#  High Resolution Polar Kerr Effect Studies of CsV3Sb5: Tests for Time
Reversal Symmetry Breaking Below the Charge Order Transition

David R. Saykin  Geballe Laboratory for Advanced Materials, Stanford
University, Stanford, CA 94305, USA.  Department of Physics, Stanford
University, Stanford, CA 94305, USA.  Stanford Institute for Materials and
Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road,
Menlo Park, CA 94025, USA.  Camron Farhang  Department of Physics and
Astronomy, University of California, Irvine, CA 92697, USA.  Erik D. Kountz
Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA
94305, USA.  Department of Physics, Stanford University, Stanford, CA 94305,
USA.  Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA.  Dong
Chen  Max Planck Institute for Chemical Physics of Solids, 01187 Dresden,
Germany.  College of Physics, Qingdao University, Qingdao 266071, China
Brenden R. Ortiz  Materials Department, University of California, Santa
Barbara, Santa Barbara, CA 93106, USA.  Chandra Shekhar  Max Planck Institute
for Chemical Physics of Solids, 01187 Dresden, Germany.  Claudia Felser  Max
Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany.
Stephen D. Wilson  Materials Department, University of California, Santa
Barbara, Santa Barbara, CA 93106, USA.  Ronny Thomale  Institut für
Theoretische Physik und Astrophysik, Universität Würzburg, D-97074 Würzburg,
Germany.  Jing Xia  Department of Physics and Astronomy, University of
California, Irvine, CA 92697, USA.  Aharon Kapitulnik  Geballe Laboratory for
Advanced Materials, Stanford University, Stanford, CA 94305, USA.  Department
of Applied Physics, Stanford University, Stanford, CA 94305, USA.  Stanford
Institute for Materials and Energy Sciences, SLAC National Accelerator
Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA.  Department of
Physics, Stanford University, Stanford, CA 94305, USA.

###### Abstract

We report high resolution polar Kerr effect measurements on CsV3Sb5 single
crystals in search for signatures of spontaneous time reversal symmetry
breaking below the charge order transition at $T^{*}\approx 94$ K. Utilizing
two different versions of zero-area loop Sagnac interferometers operating at
1550 nm wavelength, each with the fundamental attribute that without a time
reversal symmetry breaking sample at its path, the interferometer is perfectly
reciprocal, we find no observable Kerr effect to within the noise floor limit
of the apparatus at 30 nanoradians. Simultaneous coherent reflection ratio
measurements confirm the sharpness of the charge order transition in the same
optical volume as the Kerr measurements. At finite magnetic field we observe a
sharp onset of a diamagnetic shift in the Kerr signal at $T^{*}$, which
persists down to the lowest temperature without change in trend. Since 1550 nm
is an energy that was shown to capture all features of the optical properties
of the material that interact with the charge order transition, we are led to
conclude that it is highly unlikely that time reversal symmetry is broken in
the charge ordered state in CsV3Sb5 .

###### pacs:

NaN

##  I Introduction

Symmetry-breaking is the phenomenon where an infinitesimal perturbation can
cause the system to break the underlying symmetry of the Hamiltonian. And it
is a cornerstone concept in the understanding and manipulation of quantum
materials. The state of a system can also be manipulated without explicit
symmetry-breaking by controlling topological aspects of the material in
momentum space, thereby regulating the electronic band-structure. When strong
electron correlations dominate, the system is often observed to be close to
multiple competing ordered phases with similar energies. For example, the
appearance of charge order may trigger a competition with an emerging new
electronic state, which in turn leads to the phenomenon of “intertwined order”
Fradkin  _et al._ (2015).

A particularly interesting material system in that respect are the quasi-two-
dimensional kagomé compounds AV3Sb5 (A=K, Rb and Cs) Ortiz  _et al._ (2019).
These are layered materials with ideal kagomé lattice of V ions coordinated by
Sb crystallizing in the P6/mmm space group. Cooling these materials below a
temperature $T^{*}$, the kagomé lattice distorts, undergoing a charge-order
transition ($T^{*}\approx$ 80 K, 110 K and 94 K for K, Rb and Cs respectively)
Ortiz  _et al._ (2020, 2021a); Yin  _et al._ (2021), which was shown to be
associated with a CDW superstructure modulation displaying chiral anisotropy
Jiang  _et al._ (2021); Shumiya  _et al._ (2021); Wang  _et al._ (2021).
Further investigation of the CDW transitions using hard-X-ray scattering
revealed that the observed superstructure has in fact a three-dimensional (3D)
nature with either $2\times 2\times 4$ or $2\times 2\times 2$ superstructure
depending on conditions of crystal growth, imposed strain and sample’s thermal
history Stahl  _et al._ (2022); Ortiz  _et al._ (2021b); Liang  _et al._
(2021); Li  _et al._ (2021); Stahl  _et al._ (2022); Xiao  _et al._ (2022).
Intertwined order was initially identified with the discovery of
superconductivity in this material system with $T_{c}\approx 2.8$ K for
CsV3Sb5 Ortiz  _et al._ (2020), where the charge-order transition was first
suspected to be wholly electronic in origin. However, detailed nuclear
magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) measurements
on CsV3Sb5 single crystals revealed an orbital ordering at $T^{*}\approx 94$ K
clearly induced by a first order structural transition. This is accompanied by
electronic charge order that appears to grow gradually below $T^{*}$ as a
secondary (intertwined) order Song  _et al._ (2022). In fact, a first order
characteristic of the transition can be further deduced from the anomalous
peak of the specific heat Ortiz  _et al._ (2020); Yu  _et al._ (2021a); Li
_et al._ (2021) and the abrupt large diamagnetic shift of the magnetic
susceptibility for magnetic field aligned with the $c$-axis Ortiz  _et al._
(2020); Yu  _et al._ (2021a). The absence of such diamagnetic shift at $T^{*}$
for magnetic field in the $ab$ plane for KV3Sb5 (see “Extended Data,” Fig. 3
in Mielke  _et al._ (2022)) and the much reduced effect for CsV3Sb5 Chen  _et
al._ (2022) further support the NMR/NQR findings.

Focusing on the transitions at $T^{*}$, a key issue of a possible co-
occurrence of time-reversal symmetry breaking (TRSB) has emerged through
claims of observation of anomalous Hall effect (AHE)Yang  _et al._ (2020); Yu
_et al._ (2021a); Zhou  _et al._ (2022), increases in muon spin relaxation
below $T^{*}$ Yu  _et al._ (2021b); Mielke  _et al._ (2022); Khasanov  _et
al._ (2022), and magneto-optic Kerr effects Wu  _et al._ (2022); Xu  _et al._
(2022); Hu  _et al._ (2022). From a theoretical standpoint, a CDW formation in
kagome metals can appear both through electronically and phonon-mediated
mechanisms, or a cooperative version thereof. For CsV3Sb5 , both electronic
and phonon fluctuations seems to be relevant due to the joint presence of van
Hove singularities nearby the Fermi level and prominent breathing-phonon modes
Feng  _et al._ (2021); Denner  _et al._ (2021); Tan  _et al._ (2021); Park
_et al._ (2021). The possible appearance of orbital currents has suggested the
possibility of a TRSB state below $T^{*}$, along with an enhanced propensity
to nematicity due to multiple CDW nesting vectors Denner  _et al._ (2021);
Wulferding  _et al._ (2022). Both TRSB and point group symmetry breaking via
nematicity are secondary to the translation symmetry breaking which defines
the CDW, and in terms of amplitude within the CDW phase can be highly variable
depending on theprecise microscopic setting involving temperature, disorder,
and interaction profile.

In this paper we aim to closely examine the possible TRSB in high quality
crystals of CsV3Sb5 via high resolution measurements of the polar Kerr effect.
To substantiate our findings, we use crystals grown in two different
laboratories (MPI-Dresden and UC Santa Barbara), all showing salient
attributes of the charge order transition at $T^{*}\approx 94$ K. Further we
use two different zero-area-loop Sagnac interferometers (ZALSI) Xia  _et al._
(2006a) operating at a wavelength of 1550 nm, one at Stanford University and
one at UC Irvine to measure the possible Kerr effect through the CDW
transition, both in zero and in finite magnetic fields. While we clearly
observe the abrupt onset of optical birefringence and/or dichroism associated
with the structural transition, we see no evidence for a spontaneous Kerr
effect below $T^{*}$ within the volume of the same optical beam to a
measurement limit of $\pm 30$ nanoradians (nrad), neither in zero-field cool,
nor after training with a magnetic field up to 0.34 T. Measurements in an
applied magnetic field reveal a diamagnetic Kerr shift, which onsets abruptly
below $T^{*}$ similar to the magnetic susceptibility Ortiz  _et al._ (2020),
but unlike the susceptibility, it stays diamagnetic down to low temperatures.

Figure 1: a, Crystal structure of CsV3Sb5 . b, Photo of “sample 1” from
Dresden, measured at Stanford and “sample 2” from UCSB, measured at UC Irvine.
c, Schematic cross-section of probe end optics assembly and sample (not to
scale).

Our results stand in stark contrast to previously published optical
measurements claiming finite spontaneous Kerr effects. Two experiments Wu  _et
al._ (2022); Xu  _et al._ (2022) were performed at a wavelength of $\sim 800$
nm while the third Hu  _et al._ (2022) used a similar apparatus to our ZALSI
at 1550 nm. Since neither 800 nm nor 1550 nm exhibit unique optical properties
associated with the CDW Uykur  _et al._ (2021, 2022), it is expected that the
longer wavelength, which is closer in energy to the CDW energy scale will
provide a more sensitive probe of the effect. As for the third report Hu  _et
al._ (2022), here we believe that a series of uncontrolled subtractions of
data may have been the source of a false positive result. We further discuss
these effects below.

##  II Results

###  II.1 Samples

In this study we use high quality and well-characterized single crystals of
CsV3Sb5 obtained from two different sources. Samples grown at UCSB (dubbed
sample 2) followed the previously published procedure found in Ortiz  _et al._
(2020) and related publications, while samples grown at MPI-Dresden (dubbed
sample 1) followed the previously published procedure found in Huang  _et al._
(2022) and related publications. The crystal structure of CsV3Sb5 and photos
of two example crystals from the two sources are presented in Fig. 1. Specific
heat data obtained on these crystals show the “standard” sharp first-order
transition peak at 93.5 K as is shown in Fig. 2. This charge density
transition is also marked by the kink in DC resistance and the pronounced peak
in its first derivative $dR/dT$ as shown in Fig. 8. In the $dR/dT$ curve,
especially in sample 2, there is an additional feature of a dispersive line
shape centered at $T_{A}\approx 37$ K, close to the temperatures where a kink
was observed in NMR $1/T_{1}T$ Song  _et al._ (2022) and where the electronic
magneto-chiral anisotropy (eMChA) was observed in nonlinear transport Guo  _et
al._ (2022), suggesting an additional phase transition. However, as we aim to
perform optical measurements on these crystals, with beam diameter of $\sim
10$ to $20\ \mu$m, it is particularly important to confirm that the probed
optical volume shows the transition. This is done through reflectivity
measurements (see Fig 3 below) that are taken simultaneously with the Kerr
effect data as well as in Kerr effect measurements in the presence of magnetic
field (Fig. 2).

Figure 2: Kerr signal in CsV3Sb5. a, Sample 1 measured at Stanford: Relative
Kerr angle $\Delta\theta=\theta-\theta(120K)$ (left axes) in external magnetic
fields of 20 mT directed along $\pm\hat{z}$, showing the onset of diamagnetic
shift when the specific heat $C_{V}$ (right axes) develops a peak at
$T^{*}\approx 94$ K. b, Sample 2 measured at Irvine: Absolute Kerr signal
$\theta_{K}$ (left axes) during $\pm$0.34 T cooldowns showing the same onset
of diamagnetic shift that coincides with the anomalous peak in specific heat
$C_{p}$ (right axes). c, Sample 1: Kerr angle $\Delta\theta$ measured during
zero field warmups after +30 mT and -20 mT trainings. d, Sample 2: Kerr signal
$\theta_{K}$ measured during zero field warmups after $\pm$0.34 T trainings.
There is no discernible spontaneous Kerr effect below $T^{*}$ to within $30$
nrad in either samples.

Figure 3: Birefringence and dichroism in CsV3Sb5 detected by Sagnac
interferometers. a, Normalized average (DC) power $P_{0}$ in sample 1. b,
Average (DC) power $P_{0}$ in Sample 2. c, Coherent reflection ratio
$P_{2}/P_{0}$ in sample 1. d, $P_{2}/P_{0}$ in sample 2. The abrupt change of
$P_{2}/P_{0}$ at $T^{*}\approx 94$ K in both samples indicate the onset of
birefringence and/or dichroism (linear or circular) in CsV3Sb5 .

###  II.2 Optical Measurements

High resolution Polar Kerr effect measurements presented in this paper were
performed using zero-area-loop Sagnac interferometers (ZALSI) first introduced
by Xia et al. Xia  _et al._ (2006a). By construction the ZALSI is inherently
reciprocal and thus by its symmetry it can distinguish between true TRSB Kerr
effect and optical activity when reflected from a chiral medium. The apparatus
is based on the Sagnac interferometer Sagnac (1913) where two counter-
propagating beams travel the exact same optical path before interfering at the
detector. In our implementation we use two orthogonally linearly polarized
light beams, which are then converted into right and left circular
polarizations using a quarter-wave plate. An objective lens focuses the light
onto a small interaction region on the sample, and the reflected light beams
are converted back to linear polarizations with exchanged polarization
directions. Owing to the reciprocity of the apparatus, a non reciprocal phase
shift $\phi_{nr}=2\theta_{K}$ will appear at the detector only if time
reversal symmetry is broken through the interaction of the two circularly
polarized beams with the sample Xia  _et al._ (2006a) (see supplementary
material sup ). With careful construction we can routinely achieve shot-noise-
limited senstivity of ($\sim$100 nrad$/\sqrt{Hz}$) for optical power above
$\sim 10\ \mu$W, with magneto-optic Kerr and Faraday resolutions of nrad
limited by the long-term drift of the instrument. Some of our notable
accomplishments include the study of TRSB in Sr2RuO4 Xia  _et al._ (2006b),
UPt3 Schemm  _et al._ (2014) and more recently UTe2 Hayes  _et al._ (2021),
elucidating the limit for ferromagnetism in thin SrRuO3 films Xia  _et al._
(2009a) and discovery of the inverse proximity effect in
ferromagnet/superconductor bilayers Xia  _et al._ (2009b) to name a few.
Again, to be able to reliably scrutinize the possible spontaneous polar Kerr
effect below $T^{*}$, measurements were performed in two different
apparatuses, one at Stanford University and one at UC Irvine. While both
systems were constructed in a similar way employing a broadband
superluminescent light emitting diode (SLED) with a center wavelength of 1550
nm, the details of the sample holder, windows construction and end-optics are
slightly different.

Following the crystal cleavage, the light beam of ZALSI is focused onto an
optically flat region that can be hundreds of microns in size. One such region
is illustrated in the scanning images of reflected power $P_{0}$ (Fig. 6 A) on
sample 2 at $3.1K$. As shown in Fig. 6C, in a magnetic field of -0.34 T the
Kerr signal is uniform across this flat region. The temperature dependence of
the in-field Kerr signals at fixed spots are shown in Fig. 2A for sample 1
measured at Stanford, and in Fig. 2B for sample 2 measured at Irvine,
alongside the measured specific heat $C_{V}$. In both cases, there is a clear
and relatively abrupt onset of diamagnetic shift that coincides with the
anomalous peak in the specific heat at $T_{CDW}$, confirming the CDW
transition in the probed optical volume of ZALSI.

The main result of this work, the tests for spontaneous Kerr effects in
CsV3Sb5 were obtained for zero field warmup measurements after removal of a
“training” magnetic field. Results from the two laboratories are presented in
Fig. 2C and Fig. 2D respectively, showing no discernible Kerr effect below
$T^{*}$ to within 50 nrad. This uncertainty is a factor of $\sim 1000$ smaller
than the recent MOKE measurements Wu  _et al._ (2022); Xu  _et al._ (2022); Hu
_et al._ (2022). Similarly, no spontaneous Kerr signal above 30 nrad was found
during zero field cooldowns and subsequence zero field warmups (Fig. 5). The
absence of spontaneous Kerr signal is also universally found across the sample
surface, which can be seen in the scanning Kerr image taken on sample 2 at
$3.1K$ after removal of the -0.34 T magnetic field (Fig. 6D).

##  III Discussion

We first note that the above results do not support any spontaneous Kerr
effect, whether the sample was cooled in zero magnetic field or at a finite
field through $T^{*}$, thus raising doubts to whether time reversal symmetry
is indeed broken below $T^{*}$. On the other hand, measurements at a finite
magnetic field oriented along the $c$-axis show a clear and relatively abrupt
diamagnetic shift in the Kerr effect, which near $T^{*}$ behaves similarly to
the in-field magnetic susceptibility measurements Ortiz  _et al._ (2020); Yu
_et al._ (2021a); Wang  _et al._ (2021). However, the detected average (DC)
optical power ($P_{0}$) and the coherent reflection ratio ($P_{2}/P_{0}$) (see
supplementary material sup ) measurements in either zero field or in-field
(Fig. 3), which are taken simultaneously with the Kerr measurements within the
same optical volume, consistently show an abrupt transition at $T^{*}$ with a
width of $\sim 2$ to 3 K, irrespective of the applied fields. Since a TRSB
effect is expected to be secondary to any structural or CDW transitions, and
grow gradually below $T^{*}$ Denner  _et al._ (2021), it is unlikely that the
observed abrupt changes in the differential reflectivity are associated with
such a TRSB effect. A similar scenario was previously observed in the Cuprate
superconductors where a first-order structural transition from an orthorhombic
to tetragonal phase at $T_{s}$ (so called LTO to LTT transition) in
La2-xBaxCuO4 Hücker  _et al._ (2011) was accompanied by a charge-order
transition with an order parameter that evolved gradually below $T_{s}$.

The bulk magnetic susceptibility of CsV3Sb5 has been previously shown to
develop a paramagnetic response below $\sim 50$ K Ortiz  _et al._ (2020); Yu
_et al._ (2021a); Wang  _et al._ (2021), presumably due to bulk impurities,
overwhelming the initial diamagnetic drop below $T^{*}$. The Sagnac
interferometers probe only the freshly cleaved surface region within a small
volume defined by the beam diameter and the optical penetration depth and thus
are less susceptible to these bulk impurities, particularly as we typically
use a lower magnetic field. Hence the change in the Kerr signal below $T^{*}$
stays diamagnetic and is almost temperature independent as seen in the
relative Kerr angle $\Delta\theta=\theta-\theta(120K)$ for sample 1 in Fig. 2A
and in the absolute Kerr signal $\theta_{K}$ for sample 2 in Fig. 2B. The
susceptibility measurements on CsV3Sb5 further show that the diamagnetic drop
with the field along the $ab$ plane is almost 5 times weaker than if the field
points in the $c$-direction, where earlier results on KV3Sb5 show no
diamagnetic shift for the field along the $ab$-plane Ortiz  _et al._ (2020).
These observations indicate that the diamagnetic response originates from in-
plane effect, presumably orbital effects such as non-interacting loop currents
as suggested by the NMR/NQR experiments Song  _et al._ (2022). In contrast,
the paramagnetic response observed in magnetic susceptibility is isotropic
with respect to the applied magnetic field and remains on the scale of the
Pauli paramagnetism above $T^{*}$. Hence we conclude that we have no evidence
for spontaneous ferromagnetic orderings of loop currents. However, an
antiferromagnetic ordering of loop currents is still possible within our
observations, which needs to be robust enough to avoid the effect of the
magnetic field, which will tend to align the loops response.

Our results are in stark contrast to other optical measurements that attempt
to measure Kerr effect in the AV3Sb5 system Wu  _et al._ (2022); Xu  _et al._
(2022). It is therefore important to understand the techniques and their
limitations as applied to the specific material system that we study. In
particular, we note that while the ZALSI is designed to only detect TRSB
effects in the sample as an excess signal beyond the natural reciprocity of
the apparatus, the other techniques may need to employ less controlled means
to scrutinize true TRSB effects. Assume a sample with its surface in the
$x$-$y$ plane and a linearly polarized electric field propagating along the
$z$-direction which is perpendicular to the surface of the sample. Adopting
the common convention where the sense of circular polarization is determined
with respect to a fixed axis (here the $+z$ axis), the handedness of the
reflected light with respect to the direction of propagation flips. This
convention defines four different indices of refraction: $n_{R+}$ and $n_{L+}$
are the refractive indices for right circularly polarized (RCP) light and left
circularly polarized (LCP) light propagating along $+z$ and $n_{R-}$ and
$n_{L-}$ are the refractive indices for RCP and LCP propagating along $-z$.
Where TRS is not broken $n_{R+}=n_{R-}$ and $n_{L+}=n_{L-}$ and in general
$(n_{R+}=n_{R-})\neq(n_{L+}=n_{L-})$. However, if TRS is broken
$(n_{R+}=n_{L-})\neq(n_{L+}=n_{R-})$. Focusing on a polar configuration where
light impinges on the sample at normal incidence (as it is in our ZALSI
apparatus) and using scattering theory to calculate the Kerr effect from the
difference between the reflection amplitudes for right and left circularly
polarized light Kapitulnik (2015), it can be shown that
$\theta_{K}\propto[(n_{R+}-n_{R-})-(n_{L+}-n_{L-})]$. It is then clear that
$\theta_{K}$ is finite _only_ for the case of TRSB. The above results are
robust, independent of whether the material that we study is isotropic, or it
possesses linear birefringence or natural optical activity.

A direct consequence of the above analysis is that the polarization state of a
normal incident light beam reflected from a TRS preserving birefringent or
optically active material will not change. However, if light is incident at an
angle to the normal to the surface of the sample, the reflected light will
exhibit elliptical polarization with the principal axis of the ellipse rotated
with respect to the original polarization, governed by the Fresnel
coefficients. If the material is both, a metal and birefringent or optically
active, a more complex set of Fresnel equations need to be analyzed, in
general yielding a complicated result for the final polarization state of the
reflected light Born and Wolf (1980). For example, for an optically active
material, the polarization of the reflected light at oblique incidence will
always rotate with respect to the initial state of polarization irrespective
of the relative orientation of the initial linear polarization relative to the
principal axes of the crystal Silverman (1987); Silverman  _et al._ (1992).
Furthermore, the statement that no finite optical rotation is possible for
reflected light at normal incidence is true for polar Kerr effect, but does
not apply to circular dichroism, which originates from different optical
penetration depths for the two circularly polarized components. Conversely, if
circular dichroism is detected at or near normal incidence it is not a proof
for TRSB. In fact, in most cases it will indicate circular birefringence that
respects time reversal symmetry. A notable example is reflection from the
surface of tellurium, where large rotary power and circular dichroism were
previously measured Fukuda  _et al._ (1975); Ades and Champness (1975); Stolze
_et al._ (1977), but no Kerr effect was detected using our ZALSI instrument
Fried (2014).

The above discussion suggests that at least within the technical aspect, those
other optical measurements fall short in comparison to the ZALSI, which
inherently measures only non-reciprocal effects, thus returning a zero signal
if time reversal symmetry is not broken. By contrast, the basic assumption in
Wu  _et al._ (2022); Xu  _et al._ (2022) is that if the linear polarization of
the incident light is aligned with one of the principal axes of a birefringent
material, then no polarization rotation is expected and thus any residual
rotation of polarization that is detected at that minimum must be due to a
TRSB effect. However, as we have explained in the “Results” section, this
assumption is irrelevant at normal incidence; and for oblique incidence it
usually fails, either because of material or instrumentation issues. On the
materials side we note that if the CDW is chiral along the $c$-axis of the
material, mirror symmetries are broken as was previously observed in STM Jiang
_et al._ (2021); Shumiya  _et al._ (2021); Wang  _et al._ (2021), and circular
dichroism is allowed. Moreover, a finite minimum value of polarization
rotation will be present with magnitude that depends on the pitch of the
chiral CDW, which depends on the initial symmetry of the unbroken state, as
well as the optical penetration depth at the probing photon energy. These
effects may then be incorrectly interpreted as signatures of TRSB. From the
technical point of view we also note that using linear polarization to probe
complex materials requires perfect alignments of the optical components with
respect to the plane of incidence. For example, if rotation of the incident
light polarization is performed using a perfect half-wave plate (HWP), which
is tilted by a small angle with respect to the propagation axis, even a
perfect metallic mirror will show an apparent polarization rotation Zhu
(1994). For CsV3Sb5 parameters, a 5 degrees error in the HWP alignment yields
a $\pm 1$ millirad error. Commercial zero-order HWPs have a typical retardance
error of 1$\%$ even for normal incidence, yielding further errors.

Concerning the difference in photon wavelength, we turn to the optical
properties of CsV3Sb5 Uykur  _et al._ (2021), where we note that the plasma
frequency is at $\sim 1$ eV, much closer to our measurement energy of 0.8 eV
than to 1.55 eV, but unlikely to influence the magneto-optic response of this
material. At the same time, the change in intensity of interband transitions
between 0.8 eV and 1.55 eV is not significant to account for a stronger effect
at one energy than the other. Similar optical data was also obtained in Zhou
_et al._ (2021). However, while their analysis at low energies differ from
that of Uykur  _et al._ (2021), the interband transitions seem similar, where
a marked change below the CDW transition occurs in the vicinity of a broad
interband peak near $\sim 0.7$ eV. This would suggest a strong magneto-optic
at our wavelength.

Finally we briefly comment on the reported onset of anomalous Hall effect Yang
_et al._ (2020); Yu  _et al._ (2021a); Zhou  _et al._ (2022) and change in
muon spin relaxation below $T^{*}$ Yu  _et al._ (2021b); Mielke  _et al._
(2022); Khasanov  _et al._ (2022). Indeed, the anomalous Hall effect is a
direct measure of ferromagnetic order. However, in general it implies a finite
value at zero applied magnetic field. However, all reported AHE on CsV3Sb5
show a zero effect at zero magnetic field. In that respect this AHE is not
strictly related to spontaneous TRSB effect and may come from non-TRSB chiral
effects König  _et al._ (2019). Another set of experiments which are at odds
with our results involve $\mu$-Sr measurements, and in particular the study in
Mielke  _et al._ (2022) where a TRSB is inferred from a rather small change
through a sharp onset in muons relaxation below $T^{*}$ at zero field,
followed by an almost temperature independent trend at low temperatures.
Indeed, as a purely magnetic probe, $\mu$Sr is not directly sensitive to
charge order and thus a change in relaxation might be interpreted as evidence
for TRSB in the electronic system. However, $\mu$Sr does sense the presence of
charge order via the magnetic dipolar coupling of the muons with the host
nuclei at zero or low magnetic fields (see e.g. Sonier (2015)). In particular,
since the zero and low field effect in the $\mu$Sr studies is abrupt and
follows in trend the structural transition Song  _et al._ (2022), it is likely
to induce a change in the muons’ preferred “rest positions” above and below
$T^{*}$, which will show up as change in relaxation rates. While an all-
electronic spontaneous TRSB buildup mechanism for the CDW Denner  _et al._
(2021) is in principle not inconsistent with the $\mu$Sr data deeper in the
CDW phase, the abruptness in the $\mu$Sr response at the transition cannot
solely be reconciled from this angle, and suggests a pivotal role of phonon-
mediated CDW formation related to a structural first order transition.
Furthermore, the in-field enhancement of the signal in magnetic field seem to
follow the NMR results, which again cannot directly point to a spontaneous
TRSB.

In conclusion, we have used high resolution polar Kerr effect measurements to
scrutinize previously reported submissions Wu  _et al._ (2022); Xu  _et al._
(2022); Hu  _et al._ (2022) that time reversal symmetry is spontaneously
broken below the charge order transition a in CsV3Sb5 (marked as $T^{*}$). For
these studies we used two different variation of a zero-area loop Sagnac
interferometer operating at 1550 nm wavelength, each with the fundamental
attribute that without a time reversal symmetry breaking sample at its path,
the interferometer is perfectly reciprocal and thus yield a zero Kerr signal
at the detector to within its noise floor limit of $\sim 30$ nanoradians. We
then show that high quality CsV3Sb5 single crystals obtained from two
different crystal-growth laboratories show no resolvable Kerr signal at zero
magnetic field to within the instruments noise floor level, whether the sample
was cooled at zero field, or was trained through $T^{*}$ at a finite field and
then measured at zero field upon warming up. Concurrent measurements of the
coherent reflection ratio show a sharp transition within the same optical
volume where Kerr effect was measured, thus support the robustness of our
measurements. At finite magnetic field we observe a sharp diamagnetic
transition at $T^{*}$, similar to the transition observed in magnetic
susceptibility, except that the Kerr data remain diamagnetic down to low
temperatures without change in trend. Since 1550 nm is an energy that was
shown to capture all features of the optical properties of the material that
interact with the charge order transition, we are led to conclude that it is
highly unlikely that time reversal symmetry is broken in the charge ordered
state in CsV3Sb5 .

##  IV Acknowledgements

Work at Stanford University was supported by the U.S. Department of Energy,
Office of Science, Basic Energy Sciences, Division of Materials Sciences and
Engineering, under Contract DE-AC02-76SF00515. Work at UC Irvine was supported
by the Gordon and Betty Moore Foundation through Grant GBMF10276. S.D.W. and
B.R.O. acknowledge support via the UC Santa Barbara NSF Quantum Foundry funded
via the Q-AMASE-i program under award DMR-1906325. R.T. acknowledges support
from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
through QUAST FOR 5249-449872909 (Project P3), through Project-ID
258499086-SFB 1170, and from the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter – ct.qmat Project-ID 390858490-EXC
2147.

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*

##  Appendix A Supplementary Materials

##  Appendix B Samples

In the present study two CsV3Sb5 samples grown in different laboratories were
measured in two Sagnac interferometers. First samples dubbed "sample 1" is
from Cluadia Felser group in Dresden, it was measured in Kapitulnik group at
Stanford, second sample dubbed "sample 2" grown at Stephen D. Wilson’s group
at UCSB was studied at UC Irvine in Xia group.

##  Appendix C Zero-area-loop Fiber-optic Sagnac Interferometer (ZALSI)

Samples were measured using zero loop area fiber Sagnac interferometers
(ZALSI) Xia  _et al._ (2006a) using $30$ $\mu$W optical power at 1550 nm
wavelength with phase modulation at $\omega=5$ MHz. Two low-coherence light
waves of right and left circularly polarizations were sent to the sample. And
the non-reciprocal phase difference $\varphi_{nr}=2\theta_{K}$ between the two
lights acquired upon reflection was detected with lock-in amplifiers. By
construction, unlike a standard ellipsometer, this approach fundamentally
rejects polarization rotations due to non-TRSB effects such as linear and
circular birefringence and dichroism that could mimic a TRSB Kerr signal. In
addition, by reducing the Sagnac loop to zero area within a single fiber, it
also rejects a background Sagnac signal from earth rotation, which breaks
time-reversal symmetry and is the basis for fiber gyroscopes.

Figure 4: ZALSI. Schematics of the zero-area-loop fiber-optic Sagnac
interferometer (ZALSI).

A schematics of ZALSI as well as polarization states at each point are shown
in Fig. 4. The beam of light polarized at 45o to the axis of a electro-optic
modulator (EOM), which generates $5$ MHz time-varying differential phase
shifts $\phi_{m}$ along its two major axis and split the light into two beams
of roughly equal powers. The two beams are then launched into the fast and
slow axes respectively of a polarization maintaining (PM) fiber. Upon exiting
the fiber, the two orthogonally polarized beams are converted into right- and
left-circularly polarized light by a quarter-wave ($\lambda/4$) plate, and are
then focused onto the sample. The non-reciprocal phase shift $\phi_{nr}$
between the two circularly polarized beams upon reflection from the sample is
twice the Kerr rotation ($\varphi_{nr}=2\theta_{K}$). The same quarter-wave
plate converts the reflected beams back into linear polarization, but with a
net 90o rotation of the polarization axis. The two beams then travel back
through the PM fiber and the EOM with exchanged axes before they arrive again
at the polarizer. At this point, the two beams have gone through exactly the
same path but in opposite directions, except for a phase difference of
$\phi_{nr}=2\theta_{K}$ from reflection off of the sample. The two beams are
once again coherent, and interfere to produce an elliptically polarized beam,
whose in-plane component is routed by the circulator to the photodetector.
Lock-in detection was used to measure the average (DC) power ($P_{0}$), the
first harmonics ($P_{1}$), and the second harmonics ($P_{2}$) of the detected
optical power $P(t)$:

$\centering P(t)=\frac{1}{2}P[1+\cos(\varphi_{nr}+2\phi_{m}\sin(\omega
t))]\@add@centering$ (A.1)

where $P$ is the returned power without modulation, and depends on focus of
the objective lens and sample reflectivity. $P(t)$ can be further expanded
into Fourier series of $\omega$ if we keep $\varphi_{nr}$ as a slowly time-
varying quantity compared to $\omega$:

$\displaystyle\centering P(t)/P\@add@centering$ $\displaystyle=$
$\displaystyle[1+J_{0}(2\phi_{m})]/2$ (A.2)
$\displaystyle+(\sin(\varphi_{nr})J_{1}(2\phi_{m}))\sin(\omega t)$
$\displaystyle+(\cos(\varphi_{nr})J_{2}(2\phi_{m}))\cos(2\omega t)$
$\displaystyle+2J_{3}(2\phi_{m})\sin(3\omega t)$ $\displaystyle+...$

where $J_{1}$ and $J_{2}$ are Bessel functions. Therefore, the detected powers
$P_{0}$, $P_{1}$ and $P_{2}$ are:

$\centering P_{0}/P=[1+J_{0}(2\phi_{m})]/2\@add@centering$ (A.3)

$\centering P_{1}/P=(\sin(\varphi_{nr})J_{1}(2\phi_{m}))\@add@centering$ (A.4)

$\centering P_{2}/P=(\cos(\varphi_{nr})J_{2}(2\phi_{m}))\@add@centering$ (A.5)

Hence Kerr signal $\theta_{K}=\varphi_{nr}/2$ can be obtained using equation
A.6, which is independent of optical power, sample reflectivity and focus of
the objective lens. For optimal $\theta_{K}$ sensitivity $\phi_{m}$ is often
chosen to be close to 0.92.

$\theta_{K}=\frac{1}{2}\tan^{-1}\left[\frac{J_{2}(2\phi_{m})P_{1}}{J_{1}(2\phi_{m})P_{2}}\right]$
(A.6)

##  Appendix D Detection of CDW with coherent reflection ratio $P_{2}/P_{0}$

In addition to the Kerr signal, we record the total (P0) and the coherence
(P2) parts of the reflected optical power, as their ratio serves as a measure
of the linear and/or circular birefringence. The above calculations of the
ZALSI assume perfect retardance of the quarter-wave plate and absence of
either linear or circular birefringence and dichroism of the sample. In
reality, commercial zero-order quarter-wave plates have a typical retardance
error of 1$\%$ even for normal incidence, and samples such as CsV3Sb5 display
birefringence and/or dichroism. As a result, the reflected beams, after
passing the quarter-wave plate again, become elliptical instead of being
perfectly linearly polarized. And a small fraction of the light will be
incoherent with the major beams and thus won’t participate in the
interference. These incoherent components will not be captured by $P_{1}$ or
$P_{2}$, but will still be detected as part of the average power $P_{0}$. And
pre-factors need to be added to the formulas for $P_{0}$, $P_{1}$ and $P_{2}$:

$\centering P_{0}/P=(1+A_{0})[1+J_{0}(2\phi_{m})]/2\@add@centering$ (B.1)

$\centering
P_{1}/P=(1+A_{1})(\sin(\varphi_{nr})J_{1}(2\phi_{m}))\@add@centering$ (B.2)

$\centering
P_{2}/P=(1+A_{2})(\cos(\varphi_{nr})J_{2}(2\phi_{m}))\@add@centering$ (B.3)

where $A_{0}$, $A_{1}$ and $A_{2}$ are small correction pre-factors for sample
birefringence and/or dichroism, and retardance error of the wave plate. The
Kerr signal $\theta_{K}$ can be obtained using updated equation B.4, with a
small correction to the scaling factor. There is no change to the zero point
of $\theta_{K}$, which is guaranteed by the symmetry of the interferometer.

$\theta_{K}=\frac{1}{2}\tan^{-1}\left[\frac{(1+A_{2})J_{2}(2\phi_{m})P_{1}}{(1+A_{1})J_{1}(2\phi_{m})P_{2}}\right]$
(B.4)

On the other hand, a change in sample birefringence and/or dichroism will
induce changes to $P_{0}$ and $P_{2}$. However, as previously mentioned, they
are also dependent on $P$, which changes with focus of the objective lens and
sample reflectivity.

$P_{2}/P_{0}=\frac{(1+A_{2})J_{2}(2\phi_{m})}{(1+A_{0})(1+J_{0}(2\phi_{m}))}$
(B.5)

Their ratio $P_{2}/P_{0}$ is independent of these factors and represents the
ratio between the coherent and the total optical powers, dubbed "coherent
reflection ratio". Since the wave plate retardance error is a slow varying
quantity usually dominated the slow drifts of its tilt and rotation,
$P_{2}/P_{0}$ can be used to measure that change of sample birefringence
and/or dichroism during temperature sweeps.

##  Appendix E Experimental Details

Figure 5: Zero-field cool. Kerr signal during zero-field cools and subsequent
warmups in sample 2 measured at Irvine, showing no spontaneous Kerr signal
above $30$ $nrad$. Figure 6: Sagnac scanning images. Sample 2 measured at
Irvine. a, P0 and b, P2 are detected DC and second harmonics optical powers
taken at T = 3.1 $K$ sample temperature, showing an optically flat region
colored in red to green. c, Kerr image taken at T = 3.1 $K$ and in a magnetic
field of B = -0.34 T, showing a uniform Kerr signal of - 8.16 $\ \mu$rad (-24
$\ \mu$rad/T). d, Kerr image taken after the magnetic field is removed,
showing zero spontaneous Kerr signal.

In a typical experiment, we align the focused beam to an optically flat area
of the crystal after cleaving. This can be facilitated by scanning imaging as
shown in Fig. 6 for sample 2, where a flat region of 300 $\ \mu$m in size can
be identified with high reflectivity (P0 of 10 to 30 $\ \mu$W) at T = 3.1 $K$.
Kerr signals are obtained in the same scan. At B = - 0.34 T, a uniform Kerr
signal of - 8.16 $\ \mu$rad is observed, even though the reflected light
varies by 300$\%$ across the region. And upon removing the magnetic field, the
Kerr signal is reduced to zero without any sign of spontaneous TRSB. The
optical beam is then fixed at a location temperature sweeps, as presented in
Fig. 2. During zero magnetic field warmups, an absolute Kerr resolution of
$30$ nrad is typically achieved over a wide temperature range between 0 and
300 $K$. For a smaller temperature range of a few $K$, we can detect a change
of Kerr signal as small as $10$ nrad.

Figure 7: Calibration of background offsets. Window calibration of Irvine
Sagnac setup. (A) P2 and (B) Kerr images of a polished aluminum surface
measured at T = 3.1 $K$ and B =-0.34 T, showing a uniform background Kerr
signal of 39.7 $\ \mu$rad from the optical window, even when P2 varies between
7 to 12 $\ \mu$W across the imaged region. (C) This 39.7 $\ \mu$rad optical
window background (blue) as well as an instrument offset of 1.6 $\ \mu$rad are
both independent of sample temperature. The total calibrated offset is 1.6 -
112 B $\ \mu$rad.

To reveal the diamagnetic shift below $T_{\text{CDW}}=93.5K$, it is necessary
to perform Sagnac measurements when the sample is subject to a magnetic field.
In our normal Sagnac setupsXia  _et al._ (2006b), the fiber-optic head
containing the wave-plate and focusing lens are placed inside the cryostat to
achieve the lowest sample temperatures. Under a magnetic field, the fiber-
optic head will contribute its own temperature-dependent Kerr signal, about
$50$ $\mu\text{rad}/$T at room temperature and $100$ $\mu\text{rad}/$T below 4
$K$. Due to the long thermalization time of glass at cryogenic temperatures,
the temperature of the fiber-optic head often deviates significantly from the
sample temperature readings, making accurate subtractions of this background
signal practically impossible during temperature sweeps. And we should be
cautious with such subtractions Hu  _et al._ (2022). To eliminate this source
of error, we have moved the optics outside the low temperature environment. In
this project, both Stanford and Irvine Sagnac fiber-optic heads are located
outside optical cryostats with a room temperature optical window between the
focusing lens and the sample. The Kerr offset contributed by the optical
window due fringing magnetic fields is thus independent of sample temperature,
and can be calibrated accurately. One such calibration for the Irvine
instrument is illustrated in Fig. 7, using a polished aluminum surface with
negligible Kerr effect. The P2 image at T = 3.1 $K$ and B = - 0.34 T shows
optical power variations between 7 and 12 $\ \mu$W across the surface, while
the Kerr image taken at the same time reveals a spatially uniform Kerr
background of 39.7 $\ \mu$rad due to the Faraday effect of the optical window
held at 295$\pm$0.1 $K$. And we have verified that it is indeed independent of
sample temperature, as shown in Fig. 7(C) (blue). There is a temperature and
time-independent instrument offset of 1.6 $\ \mu$rad due to minor misalignment
of optics and electronic offsets (Fig. 7(C) (red)). The total offset is thus
calibrated to be 1.6 - 112 B $\ \mu$rad, which can be subtracted to produce
accurate Kerr values for temperature sweeps (Fig. 2) and spatial scans (Fig.
6) even in a magnetic field.

Figure 8: DC resistance in CsV3Sb5. a, Resistance and derivative $dR/dT$ in
CsV3Sb5 sample 1. b, Resistance and derivative $dR/dT$ in CsV3Sb5 sample 2.
The charge density transition at $T_{CDW}$ is marked by the kink in the
resistance and the pronounced peak in $dR/dT$. The superconducting transition
is marked by the zero resistance below the critical temperature $T_{C}\approx
2.5$ K. There is an additional feature in $dR/dT$ of a dispersive line shape
centered at $T_{A}\approx 37$ K, which is more pronouced in sample 2.

##  Appendix F General Considerations for MO effects

In general Magneto-optical (MO) effects appear because in the presence of
magnetism right and left circularly polarized lights propagate differently in
solids. When a magnetic field is applied to a diamagnetic insulating solid,
magneto-optical effects will originate from the direct effect of the magnetic
field on the orbital electronic motion. On the other hand, for ferromagnetic
materials, or paramagnetic materials at low temperatures (when their Curie
susceptibility is large enough), the effect of the magnetic field on the
orbital motion is negligible compared with effects associated with spin-orbit
interaction Pershan (1967). For simple metals, far from plasma frequency
resonances we expect that the main contribution to Kerr response is dominated
by off-diagonal intraband Drude-type transitions (i.e. originating from
optical conductivity terms
$\sigma_{xy}(\omega)=\sigma_{0}(\omega_{c}\tau)/[(1-i\omega\tau)^{2}+(\omega_{c}\tau)^{2}]$,
where $\omega_{c}=eH/m^{*}c$ is the cyclotron frequency, $\sigma_{0}$ is the
DC Drude conductivity and $\tau$ is the scattering time). For example, in Al
and Ag Stern  _et al._ (1964) and nobel metals including Cu and Au McGroddy
_et al._ (1965) these effects were measured and recently calculated, showing
that for energies below $\sim 1.5$ eV the Kerr rotation is of order $\sim
10^{-9}$ rad/Oe Uba  _et al._ (2017). In the absence of magnetic polarization,
the orbital and spin Zeeman terms will contribute off-diagonal terms through
interband transitions, which for the above simple metals are at least an order
of magnitude smaller. Taking into account optical and transport measurements
on CsV3Sb5 , both effects are expected to yield an even smaller response,
which will not be detectable for the magnetic fields we used with the ZALSI
experiments.

##  Appendix G DC resistance Measurements

DC resistances measured on sample 1 and sample 2 are shown in Fig. 8. A kink
at $T_{CDW}$ is clearly visible in the resistance of both samples, marking the
charge density transition. The exact values of $T_{C}DW$ are determined by the
temperatures of the peaks in the first derivative $dR/dT$ curve, and agree
with literature data Ortiz  _et al._ (2020) within $1$ K. The superconducting
transition is marked by the zero resistance below the critical temperature
$T_{C}\approx 2.5$ K. In the $dR/dT$ curve, especially in sample 2, there is
an additional feature of a dispersive line shape centered at $T_{A}\approx 37$
K. We note that NMR experiments Song  _et al._ (2022) have observed a kink
behavior at a similar temperature of 35 K in $1/T_{1}T$, and that the
electronic magneto-chiral anisotropy (eMChA) was detected in nonlinear
transport Guo  _et al._ (2022) below $34$ K. Since we observe no spontaneous
Kerr signal across $T_{A}$, it is suggestive of a phase transition without
TRSB at $T_{A}\approx 37$ K.