Philippe Bourges
email: philippe.bourges@cea.fr

Dalila Bounoua
email: dalila.bounoua@cea.fr

Yvan Sidis
email: yvan.sidis@cea.fr

Loop currents in quantum matter

Keywords: loop-current, quantum matter, polarized neutron diffraction, high-temperature superconductors

come

Introduction

Strong electron correlations in condensed matter lead to the emergence of novel phenomena and exotic states. None of these collective phenomena is encompassed by the simple addition of individual electrons but results instead from many-body effects as it was advocated by P. W. Anderson almost 50 years ago [START_REF] Anderson | More Is Different[END_REF]. Spin liquids, Mott physics, strange metals, high-temperature superconductivity and its related pseudogap phase are all typical examples of these novel states of matter, generically labeled as quantum matter.

The study of the complex phase diagram of hole-doped superconducting cuprates highlighted the existence of a mysterious state of matter, the so-called pseudogap (PG) metal [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF][START_REF] Proust | The Remarkable Underlying Ground States of Cuprate Superconductors[END_REF][START_REF] Varma | Colloquium: Linear in temperature resistivity and associated mysteries including high temperature superconductivity[END_REF], where large portions of the Fermi surface are gapped out and only Fermi arcs survive. The PG state is characterized by a significant loss of electronic entropy [START_REF] Loram | Evidence on the pseudogap and condensate from the electronic specific heat[END_REF] below a certain temperature, socalled T * , in the phase diagram, as represented in Fig. 1. On cooling down, unconventional superconductivity with a d -wave symmetry occurs and seems to emerge out of the PG state. Furthermore, deep inside the PG state, an incipient charge density wave (CDW), with a d -wave structure factor, competes with superconductivity [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF][START_REF] Proust | The Remarkable Underlying Ground States of Cuprate Superconductors[END_REF], while strong antiferromagnetic (AF) spin fluctuations are still present in the materials. The incipient CDW breaks the lattice translation (LT) symmetry and gives rise to a new periodicity associated with its modulation wave vector q c = 0. It coexists with a set of intra-unit-cell (IUC) electronic instabilities which preserve the LT invariance (q=0) (see e.g. [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Fujita | Simultaneous Transitions in Cuprate Momentum-Space Topology and Electronic Symmetry Breaking[END_REF][START_REF] Fujita | Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates[END_REF]), but break discrete Ising-like symmetries: the fourfold rotation (R) symmetry for an electronic nematic state and additionally, time reversal (T) and parity (P) symmetries, as we shall discuss in details here. Remarkably, these different IUC orders show up concomitantly with the PG state. They are then necessarily tied up with it and should also be associated with a significant energy condensation. It should be stressed out that no additional noticeable change of the electronic entropy is reported below T * , of course apart from the one associated with superconductivity [START_REF] Loram | Evidence on the pseudogap and condensate from the electronic specific heat[END_REF]. This is, for instance, the case for states like the CDW occurring within the PG state. A central question is therefore which one of these order parameters is the primary one ? It should be also stressed that there is two known mechanisms to open gaps in metals, either it results from a finite q =0 ordering breaking LT or it is due to a particleparticle pairing as does superconductivity. Therefore, none of the IUC broken symmetries alone can induce the needed fermionic gap characterizing the PG state.

To get a better understanding about these IUC orders, it is useful to look at the basic atomic structure of superconducting cuprates. They are layered perovskite materials, made of the stacking of CuO 2 layers, whose building block is a CuO 2 squared plaquette (Fig. 3.a-c). In the insulating undoped state, there is only a single electron on the Cu site carrying a half-integer spin S=1/2. Upon hole doping, an extra hole on the O site can form a bound state with the electron on the Cu site, yielding the so-called Zhang-Rice singlet in the CuO 2 plaquette [START_REF] Zhang | Effective Hamiltonian for the superconducting Cu oxides[END_REF]. Owing to the strong on-site Coulomb repulsion which prevents a second hole to enter the plaquette, one is left with a set of three states for the CuO 2 squared plaquette, useful to build a minimum model (such as the widely studied t -J model) for cuprates, but which also eliminate the internal degrees of freedom within the CuO 2 plaquette. In contrast, the IUC electronic orders imply the emergence of non-Zhang-Rice singlet states [START_REF] Moskvin | Pseudogap phase in cuprates: Oxygen orbital moments instead of circulating currents[END_REF]. For an electronic nematic state, an unbalanced electronic density at the O sites breaks the fourfold R-symmetry of the CuO 2 plaquette, as observed in scanning tunneling microscopy (STM) [START_REF] Fujita | Simultaneous Transitions in Cuprate Momentum-Space Topology and Electronic Symmetry Breaking[END_REF][START_REF] Fujita | Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates[END_REF].

Put altogether, the IUC symmetry breakings point towards novel electronic states that have been proposed either to account for or to characterize the PG state. That takes in particular the forms of loop currents (LCs) circulating coherently between copper and oxygen orbitals [START_REF] Varma | Colloquium: Linear in temperature resistivity and associated mysteries including high temperature superconductivity[END_REF] as depicted in Fig. 2.a-b. LC states can take different shapes [START_REF] Varma | Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals[END_REF][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] Weber | Orbital Currents in Extended Hubbard Models of High-T c Cuprate Superconductors[END_REF][START_REF] Weber | Phase Diagram of a Three-Orbital Model for High-T c Cuprate Superconductors[END_REF][START_REF] Varma | Pseudogap and Fermi arcs in underdoped cuprates[END_REF], where the electronic density remains uniform, but spontaneous charge currents appear between Cu and O sites within the CuO 2 plaquette. The loop currents flow coherently among the three orbitals of copper and oxygen, spontaneously breaking T-, P-and/or R-symmetries. The search for states of matter involving charge currents is a long story in the physics of high temperature superconductors and various states of matter were proposed. Unlike IUC and LC states (both being q=0 electronic instabilities), charge current density wave (q =0 electronic instability) were also proposed [START_REF] Wang | Charge-density-wave order with momentum (2Q, 0) and (0, 2Q) within the spin-fermion model: Continuous and discrete symmetry breaking, preemptive composite order, and relation to pseudogap in hole-doped cuprates[END_REF]. For instance, within a single band Hubbard model, charge currents were predicted to flow around the CuO 2 square lattice, yielding the so called « πflux phase » or D-density wave (DDW) [START_REF] Hsu | Two observable features of the staggered-flux phase at nonzero doping[END_REF][START_REF] Chakravarty | Hidden order in the cuprates[END_REF] which doubles the unit cell. As the π-flux phase breaks the LT symmetry, additional superstructure Bragg reflections are expected at several Q-points in the reciprocal space equivalent to (π, π) [START_REF] Hsu | Two observable features of the staggered-flux phase at nonzero doping[END_REF][START_REF] Chakravarty | Hidden order in the cuprates[END_REF]. Despite various attempts in different sets of samples, neutron diffraction measurements failed to prove the existence of magnetic peaks at the planar wave vector (1/2, 1/2) ≡ (π, π) in the 2D first Brillouin zone. As a generic property, all the types of charge current phases break T-symmetry and closed current loops generate a magnetic flux and orbital magnetic moments. That is the property primarily reported to provide evidence in favor of the LC states in various quantum materials, and which people have tried to evidence by using dichroic effect using circularly polarized angle resolved photoemission (ARPES) [START_REF] Kaminski | Spontaneous breaking of time-reversal symmetry in the pseudogap state of a high-Tc superconductor[END_REF] and next by looking for the LC induced orbital magnetism using polarized neutron diffraction (PND) as we shall review here in section 2. To date, the LC state has been reported in several oxide materials either with 2D structure, in the superconducting cuprates [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Mook | Observation of magnetic order in a superconducting YBa 2 Cu 3 O 6.6 single crystal using polarized neutron scattering[END_REF][START_REF] Balédent | Evidence for competing magnetic instabilities in underdoped YBa 2 Cu 3 O 6+x[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] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF][START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor HgBa 2 CuO 4+δ[END_REF][START_REF] Li | Magnetic order in the pseudogap phase of HgBa 2 CuO 4+δ studied by spin-polarized neutron diffraction[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][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] Almeida-Didry | Evidence for intra-unitcell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF][START_REF] Mangin-Thro | Characterization of the intra-unit-cell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF] and in the iridates [START_REF] Jeong | Time-reversal symmetry breaking hidden order in Sr 2 (Ir,Rh)O 4[END_REF], or in the 1D spin ladder cuprate [START_REF] Bounoua | Loop currents in two-leg ladder cuprates[END_REF], showing the universal nature of this exotic state.

In this article, we next list in section 3 the various experimental data (other than PND) that support the emergence of such LC states built from the LC-Θ I I pattern represented in Fig. 2.a. In particular, the macroscopic torque measurements in two cuprates families, namely in bilayer YBa 2 Cu 3 O 6+x and monolayer HgBa 2 CuO 4+δ , [START_REF] Sato | Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in YBa 2 Cu 3 O[END_REF][START_REF] Murayama | Diagonal nematicity in the pseudogap phase of HgBa 2 CuO 4+δ[END_REF], although showing different results, match nicely the PND results in both materials [START_REF] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF][START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor HgBa 2 CuO 4+δ[END_REF].

The remainder of the manuscript focuses in section 4 on the interplay with the PG state, which depends on the hierarchy among the various co-existing/competing observed phases. LCs can be bound to the PG physics in two ways. First, the LC phase is degenerate and several states corresponding to distinct LC patterns are allowed [START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF]. This degeneracy can be the source of a proliferation of randomly distributed LC domains. Assuming the existence of a super-cell made of a coherent juxtaposition of LC domains [START_REF] Varma | Pseudogap and Fermi arcs in underdoped cuprates[END_REF] (as shown by Fig. 2.d-e), the LT invariance would be broken and this modulated (q =0) LC-based phase could trigger a gap opening on the Fermi surface. Following a fundamentally different approach, the LC order can also be understood in terms of a so-called "ancillary state" inseparable from the PG state [START_REF] Agterberg | Emergent loop current order from pair density wave superconductivity[END_REF][START_REF] Dai | Pair-density waves, charge-density waves, and vortices in high-T c cuprates[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF][START_REF] Sarkar | Incipient loop-current order in the underdoped cuprate superconductors[END_REF]. The ancillary state is a secondary instability associated to a mother instability which itself effectively opens the PG. For instance, it can correspond to pair-density-wave (PDW) states [START_REF] Agterberg | Emergent loop current order from pair density wave superconductivity[END_REF][START_REF] Dai | Pair-density waves, charge-density waves, and vortices in high-T c cuprates[END_REF][START_REF] Sarkar | Incipient loop-current order in the underdoped cuprate superconductors[END_REF]. Within these approaches, it was first pointed out by Agterberg et al [START_REF] Agterberg | Emergent loop current order from pair density wave superconductivity[END_REF] that the LC phase necessarily happens owing to the symmetry and therefore is essential to capture the specific nature of the PG. We conclude in section 5 with some perspectives for future experimental developments.

Polarized neutron diffraction experiments

Hunting for LCs, we have performed PND experiments on several correlated materials over the last 15 years. We here discuss layered perovskite materials with a (nearly) tetragonal structure within the ab basal plane corresponding to the (CuO 2 or IrO 2 ) layers (Fig. 3c), stacked along the c-axis. The structural unit-cell contains one or more layers, depending on the materials, yielding different c lattice parameter. Examples of the corresponding crystallographic structure are shown in Fig. 3a-b. The PND technique is a momentum-selective probe for magnetism. We commonly write, see e.g. ref. [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF], the measured momentum as Q = (H , K , L) in reduced lattice units (2π/a, 2π/b, 2π/c) with a ∼ b ∼3.85 Å. Various attempts to observe the different orbital magnetic moments associated with the charge currents (π-flux or D-density wave) phase have been already discussed in [START_REF] Sidis | Search for the existence of circulating currents in high-Tc superconductors using the polarized neutron scattering technique[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF].

The IUC (q=0) magnetism

Unlike the π-flux phase, the staggered orbital magnetism induced by the Varma's LCs [START_REF] Varma | Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals[END_REF][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] is associated with a set of pairs of charge currents looping clockwise and anticlockwise within each unit cell, Fig. 2.a [START_REF] Sidis | Evidence for Intra-Unit-Cell magnetic order in the pseudo-gap state of high-Tc cuprates[END_REF]. Such a LC state preserves the LT symmetry, so that, in a diffraction experiment its magnetic Bragg reflections are superimposed to the atomic Bragg reflections. This makes the experiments particularly difficult, as the magnetic contribution is weak, and polarized neutrons are required to discriminate between the atomic and magnetic scattering on a Bragg reflection. Indeed, any source of magnetism in the sample is able to flip the neutron spin, changing the spin polarization of the scattered neutron beam. To determine a possible orbital magnetic order, one needs to measure the variation of the neutron polarization versus temperature. A long range magnetic order would give rise to an upturn of the so-called inverse flipping ratio, 1/FR, which measures the ratio of magnetic to nuclear scattering when the neutron polarization is set parallel to the scattering wavevector [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF]. A typical result is reported in Fig. 3.e where 1/FR ratio is shown versus temperature in the bilayer cuprate YBa 2 Cu 3 O 6.6 [START_REF] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF]. Two Bragg peaks are measured: at large Q, (2,0,0), where the magnetic signal should vanish, an expected smooth thermal variation is reported whereas an upturn is observed around T 235K at low Q=(1,0,0). This corresponds to the occurrence of a spontaneous magnetic signal. Assessment of the technical difficulties and limitations related to the PND experiments have been recently reemphasized in [START_REF] Bourges | Evidence for Intra-Unit Cell Magnetism in Superconducting Cuprates: a Technical Assessment[END_REF].

Numerous PND measurements reveal the appearance of a q=0 AF order at the Bragg reflections Q=(1,0,L) [START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF], also referred to as an IUC magnetism [START_REF] Sidis | Evidence for Intra-Unit-Cell magnetic order in the pseudo-gap state of high-Tc cuprates[END_REF]. This order displays a magnetic structure factor consistent with the orbital staggered magnetism produced by the LC-Θ I I state within the CuO 2 planes. In the field of unconventional high temperature superconductors, we reported such observations in four distinct cuprate families : YBa 2 Cu 3 O 6+x [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Mook | Observation of magnetic order in a superconducting YBa 2 Cu 3 O 6.6 single crystal using polarized neutron scattering[END_REF][START_REF] Balédent | Evidence for competing magnetic instabilities in underdoped YBa 2 Cu 3 O 6+x[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] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF], HgBa 2 CuO 4+δ [START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor HgBa 2 CuO 4+δ[END_REF][START_REF] Li | Magnetic order in the pseudogap phase of HgBa 2 CuO 4+δ studied by spin-polarized neutron diffraction[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], (La,Sr) 2 CuO 4 [START_REF] Balédent | Two-Dimensional Orbital-Like Magnetic Order in the High-Temperature La 2-x Sr x CuO 4 Superconductor[END_REF], and Bi 2 Sr 2 CaCu 2 O 8+δ [START_REF] Almeida-Didry | Evidence for intra-unitcell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF][START_REF] Mangin-Thro | Characterization of the intra-unit-cell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF]. Fig. 1 groups, in a generic phase diagram, the magnetic ordering temperatures measured by PND in the monolayer HgBa 2 CuO 4+δ and bilayer YBa 2 Cu 3 O 6+x compounds, as a function of the hole doping.

Later, we also reported an IUC magnetism in the large spin-orbit coupled Mott insulators, Sr 2 (Ir, Rh)O 4 monolayer iridate [START_REF] Jeong | Time-reversal symmetry breaking hidden order in Sr 2 (Ir,Rh)O 4[END_REF], suggesting a more generic character of such LC orders in correlated electron oxides. It is worth noting that both iridates (Ir, 5d) and cuprates (Cu, 3d) are both AF insulators which can be doped (Mott or charge-transfer insulators, respectively, and S=1/2 on Cu versus J=1/2 on Ir). Likewise, they share a similar layered perovskite structure. A noticeable difference with the 2D cuprates is that the LC occurs as well in the insulating compounds at half-filling, whereas in cuprates it is so far observed in the doped metallic and superconducting materials. It is worth stressing out that the mechanism that could be invoked to produce LC state could differ in both oxides.

Despite the large body of PND data supporting an IUC magnetism in superconducting cuprates, its existence was questioned by a PND attempt [START_REF] Croft | No evidence for orbital loop currents in charge-ordered YBa 2 Cu 3 O 6+x from polarized neutron diffraction[END_REF], which failed to detect it in small YBa 2 Cu 3 O 6+x single crystals (weighing less than 18 mg in contrast with our samples which weighted at least 2000 mg [START_REF] Bourges | Comment on "No evidence for orbital loop currents in charge-ordered YBa 2 Cu 3 O 6+x from polarized neutron diffraction[END_REF]). Due to the limited flux of polarized neutron beams and the showing the intra-unit cell (IUC) magnetic order associated with the pseudogap state from polarized neutron diffraction (PND) [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Mook | Observation of magnetic order in a superconducting YBa 2 Cu 3 O 6.6 single crystal using polarized neutron scattering[END_REF][START_REF] Balédent | Evidence for competing magnetic instabilities in underdoped YBa 2 Cu 3 O 6+x[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] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF][START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor HgBa 2 CuO 4+δ[END_REF][START_REF] Li | Magnetic order in the pseudogap phase of HgBa 2 CuO 4+δ studied by spin-polarized neutron diffraction[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] and T * , the pseudogap temperature, determined from 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][START_REF] Murayama | Diagonal nematicity in the pseudogap phase of HgBa 2 CuO 4+δ[END_REF], second harmonic generation (SHG) [START_REF] Zhao | Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate[END_REF], muon spectroscopy (µSR) [START_REF] Zhang | Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa 2 Cu 3 O[END_REF] and resonant ultrasound (Rus) [START_REF] Shekhter | Bounding the pseudogap with a line of phase transitions in YBa 2 Cu 3 O 6+x[END_REF] experiments. Representative lines of transition of antiferromagnetism (AF), superconductivity (SC), pseudogap (PG) and charge density wave (CDW) (full line for YBCO and dashed line for Hg1201) are also depicted. relatively weak orbital magnetic moments, we demonstrated in [START_REF] Bourges | Evidence for Intra-Unit Cell Magnetism in Superconducting Cuprates: a Technical Assessment[END_REF][START_REF] Bourges | Comment on "No evidence for orbital loop currents in charge-ordered YBa 2 Cu 3 O 6+x from polarized neutron diffraction[END_REF] that the experimental accuracy in [START_REF] Croft | No evidence for orbital loop currents in charge-ordered YBa 2 Cu 3 O 6+x from polarized neutron diffraction[END_REF] was clearly insufficient to draw any conclusions from the data as a consequence of the one well-known drawback of neutron scattering, which is the systematic need of large enough single crystals.

The PND results prove that the T-symmetry is broken when entering the PG phase of high-T c superconducting cuprates, consistent with the LC-Θ I I phase symmetry (Fig. 2.a) originally proposed by C.M. Varma [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]. The deduced spontaneous orbital magnetic moment is typically of the order of ∼ 0.1µ B per loop of the LC-Θ I I phase [START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF] and its magnitude decreases with increasing hole doping following the trend of the pseudogap [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Mangin-Thro | Characterization of the intra-unit-cell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF]. In general, it should be stressed out that PND data can be interpreted by a large number of magnetic models as long as these models produce compensated magnetic moments within a given unit cell and respect the LT symmetry. This is the case for instance for the Dirac multipole ordering proposed in Ref. [START_REF] Lovesey | Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor[END_REF][START_REF] Lovesey | Ordered state of magnetic charge in the pseudo-gap phase of a cuprate superconductor (HgBa 2 CuO 4+δ )[END_REF][START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF], which we will discuss below. To conclude, a model accounting for the PND data needs at least two opposite magnetic moments in each unit cell, to comply with the q=0 AF structure.

Magnetic moment direction

The direction of the magnetic moments is experimentally determined through the neutron polarization analysis [START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF], from which one can deduce the magnitude of the out-of-plane component (M c ) and the in-plane component (M ab ). In the LC-Θ I I phase, the orbital magnetic moments should be perpendicular to the CuO 2 plane in which LCs are confined (Fig. 2.a-b). However, as sketched in Fig. 3.d, the outcome of the PND analysis is a Θ∼ 40 ± 20 • tilt angle of the magnetic moment with respect to the c axis. While systematically observed in all 2D cuprates [START_REF] Tang | Orientation of the intra-unit-cell magnetic moment in the high-T c superconductor HgBa 2 CuO 4+δ[END_REF], Θ can vary noticeably between the cuprates, as well as a function of the measured momentum Q-points. Interestingly, the moment tilt could also be temperature dependent [START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in YBa 2 Cu 3 O 6.85[END_REF]. Furthermore, as sketched in Fig. 3.d, the exact orientation of M ab in the ab plane has not been determined experimentally yet. More generally, the observed tilt is not well accounted for by most of the theoretical models proposed for the IUC magnetism, as we shall see below. As a matter of fact, the observed tilt is certainly a key and unexpected feature of the observed IUC magnetism, putting stringent constraints on theoretical models.

IUC spatial correlations

Before, going further in the discussion, it is worth emphasizing some experimental limitations of the neutron scattering technique. Generally speaking, the neutron scattering cross-section is convoluted with the instrumental resolution of the instrument in 4 dimensions (Q, ω) [START_REF] Hennion | La diffusion inélastique des neutrons sur monocristal. Le spectromètre 3-axes[END_REF]. The determination of the correlation length and characteristic time scale are limited by the instrument resolution. In order to detect a weak magnetic response, one usually relaxes the resolution. Likewise, a modulated structure, with a very long pitch, gives rise to satellite reflections very close to the Bragg spot. These satellite scatterings can merge with the Bragg scattering owing to the experimental resolution. The canonical example is the long period helimagnet MnSi [START_REF] Ishikawa | Helical spin structure in manganese silicide MnSi[END_REF], for which the observation of the magnetic satellites requires the use of a very good resolution for diffraction measurements or the use of small angle neutron scattering techniques. In the latter case, the measurement is not performed around a Bragg reflection at finite Q( = 0) anymore, but around Q → 0. For LC-like magnetism, whose magnetic structure factor cancels at small angle, small angle neutron scattering is unfortunately unsuited.

In three of the cuprates families, namely YBa 2 Cu 3 O 6+x , HgBa 2 CuO 4+δ and Bi 2 Sr 2 CaCu 2 O 8+δ , the observed IUC magnetism superimposes to the atomic Bragg peak [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] Almeida-Didry | Evidence for intra-unitcell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF], suggesting that the IUC order is long range and 3D as it is clearly established in underdoped YBa 2 Cu 3 O 6.6 [START_REF] Mook | Observation of magnetic order in a superconducting YBa 2 Cu 3 O 6.6 single crystal using polarized neutron scattering[END_REF]. Due the relaxed instrument resolution that was used, only an upper limit of the correlation length along the c axis ξ c ≤ 75 Å could be given [START_REF] Mook | Observation of magnetic order in a superconducting YBa 2 Cu 3 O 6.6 single crystal using polarized neutron scattering[END_REF][START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF]. As in MnSi, a situation with magnetic satellites with a long magnetic pitch, λ, could happen. Various attempts to observe a magnetic response in the reciprocal space distant from the Bragg position were however unsuccessful, giving a lower limit of a possible pitch of λ 40Å. In an nearly optimally doped YBa 2 Cu 3 O 6.85 sample only, an IUC magnetic signal could be detected broader than the in-plane momentum resolution [START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in YBa 2 Cu 3 O 6.85[END_REF]. This clearly indicates short range correlations with finite in-plane correlation lengths, ξ ab ∼ 20a in that compound. A particularly interesting case is that of La 2-x Sr x CuO 4 , a system which is known to experience a segregation of doped holes into charge stripes [START_REF] Julien | Magnetic order and superconductivity in La 2-x Sr x CuO 4 : a review[END_REF]. For Sr substitution close to x=1/12, we observed a quasi-static IUC magnetic signal that remained quasi-2D with short range correlation lengths [START_REF] Balédent | Two-Dimensional Orbital-Like Magnetic Order in the High-Temperature La 2-x Sr x CuO 4 Superconductor[END_REF]. This contrasts with observations in other cuprates where a (quasi) long range IUC magnetism develops at 3D. This short range magnetism in lightly doped La 2-x Sr x CuO 4 can be associated with very weak correlation lengths of LC-Θ I I 1D ribbons (∼ 10 Å, i.e 2-3a). The short range magnetism can be viewed as LC-Θ I I objects confined on the charge stripes separated by hole poor AF domains of Cu spins, in relation with the charge order in this material [START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF][START_REF] Julien | Magnetic order and superconductivity in La 2-x Sr x CuO 4 : a review[END_REF]. LCs can therefore interact with other electronic instabilities and persist in low dimensional systems (1D).

Following this observation and theoretical predictions [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], we recently studied the 2leg ladder cuprate Sr 14-x Ca x Cu 24 O 41 , which is an archetype of 1D spin liquids. This aperiodic system hosts a very rich phase diagram where, depending on the Ca content (which induces hole doping), it exhibits a strong spin liquid state ending into an intriguing ordered magnetic state at larger Ca content, passing through a CDW phase. Using PND, we discovered the existence of 2D short range magnetism in this material for two Ca contents [START_REF] Bounoua | Loop currents in two-leg ladder cuprates[END_REF]. This magnetism cannot be described by any type of Cu spins orders. Instead, our investigations establish the existence of LC correlations within the 2-leg ladders [START_REF] Bounoua | Loop currents in two-leg ladder cuprates[END_REF]. The results from both studies suggest that LCs could be confined within quasi-1D structures, such as 2-leg Cu ladders, imposed either by the structure of the material in Sr 14-x Ca x Cu 24 O 41 or by an electronic phase segregation yielding the formation of (bond centered-) stripes in La 2-x Sr x CuO 4 .

L-dependence of the IUC magnetic structure factor

L dependence of the magnetic scattering associated with the IUC order has been reported for wavevectors Q=(1, 0, L) in two monolayer compounds HgBa 2 CuO 4+δ [START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor HgBa 2 CuO 4+δ[END_REF] and La 2-x Sr x CuO 4 [START_REF] Balédent | Two-Dimensional Orbital-Like Magnetic Order in the High-Temperature La 2-x Sr x CuO 4 Superconductor[END_REF]. In the former, the magnetic signal from IUC order is only at integer L values, while it remains 2D (any L values) for the latter. Surprisingly, the scattered magnetic intensity displays the same fast decay as a function of L [START_REF] Almeida-Didry | Evidence for intra-unitcell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF]. In principle, in bilayer compounds, in-phase or out-of-phase coupling between layers would give rise to a magnetic structure factor modulated either by a term 4 cos 2 (πzL) or 4 sin 2 (πzL), where z (=d/c) stands for the reduced interlayer distance. Instead, the IUC magnetic pattern in YBa 2 Cu 3 O 6+x exhibits a crisscrossed stacking of the IUC patterns [START_REF] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF] (as illustrated in Fig. 5.b), so that the L-dependence associated with the bilayer vanishes in the magnetic structure factor when dealing with a twinned crystal. Accordingly, the L-dependence of the scattered magnetic intensity measured in bilayer systems like twinned YBa 2 Cu 3 O 6+x and Bi 2 Sr 2 CaCu 2 O 8+δ , matches that found for monolayer cuprates [START_REF] Almeida-Didry | Evidence for intra-unitcell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF]. Furthermore, the calibration of the magnetic intensity measured on Bragg reflections with similar |Q| and in comparable samples ( i. e. similar at T * and doping levels) indicates that the intensity is about only twice larger for system with 2 Cu per formula unit instead of 1 Cu /f.u.. This ratio is also consistent with the crisscrossed arrangement of LCs within a bilayer.

The fast decay of magnetic intensity along L observed in all cuprates is actually much faster than the one controlled by |f(Q)| 2 the squared magnetic form factor of Cu or O single ion [START_REF] Almeida-Didry | Evidence for intra-unitcell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF]. This phenomenon could then be accounted for by IUC magnetic moments being spread out within the CuO 2 layer. In this picture, LCs would not be simply confined within the CuO 2 layers. They would be more delocalized than a magnetic moment bound to a single ion, giving rise to a faster decay of the IUC structure factor along the out-of-plane direction. This interpretation seems supported by the study carried out on twin-free samples: in this case, the scattered intensities along (1, 0, L) and (0, 1, L) differ and are weighted by 4 cos 2 (πzL) and 4 sin 2 (πzL), respectively [START_REF] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF]. However, the observed ratio between the magnetic IUC intensities measured along a and b is not quantitatively reproduced with the value z=0. [START_REF] Balédent | Two-Dimensional Orbital-Like Magnetic Order in the High-Temperature La 2-x Sr x CuO 4 Superconductor[END_REF]. A larger value of z provides a better result, suggesting that the magnetic moments could be located outside the CuO 2 layer. Such a property has not been verified yet in all materials exhibiting IUC order, such as iridates or 2-leg ladder cuprates.

Symmetry considerations

In both cuprates and iridates, PND shows the existence of an IUC magnetism, whose characteristic properties are consistent with the LC-Θ I I phase pattern breaking discrete T-, P-and fourfold R-symmetries. In addition to PND, complementary evidences of such broken symmetries were reported by other techniques, ranging from dichroism in circularly polarized ARPES spectroscopy [START_REF] Kaminski | Spontaneous breaking of time-reversal symmetry in the pseudogap state of a high-Tc superconductor[END_REF], Kerr effect [START_REF] Xia | Polar Kerr-Effect Measurements of the High-Temperature YBa 2 Cu 3 O 6+x Superconductor: Evidence for Broken Symmetry near the Pseudogap Temperature[END_REF], Nernst effect [START_REF] Daou | Broken rotational symmetry in the pseudogap phase of a high-Tc superconductor[END_REF], optical birefringence measurements [START_REF] Lubashevsky | Optical Birefringence and Dichroism of Cuprate Superconductors in the THz Regime[END_REF], second harmonic generation (SHG) [START_REF] Zhao | Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate[END_REF][START_REF] Zhao | A global inversion-symmetrybroken 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][START_REF] Murayama | Diagonal nematicity in the pseudogap phase of HgBa 2 CuO 4+δ[END_REF][START_REF] Murayama | Bond Directional Anapole Order in a Spin-Orbit Coupled Mott Insulator Sr 2 (Ir 1-x Rh x )O 4[END_REF]] and recently photo-galvanic effects [START_REF] Lim | Observation of broken inversion and chiral symmetries in the pseudogap phase in single and double layer bismuth-based cuprates[END_REF], respectively. The Kerr effect was reported in cuprates and necessarily implies a global T-broken symmetry although this happens in YBa 2 Cu 3 O 6+x [START_REF] Xia | Polar Kerr-Effect Measurements of the High-Temperature YBa 2 Cu 3 O 6+x Superconductor: Evidence for Broken Symmetry near the Pseudogap Temperature[END_REF] at a slightly lower temperature than the other experimental probes [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Sato | Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in YBa 2 Cu 3 O[END_REF][START_REF] Zhao | Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate[END_REF][START_REF] Daou | Broken rotational symmetry in the pseudogap phase of a high-Tc superconductor[END_REF][START_REF] Lubashevsky | Optical Birefringence and Dichroism of Cuprate Superconductors in the THz Regime[END_REF]. Each technique brings complementary information to draw the portrait of the IUC order and to guide theoretical models.

Anapoles and quadrupoles

The LC-Θ I I state is the ground reference of the observed IUC magnetic order. It is characterized by T-and P-symmetry breaking (although their product is preserved) [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]. This view was reinforced in cuprates [START_REF] Zhao | A global inversion-symmetrybroken phase inside the pseudogap region of YBa 2 Cu 3 O[END_REF] and put forward in iridates [START_REF] Zhao | Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate[END_REF] by the observation of P-symmetry breaking in SHG experiments at the same temperature for a given doping. The LC-Θ I I magnetic pattern is described by two staggered orbital magnetic moments M i located on a diagonal of the CuO 2 plaquette and equidistant from the Cu site. Within the basal lattice and for the state described in Fig. 2.a, the magnetic moments M i = ±(0, 0, ±M c ) are then located at r i = ±(-r 0 , r 0 , 0). The LC-Θ I I exhibits, in principle, magneto-electric effects that can be described by a uniform C. R. Physique-Draft, 13th July 2021 arrangement of anapoles (also named : toroidal moments) [START_REF] Shekhter | Considerations on the symmetry of loop order in cuprates[END_REF]. The anapole Ω corresponds to a polar vector shown in Fig. 2.a constructed from the LC as,

Ω = i r i ∧ M i (1)
In general, an anapole is described using a current flowing around a solenoid bent to form a torus (Fig. 2.c). The winding current around the solenoid generates a circular magnetic field inside the torus which gives rise to a pure anapole along the axis of the torus, according to the definition of Ω. In 2D, the 3D torus reduces to 2 LCs turning clockwise and anticlockwise. Actually, this staggered orbital magnetism supports not only an anapole, but also a magnetic quadrupole tensor [START_REF] Di Matteo | Orbital currents, anapoles, and magnetic quadrupoles in CuO[END_REF]. This is not surprising as the anapole appears at the same order of the magnetic multipole expansion as the magnetic quadrupole moment, as discussed in the context of multiferroics [START_REF] Di Matteo | Orbital currents, anapoles, and magnetic quadrupoles in CuO[END_REF][START_REF] Spaldin | The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect[END_REF]. More precisely, for the LC-Θ I I state, Ω = 2r 0 M c (a + b) and the non-zero components of the magnetic quadrupole Q ac = Q bc = r 0 M c [START_REF] Di Matteo | Orbital currents, anapoles, and magnetic quadrupoles in CuO[END_REF].

LC-free models, involving Dirac multipoles, were actually developed to account for the IUC magnetic scattering reported by PND [START_REF] Lovesey | Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor[END_REF][START_REF] Lovesey | Ordered state of magnetic charge in the pseudo-gap phase of a cuprate superconductor (HgBa 2 CuO 4+δ )[END_REF][START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF]. The CuO 2 plaquette is not occupied anymore by an anapole bound to LCs flowing through Cu and O, but by magnetic quadrupoles localized on the Cu site. Symmetry arguments are used to select the allowed magnetic quadrupoles, which can further ensure the P-and T-symmetries breaking [START_REF] Lovesey | Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor[END_REF][START_REF] Lovesey | Ordered state of magnetic charge in the pseudo-gap phase of a cuprate superconductor (HgBa 2 CuO 4+δ )[END_REF][START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF]. Unlike electronic multiferroics where the charge and magnetic degrees of freedom form two distinct order parameters, these theoretical models predict the existence of new magneto-electric objects, in which the charge and electronic degrees of freedom are intimately coupled to form a single exotic order parameter, for instance due to electron-phonon coupling [START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF].

Similarly to the magnetic distribution of the LC-Θ I I state, a magnetic distribution with only in-plane magnetic moments can be considered as well [START_REF] Di Matteo | Orbital currents, anapoles, and magnetic quadrupoles in CuO[END_REF]: this corresponds to a pure magnetic quadrupole which is generated by a set of magnetic dipoles. Keeping the same notation as for the LC-Θ I I state, one could consider two magnetic moments M i = ±(-M ab , M ab , 0), again located at r i = ±(-r 0 , r 0 , 0). The non-zero components of the magnetic quadrupole are Q aa = 1 3 Q ab = - 1 2 Q cc = 2 3 r 0 M ab [START_REF] Di Matteo | Orbital currents, anapoles, and magnetic quadrupoles in CuO[END_REF] and differ from the ones of the quadrupole above. These planar magnetic moments do not generate an anapole. The existence of such a magnetic quadrupole could then explain the observed in-plane component (M ab ) of the IUC magnetism reported by PND, whereas the original LC-Θ I I state can only explain the measured out-of-plane component (M c ). As one observes experimentally a tilted IUC moment, none of these magnetic distributions alone can describe the PND results. However, a combination of these two pictures could account for the tilted IUC moment as we shall see below (section 3.3).

Degeneracy of LC-Θ I I state

It is worth emphasizing that the LC-Θ I I phase is characterized by four degenerate quantum states. The four LC patterns are deduced from each other by a 90 • rotation with the anapole pointing along all diagonals of the basal plane [START_REF] Aji | Theory of the coupling of quantum-critical fluctuations to fermions and d -wave superconductivity in cuprates[END_REF][START_REF] He | Theory of polarized neutron scattering in the loop-ordered phase of cuprates[END_REF]. They produce four types of magnetic domains that will be summed up in bulk PND experiments (Fig. 3.e). If one considers a classical LC model, each of the LC domains gives a magnetic scattering in PND corresponding to magnetic moments perpendicular to the CuO 2 layer. In contrast, if a quantum superposition of the four LC states is allowed, then an effective planar magnetic response can appear as a result of quantum interference in the neutron scattering cross-section [START_REF] He | Theory of polarized neutron scattering in the loop-ordered phase of cuprates[END_REF]. Therefore, the existence of the in-plane magnetic response could highlight the quantum effects in the LC state. The observation of a temperature dependent tilt angle in nearly optimally doped YBa 2 Cu 3 O 6.85 [START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in YBa 2 Cu 3 O 6.85[END_REF] could suggest a crossover from classical to quantum LC states, when thermal fluctuations are overcome by quantum ones. Within that scenario, quantum effects would then occur at relatively high temperature, around 200-250 K. This might be surprising unless the PG itself could impose the quantum superposition of the LC states which in turn produce the observed tilt [START_REF] He | Theory of polarized neutron scattering in the loop-ordered phase of cuprates[END_REF].

IUC point group symmetry

Considering the simple crystal structure of a tetragonal monolayer system, such as HgBa 2 CuO 4+δ (see Fig. 3.a), the LC-Θ I I state belongs to the magnetic point group m mm [START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF][START_REF] Orenstein | Optical Nonreciprocity in Magnetic Structures Related to High-T c Superconductors[END_REF] (m is a mirror plane followed by a time reversal operation). The three mirror planes are respectively: the (a+b,a-b), (a+b,c) and (a-b,c) planes. Combining the above discussed magnetic distributions, one can build the magnetic pattern reported in Fig. 4.a with two magnetic moments M i = ±(M ab , M ab , M c ). This magnetic arrangement corresponds to a rotation of the LC-Θ I I magnetic pattern around the (a-b) diagonal axis, yielding a monoclinic magnetic point group 2/m (as the rotation preserves the m mirror plane) [START_REF] Orenstein | Optical Nonreciprocity in Magnetic Structures Related to High-T c Superconductors[END_REF]. The magnetic pattern shown in Fig. 4.a differs from the one shown in Fig. 4.b where the planar magnetic component is aligned along the (ab) axis (the moments are tipped out of the plane of the M c components). The latter magnetic pattern has a magnetic point group 2 /m [START_REF] Orenstein | Optical Nonreciprocity in Magnetic Structures Related to High-T c Superconductors[END_REF]. Note that the planar magnetic components of Fig. 4.b cannot be associated with a pure magnetic quadrupole only as in Fig. 4.a. Indeed, they give a contribution to an anapole (-4r 0 M ab c), in addition to the magnetic quadrupole (Q aa = -Q bb = 2r 0 M ab ). While being distinct, the magnetic patterns reported in Fig. 4 produce hardly distinguishable neutron scattering cross-section and fully account for the observed IUC magnetism. It should be stressed out that the 2 /m point group was deduced from SHG results in YBa 2 Cu 3 O 6+x [START_REF] Zhao | A global inversion-symmetrybroken phase inside the pseudogap region of YBa 2 Cu 3 O[END_REF] based on i) the observed 2/m symmetry of the atomic structure at 300K and ii) the broken global inversion related to the absence of C 2 axis.

In order to account for the existence of an in-plane magnetic component, one could also relax the constraint that LCs have to be confined within the CuO 2 plaquette. In hole doped cuprates, the Cu site is located at the center of an O octahedron in monolayer compounds, which splits in O pyramids in bilayer ones. The LCs could then be delocalized on the faces of the CuO 6 octahedra or on the CuO 5 pyramids, yielding a natural tilt of their orbital moments. However, the PND experiments exclude the LC states occurring on the faces of the CuO 6 octahedra [START_REF] Weber | Orbital Currents in Extended Hubbard Models of High-T c Cuprate Superconductors[END_REF][START_REF] Orenstein | Optical Nonreciprocity in Magnetic Structures Related to High-T c Superconductors[END_REF][START_REF] Yakovenko | Tilted loop currents in cuprate superconductors[END_REF] or on the CuO 5 pyramids [START_REF] Lederer | Observable NMR signal from circulating current order in YBCO[END_REF] because the basal plane cannot also be a mirror plane for the inplane component M ab . Only the LC state decorating the CuO 6 octahedra with only two opposite moments, as shown in Fig. 5.a and represented in [START_REF] Agterberg | Emergent loop current order from pair density wave superconductivity[END_REF][START_REF] Yakovenko | Tilted loop currents in cuprate superconductors[END_REF], is compatible so far with the PND results in HgBa 2 CuO 4+δ [START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor HgBa 2 CuO 4+δ[END_REF][START_REF] Li | Magnetic order in the pseudogap phase of HgBa 2 CuO 4+δ studied by spin-polarized neutron diffraction[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] and belongs to the 2/m point group. It should be stressed that no LC models on the CuO 5 pyramids have been found to be consistent with the PND data in bilayer materials.

Beyond the LC approach, a (Cu-)quadrupole modelling of the PND data [START_REF] Lovesey | Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor[END_REF][START_REF] Lovesey | Ordered state of magnetic charge in the pseudo-gap phase of a cuprate superconductor (HgBa 2 CuO 4+δ )[END_REF][START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF] is also proposed with a magnetic structure belonging to m m m point group. It is not consistent actually with the data, as the basal plane cannot have a m symmetry plane for the M c component. This is due to the measurement of the magnetic structure factor at the Q-position for L = 0 where a larger tilt is typically deduced [START_REF] Mook | Observation of magnetic order in a superconducting YBa 2 Cu 3 O 6.6 single crystal using polarized neutron scattering[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]. Other possible quadrupole arrangements [START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF] are then designed to give a magnetic structure that belongs to the 2/m point group.

In the bilayer YBa 2 Cu 3 O 6+x system, the PND results in a detwinned sample [START_REF] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF] (see e.g. Fig. 5.b) are important as they rule out models [START_REF] Lovesey | Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor[END_REF][START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF][START_REF] Lederer | Observable NMR signal from circulating current order in YBCO[END_REF], which assume either an in-phase or out-ofphase coupling of IUC order within a given bilayer. This dismisses models based on a magnetic nematic order, involving spin or orbital moments located at the O sites [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Moskvin | Pseudogap phase in cuprates: Oxygen orbital moments instead of circulating currents[END_REF] as well.

Linear and circular photo-galvanic effects recently observed at the PG temperature [START_REF] Lim | Observation of broken inversion and chiral symmetries in the pseudogap phase in single and double layer bismuth-based cuprates[END_REF] in both monolayer and bilayer Bi-based cuprates suggest a mm2 point group symmetry, although monoclinic point group as 2/m or 2 /m are also possible. A very interesting outcome of these experiments is the fact that the phase coherence of the IUC order should be as large as the size of the illuminated beam (∼ 10µm).

Rotational symmetry breaking

The different IUC phases proposed break R-symmetry. This can be tested in various experiments as, for instance, via the loss of both C 4 rotation and mirror symmetry in the electronic structure of the CuO 2 plane [START_REF] Lubashevsky | Optical Birefringence and Dichroism of Cuprate Superconductors in the THz Regime[END_REF]. Indeed, the LC-Θ I I state imposes a preferential direction of the anapole along one of the diagonals of the CuO 2 plane. On the opposite, staggered LCs flowing on O sites only, as proposed in Ref. [START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF][START_REF] Chatterjee | Insulators and metals with topological order and discrete symmetry breaking[END_REF], give a pattern rotated at 45 • with respect to LC-Θ I I one and are characterized by an anapole along one of the CuO bonds (Fig. 2.b). However, owing to the formation of degenerate domains and assuming equal populations of the domains, PND cannot distinguish between these different magnetic models, see e.g. our results in 2-leg ladder cuprates [START_REF] Bounoua | Loop currents in two-leg ladder cuprates[END_REF]. Other techniques can provide information on the R-symmetry breaking. The broken fourfold R-symmetry is very often associated with the existence of electronic nematicity, which was reported using magnetic torque measurements in both cuprates [START_REF] Sato | Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in YBa 2 Cu 3 O[END_REF][START_REF] Murayama | Diagonal nematicity in the pseudogap phase of HgBa 2 CuO 4+δ[END_REF] and iridates [START_REF] Murayama | Bond Directional Anapole Order in a Spin-Orbit Coupled Mott Insulator Sr 2 (Ir 1-x Rh x )O 4[END_REF]. In the monolayer HgBa 2 CuO 4+δ compound, the torque magnetometry reports an anisotropy, χ ab = 0, along the diagonals of the CuO 2 plane [START_REF] Murayama | Diagonal nematicity in the pseudogap phase of HgBa 2 CuO 4+δ[END_REF], consistent with a distortion induced by the original LC-Θ I I state [START_REF] Shekhter | Considerations on the symmetry of loop order in cuprates[END_REF]. In monolayer iridates, torque experiments also show an electronic C. R. Physique-Draft, 13th July 2021 nematicity [START_REF] Murayama | Bond Directional Anapole Order in a Spin-Orbit Coupled Mott Insulator Sr 2 (Ir 1-x Rh x )O 4[END_REF] with a nematic director, which is rotated by 45 • from that of monolayer cuprate HgBa 2 CuO 4+δ . This implies the different LC patterns between the two systems, suggesting orbital-current configurations also turned by 45 • (Fig. 2.b) [START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF][START_REF] Chatterjee | Insulators and metals with topological order and discrete symmetry breaking[END_REF].

In the bilayer YBa 2 Cu 3 O 6+x compound, there is a weak orthorhombicity (a = b) and an electronic nematicity was reported through Nernst effect measurements [START_REF] Daou | Broken rotational symmetry in the pseudogap phase of a high-Tc superconductor[END_REF], showing the spontaneous appearance of a strong ab anisotropy. In addition, the torque magnetometry shows an anisotropy along the principal directions, with χ aa > χ bb [START_REF] Sato | Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in YBa 2 Cu 3 O[END_REF] i.e. turned by 45 • from the single layer compound. Actually, the surprising difference in torque measurement results in monolayer HgBa 2 CuO 4+δ and bilayer YBa 2 Cu 3 O 6+x compounds can be solved by a close examination of the way LC-Θ I I patterns are stacked within a bilayer (along the c-axis) as shown by previous PND results [START_REF] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF]. Indeed, using a twin-free sample of the orthorhombic YBa 2 Cu 3 O 6.6 , our PND results show a distinct ab anisotropy of the IUC magnetic structure factor [START_REF] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF], as a result of R-symmetry breaking. This highlights that the IUC order in this material breaks the mirror symmetry of the CuO 2 bilayer (Fig. 5.b). For a single CuO 2 plaquette, 4 LC patterns are allowed and, then, 4 × 4 LC patterns for a bilayer. Only 4 of them produce the magnetic structure factor matching the observed ab anisotropy found in PND measurements. They correspond to a crisscrossed arrangement of the LC patterns in the bilayer, yielding a resulting anapole, Ω 1 + Ω 2 , parallel to b axis [START_REF] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF] as shown in Fig. 5.b. Note that each of the 4 selected LC patterns could be fully identified when using both their anapole and their chiral parameter, c(Ω 1 ∧ Ω 2 ).

From this result and using the corrugation of the CuO 2 plane (usually called dimpling in the literature), corresponding to the fact that Cu and O atoms lay in slightly different (but parallel) planes, one can construct a picture to explain the origin of the torque anisotropy (Fig. 6a-d).

Assuming that LCs run exactly within the Cu-O triangle, the dimpling of the CuO 2 layer induces a tilt of the out-of-plane moment, yielding a weak ferromagnetic in-plane component M, pointing perpendicular to the anapole Ω i in each layer. As the total bilayer anapole is found to be along the b direction, the remaining weak ferromagnetic component for the bilayer is found also along b (Fig. 6.d). Accordingly, the magnetic torque τ, defined as τ ∝ M ∧ H for an applied magnetic field H along c, is then expected to give an anisotropy along the a direction as it is reported in [START_REF] Sato | Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in YBa 2 Cu 3 O[END_REF]. This simple picture gives a nice agreement between the microscopic picture deduced from PND experiment and the macroscopic torque measurements. It should be stressed that the weak ferromagnetic component would be averaged owing to the existence of the four domains of the LC-Θ I I state [START_REF] Aji | Theory of the coupling of quantum-critical fluctuations to fermions and d -wave superconductivity in cuprates[END_REF][START_REF] He | Theory of polarized neutron scattering in the loop-ordered phase of cuprates[END_REF].

Electronic nematicity was reported in the bilayer Bi 2 Sr 2 CaCu 2 O 8+δ family as well, using either STM images of the IUC states [START_REF] Fujita | Simultaneous Transitions in Cuprate Momentum-Space Topology and Electronic Symmetry Breaking[END_REF][START_REF] Fujita | Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates[END_REF][START_REF] Lawler | Intra-unit-cell electronic nematicity of the high-Tc copper-oxide pseudogap states[END_REF][START_REF] Mukhopadhyay | Evidence for a vestigial nematic state in the cuprate pseudogap phase[END_REF] or electronic Raman scattering in the B 1g symmetry [START_REF] Auvray | Nematic fluctuations in the cuprate superconductor Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF]. As in the bilayer YBa 2 Cu 3 O 6+x material, a nematic director is found along the Cu-O bonds. Therefore, the nematic state in Bi 2 Sr 2 CaCu 2 O 8+δ can be understood the same way assuming a similar crisscrossed arrangement of LCs within the CuO 2 bilayer, i.e. with a total anapole moment in the bilayer pointing along the Cu-O bonds. Alternatively, the LC-Θ I I state is a q = 0 electronic state that could co-exist with other orthogonal q = 0 electronic states, such as an electronic nematic state.

To conclude, the proposed LC-Θ I I represents the most adequate starting point to describe the IUC magnetism observed in PND experiments. It takes into account the various constraints imposed by symmetry even though additional features need to be considered. experimental spectroscopic probes in cuprates, for instance electronic Raman scattering [START_REF] Sacuto | Pseudogap in Cuprates by Electronic Raman Scattering[END_REF]. For a long time, the PG physics in 2D cuprates has been considered as a crossover phenomenon observed in various physical properties, occurring over a certain range of temperatures. No sharp anomaly was seen in the specific heat data, from which it was concluded that T * , where the PG opens in transport measurements, cannot correspond to a thermodynamic phase transition [START_REF] Loram | Evidence on the pseudogap and condensate from the electronic specific heat[END_REF]. In the recent years, as pointed out above, various symmetry breakings were identified over a much thinner range of temperatures. Independent measurements of an order parameter breaking T-, P-and R-symmetries agree on the location of a phase transition seen in experiments probing the thermodynamic properties. This is clearly established in three high-T c superconducting cuprate families (the monolayer HgBa 2 CuO 4+δ , the bilayers YBa 2 Cu 3 O 6+x and C. R. Physique-Draft, 13th July 2021 Bi 2 Sr 2 CaCu 2 O 8+δ ). Fig. 7 shows exclusively experimental results in Bi 2 Sr 2 CaCu 2 O 8+δ . That system is important to adress the issue of the PG because surface spectroscopic techniques can be performed as that material cleaves easily. Numerous ARPES [START_REF] Vishik | Phase competition in trisected superconducting dome[END_REF] and tunneling spectroscopy of junctions [START_REF] Dipasupil | Energy Gap Evolution in the Tunneling Spectra of Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF][START_REF] Ozyuzer | Absence of pseudogap in heavily overdoped Bi 2 Sr 2 CaCu 2 O 8+δ from tunneling spectroscopy of break junctions[END_REF] studies have been performed to establish the depletion of the electronic states in that particular cuprate family. These estimates agree with T * deduced from resistivity measurements [START_REF] Watanabe | Anisotropic Resistivities of Precisely Oxygen Controlled Single-Crystal Bi 2 Sr 2 CaCu 2 O 8+δ : Systematic Study on "Spin Gap" Effect[END_REF] as well with those obtained from various spectroscopies [START_REF] Toda | Rotational symmetry breaking in Bi 2 Sr 2 CaCu 2 O 8+δ probed by polarized femtosecond spectroscopy[END_REF][START_REF] Loret | Universal relationship between the energy scales of the pseudogap phase, the superconducting state, and the charge-density-C. R. Physique-Draft, 13th[END_REF]. The IUC magnetism observed using PND was among the first reported order parameters in the PG state, which starts at a temperature T mag that matches T * (see Figs. 1 and7) and agrees with the downturn of the resistivity measurements in YBa 2 Cu 3 O 6+x [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF], in HgBa 2 CuO 4+δ [START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor HgBa 2 CuO 4+δ[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] and in Bi 2 Sr 2 CaCu 2 O 8+δ [START_REF] Watanabe | Anisotropic Resistivities of Precisely Oxygen Controlled Single-Crystal Bi 2 Sr 2 CaCu 2 O 8+δ : Systematic Study on "Spin Gap" Effect[END_REF]. That suggested that the PG line corresponds to a true phase transition. Later, detailed analysis of the magnetization measurements [START_REF] Leridon | Thermodynamic signature of a phase transition in the pseudogap phase of YBa 2 Cu 3 O x high-TC superconductor[END_REF], resonant ultrasound spectroscopy [START_REF] Shekhter | Bounding the pseudogap with a line of phase transitions in YBa 2 Cu 3 O 6+x[END_REF] and torque measurements [START_REF] Sato | Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in YBa 2 Cu 3 O[END_REF][START_REF] Murayama | Diagonal nematicity in the pseudogap phase of HgBa 2 CuO 4+δ[END_REF] demonstrated in both YBa 2 Cu 3 O 6+x and HgBa 2 CuO 4+δ that the PG line is truly a thermodynamic phase transition.

Loop currents for Cuprates

Pseudogap line in cuprates: a phase transition

Although the reported features at T * are rather weak in amplitude, they are all observed due to the higher accuracy of those techniques compared with specific heat data, where a large phonon background has to be removed. This establishes that the PG state is necessarily associated with an ordered state, a point that should be present in any model relevant for the physics of high-T c cuprates. For instance, it is argued [START_REF] Varma | Specific heat and sound velocity at the relevant competing phase of high-temperature superconductors[END_REF] that these results are compatible with the thermodynamics of the Ashkin-Teller model, which corresponds to the universality class of the LC-ordered phase, as this model does not produce any strong jump in the specific heat for a wide range of parameters [START_REF] Grønsleth | Thermodynamic properties near the onset of loop-current order in high-T c superconducting cuprates[END_REF].

It has to be stressed that no other true symmetry-breaking phase than the IUC magnetism develops long range 3D correlation at T * . Contrariwise, the incipient CDW phase develops at lower temperature (Fig. 1), deep inside the PG state [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF][START_REF] Proust | The Remarkable Underlying Ground States of Cuprate Superconductors[END_REF] and remains short range and quasi-2D. Note that uniform perturbations, such as a strong external magnetic field [START_REF] Leboeuf | Thermodynamic phase diagram of static charge order in underdoped YBa 2 Cu 3 O[END_REF] or a uni-axial strain [START_REF] Kim | Uniaxial pressure control of competing orders in a high-temperature superconductor[END_REF], are required to promote a long range ordered CDW, at the expense of superconductivity. If uniform perturbations can be used to tune the balance between superconductivity and CDW inside the PG state, the substitution by a minute amount of non magnetic (Zn) impurity on Cu site is sufficient to lower strongly the onset temperature of both states [START_REF] Blanco-Canosa | Momentum-Dependent Charge Correlations in YBa 2 Cu 3 O 6+δ Superconductors Probed by Resonant X-Ray Scattering: Evidence for Three Competing Phases[END_REF]. In contrast, the PG state [START_REF] Alloul | Correlations between magnetic and superconducting properties of Zn-substituted YBa 2 Cu 3 O 6+x[END_REF] and the onset temperature of IUC magnetism [START_REF] Balédent | Evidence for competing magnetic instabilities in underdoped YBa 2 Cu 3 O 6+x[END_REF] are both extremely robust against non magnetic impurity substitution. Furthermore, the IUC magnetism is reduced in amplitude through Zn substitution [START_REF] Balédent | Evidence for competing magnetic instabilities in underdoped YBa 2 Cu 3 O 6+x[END_REF], in agreement with a picture established by NMR and STM [START_REF] Alloul | Defects in correlated metals and superconductors[END_REF] that the PG physics basically vanishes near Zn impurities.

Loop currents and pseudogap

Because of its central play in the generic phase diagram, it is compulsory to consider PG physics in any scenario aiming at a description of the physics of cuprates. Being intimately tied to the PG state, the IUC magnetism, reported by PND, has to be implemented in any theoretical approach. Although the interplay between the LCs and the PG state is rather obvious from experimental results, its theoretical importance remains unclear as the proposed LC order in cuprates cannot open a PG at the Fermi level simply because it preserves LT symmetry [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF]. Indeed, q=0 electronic instabilities in the electron-hole channel commonly split or distort the Fermi surface. This is the case for a ferromagnetic state, where two Fermi surfaces with different spin polarizations form in order to break the T-symmetry, or for an electronic nematic state where the Fermi Surface is distorted to break the R-symmetry. The only way to open a gap is then to break the LTsymmetry. Note, that superconductivity, which gaps the Fermi surface, is also a q=0 electronic instability, but in the electron-electron channel and it breaks a gauge symmetry. Therefore, one is left with two possibilities that we discuss below: (i) there is an hidden modulation of the IUC [START_REF] Almeida-Didry | Evidence for intra-unitcell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF][START_REF] Mangin-Thro | Characterization of the intra-unit-cell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF], transport [START_REF] Watanabe | Anisotropic Resistivities of Precisely Oxygen Controlled Single-Crystal Bi 2 Sr 2 CaCu 2 O 8+δ : Systematic Study on "Spin Gap" Effect[END_REF], tunnel [START_REF] Dipasupil | Energy Gap Evolution in the Tunneling Spectra of Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF][START_REF] Ozyuzer | Absence of pseudogap in heavily overdoped Bi 2 Sr 2 CaCu 2 O 8+δ from tunneling spectroscopy of break junctions[END_REF], electronic Raman scattering [START_REF] Loret | Universal relationship between the energy scales of the pseudogap phase, the superconducting state, and the charge-density-C. R. Physique-Draft, 13th[END_REF], ARPES [START_REF] Vishik | Phase competition in trisected superconducting dome[END_REF], pump probe spectroscopy (PPS) [START_REF] Toda | Rotational symmetry breaking in Bi 2 Sr 2 CaCu 2 O 8+δ probed by polarized femtosecond spectroscopy[END_REF] and photo-galvanic (PGE) [START_REF] Lim | Observation of broken inversion and chiral symmetries in the pseudogap phase in single and double layer bismuth-based cuprates[END_REF] experiments. Representative area for superconductivity (SC) and pseudogap (PG) are also depicted. magnetism probed by PND, (ii) or the PG state is a distinct state of matter, intertwined with the IUC magnetism.

A modified version of the LC state, now breaking LT-symmetry, was recently proposed [START_REF] Varma | Pseudogap and Fermi arcs in underdoped cuprates[END_REF]. It consists of a LC super-cell of the four degenerate LC domains. Importantly, the next-nearest domain of a given domain should be characterized by an anapole rotated by π/2, either turned left or right, whereas the domain with an anapole rotated by π is only joining the first domain through its corners. The domains size, counted in number of unit cells P , ranges from a few cells to dozens of them. Examples of such phases are depicted in Fig. 2.d for P=1 and in Fig. 2.e for P=2. It is shown that domains of P ∼ 10-20 unit cells give rise to both a PG and Fermi arcs [START_REF] Varma | Pseudogap and Fermi arcs in underdoped cuprates[END_REF], as experimentally evidenced by ARPES and STM over the last two decades [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF]. In this approach, the new magnetic unit cell is square, (2P × 2P ) larger than the CuO 2 unit cell, and it is characterized by the planar wave vectors q LC =(1/2P, 0) and (0, 1/2P ). In a diffraction experiment like PND or resonant X-ray diffraction, this has major consequences, as the translation symmetry is now broken. Although this super-cell LC state has not been observed experimentally yet, it cannot be excluded from the current PND data sets if P is large enough. Indeed, the diffraction pattern has no maximum at the (1, 0, L) or (0, 1, L) Bragg reflections anymore, but gives rise to satellite reflections at ±q LC away from them. If P is large enough, the diffracted peaks would C. R. Physique-Draft, 13th July 2021 occur as centered at the Bragg positions owing to the broad neutron instrument resolution, and to the limitations of the experiments with respect to the finite neutron polarization [START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF][START_REF] Bourges | Evidence for Intra-Unit Cell Magnetism in Superconducting Cuprates: a Technical Assessment[END_REF][START_REF] Bourges | Comment on "No evidence for orbital loop currents in charge-ordered YBa 2 Cu 3 O 6+x from polarized neutron diffraction[END_REF]. Furthermore, if there were a distribution of domains size P [START_REF] Varma | Pseudogap in cuprates in the loop-current ordered state[END_REF], it would lead to an effective broadening of the magnetic peaks appearing on the shoulders of the crystal Bragg peaks.

Keeping the initial uniform LC phase, the latter can be considered as ancillary to a mother state associated with the PG state. For instance, square lattice spin liquid states are predicted to possess spontaneous LCs [START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF][START_REF] Chatterjee | Insulators and metals with topological order and discrete symmetry breaking[END_REF]. In the framework of a fractionalized spin density wave, the LCs also appear, forming an ancillary order, resulting from the intertwining between a topological order and the discrete Z 2 broken symmetries. In spin liquids, the emergence of LC orders in the archetype of doped 1D spin-liquids (hole-doped spin ladders) has been proposed [START_REF] Chudzinski | Orbital current patterns in doped two-leg Cu-O Hubbard ladders[END_REF]. Our recent results in spin-ladders [START_REF] Bounoua | Loop currents in two-leg ladder cuprates[END_REF] shows that local discrete symmetries are also broken in 1D spin liquid systems, as it has been theoretically predicted. In other models, based on a PDW state [START_REF] Agterberg | Emergent loop current order from pair density wave superconductivity[END_REF][START_REF] Dai | Pair-density waves, charge-density waves, and vortices in high-T c cuprates[END_REF] or a fractionalized PDW instability [START_REF] Sarkar | Incipient loop-current order in the underdoped cuprate superconductors[END_REF], the PG is described as a highly fluctuating state, where a preemptive phase breaking both T-and P-symmetries (a LC-like phase) is expected at higher temperature. More generally, discrete symmetry breaking usually happens at higher temperature than continuous symmetry breaking. A strong influence of the LCs state on both unidirectional and bidirectional d -wave charge-density-wave/pair-density-wave (CDW/PDW) composite orders has been emphasized [START_REF] Wang | Charge-density-wave order with momentum (2Q, 0) and (0, 2Q) within the spin-fermion model: Continuous and discrete symmetry breaking, preemptive composite order, and relation to pseudogap in hole-doped cuprates[END_REF][START_REF] De Carvalho | Coexistence of Θ I I -loop-current order with checkerboard d -wave CDW/PDW order in a hot-spot model for cuprate superconductors[END_REF]. In these approaches, whatever the theoretical framework, the IUC LC order has to be intertwined with other states of matter [START_REF] Cai | Intertwined Order and Holography: The Case of Parity Breaking Pair Density Waves[END_REF]. For instance in ref. [START_REF] Sarkar | Incipient loop-current order in the underdoped cuprate superconductors[END_REF], one starts with two primary states reported at lower temperature deep in the PG state (namely, the incipient CDW and superconductivity, both carrying a d -wave symmetry). The PG state is proposed to be a composite multi-component order parameter, where the system fluctuates between the two primary states which are intertwined. Then, the LC state would emerge as a preemptive state, born out of a higher order combination of the composite order parameters, which can be observed in the PG state (as soon as it forms), whenever the primary states still fluctuate or become static on cooling down.

Another interesting question is the relationship of the IUC magnetism with other q=0 electronic instabilities. In addition to the electronic nematicity observed through the Nernst effect in YBa 2 Cu 3 O 6+x effect at T * [START_REF] Daou | Broken rotational symmetry in the pseudogap phase of a high-Tc superconductor[END_REF], enhanced electronic nematic fluctuations were observed close to the end point of the PG state at large hole doping [START_REF] Auvray | Nematic fluctuations in the cuprate superconductor Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF]. However, they deviate from a canonical quantum critical scenario and cannot be directly associated with the PG state. Finally, the electronic nematicity could be inherited from another instability as suggested, for instance, by the sub-linear thermal dependence of torque measurements [START_REF] Sato | Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in YBa 2 Cu 3 O[END_REF][START_REF] Murayama | Diagonal nematicity in the pseudogap phase of HgBa 2 CuO 4+δ[END_REF]. This other instability could actually correspond to the LC state. Alternatively, it could be a vestigial order [START_REF] Fradkin | Colloquium: Theory of intertwined orders in high temperature superconductors[END_REF] related to the CDW, which exhibits a d -wave structure factor, implying that the electronic density on the O sites is involved [START_REF] Mukhopadhyay | Evidence for a vestigial nematic state in the cuprate pseudogap phase[END_REF]. At low temperature, the CDW breaks both the LT symmetry and the fourfold R-symmetry. At high temperature, when the CDW correlation length becomes shorter than the pitch of the modulation, the LT symmetry is restored and, the electronic nematic correlation only is left inside the unit cell. Both states can be viewed as the nematic (R-broken symmetry) and smectic (LT-broken symmetry) phases of an electronic liquid crystal. The role of IUC magnetism in this last approach remains unclear.

IUC time-scale

An important aspect of the IUC magnetic order seen in PND is that the local magnetic probes like nuclear magnetic resonance (NMR) and muon-spin rotation (µSR) are basically silent [START_REF] Bourges | Novel magnetic order in the pseudogap state of high-Tc copper oxides superconductors[END_REF] because detecting the corresponding magnetic fields with magnetic resonance techniques has proved to be vastly unsuccessful, see e.g. in HgBa 2 CuO 4+δ [START_REF] Mounce | Absence of Static Loop-Current Magnetism at the Apical Oxygen Site in HgBa 2 CuO 4+δ from NMR[END_REF]. Another attempt using NMR in an ortho-II YBa 2 Cu 3 O 6+x sample concluded that any static field at T=60K cannot be larger than ∼0.3 G at apical O site and ∼4 G at the planar O(2) sites [START_REF] Wu | Incipient charge order observed by NMR in the normal state of YBa 2 Cu 3 O[END_REF]. For a static IUC magnetic order, a magnetic field of one order of magnitude larger is expected on these sites. The fact that the existence of a static long ranged IUC magnetism cannot be validated by NMR is a serious issue, which questions the outcome of the PND measurements. One possibility to reconcile NMR and PND data is to consider internal fluctuating fields at a timescale slow enough to appear static at the timescale of PND scattering (∼ 0.1 ns) but too fast to impact the NMR lines (≥ 10µs).

The situation with µSR is more subtle. Since the first discovery of IUC magnetism with PND, it has been envisaged [START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Varma | Pseudogap in cuprates in the loop-current ordered state[END_REF][START_REF] Pal | Investigation of potential fluctuating intra-unit cell magnetic order in cuprates by µSR[END_REF] that the magnetic moments could still fluctuate at the µSR timescale, meaning slowly enough to appear static to neutrons, but too fast to be observed as a magnetic order in µSR, therefore even more so for NMR, which probes even slower timescales. Several controversial results were reported by one team [START_REF] Pal | Investigation of potential fluctuating intra-unit cell magnetic order in cuprates by µSR[END_REF][START_REF] Pal | Quasistatic internal magnetic field detected in the pseudogap phase of Bi 2+x Sr 2-x CaCu 2 O 8+δ by muon spin relaxation[END_REF][START_REF] Gheidi | Absence of µSR evidence for magnetic order in the pseudogap phase of Bi 2+x Sr 2-x CaCu 2 O 8+δ[END_REF] but as a matter of fact, this interesting picture has been experimentally approved, as slow magnetic fluctuations were discovered using a longitudinal field µSR technique in YBa 2 Cu 3 O 6+x compounds [START_REF] Zhang | Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa 2 Cu 3 O[END_REF][START_REF] Zhu | Muon spin relaxation and fluctuating magnetism in the pseudogap phase of YBa 2 Cu 3 O[END_REF] and in iridates [START_REF] Tan | Slow magnetic fluctuations and critical slowing down in Sr 2 Ir 1-x Rh x O 4[END_REF].

In several YBa 2 Cu 3 O 6+x underdoped samples [START_REF] Zhang | Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa 2 Cu 3 O[END_REF][START_REF] Zhu | Muon spin relaxation and fluctuating magnetism in the pseudogap phase of YBa 2 Cu 3 O[END_REF], at 80K just above the superconducting temperature, the µSR measurements report magnetic correlations at a finite timescale of ∼ 10 ns which fluctuate slowly enough to give rise to resolution-limited magnetic peaks in a PND experiment. Furthermore, this time-scale does not appear to be strongly temperature dependent. Such a finite time-scale associated with finite spatial domains was proposed to happen if defects limit the magnetic correlation length [START_REF] Varma | Pseudogap in cuprates in the loop-current ordered state[END_REF]. Short range correlations with finite in-plane correlation lengths, ξ ∼ 20a, were actually detected in nearly optimally doped YBa 2 Cu 3 O 6+x using PND [START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in YBa 2 Cu 3 O 6.85[END_REF]. As discussed above, shorter in-plane IUC correlation lengths are even observed in other cuprates, limited either by the competition with charge stripes in (La,Sr) 2 CuO 4 [START_REF] Balédent | Two-Dimensional Orbital-Like Magnetic Order in the High-Temperature La 2-x Sr x CuO 4 Superconductor[END_REF] or by dimension in two-leg spin ladder compounds [START_REF] Bounoua | Loop currents in two-leg ladder cuprates[END_REF]. Clearly, dimension, disorder and proliferation of magnetic domains associated with degenerate LC states all can play a role.

The magnetic correlations seen by µSR can therefore be associated with slowly fluctuating magnetic domains. Interestingly, these fluctuations are maximum at the same temperature T mag at which the neutron magnetic signal sets in [START_REF] Zhang | Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa 2 Cu 3 O[END_REF][START_REF] Zhu | Muon spin relaxation and fluctuating magnetism in the pseudogap phase of YBa 2 Cu 3 O[END_REF]. This suggests a critical slowing down of these magnetic fluctuations at T * , associated with the PG onset. Such dynamical fluctuations at T * have also been detected by PND in nearly optimally doped YBa 2 Cu 3 O 6+x [START_REF] Mangin-Thro | Intra-unit-cell magnetic correlations near optimal doping in YBa 2 Cu 3 O 6.85[END_REF] and in HgBa 2 CuO 4+δ at two hole doping levels. [START_REF] Tang | Orientation of the intra-unit-cell magnetic moment in the high-T c superconductor HgBa 2 CuO 4+δ[END_REF].

Possible roles of loop currents

As it does not affect the LT symmetry, the original LC states are usually not considered as a key player for the physics of high-T c superconducting cuprates. It is typically much less discussed than CDW, stripes physics or AF fluctuations [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF][START_REF] Proust | The Remarkable Underlying Ground States of Cuprate Superconductors[END_REF]. This might be for two main reasons. First, it is a more exotic phenomenon with, so far, a limited number of realizations in nature. The recent observation of P- [START_REF] Zhao | Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate[END_REF], T- [START_REF] Jeong | Time-reversal symmetry breaking hidden order in Sr 2 (Ir,Rh)O 4[END_REF] and R- [START_REF] Murayama | Bond Directional Anapole Order in a Spin-Orbit Coupled Mott Insulator Sr 2 (Ir 1-x Rh x )O 4[END_REF] symmetry breaking in (Sr,Rh) 2 IrO 4 iridates clearly opens new frontiers as it basically corresponds to a LCs picture [START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF][START_REF] Chatterjee | Insulators and metals with topological order and discrete symmetry breaking[END_REF]. Second, it is a state which involves not only copper orbitals but also oxygen orbitals. This mixing with an electronic phase running among the three d and p orbitals is a central feature, which imposes a three-band Hubbard model as a starting point [START_REF] Varma | Theory of the pseudogap state of the cuprates[END_REF][START_REF] Weber | Phase Diagram of a Three-Orbital Model for High-T c Cuprate Superconductors[END_REF][START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF]. A large majority of the models built for cuprates usually neglects the possible role of oxygen orbitals, as the band structure is nicely reproduced by a single renormalized band electronic structure observed in ARPES and STM [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF].

Numerical calculations using different methods still give contrasted pictures, even for threeband Hubbard model. On the one hand, variational Monte Carlo calculations on asymptotically large lattices and exact diagonalization on a 24-site cluster shows a stabilization of the LC-Θ I I for a wide range of parameters [START_REF] Weber | Phase Diagram of a Three-Orbital Model for High-T c Cuprate Superconductors[END_REF]. On the other hand, no instance of spontaneous LCs has C. R. Physique-Draft, 13th July 2021 been found by applying cluster dynamical mean field theory in ladder cuprates [START_REF] Lu | Loop currents in ladder cuprates: A dynamical mean field theory study[END_REF] in materials where they have been observed [START_REF] Bounoua | Loop currents in two-leg ladder cuprates[END_REF]. As a matter of fact, variational Monte Carlo is a method to determine ground states at zero temperature, in contrast to the dynamical mean field theory, which is typically more efficient at finite temperature.

Generally speaking, LCs could be of great fundamental interest for several reasons. First, their fluctuations could be involved in the pairing mechanism and thus explain high-temperature superconductivity [START_REF] Aji | Theory of the coupling of quantum-critical fluctuations to fermions and d -wave superconductivity in cuprates[END_REF]. Within that scenario, the electrons are paired through the coupling of their local angular momentum to an operator defined as the generator of the rotations in the space of the four degenerate states discussed above. The corresponding pairing vertex favors d -wave superconductivity and is also proportional to the LC order susceptibility. Along similar lines, a novel spin-fluctuation-driven charge loop current mechanism based on the functional renormalization group theory has been proposed [100]. The discussed mechanism leads to ferro-LC order in a simple frustrated chain Hubbard model, which indicates that the LC can be universal in strongly correlated electron metals near magnetic criticality with geometrical frustration.

In the normal state, the related density susceptibility takes the functional form hypothesized for the Marginal Fermi liquid or strange metals [START_REF] Varma | Colloquium: Linear in temperature resistivity and associated mysteries including high temperature superconductivity[END_REF]. Namely, it should be rather independent of momentum and with almost structure-less energy dependence. Such a charge response has been recently measured in cuprates using a new technique probing the charge-charge correlation function [START_REF] Mitrano | Anomalous density fluctuations in a strange metal[END_REF]. It is argued that it corresponds to the density susceptibility expected for the LC fluctuations in the normal state [START_REF] Varma | Colloquium: Linear in temperature resistivity and associated mysteries including high temperature superconductivity[END_REF]. Interestingly, LC states are thought to be an emergent phenomenon, important for the establishment of the strange metals physics [START_REF] Else | Strange metals as ersatz Fermi liquids[END_REF]. More specifically, if the scaling behavior of the conductivity is observed independent of impurities, they must be due to fluctuations of an order parameter which is a vector that breaks T and P symmetries and preserves their product [START_REF] Else | Strange metals as ersatz Fermi liquids[END_REF]. That is a precisely the statement for LC order.

LC orders could also be of particular importance in certain theories of the PG state in cuprates in relation with topological orders [START_REF] Scheurer | Orbital currents in insulating and doped antiferromagnets[END_REF] or emanating from the fluctuations of a vectorial order parameter [START_REF] Sarkar | Incipient loop-current order in the underdoped cuprate superconductors[END_REF][START_REF] Morice | Pseudo-spin skyrmions in the phase diagram of cuprate superconductors[END_REF] as discussed above.

Conclusions and Perspectives

In this brief overview, arguments were listed in favor of the physics of LCs in quantum materials. Numerous experiments that include PND point towards the existence of an exotic LC order parameter that does not break the LT-symmetry. At a microscopic level, this corresponds to an electronic state phasing d orbitals of the transition metals with the ligand p orbitals. To some extent, the observed phenomenon could also be described by the ordering of magnetic quadrupoles of the transition metals [START_REF] Lovesey | Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor[END_REF][START_REF] Fechner | Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors[END_REF]. All these order parameters, anapoles, quadrupoles or orbital currents curls have in common the breaking of both T-and P-symmetries and the conservation of their product. These objects have dual (magnetic and electric) character order parameter, leading to interesting new applications for magneto-electricity [START_REF] Fiebig | Revival of the magnetoelectric effect[END_REF]. For instance, they are proposed for potential applications in the field of data storage, where it would be possible to control magnetization via the application of an electric field, and thus use these toroidal domains as magnetic qubit [START_REF] Spaldin | The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect[END_REF].

In this context, future developments will aim at exploring and generalizing the observation of these new exotic objects in a wider range of materials. That will follow several directions: for instance, the Fe-based superconducting materials could host as well such a phase. It is actually proposed that the spin-orbit coupling present in the pnictides enforces the emergence of orbital loop current order inside the usual stripe-type spin-density wave state [START_REF] Klug | Orbital loop currents in iron-based superconductors[END_REF]. In any case, the investigation of LCs in other families of cuprate compounds using PND, namely in CuO, Sr 2 CuO 2 Cl 2 and La 2-x Ba x CuO 4 should be pursued. In CuO, LCs were reported by a resonant x-ray scattering study [START_REF] Scagnoli | Observation of Orbital Currents in CuO[END_REF], though this interpretation is not necessarily supported by later experiments and simulations [START_REF] Di Matteo | Orbital currents, anapoles, and magnetic quadrupoles in CuO[END_REF][START_REF] Joly | Birefringence and polarization rotation in resonant x-ray diffraction[END_REF]. In Sr 2 CuO 2 Cl 2 , a recent optical SHG experiment points towards a magneto-chiral state [START_REF] De La Torre | Mirror symmetry breaking in a model insulating cuprate[END_REF]. Following previous PND results [START_REF] Balédent | Two-Dimensional Orbital-Like Magnetic Order in the High-Temperature La 2-x Sr x CuO 4 Superconductor[END_REF], studying La 2-x Ba x CuO 4 will allow one to the study of the competition between the IUC magnetic order and other electronic instabilities, in that particular case the long range ordering of charge stripes [START_REF] Hücker | Stripe order in superconducting La 2-x Ba x CuO 4 (0.095 x 0.155)[END_REF]. All these examples should address the important question of the coexistence of LC phases with AF order in highly hole doped cuprates.

Another route to explore the magnetism related to the anapoles or quadrupoles is to use resonant X-ray diffraction at the appropriate atomic edge [START_REF] Lovesey | Ordered state of magnetic charge in the pseudo-gap phase of a cuprate superconductor (HgBa 2 CuO 4+δ )[END_REF][START_REF] Di Matteo | Orbital currents, anapoles, and magnetic quadrupoles in CuO[END_REF][START_REF] Scagnoli | Observation of Orbital Currents in CuO[END_REF][START_REF] Joly | Birefringence and polarization rotation in resonant x-ray diffraction[END_REF]. Unlike usual nonresonant X-ray diffraction, the magnetic signal is enhanced at the resonant energy threshold of atomic electronic transitions. This technique, sensitive to both T-and P-symmetries, allows the separation of the magnetic signal from the atomic signal, by the use of linear or circular polarization analysis of the scattered beam. It allows one to selectively probe dipole, quadrupole transitions or the interaction between these two processes. A key advantage is the ability to select the ion whose electronic transitions are induced by proper selection of the incident X-ray energy. For cuprates, measurements need to be carried at the absorption K-edge of Cu in order to access the terms corresponding to the anapoles, as derived from the multipolar expansion series. The IUC magnetic ordering observed in PND should induce an additional contribution to the anomal structure factor. For example, at the Cu K-edge (8985 eV), a dipolar-quadrupolar interference term (E1-E2) should occur with a characteristic dependence as a function of the azimuthal angle. This experiment certainly represents an important challenge.

Finally, the LC story has not been completed yet. The exact interplay between LC states and high-T c superconductivity needs more experimental results, which will contribute to consolidate the theoretical models and to complete them. We hope our discovery will open up a new field of study when addressing the physical properties of highly correlated electron materials.

Figure 1 .

 1 Figure 1. High-T C superconducting cuprates phase diagram versus hole concentration for two cuprates families YBa 2 Cu 3 O 6+x (YBCO) (bottom and right scales) and HgBa 2 CuO 4+δ (Hg1201) (top and left scales)showing the intra-unit cell (IUC) magnetic order associated with the pseudogap state from polarized neutron diffraction (PND)[START_REF] Fauqué | Magnetic Order in the Pseudogap Phase of High-T C Superconductors[END_REF][START_REF] Mook | Observation of magnetic order in a superconducting YBa 2 Cu 3 O 6.6 single crystal using polarized neutron scattering[END_REF][START_REF] Balédent | Evidence for competing magnetic instabilities in underdoped YBa 2 Cu 3 O 6+x[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] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF][START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor HgBa 2 CuO 4+δ[END_REF][START_REF] Li | Magnetic order in the pseudogap phase of HgBa 2 CuO 4+δ studied by spin-polarized neutron diffraction[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] and T * , the pseudogap temperature, determined from 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][START_REF] Murayama | Diagonal nematicity in the pseudogap phase of HgBa 2 CuO 4+δ[END_REF], second harmonic generation (SHG)[START_REF] Zhao | Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate[END_REF], muon spectroscopy (µSR)[START_REF] Zhang | Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa 2 Cu 3 O[END_REF] and resonant ultrasound (Rus)[START_REF] Shekhter | Bounding the pseudogap with a line of phase transitions in YBa 2 Cu 3 O 6+x[END_REF] experiments. Representative lines of transition of antiferromagnetism (AF), superconductivity (SC), pseudogap (PG) and charge density wave (CDW) (full line for YBCO and dashed line for Hg1201) are also depicted.

Figure 2 .

 2 Figure 2. (a) Loop currents (LC) in CuO 2 plane, so-called LC-Θ I I phase, as proposed by C.M. Varma in superconducting cuprates [12,13], with the corresponding toroidal moment, Ω, in black. (b) Model of loop currents proposed in Ref. [37]. The LCs now only run between oxygen sites, in contrast with the LC-Θ I I phase. (c) Anapole or polar toroidal moment Ω formed by winding current around a solenoid. A 2D section in the torus reveals two loop currents turning clockwise (blue) and anti-clockwise (red). (d) (2P × 2P ) LC-Θ I I super-cell [16] formed by a π/2 rotation of the anapole moment (in black) between neighbouring unitcells. The magnetic cell encompasses the 4 possible domains for a LC pattern separated by domain walls (represented by the shaded lines). Here, the domains size as a function of the unit-cell is P = 1 . (e) (2P × 2P ) magnetic supercell for P = 2.

Figure 3 .

 3 Figure 3. Crystal structure of layered perovskites (a) HgBa 2 CuO 4+δ and (b) YBa 2 Cu 3 O 6+x . (c) CuO 2 plane with copper (blue circles) and oxygen (red circles) atoms. (d) Tilt of the observed magnetic moment. The magnetic moment is tilted by an angle θ from the c-axis direction, perpendicular to the CuO 2 plane. (e) Temperature dependencies of the inverse of the flipping ratio 1/FR (see text) measured at Q = (1,0,0) (purple) and at Q = (2,0,0) (gray) from [24].

Figure 4 .

 4 Figure 4. Representation of two possible magnetic moment arrangements which are compatible with the PND results. They correspond to a superposition of out-of-plane moment as the LC-Θ I I phase [12, 13] with in-plane moments either a) along the diagonal joining up the moments, or b) along the perpendicular diagonal. This yields to different symmetries with a) 2/m point group and b) 2 /m point group. Note that, as in Fig. 2.a, only one of the four possible LC states is shown.

  Describing the generic experimental phase diagram shown in Fig.1, one realizes that the IUC order occurs at T * , the PG onset temperature. The PG energy scale impacts a large number of C. R. Physique-Draft, 13th July 2021

Figure 5 .

 5 Figure 5. (a) Loop currents decorating a CuO 6 octahedra with two circulating loops[START_REF] Agterberg | Emergent loop current order from pair density wave superconductivity[END_REF][START_REF] Yakovenko | Tilted loop currents in cuprate superconductors[END_REF] which can describe the PND results in HgBa 2 CuO 4+δ[START_REF] Li | Unusual magnetic order in the pseudogap region of the superconductor HgBa 2 CuO 4+δ[END_REF][START_REF] Li | Magnetic order in the pseudogap phase of HgBa 2 CuO 4+δ studied by spin-polarized neutron diffraction[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]. (b) Model of magnetic moments for a bilayer cuprate like YBa 2 Cu 3 O 6+x built from the magnetic moment arrangement of a single layer shown in 4.a and compatible with the observed neutron structure factor[START_REF] Mangin-Thro | ab Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa 2 Cu 3 O 6.6[END_REF].

Figure 6 .

 6 Figure 6. (a) Crystal structure of YBa 2 Cu 3 O 6+x , showing the corrugation or dimpling of the CuO 2 planes. (b) Top view of the loop currents arrangements of the LC-Θ I I -type in each CuO 2 layer (from [24]) where the toroidal moment is shown in black. The summed toroidal moment for a bilayer is pointing along the b direction. (c) Cut of the unit cell along the diagonal showing the orbital moments perpendicular to the O-Cu-O triangle. The dimpling of the O-Cu-O triangle gives rise to an horizontal magnetic component along the diagonal direction. (d) Top view of the summed magnetic moment, also pointing along b.

Figure 7 .

 7 Figure 7. High-T C superconducting cuprates phase diagram versus hole concentration for the cuprate family Bi 2 Sr 2 CaCu 2 O 8+δ showing the intra-unit cell (IUC) magnetic order associated with the pseudogap state from polarized neutron diffraction (PND)[START_REF] Almeida-Didry | Evidence for intra-unitcell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF][START_REF] Mangin-Thro | Characterization of the intra-unit-cell magnetic order in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF], transport[START_REF] Watanabe | Anisotropic Resistivities of Precisely Oxygen Controlled Single-Crystal Bi 2 Sr 2 CaCu 2 O 8+δ : Systematic Study on "Spin Gap" Effect[END_REF], tunnel[START_REF] Dipasupil | Energy Gap Evolution in the Tunneling Spectra of Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF][START_REF] Ozyuzer | Absence of pseudogap in heavily overdoped Bi 2 Sr 2 CaCu 2 O 8+δ from tunneling spectroscopy of break junctions[END_REF], electronic Raman scattering[START_REF] Loret | Universal relationship between the energy scales of the pseudogap phase, the superconducting state, and the charge-density-C. R. Physique-Draft, 13th[END_REF], ARPES[START_REF] Vishik | Phase competition in trisected superconducting dome[END_REF], pump probe spectroscopy (PPS)[START_REF] Toda | Rotational symmetry breaking in Bi 2 Sr 2 CaCu 2 O 8+δ probed by polarized femtosecond spectroscopy[END_REF] and photo-galvanic (PGE)[START_REF] Lim | Observation of broken inversion and chiral symmetries in the pseudogap phase in single and double layer bismuth-based cuprates[END_REF] experiments. Representative area for superconductivity (SC) and pseudogap (PG) are also depicted.

GlossaryµSR:

  muon spectroscopy/muon-spin rotation. 5, 17, 18 AF: antiferromagnetic/antiferromagnetism. 2, 4-7, 18, 20 ARPES: angle resolved photoemission. 3, 9, 15, 16, 18 CDW: charge density wave. 2, 5, 8, 15, 17, 18 IUC: intra-unit-cell. 2-13, 15-18, 20 C. R. Physique-Draft, 13th July 2021 LC: loop current. 3-20 LT: lattice translation. 2-4, 6, 15-19 NMR: nuclear magnetic resonance. 15, 17, 18 P: parity. 3, 8-10, 14, 17-20 PDW: pair density wave. 3, 17 PG: pseudogap. 2, 3, 5, 10, 12-19 PND: polarized neutron scattering. 3-6, 8, 10-20 R: rotational. 2, 3, 8, 12-15, 17, 18 SHG: second harmonic generation. 5, 9, 11, 20 STM: scanning tunneling microscopy. 2, 13, 15, 16, 18 T: time. 3, 5, 8-10, 14, 15, 17-20

C. R. Physique-Draft, 13th July 2021

Acknowledgements

The authors are deeply indebted to C. M. Varma who first proposed to search for a loop currents phase in high-T c cuprates and has been suggesting appealing novel ideas ever since. We acknowledge also the many students and colleagues who contributed to this work either in France, V. Balédent, S. de Almeida, B. Fauqué, Jaehong Jeong, L. Mangin-Thro, S. Pailhès or within our international collaborations, M. Greven, Yuan Li and all our collaborators whose names appear in Ref. [6,[21][22][23][24][25][26][27][28][29][30][31][32]. We wish to thank F. Damay, A. Georges, T. Giamarchi, S. Lovesey, C. Pépin, E. Tailleur and A. Shekhter for stimulating discussions on various aspects related to this work. We acknowledge financial supports from the project NirvAna (Contract ANR-14-OHRI-0010) of the Agence National de la Recherche (ANR) French agency.

Funding. We acknowledge financial supports from the project NirvAna (Contract ANR-14-OHRI-0010) of the Agence National de la Recherche (ANR) French agency.

Prix CEA "science et innovation" de l'Académie des sciences 2020.

This article is a draft (not yet accepted!)