Dalila Bounoua
email: dalila.bounoua@cea.fr

Lucile Mangin-Thro

Jaehong Jeong

Romuald Saint-Martin

Loreynne Pinsard-Gaudart

Yvan Sidis

Philippe Bourges
email: philippe.bourges@cea.fr

Loop Currents in Two-leg Ladder Cuprates

New phases with broken discrete Ising symmetries are uncovered in quantum materials with strong electronic correlations. The two-leg ladder cuprate 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. Using polarized neutron diffraction, we report here the existence of short range magnetism in this material for two Ca contents, whose origin cannot be ascribed to Cu spins. 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 a one dimensional spin liquid system as theoretically predicted. It further suggests that a loop current-like phase could trigger the long 1

Introduction

In the recent years, the study of doped Mott insulators, such as superconducting (SC) cuprates [START_REF] Fauqué | Magnetic order in the pseudogap phase of high-t c superconductors[END_REF][START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor hgba 2 cuo 4+δ[END_REF][START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-tc copper oxides superconductors[END_REF][START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in yba 2 cu 3 o 6.85[END_REF] or iridates [START_REF] Jeong | Time-reversal symmetry breaking hidden order in sr 2 (ir, rh)o 4[END_REF], raised the question of the existence of other kinds of magnetism.

Beyond conventional spin magnetism, a new form of magnetism may originate from magnetoelectric loop currents (LCs) [START_REF] Simon | Detection and implications of a time-reversal breaking state in underdoped cuprates[END_REF][START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF][START_REF] Agterberg | Emergent loop current order from pair density wave superconductivity[END_REF][START_REF] Chatterjee | Insulators and metals with topological order and discrete symmetry breaking[END_REF][START_REF] Chatterjee | Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF][START_REF] Sarkar | Incipient loop current order in the under-doped cuprate superconductors[END_REF] or Dirac multipoles [START_REF] Lovesey | Neutron diffraction and the electronic properties of bafe 2 se 3[END_REF][START_REF] Lovesey | Neutron scattering by dirac multipoles[END_REF][START_REF] Spaldin | The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect[END_REF][START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF]. 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 [START_REF] Chudzinski | Orbital current patterns in doped two-leg cu-o hubbard ladders[END_REF][START_REF] Chudzinski | Spin rotational symmetry breaking by orbital current patterns in two-leg ladders[END_REF][START_REF] Nishimoto | Current-current correlations in the threeband model for two-leg cuo ladders: Density-matrix renormalization group study[END_REF]. 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. This study is motivated by the recent theoretical proposal that spin liquids and topological order could be dressed with ancillary phases, such as LCs, that highlight their intrinsic nature [START_REF] Chatterjee | Insulators and metals with topological order and discrete symmetry breaking[END_REF][START_REF] Chatterjee | Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF]. It represents a promising candidate for LCs hunting in the context of low-dimensional spin liquids [START_REF] Chudzinski | Orbital current patterns in doped two-leg cu-o hubbard ladders[END_REF][START_REF] Chudzinski | Spin rotational symmetry breaking by orbital current patterns in two-leg ladders[END_REF][START_REF] Nishimoto | Current-current correlations in the threeband model for two-leg cuo ladders: Density-matrix renormalization group study[END_REF].

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-reversal, parity and rotation symmetries, provided by polarized neutron diffraction (PND) [START_REF] Fauqué | Magnetic order in the pseudogap phase of high-t c superconductors[END_REF][START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor hgba 2 cuo 4+δ[END_REF][START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-tc copper oxides superconductors[END_REF][START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in yba 2 cu 3 o 6.85[END_REF][START_REF] Almeida-Didry | Evidence for intra-unit-cell magnetic order in bi 2 sr 2 cacu 2 o 8+δ[END_REF][START_REF] Mangin-Thro | a-b anisotropy of the intra-unit-cell magnetic order in yba 2 cu 3 o 6.6[END_REF][START_REF] Tang | Orientation of the intra-unit-cell magnetic moment in the high-t c superconductor hgba 2 cuo 4+δ[END_REF], muon spin spectroscopy [START_REF] Zhang | Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of yba 2 cu 3 o[END_REF], second harmonic generation [START_REF] Zhao | A global inversion-symmetry-broken phase inside the pseudogap region of yba 2 cu 3 o[END_REF], torque [START_REF] Sato | Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in yba 2 cu 3 o[END_REF] and optical birefringence measurements [START_REF] Lubashevsky | Optical birefringence and dichroism of cuprate superconductors in the thz regime[END_REF]. Further, second harmonic generation [START_REF] Zhao | Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate[END_REF] and PND [START_REF] Jeong | Time-reversal symmetry breaking hidden order in sr 2 (ir, rh)o 4[END_REF] observations in iridates provide evidence for the universality of the LCs phase in correlated electron systems such as layered oxides.

In most SC cuprates, the LC state fully develops in the CuO 2 planes, yielding a 3D long range order. In lightly doped La 2-x Sr x CuO 4 , known to host a spontaneous charge segragation in the form of quasi-1D charge stripes, the LC order is frustrated. As a result, the LC magnetism remains quasi-2D and at very short range [START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF], suggesting a possible confinement of LCs within bond centrered stripes, taking the form of two-leg ladders. However, the evidence for such a kind of LC-like magnetism in low-dimension remains untackled up to date and can be investigated in the model quasi-1D system (Sr, Ca) 14 Cu 24 O 41 . Sr 14-x Ca x Cu 24 O 41 , hereafter SCCO-x, crystallizes with an aperiodic nuclear structure consisting of an alternating stack of 1D CuO 2 chains and quasi-1D Cu 2 O 3 two-leg ladder layers. It realizes an intrinsically hole-doped compound with an effective charge of 2.25 + per Cu ion where, in the pure compound (x = 0), holes are located within the chains subsystem. Substitution with Ca 2+ on the Sr 2+ site results in a charge transfer of the holes from the chains to the ladders [START_REF] Gotoh | Structural modulation, hole distribution, and hole-ordered structure of the incommensurate composite crystal (sr 2 cu 2 o 3 ) 0.70 cuo 2[END_REF], 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 phases within the ladders (∆ gap ∼ 32meV), short-range 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 [START_REF] Faye | d-wave superconductivity in coupled ladders[END_REF], and even pseudogap-like behavior for x ≥ 9 [START_REF] Vuletić | The spin-ladder and spin-chain system (la, y, sr, ca) 14 cu 24 o 41 : Electronic phases, charge and spin dynamics[END_REF][START_REF] Uehara | Superconductivity in the ladder material sr 0.4 ca 13.6 cu 24 o 41.84[END_REF]. 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 [START_REF] Osafune | Optical study of the sr 14-x ca x cu 24 o 41 system: Evidence for hole-doped cu 2 o 3 ladders[END_REF].

Using polarized neutron diffraction, we here report a short range magnetism in SCCO-x below ∼50K and 80 K for two Ca contents. This magnetism is associated exclusively with the two-leg ladder layers as the diffracted magnetic intensity occurs at forbidden Bragg positions of the ladder sub-system. Its origin cannot be related to Cu spins but rather may lie in the magnetoelectric loop currents, as the ones previously reported in superconducting 2D cuprates [START_REF] Fauqué | Magnetic order in the pseudogap phase of high-t c superconductors[END_REF][START_REF] Tang | Orientation of the intra-unit-cell magnetic moment in the high-t c superconductor hgba 2 cuo 4+δ[END_REF].

The calculated magnetic structure factors, that satisfactorily reproduce our data, correspond to two distinct loop current patterns proposed theoretically [START_REF] Simon | Detection and implications of a time-reversal breaking state in underdoped cuprates[END_REF][START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF][START_REF] Chatterjee | Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF]. We further suggest that the long range magnetic order reported below 5K at larger doping in two-leg ladder cuprates could be induced by the observed short range magnetism.

Results

Atomic structure Sr 14-x Ca x Cu 24 O 41 exhibits an aperiodic atomic structure (shown in Fig. 1.a-b) with two, chains and ladders, incommensurate sub-lattices [START_REF] Deng | Structural evolution of one-dimensional spin-ladder compounds sr 14-x ca x cu 24 o 41 with ca doping and related evidence of hole redistribution[END_REF], which are described by the orthorhombic space groups Amma and Fmmm, respectively. The crystal lattice is incommensurate along the c-axis, with an incommensurability parameter between the chains and the ladders, 1 γ = c Ladders c chains ≈ 1.43. Upon Ca-doping(x>3), the chains sub-space group changes from Amma to F mmm such that the whole structure is described using 4D crystallography in Ref. [START_REF] Deng | Structural evolution of one-dimensional spin-ladder compounds sr 14-x ca x cu 24 o 41 with ca doping and related evidence of hole redistribution[END_REF] as belonging to Xmmm(00g)ss0 superspace group, where X stands for nonstandard centering (0, 0, 0, 0), (0, 1 2 ,

1 2 , 1 2 ), ( 1 2 , 1 2 , 0, 0) and ( 1 2 , 0, 1 2 , 1 2
). In principle, Bragg peaks need then be indexed in the 4D superspace indexes 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 to the Bragg positions as (H,K,L), in units of: Short range magnetism in SCCO-8 shows that the magnetic correlations set-in below T mag ∼ 80K.

A new magnetic signal is clearly observed at (H, 0, L) with integer values of H and L, corresponding to the ladder subsystem. This raises the question of its possible existence (or fingerprint) also in the chains subsystem. In order to answer that question, one needs to look at Bragg positions (H, 0, 0, 1) using superspace notations, corresponding to (H, 0, 1.43) in the ladder subsystem units. In Fig. 1.c, no magnetic intensity is sizeable at these positions. Furthermore, Fig 3 .a, reporting the full scattered magnetic intensity as deduced from XYZ-PA (4F1) at two Bragg positions, shows no magnetism. That reveals the absence of any chains magnetic response at commensurate chains Q-positions. A scan across the (1, 0, L) direction in SCCO -8 was also performed in order to crosscheck the absence of signal at positions of the form (H, 0, 0) (common to both ladders and chains), as shown in Fig. 3.b. All scans reported in Fig. 3, emphasize that the SRM detected within the Cu 2 O 3 ladders is not transferred to the underlying CuO 2 chain subsystem.

However, a correlated magnetic signal appears on top of satellite reflections mixing both ladders and chains subsystems, along the (H, 0, 0.43) line, with H-even (blue arrows Fig. 1.c), in r.l.u of the ladders, corresponding to the (H,0,-1,1) line in the superspace notations. That unexpected result does not necessarily mean that the chain subsystem carries magnetic moments which would be inconsistent with the absence of a magnetic signal on the chains Bragg positions. Instead, that observation suggests that the magnetic moments of the ladder could actually be magnetically coupled via the chain susbsytem, leading to a non-zero magnetic structure factor at the satellite. That interpretation is inline with the inter-ladder correlations of the magnetic signal.

Lost inter-ladder correlations at lower Ca-substitution: SCCO-5

Fig. 4.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 (after deconvolution from the instrumental resolution) a correlation length of ξ c ∼ 20 ± 6 Å 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. 4.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-range magnetism, confined within a single ladder, and the loss of inter-ladder magnetic correlations when lowering the Ca-content. Two features, found in SCCO-8, have been similarly observed. First, the XYZ-PA along (3,K,1) reveals no maximum intensity at K = 0. In light of the results for x=8 sample (Fig. 2.d), this is consistent with the absence of correlations along the b-axis (Fig. 4.c), the inter-planes direction. Second, the XYZ-PA within the chains subsystem confirms the absence of chains magnetism (Supplementary Note 4). However, in contrast to SCCO-8, no magnetic intensity is observed at the satellite position corresponding to the (H,0,-1,1) line in the superspace notations, confirming the assumption that the CuO chain could bridge inter-ladder coupling in SCCO-8.

Fig. 4.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-range, 2D and exclusively carried by the ladder subsystems with weak inter-ladders correlations. ii) The magnetic scattering 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 [START_REF] Deng | Structural evolution of one-dimensional spin-ladder compounds sr 14-x ca x cu 24 o 41 with ca doping and related evidence of hole redistribution[END_REF].

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 [START_REF] Fauqué | Magnetic order in the pseudogap phase of high-t c superconductors[END_REF][START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-tc copper oxides superconductors[END_REF][START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in yba 2 cu 3 o 6.85[END_REF][START_REF] Jeong | Time-reversal symmetry breaking hidden order in sr 2 (ir, rh)o 4[END_REF], which is usually interpreted in terms of LCs. These first two points concern both Ca contents. 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 [START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF], with only a minimum intensity at H=1.5. Similarly, the observation of magnetic intensity at the satellite position (H,0,-1,1) differs noticeably between both Ca-concentrations inline with the loss of the inter-ladder correlations at low Ca-substitution.

Amplitude of the SRM

The scattering intensity can be converted in absolute units (barn), after a calibration using a reference vanadium sample (Supplementary Note 5). 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 (insets of Fig. 5.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) [START_REF] Fauqué | Magnetic order in the pseudogap phase of high-t c superconductors[END_REF][START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-tc copper oxides superconductors[END_REF]. To date, we report the largest intensity of the 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 Note 6) and m 2 ac = m 2 a + m 2 c . Both components, m ac and m b , can be derived from a full XYZ-PA but not the values m a and m c which are interdependent.

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 [START_REF] Fauqué | Magnetic order in the pseudogap phase of high-t c superconductors[END_REF][START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in yba 2 cu 3 o 6.85[END_REF][START_REF] Tang | Orientation of the intra-unit-cell magnetic moment in the high-t c superconductor hgba 2 cuo 4+δ[END_REF].

Discussion

Phase diagram: 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. 5.a). We note as well the absence of a structural distorsion associated with a charge density wave instability in our samples (see Supplementary Note 7). According to magnetic susceptibility and specific heat data, no phase transition occurs in this region of the phase diagram [START_REF] Vuletić | The spin-ladder and spin-chain system (la, y, sr, ca) 14 cu 24 o 41 : Electronic phases, charge and spin dynamics[END_REF]. However at larger Ca doping (x ≥ 9), an AFM LRO phase is reported, but only below 4.2K (Fig. 5.a) whose antiferromagnetic nature has been basically deduced through the cusp in the temperature dependence of the macroscopic susceptibility [START_REF] Isobe | Antiferromagnetic transition in sr 14-x ca x cu 24 o 41 (12.5< x <13. 6) observed by magnetic measurements[END_REF][START_REF] Nagata | Antiferromagnetic ordering in the spin singlet state of the ladder/chain material: Sr 2.5 ca 11.5 cu 24 o 41[END_REF][START_REF] Nagata | Antiferromagnetic ordering in the spin ladder compound; sr 14-x ca x cu 24 o 41[END_REF][START_REF] Deng | Coexistence of long-range magnetic ordering and singlet ground state in the spin-ladder superconductor srca 13 cu 24 o 41[END_REF]. Interestingly, the diffraction patterns of the SRM and the AFM order are located at the same wave vector in momentum space, 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 suggests.

Meanwhile, the instrinsic nature of the 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 [START_REF] Nagata | Antiferromagnetic ordering in the spin singlet state of the ladder/chain material: Sr 2.5 ca 11.5 cu 24 o 41[END_REF][START_REF] Nagata | Antiferromagnetic ordering in the spin ladder compound; sr 14-x ca x cu 24 o 41[END_REF][START_REF] Deng | Coexistence of long-range magnetic ordering and singlet ground state in the spin-ladder superconductor srca 13 cu 24 o 41[END_REF] 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 Nagata et al [START_REF] Nagata | Antiferromagnetic ordering in the spin singlet state of the ladder/chain material: Sr 2.5 ca 11.5 cu 24 o 41[END_REF][START_REF] Nagata | Antiferromagnetic ordering in the spin ladder compound; sr 14-x ca x cu 24 o 41[END_REF], 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. Deng et al [START_REF] Deng | Coexistence of long-range magnetic ordering and singlet ground state in the spin-ladder superconductor srca 13 cu 24 o 41[END_REF] are reporting two different highly non trivial magnetic structures with 96 independent spins in a large super cell (corresponding to 24 SCCO unit cells) to accommodate the measured structure factors of 36 magnetic Bragg peaks. Obviously, such a model fails to produce the SRM of SCCO-5 and 8 as the reported correlation lengths are much shorter than the super-cell size necessary to describe the complex spin structure. Additionally, the orientation of the magnetic moments in the AFM LRO phase, assumed to be related to spin moments, is found to be strictly in-plane (in the [a,c] plane) according to neutron diffraction data [START_REF] Nagata | Antiferromagnetic ordering in the spin singlet state of the ladder/chain material: Sr 2.5 ca 11.5 cu 24 o 41[END_REF][START_REF] Nagata | Antiferromagnetic ordering in the spin ladder compound; sr 14-x ca x cu 24 o 41[END_REF]. That contrasts with the observed SRM and suggests that both phases are distinct although related.

Modelisation: We next propose various magnetic models which could account for the observed magnetic intensities as shown in Figs.5.b-c. One then needs to calculate the magnetic structure factor expected for given magnetic patterns. This is discussed in details in the Supplementary Note 8 for various magnetic configurations; we here recall the main conclusions. We consider various decorations of the ladder unit cell as shown in Fig. 6. Some models have been proposed theoretically, others are simplified magnetic decorations of the unit cell.

First, one considers a model of periodic antiferromagnetic Cu spins on a square lattice (Fig. 6.a), that can decorate an isolated ladder. In principle, this model can be directly applied to SCCO-5 sample where one observes a magnetism confined to a single ladder. However, it completely fails to reproduce our experimental data as it breaks the translation symmetry of the lattice and would lead to SRM with scattering only at half integer H and L values at odds with our observation. For the same reason, any model built of such an AFM coupled ladder does not describe the data whatever the coupling between ladders. Neither does a model of magnetic (spin or orbital) moments on oxygen sites as considered in [START_REF] Fauqué | Magnetic order in the pseudogap phase of high-t c superconductors[END_REF][START_REF] Sun | Topological insulators and nematic phases from spontaneous symmetry breaking in 2d fermi systems with a quadratic band crossing[END_REF][START_REF] Fischer | Mean-field analysis of intra-unit-cell order in the emery model of the cuo 2 plane[END_REF][START_REF] Moskvin | Pseudogap phase in cuprates: oxygen orbital moments instead of circulating currents[END_REF] (Fig. 6.b). That model does respect the translation of the lattice but leads to an extinction of the structure factor at (3,0,1) where the observed intensity is maximum. Therefore, one is left to find out alternative models to give an explanation for the SRM and a possible link with the AFM LRO.

Loop Currents (LCs) modeling: In a marked contrast with highly complex magnetic arrangements of Cu spins used to describe the LRO AFM state, we propose a comprehensive interpretation of our PND measurements in the framework of LCs in two-leg ladders. Following theoretical proposals [START_REF] Simon | Detection and implications of a time-reversal breaking state in underdoped cuprates[END_REF][START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF][START_REF] Chatterjee | Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF], 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 (Fig. 6.c usually referred to as CCθ I phase) [START_REF] Chudzinski | Orbital current patterns in doped two-leg cu-o hubbard ladders[END_REF][START_REF] Chudzinski | Spin rotational symmetry breaking by orbital current patterns in two-leg ladders[END_REF][START_REF] Nishimoto | Current-current correlations in the threeband model for two-leg cuo ladders: Density-matrix renormalization group study[END_REF] give rise to different magnetic scattering selection rules that do not satisfy the measured structure factor with extinctions at Bragg positions where the observed intensities are maximum.

CC-Θ II model: The first model consists in a CC-Θ II like pattern of LCs (Fig. 6.d) [START_REF] Simon | Detection and implications of a time-reversal breaking state in underdoped cuprates[END_REF][START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF] within the ladders. One needs to decorate each ladder unit cell (insets of Fig. with the two opposite LCs around each Cu atom. Note that only the out-of-plane magnetic component m b perpendicular to the LCs, ≡ m LC , can be considered for modeling [START_REF] Mangin-Thro | a-b anisotropy of the intra-unit-cell magnetic order in yba 2 cu 3 o 6.6[END_REF]. Further, one considers equal contribution from the 4-fold degenerate domains given by a 90 • rotation of LCs about the Cu-site [START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-tc copper oxides superconductors[END_REF]. The Cu magnetic form factor was used to fit the experimental data shown in Fig. 5.b-c. Note that the data could be as well described by using an oxygen form factor (see Supplementary Note 9).

First, the model of isolated ladders with the same pattern around each Cu atom (inset A of 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 [START_REF] Fauqué | Magnetic order in the pseudogap phase of high-t c superconductors[END_REF][START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-tc copper oxides superconductors[END_REF][START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in yba 2 cu 3 o 6.85[END_REF] due to more complex magnetic structure factor with interferences related to the larger magnetic unit cell and, as well, because only m b is here considered.

CC-Θ III model: Next, we test the model proposed in refs. [START_REF] Chatterjee | Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF] for the case of twodimensional spin-liquids, here labeled CC -Θ III constructed from patterns shown in Fig. 6.e.

As for CC-Θ II , there are two counter-propagating LCs, but the currents now flow only between O-sites (insets of Fig. 5.a). Typically, the LCs pattern is rotated by 45 • with respect to CC-Θ II . Similarly, two orientational domains rotated by 90 • about Cu atoms, with a two-fold degeneracy each, could be considered for this phase. However, only one orientation is found to give agreement with the experimental data along the (H,0,1) line of uncorrelated SCCO- Interestingly, for isolated ladders, only one orientational domain of CC -Θ III can account for the experimental data in contrast with the CCθ II model which has no preferred domain orientation. In terms of the observed magnetic structure factors, magnetic dipoles (as Cu spins or moments at the oxygen sites) fail to describe our data. This is a clear indication that higher multipoles, ordered at short range, are necessary. It can correspond to anapoles (such as the LCs phases) or magnetic quadrupoles as both are intimately connected as they occur at the same level of the multipole expansion [START_REF] Di Matteo | Orbital currents, anapoles, and magnetic quadrupoles in cuo[END_REF]. Both types of order parameters break both parity and time-reversal symmetries and would, in principle, cause the measured magnetism.

For instance, and from purely symmetrical considerations, it is likely that models built from magneto-electric quadrupoles, as the ones proposed in 2D cuprates [START_REF] Lovesey | Neutron diffraction and the electronic properties of bafe 2 se 3[END_REF][START_REF] Lovesey | Neutron scattering by dirac multipoles[END_REF][START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF], could describe the measured magnetism if proper couplings between adjacent quadrupoles in the ladder unit cell are considered. However, at present, it is unclear what type of microscopic couplings between quadrupoles would correlate adjacent ladders. We here stressed that LCs phases offer more documented microscopic models where magnetic structure factors can be readily computed.

Our report is inline with the observation of loop currents in 2D superconducting cuprates [START_REF] Fauqué | Magnetic order in the pseudogap phase of high-t c superconductors[END_REF][START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor hgba 2 cuo 4+δ[END_REF][START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-tc copper oxides superconductors[END_REF][START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in yba 2 cu 3 o 6.85[END_REF]. Particularly, it shows that LCs phases occur as well in 1D spin liquids systems as it has been theoretically anticipated [START_REF] Chatterjee | Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF][START_REF] Chudzinski | Orbital current patterns in doped two-leg cu-o hubbard ladders[END_REF][START_REF] Chudzinski | Spin rotational symmetry breaking by orbital current patterns in two-leg ladders[END_REF][START_REF] Nishimoto | Current-current correlations in the threeband model for two-leg cuo ladders: Density-matrix renormalization group study[END_REF]. Consistently, it is interesting to notice that short-range orbital-like magnetic order has been as well reported in La 2-x Sr x CuO 4 once the doped charges are confined in two-leg ladders. [START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF], bridging the occurence of loop currents in various copper oxides. The quasi-1D structure of Cu 2 O 3 ladder enforces geometrical constraints that allow us to be more specific about the LCs models because the ladder unit-cell is anisotropic and it exhibits a different atomic structure. For instance, although there is an oxygen atom above Cu atoms on the ladders [START_REF] Isobe | Structural modulation and charge distribution in the spin-ladder ca 13.6 sr 0.4 cu 24+y o 41+z[END_REF] at about the same distance, typically ∼ 2.7 Å, as the apical oxygen in hole doped mono-or bi-layer cuprates where copper is located within a CuO 6 octahedron or CuO 5 pyramid, the atomic structure is aperiodic and then that oxygen does not primarily belong to the same atomic sub-system than the Cu atom on the ladder. Therefore, one can consider that LCs are established only in the planar CuO 2 plane, that is the first time LCs correlations are observed without apical oxygen. That point is particularly the case in the SCCO -5 sample of decorrelated ladders where no magnetic intensity occurs at the satellite position, (0,0,-1,1), mixing both sublattices. The occurrence of LCs in a system without specific apical oxygen has stringent consequences to explain the observed tilt of the q=0 magnetism [START_REF] Tang | Orientation of the intra-unit-cell magnetic moment in the high-t c superconductor hgba 2 cuo 4+δ[END_REF] (see Supplementary Note 11).

Both, CC-Θ II and CC-Θ III , LCs models that we propose capture the most salient observation of the Q-dependence of the magnetic scattered intensity as reported in Fig. 5.b and c whereas other models based on magnetic moments on Cu or oxygen sites fail to account for the experimental results. 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 [START_REF] Deng | Coexistence of long-range magnetic ordering and singlet ground state in the spin-ladder superconductor srca 13 cu 24 o 41[END_REF]. In our previous work on LSCO, the appearence of the LC-like magnetism also coincides with a net anomaly in the spin dynamics, suggesting an interplay between AFM spin correlation and LC-like electronic instability [START_REF] Balédent | Two-dimensional orbital-like magnetic order in the high-temperature la 2-x sr x cuo 4 superconductor[END_REF]. A similar interplay could be present in two-leg ladders, so that LC-like phase triggers at low temperature the AFM order at large Ca content. 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 [START_REF] Chatterjee | Insulators and metals with topological order and discrete symmetry breaking[END_REF][START_REF] Chatterjee | Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF]. manuscript with further contributions from all authors. All authors contributed to this work, read the manuscript and agree to its contents. Such scans allowed the mapping of 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. Typical instrumental resolutions (full width at half maximum: FWHM) are 0.07 r.l.u. and 0.03 r.l.u. along the H and L directions, respectively. 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.

On 4F1, L and H-scans of the form (H,0,1) or (1,0,L) were done across reciprocal positions corresponding to the ladder scattering ridge i.e. with integer L and H values. Typical instrumental resolutions (FWHM) are 0.1 r.l.u. and 0.04 r.l.u. along the H and L directions, respectively.

Both SF and NSF scans were done in order to crosscheck the absence of nuclear scattering at magnetic positions. The PA across (H,0,1) was performed for positive and negative H-values and the corresponding intensities were then symmetrized. Our experiments also include scans along K. Such scans were performed by tilting the sample out of plane using goniometers. [START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF] showing two possible patterns breaking rotational symmetry along the diagonals. (e) LC state CCθ III [START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF] showing Horizontal and Vertical patterns. For all models (b-e), the same pattern is assumed around each Cu atom on the square lattice of a single ladder.

SUPPLEMENTARY NOTE 1: POLARIZED NEUTRON DIFFRACTION (PND) AND XYZ-POLARIZATION ANALYSIS (XYZ-PA)

Neutron diffraction

The PND experiments were carried out on two instruments: the 4F1 triple axis spectrometer (Orphée reactor, Laboratoire Léon Brillouin, Saclay) and the D7 multidetector diffractometer (Institut Laue Langevin, Grenoble). On each instrument, measurements were carried out with a neutron wavevector of 2.57 Å-1 and 2.02 Å-1 , respectively. Sr 14-x Ca x Cu 24 O 41 (SCCO-x) exhibits an aperiodic atomic structure with two, ladders (ld) and chains (ch), incommensurate sub-lattices. Therefore, Bragg peaks need to be indexed in the 4D superspace as (H, K, L ld , L ch ). Both sublattices are incommensurate along the c-axis with an incomensurability parameter 1 γ = c ch c ld = 1.42. In the experiments, we used the ladder sub-lattice parameters as a reference: H, K and L coordinates are expressed in reduced lattice units (r.l.u) of the ladder (ld) subsystem: 2π a = 0.55 Å-1 , 2π b = 0.48 Å-1 and 2π c ld = 1.61 Å-1 , respectively. The samples were always aligned in the (H,0,0)/(0,0,L) scattering plane, in order to probe the [a,c] plane of the ladders. Within that scattering plane, a transferred momentum Q of the form (H,0,L) is accessible. Within these units, the Bragg peaks associated with the chains subsystem contribute at L chains = 1.42L as 2π c ch = 2.28 Å-1 . Additionally, tilting the sample out of the scattering plane using goniometers allowed us to access wavevectors of the form Q(H,K,L) on 4F1.

Considering a magnetic sample, the magnetic scattering cross-section [S9], I mag ,reads:

I mag = Φ S N cell r 2 0 f (Q) 2 |F (Q)| 2 m 2 ⊥ ( 1
)
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:

m 2 ⊥ = m 2 -(m.Q) 2 = i,j=a,b,c (δ ij - Q i Q j Q 2 ) m i m j (2)
For a magnetic moment, m(±m a , ±m b , ±m c ) with n non zero components, 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:

I mag = I ac + I b (3)
With

I ac ∝ m 2 ac = [(1 -| H Q | 2 )m 2 a + (1 -| L Q | 2 )m 2 c ] and I b ∝ m 2 b (because K = 0).
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.

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 [S1]. 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 (Supplementary Figure S1.a)

Polarized neutron diffraction cross sections

Polarization analysis of PND allows us to distinguish between the different contributions to the scattered intensity [S9]. 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 (Supplementary Figure S1.b). Therefore, one needs to estimate the scattered intensities in directions of P corresponding to unitary vectors (X , Y , Z ), as shown in Supplementary Figure S1.b, given by:

  X Y Z   =   cosα sinα 0 -sinα cosα 0 0 0 1     X Y Z   (4)
Then, the full magnetic scattering, I mag , given in Eq. 3, splits in two terms, I SF mag (P) and I N SF mag (P)

               I SF mag (X ) = I ac cos 2 α + I b I N SF mag (X ) = I ac sin 2 α I SF mag (Y ) = I ac sin 2 α + I b I N SF mag (Y ) = I ac cos 2 α I SF mag (Z ) = I ac I N SF mag (Z ) = I b (5)
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:

    
I N SF (P) ∼ Bg N SF + I nucl + I N SF mag (P)

I SF (P) ∼ Bg SF + I SF mag (P) (6)
As discussed in [S10], due to imperfect polarizations, the measured neutron intensities are mixing the cross-sections of Eq. 6.

In each channel, it can be actually written,

     I N SF meas (P) ∝ Bg N SF + I nucl + I N SF mag (P) + 1 F R(P) [Bg SF + I SF mag (P)] I SF meas (P) ∝ Bg SF + I SF mag (P) + 1 F R(P) [Bg N SF + I nucl + I N SF mag (P)] (7)
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 cross-sections, Eq. 6, from the measured ones, Eq. 7 [S10]. 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 [S2, S3] 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 provides 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 = 90θ + 41.6 • (Supplementary Figure 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 :

               I ac = [I SF (X ) -I SF (Y )]/[cos 2 α -sin 2 α] I b = [I SF (X ) -I SF (Z )] + sin 2 α I ac I mag = [I SF (X ) -I SF (Z )] + g(α)[I SF (X ) -I SF (Y )] (8)
For I mag , the second term is weighted by g(α) = [1 + sin 2 α]/[cos 2 αsin 2 α], which goes to unity for α=0.

• On 4F1, Triple axis spectrometer: α = 0, one obtains the usual relations:

               I ac = I SF (X) -I SF (Y) I b = I SF (X) -I SF (Z) I mag = 2 I SF (X) -I SF (Y) -I SF (Z) (9)
SUPPLEMENTARY NOTE 2: L-DEPENDENCE OF THE SHORT RANGE MAGNETISM (SRM)

Supplementary Figure S2.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 (Supplementary Figure S2.b).

SUPPLEMENTARY NOTE 3: TEMPERATURE 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 (Supplementary Figure S3.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 the main text), the SRM signal centered at L=1 vanishes at L=0.8 and 1.2. Two additional Tdependencies were measured at those L values and averaged to provide the T-dependence of the non magnetic background. The comparison of both T-dependencies indicates that 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 (Supplementary Figure S3.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 SRM sets in. XYZ-PA has been also employed at ∼ 100K at (0,0,1), showing the vanishing of the SRM at high temperature.

I(0, 0, S1. Summary of the measured magnetic intensities for SCCO -5 and SCCO -8 on D7 and 4F 1.

I b 4 ± 1.7 1.8 ± 1.7 - 5.4 ± 1.4 - I ac 7.4 ± 1.7 5.3 ± 1.7 - 1.9 ± 1.4 - Supplementary Table

SUPPLEMENTARY NOTE 7: ABSENCE OF CHARGE DENSITY WAVE-LIKE INSTABILITY

A charge density wave has been reported in the two-leg ladders system, SCCO-x, from conductivity measurements that established a continuous phase diagram of the doped ladders [S11]. However, a contradictory result has been reported as resonant X-ray diffraction concluded that CDW order dominates at two particular x=0 and x=11 [S12], and is supposed to melt for other x, including the present x = 5 and x=8.

In the pure compound SCCO, both ladders (ld) and chains (ch) display charge density waves (CDW), as reported by RXD and Neutron Diffraction studies [S13, S14]. 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 [S11]. For SCCO -5 and SCCO -8 compositions, optical conductivity measurement [S15, S16] 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 [S11], T CDW should be at ∼90 K for the SCCO -5 sample. We performed two L-scans below and above that temperature. Supplementary Figure S6.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 (Supplementary Figure S6.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 Supplementary Figure S6.c shows the same measurement within the chain subsystem, where one expects CDW scattering at L =0.8, corresponding to L=1.14 using the ladder lattice parameter. Additionally, 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 (Supplementary Figure S6.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.

SUPPLEMENTARY NOTE 8: MAGNETIC PATTERNS AND RELATED STRUCTURE FACTORS

To describe the observed magnetic intensities, one needs to calculate the momentum dependence of the magnetic cross-sections (Eq. 1) for given magnetic patterns. 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.

Supplementary Figure S7.a shows the ladder unit cell in the a-c plane. The building block is given by a CuO 2 square plaquette with a characteristic length of the square lattice, a s = c. Hereafter, all positions in real space are given in units of a s . 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 Supplementary Figure S7.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 are 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 (Supplementary Figure S7.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 (Supplementary Figure S8.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 (Supplementary Figure S8.a). In this limit, one would actually expect the individual ladders to be in a non-classical state of singlets on each rung, forming spin dimers [S17]. This corresponds to a non magnetic ground state, without any magnetic static fingerprint observable. Here, our purpose is to look for the orign of a short range magnetism located at bragg positions. We therefore simply ask the question whether or not the Cu spins on a single ladder may lead to magnetic signal observed in neutron diffraction. Thus is at least valid for the isolated ladders limit of SCCO-5. As a consequence, one considers a set of independent antiferromagnetic ladders to compute the squared magnetic structure factor present in Eq. 1:

|F (Q)| 2 = |4 sin(πL) sin(π H 3 )| 2 (12)
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 (Supplementary Figure S8.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 [S3, S18-S20] where O sites within a CuO 2 plaquette carry magnetic moments (spin or orbital) pointing in opposite directions for oxygen atoms located either along a or c directions with respect to the Cu site (Supplementary Figure S9). Once the ladders are decorated with such a magnetic nematic patterns, one observes 3 spins 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:

|F (Q)| 2 = |2 sin( π 2 ( H 3 + L)) (1 + 2 cos(2π H 3 ))| 2 (13)
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) (Supplementary Figure S9.b). The model with magnetic moments on O sites (Supplementary Figure S9) 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 Supplementary Figure S7.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 Supplementary Figure S7.d-f. For the three LC models, there are 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. 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 r j its position with respect to the Cu site at the center of the CuO 2 plaquette [S21-S23]. Taking each of the four φ values, Ω i is pointing along the diagonal separating the clockwise and anticlockwise LCs. We consider three distinct LC states:

• CCθ I (Supplementary Figure S7.d): Theoretical works on copper oxide ladders predicted the appearance of a CCθ I phase with LCs in hole-doped SCCO [S4, S5, S24]. 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) = ±4 sin(2πx 0 H 3 )sin(2πx 0 L) (14)
|A φ (Q)| 2 for CCθ I is shown on Supplementary Figure S10.a (green line).

• CCθ II (Supplementary Figure S7.e): This state [S6, S25] breaks the inversion and time-reversal symmetry. In each square plaquette, it is made of two LCs turning clockwise and anti-clockwise and aligned along one diagonal ( = +1) or the other ( = -1). This state is fourfold degenerate with :

A φ± π 2 (Q) = -A φ (Q) and A φ± π 2 (Q) = A φ (Q)
, corresponding to both orientations or domains ( = ±1). The two LCs produce 2 staggered orbital moments located with respect to the Cu site again at positions: ±(x 0 , x 0 ). The magnetic structure factor reads:

A φ (Q) = 2 sin(2πx 0 ( H 3 + L)) (15)
|A φ (Q)| 2 for the two different CCθ II orientations ( = ±1) are shown on Supplementary Figure S10.a (dashed blue lines). The full blue line represents an averaged of both domains with equal population.

• CCθ III (Supplementary Figure S7.f): this state corresponds to the LC pattern proposed to describe an ancillary phase associated within a spin liquid (mother) state [S7, S8], that for convenience we labeled here CCθ III . At variance with the CCθ II state, the two LCs are rotated by π 4 and circulate between O sites only (Supplementary Figure S7.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 CCθ III state reduces to twofold only, with A φ+π (Q) = -A φ (Q). For both configurations, the magnetic structure factor is given by :

Horizontal : A φ (Q) = 2 sin(2πx 0 H 3 ) (16)
V ertical :A φ+ π 2 (Q) = 2 sin(2πx 0 L) (17)
|A φ (Q)| 2 are shown on Supplementary Figure S10.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 (Supplementary Figure S7.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 = φφ 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) = [A φ (Q) + A φ (Q)]cos(π H 3 ) + i[A φ (Q) -A φ (Q)]sin(π H 3 ) (18)
|B φφ (Q)| 2 is shown on Supplementary Figure S10.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:

B φ (Q) = 2A φ (Q)cos(π H 3 ) (19)
• 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 = φφ 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) = [B φ (Q) + B φ (Q)]cos( π 2 (H + L)) + i[B φ (Q) -B φ (Q)]sin( π 2 (H + L)) (20)
|C φφ (Q)| 2 is shown on Supplementary Figure S10.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 -θI 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-ofphase as shown in Supplementary Figure S11.a as they share a current link along the diagonal of the inter-ladder small square. This gives the following |F (Q)| 2 :

|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 (Supplementary Figure S11.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 -θII like phases of LCs

• Uncorrelated ladders: SCCO -5

We first discuss the case of the independent ladders using the CCθ II pattern (Supplementary Figure S12.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:

|F (Q)| 2 = =±1 1 2 |4 cos(π H 3 ) sin(2πx 0 ( H 3 + L))| 2 (22)
Here, we assumed four possible domains with equal population. This structure factor (shown in Supplementary Figure S12.d) reproduces the SRM along the (H,0,1) line (see Fig. 5.b of the manuscript), while the magnetic correlations along a (perpendicular to 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) (Supplementary Figure S12.b), out-of-phase (ψ = π) or exhibit a crisscrossed coupling (ψ = ± π 2 ) (Supplementary Figure S12.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). Using the toroidal moment formalism (see section above), one can define Ω = 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 factors, |F (Q)| 2 = |C φφ (Q)| 2 , are:

ψ = 0 : |F (Q)| 2 = =±1 1 2 |8

  2π c Ladders = 1.61 Å-1 , with a = 11.4 Å, b = 12.9 Å and c = 3.91 Å, where L chains = 0 (except where it is explicitly indicated otherwise). Two large SCCO-x single crystals have been grown (see Methods) to perform the PND experiments, which are described in more details in the Supplementary Note 1. During these experiments, SCCO-x samples were aligned with the ladder sub-lattice parameters within 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 Fig.1.a-b).

Fig. 1 . 2 ×

 12 Fig. 1.c shows a Q-map of the full magnetic scattering at T = 5K in SCCO-8, measured on the D7 diffractometer extracted from longitudinal XYZ-PA (see Methods). A magnetic signal is systematically observed along (H,0,1) for odd integer H values and where L=1 corresponds to the ladders sublattice (crosses on Fig.1.c). These Q-positions correspond to nuclear extinctions according to the atomic space group symmetry selection rules as shown by the absence of nuclear scattering at the same Q-positions in the NSF channel Fig.1.d. However, the magnetic scattering appears significantly broader than the instrumental resolution, the hallmark of only short range magnetism.All of these results, obtained on the D7 diffractometer, are confirmed on the TAS-4F1 (T = 10K). Fig.2.a show 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 short range magnetism (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) (raw data are shown in Supplementary Note 2), and a H-scan across (1,0,1) as extracted from the D7 XYZ-PA map (T = 5K), are reported in Fig.2.b and Fig.2.c, respectively. The scans show peaked signals with momentum widths (FWHM), ∆ H = 0.5 r.l.u and ∆ L = 0.12 r.l.u, which are much broader than the instrumental resolution. All along the manuscript, the correlation lengths

  5.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)

Fig. 5 .

 5 Fig.5.a) nicely reproduces the main features of the SCCO-5 data along the (H,0,1) line (Fig.5.b) accounting for lost inter-ladders correlations. It explains also why the magnetic correlations along a (perpendicular to the ladders) are confined within a single ladder. The model gives a m LC = 0.05 ± 0.01µ B estimate for the magnetic moment amplitude (see Supplementary Note 10). Next, we add inter-ladder correlations which are typically imposed by the structure geometry and currents continuity as shown in the inset B of Fig.5.a. Again, the same model

5

  Fig.5.a. The Horizontal-CCθ III pattern cannot explain the measured data at H=0. In the case of correlated ladders SCCO-8, both patterns can describe the data of Fig.5.c. The fits give good agreement to the experimental data yielding m LC magnitudes of m LC = 0.04 ± 0.01µ B and m LC = 0.04 ± 0.01µ B for SCCO -8 and SCCO -5, respectively.

Fig. 1 :

 1 Fig. 1: Crystal structure and magnetic map in Sr 6 Ca 8 Cu 24 O 41 , sample: (a) Crystal structure of Sr 14 Cu 24 O 41 showing the alternating stack of CuO 2 chains (Cu in blue and O in red) and Cu 2 O 3 ladder planes, separated by Sr ions (in green) along the b-axis. (b) [a,c] plane projection of the ladders subsystem including one two-leg ladder. The ladders are formed by edge sharing CuO 2 squares. (c-d) Mapping in momentum space of the full magnetic scattering deduced from XYZ-PA (c), and nuclear intensity measured in the non spin-flip (NSF) channel (d) (measured on D7, T = 5K). The maps are given in reduced lattice unit (r.l.u.) of the ladders subsystem and the intensities in mbarn. The area bounded by dashed lines indicates the ladder scattering ridge along (H,0,1). The solid lines are associated with the chains nuclear response and blue crosses at integer H and L values correspond to the nuclear Bragg scattering associated with the ladders : (c) The ladders and satellite reflections magnetic spots are located by crosses and blue arrows, respectively. The magnetic satellite reflections are of the form (H, 0, -1, 1) using 4D superspace notations. In (d), the halos correspond to aluminum powder scattering from the sample holder. Sharper red spots in (c) located at positions of strong nuclear Bragg peaks in the NSF map (d) are not of magnetic origin but corresponds to polarization leakages from the NSF channel.

Fig. 2 :

 2 Fig. 2: Magnetic scans in Sr 6 Ca 8 Cu 24 O 41 : (a) Magnetic intensity of (H,0,1) points for integer H, i.e. along the inter-ladders direction (rungs) as extracted from XYZ polarisation analysis (XYZ-PA). The inset shows real-space magnetic correlation lengths within the ladders planes. (b) Background substracted L-scan across (3,0,1) in the spin-flip (SF X ) channel. The magnetic intensity appears as a Gaussian signal centered at (3,0,1) (Raw data given in Supplementary Note 2). (c) H-scan across (1,0,1) direction, along the inter-ladders direction (rungs) extracted from (XYZ-PA). (d) K-scan across (3,0,1) showing the magnetic intensity along the inter-plane direction as deduced from XYZ-PA (black) and SF X measurements after subtraction of a background intensity taken at (3,K,0.8) (red). (e) Temperature dependence of the magnetic intensity at (3,0,1), measured in SF X and obtained after background subtraction (Raw data given in Supplementary Note 3). Data in (c) were measured on D7 at 5K and the others on 4F 1 at 10K. Lines are guide to the eye. Error bars represent one standard deviation.

Fig. 3 :

 3 Fig. 3: Absence of magnetic signal on chains in Sr 6 Ca 8 Cu 24 O 41 : Full magnetic intensity deduced from XYZ polarisation analysis on (a) (4F1, T=10K) within the chain subsystem along (H,0,0,1) in superspace reduced lattice unit (r.l.u.) or (H,0,1.43) in ladders r.l.u. (b) (D7, T=5K) along (1,0,L). Error bars represent one standard deviation.

Fig. 4 :

 4 Fig. 4: Magnetic scans in Sr 9 Ca 5 Cu 24 O 41 : (a) L-scan across (0,0,1) in the spin-flip (SF X ) channel, showing the magnetic scattering along the ladders legs. The magnetic scattering appears as a Gaussian signal centered at (0,0,1). (Raw data given in Supplementary Note 2). (b) H-scan of the (H,0,1) rod, the inter-ladders direction (rungs), showing the modulation of the magnetic intensity extracted from XYZ polarisation analysis (XYZ-PA) (squares). The inset shows real-space magnetic correlation lengths within the ladders planes. (c) K-scan across (3,0,1) after full XYZ-PA, indicating the absence of inter-plane magnetic correlations. (d) Temperature dependence of the magnetic signal, after background subtraction (Raw data given in Supplementary Note 3). The figure combines polarized neutron data (full symbols) measured at (3,0,1) in the SF X channel and unpolarized neutron data (open symbols) measured at (1,0,1). All Q-scans were performed on the instrument 4F 1 at 10K. Lines are guide to the eye. Error bars represent one standard deviation.

Fig. 5 :Fig. 6 :

 56 Fig. 5: Phase diagram, modelisation and structure factors: (a) Schematic phase diagram in Sr 14-x Ca x Cu 24 O 41 (SCCOx) showing the evolution of the loop current (LC) pattern as a function of the Ca content (x Ca ). At large doping, magnetic correlations develop between ladders at the onset temperature T mag (red crosses). In heavily doped samples, a magnetic long range order (LRO) further develops below a Néel temperature T N of a few K [31]. Insets: (A) (top) CC -Θ II [7, 8] and (down) CC -Θ III [10, 11, 12] models built on single ladder with two staggered Cu -O orbital currents per Cu site flowing clockwise (red triangles) and anticlockwise (blue triangles). Both models described nicely the experimental results shown in panel (b). (B) CC -Θ II and (down) CC -Θ III models with adddtional interladders correlations within the ladder unit cell. Both models described nicely the experimental results shown in panel (c).(b-c) H-scan of the (H,0,1) rod, the inter-ladders direction (rungs) showing the modulation of the out-of-plane magnetic intensity, I b extracted from XYZ polarisation analysis (PA) (squares) compared with magnetic structure factors deduced from 2 loop currents models, labeled CC -Θ II (solid line)[START_REF] Simon | Detection and implications of a time-reversal breaking state in underdoped cuprates[END_REF][START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF] and CC -Θ III (dashed line)[START_REF] Chatterjee | Insulators and metals with topological order and discrete symmetry breaking[END_REF][START_REF] Chatterjee | Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF] in (b) SCCO -5 and (c) SCCO -8. Error bars represent one standard deviation.

  I mag (mbarn) 0 ± 5 6 ± 4.0 16 ± 10 27.9 ± 4.[START_REF] Bourges | Novel magnetic order in the pseudogap state of high-tc copper oxides superconductors[END_REF] 36 ± 15 I b -1 ± 2 3.1 ± 1.7 10 ± 1 12.7 ± 1.5 21 ± 13 I ac 1 ± 2 2.9 ± 1.8 6 ± 1 15.2 ± 1.5 15 ± 8

		4F1	4F1	D7	4F1	D7
	SCCO -8
	SCCO -5	I mag (mbarn) 11.4 ± 2.9 7.1 ± 2.9	-	7.4 ± 2.4	-

We report a PND study of two SCCO-x single crystals. The single crystals of Sr 14-x Ca x Cu 24 O 41 with (x = 5 and 8) were grown by the traveling solvent floating zone method[START_REF] Revcolevschi | Crystal growth of pure and substituted lowdimensionality cuprates cugeo 3 , la 2 cuo 4 , srcuo 2 , srcuo 3 and sr 14 cu 24 o 41 by the floating zone and travelling solvent zone methods[END_REF] using a four mirrors image furnace at SP2M-ICMMO. The crystals were grown from polycrystalline feed rods of the corresponding compounds, obtained by solid state reaction of stoichiometric amounts of CuO, SrO and CaO precursors. The growth was carried out under an oxygen pressure of 5 and 8 bars for the x= 5 and 8 Ca-doped Sr 14 Cu 24 O 41 respectively, in order to avoid the formation of secondary phases and favor the constrained structure resulting from Cadoping. The growth was initiated using a solvent pellet containing 30% (Sr, Ca)O and 70%CuO and carried out with a rate of 1mm.h -1 .The crystals weigh 2.1g and 2.5g for the x = 5 and 8 compositions, respectively.

Acknowledgments

The authors would like to acknowledge B. Fauqué, M. Greven, A. Goukassov, C.M. Varma and A. Wildes for fruitful discussions. We acknonwledge F. Maignen and C. Meunier for their valuable technical assistance on 4F1. We acknowledge supports from the project NirvAna (contract ANR-14-OHRI-0010) of the French Agence Nationale de la Recherche (ANR). The open access fee was covered by FILL2030, a European Union project within the European Commission's Horizon 2020 Research and Innovation programme under grant agreement N • 731096.

Data Availability. Data collected on D7 are available at https://doi.ill.fr/10.5291/ILL-DATA.5-53-279. The rest of the data that support the findings of this study is available from the corresponding authors upon request.

Methods

Crystal growth

Polarized neutron diffraction

The PND experiments of SCCO-x single crystals, described in more details in the Supplementary Note 1, were carried out on two instruments: the triple axis spectrometer (TAS) 4F1 (Orphée reactor, Saclay) and the multidetector diffractometer D7 (Institut Laue Langevin, Grenoble). 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. In a PND experiment, the quantization axis of the neutron spin polarization, P , is given within a (X,Y,Z) Cartesian referential. For a fixed neutron spin polarization, one can selectively probe the scattered intensity in the spin flip channel (SF P ) where the neutron spin polarization is reversed after interaction with the sample, and the non-spin-flip channel (N SF P ), where the neutron spin is conserved. The amount of the magnetic scattering in the SF P channel varies as a function of the polarization P . The combination of measurements in the SF channel for different P is called the polarization analysis (PA). It allows a full determination of the magnetic intensity I mag where the non-magnetic background is removed.

We used incident neutron wavevectors of k i = 2.57 Å-1 on 4F1 and k i = 2.02 Å-1 on D7. The longitudinal XYZ-PA was performed using Helmoltz-like (4F1) and a quadrupolar assembly (D7) coils [START_REF] Fennell | Wavevector and energy resolution of the polarized diffuse scattering spectrometer d7[END_REF] allowing to choose the polarization of the neutron either along Z always perpendicular to the scattering plane. The X and Y polarizations correspond to arbitrary directions within the scattering plane on D7, whereas X is always set to be parallel to the scattering vector Q on 4F1 and Y is perpendicular to Q but still within the scattering plane. In all experiments, the samples were aligned in the (1,0,0)/(0,0,1) scattering plane. The data are all reported in reduced lattice units (r.l.u) and the measurement procedure follows refs. [START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in yba 2 cu 3 o 6.85[END_REF][START_REF] Tang | Orientation of the intra-unit-cell magnetic moment in the high-t c superconductor hgba 2 cuo 4+δ[END_REF].

On D7, the data were collected by performing ±10 • rocking trajectories around the (1,0,1) and (-1,0,0) positions. We then performed data reduction adapting the standard procedure [START_REF] Stewart | Disordered materials studied using neutron polarization analysis on the multi-detector spectrometer, d7[END_REF].

1 Supplementary Material

SUPPLEMENTARY NOTE 4: ABSENCE OF MAGNETIC SIGNAL WITHIN THE CHAINS

A new magnetic signal is clearly observed at (H, 0, L) with integer values of H and L, corresponding to the ladder subsystem. However, no magnetic signal is observed for the chain subsystem at Bragg positions (H, 0, 0, 1) using superspace notations, corresponding to (H, 0, 1.43) in the ladder subsystem units. A scan performed across (H, 0, 1.43), shown in Supplementary Figure S4, shows the full scattered magnetic intensity as deduced from XYZ-PA (4F1) in SCCO -5 (which complements a similar scan in SCCO -8 in the Fig. 3.a of the main text). In both samples, the XYZ-PA reveals the absence of scattered magnetic intensity at positions corresponding to the chains subsystem.

SUPPLEMENTARY NOTE 5: 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:

Taking into account the instrument energy resolution : ∆ ω = 1.25 meV for k i = k f =2.57 Å-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 [S10].

SUPPLEMENTARY NOTE 6: 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 Supplementary Figure S5. From these maps, one sees that both magnetic components, in-plane and out-of-plane of the ladder a-c plane, are sizeable. For results obtained on both instruments, Supplementary 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 given above for both 4F1 & D7 (we remind that the procedure is also described in [S10] for D7) . 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 Supplementary 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 planes as it is observed in all superconducting cuprates [S2, S3].

Among all |F (Q)| 2 (Supplementary Figure S12.d), the squared structure factor for the in-phase case (Supplementary Figure S12.b) is the one which reproduces the main features of our experimental results, namely: 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. The crisscrossed CCθ II with the effective toroidal moment along the ladder (ψ=± π 2 , δ = -1), shown Supplementary Figure S12.c, is also consistent for the observed SRM, 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 [S23].

CC -θIII like phases of LCs

• Uncorrelated ladders: SCCO -5 Supplementary Figure S13.a,d show SCCO unit cell when the first ladder is decorated with the same CCθ III pattern on each Cu site of the first ladder, while the second remains non magnetic. |F (Q)| 2 reads:

With

3 ) for the Horizontal pattern or A(Q) = 2 sin(2πx 0 L) for the Vertical one. Along (H,0,1) direction, the Vertical-CCθ 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-CCθ III pattern for which the scattering intensity cancels at H=0 which cannot explain the measured data.

• Correlated ladders: SCCO -8 Supplementary Figure S13.b,e show the case where the 2 ladders are decorated with the same CCθ 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 CCθ III pattern.

SUPPLEMENTARY NOTE 9: 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. [S26]). 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 [S27] suggested that both magnetic Cu-and O-form factor could be used. Supplementary Figure S14.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, Supplementary Table S2. As discussed above, among the different magnetic patterns of Supplementary Figure S7.b-f, some of the LCs-like phases can describe 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 (Supplementary Table S1). Using I b ∝ m 2 b , one can deduce the out-of-plane magnetic moment, m b ≡ m LC . Supplementary Figure S14.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 (Supplementary Figure S14.d) and SCCO-8 (Supplementary Figure S14.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:

The Figs. S2.

• In-plane magnetic moment: mac

One can perform the same analysis for the full measured intensities I mag although, in principle, the classical LCs phase cannot account for the in-plane magnetic intensities reported in Supplementary Table S1.

As it combines magnetic moments along the ladder and along the rung, m ac varies with Q due to the orientation factors. For a H-scan along (H,0,1), the variations of these orientation factors, weighting m 2 a and m 2 c are given in Supplementary Figure S15. 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 . Supplementary Figure S15.b-c show the fit of the PND data for samples SCCO -5 and SCCO -8 using the same functions as in Supplementary Figure S14.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 Supplementary Table S2. 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. 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 CCθ II states, but could rather emerge by their quantum superposition [S21, S22]. Within that framework, the degree of quantum admixture shows up in PND measurements in the form of an 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 [S2, S28-S30]. 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-leg 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.

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