
though is not proof, that the incommensurate character of the CDW in UTe2
occurs only in the $(1,0,0)$ reciprocal lattice direction.

Combining Eq. 6 and 7, we find the expected locations of the three CDWs in
UTe2, such as can be inferred from STM measurements, to be
$\textbf{q}^{CDW}_{1}=(0.43,l^{CDW}_{1}-1,l^{CDW}_{1})$,
$\textbf{q}^{CDW}_{2}=(0.43,l^{CDW}_{2}+1,l^{CDW}_{2})$, and
$\textbf{q}^{CDW}_{3}=(0.57,l^{CDW}_{3},l^{CDW}_{3})$, where the values of
$l^{CDW}_{i}$ are unknown. These expressions constrain the locations of the
CDWs to one-dimensional cuts through 3D reciprocal space. Note that these
vectors are defined relative to any crystalline Bragg peak, implying a large
number of symmetry-related CDW satellites that could potentially be visible in
any Brillouin zone.

## Acknowledgements

We thank P.F.S. Rosa for helpful discussions. This work was primarily
supported by the Center for Quantum Sensing and Quantum Materials, an Energy
Frontier Research Center funded by the US Department of Energy (DOE), Office
of Science, Basic Energy Sciences (BES), under award no. DE-SC0021238. Growth
of UTe2 crystals was supported by National Science Foundation grant NSF-DMR
2105191. STM measurements were supported by DOE BES award no. DE-SC0022101.
Sample characterization was partly supported by DOE BES award no. DE-
SC-0019154 and the Maryland Quantum Materials Center. E.F. was partly
supported by NSF grant NSF-DMR 2225920. PA, VM, and JPP gratefully acknowledge
additional support from the EPiQS program of the Gordon and Betty Moore
Foundation, grant nos. GBMF9452, GBMF9465, and GBMF9071, respectively. This
research used resources of the Advanced Photon Source, a DOE User Facility
operated Argonne National Laboratory under contract no. DE-AC02-06CH11357.
Identification of commercial equipment does not imply recommendation or
endorsement by NIST.

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