# Universality of the q=1/2 Orbital Magnetism in the Pseudogap Phase of the
High-$T_{c}$ superconductor $\rm YBa_{2}Cu_{3}O_{6+x}$

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  Martin
Boehm  Institut Laue-Langevin, 71 avenue des Martyrs, Grenoble 38042, France
Paul Steffens  Institut Laue-Langevin, 71 avenue des Martyrs, Grenoble 38042,
France  Toshinao Loew  Max Planck Institute for Solid State Research,
Heisenbergstrasse 1, 70569 Stuttgart, Germany  Lin Shan Guo  Shanghai Jiao
Tong University, 800 Dong Chuan Road, Shanghai 200240, China  Jun Qian
Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China
Xin Yao  Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240,
China  Philippe Bourges  Université Paris-Saclay, CNRS, CEA, Laboratoire Léon
Brillouin, 91191, Gif-sur-Yvette, France

###### Abstract

Several decades of debate have centered around the nature of the enigmatic
pseudo-gap state in high temperature superconducting copper oxides. Recently,
we reported polarized neutron diffraction measurements that suggested the
existence of a magnetic texture bound to the pseudo-gap phase [Bounoua, et al.
Communications Physics 5, 268 (2022)]. Such a magnetic texture is likely to
involve the spontaneous appearance of loop currents within the CuO2 unit
cells, which give birth to complex correlated patterns. In the underdoped
${\rm YBa_{2}Cu_{3}O_{6.6}}$, the magnetic structure factor of such an orbital
magnetic texture gives rise to two distinct magnetic responses at q=0 and
q=1/2. As this pattern alters the lattice translation invariance, such a state
of matter could contribute to an instability of the Fermi surface. Here, we
report polarized neutron scattering measurements on a nearly optimally doped
high quality single crystal of ${\rm YBa_{2}Cu_{3}O_{6.9}}$ that exhibits the
same q=1/2 magnetism and a weakly overdoped ${\rm YBa_{2}Cu_{3}O_{7}}$ sample
where this signal is no longer sizeable. The in-plane and out-of-plane
magnetic neutron scattering intensities in ${\rm YBa_{2}Cu_{3}O_{6.9}}$ (at
q=1/2) and ${\rm YBa_{2}Cu_{3}O_{6.85}}$ (at q=0), reported previously,
display the same temperature dependent hallmarks. The magnitudes of both q=0
and q=1/2 magnetic signals further exhibit the same trends upon doping in
${\rm YBa_{2}Cu_{3}O_{6+x}}$, confirming that they are likely intertwined.

##  I  Introduction

The generic phase diagram of hole doped cuprate superconductors shows various
electronic instabilities [1, 2, 3]. Beyond unconventional superconductivity,
evidence for (spin, charge and pair) density waves were reported, in addition
to intra-unit-cell orders breaking local Ising symmetries (time reversal,
parity, rotation). The pairing mechanism that permits superconductivity in
copper oxides remains unknown, although a mechanism based on magnetic
interactions is generally favoured. For instance, the correspondence between
the changes of the charge-transfer-gap (controlling the magnetic superexchange
interaction) and the Cooper pair density demonstrates a strong and compelling
correlation, hence perhaps constitutes the clearest evidence yet of a
mechanism that underlies superconductivity in cuprates[4].

At present, there is a lack of consensus about the pseudogap state of matter
that appears below a temperature T* and from which the superconductivity
emerges in a wide range of its phase diagram. In the underdoped regime, the
$d$-wave superconducting (SC) state competes with an incipient bi-axial charge
density wave (CDW). Both states appear deep inside the so-called pseudogap
(PG) phase, which creates gaps in extended portions of the Fermi surface and
gives rise to Fermi arcs only. Concomitantly, an intra-unit-cell (IUC) q=0
antiferromagnetism develops with the PG state [5, 6, 7, 8, 9, 10, 11], without
giving a net magnetization in the CuO2 plaquette that would be expected for a
(q=0) ferromagnetism. This state is usually associated with magneto-electric
loop currents (LC) running through copper and oxygen orbitals, and whose order
parameter can be conveniently described by a polar anapole vector [3, 11]. The
observed magnetism then corresponds to orbital moments, a picture supported by
the observation of an unusual form factor [8]. It breaks both time and parity
symmetries, but preserves the lattice translation invariance [12, 3, 13, 14].
Supporting these observations from polarized neutron measurements and muon
spin spectroscopy confirms the existence of a slowly fluctuating magnetism in
the PG state [15]. The corresponding magnetic fluctuations are characterized
by a finite timescale that does not evolve with temperature [16]. An
interesting characteristic of the IUC orbital magnetism is that it survives
upon dimensional confinement, and its correlation lengths become short-ranged,
as observed in the quasi-1D ${\rm(Sr,Ca)_{14}Cu_{24}O_{41}}$ two leg-ladder
cuprates [17], or in ${\rm La_{2-x}Sr_{x}CuO_{4}}$ [18] due to charge
segregation (leading to charge stripes). However, it is worth emphasizing that
the breaking of the discrete symmetries cannot alone open the PG [19].

Recently, polarized neutron diffraction studies uncovered the existence of a
novel form of magnetism, labelled q=1/2, hidden in the pseudogap phase of
superconducting cuprates [20]. In two distinct ${\rm YBa_{2}Cu_{3}O_{6.6}}$
underdoped samples, a static short-range magnetic response is observed at the
planar wave vector (0.5,0). Gathering all the information available so far,
one can state that the q=1/2 magnetism belongs to the CuO2 planes and is
observed at both (0.5,0) and (0,0.5) in a detwinned sample [20]. Breaking the
lattice symmetry, the novel magnetism with a propagation vector
$(\pi,0)\equiv(0,\pi)$ can then be described by a 2x2 larger unit cell, which
is inconsistent with known spin structures in cuprates, rather suggesting an
orbital magnetism related to loop currents. Similarly to the q=0 IUC
magnetism, the q=1/2 magnetic signal typically sets in at the PG onset
temperature on cooling down from room temperature, suggesting that both
magnetic responses are intertwined and are directly related to the PG physics.
The deduced magnetic moment of the q=1/2 magnetism is further found to point
predominantly along the c-axis [20], in contrast to the q=0 IUC magnetism
whose ordered moment exhibits a systematic tilt of about 45∘ from the c-axis
[10]. Even though there is no definitive explanation for the origin of the
tilt [11], the magnetic response was found to reorient from fully out-of-plane
to tilted on cooling in ${\rm YBa_{2}Cu_{3}O_{6.85}}$[9] supporting a possible
scenario of an evolution from classical (above T*) to quantum LC states [21],
the classical LC state being characterized by the existence of a magnetic
moment strictly perpendicular to the circulating loop currents. The out-of-
plane c-polarized magnetic response favors interpretations related to LC
running within the CuO2 planes, naturally leading to orbital magnetic moments
pointing along the c-axis. One can nevertheless wonder why the q=1/2 magnetism
should not exhibit an in-plane component as does the q=0 IUC magnetism if both
features were intertwined.

The q=1/2 magnetism remains at short range at all temperatures with a
correlation length of about 25 Å range, corresponding to the formation of
clusters of 5-6 unit cells within the CuO2 planes [20]. Together with the
previous reports of the q=0 IUC magnetism in cuprates [5, 10, 11] (that
develops at longer distance), the novel magnetism at $(\pi,0)$ could actually
belong to a unique complex bi-axial magnetic texture of the CuO2 unit cells
hosting loop currents/anapoles [20]. This picture is very similar to the case
of charge density instabilities where a q=0 nematic charge order co-exists
with the q$\neq$0 charge density wave with shorter correlation lengths [22]
transposed in the orbital magnetism sector. While various magnetic patterns
could account for the magnetic diffraction data available so far, one can for
instance visualize such a magnetic texture as a supercell made of 4 large
domains with q=0 LC order domains (where loop current patterns/anapoles rotate
at 90∘ from one domain to the other) and, at their junction, a bubble of
intertwined 2x2 loop currents related to the local $(\pi,0)$ magnetic response
[20]. In both ${\rm YBa_{2}Cu_{3}O_{6.6}}$ samples, the q=1/2 magnetic signal
is weakly correlated along the c-axis, perpendicular to the CuO2 planes, with
a maximum structure factor centered at $L$=0 [20]. If its observation is
ubiquitous in the PG phase, the q=1/2 magnetism is likely important for the
understanding of the instrinsic nature of the PG state since it breaks the
lattice translational symmetry and develops at the PG onset temperature.
However, it was only reported for a single hole doping level, so far.

Here, using polarized neutron diffraction, we investigated the existence of
the q=1/2 magnetism in two high quality single crystals at higher doping
levels than in the former study (p$\simeq$ 0.11 for ${\rm
YBa_{2}Cu_{3}O_{6.6}}$ [20]). In particular, we report the existence of the
q=1/2 magnetism in a nearly optimally doped ${\rm YBa_{2}Cu_{3}O_{6.9}}$
single crystal, with a hole-doping of p$\simeq$ 0.16, where the q=1/2
magnetism displays the same temperature dependence as the q=0 IUC magnetism
reported for a similar doping [9]. Compared to the ${\rm
YBa_{2}Cu_{3}O_{6.6}}$ samples, the q=1/2 magnetism is reduced in amplitude
upon increasing doping in a similar way to the q=0 IUC magnetism.
Interestingly, its c-axis dependence exhibits a complex behaviour with a
structured magnetic response along Q=$(0.5,0,L)$, giving rise to two well-
defined peaks at $L$=0 and $L$=1/2, suggesting a magnetic doubling of the unit
cell in the stacking direction of CuO2 planes along the c-axis. The magnetic
scattering associated with the $L$=1/2 peak is found to be mainly related to
the in-plane component in contrast to the $L$=0 peak reported so far [20],
which arises predominantly from the out-of-plane magnetic component. The
L-dependent orientation of the magnetic response indicates that the in-plane
and out-of-plane magnetic components at q=1/2 occur at different $L$ positions
along Q=$(0.5,0,L)$, in contrast with the q=0 IUC magnetic response for which
both types of components are superimposed in momentum space.

##  II  Experimental details

####  II.0.1  Polarized neutron scattering experiment

The polarized neutron measurements were performed on two different neutron
Triple Axis Spectrometers (TAS) with different momentum and energy
resolutions. In both cases, the measurements were carried out in elastic
conditions with the final neutron wavevector (kf) equal to the incident one
(ki). On the instrument 4F1 located at the LLB-Orphée reactor in Saclay
(France), the neutron wavevector was kf=2.57 Å-1; a Pyrolitic Graphite (PG)
filter installed on the incident beam was used to remove higher order
harmonics. A double PG monochromator was used to select the incident neutron
wavelength. The beam was polarized using a bender supermirror located between
the second monochromator and the sample position and the final polarization
state was analysed with an Heusler crystal Cu2MnAl using the (1,1,1) Bragg
reflection. On Thales located at Institut Laue Langevin (ILL) in Grenoble
(France), the neutron wavevector was much lower, kf=1.5 Å-1, yielding a better
energy resolution. A velocity selector in the incoming and a Be filter in the
outgoing beam were used to remove higher order harmonics. The incoming and
outgoing beam polarizations were realized using Heusler crystals.

Both TAS instruments were equipped with longitudinal XYZ polarization analysis
(XYZ-PA) (see [20]), a powerful technique to selectively probe and disentangle
the magnetic response from the nuclear one with no assumptions on the
background. Both spin-flip (SF) and non-spin-flip (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.
Throughout the article, the total scattered magnetic intensity, $I_{mag}$, is
extracted such as (See [17, 20] for more details about the analysis of the
neutron data):

$I_{mag}=2\,SF_{\bf X}-{SF}_{\bf Y}-{SF}_{\bf Z}$ (1)

The flipping ratios FR=$(\frac{NSF_{X,Y,Z}}{SF_{X,Y,Z}})$ on both instruments
were measured on the sample Bragg peaks as well as on a quartz speciment. They
were found to be homogeneous along $\bf X,Y$ and $\bf Z$ polarizations, such
as (FR=15) on 4F1 TAS and (FR=40) on Thales TAS (see Supplemental Material
[23] for more details). On Thales, the sample was mounted in a spherical
polarization analysis device, CryoPAD[24], with zero magnetic field at the
sample chamber that allows one to perform XYZ-PA even in the superconducting
state without risks of neutron spin depolarization. On 4F1, the neutron
polarization at the sample position was controlled using MuPad[25], another
type of spherical polarization analysis device where the sample is installed
inside a zero magnetic field chamber. As a result, the polarisation is very
homogeneous with tiny variations of the flipping ratio upon turning the
polarization. Therefore, corrections from imperfect neutron polarization have
negligible effects in Eq. 1 (see an example in Supplemental Material [23]).

####  II.0.2  Sample preparation and characterization

Two different ${\rm YBa_{2}Cu_{3}O_{6+x}}$ (YBCO) samples near optimal hole
doping were studied (Fig. 1). Both samples exhibit the ortho-I orthorhombic
structure where the Cu-O chains fully align along the $b$ axis for x$\geq$0.85
[26]. The first sample is a large ${\rm YBa_{2}Cu_{3}O_{6.9}}$ single crystal,
hereafter sample YBCO69 (Fig. 1.a). It was grown at Shanghai Jiao Tong
University using a polythermal method of top-seeded solution-growth to get
large ${\rm YBa_{2}Cu_{3}O_{6+x}}$ single crystals [27]. The YBCO69 sample was
never studied with polarized neutron diffraction before, however YBCO single
crystals grown with this growth technique have been already studied with
inelastic neutron scattering [28] and quantum oscillations [29] among other
technique. The sample is twinned with a mass of $\simeq$7 g and contains no
impurity such as the so-called green phase ${\rm Y_{2}BaCuO_{5}}$ (Fig. 1.c.)
or cupric oxide CuO (see Supplemental Material [23]). The as-grown single
crystal was annealed for 18 days at 400∘C, followed by 40 days heat treatment
at 520∘C under oxygen flow [30] to ensure the diffusion of oxygen into the
bulk, then quenched in liquid nitrogen. After annealing, the sample exhibits a
sharp superconducting transition temperature at $T_{c}$ = 91.9 K (Fig. 1.d).
The c-axis parameter was further measured using neutron diffraction. Using the
standard determination of hole doping [31], one obtains p$\simeq$ 0.16.

The second sample is an array of detwinned ${\rm YBa_{2}Cu_{3}O_{7}}$ crystals
(hereafter sample YBCO7 shown on Fig.1.b) , grown at the Max Planck Institute
for Solid State Research in Stuttgart. This sample was previously used for an
inelastic neutron scattering experiment to study phonon anomalies [32]. The
detwinning ratio of the array is 86%. It has a mass of 2.6 g and exhibits a
superconducting transition temperature at 90K [32], leading to p$\simeq$ 0.19.

The YBCO69 sample was studied on both the 4F1 and Thales instruments in the
(1,0,0)/(0,0,1) scattering plane such that wave-vectors of the form $(H,0,L)$
were accessible. The YBCO7 sample (only measured on 4F1) was instead mounted
in the (1,0,0)/(0,1,0) scattering plane to study wave-vectors of the form
$(H,K,0)$. In the following, the wave-vectors are indexed in reduced lattice
units (r.l.u.), in units of $(\frac{2\pi}{a},\frac{2\pi}{b},\frac{2\pi}{c})$
where $a$, $b$ and $c$ stand for the lattice parameters of the sample.

##  III  Experimental results

####  III.0.1  Short-range magnetic correlations

Figure 2 shows momentum scans along the H-direction in the SF channels around
Q=(0.5,0,0) in the YBCO69 sample. On both instruments, a peak is observed at
low temperature in the superconducting state around H=0.5 (Fig. 2.a,d). The
polarization analysis further proves the magnetic origin of the signal as the
SF peak intensity is reduced when one turns the polarization from $\bf X$ to
$\bf Y$ and $\bf Z$. The magnetic signal can be described by a Gaussian
profile with a finite momentum distribution, broader than the instrumental
resolution. After deconvolution, its intrinsic linewidth - half width at half
maximum (HWHM) - is $\Delta_{H}$=$0.02\pm 0.005$ r.l.u. This corresponds to a
finite correlation length along the $a$-axis of
$\xi_{a}=\frac{a}{2\pi\Delta_{H}}$= $\sim 30\pm 3\ \text{\AA}$, corresponding
to $\sim 7-8$ planar unit-cells. These measurements were performed using two
instrument configurations (different neutron wavevectors) indicating that the
signal is static within the lowest energy resolution of Thales of $\sim$ 0.2
meV, implying a characteristic time-scale necessarily much longer than 0.02
ns. This timescale is consistent with the one extracted from muon spin
spectroscopy, which indicates slow magnetic fluctuations in the pseudogap
state of about 25 ns at this doping level [15]. Upon increasing temperature,
the short range magnetic peak at q=1/2 is still present in the normal state,
in the 100-150K range (Fig. 2.b,e). However, the peak is significantly reduced
at 300K (Fig. 2.c,f) being hardly visible.

Similar measurements were performed in the second sample, YBCO7. As the sample
YBCO7 is detwinned, measurements needed to be performed along both a* and b*
directions. Fig. 3.a-b represent the H-scan and K-scan at 100K in the SF
channel. No peak occurs along both directions. Similarly, XYZ-PA confirms the
absence of a peak around H=0.5 at 100K (Fig. 3.c-d). The polarization analysis
performed at 3K shows no sizeable feature as well (Fig. 3.c-d). At such a high
doping, the short range magnetism (if any) at q=1/2 falls below our threshold
of detection.

As was discussed in ${\rm YBa_{2}Cu_{3}O_{6.6}}$ [20], a magnetic signal at
q=1/2 is not compatible with known spin structures in the cuprates. Indeed,
the antiferromagnetic spin ordering in cuprates is always found near the
planar wavevector q=(1/2,1/2). At higher doping, the reported spin
fluctuations are also systematically observed close to that wave vector. That
corresponds to strong equal first neighbour antiferromagnetic interactions
along both a and b. The q=1/2 order actually corresponds to a stripe-like spin
order as it is observed in the parent compounds of Fe pnictides
superconductors, where the magnetic exchange along the a axis $J_{1a}$ differs
from that along the b axis $J_{1b}$ [33], a situation never encountered in
cuprates. Therefore, as at lower hole-doping [20], our results suggest the
existence of an orbital magnetism breaking the lattice symmetry, showing that
it occurs over a wide doping range that corresponds to the pseudogap regime.

####  III.0.2  $L$ dependence

Next, we discuss the $L$-dependence of the magnetic signal in YBCO69
corresponding to the correlations of the q=1/2 magnetism along the c-axis
perpendicular to the CuO2 planes. Fig. 4.a and b show the neutron intensities
at 150K and 280K, respectively. At 150K, two peaks occur with distinct maxima
at $L$=0 and $L$=0.5. Both peaks display similar momentum widths, typically
about $\Delta_{L}$=$0.2\pm 0.05$ r.l.u. at 150K. Contrary to the ${\rm
YBa_{2}Cu_{3}O_{6.6}}$ sample [20], we cannot prove here that the magnetism
reported here is biaxial as the YBCO69 sample is twinned. The existence of two
peaks along c contrasts with the single broad peak measured along c in twinned
${\rm YBa_{2}Cu_{3}O_{6.6}}$ at 80K [20]. Considering that the peaks of Fig.
4.a exhibit c-axis correlation lengths that are different from the one found
in the twinned ${\rm YBa_{2}Cu_{3}O_{6.6}}$ sample, one can speculate that the
arrangement of magnetic CuO2 layers along c could be modified by a sample
dependent disorder (defects, inhomogeneities, …). This $L$-dependence also
differs from the ones measured in the detwinned ${\rm YBa_{2}Cu_{3}O_{6.6}}$
sample [20] where a sharp peak was only observed at $L$=0 without any
intensity at (0.5,0,0.5) at 5K. Note that although the existence of a magnetic
signal at (0,0.5,0) was reported[20], the corresponding information at
(0,0.5,0.5) was still missing. This disparity pinpoints a more complex
behaviour than thought at first that could be related to the twinning of our
YBCO69 sample. Indeed, if the signal only occurs at Qb=(0,0.5,0.5) but not at
Qa=(0.5,0,0.5) in a detwinned sample, the twining may lead to the observed
scattering at (0.5,0,0.5). The temperature dependence of the $L$=0.5 signal
may then be different at Qa and Qb; an information that was left unnoticed in
our previous study. At 280K, the intensity of both peaks along c strongly
decreases (Fig. 4.b). This shows that the magnetic signal is nearly vanishing
near room temperature in agreement with the momentum scan at L=0 (Fig. 2.f),
and in line with previous results in ${\rm YBa_{2}Cu_{3}O_{6.6}}$ [20].

One valid question about the c-axis magnetic correlations in YBCO69 is whether
they are related to the CuO chains or to the CuO2 planes. In our previous
investigation of ${\rm YBa_{2}Cu_{3}O_{6.6}}$ [20], we could resolve that
issue as we studied two different samples with different oxygen ordering in
the CuO chains that allowed demonstrating that the q=1/2 magnetism was not
bound to the oxygen superstructure. Additionally, the observed magnetism in
the detwinned sample was bi-axial. The q=1/2 magnetism could thus only
originate from the CuO2 planes. Here, the sample structure is Ortho-I with no
oxygen between copper spins along $a$ that precludes a magnetic response from
the chains at q=1/2 along that direction. However, magnetic correlations
running only along the $b$ direction could also be considered. Indeed,
quasi-1D fluctuations were reported in ${\rm YBa_{2}Cu_{3}O_{6.93}}$ near
optimal doping. They are characterized by a net incommensurate modulation [34]
that agrees with band structure calculations associated to nesting effects at
2${\bf k}_{F}$ of the CuO chains Fermi surface. Related magnetic dynamical
correlations around $K\simeq 0.7-0.75$ were tentatively reported [35, 36] but
do not correspond to our findings. Hence, here again, one can ascribe the
origin of the q=1/2 magnetism to the CuO2 planes.

The scattered magnetic intensity of the q=1/2 magnetism at Q=(H,0,L),
$I_{mag}$, includes the neutron orientation factors related to each magnetic
component. It reads:

$I_{mag}=r_{0}^{2}\,\,f(Q)^{2}\,\left[{\bf m}_{b}({\bf
Q})^{2}+(1-|2\pi\frac{H/a}{\bf Q}|^{2}){\bf m}_{a}({\bf
Q})^{2}+(1-|2\pi\frac{L/c}{\bf Q}|^{2}){\bf m}_{c}({\bf Q})^{2}\right]$ (2)

where $r_{0}$ stands for the neutron magnetic scattering length,
$r_{0}^{2}=290$ mbarn, $f(Q)$ is the magnetic form factor. $|{\bf
Q}|=2\pi\sqrt{(H/a)^{2}+(L/c)^{2}}$ is the wave vector modulus expressed in
Å-1 and $a$=3.82 Å and $c$=11.7 Å are the lattice parameters along a and c,
respectively. ${\bf m}_{a}({\bf Q})$, ${\bf m}_{b}({\bf Q})$ and ${\bf
m}_{c}({\bf Q})$ correspond to the Fourier components of magnetic ordered
moments along the three crystal directions. In general, the Q-dependence of
the magnetic structure factor relates to the specific magnetic arrangements
within the unit cell, in particular here within the CuO2 bilayers on which we
do not have sufficient experimental information. To reduce the amount of
notations, we omit the Q dependence of the magnetic components hereafter.

As the YBCO69 sample is twinned, the $(H,0,L)$ scattering plane is
indistinguishable from the $(0,K,L)$ scattering plane. The contributions
coming from each domain superpose in the measurements. Further, as the in-
plane components along $a$ and $b$ cannot be distinguished, it is generally
assumed that ${\bf m}_{a}={\bf m}_{b}$. We here rather define an in-plane
magnetic moment component, ${\bf m}_{ab}$, as ${\bf m}_{ab}^{2}={\bf
m}_{a}^{2}+{\bf m}_{b}^{2}$. As a result of these considerations, Eq. 2 can be
re-written as,

$I_{mag}=r_{0}^{2}\,\,f(Q)^{2}\,\left[\frac{1}{2}(1+|2\pi\frac{L/c}{{\bf
Q}}|^{2}){\bf m}_{ab}^{2}+(1-|2\pi\frac{L/c}{\bf Q}|^{2}){\bf
m}_{c}^{2}\right]$ (3)

In the spin-flip channel, Eq. 3 corresponds to the full intensity measured for
a neutron polarization along X. Fig. 4.c shows the total scattered magnetic
intensity $I_{mag}$ deduced by XYZ polarization analysis (Eq. 1, see ref. [17,
20] for more details). Eq. 3 includes the neutron orientation factor for each
magnetic component as $(1-|2\pi\frac{L/c}{\bf Q}|^{2})$ for the out-of-plane
(${\bf m}_{c}$) component and $(1+|2\pi\frac{L/c}{\bf Q}|^{2})$ for the in-
plane (${\bf m}_{ab}$) component. Further, to determine separately the
different magnetic components, one needs to apply the polarization analysis.
Following our previous reports [17, 20], the difference between the measured
intensities for different polarizations are related to the magnetic components
as:

$\begin{cases}{SF}_{\bf X}-{SF}_{\bf Y}=&r_{0}^{2}\,f(Q)^{2}\,{1\over 2}{\bf
m}_{ab}^{2}\\\ {SF}_{\bf X}-{SF}_{\bf
Z}=&r_{0}^{2}\,f(Q)^{2}\,[(1-|2\pi\frac{L/c}{\bf Q}|^{2}){\bf
m}_{c}^{2}+|2\pi\frac{L/c}{{\bf Q}}|^{2}{1\over 2}{\bf
m}_{ab}^{2}]\end{cases}$ (4)

Next, one defines the quantities $I_{c}$ and $I_{ab}$ as:
$I_{c}=(1-|2\pi\frac{L/c}{\bf Q}|^{2})f(Q)^{2}{\bf m}_{c}^{2}$,
$I_{ab}=\frac{1}{2}(1+|2\pi\frac{L/c}{{\bf Q}}|^{2})f(Q)^{2}{\bf m}_{ab}^{2}$.
In the absence of magnetic chirality, the polarization sum rule holds:
$I_{mag}=I_{ab}+I_{c}$ [37]. Using XYZ-PA (Eq. 4), one determines the magnetic
in-plane ($I_{ab}$) and out-of-plane ($I_{c}$) components. In addition to the
results at 150K shown in Fig. 4.a, we studied the polarization dependence of
both peaks over a wide temperature range (4-150K) and averaged out the data to
improve statistics since no significant variation of the ratio
$\frac{I_{ab}}{I_{c}}$ was noticed in temperature. $I_{mag}$, $I_{ab}$ and
$I_{c}$ are represented in Fig. 4.c in units of $\frac{2\pi}{c}L$. At $L$=0,
the magnetic signal indicates a predominant component along the c-axis as in
the underdoped sample [20]. However, the ratio between in-plane and out-of-
plane components, $\frac{I_{ab}}{I_{c}}$, is larger than for ${\rm
YBa_{2}Cu_{3}O_{6.6}}$. The result at $L$=0 contrasts with the polarization
analysis of the second peak at $L$=1/2 with a much weaker out-of-plane
contribution, resulting in a dominant in-plane component. This actually
supports a non-collinear alignment of moments (pointing predominantly along
c), which gives rise to distinct magnetic structure factors at $L$=0 and
$L$=1/2.

At larger $L$, the polarization analysis shows that the magnetic signal
vanishes rapidly. On Fig. 4.c, we report Eq. 3 where the magnetic form factor
is the one expected for a spin moment either of Cu or oxygen atoms (dashed and
dotted lines, respectively) normalized to 1 at $L$=0. Clearly, this does not
describe our data as that has been previously shown for the q=0 IUC magnetic
intensity that also decreases rapidly at large $L$ [8]. Beyond the weak $L$
dependence coming from the orientation factors, the fast decay of the magnetic
intensity cannot be accounted for by the Cu and/or oxygen spin form factors.
This suggests an orbital origin for the measured magnetism. Along this line,
we calculate the measured magnetic intensity in Eq. 3 using the
phenomenological form factor along $L$, $|f(L)|^{2}$, found for the q=0 IUC
magnetism in 4 distinct cuprate families [8]. In Fig. 4.c, $|f(L)|^{2}$ also
normalized to 1 at $L$=0 describes better the measured $L$-dependence of the
q=1/2 magnetic intensity. In Fig. 4.c, the full lines represent the product
$|f(L)|^{2}(1-|2\pi\frac{L/c}{\bf Q}|^{2}){\bf m}_{c}^{2}$ for the out-of-
plane component or the product $|f(L)|^{2}\frac{1}{2}(1+|2\pi\frac{L/c}{\bf
Q}|^{2}){\bf m}_{ab}^{2}$ for the in-plane component, respectively. At
variance to the spin form factors, these relations with $f(L)$ match the
characteristic decrease of the q=1/2 magnetism for each magnetic component, as
it was the case for the q=0 IUC magnetism [8]. One can nevertheless notice
that the measured out-of-plane intensity is slightly lower than the
expectation whereas the opposite trend is found for the in-plane component.
That may suggest a different form factor for both components as it has been
proposed for loop currents [21]. On general grounds, it is worth stressing out
that the orbital magnetic form factor can be very anisotropic for the case of
loop currents due to their extended geometry, delocalized over the Cu and O
sites. The fact that both q=0 and q=1/2 magnetisms exhibit the same fast decay
along $L$ strongly supports a common orbital origin.

####  III.0.3  Intensity in absolute units

Next, we estimate the intensity of the q=1/2 magnetism measured in the
different samples. To get the magnetic signal in absolute units versus doping,
we need to calibrate the magnetic signal by comparison with Bragg peak
intensities. We follow the method presented in our previous study in ${\rm
YBa_{2}Cu_{3}O_{6.6}}$ [20] where the reported magnetic signal at $L$=0 is
$q$-integrated in the [a,b] plane. Its hole doping dependence is shown in Fig.
4.d. Typically, the amplitude at q=1/2 is about three times weaker in the
YBCO69 sample compared to ${\rm YBa_{2}Cu_{3}O_{6.6}}$ at lower doping. One
clearly sees that the signal at q=1/2 decreases upon increasing doping.
Interestingly, this decay matches that of the q=0 IUC magnetic scattering as
it scales with the magnetic intensity at the Bragg peak Q=(1,0,1) reported in
the supplementary information of ref. [38]. The signal at (0.5,0,0) is about 5
times weaker than the magnetic intensity at Q=(1,0,1) by comparing both
vertical scales in Fig. 4.d. It is then typically 10-15 weaker than the
magnetic intensity at Q=(1,0,0) [20]. The magnetic moment evolves as the
squared root of the neutron intensity. Therefore, using our previous
estimation of the magnetic moment in ${\rm YBa_{2}Cu_{3}O_{6.6}}$ [20], a
magnetic moment per loop current triangle of $m_{tr}\sim$ 0.015 $\mu_{B}$ is
found using the assumption that all unit cells contribute uniformly to the
magnetic intensity. We stress out that this assumption is misleading as the
q=1/2 magnetism is at short-range [20].

####  III.0.4  Temperature dependence

The temperature dependence of the magnetic intensity is further represented in
Fig. 5. The raw neutron intensity at $L$=0 (Fig. 5.a) decreases upon warming
up faster than the background determined by the polarization analysis. Using
the XYZ-PA, one can also decompose the signal at $L$=0 into the two different
magnetic components as above. The background subtracted total magnetic
intensity, $I_{mag}$ is depicted in Fig. 5.b and the temperature dependencies
of the out-of-plane, $I_{c}$, and in-plane, $I_{ab}$, magnetic components in
Fig. 5.c and d, respectively. As for ${\rm YBa_{2}Cu_{3}O_{6.6}}$ samples
[20], the in-plane component at $L$=0 (Fig. 5.c) exhibits no change in
temperature whereas the out-of-plane component (Fig. 5.d) captures the whole
decrease in temperature where the signal disappears above $\sim$ 250K. The
temperature dependence of the background subtracted magnetic intensity at
$L$=1/2 is shown in Fig. 5.e in comparison with the magnetic intensity
obtained by polarization analysis. Due to the weaker amplitude at $L$=1/2, the
decomposition of the magnetic signal into the components cannot be performed
with confidence. One can nevertheless use these data to determine the
temperature where the magnetic response sets in, which is found to be about
$T_{mag}\sim 200K$ where the intensity at $L$=0.5 drops. This temperature is
reported in the phase diagram (Fig. 4.e). As in ${\rm YBa_{2}Cu_{3}O_{6.85}}$,
$T_{mag}$ occurs at higher temperature than the appearance of the CDW that are
hardly visible using resonant elastic X-ray scattering in ${\rm
YBa_{2}Cu_{3}O_{6+x}}$ above a doping of $\sim$ 0.17 [39].

The q=0 IUC magnetism was not measured in the YBCO69 sample but our data can
be compared to the q=0 magnetism reported in ${\rm YBa_{2}Cu_{3}O_{6.85}}$ [9]
that had a close hole doping level. We then compare the evolution of
temperature dependencies of the q=1/2 magnetism with the one reported at q=0
in the nearly optimally doped ${\rm YBa_{2}Cu_{3}O_{6.85}}$ [9].
Interestingly, the temperature dependencies of the out-of-plane magnetic
components at (0.5,0,0), in YBCO69, and at q=0, in ${\rm
YBa_{2}Cu_{3}O_{6.85}}$, agree remarkably (Fig. 5.d). $T_{mag}$ corresponds to
the inflexion point of the temperature dependence of the out-of-plane
component in Fig. 5.d. By contrast, the in-plane component at q=0 (Fig. 5.f)
compares well with the temperature evolution at (0.5,0,0.5) where the magnetic
signal is predominantly in-plane in character according to the polarization
analysis (Fig.4.c). However, in this comparison, an additional magnetic
component at (0.5,0,0.5) is present at all temperature similar to the one
found at $L$=0 (Fig. 5.c). Although the amplitude at q=1/2 is much weaker, the
temperature dependences of the in-plane ($L$=0.5) and out-of-plane ($L$=0)
magnetic responses display striking similarities with the q=0 IUC response
[9]. There is however a noticeable difference. At q=0, both components occur
at the same $L$ values yielding an apparent tilt of the magnetic moment [10,
11], as expected for collinear and canted magnetic moments. At q=1/2, the
components occur at two different $L$ positions suggesting a more complex
stacking of the magnetic correlations along the c-axis, more consistent with a
non-collinear arrangement of canted magnetic moments. Typically, the
alternating in-plane component is related to a doubling of the unit-cell along
c, which does not occur for the q=0 IUC magnetism. As the doubling of the unit
cell along c affects the magnetism of 4 CuO2 layers (i.e. 2 bilayers), an in-
plane magnetic moment turning at 90∘ from one plane to the next along c could
for instance produce such a doubling of the periodicity along c.

The phase diagram of Fig. 4.e shows the universal character of the q=1/2
magnetism. It corresponds to the region in doping associated with the
existence of the pseudogap in YBCO. The ordering temperature for both the long
range q=0 IUC magnetism and the shorter range q=1/2 magnetism decreases first
linearly in the underdoped region, then flattens near optimal doping, and
vanishes beyond, concomitantly with the PG.

##  IV  Discussion and concluding remarks

Before going into a detailed interpretation of the data in terms of orbital
moments, it is important to dismiss extrinsic contributions, for instance due
to magnetic peak in cupric oxide (CuO) impurities. Indeed, CuO is
antiferromagnetic (AF) [40, 41] with an ordered moment of 0.65 $\mu_{B}$ and
T${}_{N}=$230 K, its strongest (1/2,0,-1/2) AF peak has a $|Q|$ very close to
the (0.5,0,0.5) momentum of YBCO69. The same issue was discussed and dismissed
in our previous report of $q=1/2$ magnetism [20]. As we discuss extensively in
section 2 of the Supplemental Material [23], CuO powder and crystallites
impurities can be ruled as well in our YBCO69 sample as the origin of the
reported magnetic peaks. In particular, to be able to explain the neutron
data, the CuO impurities have to be nano-size grains of $\sim$ 25 Å. As
discussed in ref. [42], the Néel temperature in such nano-size grains of CuO
is below 50K which is inconsistent with our reported temperature dependence.

As discussed in our previous report [20], these results can be described by an
hidden magnetic texture mixing small regions in the CuO2 plane having loop
currents that break the lattice translational symmetry with larger regions in
the CuO2 plane where the lattice translation symmetry is preserved. One can
then build a real space picture of 30x30 unit cells paved by anapoles
representing loop current states (represented in Fig. 6.a), with a bubble of
loop currents breaking the lattice translational symmetry (”2x2” phase)
connecting larger domains of the uniform loop currents phase as it was
theoretically proposed [43]. The Fig. 6.b shows a realistic real-space example
of such a 2D pattern where the arrows represent the anapole polar vectors in
each unit cell. The central pattern displays a correlation length of 7-8 units
cells consistent with our measurements. One can then calculate the structure
factor of this pattern in reciprocal space. The Fig. 6.c shows in log-scale
such a structure factor convoluted by the neutron 2D momentum resolution
(FWHM=$\Delta_{H}$=$\Delta_{K}$=0.05 r.l.u. ) corresponding to our
measurements on 4F1. Two main features are found. First, the strongest
magnetic peaks are the ones at the Bragg positions (1,0) and (0,1) of the
structure corresponding to the observed q=0 IUC magnetism [5, 6, 7, 9, 10,
11]. Due to the supercell size of 60 units cells of Fig. 6.a, a set of
satellites magnetic peaks located at $\pm{1\over{60}}$ from the atomic Bragg
peak positions should show up in the structure factor. However, they are
blurred by the instrumental momentum resolution to give rise to an elongated
shape in the longitudinal direction. Second, a diffuse scattering is observed
at both (0.5,0) and (0,0.5) corresponding to the q=1/2 magnetism. The size of
the ”2x2” bubble over the total size of the pattern determines the ratio of
intensities between both features. Finally, our polarized neutron diffraction
data qualitatively support the existence of a complex magnetic pattern.
However, the accuracy of the data does not allow one to pinpoint the exact
magnetic pattern and a multitude of other real space patterns can be
considered to give rise to an intensity distribution in momentum space similar
to Fig. 6.c. The picture discussed in Fig. 6.b is just an attempt to give a
coherent picture of all the neutron diffraction measurements done so far.
Interestingly, that picture got very recently some momentum by the observation
of a topological magnetic texture in the PG state of YBCO by Lorentz
transmition electronic microscope (LTEM) [44]. A vortex-like magnetization
density was reported in the CuO2 sheets, with a magnetic texture of total size
of about 100 nm. Its intensity distribution shows a maximum around 25 nm,
corresponding roughly to 60 unit cells. It is tempting to relate this magnetic
texture to large domains such as the one represented here in Fig. 6.b. More
work is necessary to quantify the exact relation.

To conclude, our polarized neutron scattering measurements on a high quality
single crystal of ${\rm YBa_{2}Cu_{3}O_{6.9}}$ generalize the existence of
magnetic correlations at q=1/2 that break the lattice translational symmetry,
over a wide hole doping range matching the pseudogap region of cuprate
superconductors. Considering the location of the q=1/2 magnetism in momentum
space and the $L$-dependence of its amplitude, decreasing at large $L$, this
magnetism is likely to be orbital in origin. Its amplitude is further reduced
at larger doping near optimal doping compared to underdoped ${\rm
YBa_{2}Cu_{3}O_{6.6}}$ cuprates [20]. It is interesting to notice that the
size of the q=1/2 magnetism does not change significantly with doping with
similar measured correlations lengths within the CuO2 planes. In addition, we
find here a dichotomy of the components of the magnetic response along the
c-axis. The out-of-plane component, naturally associated with loop currents
circulating within the CuO2 plane, respects the c-axis stacking whereas the
in-plane component is doubling the unit cell. This is surprising as this
dichotomy is not observed for the q=0 IUC magnetic response, yielding an
apparent tilt of the magnetic moment [5, 6, 7, 9, 10, 11], and deserves
further investigations.

## References

  * Keimer  _et al._ [2015] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature 518, 179 (2015).
  * Proust and Taillefer [2019] C. Proust and L. Taillefer, The remarkable underlying ground states of cuprate superconductors, Annual Review of Condensed Matter Physics 10, 409 (2019).
  * Varma [2020] C. M. Varma, Colloquium: Linear in temperature resistivity and associated mysteries including high temperature superconductivity, Rev. Mod. Phys. 92, 031001 (2020).
  * O’Mahony  _et al._ [2022] S. M. O’Mahony, W. Ren, W. Chen, Y. X. Chong, X. Liu, H. Eisaki, S. Uchida, M. H. Hamidian, and J. C. S. Davis, On the electron pairing mechanism of copper-oxide high temperature superconductivity, Proceedings of the National Academy of Sciences 119, e2207449119 (2022).
  * 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).
  * Mook  _et al._ [2008] H. A. Mook, Y. Sidis, B. Fauqué, V. Balédent, and P. Bourges, Observation of magnetic order in a superconducting YBa2Cu3O6.6 single crystal using polarized neutron scattering, Phys. Rev. B 78, 020506 (2008).
  * Balédent  _et al._ [2011] V. Balédent, D. Haug, Y. Sidis, V. Hinkov, C. T. Lin, and P. Bourges, Evidence for competing magnetic instabilities in underdoped YBa2Cu3O6+x, Phys. Rev. B 83, 104504 (2011).
  * De Almeida-Didry  _et al._ [2012] S. De Almeida-Didry, Y. Sidis, V. Balédent, F. Giovannelli, I. Monot-Laffez, and P. Bourges, Evidence for intra-unit-cell magnetic order in Bi2Sr2CaCu2O8+δ, Phys. Rev. B 86, 020504(R) (2012).
  * Mangin-Thro  _et al._ [2015] L. Mangin-Thro, Y. Sidis, A. Wildes, and P. Bourges, Intra-unit-cell magnetic correlations near optimal doping in YBa2Cu3O6.85, Nature Communications 6, 7705 (2015).
  * Tang  _et al._ [2018] Y. Tang, L. Mangin-Thro, A. Wildes, M. K. Chan, C. J. Dorow, J. Jeong, Y. Sidis, M. Greven, and P. Bourges, Orientation of the intra-unit-cell magnetic moment in the high-Tc superconductor HgBa2CuO4+δ, Phys. Rev. B 98, 214418 (2018).
  * Bourges  _et al._ [2021] P. Bourges, D. Bounoua, and Y. Sidis, Loop currents in quantum matter, Comptes Rendus Physique 22, 7 (2021).
  * Varma [2006] C. M. Varma, Theory of the pseudogap state of the cuprates, Phys. Rev. B 73, 155113 (2006).
  * Scheurer and Sachdev [2018] M. S. Scheurer and S. Sachdev, Orbital currents in insulating and doped antiferromagnets, Phys. Rev. B 98, 235126 (2018).
  * Sarkar  _et al._ [2019] S. Sarkar, D. Chakraborty, and C. Pépin, Incipient loop-current order in the underdoped cuprate superconductors, Phys. Rev. B 100, 214519 (2019).
  * 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), https://advances.sciencemag.org/content/4/1/eaao5235.full.pdf .
  * Zhu  _et al._ [2021] Z. H. Zhu, J. Zhang, Z. F. Ding, C. Tan, C. S. Chen, Q. Wu, Y. X. Yang, O. O. Bernal, P.-C. Ho, G. D. Morris, A. Koda, A. D. Hillier, S. P. Cottrell, P. J. Baker, P. K. Biswas, J. Qian, X. Yao, D. E. MacLaughlin, and L. Shu, Muon spin relaxation and fluctuating magnetism in the pseudogap phase of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{y}$, Phys. Rev. B 103, 134426 (2021).
  * 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).
  * Balédent  _et al._ [2010] V. Balédent, B. Fauqué, Y. Sidis, N. B. Christensen, S. Pailhès, K. Conder, E. Pomjakushina, J. Mesot, and P. Bourges, Two-dimensional orbital-like magnetic order in the high-temperature La2-xSrxCuO4 superconductor, Phys. Rev. Lett. 105, 027004 (2010).
  * Chatterjee  _et al._ [2017] S. Chatterjee, S. Sachdev, and M. S. Scheurer, Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves, Phys. Rev. Lett. 119, 227002 (2017).
  * 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).
  * He and Varma [2012] Y. He and C. M. Varma, Theory of polarized neutron scattering in the loop-ordered phase of cuprates, Phys. Rev. B 86, 035124 (2012).
  * Fujita  _et al._ [2014] K. Fujita, C. K. Kim, I. Lee, J. Lee, M. H. Hamidian, I. A. Firmo, S. Mukhopadhyay, H. Eisaki, S. Uchida, M. J. Lawler, E.-A. Kim, and J. C. Davis, Simultaneous transitions in cuprate momentum-space topology and electronic symmetry breaking, Science 344, 612 (2014), https://science.sciencemag.org/content/344/6184/612.full.pdf .
  * [23] _see Supplemental Material athttp://link.aps.org/supplemental/... for supporting information_.
  * Tasset [1989] F. Tasset, Zero field neutron polarimetry, Physica B: Condensed Matter 156–157, 627 (1989).
  * Janoscheck  _et al._ [2007] M. Janoscheck, S. Klimko, R. Gähler, B. Roessli, and P. Böni, Spherical neutron polarimetry with MuPAD, Physica B: Condensed Matter 397, 125 (2007).
  * Zimmermann  _et al._ [2003] M. v. Zimmermann, J. R. Schneider, T. Frello, N. H. Andersen, J. Madsen, M. Käll, H. F. Poulsen, R. Liang, P. Dosanjh, and W. N. Hardy, Oxygen-ordering superstructures in underdoped ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ studied by hard x-ray diffraction, Phys. Rev. B 68, 104515 (2003).
  * Xiang  _et al._ [2016] H. Xiang, L. Guo, H. Li, X. Cui, J. Qian, G. Hussain, Y. Liu, X. Yao, Q. Rao, and Z. Zou, Polythermal method of Top-Seeded Solution-Growth for large-sized single crystals of ${\mathrm{REBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7-\delta}$, Scripta Materialia 116, 36 (2016).
  * Li  _et al._ [2008] S. Li, Z. Yamani, H. J. Kang, K. Segawa, Y. Ando, X. Yao, H. A. Mook, and P. Dai, Quantum spin excitations through the metal-to-insulator crossover in $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+y}$, Phys. Rev. B 77, 014523 (2008).
  * Singleton  _et al._ [2010] J. Singleton, C. de la Cruz, R. D. McDonald, S. Li, M. Altarawneh, P. Goddard, I. Franke, D. Rickel, C. H. Mielke, X. Yao, and P. Dai, Magnetic Quantum Oscillations in ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathbf{O}}_{6.61}$ and ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathbf{O}}_{6.69}$ in Fields of Up to 85 T: Patching the Hole in the Roof of the Superconducting Dome, Phys. Rev. Lett. 104, 086403 (2010).
  * Gao  _et al._ [2006] H. Gao, C. Ren, L. Shan, Y. Wang, Y. Zhang, S. Zhao, X. Yao, and H.-H. Wen, Reversible magnetization and critical fluctuations in systematically doped YBa2Cu3O7-δ single crystals, Physical Review B 74, 020505 (2006).
  * Liang  _et al._ [2006] R. Liang, D. A. Bonn, and W. N. Hardy, Evaluation of ${\mathrm{CuO}}_{2}$ plane hole doping in ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ single crystals, Phys. Rev. B 73, 180505 (2006).
  * Raichle  _et al._ [2011] M. Raichle, D. Reznik, D. Lamago, R. Heid, Y. Li, M. Bakr, C. Ulrich, V. Hinkov, K. Hradil, C. T. Lin, and B. Keimer, Highly Anisotropic Anomaly in the Dispersion of the Copper-Oxygen Bond-Bending Phonon in Superconducting ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7}$ from Inelastic Neutron Scattering, Phys. Rev. Lett. 107, 177004 (2011).
  * Dai [2015] P. Dai, Antiferromagnetic order and spin dynamics in iron-based superconductors, Rev. Mod. Phys. 87, 855 (2015).
  * Mook  _et al._ [1996] H. A. Mook, P. Dai, K. Salama, D. Lee, F. Doǧan, G. Aeppli, A. T. Boothroyd, and M. E. Mostoller, Incommensurate One-Dimensional Fluctuations in ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6.93}$, Phys. Rev. Lett. 77, 370 (1996).
  * Kulda  _et al._ [1999] J. Kulda, W. Heil, H. Humblot, F. Tasset, A. Wildes, V. Plakhty, E. Moskvin, P. Burlet, J. Dreyer, and B. Fak, Use of the 3He filter in search of the in-chain spin correlations in YBa2Cu3O6+x, Physica B: Condensed Matter 267-268, 252 (1999).
  * Gavrilov  _et al._ [2001] S. Gavrilov, V. Plakhty, P. Burlet, J. Kulda, and H. Humblot, Investigation of in-chain spin correlations in YBa2Cu3O6+x by neutron polarization analysis, Physica B: Condensed Matter 297, 75 (2001), proceeding of the Third International Workshop on Polarised Neutrons.
  * Bourges and Sidis [2011] P. Bourges and Y. Sidis, Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors, Comptes Rendus Physique 12, 461 (2011).
  * Bourges  _et al._ [2018] P. Bourges, Y. Sidis, and L. Mangin-Thro, Comment on “No evidence for orbital loop currents in charge-ordered YBa2Cu3O6+x from polarized neutron diffraction”, Phys. Rev. B 98, 016501 (2018).
  * Blanco-Canosa  _et al._ [2014] S. Blanco-Canosa, A. Frano, E. Schierle, J. Porras, T. Loew, M. Minola, M. Bluschke, E. Weschke, B. Keimer, and M. Le Tacon, Resonant x-ray scattering study of charge-density wave correlations in ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$, Phys. Rev. B 90, 054513 (2014).
  * Forsyth  _et al._ [1988] J. B. Forsyth, P. J. Brown, and B. M. Wanklyn, Magnetism in cupric oxide, J. Phys. C: Solid State Phys. 21, 2917 (1988).
  * Ain  _et al._ [1992] M. Ain, A. Menelle, B. Wanklyn, and E. Bertaut, Magnetic structure of CuO by neutron diffraction with polarization analysis, J. Phys.: Condens. Matter 4, 5227 (1992).
  * Zheng  _et al._ [2005] X. G. Zheng, C. N. Xu, K. Nishikubo, K. Nishiyama, W. Higemoto, W. J. Moon, E. Tanaka, and E. S. Otabe, Finite-size effect on Néel temperature in antiferromagnetic nanoparticles, Phys. Rev. B 72, 014464 (2005).
  * Varma [2019] C. M. Varma, Pseudogap and Fermi arcs in underdoped cuprates, Phys. Rev. B 99, 224516 (2019).
  * Wang  _et al._ [2023] Z. Wang, K. Pei, L. Yang, C. Yang, G. Chen, X. Zhao, C. Wang, Z. Liu, Y. Li, R. Che, and J. Zhu, Topological spin texture in the pseudogap phase of a high-Tc superconductor, Nature 615, 405 (2023).

Acknowledgments

We thank F. Maignen, C. Meunier and S. Klimko for their valuable technical
assistance and installation of MuPaD on 4F1. We thank the group Synthèse
Propriétés et Modélisation des Matériaux de l’Institut de Chimie Moléculaire
et des Matériaux d’Orsay de l’Université Paris-Saclay for the sample
preparation and characterization. The data obtained on Thales at ILL are
available at https://doi.org/10.5291/ILL-DATA.4-02-600. We acknowledge
supports from the project NirvAna (contract ANR-14-OHRI-0010) of the French
Agence Nationale de la Recherche (ANR) and from the GenLoop project of the
LabEX PALM (contract ANR-10-LABX-0039-PALM).

Corresponding authors: Dalila Bounoua or Philippe Bourges.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Fig.  1:  YBCO69 and YBCO7 samples: (a) Single crystal of YBCO69 and (b) co-
aligned detwinned single crystals of YBCO7 used for the polarized neutron
study. (c) Diffraction pattern recorded at $k_{i}$=2.69 Å-1 on the 3T1
diffractometer at LLB-Orphée for YBCO69 aligned in an arbitrary orientation to
avoid the main Bragg peaks. The green lines represent the expected powder
diffraction from the so-called green phase ${\rm Y_{2}BaCuO_{5}}$. None of the
lines from the green phase are present in the YBCO69 diffractogram where the
two peaks correspond to scattering from tails of (1,0,3) and (0,0,7) Bragg
reflections of YBCO69. (d) Field cooled magnetization versus temperature data
recorded for YBCO69 under an applied field of 10 Oe showing the occurrence of
the superconducting transition around $T_{c}$=91.9K.

Fig.  2:  q=1/2 orbital magnetism in twinned $\rm YBa_{2}Cu_{3}O_{6.9}$
(YBCO69): Scans along (H,0,0) in the spin-flip channel $SF_{X,Y,Z}$. The
magnetic intensity appears as a commensurate peak centered at (0.5,0,0) in
(a-f). Data in (a-c) were measured on the 4F1 TAS at 3, 150, and 280K,
respectively. Data in (d-f) were measured on the Thales TAS at 10, 100, and
300K, respectively. The sample was aligned in the (1,0,0)/(0,0,1) scattering
plane in (a-f). Blue lines are fits by Gaussian functions to the data on top
of a background (in grey). The gray dots correspond the SF background (Bgr)
deduced from XYZ-PA, and then fitted by the gray lines. On both instruments,
the data have been obtained during two different experimental runs where
different sample environments have been used which explains different levels
of background. As for all figures showing raw data, the counting time of the
reported intensity is 5 minutes/point on Thales and 15 minutes/point on 4F1.
Error bars represent one standard deviation.

Fig.  3:  Absence of q=1/2 orbital magnetism in detwinned $\rm
YBa_{2}Cu_{3}O_{7}$: (a-b) Scans in the spin-flip channel $SF_{X,Y,Z}$ along
(a) The (H,0,0) direction and (b) The (0,K,0) direction. The background (Bgr)
is extracted from XYZ-PA fitted to the gray lines. Scan along (c) (H,0,0) and
(d) (0,K,0) of the magnetic intensity, $I_{mag}$, as extracted from XYZ-PA at
3K and 100K. Data in (a-d) were measured on the 4F1 TAS. The sample was
aligned in the (1,0,0)/(0,1,0) scattering plane. Error bars (smaller than the
points size) represent one standard deviation.  Fig.  4: (Caption next page.)
Fig.  4:  $L$-dependence of the q=1/2 orbital magnetism in $\rm
YBa_{2}Cu_{3}O_{6.9}$: (a-b) L-Scans along (0.5,0,L) in the spin-flip channel
$SF_{X,Y,Z}$ collected at (a) 150K and (b) 280K. The magnetic intensity in (a)
appears as a Gaussian signal centered at Q=(0.5,0,0) and Q=(0.5,0,0.5). Blue
lines are fits by Gaussian functions to the data. The background (Bgr) is
extracted from XYZ-PA fitted to the gray lines. (c) (lower part)
$L$-dependence along (0.5,0,L) of total magnetic scattering (left), out-of
plane magnetic contribution $I_{c}$ (center) and in-plane magnetic
contribution $I_{ab}$ (right), as extracted from XYZ-PA. The data were
measured at various temperatures in the range 5-150K and averaged out. In the
left upper panel, the form factor for spins from Cu and O are shown with the
phenomenological form factor, $|f(L)|^{2}$ (black dashed line), found in ref.
[8] for the q=0 magnetism. All form factors are normalized to 1 at $L$=0. The
two other upper panels show the orientation factor discussed in the text for
each magnetic component. In the lower panels, the full lines describing
$I_{c}$ (center) and $I_{ab}$ (right) are obtained from the magnetic
orientation factor multiplied to the form factor $|f(L)|^{2}$ (see text). The
full line in the left panel is a fit of Eq. 3. The light grey curves are guide
to the eye. Data in (a-c) were measured on the 4F1 TAS with the sample aligned
in the (1,0,0)/(0,0,1) scattering plane. (d) Comparison of the scattered
intensity in absolute units (mbarn) versus doping observed for the q=1/2
magnetism with the q=0 signal measured in $\rm YBa_{2}Cu_{3}O_{6+x}$ [5, 6, 7,
9, 38, 20]. The line is a guide to the eye. (e) Summary phase diagram of the
observed onset temperature, Tmag, for the q=0 and q=0.5 magnetism versus
doping [38, 20].  Fig.  5: (Caption next page.) Fig.  5:  Temperature
dependence of the q=1/2 orbital magnetism in $\rm YBa_{2}Cu_{3}O_{6.9}$: (a)
Temperature dependence of the magnetic signal at Q=(0.5,0,0) measured in the
$SF_{X,Y,Z}$ channels and the corresponding sloping background (Bgr) as
extracted from XYZ-PA. The dark and light blue points correspond to two
distinct datasets. (b) Temperature dependence of the background subtracted
magnetic scattering at Q=(0.5,0,0): $SF_{X}$-Bgr from panel (a) for the blue
points and as extracted from XYZ-PA for the purple points. The in-plane
magnetic response corresponding to $I_{ab}$ (shown independently in (c)) is
indicated by the orange area while the out-of-plane magnetic response
corresponding to $I_{c}$ (shown independently in (d)) is indicated by the
green area. The blue and violet symbols correspond to the full magnetic
scattering ($I_{c}$ \+ $I_{ab}$). The black open symbols in (d) correspond to
the temperature dependence of the out-of-plane magnetic response for the q=0
magnetism in $\rm YBa_{2}Cu_{3}O_{6.85}$ [9]. (e-f) Temperature dependence of
the magnetic intensity at Q=(0.5,0,0.5): $SF_{X}$-Bgr of panel for the blue
points and as extracted from XYZ-PA for the purple points. (f) Comparison
between the current work with the temperature dependence of the in-plane
magnetic response (open symbols, right scale) for the q=0 magnetism in $\rm
YBa_{2}Cu_{3}O_{6.85}$ [9]. The q=0 magnetism is shifted by a constant (dashed
grey area) for the sake of comparison with [9]. Data in (a-d) were measured on
the 4F1 TAS and data in (e-f) were obtained on the Thales TAS. The sample was
aligned in the (1,0,0)/(0,0,1) scattering plane. Error bars represent one
standard deviation. Lines are guide to the eye.  Fig.  6: (Caption next page.)
Fig.  6:  Structure factor calculation for the hidden magnetic structure
including q=1/2 and q=0 magnetism in $\rm YBa_{2}Cu_{3}O_{6.9}$: (a) (from
[20]) The spontaneously circulating LC state on a CuO2 unit cell comprising
two loops circulating clockwise (in gray) and anti-clockwise (in purple)
leading to two magnetic moments perpendicular to the plane. The gray and
purple arrows represent magnetic moments along the c axis. The 4 possible
patterns are characterized by horizontal colored arrows, corresponding to 4
distinct anapole polar vectors centered at the Cu-site. (b) Proposition of a
2D magnetic texture with 30x30 unit cells paved by the 4 anapoles (LC states)
shown in panel (a). The 4 different anapole orientations correspond to rows
with different colors. The central cluster reproduces the q=1/2 magnetism with
2x2 LC patterns binding large ferro-anapolar domains corresponding to the IUC
magnetism. (c) Structure factor calculation in log-scale for the LC magnetic
pattern reported in (b), convoluted with the instrumental resolution. The
$(H,K)$ intensity map shows magnetic scatterings at (1,0), (0,1) and (1,1)
associated with the q=0 magnetism and those at (0.5,0) and (0,0.5) related to
the q=1/2 magnetism.

See pages - of Supple-nov23-arxiv.pdf