Loop Currents in Two-leg Ladders Cuprates

New phases with broken discrete Ising symmetries are uncovered in quantum materials with strong electronic correlations or indicative of topological order. The two-leg ladder cuprate \textbf{$Sr_{14-x}Ca_{x}Cu_{24}O_{41}$} hosts a very rich phase diagram where, upon hole doping, the system exhibits a spin liquid state ending to an intriguing ordered magnetic state at larger $Ca$ content, passing through a charge density wave phase. Using polarized neutron diffraction, we discovered the existence of a new kind of short range magnetism in this material for two $Ca$ contents. This magnetism develops exclusively within the two-leg ladders with a diffraction pattern at forbidden Bragg scattering which is the hallmark of loop current-like magnetism breaking both time-reversal and parity symmetries. Our discovery shows local discrete symmetry breaking in one-dimensional spin liquid systems as theoretically predicted. It further suggests that a loop current-like phase could trigger the long range magnetic order reported at larger doping in two-leg ladder cuprates.

symmetries. Our discovery shows local discrete symmetry breaking in onedimensional spin liquid systems as theoretically predicted. It further suggests that a loop current-like phase could trigger the long range magnetic order reported at larger doping in two-leg ladder cuprates.
While most of these states are usually discussed for hole doped quasi-2D transition metal oxides such as cuprates and iridates, the existence of LCs was also addressed in quasi-1D spin ladder cuprates [18,19,20]. Sr 14−x Ca x Cu 24 O 41 is a prototype two-leg ladder system whose hole doping can be tuned by Ca for Sr substitution. It represents a promising candidate for LCs hunting in the context of low-dimensional spin liquids [10,11,12,18,19,20].
In cuprates, the LCs are expected to develop in the CuO 2 plaquettes, the building blocks of the materials. In a 3-band Hubbard model, they originate from the frustration of the electronic hopping and interaction parameters and generate staggered orbital moments within the CuO 2 plaquettes. Once ordered, they are expected to preserve the lattice translational invariance (q=0 magnetism) and break time reversal symmetry. There may exist different LCs patterns with a single CuO 2 plaquette, which can further break other Z 2 symmetries, such as parity and rotation. In the so-called pseudo-gap phase of SC cuprates, there are experimental evidences of a breaking of time, parity and rotation symmetries, provided by polarized neutron diffraction (PND) [1,2,3,4,5,21,22,23], muon spin spectroscopy [24], second harmonic generation [25], torque [26] and optical birefringence measurements [27]. Further, second harmonic generation [28] and PND [6] observations in iridates provide evidence for the universality of the LCs phase in correlated oxides.
Additionally, while in most SC cuprates, the LCs magnetism seem to be at long range, in lightly doped La 2−x Sr x CuO 4 , it remains quasi-2D and at short range [3]. This was ascribed to the segregation of doped holes into bond centered stripes forming arrays of two-leg ladders where LCs could be confined. However, the evidence for such a kind of q=0 magnetism in lowdimension remains untackled up to date and can be investigated in the model quasi-1D system Substitution with Ca 2+ on the Sr 2+ site results in a charge transfer of the holes from the chains to the ladders [29], due to chemical pressure. Owing to strong electronic correlations, hole doping strikingly changes the electronic properties of SCCO-x, and the corresponding phase diagram includes insulating gapped spin liquid states within the ladders (∆ gap ∼ 32meV), shortranged dimer antiferromagnetic orders within the chains, charge density wave in both chains and ladders, magnetic long range order (LRO) -assumed to be antiferromagnetic (AFM) -at large Ca-content, pressure induced superconductivity for x≥10, with a predicted d-wave character in one-band Hubbard model [30], and even pseudogap-like behavior for x ≥ 9 [31,32]. This work addresses the issue of the existence of LCs-like magnetism in the archetypal hole-doped spin-ladders compounds, SCCO-5 and SCCO-8 with ∼ 13 and ∼ 17% hole doping per Cu ion respectively, according to [33].
We report a PND study of SCCO-x single crystals, carried out on the 4F1 triple axis spectrometer (TAS) and the D7 diffractometer (See Methods). These instruments are equipped with distinct neutron polarization set-ups and were operating with two distinct neutron wavelengths, to guarantee the reproducibility of the measurements. SCCO exhibits an aperiodic atomic struc-ture with two, ladders and chains, incommensurate sub-lattices [34] (see Methods). Samples were aligned with the ladder sub-lattice parameters with the (100)/(001) scattering plane so that wavevectors Q of the form (H, 0, L) were accessible. Wavevectors are given in reduced lattice units (2π/a, 2π/b, 2π/c) where a, b, c stand for the lattice parameter of the ladder subsystem (c along the legs, a, along the rungs and b perpendicular to the ladder planes (ac), see All of these results, obtained on the D7 diffractometer, are confirmed on the TAS-4F1 (T = 10K). Fig.2.a shows the measured magnetic scattering, as extracted from XYZ-PA across the inter-ladder direction (H,0,1) in agreement with the results from D7. The Q-dependence of the magnetic intensity exhibits a peculiar structure factor with the absence of scattering for H = 0 and an enhanced intensity at H = 3, along the ladders scattering ridge. To better characterize this SRM, we performed scans across selected positions of the (H,0,1) rod. A L-scan across (3,0,1) position (SF X , 4F1, T = 10K) and a H-scan across (1,0,1) as extracted from the D7 XYZ-PA map (T = 5K), are reported in Fig.2.b-c, respectively. The scans show peaked signals with momentum widths (full width at half maximum : F W HM ), ∆ H = 0.5 r.l.u and ∆ L = 0.13 r.l.u, which are significantly broader than the instrumental resolution. The corresponding correlation lengths along the inter-ladder direction is ξ a = a π .F W HM ∼ 6 ± 2.5Å, equivalent to 1 2 a. ξ a corresponds to the size of one ladder rung (2 × Cu-O bond lengths) plus two interladders spacings (2 × Cu-O bond length) as shown on Fig.2.a. ξ c along the ladder legs is found to be ξ c ∼ 11±3Å, or correspondingly ∼ 3c. The [a,c] in-plane correlation lengths are very short ranged and indicative of the formation of magnetic clusters within the ladders.
Next, we performed a survey of the K-dependence of the magnetic scattering, along the inter-planes direction (Fig.2.d). We collected a SF X scan for a trajectory of the form (3,K,L) with L = 0.8 and 1. The scan at L = 0.8 stands for a nonmagnetic background according to Fig.2.b. Subtracting L = 0.8 from L = 1 data unveils a roughly constant level of magnetic intensity over the measured K-range, in agreement with XYZ-PA data (Fig.2.e). This indicates very weak inter-plane correlations, if any, emphasizing a 2D confinement of the measured mag-5 netism within the ladder planes. The temperature dependence of the magnetic signal ( Fig.2.e), measured at (3,0,1), in the SF X channel (4F1), shows that the magnetic correlations set-in below T mag ∼ 80K.
Lost inter-ladder correlations at lower Ca-doping: SCCO-5. Fig.3.a shows a L-scan in the SF channel across (0,0,1) along the ladder-legs direction (SF X , 4F1, T = 5K), where no magnetic signal is observed for SCCO-8. The scan shows a clear magnetic signal centered at (0,0,1), where nuclear scattering is forbidden. The FWHM of the measured signal gives a correlation length of ξ c ∼ 20 ± 6Å along the ladders or equivalently ∼ 6c, which is enhanced as compared to the SCCO-8 compound. We further performed an XYZ-PA on 4F1 along the inter-ladder direction (H,0,1). The XYZ-PA reported in Fig.3.b reveals a diffuse magnetic scattering along H indicating a vanishing ξ a with only a minimum of the magnetic intensity for H = 1.5. These results highlight an even shorter-ranged magnetism, confined within a single ladder, and the loss of inter-ladder magnetic correlations when lowering the Ca-content.
The XYZ-PA along (3,K,1) reveals a flat magnetic intensity consistent with the absence of correlations along the b-axis (Fig.3.c), the inter-planes direction. The XYZ-PA within the chains subsystem confirms the absence of chains magnetism (Supplementary Materials 2.1), consistent with SCCO-8 data. Fig.3.d shows the temperature dependencies of the magnetic intensity at (3,0,1) and (1,0,1), respectively (4F1). The signal at (3,0,1) was measured in the SF X channel and corrected from a background intensity measured at (3,0,0.8). The magnetic signal at (1,0,1) was tracked as a function of the temperature using unpolarized neutrons. Both datasets give an onset temperature T mag ∼ 50K.
Put together, all these experimental observations allow one to get a rather accurate description of the observed magnetic patterns, especially thanks to the large set of collected magnetic instensites at various Q points: i) The magnetic signal is short-ranged, 2D and exclusively carried by the ladders subsystems with weak inter-ladders correlations. ii) The magnetic scattering 6 appears on wavevectors of the form (H,0,L) with integer and odd H and L values, which are forbidden for the atomic structure due to additional symmetries of the 3D crystal structure [34].
That indicates that the translational invariance of the ladders sub-lattice is preserved with the same magnetic unit cell as the atomic one (q=0 magnetism), as reported for the superconducting cuprates and iridates [1,3,4,5,6], which is usually interpreted in terms of LCs. These first two points concern both Ca contents. iii) In contrast to the SCCO-8 compound where the magnetic intensity exhibits a pronounced maximum at (3,0,1), the SRM remains confined to a single two-leg ladder for SCCO-5, as reported for the (La, Sr) 2 CuO 4 cuprate [3], with only a minimum intensity at H=1.5.
Amplitude of the SRM: The scattering intensity can be converted in absolute units (barn), after a calibration using a reference vanadium sample (Supplementary Materials 2.4) . This leads to a full magnetic scattered intensity of I mag ∼ 28 ± 4 mbarn and I mag ∼ 36 ± 15 mbarn for SCCO-8, on 4F1 and D7 respectively, at (3, 0, 1), where the structure factor is maximum.
Correspondingly, the full magnetic intensity was found to be I mag ∼ 7 ± 2 mbarn for SCCO-5 at the same wavevector. These amplitudes correspond to the scattered magnetic intensity of one (Sr, Ca) 14 Cu 24 O 41 formula unit (f.u), namely, three CuO 2 square unit cells with 4 Cu/f.u ( Fig.4.a). Once normalized to a single Cu site, these amplitudes remain larger than those reported in superconducting cuprates (I mag ∼1-2 mbarn per Cu) [1,3,4]. To date, we report the most intense SRM respecting invariance symmetry (q=0) in cuprates.
Orientation of the magnetic moments: The magnitude and orientation of the measurable SRM magnetic moment m is defined as m 2 = m 2 ac + m 2 b , where m ac and m b denote the ladders in-plane and out-of-plane magnetic moment, respectively (Supplementary Materials 4) and m 2 ac = m 2 a + m 2 c . Both components can be derived from a full XYZ-PA. Supposing that m a = m c , we reproducibly estimate the ratio ( m b mac ) 2 ∼ 1 for both compounds. Consistently, 4F1 and D7 data show that 50% of the magnetic moment lies out of the ladder planes, with a tilt of the out-of plane magnetic moment to an angle Θ = Atan( m b /mac) ∼ 55 • . This is in agreement with previous estimates in SC cuprates where the magnetic moment associated with the LCs magnetism exhibits a similar tilt [1,5,23].
Possible interplay between ladders SRM and long range AFM ordering: Our experiments highlight the systematic onset of SRM within the Cu-O planes of lightly hole-doped spin ladders, with growing correlations upon increasing the hole content ( Fig.4.a). According to magnetic susceptibility and specific heat data, no phase transition occurs in this region of the phase diagram [31]. However at larger Ca doping (x ≥ 9), an AFM LRO phase is reported, but only below 4.2K ( Fig.4.a) whose antiferromagnetic nature has been basically deduced through the cusp in the temperature dependence of the macroscopic susceptibility [35,36,37,38]. Interestingly, the SRM is quite remarkably located, in momentum space, at exactly the same wavevectors as the AFM LRO, suggesting a related origin. As the SRM occurs at higher temperature, it is tempting to propose that the reported SRM could act as a preemptive state of the AFM LRO as the Ca-doping evolution of the correlations suggest.
The origin of AFM-LRO remains under debate as the reported locations of the magnetic Bragg peaks do not correspond to any simple model of antiferromagnetically interacting Cu spins within the ladder legs or the chains. Neutron diffraction data on single crystals indicate that the LRO involves magnetic moments both in the ladders and the chains subsystems [36,37,38] as it gives magnetic scattering at integer H and L of both ladders and chains sublattices. Therefore, complex Cu spin structures, which typically require to consider large super cells with a considerable number of independent spins, has to be invoked to describe the magnetic diffraction patterns. As recognized by the authors of refs. [36,37], the model although reproducing the experimental data gives rise to an unlikely situation where the magnetic interaction between nearest neighbor Cu spins are either ferromagnetic or antiferromagnetic and these interactions are mixed with some periodicity. In ref. [38], two different highly non trivial Proposed Loop Currents (LCs) modeling: In a marked contrast with highly complex magnetic arrangements of Cu spins, we then propose a comprehensive interpretation of our PND measurements in the framework of LCs in two-leg ladders. Following theoretical proposals [7,8,11,12], we calculate two magnetic structure factors corresponding to two distinct LCs patterns that satisfactorily reproduce our data. These two patterns are based on a set of two counter-propagating LCs per Cu site. At variance, other patterns with a set of four LCs (usually referred to as CC − θ I phase) [18,19,20] give rise to different magnetic scattering selection rules that do not satisfy the measured structure factor (Supplementary Materials 3.4), neither does a model of magnetic (spin or orbital) moments on oxygen sites as considered in [1,39,40,41] (Supplementary Materials 3.2).
The first model consists in a CC-Θ II like pattern of LCs [7,8] within the ladders (Supplementary Materials 3.5). One needs to decorate each ladder unit cell ( Fig. 4.a) of ∼ 3 Cu-O plaquettes (each plaquette has an averaged cell parameter of a s = c ∼ a/3 as shown in Fig.1.b) with the two opposite LCs around each Cu atom. Note that only the out-of-plane magnetic component m b perpendicular to the LCs can be considered for modeling [22] (Supplementary Materials 4.3). Further, one considers equal contribution from the 4-fold degenerate domains given by a 90 • rotation of LCs about the Cu-site [8,4]. The Cu magnetic form factor was used to fit the experimental data of SCCO-8 as shown by Fig.4.b. This model reasonably reproduces the data with a magnetic moment amplitude of m LC = 0.05 ± 0.01µ B . Accounting for lost inter-ladders correlations (shown as well in Fig.4.a), the same model nicely reproduces the main features of the SCCO-5 data Fig.4.c. The model gives a comparable m LC = 0.05 ± 0.01µ B estimate for the magnetic moment amplitude. For both samples and although the magnetic cross section is larger than in superconducting cuprates, the LCs magnetic moment is of the same order of magnitude [1,3,4,5] due to more complex magnetic structure factor related to the larger magnetic unit cell (Supplementary Materials 4.2) and because only m b is here considered.
Next, we test the model proposed in refs. [11,12] for the case of two-dimensional spin- The models that we propose capture the most salient observation of the Q-dependence of the magnetic scattered intensity as reported in Fig. 4.b and c. This is the first report of LCs in a system without apical oxygen with important consequences to explain the observed tilt of the q=0 magnetism [23] (see (Supplementary Materials 4.3). Considering either structure factors of LCs yields a good fit to our experimental data even-though the CC − θ II model seems to be more robust against hole doping with no preferred domain orientation. In all cases, and from purely symmetrical considerations, one cannot as well rule out the possibility of magneto-electric quadrupolar origin of the measured magnetism [14,15,17]. Note that both LCs patterns can simply be generated from a single LC orientation and considering the lattice symmetry at variance with the proposed magnetic super-cell with 96 spins to describe LRO AFM phase [38].
Coming back to the interplay with the LRO magnetic phase at larger Ca content, one can speculate that the LCs correlations could induce spin moments at low temperature. This would be consistent with a picture of a fluctuating Néel state (spin liquid state) carrying preemptive LCs orders, by analogy to the LCs order parameter resulting from the intertwining between a topological order and discrete broken symmetries in 2D spin liquids [10,11,12]. ATOMIC STRUCTURE: Sr 14−x Ca x Cu 24 O 41 exhibits an aperiodic atomic structure described in 4D crystallography with chains and ladders sublattices, which are described by the orthorhombic space groups Amma and Fmmm, respectively. Both sublattices are incommensurate along the c-axis with an incomensurability parameter 1 γ = c chains c Ladders = 1.43. Upon Ca-doping, the chains sub-space group changes from Amma to F mmm such that the whole structure is described in Ref. [34] as belonging to Xmmm(00g)ss0 superspace group, where X stands for

Methods
. In principle, Bragg peaks need then be indexed in the 4D superspace as (H, K, L ladders, L chains ). However, as the reported magnetism is basically related to the ladders sub-lattice, we refer through this manuscript the where H is integer. We then performed data reduction adapting the standard procedure [42].
Such scans allowed us to map out a wide Q-region spanning reflections of the form (H,0,1) and satellite reflections (H,0,0.43) in r.l.u of the ladders. SF and NSF data were collected in the three channels X,Y,Z. All data were corrected for the flipping ratio using a quartz sample and the conversion to absolute units is done using a vanadium sample. The corresponding data DOI       Considering a magnetic sample, ithe magnetic scattering cross-section [1], I mag , writes: Φ S corresponds to the neutron flux at the sample in n/s/barns. N cell is the number of unit cells in the sample. r 0 stands for the neutron magnetic scattering length, r 2 0 = 290 mbarn. f (Q) is to the magnetic form factor and |F (Q)| the magnetic structure factor. Owing to the dipolar nature of the interaction between neutron spin and the magnetic moments,m, I mag probes m ⊥ the magnetic components perpendicular to Q, only.
The squared modulus of m ⊥ can be expressed using the regular Cartesian coordinates of the lattice: For a magnetic moment, m(±m a , ±m b , ±m c ) with n non zero component, there are 2 n equivalent magnetic domains. The cross-terms (i = j) cancel out when summing over all domains, at variance with the diagonal terms (i = j). In the case of the present study with Q=(H,0,L), I mag reduces to the sum of in-and out-of plane terms, so that: With It is also convenient to use the so-called {XYZ} referential, where X is the unitary vector parallel to Q. Y and Z are the two unitary vectors orthogonal to X, within the scattering plane and perpendicular to the scattering plane, respectively. So that m ⊥ = m Y Y + m Z Z, with the in-and out-of (scattering) plane components. Figure S1: (a) Layout of the multidetector diffractometer D7 [2]. (b) Definition of the X' and Y' polarization directions within the scattering plane. γ = 41.6 • is the angle between the incident wave vector k i and X', set by the instrument configuration, 2θ is the scattering angle, and α is defined as the angle between the wavevector Q and X'. Reproduced from [3].

Polarized neutron diffraction setup
On the incoming neutron beam, a bender (polarizing super-mirror) can polarize the neutron spin and a Mezei flipper can adiabatically flip the neutron spin. A pyrolitic graphite filter is further added before the bender to remove high harmonics on 4F1. The neutron spin polarization is maintained using a homogeneous guide field of a few Gauss.The neutron spin polarization direction, P is controlled on the sample by Helmholtz coils on 4F1 and a quadrupolar assembly on D7 [2]. On the scattered neutron beam, the final neutron spin polarization is analyzed using either an analyzer made of co-aligned Heusler single crystals (on 4F1) or polarizing benders (on D7) placed in front of the multidetector bank ( Fig. S1.a)

Polarized neutron diffraction cross sections
Polarization analysis of PND allows us to distinguish between the different contributions to the scattered intensity [1]. For a nuclear scattering, the neutron spin remains unchanged and the scattered intensity is measured in the Non Spin-Flip (NSF) channel. Since the spin polarization of the neutron beam is not perfect, a small amount of the nuclear scattering can nevertheless leak into the Spin-Flip (SF) channel. The ratio between scattered intensities in the N SF and the SF channels is called the flipping ratio F R and characterizes the quality of spin polarization of the neutron beam. For a magnetic scattering, the scattered intensities in each channel strongly depends on P.
Indeed, neutron spins are described using Pauli matrices, whose quantization axis is given by P. The magnetic intensity I N SF mag (P) ∝ (m ⊥ . P) 2 does not flip the neutron spin and remains in the NSF channel. The remaining magnetic I SF mag (P) ∝ (m ⊥ ) 2 − (m ⊥ . P) 2 appears in the SF channel.
On D7, the direction of polarization X is not parallel to Q but along a direction X turned by an angle α (Fig. S1.b). Therefore, one needs to estimate the scattered intensities in directions of P corresponding to unitary vectors (X , Y , Z ), as shown in Fig. S1.b, given by: Then, the full magnetic scattering, I mag , given in Eq. 3, splits in two terms, I SF mag (P) and I N SF In addition to the polarization magnetic cross-sections of Eq. 5, I mag (P), one should consider the nuclear intensity, I nucl and a background, Bg, in both SF and NSF channels. Both SF and NSF cross-sections read: As discussed in [4], due to imperfect polarizations, the measured neutron intensities are mixing the cross-sections of Eq. 6. In each channel, it can be actually written, where F R(P) is the polarization dependent flipping ratio of the experiment. For both instruments, 4F1 and D7, F R(P) were measured for all relevant scattering angle using a quartz sample. One can deduce the PND crosssections, Eq. 6, from the measured ones, Eq. 7 [4]. In the present study, although the nuclear and magnetic scattering preserve the lattice translation symmetry, the short range magnetism (SRM) occurs at Q values where there is additional extinction of the nuclear peaks due to the 3D atomic structure. Therefore, the effects of imperfect polarizations of the nuclear term and background terms of Eq. 7 are relatively weak (although sizeable) and readily corrected. It should be emphasized that this is a very different situation from the case of most superconducting cuprates [3,5] where the q=0 magnetism occurs at the same Bragg position of the nuclear structure.

XYZ-Polarization analysis
We systematically performed a longitudinal XYZ polarization analysis (XYZ-PA), that allows the determination of the full magnetic intensity I mag = I ac + I b ( Eq. 3) from a set of measurements in the SF channel with P along each of the 3 unitary vectors (X , Y , Z ). The combination of all those measurements provide access to the in-(// ac) and out-of-(// b) scattering plane magnetic scattering and the SF background. Two different situations occur for both instruments: • On D7, Diffractometer: α = 90 − θ + 41.6 • (Fig. S1), with θ the Bragg angle that depends on Q and the neutron wave length. From X', Y' and Z' measurements, the full measurable magnetic intensity is deduced from Eq. 5 as follows : For I mag , the second term is weighted by g(α) = [1 + sin 2 α]/[cos 2 α − sin 2 α], which goes to unity for α=0.
• On 4F1, Triple axis spectrometer: α = 0, one obtains the usual relations: 2 PND study : complementary information  Fig. S2.c. All scans reported in Fig. S2. emphasize the absence a new magnetic signal equivalent to that observed in the ladder subsystem.

L-dependence of the short range magnetism (SRM)
Fig. S3.a shows the SF intensity measured along (3,0,L) for P//X at 10 K on sample SCCO − 8. On the Lscan, the magnetic signal at L=3, exhibits a Gaussian line-shape and appears on top of a sloping background. The determination of such a sloping background has been confirmed by XYZ-PA. A qualitatively similar type of signal is observed for sample SCCO − 5 (Fig. S3.b).

T-dependence of the short range magnetism
• SCCO − 8: the T-dependence of the scattered intensity at (3,0,1) was measured in the SF X channel on 4F1 (Fig.S4.a). It displays a net enhancement at low temperature. According to the L-scan across (3,0,1) performed at low temperature (Fig.2.b in main Text), the short ranged magnetic signal centered at L=1 vanishes at L=0.8 and 1.2. Two additional T-dependencies were measured at those L values and averaged to provide the T-dependence of the non magnetic background. The comparison both T-dependencies indicates the magnetic signal starts developing below an onset temperature T mag 80 K.
• SCCO − 5: The raw temperature dependencies of the magnetic scattering at (1,0,1) and (3,0,1), were measured on 4F1 (Fig.S4.b-c). Using an unpolarized neutron beam, the scattered intensity at (1,0,1) exhibits a linear increase on cooling down to a temperature where an extra enhancement of the intensity becomes visible. Using a polarized neutron beam, the signal at (3,0,1) was measured in SF X and compared to the scattered intensity at (3,0,0.8), a background position according to measurements in SCCO − 8. The comparison between polarized and unpolarized neutron data highlights an onset temperature T mag 50 K below which the short ranged magnetism sets in. XYZ-PA has been also employed at ∼ 100K at (0,0,1), showing the vanishing of the SRM at high temperature.

Calibration in absolute units
We converted the measured intensities in absolute units using a vanadium sample, which allows the determination of the neutron flux at the sample position Φ S . A vanadium sample is a pure incoherent scatterer. For PND measurements, 2/3 of its intensity shows up in the SF channel (1/3 in the NSF channel) and its energy (ω) integrated intensity reads: Where ( dσ dΩ ) inc = 0.394 barns stands for the vanadium incoherent cross section. For a vanadium sample mass, m V = 1 g, and a molar mass, M V = 50.94 g.mol −1 , one obtains N Cell = 0.6023. m V M V = 0.0118 cells/mol.
The incoherent scattering for vanadium is purely elastic and is described by a Dirac distribution in energy, δ(ω). The measured ω-dependence is obtained after convolution by the Gaussian instrumental energy resolution, characterized by a full width at half maximum (FWHM) ∆ ω : the measured intensity I meas (ω) acquires a Gaussian profile as, Integrating over energy, one obtains: I SF V ana = Imax∆ω 2 π ln (2) .
Taking into account the instrument energy resolution : ∆ ω = 1.25 meV for k i = k f =2.57Å −1 gives Φ S = 1678 n/s/barns. This value of Φ S holds for both experiments on SCCO − 5 and SCCO − 8 on the spectrometer 4F1. A similar procedure using a vanadium standard sample was used for data calibration on D7 [4].

In-plane and out-of-plane magnetic scattering amplitudes
The in-plane I ac and out-of-plane I b magnetic intensities for SCCO − 8, as extracted from XYZ-PA on D7 data using Eq. 8, are shown in Fig.S5. From these maps, one sees that both magnetic components, in-plane and out-ofplane of the ladder a-c plane, are sizeable. For results obtained on both instruments, Table.S1 gives a summary of the measured magnetic intensities (as extracted from XYZ-PA) at different reciprocal space positions in SCCO−5 and SCCO − 8 and the corresponding I ac and I b intensities in absolute units. The data were also systematically corrected by the quartz flipping ratios following the procedure described in [4] for D7 and the procedure given above for 4F1. Note that, on 4F1, the measured Q-dependencies were systematically measured for negative and positive H values, and symmetrized by averaging the values of the magnetic intensity. The results in Table.S1 show that both in-plane I ac and out-of-plane I b magnetic components are not zero, leading systematically to a magnetic moment which is not pointing along a high symmetry direction, but typically is making a tilt with the direction perpendicular to the CuO 2 as it is observed in all superconducting cuprates [3,5].

Magnetic patterns and related structure factors
To describe the observe magnetic intensities, one needs to calculate the momentum dependence of the magnetic cross-sections (Eq. 1) for given magnetic patterns. That Eq. 1 contains essentially two terms depending on Q: the magnetic form factor, f (Q), which depends on the nature of the magnetic moments and the magnetic structure factor F (Q) which is the Fourier transform of the given magnetic pattern. We discuss in this section several magnetic models and calculate F (Q) for each of them. Note that as a ∼ 3a s , the size of ladder unit cell is approximately the size of 3 square plaquettes, although the ladder unit cell contains 4 inequivalent Cu atoms. Indeed, as shown in Fig. S6.a, the Cu 4 O 6 unit cell is made of a first Cu 2 O 3 ladder with two Cu sites on a rung at coordinates of (0,0) and (1,0). The second ladder is obtained by a translation of these coordinates by (3/2,1/2). This gives 4 distinct Cu sites distributed on 2 ladders. Each Cu site is at the center of a CuO 2 plaquette, with O sites at (±1/2,0) and (0, ±1/2) around the Cu site. Note that there is only 6 distinct oxygen sites, since the two CuO 2 plaquettes on the same rung share one oxygen along the rung.
Using the CuO 2 plaquette as a building block, one can decorate it with various magnetic patterns (Fig. S6.b-f). The magnetic dipoles can be related to a spin moment on Cu sites, spin or orbital moments on O sites, or orbital moments produced by loop currents (LC) between Cu and O sites or O sites only.

Antiferromagnetic Cu spins
We consider a ladder where Cu spins are coupled antiferromagnetically (Fig.S7.a). This model comprises antiferromagnetic interactions along the ladder legs and rungs (due to superexchange interaction across the 180 • oxygen  bridge between Cu ions). The existence of an antiferromagnetic order at long range is questionable. Indeed such a spin arrangement within ladders generates a frustration of the interladder magnetic interaction, due to the 90 • oxygen bridges between neighboring ladders ( Fig.S7.a). As a consequence, one considers a set of independent antiferromagnetic ladders to compute the squared magnetic structure factor present in Eq. 1: The two terms describe the antiferromagnetic coupling between 2 Cu spins along the leg and along the rung, respectively. This model breaks the lattice translation symmetry and should give a net magnetic contribution at half integer values of H and L which we did not observe during our experiment in SCCO − 8 (Fig.S7.b). It further rules out any magnetic scattering for integer H or L values which is at odds with our experimental measurements, where the magnetic scattering was observed at Q-positions of the form (H,0,1) with integer H. One can therefore eliminate a conventional Cu spin antiferromagnetism as the origin of the observed magnetic scattering.

Magnetic moments on oxygen sites
Next, we consider a magnetic nematic model [5,10,11,12] where O sites within a CuO 2 plaquette carry magnetic moments (spin or orbital) pointing in opposite directions when oxygens are distributed either along a or c directions with respect to the Cu site (Fig. S8). Once the ladders are decorated with such a magnetic nematic patterns, one observes 3 spin on O sites coupled ferromagnetically along a. They are coupled antiferromagnetically with 3 other spins translated by (1/2,1/2). This gives a squared magnetic structure factor, as follows: The last term accounts for the ferromagnetic lines with 3 spins and the first term gives the Q-space relationship between neighboring lines with opposite spin directions. Such a magnetic pattern gives an extinction at (3,0,1) ( Fig.S8.b). The model with magnetic moments on O sites (Fig.S8) then fails to account for the observation of a magnetic signal at (3,0,1).

Loop Current Phases
We now discuss different magnetic patterns based on three distinct loop current models, shown in Fig. S6.d-f, all preserving the lattice translation symmetry.

Single LC pattern
Each CuO 2 square plaquette can be decorated by a single loop current (LC) pattern as shown in Fig. S6.d-f. For the three LC models, there is 4 different possibilities of putting the LC pattern around a given Cu site by making π 2 rotations. One can then write down the structure factor of a single LC pattern F (Q) ≡ A φ (Q) with φ={0, π 2 , π, 3π 2 }, where φ denotes the angle of rotation for each pattern. We consider three distinct LC states: • CC − θ I (Fig. S6.d): Theoretical works on copper oxide ladders predicted the appearance of a CC − θ I phase with LCs in hole-doped SCCO [6,7,13]. On each CuO 2 plaquette, there are 4 LCs which generate staggered orbital moment at positions: (±x 0 , ±x 0 ) with respect to a Cu site. x 0 ∼ 0.146 is the coordinate of the triangle center of mass, where the orbital moment is assumed to be. This state is twofold degenerate with: A φ± π 2 (Q) = −A φ (Q). The magnetic structure factor is independent of φ as: |A φ (Q)| 2 for CC − θ I is shown on Fig. S9.a (green line).
full blue line represents an averaged of both domains with equal population.
• CC − θ III (Fig. S6.f): this state corresponds to the LC pattern proposed to describe an ancillary phase associated within a spin liquid (mother) state [9,15], that for convenience we labeled here CC − θ III . At variance with the CC − θ II state, the two LCs are rotated by π 4 and circulate between O sites only ( Fig. S6.f). The two LCs produce 2 staggered orbital moments located with respect to the Cu site at positions: ±(x 0 , 0) for the Horizontal pattern or ±(0, x 0 ) for the Vertical pattern, with x 0 = 0.5 − 2.x 0 = 0.208 (x 0 is again the center of mass of the LC triangle). For a ladder, the physics along the rung (Horizontal) and along the leg (Vertical) can be different, so that the fourfold degeneracy of the CC − θ III state reduces to twofold only, with A φ+π (Q) = −A φ (Q). For both configurations, the magnetic structure factor is given by : |A φ (Q)| 2 are shown on Fig. S9.a for both configurations (dashed red lines).

LCs correlations
The next step is to establish how to correlate these patterns over the different sites of the ladder unit cell (Fig. S6.a). As discussed before, the Cu 2 O 3 ladder contains 2 CuO 2 plaquette (with a diamond shape) which share an oxygen on a rung. For SCCO, the ladder unit cell contains 2 Cu 2 O 3 ladders, with a translation from one to the other given by (3/2,1/2). Therefore, two types of LC correlations should be considered: i) the intra-rung correlations with the site shifted by (1,0) ii) the inter-ladder correlations with the site translated by (3/2,1/2). Both types of correlations contribute to the structure factor.
• Intra-rung correlations: For the intra-rung, one considers two patterns on each of both Cu sites of a rung, A φ (Q) and A φ (Q). Defining ϕ = φ − φ and depending on the correlations, the pattern shifted by (1,0) is either identical (ϕ = 0 and A φ (Q)= A φ (Q)) or opposite (ϕ = π and A φ (Q)= −A φ (Q)). In general, the intra-rung structure factor, B φφ (Q), can be written as: |B φφ (Q)| 2 is shown on Fig. S9.b for both correlations. Interestingly, this term gives zero structure factor at (3,0,1) for the opposite patterns (ϕ = π) at variance with the experimental results. This implies that both patterns are identical (ϕ = 0) and the intra-rung structure factor can be always simplified as: • inter-ladder correlations: The case of the inter-ladder is obtained in the same way defining two coupled LCs, B φ (Q) and B φ (Q), shifted by (3/2,1/2). Again, defining ψ = φ − φ and depending on the correlations, the pattern shifted by (3/2,1/2) is either identical (ψ = 0 and B φ (Q)= B φ (Q)) or opposite (ψ = π and B φ (Q)= -B φ (Q)) as for the intra-rung correlations. However, it is also possible that the LC shifted by (3/2,1/2) is aligned along a different diagonal than the one of the first ladder, ψ = ± π 2 and B φ± π 2 (Q) = B φ (Q). The inter-ladder structure factor can then be written as C φφ (Q) as: |C φφ (Q)| 2 is shown on Fig. S9.c for both correlations ψ = 0 and ψ = π. For in-phase ladders, one obtains the selection rule: H + L = 2n. The out-of-phase case (ψ = π) is ruled out by the experiments as it gives zero structure factor at (3,0,1).
Finally, the LCs magnetic structure factor F (Q), present in Eq. 1, corresponds to (i) A φ (Q) for an independent single pattern, (ii) B φ (Q) for an independent ladder and (iii) C φφ (Q) for coupled ladders. It is worth to recall that the case (ii) corresponds to the results of the sample SCCO-5 and (iii) to SCCO-8, respectively .

CC − θ I like phase of LCs
The CC − Θ I intra-rung pattern can be taken in-phase (ϕ = 0) as requested but the inter-ladder patterns are necessary out-of-phase as shown in Fig. S10.a as they share a current link along the diagonal of the inter-ladder small square. This gives the following |F (Q)| 2 : According to the previous section, the third term (in brackets) corresponds to the CC −Θ I pattern, the second term accounts for the ordering in-phase within the ladder and the first term the out-of-phase coupling between ladders. Such a structure factor gives rise to magnetic scattering extinction rules that do not account for our experimental observations ( Fig.S10.b). For instance, it prohibits scattering when H and L are both odd, at variance with our observed magnetic scatterings at (1,0,1) and (3,0,1). Even in the case of SCCO-5 (independent ladder), it does not correspond to the results as |F (Q)| 2 = 0 for (0,0,1) where the magnetic signal is observed.

CC − θ II like phases of LCs
• Uncorrelated ladders: SCCO − 5 We first discussed the case of the independent ladders using the CC − θ II pattern ( Fig. S11.a). Within the ladder unit cell, the first ladder is decorated with in-phase pattern, whereas no LCs occur for the second ladder. |F (Q)| 2 reduces to a product of the in-phase intra-rung term times the magnetic pattern of Eq. 15: Here, we assumed four possible domains with equal population. This structure factor (shown in Fig. S11.d) reproduces the SRM along the (H,0,1) line (see Fig. 4.c of the manuscript), while the magnetic correlations along a (perpendicular of the ladders) are confined within a single ladder.
• Correlated ladders: SCCO − 8 The magnetic structure factor is now given by Eq. 20. There are three different ways to couple the first and second ladders within the SCCO unit cell, corresponding to different phase shift ψ. The ladders couple in-phase (ψ=0) (Fig. S11.b), out-of-phase (ψ = π) or exhibit a crisscrossed coupling (ψ = ± π 2 ) (Fig. S11.c). For the last case where |B φ (Q)| = |B φ (Q)| in Eq. 20, two different situations are possible to orient the magnetic patterns shifted by (3/2,1/2) denoted (δ = +1) and (δ = −1). One can conveniently describe all the different situations by introducing a local toroidal moment for a given CuO 2 plaquette i : Ω i = j m j × r j , with m j a magnetic moment and and r j its position with respect to the Cu site at the center of the CuO 2 plaquette [16]. Ω i is pointing along the diagonal separating the clockwise and anticlockwise LCs. One can defineΩ = i Ω i the effective toroidal moment for the full SCCO unit cell. For the in-phase ladders (ψ=0) ,Ω remains along the same diagonal, whereas this vector is null for out-of-phase ladders. Interestingly, for the two crisscrossed cases,Ω points either along a rung, i.e the direction a (δ = +1) or along a leg, i.e along the direction c (δ = −1). The related squared structure Figure S11: CC − Θ II state [8]: 2 LCs per CuO 2 flowing clockwise (blue triangles) and anticlockwise (red even if the difference of intensities between H=3 and H=1 is less pronounced in that case. Such a crisscrossed structure indicate that the effective toroidal moment should be along the ladder. Note that in bilayer cuprates Y Ba 2 Cu 3 O 6+x , the effective toroidal moment is found parallel to the underlying CuO chains [16]. With A(Q) = 2 sin(2πx 0 H 3 ) Horizontal pattern or A(Q) = 2 sin(2πx 0 L) for the Vertical one. Along (H,0,1) direction, the Vertical-CC − θ III pattern accounts for the SRM observed in sample SCCO-5, with a typical modulation given by the term |2cos(π H 3 )| 2 . This at variance, with the Horizontal-CC − θ III pattern for which the scattering intensity cancels at H=0 which cannot explain the measured data.
• Correlated ladders: SCCO − 8 Fig. S12.b,e show the case where the 2 ladders are decorated with the same CC − θ III pattern (ψ = 0). This introduces in the squared structure factor an extra term |2 cos( π 2 (H + L))| 2 with respect to the uncorrelated case: The main features of our experimental results are: i) the absence of scattering at (0,0,1), ii) a scattering at odd H and L, iii) a stronger scattering at H=3 than at H=1. All these features are reproduced by both the Horizontal and the Vertical CC − θ III pattern.
4 Magnetic moment amplitudes as extracted from data modeling

Magnetic form factor
We have considered various magnetic pattern within the CuO 2 plaquette involving spin on the Cu site, or spin/orbital moment on the oxygen sites and orbital moments originating from LCs. Another factor present in Eq. 1 is the magnetic form factor. Two different form factors can be considered here, either the isotropic magnetic Cu-form factor or the oxygen-one (which can be estimated from ref. [17]). For LC states, electron are delocalized between several Cu and O sites or O sites only, but the exact form factor associated with the induced orbital moment remains unknown. Previous PND measurements in 2D cuprates [18] suggested that both magnetic Cu-and O-form factor could be used. Fig.S13.a shows the Q-dependencies of f (Q) 2 for copper and oxygen. The Q-range of interest for our study is indicated by a shaded area, where f 2 O (Q) varies of 47 % against 21% for f 2 Cu (Q). In principle, since electrons are likely to be more delocalized for LCs, the magnetic O-form factor could be best suited to describe a fast decay of LCs magnetic signal as compared to the magnetic Cu-form factor. However, fitting the Q-dependence of the magnetic signal with either form factors gives good agreement with our data and the extracted magnetic moment amplitudes are, although different, of the same order of magnitude when considering the O or Cu form factors, as will be shown in the next section, Tab.S2.

Extracted magnetic moment amplitude
• Out-of-plane magnetic moment: m b As discussed above, among the different magnetic patterns of Fig. S6.b-f, some of the LCs-like phases can described our experimental data. In principle, being confined in the (a,c) plane, classical LCs produce an orbital moment, m b ≡ m LC , pointing perpendicular to the LC plane. That corresponds to the magnetic intensity, I b , that we have extracted from XYZ-PA (Tab. S1). Using I b ∝ m 2 b , one can deduce the out-of-plane magnetic moment, m b ≡ m LC . Fig. S13.d-e show I b calibrated in mbarns fitted by different models. The evolution of the magnetic intensity along (H,0,1) is rather different for SCCO-5 (Fig. S13.d) and SCCO-8 (Fig. S13.e). No correlation develops between the ladders in the former case and the magnetic scattering remains diffusive. For the latter case, magnetic correlations develop between the ladders. The scattered intensity I b can be described as: • In-plane magnetic moment: m ac One can perform the same analysis for the full measured intensities I mag although the classical LCs phase cannot account in principle for the in-plane magnetic intensities reported Table.S1. I mag is proportional to Fig S14.b-c show the fit of the PND data for samples SCCO − 5 and SCCO − 8 using the same functions as in Fig. S13.d-e. One sees that the same LCs structure factor properly account for the full measured magnetic intenties. The resulting magnetic moments m ⊥ are listed in Tab. S2.
Combining magnetic moments along the ladder et along the rung, m ac varies with Q due to the orientation factors. For a H-scan around (3,0,1), the variations of these orientation factors, weighting m 2 a and m 2 c are given in Fig S14. Unfortunately, the set of collected data for both samples remains insufficient to determine independently m a and m c . As a consequence, one further constrains the fits by enforcing m a = m c . This simple assumption eliminates the Q-dependence of m ac . From these results, for both samples, one finds m = (m a , m b , m a ) with typically |m| 0.09µ B : that value depends on the specific LCs pattern considered via |A(Q)| 2 . That gives as well a tilt angle of 55 • of the magnetic moment with respect to the b axis.

Origin of the planar component
In the original model proposed by C.M. Varma, LCs are confined within the CuO 2 planes. This should generate only orbital magnetic moments perpendicular to the ladder planes, which is at variance with the experimental observation where an extra in-plane magnetic scattering is reported. Then, it was suggested that the ground state could not be solely made of one of the four orthogonal CC − θ II states, but could rather emerge by their quantum superposition [19,20]. Within that framework, the degree of quantum admixture shows up in PND measurements in the form of a extra magnetic scattering that looks like that originating from an effective magnetic planar component. In Cu 2 O 3 ladders, LCs settle in at lower temperature where quantum effects might be larger than thermal fluctuations. This makes the proposal of quantum effect at the origin of the planar magnetic scattering an interesting scenario. Alternatively, it was proposed in superconducting cuprates that the planar component arises from LCs running over the CuO 6 octahedron [3,21,22,23]. Indeed, the cuprates, where the q=0 magnetism was observed in monolayer and bilayer materials, are all containing CuO 6 octahedron with an apical oxygen site. In the CuO 2 layers, the Cu site is located at the center of either a CuO 6 octahedron or a CuO 5 pyramid. It was therefore proposed that LCs could delocalize on opposite edges of O-octahedra or O-pyramids, yielding a magnetic planar component. Whatever is the relevance of such a proposal for superconducting cuprates, it cannot hold for two-led ladder cuprates, since there is no apical oxygen above the CuO 2 plaquette. For that kind of materials, LCs have to remain confined with the Cu 2 O 3 ladders. Figure S15: SCCO − 5: series of scans performed on 4F1 in the N SF X channel at 8 K (red symbols) and 150 K (blue symbols). (a) scans along (3,1,L) crossing both L=1.2 and L=1.14 positions in r.l.u of the ladders, corresponding to q CDW reported for ladders and chains, respectively. (c) K-scans across (3,K,1.14) in r.l.u of the ladders, corresponding to q CDW reported for chains. The increase of the intensity in both large L or K is due scattering of Al from the sample holder. (b,d) Differential intensities 8 K-150 K, from scans reported (a,c) that show the absence of any CDW -induced structural distortion.

Absence of charge density wave-like instability
In the pure compound SCCO, both ladders (ld) and chains (ch) display charge density waves (CDW), as reported by RXD and Neutron Diffraction studies [24,25]. For CDW ld and CDW ch , the incommensurate propagation wavevector q CDW is given in super-space notations by (H, K, L, L ) with: -L = L integer ± 0.2 r.l.u within the ladders -L = L integer ± 0.2 r.l.u, within the chains Both CDW are characterized by a similar onset [26] that we reproduced in Fig. 4 of the main manuscript. For SCCO − 5 and SCCO − 8 compositions, optical conductivity and resonant X-ray diffraction measurements [24,27,28] report an onset of charge ordering within the ladders T CDW at ∼90K for SCCO − 5 and 10K for SCCO − 8.
In a neutron diffraction experiment, the charge order is detected owing the lattice distortion it induces. To detect the hallmark of a CDW, one has to look in the NSF channel in a PND study. During our PND experiments, we actually did not detect the hallmark of any CDW instability.

Search for charge density wave in the ladders
According to literature [26], T CDW should be at ∼90 K for the SCCO − 5 sample. We performed two L-scans below and above that temperature. Fig. S15.a shows two scans along (3, 1, L, 0), measured on 4F1 in the N SF X channel at T=8K and 150K. The difference between the two sets of measurements (Fig.S15.b) exhibits a featureless flat L-dependence only, pointing out the absence of any extra signal at L=1.2. Thus, the signal associated with a CDW ld (if any) falls below the threshold of detection of our measurement.

Search for charge density wave in the chains
Fig. S15.c shows the same measurement within the chain subsystem, where one expect CDW scattering at L =0.8, corresponding to L=1.14 using the ladder lattice parameter. Additionaly, we collect a K-scan across (3, 1, 0, 0.8), corresponding to (3, 1, 1.14) in ladder notations. The differential intensity between 8K and 150K ( Fig. S15.d) does not reveal any signature of CDW ch .
The absence of evidence for a CDW instability could also originate from the hole redistribution between the chains and the ladders upon Ca-doping, which leads to a change of the chains and ladders nuclear space group symmetry. These space group symmetry changes could affect q CDW .
In addition, it is worth noticing that T CDW decreases with increasing the Ca content, while the ordering temperature T mag of the new short ranged magnetic signal keeps growing. This may highlight an interesting competition between the q = 0 magnetism and the CDW , which could affect the development of the magnetic correlations.