# Search for orbital magnetism in the kagome superconductor ${\rm
CsV_{3}Sb_{5}}$ using neutron diffraction

William Liège  Université Paris-Saclay, CNRS, CEA, Laboratoire Léon Brillouin,
91191, Gif-sur-Yvette, France  Yaofeng Xie  Department of Physics and
Astronomy, Rice University, Houston, Texas 77005, USA  Dalila Bounoua
Université Paris-Saclay, CNRS, CEA, Laboratoire Léon Brillouin, 91191, Gif-
sur-Yvette, France  Yvan Sidis  Université Paris-Saclay, CNRS, CEA,
Laboratoire Léon Brillouin, 91191, Gif-sur-Yvette, France  Frédéric Bourdarot
Institut Laue-Langevin, 71 avenue des Martyrs, Grenoble 38042, France  Yongkai
Li  Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic
Quantum Architecture and Measurement (MOE), School of Physics, Beijing
Institute of Technology, Beijing, China.  Beijing Key Lab of Nanophotonics and
Ultrafine Optoelectronic Systems, Beijing Institute of Technology, Beijing,
China.  Zhiwei Wang  Centre for Quantum Physics, Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics,
Beijing Institute of Technology, Beijing, China.  Beijing Key Lab of
Nanophotonics and Ultrafine Optoelectronic Systems, Beijing Institute of
Technology, Beijing, China.  Jia-Xin Yin  Department of Physics, Southern
University of Science and Technology, Shenzhen, Guangdong, China.  Pengcheng
Dai  Department of Physics and Astronomy, Rice University, Houston, Texas
77005, USA  Philippe Bourges<EMAIL_ADDRESS>Université Paris-Saclay,
CNRS, CEA, Laboratoire Léon Brillouin, 91191, Gif-sur-Yvette, France

###### Abstract

As many Kagome metals, the topological superconductor AV3Sb5 with (A =
K,Rb,Cs) hosts a charge density wave . A related chiral flux phase that breaks
the time-reversal symmetry has been further theoretically predicted in these
materials. The flux phase is associated with loop currents that produce
ordered orbital magnetic moments, which would occur at the momentum points,
$\bf M$, characterizing the charge-density wave state. Polarized neutron-
diffraction experiments have been performed on an assembly of single crystals
of ${\rm CsV_{3}Sb_{5}}$ to search for such orbital magnetic moments. No
evidence for the existence of a three-dimensionally ordered moment is found at
any temperature at the first M1=(1/2,0,0) point in the Brillouin zone within
an excellent experimental uncertainty, i.e. ${\bf m}=0\pm 0.01\mu_{B}$ per
vanadium atom. However, a hint to a magnetic orbital moment is found in the
second Brillouin zone at ${\bf M_{2}}$=(1/2,1/2,0) at the detection limit of
the experiment. Some loop currents patterns flowing only on vanadium triangles
are able to account for this finding suggesting an ordered orbital magnetic
moment of, at most, $\sim 0.02\pm 0.01\mu_{B}$ per vanadium triangle.

##  I  Introduction

The search for orbital loop currents (LCs) is a long-standing quest in both
topological and correlated quantum matter. This exotic state of matter has
been put forward to explain the enigmatic pseudogap phase of high-temperature
cuprate superconductors [1, 2]. Over the last years, this idea has gained in
momentum primarily through the observation of an intra-unit-cell magnetism
using polarized neutron diffraction (PND) [3, 4]. The observed hidden order
parameter and its associated thermodynamic signatures [5, 6] correspond to a
magneto-electric state that breaks both parity and time-reversal symmetries as
evidenced by different experiments [3, 7, 8]. Similar measurements in the
iridates indicate the existence of an orbital magnetism consistent with loop
currents in SrIr2O4 [9]. It is thought to exist in a wider class of quantum
materials beyond superconducting cuprates and oxides. The clearest example in
favor of orbital magnetism is the observation of anomalous Hall effect in
twisted bilayer graphene [10], in the absence of any spin magnetism. Other
examples mix the orbital response with the spin magnetism as it is recently
established in ${\rm Mn_{3}Si_{2}Te_{6}}$, where the control of the colossal
magnetoresistence was demonstrated to arise from an exotic quantum state that
is driven by ab plane chiral orbital currents flowing along the edges of MnTe6
octahedra[11]. In honeycomb lattices, such flux phases were further suggested
to interact with Dirac ferimions and produce the quantum anomalous hall effect
[12].

The quasi-2D kagome lattice materials have gained the spotlight over the last
decade because of their rich ground states stemming from their non-trivial
band structure. In particular, the quasi-2D kagome lattice compounds AV3Sb5
with (A = K,Rb,Cs) represent a new family of kagome metals [13] with
intertwined topological charge density wave (CDW) order around T${}_{CDW}\sim$
93K in the Cs compound and superconductivity at low temperature (below $\sim$
2.5 K) [14]. The topological CDW is associated with a Fermi surface
instability at the M-point of the hexagonal structure as observed by scanning
tunneling microscopy (STM) [15, 16]. Using X-ray diffraction, the CDW in
CsV3Sb5 was later proven to be three-dimensional, with a 2 x 2 x 2
superstructure [17] larger than the 2 x 2 x 1 superstructure inffered from
STM. There are however some recent claims that the three dimensional ordering
of the CDW can be sensitive to stacking along the c-axis [18, 19] and possibly
to the way samples are prepared. Further, resonant X-ray studies at high
pressure demonstrate that the 2 x 2 x 1 CDW associated with the vanadium
kagome sublattice coexists with a Sb $5p$-electron assisted 2 x 2 x 2 CDW[20].
Interestingly, neutron diffraction on an assembly of CsV3Sb5 single crystals
reported an associated lattice distortion in the CDW state where the magnitude
of the 2 x 2 x 1 superstructure was found to be larger [21] in contrast to the
X-ray diffraction result. Additionally, rotational symmetry breaking from
$C_{6}$ to $C_{2}$ symmetry in the CDW state, suggesting a nematic phase, has
been reported through spectroscopic-imaging STM [22] and birefringence
measurements [23]. However, recent transport elastoresistivity measurements
indicate no breaking of the 2-fold rotational symmetry below the CDW state,
meaning no sign of nematicity [24].

These materials are further characterized by a large anomalous Hall
conductivity[25]. Accordingly, a chiral CDW has been first reported using STM
in KV3Sb5 with modulated intensity of the different charge peaks [16].
However, further scanning tunneling measurements that used spin-polarized tips
show no trace of the chiral flux current phase in CsV3Sb5 [26] within their
experimental uncertainty. Despite these experiments, these materials represent
new platforms to investigate the interplay between topology, unconventional
superconductivity and strong electron correlations where localized spin
magnetism from vanadium atoms is readily thought to be absent as concluded
from the macroscopic magnetic susceptibilty [13]. Consistently, a muon spin
spectroscopy ($\mu$SR) experiment in polycrystalline KV3Sb5 samples does not
show any static long-range magnetism in KV3Sb5 [27] and neutron powder
measurements concluded that short range antifferromagnetic ordering is absent
[13]. Therefore, local moment spin magnetism from $d$-electrons does not show
up in any magnetic measurements leaving open the possibility of weaker orbital
magnetism.

Different approaches were employed to study the mechanism of electronic
instabilities and explain the charge ordering and superconductivity, leading
to the emergence of nematic chiral charge order, charge bond order and the
appearance of orbital moments [28, 29, 30, 31, 32, 33, 34]. Chiral flux
phases, with orbital loop currents, have been theoretically predicted in these
materials [28, 31, 32, 33, 34]. The proposed chiral flux state breaks time-
reversal symmetry (TRS) and results in an anomalous Hall effect. That is
consistent with the observation of a huge enhancement of the anomalous Hall
effect at the CDW temperature[25], which in principle, requires TRS breaking
in the CDW state. A few recent $\mu$SR studies report time-reversal symmetry
breaking below the charge order temperature (around 80 -95K depending on the
material)[35, 36, 37, 38]. This is actually interpreted as originating from
loop current states and is reminiscent of those reported in cuprates [2, 4].
For instance, Mielke et al [35] propose a current pattern among the triangles
of vanadium atoms but the exact LC pattern can only be deduced from
diffraction techniques. Additionally, electronic magneto chiral anisotropy
gives convincing evidence for the TRS breaking phase, although its onset
temperature is still under debate [39]. Even more recently, an optical
manipulation of the CDW state RbV3Sb5 suggests an unusual piezo-magnetic
response that requires TRS breaking [40].

Consistently, torque magnetometry experiments report the existence of an odd-
parity order parameter appearing at a temperature T∗=130K $>$TCDW above the
CDW order temperature [41], suggesting that TRS is broken below T∗. Using
symmetry arguments, hidden Dirac multipoles, involving anapoles, have been
also predicted to occur in AV3Sb5 [42] to explain the muSR results. However,
recent resonant X-ray experiments [20, 43] only show resonant enhancement at
the Sb L-edge at the 2 x 2 x 2 superstructure and no resonant feature at the V
K-edge. The resonant effect at the Sb L-edge is attributed to anisotropic
contributions from the different Sb sites further away from the kagome
lattice. As a result, for both CDW superstructures, no particular sensitivity
to magnetic contributions (which would lead to time-reversal breaking) was
reported [43], possibly due to the weakeness of the expected feature.
Spectroscopic-imaging STM and angle-resolved photoemission spectroscopy reveal
as well the existence of small reconstructed Fermi pockets[44]. They are due
to Fermi surface reconstruction that is induced by the 2 x 2 x 1 CDW state
that could acquire orbital magnetic moments if time-reversal symmetry is
broken in the CDW state[33].

The orbital magnetic moments can be detected using polarized neutron
diffraction although the experiment is challenging due to the weakness of the
expected orbital magnetism. Indeed, closed in-plane loops lead to magnetic
moments pointing perpendicular to the hexagonal plane A structure factor
calculation suggests magnetic intensity at different points in Q-space
depending on the correlations between loop currents in neighboring unit cells.
For the chiral flux phase proposed in these materials[16, 31, 33], the CDW
modulates the interactions between unit cells, with a magnetic intensity
expected at half-integer $H$ such as the M-points where the CDW occurs [17,
21].

In this paper, we report a quantitative study of CsV3Sb5 single crystals with
polarized neutron diffraction. Despite numerous efforts, we could not observe
any reliable magnetic signal at the $\bf M$-point in the first brillouin zone,
$\bf M_{1}$=(1/2,0,0), corresponding to the propagation wavevector of the
unconventional chiral CDW in the frustrated kagome lattice. From the threshold
of our statistical error bars, one can determine an upper limit of the
magnetic moment which has to be lower than the experimental uncertainty of
$\sim$ 0.01 $\mu_{B}$ per vanadium at this Q-position. Other magnetic patterns
that preserve the lattice symmetry have been investigated as well with no
indication of magnetism. However, we found a hint of a weak magnetic signal at
the M-point, $\bf M_{2}$=(1/2,1/2,0), in the second brillouin zone. A
conservative estimate of the magnetic signal suggests a magnetic moment of, at
most, $\sim$ 0.02$\pm 0.01\mu_{B}$ per formula unit. Such a low value sounds
consistent with a few Gauss internal field estimated at the muon stopping site
from $\mu$SR experiments [35]. Our finding questions the different loop
current models that have been discussed in the kagome superconductors. In
particular, a certain number of loop currents patterns flowing only on
vanadium triangles can account for our polarized neutron data, in contrast to
LCs patterns flowing on hexagons.

##  II  Background

###  II.1  Neutron cross section and polarization

It is straightforward to consider that LCs phases can be tested by neutron
diffraction as the closed circulating charge loops give rise to a magnetic
field, in principle perpendicular to the loop. In addition to the interaction
between neutron and atomic nuclei, the neutron spin interacts with any source
of magnetic fields present in the materials. The only experimental limits are
the available neutron flux and the sample mass. Usually, one considers the
magnetic field ${\bf B}$ generated by unpaired electron spins ($S$). However,
orbital currents contribute equally. The magnetic field probed by the neutron
spin at any point ${\bf R}$ can actually be decomposed in two terms[45, 46]
as:

$\displaystyle{\bf B(\bf{R})}$ $\displaystyle={\bf B_{S}}+{\bf B_{L}}$
$\displaystyle=\frac{\mu_{0}}{4\pi}\large\\{2\mu_{B}{\bf\nabla}\Big{(}\frac{{\bf
S}\times{\bf\hat{R}}}{R^{2}}\Big{)}-I\frac{d{\bf
l}\times{\bf\hat{R}}}{R^{2}}\large\\}$ (1)

$\mu_{0}$ is the vacuum permeability, ${\bf\hat{R}}$ is a unit vector in the
direction of ${\bf R}$ and $I$ is current intensity along the current path
$d{\bf l}$. The first term comes from the spin of unpaired electrons and the
second one from their orbital motion. The scattering due to the loop currents
belongs to the second category.

The neutron intensity is given by the neutron cross section, which in case of
elastic magnetic scattering and polarized neutron can then be written, in
general, as [45, 46]:

$\displaystyle{\frac{d\sigma}{d\Omega}}\Big{|}_{mag}=$
$\displaystyle\sum_{\tau}|F_{M}({\bf Q})|^{2}\delta({\bf Q}-{\bf\tau})$
$\displaystyle|F_{M}({\bf Q})|^{2}=$
$\displaystyle{r_{0}^{2}}f(\bf{Q})^{2}|<\pm|{\bf\sigma}.{\bf B}({\bf
Q})|\pm,\mp>|^{2}$ (2)

$F_{M}({\bf Q})$ is the magnetic structure factor at the momentum tranfer
${\bf Q}$. ${\bf\tau}$ represents the propagation wavevector of the LCs state
where the magnetic signal is expected in momentum space. ${\bf\sigma}$ are
Pauli matrices describing the neutron spin which can take the two states:
$|+>$ and $|->$. Spin-flip (SF) and non-spin-flip (NSF) cross-sections are
defined when the neutron spin is flipped or conserved after interaction with
the sample, respectively. The Pauli matrices are related to neutron moment as
${\bf\mu_{N}}=-\gamma\mu_{N}{\bf\sigma}$ where $\mu_{N}$ is the nuclear
magneton and $\gamma$=1.913 is the gyromagnetic ratio for the neutron spin.
The prefactor $r_{0}=0.54\ 10^{-12}$ cm includes all multiplying factors
discussed so far and corresponds to the neutron magnetic scattering length for
a magnetic moment of 1 Bohr magneton ($\mu_{B}$) ($r_{0}/\gamma$ is the
classical radius of the electron).

$f(\bf{Q})$ in Eq. 2 is the atomic magnetic form factor (by definition,
$f(0)=1$). In the present case, it corresponds to the Fourier transform of
magnetic atomic orbitals involved in the loop currents state, here associated
with the vanadium orbitals on which the LCS are assumed to flow. Because the
kagome AV3Sb5 materials exhibits no spin magnetism, one considers the magnetic
form factor of V4+ atom which carries no spin. That form factor is tabulated
in the crystallographic tables. We show in table 1 the values of that form
factor for relevant ${\bf Q}$-points of the hexagonal reciprocal lattice of
CsV3Sb5.

Momentum position: $(H,K,L)$ |  M1=(1/2,0,0) |  M2=(1/2,1/2,0) | (1,0,0) | (1,1,0)
---|---|---|---|---
Q (Å-1) | 0.66 | 1.143 | 1.32 | 2.287
$f({\bf Q})$ | 0.97 | 0.92 | 0.89 | 0.72
$f({\bf Q})^{2}$ | 0.95 | 0.85 | 0.80 | 0.52
Table 1: Magnetic form factor of V4+ orbitals at different key positions in
${\bf Q}$-points of the CsV3Sb5 hexagonal reciprocal lattice

The key term in Eq. 2 is ${\bf B}({\bf Q})$, usually called the interaction
vector [45, 46], which is proportional to the Fourier transform of the real
space magnetic field distribution ${\bf B({\bf R})}$, at the wave vector Q.
Following Eq. 1, ${\bf B}({\bf Q})$ can be, in general, written as a sum of
the spin part with the spin moment, ${\bf M_{S}}=2\mu_{B}\bf{S}$, and the
orbital part (for which we keep on purpose the notation in terms of currents),

${\bf B}({\bf Q})=\sum_{j}\exp(-i{\bf Q.r}_{j})\big{\\{}{\bf\hat{Q}}\times{\bf
M_{S}}\times{\bf\hat{Q}}-i{I\over{2\mu_{B}}}\frac{{\bf\hat{Q}}\times d{\bf
l}_{j}}{Q}\big{\\}}$ (3)

where ${\bf\hat{Q}}={\bf Q}/Q$. The sum is made over all magnetic sites, $j$,
within each magnetic unit cell. There are a few different ways to express the
neutron cross section arising from the orbital part. It is generally assumed
in neutron textbooks[45, 46], that one can define an orbital moment, ${\bf
M_{L}}$, associated to the magnetic field distribution of the orbital part. In
the low $\bf q$ limit, the orbital moment generally corresponds to the atomic
orbital moment[46]. This is misleading for the case of loop currents because
the currents flow on more than one atomic orbital. Here, in particular, one
would consider the vanadium hexagons or triangles of the kagome lattice. In
cuprates [2, 4], the LCs involve hybridizations of copper $d-$ and oxygen
$p-$orbitals. Clearly, the currents geometry plays an important role that
impacts the momentum dependencies of the structure factor. It is however
useful to describe the LCs contribution by an orbital moment located at the
geometric center of the closed loops. In such a case, the magnetic cross
section is taken to be of the same general form whatever the source of
magnetism. One then arrives to the simple form that the neutron interacts with
a moment, ${\bf M=M_{S}+M_{L}}$, which is simply the sum of the spin moment
and the orbital moment, and ${\bf B}({\bf Q})$ takes the simple form,

${\bf B}({\bf Q})=\sum_{j}\exp(-i{\bf q.r}_{j})({\bf\hat{Q}}\times{\bf
M_{j}}\times{\bf\hat{Q}})$ (4)

That expression fully defines the magnetic structure factor for a set of
magnetic moments, ${\bf M_{j}}$, in the unit cell. One can define ${\bf
m_{\perp}(Q)}={\bf{\hat{Q}}}\times{\bf M_{j}}\times{\bf{\hat{Q}}}$ which
stands for the Fourier transform of the magnetic moment distribution
perpendicular to the unity vector ${\bf{\hat{Q}}}={\bf Q}/|\rm{\bf
Q}|=(H,K,L)/|\rm{\bf Q}|$. ${\bf m_{\perp}(Q)}$ includes the neutron
orientation factor [46, 45], that implies, as usual, that one measures in
neutron diffraction only the magnetic component perpendicular to the wave
vector ${\bf\hat{Q}}$, owing to the dipolar nature of the magnetic scattering
potential.

###  II.2  Calculation of the structure factor for different loop current
patterns

A certain number of loop current states have been proposed to exist in the
chiral charge density wave states [28, 31, 32, 33, 34] of CsV3Sb5. These
states break time-reversal symmetry and lattice translational symmetry as they
all quadruple (2x2) the hexagonal unit cell shown in Fig. 1.a. Examples of
such LCs states are shown in Figs. 2-5. We here evaluate the magnetic
structure factor, ${\bf B}({\bf Q})$, for these different LCs configurations.
Classically, the closed loops are forming an orbital moment pointing in
principle perpendicularly to the hexagonal plane. This is the quantity that is
experimentally probed in neutron diffraction experiments.

Before going into the detailed description of the LCs patterns, let us stress
that they are all supposed to be 2D with no correlation between the hexagonal
planes. The structure factor is then assumed to be $L$-independent in these
calculations. The correlation between the vanadium planes stacked along the
c-axis would, of course, also play a role (See Fig. 1.a). It may be either in-
phase or out-of-phase leading to magnetic contributions at L integer (L=0) or
half-integer (L=1/2) as argued from $\mu$SR data [36]. In this section, we
consider only the scattered intensities in the 2D hexagonal plane.

These 2D patterns can be classified in different types depending on how the
different vanadium atoms are connected by charge currents within the hexagonal
plane in the 2x2 unit cell. There are many possible orbital currents
configurations in the kagome lattice, considering either the vanadium atoms
form hexagon or triangle blocks. Loop currents can develop on the different
links between vanadium atoms. The only constraint is that the current
distribution within the 2x2 unit cell is such that there is a conservation of
currents at each lattice node, so that there is no net magnetic moment within
the 2x2 unit cell. For each moment distribution, one can estimate the magnetic
neutron structure factor in the 2D $(H,K)$ hexagonal plane. The 2D reciprocal
space of the atomic hexagonal cell is shown in Fig. 1.b. The zone boundary
M-points where the CDW occurs are represented. There are 6 equivalent points,
${\bf M_{1}}=\pm(1/2,0,0)\equiv\pm(0,1/2,0)\equiv\pm(1/2,-1/2,0)$ in the first
Brillouin zone and 6 other points ${\bf
M_{2}}=\pm(1/2,1/2,0)\equiv\pm(1,-1/2,0)\equiv\pm(1/2,-1,0)$ in the next
Brillouin zones. In the following, we consider various loop current models
involving either the vanadium hexagons or mixing triangles and hexagons.

####  II.2.1  Loop current patterns involving vanadium hexagons

The first example to be considered is when loop currents are running only on
vanadium hexagons or only on vanadium triangles although this has not been
proposed in any theory, yet. We here explicitly consider the cases of currents
running around the hexagons for a sake of clarity. Similar structure factors
can be obtained when loop currents concern only the vanadium triangles that
will be discussed in section V. There are 4 hexagons in the 2x2 unit cell
(Fig. 2.a): one can assume that two hexagons exhibit a current running
clockwise and two others with currents running anti-clockwise: two hexagons
then carry a moment $+{\bf m}_{hex}$ and the two others $-{\bf m}_{hex}$. This
model somehow reminds the antiferromagnetic spin ordering in cuprates. A
simple calculation leads to a structure factor:

${\bf B(Q)}=4{m}_{hex}\sin\pi H\sin\pi K$ (5)

The magnetic intensity map obtained from this structure factor multiplied by
the vanadium form factor $f(Q)^{2}$ is shown in Fig. 2.b. For that phase, the
$C_{6}$ axis symmetry of the hexagonal lattice is broken: in reciprocal space,
meaning that only one of the three independent M-points leads to magnetic
scattering. The magnetic intensity is found to be maximum at M1=(1/2,-1/2,0)
and at M2=(1/2,1/2,0) whereas zero intensity is found at equivalent points. It
is worth stressing right away that one cannot distinguish these three
different orientations experimentally, since they are equivalent from the
atomic structure point of view. Indeed, for the magnetic scattering, this
would result in an averaged intensity map of the three equivalent points due
to the existence of magnetic domains.

Inspired by the model of Lin and Nandkishore [32], other LCs distribution on
hexagons that respect the $C_{6}$ axis symmetry are also possible. Quite
similar to the theoretical approaches [32, 33], one hexagon can carry a moment
${\bf m}_{hex}$ whereas the 3 others have only $-{1\over 3}{\bf m}_{hex}$
(Fig. 3.a) . In the 2D $(H,K)$ hexagonal plane, this leads to a magnetic
neutron structure factor that can be written as,

${\bf B(Q)}={4\over 3}{m}_{hex}(1-\exp^{i\pi(H+K)}\cos\pi H\cos\pi K)$ (6)

From this moment distribution, the neutron magnetic intensity map obtained
from the squared structure factor multiplied by the vanadium form factor
$f(Q)^{2}$ is represented in Fig. 3.b. It exhibits again maxima at both ${\bf
M_{1}}$=(1/2,0,0) and ${\bf M_{2}}$=(1/2,1/2,0) which only differ by the form
factor term. All the six equivalent ${\bf M_{1}}$ and ${\bf M_{2}}$ show the
same intensity respecting the $C_{6}$ axis symmetry unlike the previous model.

####  II.2.2  Loop current patterns involving vanadium hexagons and triangles

The second type of patterns mix both hexagons and triangles sub-blocks. There
were originally proposed by Feng et al [31]: they considered LCs with currents
running around either given vanadium triangles or hexagons (Fig. 4.a). There
is a complete cancellation between the hexagon magnetic moment ${\bf m}_{hex}$
and the magnetic moment on each triangle ${\bf m}_{tr}$: ${\bf m}_{hex}+2{\bf
m}_{tr}=0$. From this moment distribution, one can also calculate the magnetic
neutron structure factor in the 2D $(H,K)$ scattering plane. One simply
obtains:

${\bf B(Q)}={m}_{hex}[1-\cos({4\pi\over 3}(H-K))]$ (7)

The squared structure factor, again multiplied by the vanadium form factor
$f(Q)^{2}$ to get the neutron intensity map, gives the neutron intensity shown
in Fig. 4.b where the strongest magnetic intensity is found at ${\bf
M_{1}}$=(1/2,0,0) whereas no intensity is expected at ${\bf
M_{2}}$=(1/2,1/2,0). All the six equivalent ${\bf M_{1}}$ show the same
intensity respecting the $C_{6}$ axis symmetry.

A third example is when loop currents occur around all sub-blocks either the
hexagons or the triangles [32, 33] (Fig. 5). A flux exists even if the
currents are not closing in all sub-patterns. This is allowed in quantum
mechanics although that would give zero flux in a classical picture. In such a
case, the currents on the different arms of the triangles/hexagons are
different, and the line integral of the current around any given
triangle/hexagon does not vanish even when the arrows do not form a loop. For
instance in ref. [32], when the currents, indicated by coloured arrows, are
not the same in the different links and do not seem to circulate, there is
still a non-zero flux, although it gives rise to a weaker flux. In the 2x2
plaquette, the total flux over all sub-blocks is always zero. Interestingly,
in Lin and Nandkishore [32], the fluxes can be reassembled in four different
groups, each having the same pattern as the model discussed above of Feng et
al [31] (Fig. 4.a). As a result, this leads to the same structure factor as
the one shown in Fig. 4.b. In ref. [33], the null sum of fluxes translates
into $\Phi_{1}+3\Phi_{2}+2\Phi_{3}+6\Phi_{4}$=0 for all hexagons and triangles
of the 2x2 plaquette (Fig. 5.a). The resulting neutron intensity map, plotted
in Fig. 5.b, is similar to the previous example with stronger peaks at the
${\bf M_{1}}$ points and very weak intensity at the ${\bf M_{2}}$ positions.

##  III  Experimental details

###  III.1  Sample preparation and characterization

Single crystals of ${\rm CsV_{3}Sb_{5}}$ of less than half a mm3 size each
used in the present study were grown by a self-flux method as described
elsewhere[15]. Transport data for ${\rm CsV_{3}Sb_{5}}$, confirms the
existence of CDW order below TCDW = 95K and superconductivity below Tc
$\simeq$ 2.5K [21]. The single crystals were next glued and co-aligned onto
aluminum plates with CYTOP (Amorphous Fluoropolymers) glue. About 400
individual single crystals were co-aligned on four aluminum plates to form an
assembly with a volume of 0.11 cm3 corresponding to a resulting total mass of
$\sim$ 0.7 gram. The same sample assembly was previously used in our phonon
study with inelastic neutron scattering [21].

Two different mounting of the ${\rm CsV_{3}Sb_{5}}$ assembly were used to
access the different M-points of the Brillouin zone: ${\bf M_{1}}$=(1/2,0,0)
or ${\bf M_{2}}$=(1/2,1/2,0), allowing us to investigate different loop
currents models. In both cases, we could access the $L$-dependencies to test
possible doubling of the unit cell along $c^{*}$. First, the whole sample
array was oriented with $[H,0,0]$ and $[0,0,L]$ reciprocal lattice directions
of the hexagonal crystal structure in the horizontal scattering plane to
measure the ${\bf M_{1}}$=(1/2,0,0) point. In this scattering plane, the full
crystal assembly was fitting inside a He cryostat. The sample mosaic spread
was found to be 3.5deg as in our previous experiment [21]. In a second
experiment, it was not possible to fit the whole assembly into the cryostat as
we had to turn the aluminum plates by 30 deg to get the $[H,H,0]$ and
$[0,0,L]$ reciprocal lattice directions in the horizontal scattering plane to
be able measure the ${\bf M_{2}}$=(1/2,1/2,0) point. Only one aluminum plate
with the more single crystals glued on it was used and turned by 30 deg (see
sample photograph in Fig. 1.c). The resulting mosaic spread was a bit better
of $\sim$ 2.5 deg for a mass of about 40% of the previous one. As the sample
is an assembly of smaller single crystals, the magnetic domains discussed
above would be equally populated. In the following, the momentum transfer
${\bf Q}$ = H${\bf a^{*}}$ \+ K${\bf b^{*}}$ \+ L${\bf c^{*}}$ is denoted as
$(H,K,L)$ in reciprocal lattice units (r.l.u.) of the hexagonal lattice with a
= b = 5.495 Å, and c = 9.309 Å.

###  III.2  Polarized neutron scattering experiments

The polarized neutron measurements were performed on the neutron Triple Axis
Spectrometers (TAS) IN22 located at Institut Laue Langevin (ILL) in Grenoble
(France). The measurements were carried out in elastic conditions with the
final neutron wavevector (kf) equal to the incident one (ki). The neutron
wavevector was kf=2.662 Å-1, yielding an energy resolution of about $\sim$1
meV. To get a clean beam, two Pyrolitic Graphite (PG) filters, installed on
the scattered beam, were used to remove higher order harmonics. In particular,
we put two PG filters to avoid second order harmonic from the nuclear peaks as
we expect magnetic intensity at the zone boundary ($\bf M$-points) of the
Brillouin zones.

The beam was polarized and analyzed using Heusler crystals, Cu2MnAl, with the
(1,1,1) Bragg reflection. The instrument is equipped with longitudinal XYZ
polarization analysis (XYZ-PA) (see ref. [47]), a powerful technique to
selectively probe and disentangle the magnetic response from the nuclear one
with no assumptions on the background. Both SF and NSF scans were done in
order to crosscheck the absence of nuclear scattering at magnetic positions.
According to conventional notations, $\bf X$ stands for the direction of the
neutron polarization parallel to the transferred momentum $\bf Q$, $\bf Y$ and
$\bf Z$ polarization directions are both perpendicular to $\bf Q$. $\bf Y$ is
the in-plane orthogonal direction and $\bf Z$ is perpendicular to the
scattering plane. The spin-flip, $SF_{X,Y,Z}$, and non-spin-flip,
$NSF_{X,Y,Z}$, intensities for the different neutron polarizations were
measured in order to be able to perform the XYZ polarization analysis.

Setup | Sample | Bragg peak$\equiv(H,K,L)$ | FRX | FRY | FRZ
---|---|---|---|---|---
CRYOPAD | CsV3Sb5 | (2,0,0) | ${16\pm 0.15}$ | $16.1\pm 0.2$ | $16.1\pm 0.2$
Helm | CsV3Sb5 | (1,1,0) | $15.9\pm 0.1$ | $15.6\pm 0.1$ | $16.1\pm 0.1$
Helm | Quartz | (1/2,1/2,0) | $14\pm 0.3$ | $13.8\pm 0.4$ | $14.6\pm 0.4$
Table 2: Summary of flipping ratios (FR) measured on IN22 along the 3
directions for both setups: Helm (meaning using the Helmholtz-like coils) or
CRYOPAD. The flipping ratios were either measured on Bragg peaks of the sample
or on the coherent scattering of a quartz specimen, where $(H,K,L)$ units are
still given within the reciprocal lattice of CsV3Sb5. Typical error bars on
the flipping ratios are given for each setup.

The experiments were performed in three different experimental runs using two
different setups to control the neutron polarization at the sample position.
We utilized either an Helmholtz-like sets of coils or a spherical polarization
analysis device, CRYOPAD[48], with zero magnetic field at the sample chamber.
In principle, CRYOPAD offers a better neutron polarization homogeneity when
turning the polarization along the three different directions. However, both
setups give very similar neutron polarization in all three directions. The
flipping ratios FR=$(\frac{NSF_{X,Y,Z}}{SF_{X,Y,Z}})$ for both setups were
measured on the sample Bragg peaks as well as on a quartz specimen. The
polarization is then found to be quite homogeneous along the three directions
(FR=16) on Bragg peaks for both setups (see the Table 2 for actual values).
However, as usual, a slighly better polarization homogeneity is obtained with
CRYOPAD compared to the Helmholtz-like coils setup. Lower flipping ratios are
also found using the quartz specimen. This is expected to be the case as the
amorphous quartz scatters at all scattering angles corresponding to a broader
${\bf Q}$-resolution compared to the Bragg peaks of CsV3Sb5, then leading to a
more imperfect polarization. Even if the polarization is very homogeneous with
tiny variations of the flipping ratio upon turning the polarization (see Tab.
2), we apply corrections from imperfect neutron polarization as discussed in
[49]. This correction has negligible effects on the deduced magnetic
intensity.

###  III.3  Polarization analysis

XYZ-polarization analysis (XYZ-PA) of polarized neutron diffraction allows us
to distinguish between the nuclear and the magnetic contributions in the
scattered intensity [45, 46]. For a nuclear scattering, the neutron spin
remains unchanged and the scattered intensity is always measured in the NSF
channel. For the magnetic scattering, the intensity in each channel strongly
depends on the direction of the neutron polarization $\bf{P_{n}}$ because
neutron spins are described by Pauli matrices ${\bf\sigma}$ in Eq. 2, whom
quantization axis is set by $\bf{P_{n}}$. The scattered magnetic intensity is
thus proportional to $\rm|{\bf\sigma}.{\bf m}_{\perp}(Q)|^{2}$. As a result of
the Pauli matrices, only the components of ${\bf m_{\perp}(Q)}$ perpendicular
to the neutron spin polarization vector $\rm{\bf P_{n}}$ are Spin-Flip,
$I_{mag}^{SF}({\bf P_{n}})\propto({\bf m}_{\perp})^{2}-({\bf m}_{\perp}.\,{\bf
P_{n}})^{2}$, whereas the component $I_{mag}^{NSF}({\bf P_{n}})\propto({\bf
m}_{\perp}.\,{\bf P_{n}})^{2}$ does not flip the neutron spin and remains in
the NSF channel. To perform a standard XYZ polarization analysis, one then
sets $\rm{\bf P_{n}}$ in the laboratory referential with the three orthogonal
unitary directions (X,Y,Z) defined above.

For the X polarization, ${\bf P_{n}}$ parallel to $\rm{\bf{\hat{Q}}}$, the
full magnetic scattering is SF only. Let us consider the case of a generic
magnetic moment with components along three cartesian directions. In the case
of hexagonal lattice of the kagome metals, these directions correspond to
either $\bf(a,b^{*},c)$, meaning i.e. $\rm{\bf m}=({\bf m}_{a},{\bf
m}_{b^{*}},{\bf m}_{c})$, or $\bf(a^{*},b,c)$. For LCs phases where there is
no net magnetic moment, we are in the case where all magnetic moments are
antiparallel to a single direction. In such a case, the magnetic intensity
measured in the SF channel, at a momentum position ${\bf Q}=(H,K,L)$, for the
X polarization, $I_{mag}^{SF}(X)$, can be expressed [46] as:

$I^{SF}_{mag}(X)\propto\left[1-|\frac{2\pi}{a}\frac{H}{{\bf
Q}}|^{2}\right]{\bf
m}_{a}^{2}+\left[1-\frac{4}{3}|\frac{2\pi}{a}\frac{H/2+K}{{\bf
Q}}|^{2}\right]{\bf m}_{b^{*}}^{2}+\left[1-|\frac{2\pi}{c}\frac{L}{\bf
Q}|^{2}\right]{\bf m}_{c}^{2}$ (8)

In contrast, the scattered intensities in the two complementary Y and Z
polarizations probe only sub-parts of this cross-section (Eq. 8), depending on
both i) the direction of the polarization and ii) the scattering plane of the
given experiment, i.e. how the sample is oriented relative to the (X,Y,Z)
referential. In the limit $L=0$, which is relevant for the discussed
experiments, $I^{SF}_{mag}(Z)$ probes the $c$-axis magnetic contribution,
${\bf m}_{c}^{2}$, which corresponds to the expected moment orientation for
the LCs states discussed above.

All these cross-sections are measured fot each polarization on top of a
background as $I^{SF}_{meas}({\bf X,Y,Z})=I^{SF}_{mag}({\bf
X,Y,Z})+I_{BG}({\bf X,Y,Z})$. $I_{BG}({\bf X,Y,Z})$ is assumed at first
approximation to be independent of the polarization. The total scattered
magnetic intensity, $I_{mag}$, is next extracted such as (see [50, 51, 47] for
more details about the analysis of the neutron data):

$I_{mag}=2\,I^{SF}_{meas}({\bf X})-I^{SF}_{meas}({\bf Y})-I^{SF}_{meas}({\bf
Z})$ (9)

This expression is very useful as it gives access to the intrinsic magnetic
intensity by removing the background contribution, simply by assuming the same
background for the three polarizations. In practice, however, the background
in the three polarizations can be sligthly different and the background
contribution to $I_{mag}$ does not completely cancel; it then may give rise to
a shift from zero of the baseline even in the absence of magnetism. This
effect can be particularly sizeable when the magnetic signal is very weak as
we face in the present study. This is one of the most important challenges of
such PND experiments.

###  III.4  Intensity calibration

To determine the absolute amplitude of the magnetic intensity, we need i) to
estimate the magnetic structure factor in absolute units and ii) to take into
account the instrument resolution. First, the magnetic scattering cross-
section [45, 46], $I_{mag}$, reads in absolute units:

$I_{mag}=\Phi_{S}\frac{N_{cell}}{V_{0}}R_{0}(Q_{m})I_{M}({\bf
Q})=\Phi_{S}\frac{N_{cell}}{V_{0}}R_{0}(Q_{m})r_{0}^{2}\,f(Q)^{2}\,\frac{|B({\bf
Q})|^{2}}{4}$ (10)

$\Phi_{S}$ corresponds to the neutron flux at the sample position in
neutron/s/barns (1 barn= $10^{-24}$cm2), $N_{cell}$ is the number of unit
cells (formula units) in the sample, $V_{0}$ is the volume of the unit cell,
$I_{M}({\bf Q})=r_{0}^{2}f(Q)^{2}|{\bf B({Q})}|^{2}/4$ where $r_{0}$ stands
for the neutron magnetic scattering length, $r_{0}^{2}=290$ mbarn, $f(Q)$ is
the magnetic form factor, ${\bf B(Q)}$ is the magnetic structure factor
defined in Eq. 4 and calculated in section II.2 for various LCs patterns in
the 2x2 plaquette. As a result, one needs to divide $|B({\bf Q})|^{2}$ by 4
because $I_{M}$ is given per formula unit. The expected Q-dependence of
$I_{M}({\bf Q})$ is represented in Figs. 2-5. Assuming that the magnetic order
of LCs is long-range, corrections from resolution effects are also included ar
each ${\bf Q}$ position through the factor $R_{0}(Q_{m})$ which normalised the
triple-axis resolution function[52] for a Bragg peak. $R_{0}(Q_{m})$ is
readily calculated from instrument geometry and sample mosaicity.

To estimate $I_{M}({\bf Q})$ in absolute units, one needs to calibrate the
magnetic intensity by the nuclear intensity of a reference Bragg peak whose
intensity $I_{B}$ can be written as a function of its nuclear structure
factor, $F_{N}$, as

$I_{B}=\Phi_{S}\frac{N_{cell}}{V_{0}}R_{0}(Q_{N}){|F_{N}|^{2}}$ (11)

$|F_{N}|^{2}$ can be obtained by a simple calculation of the structure factor
or using structure factor calculation software like VESTA. It is expressed in
barns. The magnetic intensity in barns, $I_{M}({\bf Q})$, is simply deduced
from Eqs. 10 and 11, as

$I_{M}({\bf
Q})=\frac{I_{mag}}{I_{B}}\frac{R_{0}(Q_{N})}{R_{0}(Q_{m})}{|F_{N}|^{2}}=r_{0}^{2}f(Q)^{2}\frac{|B({\bf
Q})|^{2}}{4}$ (12)

The left hand of Eq. 12 is determined experimentally whereas the right part is
computed for the different LCs models. This calculated magnetic intensity is
the quantity shown in all color map figures of the manuscript with a color
scale given in mbarns assuming a magnetic moment of 0.02 $\mu_{B}$.

##  IV  Results

We performed a series of polarized neutron diffraction on IN22 experiments to
cover the various loop currents patterns discussed above in section II.2. We
had to mount the sample in different geometries to access the various points
in ${\bf Q}$-space.

###  IV.1  Absence of magnetic scattering at the $\bf M_{1}$-point

Following the proposed LCs models [28, 31, 32, 33, 34], we first investigated
the full ${\rm CsV_{3}Sb_{5}}$ sample in the $(H,0,L)$ scattering plane as a
magnetic intensity is systematically expected at $\bf M_{1}$ in the
topological CDW state. The experiment has been performed in two separated
runs. The run-1 was a quick measurement to cover different points in ${\bf
Q}$-space whereas the second one focused on the possible magnetic scattering
at the ${\bf M_{1}}$=(1/2,0,0) and ${\bf Q}$=(1/2,0,1/2) points. The counting
time for the second run was noticeably longer: typically 30 minutes/point for
run-1 and up to 5 hours/point for run-2. The Figs. 6.a and b show for Run-1
the $L$-dependence at $H=0.5$ of both the non-spin-flip and spin-flip
intensity for the three polarization states. A broad peak maximum at $L=0$ is
observed in both channels. The spin flip intensity does not change with the
polarization which indicates that this signal is not magnetic in origin within
the error bars at any of these ${\bf Q}$ (at any $L$ value), in particular at
(1/2,0,0), (1/2,0,1/2) and (1/2,0,1) where a magnetic contribution could have
been expected in the models. Similar measurements for $H$=1.5 (not shown)
indicate the same trend.

In Run-2, we repeated these measurements to improve the statistics, namely the
$L$-scan at $H=0.5$ but over a much limited L-range. The result in the spin-
flip channel for a polarization along $X$ where the magnetic signal should be
maximum is shown in Fig. 6.c. The scan performed at 5K did not show any
structure. The scan exhibits a weak sloping background due to the sample
geometry as the sample is made from an assembly of 4 different Al plates.
Further, polarization analysis using Eq. 9 is shown in Fig. 6.d from which the
magnetic intensity is readily extracted without assumptions on the background.
These two figures indicate no sign of a magnetic signal at $\bf M_{1}$.
However, it should be stressed that the baseline for the background in Fig.
6.d is found negative. As discussed above in section III.3, this is related to
small differences in the background contribution in the three polarizations
that Eq. 9 does not remove. In order to be able to give an upper limit of the
magnetic intensity, we perform a statistical analysis of Fig. 6.c and Fig.
6.d. We fit these curves along $L$ by the following simple function,

$I=A\exp(-aL^{2})+B+sL$ (13)

The first term represents the possible magnetic intensity with an amplitude
$A$ and a broadening factor $a$. $a$ cannot be determined experimentally due
to the limited number of points and is set to a value constrained by the
sample mosaicity. The other terms are the background where the slope $s$ (not
zero only for the $SF_{X}$ intensity of Fig. 6.c) is also constrained by the
difference of the extreme points at $L=\pm 0.5$. In the fit shown in Fig. 6,
there is therefore only two free parameters: $A$ and $B$. Fitting Fig. 6.c by
Eq. 13 gives $A=0.2\pm 3$. Within error bars, no magnetic signal is observed
in these two experiments at the $\bf M_{1}$-point as well as at the point
$H=1/2$ and $L$=1/4 or 1/2. This contrasts with the LCs theoretical
predictions[31, 32, 33] for which one expects the largest magnetic
contribution at the $\bf M_{1}$-point (see Figs. 3-5).

$(H,0,L)$ |  $|F_{N}|^{2}$ (barns) | $R_{0}(Q_{N})$ | $I_{B}$ | $R_{0}(Q_{M})$ |  $I_{mag}$ (5K) |  $I_{M}$ (mbarns)
---|---|---|---|---|---|---
(0,0,4) | 10.1 | 0.54 | 154000 | 1.78 |  $<$3 |  $<$ 0.06
(3,0,0) | 11.3 | 0.35 | 71000 | 1.78 |  $<$3 |  $<$0.09
Table 3: Calibration of the magnetic intensity at the ${\bf M_{1}}$=(1/2,0,0)
point deduced from the nuclear Bragg peak $(H,0,L)$. The intensity in absolute
units is given in barns per formula unit of ${\rm CsV_{3}Sb_{5}}$ as deduced
from Eq. 12. The intensities are all expressed for a monitor M=1e6
corresponding to a counting time of 317 sec. The upper limit of the
experimental magnetic intensity is estimated from the fit using Eq. 13 of Fig.
6.c.

Although no magnetism is observed, one can give an upper value of the magnetic
moment from these measurements. For that, we need to calibrate the neutron
intensities by using nuclear Bragg peaks (see section III.4). In the $(H,0,L)$
plane, one can use two different strong Bragg peaks for that purpose as listed
in the table 3. The calibration of the intensity gives an upper limit for the
magnetic intensity to be 0.075 mbarns on average that one conservatively
rounds up to $\sim$ 0.1 mbarns and from which one can make in section V an
estimate of the upper limit of the magnetic moment for different models.

###  IV.2  Possible weak magnetic scattering at the $\bf M_{2}$-point

Next, the sample was installed on IN22 in a different scattering plane,
$(H,H,L)$, in order to access $\bf M_{2}$=(1/2,1/2,0) and related points. The
$L$-dependence of the spin-flip intensity is shown in Fig. 7.a at 5K. As in
the other scattering plane the data has been corrected for imperfect
polarization. Although the counting time is similar to the data in the
$(H,0,L)$ plane (Fig. 6.c), the counts are weaker as only one Al plate (Fig.
1.c), with ${\rm CsV_{3}Sb_{5}}$ samples glued on both faces, was mounted with
a total sample mass of about $\sim$ 2/5 of the whole sample. We found a hint
of a weak magnetic signal at the ${\bf M_{2}}$-point (1/2,1/2,0) in the SFX
$L$-scan at 5K (Fig. 7.a). The reported magnetic signal is consistent with a
long range three-dimentional ordering although one cannot prove that point due
to the scarcity of data points. The polarization analysis at 5K still using
Eq. 9 for the same scan (shown in Fig. 7.c) suggests as well a magnetic
contribution at the same point confirming the SFX scan. Further, the same scan
SFX at 110K (Fig. 7.b) shows a reduction of the magnetic feature at $\bf
M_{2}$=(1/2,1/2,0). Again, this trend is confirmed by the polarization
analysis at 110K where no feature at $L=0$ is observed (Fig. 7.d). A long
counting time was necessary to achieve this result (about 5 hours/point for
each polarization) underlying the limit and challenge of the experiment. At
both temperatures, the reference line for the background is consistently
shifted by a negative constant for all $L\neq 0$ similar to what is found
above for the $\bf M_{1}$ study (Fig. 6.d). Relative to the background level,
we are a bit more sensitive to such small effects here because the Helmholtz-
like coils was used to control the neutron polarization where the background
is less homogenous when turning the polarization compared to CRYOPAD. Due to
the weakness of the magnetic signal and the shift of the reference line, it is
unfortunately not possible to extract with enough confidence the magnetic
moment direction.

$(H,H,L)$ |  $|F_{N}|^{2}$ (barns) | $R_{0}(Q_{N})$ | $I_{B}$ | $R_{0}(Q_{M})$ |  $I_{mag}$ (5K) |  $I_{M}$ (mbarns)
---|---|---|---|---|---|---
(0,0,4) | 10.1 | 0.54 | 77785 | 1.09 | 3.6 $\pm$ 1.5 | 0.23 $\pm$ 0.11
(1,1,0) | 11.3 | 0.62 | 89065 | 1.09 | 3.6 $\pm$ 1.5 | 0.26 $\pm$ 0.11
Table 4: Calibration of the magnetic intensity at the ${\bf
M_{2}}$=(1/2,1/2,0) point deduced from the nuclear Bragg peak $(H,H,L)$. The
intensity in absolute units, given in barns per formula unit of ${\rm
CsV_{3}Sb_{5}}$, is deduced from Eq. 12. The intensities are expressed for a
moniteur M=1e6 corresponding to a couting time of 235 sec. The magnetic
intensity estimation is obtained from the fits using Eq. 13 of Figs. 7.a and
c.

Following the procedure discussed above for ${\bf M_{1}}$, we fit the data
with Eq. 13 with only two free parameters. Fitting Fig. 7 gives on average
$A=3.6\pm 1.5$ at 5K and $A=1\pm 1.5$ at 110K. Again, one can calibrate the
neutron intensities to estimate the absolute amplitude of the possible
magnetic intensity using strong nuclear Bragg peaks and in the hypothesis of a
long range three-dimentional magnetic ordering. In the $(H,H,L)$ plane, one
can use two different strong Bragg peaks for that purpose as listed in the
table 4. A conservative value of the magnetic scattering is then about 0.25
$\pm$ 0.11 mbarns from which one can make in section V an estimate of a
magnetic moment for the different models.

###  IV.3  Temperature dependence of the spin flip intensity $\bf M_{1}$-point
and $\bf M_{2}$-point.

To better characterize the weak magnetic ordering, we study the temperature
dependencies of the spin flip intensity at both $\bf M_{1}$ and $\bf
M_{2}$-points. In particular, according to torque magnetometry experiments,
TRS is broken at a temperature T∗=130K $>$TCDW with a maximum effect above
$\sim$ 80K [41] leaving open the possibility of a magnetic intensity maximum
around 100K at the $\bf M_{1}$-position which would allow us to reconcile with
the theoretical predictions of a stronger magnetic signal at ${\bf M_{1}}$.
Fig. 8.a represents the SF and NSF intensity for a polarization along $X$ at
the ${\bf M_{1}}=(1/2,0,0)$ as a function of temperature. One notices a
decreasing of the raw intensity upon warning similar in both channels which
are simply scaled by a factor 2 at all temperatures. Additional measurements
for the polarizations along $Y$ and $Z$ (not shown) indicate that the signal
is not magnetic at any temperature. One then does not observe any maximum in
the SF intensity in the range 80-130K as it would be expected from the
reported torque magnetometry data [41].

Fig. 8.b shows the temperature evolution in the spin-flip channel at ${\bf
M_{2}}=(1/2,1/2,0)$ for a polarization along $X$ where the magnetism is
maximum. One may notice a decreasing of the intensity above $T_{CDW}$. For a
comparison, we plotted the nuclear intensity due to the lattice distortion
related with the CDW that was observed at large Q on the same sample with
unpolarized neutron at the position ${\bf Q}$=(3.5,0,0) below $T_{CDW}$ [21].
Unfortunately, our current temperature dependence is not accurate enough to
make any statements concerning any anomaly at T’$\sim$ 30K as it has been
reported in the $\mu$SR experiments[35, 36, 37, 38] or a continuation of the
intensity up to T∗as seen in torque magnetometry measurements [41].

###  IV.4  No magnetic scattering at the $\Gamma$-point

All the examples discussed so far are LC states corresponding to q = 1/2
antiferromagnetic (AFM) order, i.e., magnetic order where the ordering
propagation vector is the same as the CDW state. So, for completeness, we
checked as well the possibility of a magnetic scattering at q = 0 which
respects the lattice symmetry, like the LCs phase predicted and observed in
cuprates [2, 3, 4]. We then consider patterns which respect the hexagonal
lattice symmetry. It should be stressed that this has not been proposed
theoretically in kagome metals. Anyway, it corresponds to the case of orbital
currents decorating all triangles of the kagome lattice with currents running
on two vanadium triangles of a single unit cell in clockwise or anticlockwise
directions (Fig. 9.a) quite similarly to the case of cuprates [4]. In such a
case, the structure factor reads ${\bf B(Q)}=2{m}_{tr}\sin({\pi\over
3}(H-K))$. In this case, the magnetic intensity, shown in Fig. 9.b, is zero at
all M-points and is maximum at Bragg positions with integer $(H,K,L)$, like
${\bf Q}=(1,0,0)$.

Fig. 9.c shows the temperature dependence of both SF and NSF intensities at
the very weak nuclear Bragg peak ${\bf Q}$=(1,0,0) (about 10-4 weaker than the
strong Bragg peaks listed in Tab. 3). No sizeable additional intensities are
observed in the SF channel at low temperature compared to the reference
intensity at 100K (above the CDW ordering temperature which is about 94K in
that sample[21]). This indicates no magnetic contribution in agreement with a
polarization analysis (Fig. 9.c) where also no difference of intensities is
observed between the three different polarizations, XYZ. Other momentum
positions like (1,0,1/2), (1,0,1) or even ${\bf Q}$=(1,1,1) in the $(H,H,L)$
scattering plane were also measured using XYZ polarization analysis. No sign
of magnetism was evidenced at any of these momenta within our experimental
accuracy. As the Bragg peaks at Q=(1,0,0) and Q=(1,1,1) are very weak, the
accuracy of these measurements is quite good due to the absence of any
polarization leakage. For the Bragg position ${\bf Q}$=(1,0,0), $R_{0}=0.97$
and with the estimate $I_{mag}<5$cnts for a monitor M=1e6 (Fig. 9.c), one gets
an average of $I_{M}<0.25$ (mbarns) to be compared with about 1-2 mbarns
observed in cuprates [50]. This leads to a moment of less than 0.03 $\mu_{B}$
per vanadium triangle corresponding to the model discussed above and using the
form factor given in Tab. 1.

##  V  Discussion

The main finding of our polarized neutron diffration experiments in ${\rm
CsV_{3}Sb_{5}}$ is that the magnetic signal, if it exists, is unfortunately
very weak, being actually at the limit of the current best experimental PND
capabilities. Although the reported magnetic intensity is very low, it is
worth emphasizing that we do not meet here the experimental complications,
faced for the study of q=0 intra-unit-cell magnetism in cuprates[50], related
to the neutron polarization inhomogeneities, as the magnetic signal is
expected in kagome metals at the Brillouin zone boundaries where no nuclear
Bragg peaks are present. The experimental challenges are instead rather the
available neutron flux and sample mass. Therefore, the fact that we observe a
noticeable difference between both scattering planes enables one to draw
meaningfull conclusions.

First, no sizeable magnetic signal is observed at ${\bf M_{1}}$ compared to
${\bf M_{2}}$. If we are likely able to determine a weak magnetic scattering
at ${\bf M_{2}}$ of $\sim$ 0.25 $\pm$ 0.1 mbarns, a similar magnetic amplitude
at ${\bf M_{1}}$=(1/2,0,0) can be dismissed as the experimental limits are
similar. Clearly, the magnetic signal is absent at the ${\bf M_{1}}$-points in
the first Brillouin zone questioning the theoretical LCs models in the kagome
metals [31, 32, 33] as none of the patterns of Figs. 2-5 are consistent with
the neutron data as a stronger intensity is always expected at ${\bf M_{1}}$
compared to ${\bf M_{2}}$. This suggests that the orbital moments, if they
exist, do not occur on vanadium hexagons. This is for example the case in
simpler models of moments on hexagons shown in Figs. 2-3 where the magnetic
intensity at ${\bf M_{1}}$ and ${\bf M_{2}}$ should be similar. At least, it
gives an upper limit for the putative magnetic moment. That value depends on
the considered model, mainly depending on how many orbital moments are present
in the 2x2 unit cell. Using that estimate and with the form factor given in
the table 1 which is $|f(Q)|^{2}$ =0.95 at ${\bf M_{1}}=(1/2,0,0)$, applying
Eq. 7, one gets for $I_{m}<0.1$ mbarns at 5K $m_{hex}<0.02\mu_{B}$ for the
model of Feng et al[31] (Fig. 4). For the case of [32, 33] (Fig. 5), 12
orbital moments contribute even if only three of them are actually clearly
larger. In the model of Lin and Nandkishore [32], one would get
$m_{hex}<0.012\mu_{B}$ as there are more magnetic fluxes contributing to the
structure factor. For the magnetic flux distribution proposed by Zhou and Wang
[33], a similar orbital moment upper limit is obtained for the strongest
orbital moment associated with the $\Phi_{1}$ flux. In the simpler model of
Fig. 2, one needs further to consider the three magnetic domains: that leads
to $m_{hex}<0.016\mu_{B}$ per formula unit in that case.

Among the different models discussed so far, the model of Feng et al [31]
(Fig. 4) is giving the higher upper limit of $m_{hex}<0.02\mu_{B}$ per formula
unit which is well below the experimental limit of STM with spin-polarized
tips [26] of about $\leq 0.2\mu_{B}$. Making the assumption that the magnetic
moments are only carried out by the vanadium atoms, it gives a tiny orbital
moment of less than $0.01\mu_{B}$ per vanadium atom. This value is
substantially lower than what the different theories have been expecting[31,
32, 33]. In particular, Zhou and Wang [33] estimated an intrinsic
thermodynamic orbital magnetization of 0.022 $\mu_{B}$ per vanadium atom
whereas the experimental limit is $\sim 0.005\mu_{B}$ per vanadium for this
model. However, to determine the amplitude of the magnetic moment through the
article, we assume a long range magnetic ordering. It is worth stressing that
in case of finite magnetic correlation lengths, the extracted magnetic moment
could be larger. The low value of the measured orbital moment might be as well
related to the presence of impurities. It was indeed recently theoretically
argued [53] that the LCs state might be extremely sensitive to a small number
of impurities and that the amplitude reduction originates from the non-local
contribution from the itinerant circulation of electrons.

Second, another conclusion is that no magnetic signal occurs at various non
zero $L$ values in both scattering planes. The only $L$ location where a
signal can be extracted is at $L=0$ at the position ${\bf M_{2}}=(1/2,1/2,0)$.
In particular, nothing is observed at $L$=1/2 that suggests no doubling of the
unit cell along the $\bf c$-axis contrary to the discussion made from the
$\mu$SR data [36]. We also do not see any magnetic signal at $L$=1/4 as it has
been suggested for the CDW[18]. If measurable, the magnetic signal is
correlated in-phase from one hexagonal plane to the next one along the $\bf
c$-axis. The magnetic ordering, if it exists, exhibits therefore in-phase
three-dimensional correlations.

Third, we nevertheless detect a hint of orbital magnetism at the ${\bf
M_{2}}$-points in the second Brillouin zones, suggesting a TRS breaking at low
temperature in agreement with $\mu$SR experiments[35, 36, 37, 38]. However,
this should not be confused with the controversial reports of TRS breaking by
Kerr effect [23, 54] which is characteristic of a global TRS breaking whereas
neutron diffraction data as well as muons are indicative of a local TRS
breaking. Our finding of the strongest magnetic intensity at the $\bf M_{2}$
points in the second Brillouin zone might be related to the recent ARPES
observation of pockets [44] (confirmed by quasi-particle interference in STM)
in the CDW state not observed in the first zone, but that appears at some (not
all) of the predicted locations in the second zone in agreement with the
theoretical prediction of the location of the pockets with a largest spectral
weight [44].

Actually, such a result of an orbital magnetism at ${\bf M_{2}}=(1/2,1/2,0)$
can be explained by LCs patterns flowing only on the triangles, as for
instance it has been discussed in the context of $\mu$SR experiments [35]. To
be more explicit, one first considers the simplest LCs phase breaking $C_{6}$
discussed above for the hexagons (Fig. 2). Instead of putting moment at the
center of the hexagons, one now decorates all vanadium triangles of the 2x2
unit cell by magnetic moments with the symmetry of Fig. 2 and still with the
constraint that the moment sum is zero ($4\Phi_{1}+4\Phi_{2}=0$). Two
situations, shown on Fig. 10.a and 10.c, are possible. One can repeat the
calculation of the magnetic structure factor: for the configuation of Fig.
10.a, Eq 5 becomes,

${\bf B(Q)}=8{m}_{tr}\sin\pi H\sin\pi K\sin(\frac{\pi}{3}(2H+K))$ (14)

For the configuration of Fig. 10.c, the factor $\sin(\frac{\pi}{3}(2H+K))$ in
Eq. 14 is changed by $\cos(\frac{\pi}{3}(2H+K))$. The calculated neutron
intensity is reported in Fig. 10.b and Fig. 10.d for both configurations,
respectively. One sees in Fig. 10.d that the pattern of Fig. 10.c is again
inconsistent with our experimental data whereas Fig. 10.b gives a noticeably
larger intensity at ${\bf M_{2}}=(1/2,1/2,0)$ compared to ${\bf
M_{1}}=(1/2,-1/2,0)$ that is favored by the pattern of Fig. 10.a due to an
additional decoration between triangles. This is clearly compatible with the
experiment where only the magnetic intensity at ${\bf M_{2}}$ is sizeable.

Using the above estimate of 0.25 mbarns and with the form factor given in the
table 1 which is $|f(Q)|^{2}$ =0.85 at ${\bf M_{2}}$, one gets
$m_{tr}=0.014\pm 0.009\mu_{B}$ per triangle at 5K for the model of Eq. 14
where one has further to consider that we probe at ${\bf M_{2}}=(1/2,1/2,0)$
only 1/3 of the magnetic intensity due to the existence of magnetic domains.
Actually, another LCs configuration, based on the proposed models [32, 33] of
Fig. 5, can also account for the neutron data where only moments on triangles
are considered with the constraint $6\Phi_{3}+2\Phi_{4}=0$, that is akin to
the structure shown in Fig. 11.a. The related calculated neutron intensity of
11.b suggests a map also compatible with the neutron data from which one can
estimate at 5K $m_{tr}=0.024\pm 0.013\mu_{B}$ per triangle associated with the
magnetic flux $\Phi_{4}$. That value is calculated from the related magnetic
structure factor at ${\bf M_{2}}$, ${\bf B(\bf
M_{2})}=\frac{8}{3}{\bf{m}_{tr}}$. For the models of Fig. 10.a and Fig. 11.a,
the calculated intensity at ${\bf M_{1}}$ is $\sim$ 3.6 weaker than at ${\bf
M_{2}}$ which is then compatible with the measured upper limit at ${\bf
M_{1}}$. This small magnetic moment, deduced from the neutron data, sounds
consistent with the $\mu$SR data [35, 36, 37, 38] about the existence of a
weak orbital moment ordering in kagome metals. For instance, only a field of 5
Gauss is extracted in ${\rm CsV_{3}Sb_{5}}$[35] although we have not made any
precise estimate as to how to convert the field seen at muon stopping site
from our small reported magnetic moment.

In conclusion, we searched for time-reversal symmetry breaking in the kagome
superconductor ${\rm CsV_{3}Sb_{5}}$, using polarized neutron diffraction. At
low temperature and particularly below the CDW temperature, we found within
our experimental limit no magnetic scattering at ${\bf M_{1}}=(1/2,0,0)$-point
in the first Brillouin zone although we have an indication of a weak signal at
${\bf M_{2}}=(1/2,1/2,0)$-point. This puts a very low limit on the expected
magnetic orbital moment of less than 0.01 $\mu_{B}$ for the most popular
models [31, 32, 33] proposed for the chiral flux state breaking time-reversal
symmetry that are discussed in Fig. 4-5. These results are compatible with
loop currents flowing only on the vanadium triangles. From these data, we
could determine a very weak orbital magnetic moment of the order of
$m_{tr}\sim 0.02\pm 0.01\mu_{B}$ per vanadium triangle. Improving the
experiment in the future would require a sample with a much better mosaicity
and a larger mass. The kagome metals are a nice opportunity to show that loop
currents can exist over a wider range of quantum materials. Many other
materials should belong to that category, particularly low dimensional and
unconventional superconductors.

Acknowledgments

We thank Chandra Varma and Ziqiang Wang for stimulating discussions on loop
currents in kagome metals. We acknowledge private communications from Yu Ping
Lin and Rahul Nankishore who provide the detailed flux patterns corresponding
to their model [32]. We thank Victor Balédent, Quentin Faure and Paul McClarty
for critical reading of the manuscript. We are grateful to the full
IRIG/D-phy/MEM/MDN group of CEA Grenoble who helped us to mount the different
experimental setups. We also thank Wolfgang Schmidt for assistance during one
experiment. We also acknowledge financial support from the Fédération
Française de Diffusion Neutronique (2FDN). The neutron scattering and basic
materials synthesis work at Rice University are supported by the US DOE, Basic
Energy Sciences (BES), undergrant no. DE-SC0012311 and by the Robert A. Welch
Foundation under grant no. C-1839, respectively.

## References

  * Chakravarty  _et al._ [2001] S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Hidden order in the cuprates, Phys. Rev. B 63, 094503 (2001).
  * Varma [2020] C. M. Varma, Colloquium: Linear in temperature resistivity and associated mysteries including high temperature superconductivity, Rev. Mod. Phys. 92, 031001 (2020).
  * Fauqué  _et al._ [2006] B. Fauqué, Y. Sidis, V. Hinkov, S. Pailhès, C. T. Lin, X. Chaud, and P. Bourges, Magnetic order in the pseudogap phase of high-${T}_{C}$ superconductors, Phys. Rev. Lett. 96, 197001 (2006).
  * Bourges  _et al._ [2021] P. Bourges, D. Bounoua, and Y. Sidis, Loop currents in quantum matter, Comptes Rendus Physique 22, 7 (2021).
  * Shekhter  _et al._ [2013] A. Shekhter, B. J. Ramshaw, R. Liang, W. N. Hardy, D. A. Bonn, F. F. Balakirev, R. D. McDonald, J. B. Betts, S. C. Riggs, and A. Migliori, Bounding the pseudogap with a line of phase transitions in YBa2Cu3O6+x, Nature 498, 75 (2013).
  * Sato  _et al._ [2017] Y. Sato, S. Kasahara, H. Murayama, Y. Kasahara, E.-G. Moon, T. Nishizaki, T. Loew, J. Porras, B. Keimer, T. Shibauchi, and Y. Matsuda, Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in YBa2Cu3Oy, Nature Physics 13, 1074 (2017).
  * Zhao  _et al._ [2017] L. Zhao, C. A. Belvin, R. Liang, D. A. Bonn, W. N. Hardy, N. P. Armitage, and D. Hsieh, A global inversion-symmetry-broken phase inside the pseudogap region of YBa2Cu3Oy, Nature Physics 13, 250 (2017).
  * Zhang  _et al._ [2018] J. Zhang, Z. Ding, C. Tan, K. Huang, O. O. Bernal, P.-C. Ho, G. D. Morris, A. D. Hillier, P. K. Biswas, S. P. Cottrell, H. Xiang, X. Yao, D. E. MacLaughlin, and L. Shu, Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa2Cu3Oy, Science Advances 4, eaao5235 (2018).
  * Jeong  _et al._ [2017] J. Jeong, Y. Sidis, A. Louat, V. Brouet, and P. Bourges, Time-reversal symmetry breaking hidden order in Sr2(Ir,Rh)O4, Nature Communications 8, 15119 (2017).
  * Sharpe  _et al._ [2019] A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene, Science 365, 605 (2019).
  * Zhang  _et al._ [2022] Y. Zhang, Y. Ni, H. Zhao, S. Hakani, F. Ye, L. DeLong, I. Kimchi, and G. Cao, Control of chiral orbital currents in a colossal magnetoresistance material, Nature 611, 267 (2022).
  * Haldane [1988] F. D. M. Haldane, Model for a quantum hall effect without landau levels: Condensed-matter realization of the ”parity anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  * Ortiz  _et al._ [2019] B. R. Ortiz, L. C. Gomes, J. R. Morey, M. Winiarski, M. Bordelon, J. S. Mangum, I. W. H. Oswald, J. A. Rodriguez-Rivera, J. R. Neilson, S. D. Wilson, E. Ertekin, T. M. McQueen, and E. S. Toberer, New kagome prototype materials: discovery of ${\mathrm{KV}}_{3}{\mathrm{Sb}}_{5},{\mathrm{RbV}}_{3}{\mathrm{Sb}}_{5}$, and ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$, Phys. Rev. Mater. 3, 094407 (2019).
  * Ortiz  _et al._ [2020] B. R. Ortiz, S. M. L. Teicher, Y. Hu, J. L. Zuo, P. M. Sarte, E. C. Schueller, A. M. M. Abeykoon, M. J. Krogstad, S. Rosenkranz, R. Osborn, R. Seshadri, L. Balents, J. He, and S. D. Wilson, $\mathrm{Cs}{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$: A ${\mathbb{Z}}_{2}$ Topological Kagome Metal with a Superconducting Ground State, Phys. Rev. Lett. 125, 247002 (2020).
  * Wang  _et al._ [2021] Z. Wang, Y.-X. Jiang, J.-X. Yin, Y. Li, G.-Y. Wang, H.-L. Huang, S. Shao, J. Liu, P. Zhu, N. Shumiya, M. S. Hossain, H. Liu, Y. Shi, J. Duan, X. Li, G. Chang, P. Dai, Z. Ye, G. Xu, Y. Wang, H. Zheng, J. Jia, M. Z. Hasan, and Y. Yao, Electronic nature of chiral charge order in the kagome superconductor $\mathrm{Cs}{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$, Phys. Rev. B 104, 075148 (2021).
  * Jiang  _et al._ [2021] Y.-X. Jiang, J.-X. Yin, M. M. Denner, N. Shumiya, B. R. Ortiz, G. Xu, Z. Guguchia, J. He, M. S. Hossain, X. Liu, J. Ruff, L. Kautzsch, S. S. Zhang, G. Chang, I. Belopolski, Q. Zhang, T. A. Cochran, D. Multer, M. Litskevich, Z.-J. Cheng, X. P. Yang, Z. Wang, R. Thomale, T. Neupert, S. D. Wilson, and M. Z. Hasan, Unconventional chiral charge order in kagome superconductor KV3Sb5, Nature Materials 20, 1353 (2021).
  * Li  _et al._ [2021] H. Li, T. T. Zhang, T. Yilmaz, Y. Y. Pai, C. E. Marvinney, A. Said, Q. W. Yin, C. S. Gong, Z. J. Tu, E. Vescovo, C. S. Nelson, R. G. Moore, S. Murakami, H. C. Lei, H. N. Lee, B. J. Lawrie, and H. Miao, Observation of Unconventional Charge Density Wave without Acoustic Phonon Anomaly in Kagome Superconductors ${A\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ ($A=\mathrm{Rb}$, Cs), Phys. Rev. X 11, 031050 (2021).
  * Ortiz  _et al._ [2021] B. R. Ortiz, S. M. L. Teicher, L. Kautzsch, P. M. Sarte, N. Ratcliff, J. Harter, J. P. C. Ruff, R. Seshadri, and S. D. Wilson, Fermi Surface Mapping and the Nature of Charge-Density-Wave Order in the Kagome Superconductor ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$, Phys. Rev. X 11, 041030 (2021).
  * Xiao  _et al._ [2023] Q. Xiao, Y. Lin, Q. Li, X. Zheng, S. Francoual, C. Plueckthun, W. Xia, Q. Qiu, S. Zhang, Y. Guo, J. Feng, and Y. Peng, Coexistence of multiple stacking charge density waves in kagome superconductor ${\mathrm{csv}}_{3}{\mathrm{sb}}_{5}$, Phys. Rev. Res. 5, L012032 (2023).
  * Li  _et al._ [2022a] H. Li, G. Fabbis, A. Said, J. Sun, Y.-X. Jiang, J.-X. Yin, Y.-Y. Pai, S. Yoon, A. R. Lupini, C. S. Nelson, Q. W. Yin, C. S. Gong, Z. J. Tu, H. C. Lei, J.-G. Cheng, M. Z. Hasan, Z. Wang, B. Yan, R. Thomale, L. H. N., and H. Miao, Discovery of conjoined charge density waves in the kagome superconductor CsV3Sb5, Nature Communications 13, 6348 (2022a).
  * Xie  _et al._ [2022] Y. Xie, Y. Li, P. Bourges, A. Ivanov, Z. Ye, J.-X. Yin, M. Z. Hasan, A. Luo, Y. Yao, Z. Wang, G. Xu, and P. Dai, Electron-phonon coupling in the charge density wave state of ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$, Phys. Rev. B 105, L140501 (2022).
  * Li  _et al._ [2022b] H. Li, H. Zhao, B. R. Ortiz, T. Park, M. Ye, L. Balents, Z. Wang, S. D. Wilson, and I. Zeljkovic, Rotation symmetry breaking in the normal state of a kagome superconductor KV3Sb5, Nature Physics 18, 265 (2022b).
  * Xu  _et al._ [2022] Y. Xu, Z. Ni, Y. Liu, B. R. Ortiz, Q. Deng, S. D. Wilson, B. Yan, L. Balents, and L. Wu, Three-state nematicity and magneto-optical kerr effect in the charge density waves in kagome superconductors, Nature Physics 18, 1470 (2022).
  * Liu  _et al._ [2024] Z. Liu, Y. Shi, Q. Jiang, E. W. Rosenberg, J. DeStefano, J. Liu, C. Hu, Y. Zhao, Z. Wang, Y. Yao, D. Graf, P. Dai, J. Yang, X. Xu, and J.-H. Chu, Absence of nematic instability in the kagome metal CsV3Sb5, preprint: arXiv/2309.14574 (2024).
  * Yu  _et al._ [2021a] F. H. Yu, T. Wu, Z. Y. Wang, B. Lei, W. Z. Zhuo, J. J. Ying, and X. H. Chen, Concurrence of anomalous hall effect and charge density wave in a superconducting topological kagome metal, Phys. Rev. B 104, L041103 (2021a).
  * Li  _et al._ [2022c] H. Li, S. Wan, H. Li, Q. Li, Q. Gu, H. Yang, Y. Li, Z. Wang, Y. Yao, and H.-H. Wen, No observation of chiral flux current in the topological kagome metal ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$, Phys. Rev. B 105, 045102 (2022c).
  * Kenney  _et al._ [2021] E. M. Kenney, B. R. Ortiz, C. Wang, S. D. Wilson, and M. J. Graf, Absence of local moments in the kagome metal KV3Sb5 as determined by muon spin spectroscopy, Journal of Physics: Condensed Matter 33, 235801 (2021).
  * Park  _et al._ [2021] T. Park, M. Ye, and L. Balents, Electronic instabilities of kagome metals: Saddle points and landau theory, Phys. Rev. B 104, 035142 (2021).
  * Christensen  _et al._ [2021] M. H. Christensen, T. Birol, B. M. Andersen, and R. M. Fernandes, Theory of the charge density wave in $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ kagome metals, Phys. Rev. B 104, 214513 (2021).
  * Denner  _et al._ [2021] M. M. Denner, R. Thomale, and T. Neupert, Analysis of Charge Order in the Kagome Metal $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ ($A=\mathrm{K},\mathrm{Rb},\mathrm{Cs}$), Phys. Rev. Lett. 127, 217601 (2021).
  * Feng  _et al._ [2021] X. Feng, K. Jiang, Z. Wang, and J. Hu, Chiral flux phase in the Kagome superconductor AV3Sb5, Science Bulletin 66, 1384 (2021).
  * Lin and Nandkishore [2021] Y.-P. Lin and R. M. Nandkishore, Complex charge density waves at Van Hove singularity on hexagonal lattices: Haldane-model phase diagram and potential realization in the kagome metals $A{V}_{3}{\mathrm{Sb}}_{5}$ ($A$=K, Rb, Cs), Phys. Rev. B 104, 045122 (2021).
  * Zhou and Wang [2022] S. Zhou and Z. Wang, Chern fermi pocket, topological pair density wave, and charge-4e and charge-6e superconductivity in kagome superconductors, Nature Communications 13, 7288 (2022).
  * Christensen  _et al._ [2022] M. H. Christensen, T. Birol, B. M. Andersen, and R. M. Fernandes, Loop currents in $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ kagome metals: Multipolar and toroidal magnetic orders, Phys. Rev. B 106, 144504 (2022).
  * Mielke  _et al._ [2022] C. Mielke, D. Das, J.-X. Yin, H. Liu, R. Gupta, Y.-X. Jiang, M. Medarde, X. Wu, H. C. Lei, J. Chang, P. Dai, Q. Si, H. Miao, R. Thomale, T. Neupert, Y. Shi, R. Khasanov, M. Z. Hasan, H. Luetkens, and Z. Guguchia, Time-reversal symmetry-breaking charge order in a kagome superconductor, Nature 602, 245 (2022).
  * Yu  _et al._ [2021b] L. Yu, C. Wang, Y. Zhang, M. Sander, S. Ni, Z. Lu, S. Ma, Z. Wang, Z. Zhao, H. Chen, K. Jiang, Y. Zhang, H. Yang, F. Zhou, X. Dong, S. L. Johnson, M. J. Graf, J. Hu, H.-J. Gao, and Z. Zhao, Evidence of a hidden flux phase in the topological kagome metal CsV3Sb5, preprint: arXiv/2107.10714 (2021b).
  * Khasanov  _et al._ [2022] R. Khasanov, D. Das, R. Gupta, C. Mielke, M. Elender, Q. Yin, Z. Tu, C. Gong, H. Lei, E. T. Ritz, R. M. Fernandes, T. Birol, Z. Guguchia, and H. Luetkens, Time-reversal symmetry broken by charge order in ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$, Phys. Rev. Res. 4, 023244 (2022).
  * Guguchia  _et al._ [2023] Z. Guguchia, C. Mielke, D. Das, R. Gupta, J.-X. Yin, H. Liu, Q. Yin, M. H. Christensen, Z. Tu, C. Gong, N. Shumiya, M. S. Hossain, T. Gamsakhurdashvili, M. Elender, P. Dai, A. Amato, Y. Shi, H. C. Lei, R. M. Fernandes, M. Z. Hasan, H. Luetkens, and R. Khasanov, Tunable unconventional kagome superconductivity in charge ordered RbV3Sb5 and KV3Sb5, Nature Communications 14, 153 (2023).
  * Guo  _et al._ [2022] C. Guo, C. Putzke, S. Konyzheva, X. Huang, M. Gutierrez-Amigo, I. Errea, D. Chen, M. G. Vergniory, C. Felser, M. H. Fischer, T. Neupert, and P. J. W. Moll, Switchable chiral transport in charge-ordered kagome metal CsV3Sb5, Nature 611, 461 (2022).
  * [40] Y. Xing, S. Bae, E. Ritz, F. Yang, T. Birol, A. N. Capa Salinas, B. R. Ortiz, S. D. Wilson, Z. Wang, R. M. Fernandes, and V. Madhavan, Optical Manipulation of the Charge Density Wave state in RbV3Sb5, preprint: arXiv/2308.04128 (2023) .
  * Asaba  _et al._ [2024] T. Asaba, A. Onishi, Y. Kageyama, T. Kiyosue, K. Ohtsuka, S. Suetsugu, Y. Kohsaka, T. Gaggl, Y. Kasahara, H. Murayama, K. Hashimoto, R. Tazai, H. Kontani, B. R. Ortiz, S. D. Wilson, Q. Li, H.-H. Wen, T. Shibauchi, and Y. Matsuda, Evidence for an odd-parity nematic phase above the charge-density-wave transition in a kagome metal, Nature Physics 20, 40 (2024).
  * Scagnoli  _et al._ [2022] V. Scagnoli, D. D. Khalyavin, and S. W. Lovesey, Hidden magnetic order on a kagome lattice for ${\mathrm{KV}}_{3}{\mathrm{Sb}}_{5}$, Phys. Rev. B 106, 064419 (2022).
  * Scagnoli  _et al._ [2024] V. Scagnoli, L. J. Riddiford, S. W. Huang, Y.-G. Shi, Z. Tu, H. Lei, A. Bombardi, G. Nisbet, and Z. Guguchia, Resonant x-ray diffraction measurements in charge ordered kagome superconductors KV3Sb5 and RbV3Sb5, Journal of Physics: Condensed Matter 36, 185604 (2024).
  * Li  _et al._ [2023] H. Li, D. Oh, M. Kang, H. Zhao, B. R. Ortiz, Y. Oey, S. Fang, Z. Ren, C. Jozwiak, A. Bostwick, E. Rotenberg, J. G. Checkelsky, Z. Wang, S. D. Wilson, R. Comin, and I. Zeljkovic, Small fermi pockets intertwined with charge stripes and pair density wave order in a kagome superconductor, Phys. Rev. X 13, 031030 (2023).
  * Squires [1978] G. L. Squires,  _Thermal Neutron Scattering_ , edited by C. U. Press (1978).
  * Lovesey [1981] S. W. Lovesey,  _Theory of Neutron Scattering from Condensed Matter, Vol II_ , edited by O. S. Publications (1981).
  * Bounoua  _et al._ [2022] D. Bounoua, Y. Sidis, T. Loew, F. Bourdarot, M. Boehm, P. Steffens, L. Mangin-Thro, V. Balédent, and P. Bourges, Hidden Magnetic Texture in the Pseudogap Phase of High-Tc $\rm{YBa_{2}Cu_{3}O_{6.6}}$, Communications Physics 5, 268 (2022).
  * Tasset [1989] F. Tasset, Zero field neutron polarimetry, Physica B: Condensed Matter 156–157, 627 (1989).
  * Stewart  _et al._ [2009] J. Stewart, P. Deen, K. Andersen, H. Schober, J.-F. Barthélémy, J. Hillier, A. Murani, T. Hayes, and B. Lindenau, Disordered materials studied using neutron polarization analysis on the multi-detector spectrometer, d7, Journal of Applied Crystallography 42, 69 (2009).
  * Bourges  _et al._ [2018] P. Bourges, Y. Sidis, and L. Mangin-Thro, Comment on “No evidence for orbital loop currents in charge-ordered ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ from polarized neutron diffraction”, Phys. Rev. B 98, 016501 (2018).
  * Bounoua  _et al._ [2020] D. Bounoua, L. Mangin-Thro, J. Jeong, R. Saint-Martin, L. Pinsard-Gaudart, Y. Sidis, and P. Bourges, Loop currents in two-leg ladder cuprates, Communications Physics 3, 123 (2020).
  * Dorner [1972] B. Dorner, The normalization of the resolution function for inelastic neutron scattering and its application, Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography 28, 319 (1972).
  * Nakazawa  _et al._ [2024] S. Nakazawa, R. Tazai, Y. Yamakawa, S. Onari, and H. Kontani, Giant Impurity Effects on Charge Loop Current Order States in Kagome Metals, preprint: arXiv/2405.12141 (2024).
  * Saykin  _et al._ [2023] D. R. Saykin, C. Farhang, E. D. Kountz, D. Chen, B. R. Ortiz, C. Shekhar, C. Felser, S. D. Wilson, R. Thomale, J. Xia, and A. Kapitulnik, High Resolution Polar Kerr Effect Studies of ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$: Tests for Time-Reversal Symmetry Breaking below the Charge-Order Transition, Phys. Rev. Lett. 131, 016901 (2023).

Fig.  1:  (a) Stacking along the $c$-axis and projection in the $\bf(a,b)$
basal plane of the hexagonal structure of ${\rm CsV_{3}Sb_{5}}$. (b) 2D
Reciprocal space of the hexagonal latttice. The real space directions
$\bf(a,b)$ and reciprocal directions $\bf(a^{*},b^{*})$ are represented as
well as the equivalent ${\bf M_{1}}$ (in blue) and ${\bf M_{2}}$ (in green)
points. (c) Photograph of the sample assembly mounted in the $(H,H,L)$ plane.
${\rm CsV_{3}Sb_{5}}$ single crystals are glued on both faces of the Al plate.

Fig.  2:  (a) LCs with only moment on hexagons in the 2x2 unit cell for an
antiferromagnetic model that breaks $C_{6}$ rotational symmetry with
$\Phi_{1}+\Phi_{2}$=0. (b) Calculated magnetic intensity in reciprocal space
with similar intensities at both ${\bf M_{1}}$ and ${\bf M_{2}}$ point. Due to
the broken $C_{6}$ rotational symmetry, magnetic intensities are not expected
to be the same at all equivalent ${\bf M}$-points. As for all figures where
the magnetic intensity is calculated, it is given in absolute units (mbarn)
for a single unit cell using Eq. 12 assuming an ordered moment associated with
$\Phi_{1}$ of 0.02 $\mu_{B}$.

Fig.  3:  (a) LCs configuration with moments only on hexagons in the 2x2 unit
cell with a symmetry similar as the models proposed [32, 33] for the LCs in
kagome metals with $\Phi_{1}+3\Phi_{2}$=0. (b) Calculated magnetic intensity
in reciprocal space with similar intensities at both ${\bf M_{1}}$ and ${\bf
M_{2}}$ points. To calculate the magnetic intensity in absolute units, the
magnetic moment associated with $\Phi_{1}$ is set to 0.02 $\mu_{B}$.

Fig.  4:  (a) LCs pattern in the 2x2 unit cell proposed in the model of Feng
et al [31] with $\Phi_{1}+2\Phi_{2}$=0. (b) Calculated magnetic intensity in
reciprocal space with the maximum intensity at ${\bf M_{1}}$ and zero
intensity ${\bf M_{2}}$. Additional intensities are observed at the Bragg
position such as (1,0,0) and equivalent points. To calculate the magnetic
intensity in absolute units, the magnetic moment associated with $\Phi_{1}$ is
set to 0.02 $\mu_{B}$.

Fig.  5:  (a) LCs configuration with moments only on hexagons in the 2x2 unit
cell with a symmetry corresponding to the proposed models [32, 33] for the LCs
in kagome metals. (b) Calculated magnetic intensity in reciprocal space with
the maximum intensity at ${\bf M_{1}}$ and nearly zero intensity ${\bf
M_{2}}$. To calculate the magnetic intensity in absolute units, the largest
magnetic moment associated with $\Phi_{1}$ is set to 0.02 $\mu_{B}$.

Fig.  6:  L-dependence of the neutron intensity around ${\bf M_{1}}$ at T=4 K:
(a) (Run-1) Scan along ${\bf Q}=(0.5,0,L)$ in the non-spin-flip channel
$NSF_{X}$ (b) (Run-1) Scans along ${\bf Q}=(0.5,0,L)$ in the spin-flip channel
for the 3 polarizations. (c) (Run-2) Scans along ${\bf Q}=(0.5,0,L)$ in the
spin-flip channel for the $X$ polarization. (d) (Run-2) Magnetic intensity
obtained by polarization analysis (Eq. 9). The data in (c-d) have been
corrected from imperfect polarization. The lines in (c-d) are fits using Eq.
13 as explained in the text. The intensities are all reported for a monitor
M=1e6 corresponding to a counting time of 317 sec although the counting time
of the Run-1 experiment was about 30 minutes/points whereas it was about 5
hours/point in the Run-2 experiment. In (d), the reference line for zero
magnetic signal is negative (-4.8 cnts) due to inhomogeneous background for
the different polarizations as explained in the main text.

Fig.  7:  L-dependence of the magnetic intensity at Q=(1/2,1/2,L): (a-b) Scans
along ${\bf Q}=(0.5,0.5,L)$ in the spin-flip channel $SF_{X}$ in ${\rm
CsV_{3}Sb_{5}}$ at (a) T=5K and (b) T=110K. (c-d) Magnetic intensity obtained
by polarization analysis (Eq. 9) at (c) T=5K and (d) T=110K. The data in (a-d)
have been corrected from imperfect polarization. The lines in (a-d) are fits
using Eq. 13 as explained in the text. The intensities are all reported for a
monitor M=1e6 corresponding to a counting time of 235 sec although the total
counting time reaches about 5 hours/point for each polarization. In (c-d), the
reference line for zero magnetic signal is negative due to inhomogeneous
background for the different polarizations as explained in the main text. That
corresponds to a consistent shift by -3.6 cnts at both temperatures for $L\neq
0$ for zero magnetic signal.

Fig.  8:  (a) Temperature dependence of the SF (left scale) and NSF (right
scale) intensities at ${\bf M_{1}}$=(0.5,0,0) for a polarization along $X$.
(b) Temperature dependence of the spin flip intensity at ${\bf
M_{2}}$=(0.5,0.5,0) for a polarization along $X$. The fit of the temperature
evolution of the CDW signal observed in neutron diffraction on the same
sample[21] is scaled for a comparison.

Fig.  9:  (a) LCs configuration respecting the lattice symmetry (b) Related
calculated intensity with maxiumum at the Bragg position ${\bf Q}=(1,0,0)$ (c)
Temperature dependence of The Bragg peak ${\bf Q}=(1,0,0)$ of both spin-flip
(with the three XYZ polarizations) (left scale) and non-spin-flip (right
scale) intensities. The intensities are all reported for a monitor M=1e6
corresponding to a counting time of 317 sec although the counting time could
reach 30 minutes/points for the most counted points.

Fig.  10:  (a) and (c) Two LCs patterns with only moments on triangles in the
2x2 unit cell for an antiferromagnetic model that breaks $C_{6}$ rotation
symmetry with $\Phi_{1}+\Phi_{2}$=0. (b) and (d) Calculated magnetic intensity
map in reciprocal space for both models. In (b), a larger intensity is found
at ${\bf M_{2}}$ compared to ${\bf M_{1}}$. To calculate the magnetic
intensity in absolute units, the magnetic moment associated with $\Phi_{1}$ is
set to 0.02 $\mu_{B}$.

Fig.  11:  (a) LCs configuration with moments only on triangles in the 2x2
unit cell with a symmetry similar as the proposed models [32, 33] for the LCs
in kagome metals. It actually corresponds to the model of Zhou and Wang [33]
shown in Fig. 5 but with $\Phi_{1}=0$, $\Phi_{2}=0$ and
$6\Phi_{3}+2\Phi_{4}$=0. (b) Calculated magnetic intensity map in reciprocal
space with larger intensity at ${\bf M_{2}}$ compared to ${\bf M_{1}}$. To
calculate the magnetic intensity in absolute units, the magnetic moment
associated with $\Phi_{3}$ is set to 0.02 $\mu_{B}$.