J Q Lin

P Villar Arribi

G Fabbris

A S Botana

D Meyers

H Miao

Y Shen

D G Mazzone

J Feng

S G Chiuzbȃian

A Nag

A C Walters

M García-Fernández

Ke-Jin Zhou

J Pelliciari

I Jarrige

J W Freeland

Junjie Zhang

J F Mitchell

V Bisogni

X Liu

M R Norman
email: norman@anl.gov

M P M Dean
email: mdean@bnl.gov

Strong Superexchange in a d 9-δ Nickelate Revealed by Resonant Inelastic X-Ray Scattering

The discovery of superconductivity in a d 9-δ nickelate has inspired disparate theoretical perspectives regarding the essential physics of this class of materials. A key issue is the magnitude of the magnetic superexchange, which relates to whether cuprate-like high-temperature nickelate superconductivity could be realized. We address this question using Ni L-edge and O K-edge spectroscopy of the reduced d 9-1/3 trilayer nickelates R4Ni3O8 (where R=La,Pr) and associated theoretical modeling. A magnon energy scale of ∼ 80 meV resulting from a nearest-neighbor magnetic exchange of J = 69(4) meV is observed, proving that d 9-δ nickelates can host a large superexchange. This value, along with that of the Ni-O hybridization estimated from our O K-edge data, implies that trilayer nickelates represent an intermediate case between the infinite-layer nickelates and the cuprates. Layered nickelates thus provide a route to testing the relevance of superexchange to nickelate superconductivity.

Ever since the discovery of superconductivity in the cuprates [START_REF] Bednorz | Possible high Tc superconductivity in the Ba-La-Cu-O system[END_REF], researchers have been searching for related unconventional high-temperature (T c ) superconductors based on different transition metal ions [2][3][4]. Nickel, given its proximity to copper in the periodic table, represents an obvious target element. A popular concept has been to try to realize materials with Ni 1+ : 3d 9 ions with planar oxygen coordination residing in layers, as it was conjectured that this would mimic the strong magnetic superexchange that was proposed to be important for cuprate superconductivity [5]. The appropriateness of this assumption in layered RNiO 2 materials (R= La, Pr, Nd) was, however, questioned as the predicted increase in charge-transfer energy in RNiO 2 , with respect to cuprates, would be expected to reduce the superexchange [6]. Superconductivity at a relatively modest T c ≈ 15 K in Nd 1-x Sr x NiO 2 was nonetheless reported [7]. This has motivated many studies, often conflicting, concerning the nature of the normal-state electronic structure and correlations in these and related materials . RNiO 2 materials are the infinite-layer members of a series of low-valence (with d 9-δ filling) layered nickelates R n+1 Ni n O 2n+2 where n represents the number of NiO 2 layers per formula unit [START_REF] Crespin | Reduced forms of LaNiO3 perovskite. part 1.-evidence for new phases: La2Ni2O5 and LaNiO2[END_REF][START_REF] Hayward | Sodium hydride as a powerful reducing agent for topotactic oxide deintercalation: Synthesis and characterization of the Nickel(I) oxide LaNiO2[END_REF][START_REF] Hayward | Synthesis of the infinite layer Ni(I) phase NdNiO2+x by low temperature reduction of NdNiO3 with sodium hydride[END_REF][START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF]. Given the important role of charge-transfer and superexchange in many theories of unconventional superconductivity, determining trends for these quantities is highly important for understanding nickelate superconductivity and potentially discovering new nickelate superconductors [START_REF] Anderson | The resonating valence bond state in La2CuO4 and superconductivity[END_REF][START_REF] Scalapino | A common thread: The pairing interaction for unconventional superconductors[END_REF]. Among the known members of this nickelate family, trilayer materials shown in Fig. 1(a) are ideal for testing the fundamental aspects of the analogy between layered nickelates and cuprates. This is because complications from rare-earth self-doping, c-axis coupling and inhomogeneous samples are less severe in R 4 Ni 3 O 8 than in RNiO 2 [START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF][START_REF] Poltavets | Bulk magnetic order in a two-dimensional Ni 1+ /Ni 2+ (d 9 /d 8 ) nickelate, isoelectronic with superconducting cuprates[END_REF][START_REF] Botana | Charge ordering in Ni 1+ /Ni 2+ nickelates: La4Ni3O8 and La3Ni2O6[END_REF].

In this Letter, we combine Resonant Inelastic X-ray Scattering (RIXS) with first principles calculations and theoretical modeling to characterize the magnetic exchange in trilayer R 4 Ni 3 O 8 that fall in the overdoped regime of cuprates in terms of electron count. We find a near-neighbor exchange of J = 69(4) meV, in good accord with our first principles calculations. This demonstrates that these reduced nickelates indeed have a strong superexchange (within a factor of two of the cuprates). By comparing the O K-edge pre-peak intensity to that of cuprates and infinite-layer nickelates, we argue that these trilayer materials are intermediate between cuprates and infinite-layer nickelates. Based on this, we suggest that electron-doping R 4 Ni 3 O 8 materials provides a compelling route to testing the relevance of superexchange for nickelate superconductivity. R 4 Ni 3 O 8 (R =La, Pr) single crystals were prepared by synthesizing their parent Ruddlesden-Popper phases and reducing them in H 2 /Ar gas as described previously [START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF]41]. The resulting samples are single-phase crystals with a tetragonal unit cell (I4/mmm space group) and lattice constants of a = b = 3.97 Å, c = 26.1 Å. The trilayer R 4 Ni 3 O 8 phase is shown in Fig. 1(a); panel (b) zooms on the Ni-O planes. These samples have an effective hole-doping of δ = 1/3. Reciprocal space is indexed in terms of scattering vector Q = (2π/a, 2π/a, 2π/c). Both La and Pr materials are rather similar regarding their high-and medium-energy physics such as spin states and orbital polarization [START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF]. The primary difference is that La 4 Ni 3 O 8 (which exhibits strong antiferromagnetic spin fluctuations [START_REF] Aproberts-Warren | Critical spin dynamics in the antiferromagnet la4ni3o8 from 139 La nuclear magnetic resonance[END_REF]) has stripe order than opens up a small insulating gap [START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF], whereas Pr 4 Ni 3 O 8 remains metallic without long-range order. Since the more ordered and insulating nature of La 4 Ni 3 O 8 compared to Pr 4 Ni 3 O 8 is expected to give sharper magnetic RIXS spectra, we focus on the former material for this paper.

We used XAS to confirm the expected electronic properties of the La 4 Ni 3 O 8 samples. The Ni L-edge spectrum from 846-878 eV is shown in Fig. 1(c). The strongest feature around 850 eV is the La M 4 edge, which is followed by the Ni L 3 and L 2 edges at 852 and 870 eV respectively. Substantial linear dichroism is apparent, especially at the L 2 edge where the spectrum is not obscured by the La M 4 edge, indicating that the unoccupied 3d states are primarily x 2y 2 in character [START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF]. The overall spectral shape is very similar to that seen in cuprates [START_REF] Chen | Electronic states in La2-x Srx CuO 4+δ probed by soft-x-ray absorption[END_REF][START_REF] Chen | Out-of-plane orbital charac-ters of intrinsic and doped holes in La2-x Srx CuO4[END_REF][START_REF] Brookes | Stability of the zhang-rice singlet with doping in lanthanum strontium copper oxide across the superconducting dome and above[END_REF], consistent with a d 9 L configuration, with no indication for a high-spin d 8 component of the holes [START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF]. This is reasonable, since the planar coordination of Ni leads to a large splitting between the x 2 -y 2 and 3z 2 -r 2 states, which is expected to out-compete the Hunds exchange coupling, thus favoring a low-spin ground state [START_REF] Fabbris | Orbital engineering in nickelate heterostructures driven by anisotropic oxygen hybridization rather than orbital energy levels[END_REF]. The O K-edge spectrum around 525-545 eV in Fig. 1 shows a pre-peak feature around 532 eV, which is known to indicate hybridization between the Ni 3d and O 2p states [START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF][START_REF] Chen | Electronic states in La2-x Srx CuO 4+δ probed by soft-x-ray absorption[END_REF][START_REF] Chen | Out-of-plane orbital charac-ters of intrinsic and doped holes in La2-x Srx CuO4[END_REF]. Our measurements find that this pre-peak has a strong linear dichroism as well, as observed in cuprates [START_REF] Chen | Out-of-plane orbital charac-ters of intrinsic and doped holes in La2-x Srx CuO4[END_REF].

We then performed RIXS to study the low-energy degrees of freedom. High energy-resolution RIXS measurements were performed at I21 at the Diamond Light Source with a resolution of 45 meV and at NSLS-II with a resolution of 30 meV. All RIXS data shown were taken at a temperature of 20 K using a fixed horizontal scattering angle of 2θ = 154 • and x-ray polarization within the horizontal scattering plane (π polarization). Different momenta were accessed by rotating the sample about the vertical axis, such that the projection of the scattering vector varies. (H, 0) and (H, H) scattering planes were accessed by rotating the sample about its azimuthal angle. Figure 2 plots low-energy RIXS spectra of La 4 Ni 3 O 8 as a function of Q. A relatively strong elastic line is present for all Q likely arising from apical oxygen removal during sample preparation, which induces internal strain in the samples. In the 70-90 meV energy range, a weakly dispersive, damped feature is observed. Based on the energy of the feature, this could either be magnetic or the bond-stretching phonon common to complex oxides 0 100 200 (0.12, 0) (0.17, 0) (0. 2. RIXS spectra of La4Ni3O8 as a function of Q at the resonant energy of the magnon 852.7 eV [41]. Data are shown as red points and the fit is shown as a black line, which is composed of the magnetic excitation in orange and the elastic line in blue. The in-plane Q of the measured spectrum is denoted in the top right of each panel. [START_REF] Fabbris | Doping dependence of collective spin and orbital excitations in the spin-1 quantum antiferromagnet La2-xSrxNiO4 observed by x rays[END_REF][START_REF] Betto | Three-dimensional dispersion of spin waves measured in NiO by resonant inelastic x-ray scattering[END_REF][START_REF] Nag | Many-body physics of single and double spin-flip excitations in NiO[END_REF][START_REF] Lin | Strongly correlated charge density wave in La2-xSrxCuO4 Evidenced by Doping-Dependent Phonon Anomaly[END_REF][START_REF] Devereaux | Directly characterizing the relative strength and momentum dependence of electron-phonon coupling using resonant inelastic xray scattering[END_REF]. It is known, however, that the intensity of the bond-stretching phonon increases like |Q| 2 , inconsistent with what we find [START_REF] Lin | Strongly correlated charge density wave in La2-xSrxCuO4 Evidenced by Doping-Dependent Phonon Anomaly[END_REF][START_REF] Devereaux | Directly characterizing the relative strength and momentum dependence of electron-phonon coupling using resonant inelastic xray scattering[END_REF] (see Fig. 2). The peak also resonates slightly above the Ni L edge [41], which is also consistent with a magnetic origin [START_REF] Fabbris | Doping dependence of collective spin and orbital excitations in the spin-1 quantum antiferromagnet La2-xSrxNiO4 observed by x rays[END_REF]. On the basis of these observations, we assign this feature to magnetic excitations. Further supporting this assignment we note that Pr 4 Ni 3 O 8 spectra exhibit a weaker, more damped paramagnon excitation, which is expected as this compound is metallic with spin-glass behavior [41,[START_REF] Huangfu | Short-range magnetic interactions and spin-glass behavior in the quasi-two-dimensional nickelate Pr4Ni3O8[END_REF]. Below, we will demonstrate a consistency between this mode and analytical modeling and density functional theory (DFT).

In order to analyze the magnetic dispersion, we fit the low-energy RIXS spectra with the sum of a zero-energy Gaussian fixed to the experimental energy resolution in order to account for the elastic line and a damped harmonic oscillator to capture the magnetic excitation [41]. To model the magnetic interactions, we expect a leading contribution from the nearest-neighbor in-plane Ni-O-Ni superexchange, J. We further know that La 4 Ni 3 O 8 , like some other nickelates and cuprates, has a striped ground state with both spin and charge character, with an inplane wavevector of Q = (1/3, 1/3) [START_REF] Zhang | Stacked charge stripes in the quasi-2D trilayer nickelate La4Ni3O8[END_REF][START_REF] Zhang | Spin stripe order in a square planar trilayer nickelate[END_REF]. This diagonal stripe order is illustrated in Fig. 1(b) [41]. In each plane, we have two antiparallel spin rows (corresponding to d 9 ) separated by an anti-phase domain wall (corresponding to non-magnetic d 8 ). This gives rise to six spins in the magnetic unit cell in a given trilayer, which we label as spins 1-6. Nearest-neighbor spins within the planes in a given stripe are coupled by the superexchange J, which we expect to be the strongest interaction. The antiphase domain wall is due to coupling between the magnetic stripes. There are two potential couplings (supersuperexchange), but we only expect one of them (the one along the tetragonal axes) to be significant, as the other involves a 90 degree pathway [START_REF]Diagrams for the exchange pathways are shown[END_REF]. As the planes are antiferromagnetically coupled [START_REF] Zhang | Spin stripe order in a square planar trilayer nickelate[END_REF], this gives rise to a positive J z coupling between successive layers (there is no evidence for magnetic coupling between the trilayers, so our model deals with only a single trilayer). We solved the resulting Heisenberg model in the spin-wave approximation [START_REF] Carlson | Our model is essentially a trilayer generalization of[END_REF], which yields three dispersive modes (split by J z ), which we term the acoustic, middle and optic modes [41]. The energy of each of these three modes changes with in-plane momentum and the relative intensity of the modes is modulated by the out-of-plane momentum, which varies with in-plane momentum due to our fixed-scattering-angle configuration. From cuprates, we anticipate that the interlayer coupling will be of order 10 meV, below our energy resolution [START_REF] Reznik | Direct observation of optical magnons in YBa2Cu3O6.2[END_REF][START_REF] Dean | Itinerant effects and enhanced magnetic interactions in bi-based multilayer cuprates[END_REF]. On this basis, we analyzed our data in terms of the sum of the three magnon modes. The RIXS intensity for a particular acoustic, middle or optic magnetic mode n in the πσ polarization channel is given by [START_REF] Haverkort | Theory of resonant inelastic x-ray scattering by collective magnetic excitations[END_REF]

I n (Q) = i k in • M n,Q (r i ) 2 (1)
where k in is the incident wavevector and M n,Q (r i ) is magnetization vector at site i (i.e., the eigenvector of the n th spin-wave mode at Q). This vector is in-plane since the ground-state moments are along c [START_REF] Zhang | Spin stripe order in a square planar trilayer nickelate[END_REF]. The final element of our model is to sum over the two tetragonal domains given the known magnetic twinning in La 4 Ni 3 O 8 : (H, K) → (H, -K) [START_REF] Zhang | Stacked charge stripes in the quasi-2D trilayer nickelate La4Ni3O8[END_REF][START_REF] Zhang | Spin stripe order in a square planar trilayer nickelate[END_REF]. We determined the energies and eigenvectors of these modes from the resulting 12 by 12 secular matrix [41], and computed the weighted sum of the three modes at each Q. [START_REF]Our spin-wave approach does not include any quantum renormalization factor, Zc, of the spin wave energy, and[END_REF] To estimate the magnetic exchange parameters in La 4 Ni 3 O 8 , we computed the energy of four different spin configurations in the above-mentioned magnetic cell [41], and then mapped these energies to a Heisenberg model. This was done using DFT in the generalized gradient approximation (GGA) approximation as implemented in the WIEN2k code [START_REF] Blaha | WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties[END_REF][START_REF] Blaha | WIEN2k: an APW+lo program for calculating the properties of solids[END_REF]. The experimentally determined insulating charge and spin stripe-ordered ground state (Fig. 1b) is obtained even at the GGA level given that the exchange splitting is larger than the bandwidth in this state. Adding a U simply increases the size of the gap with respect to the GGA solution, but the nature of the ground state remains the same [START_REF] Poltavets | Bulk magnetic order in a two-dimensional Ni 1+ /Ni 2+ (d 9 /d 8 ) nickelate, isoelectronic with superconducting cuprates[END_REF]. Results presented here are for GGA, but GGA+U results are presented in [41]. This yields J = 71 meV, J z = 13.6 meV, and J 1 = 10.6 meV. We fix J z = 13.6 meV in our model since, because of our resolution and contamination from the elastic line, we cannot accurately estimate it from experiment. We then vary J and J 1 to get the best fit. This fit yields J = 69(4) meV and J 1 = 17(4) meV in good agreement with experiment, although the small difference in J (2 meV) is likely coincidental. These exchange values can be rationalized from the single-layer analytic relation (i.e., ignoring J z ) that E mag ∼ 4S √ JJ 1 where E mag is the zone-boundary magnon energy. We overplot the magnetic dispersion with our theory analysis in Fig. 3 showing a good level of agreement. The model also captures the observed softening that occurs as Q approaches (- 1 3 , -1 3 ). We also measured Pr 4 Ni 3 O 8 [41], which is similar to La 4 Ni 3 O 8 , but metallic rather than insulating; the results show a lower-intensity damped magnetic excitation, which is expected in view of its metallicity [START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF] and the spin-glass behavior reported for this material [START_REF] Huangfu | Short-range magnetic interactions and spin-glass behavior in the quasi-two-dimensional nickelate Pr4Ni3O8[END_REF]. The paramagnon energy in Pr 4 Ni 3 O 8 is only slightly reduced compared to La 4 Ni 3 O 8 . Again, this is similar to cuprates where magnon-like excitations are seen for paramagnetic dopings [START_REF] Dean | Persistence of magnetic excitations in La2-xSrxCuO4 from the undoped insulator to the heavily overdoped non-superconducting metal[END_REF].

Our rather large value of J = 69(4) meV is the principal result of this Letter. This magnetic exchange is 2.5 times larger than that of the 1/3 doped nickelate La 2-x Sr x NiO 4 which also has a diagonal stripe state (with S = 1 d 8 magnetic rows and S = 1/2 d 7 domain walls), though the two have comparable J 1 [START_REF] Fabbris | Doping dependence of collective spin and orbital excitations in the spin-1 quantum antiferromagnet La2-xSrxNiO4 observed by x rays[END_REF][START_REF] Boothroyd | Spin dynamics in stripe-ordered la 5/3 sr 1/3 nio4[END_REF][START_REF] Woo | Mapping spin-wave dispersions in stripe-ordered La2-xSrxNiO4 (x = 0.275, 0.333[END_REF]. J for La 4 Ni 3 O 8 is, in fact, within a factor of two of cuprates, which have among the largest superexchange of any known material [START_REF] Dean | Persistence of magnetic excitations in La2-xSrxCuO4 from the undoped insulator to the heavily overdoped non-superconducting metal[END_REF][START_REF] Coldea | Spin waves and electronic interactions in La2CuO4[END_REF][START_REF] Le Tacon | Intense paramagnon excitations in a large family of high-temperature superconductors[END_REF][START_REF] Peng | Influence of apical oxygen on the extent of in-plane exchange interaction in cuprate superconductors[END_REF]. This suggests, along with our XAS results, that these nickelates are strongly correlated charge-transfer materials. Two questions are apparent: Why is the superexchange in La 4 Ni 3 O 8 so large? And what is the relationship between trilayer nickelates and their infinite-layer counterparts?

Given the 180 degree Ni-O-Ni bonds in the d 9 nickelates, superexchange is the most likely mechanism for generating their exchange interactions, as in the cuprates. In the charge-transfer limit, the strength of this interaction scales as t 4 pd /∆ 3 where t pd is the hopping between the transition metal x 2 -y 2 and oxygen pσ orbitals, and ∆ ≡ E d -E p is the energy difference between them. Large pd hopping and a small ∆ implies a large ligand-hole character for the doped holes, as this is controlled by the ratio t pd /∆. We therefore fit the O K pre-peak intensity to compare to the literature for La 2-x Sr x CuO 4 [START_REF] Chen | Electronic states in La2-x Srx CuO 4+δ probed by soft-x-ray absorption[END_REF] and Nd 1-x Sr x NiO 2 [8] and show the results in Fig. 4. The pre-peak in Nd [START_REF] Nica | Theoretical investigation of superconductivity in trilayer square-planar nickelates[END_REF][START_REF] Mcmahan | Cuprate parameters from numerical wannier functions[END_REF]. The situation in NdNiO 2 is less certain given the large difference of the ratios (i.e., whether one considers the maximum intensity or the integrated weight). This will likely only be solved when higher-quality more homogeneous Nd 1-x Sr x NiO 2 samples are prepared and studied in detail. Still, it seems likely that the superexchange in NdNiO 2 is smaller, but still large enough that its potential contribution to superconductivity deserves serious consideration. Recent Raman scattering measurements estimate J ≈ 25 meV [START_REF] Fu | Core-level x-ray photoemission and Raman spectroscopy studies on electronic structures in mott-hubbard type nickelate oxide NdNiO[END_REF], which is consistent with this and which likely arises from ∆ being larger, though the enhanced c-axis coupling and the screening from the R 5d states, which are predicted to be partially occupied NdNiO 2 , could be playing a role as well.

In conclusion, we report the presence of a large superexchange J = 69(4) meV in La 4 Ni 3 O 8 -the first direct measurement of superexchange in a d 9-δ nickelate. This superexchange value is within a factor of two of values found in the cuprates, and this, coupled with a substantial O K pre-peak, establishes the charge-transfer nature of this d 9-δ nickelate with substantial d-p mixing. By comparing the O K-edge XAS spectra of La 4 Ni 3 O 8 to that of the cuprates and the infinite-layer nickelate, we establish that trilayer nickelates represent a case that is intermediate between them. This result is interesting in view of the widespread belief that increasing magnetic superexchange might promote higher-T c superconductivity [4,[START_REF] Zhang | Large orbital polarization in a metallic square-planar nickelate[END_REF][START_REF] Botana | Electron doped layered nickelates: Spanning the phase diagram of the cuprates[END_REF]. Studying a series of layered nickelates would also provide a route to testing the relevance of superexchange to nickelate superconductivity given the variation in their nominal Ni valence.

Work S2. RIXS spectra of Pr4Ni3O8 as a function of Q at the resonant energy of the magnon 852.7 eV. Data are shown as red points and the fit is shown as a black line, which is composed of the magnetic excitation in orange and the elastic line in blue. The in-plane Q of the measured spectrum is denoted in the top right of each panel. Note that the scale of the y-axis is half of that in Fig. 2 of the main text. and f (x) = a * x 2 + b for x > 0, a and b are free parameters. Prior to computing the final fit, we performed an initial fit in which the elastic energy was allowed to vary, which we used to shift the spectra in energy such that the elastic energy is set to exactly zero.

IV. COMPARISON OF La4Ni3O8 AND Pr4Ni3O8

The difference between La 4 Ni 3 O 8 and Pr 4 Ni 3 O 8 has been studied in prior x-ray absorption and density functional theory (DFT)-based work [2]. This study concluded that both materials are rather similar regarding their high-and medium-energy physics such as spin states, orbital polarization, etc. The primary difference is that stripe order opens a small insulating gap in La 4 Ni 3 O 8 , whereas Pr 4 Ni 3 O 8 remains metallic without long-range order. Pr 4 Ni 3 O 8 was later reported to have spin-glass behavior likely coming from short-range stripe correlations [4]. Since the more ordered and insulating nature of La 4 Ni 3 O 8 compared to Pr 4 Ni 3 O 8 is expected to give sharper magnetic RIXS spectra, we focused on the former material for this paper, but we also took data on Pr 4 Ni 3 O 8 as shown in Fig. S2. A similar energy peak is observed which is slightly broadened and less intense compared to La 4 Ni 3 O 8 . Fitting the spectra for Pr 4 Ni 3 O 8 in the same way as was done for La 4 Ni 3 O 8 yields a value of the near-neighbor exchange perhaps 10% lower, but overall the two materials are very similar (Fig. S3).

V. THEORY OF MAGNETIC EXCITATIONS IN THE STRIPE-ORDERED STATE

In this Section, we compute the dispersion relation and RIXS intensity for the magnons in a diagonal stripe state. To reproduce the stripe order shown in Fig. 1(b) of the main text, we consider a model with the following interactions illustrated in Fig. S4. J couples nearest-neighbor spins within the same stripe, J 1 couples spins across the stripes in the [1, 0, 0] direction, and J z couples spins between layers within the trilayer in the [0, 0, 1] direction. A further J 2 coupling across the stripes along the [1, 1, 0] direction was also considered, but its effect could not be distinguished in the measured RIXS spectra, so it was omitted. This is expected as this super-superexchange contribution is weak given the 90 degree Ni-Ni-Ni pathway that is involved. We also ignore any single-ion anisotropy, again because it would be difficult to detect given the width of the elastic line. The in-plane lattice vectors for the structural unit cell are a 1 = (a, 0, 0), a 2 = (0, a, 0) [START_REF] Bednorz | Possible high Tc superconductivity in the Ba-La-Cu-O system[END_REF] and for the magnetic unit cell are where, for simplicity, we have assumed a primitive magnetic cell with non-orthogonal lattice vectors. Along the c direction, we have trilayers separated by the body-centered translation (a/2, a/2, c/2). Since there are no observable correlations between trilayers [5], we consider only a single trilayer here. The layers within a trilayer are in registry along c, with each layer separated by the interlayer distance d ≈ c/8. The scattering vector Q is presented in normalized units with a = c = 1.

a mag 1 = (3a, 0, 0), a mag 2 = (-a, a, 0). ( 2
)
( 1/3, 1/3) (0, 0) (1/2, 0) Q 0 20 40

A. Dispersion relation

We proceed to calculate the magnon dispersion for the diagonal stripe state by generalizing the torque equation formalism of Carlson et al. [6] to the trilayer case (equivalent results can be obtained using the less transparent Holstein-Primakoff treatment [6]). According to neutron scattering data, the spins in the ground state are oriented along c [5]. Therefore, the generalized torque equations for the spins reduce to:

dS x r,i dt = - 1 h   S y r,i r ,j J ij rr S z r ,j -S z r,i r ,j J ij rr S y r ,j   , dS y r,i dt = - 1 h   S z r,i r ,j J ij rr S x r ,j -S x r,i r ,j J ij rr S z r ,j   , dS z r,i dt ≈ 0, (3)
where r, r label the positions of the spins in different magnetic unit cells and the indices i, j label the spins within each magnetic unit cell (i, j = 1, ..., 6) as shown in Fig. S2. We seek sinusoidal solutions of the form

S x r,i = S x i exp[i(Q • r -ωt)] , S y r,i = S y i exp[i(Q • r -ωt)], (4)
and we set S z r,i = ±S with the sign given by the orientation of the spin in the ground state. To start, we identify the couplings J ij rr that connect the spins at different lattice positions, with the origin taken to be the location of spin 1. We distinguish two groups of spins, (S 1 , S 3 , S 5 ) and (S 2 , S 4 , S 6 ), with each group having equivalent in-plane locations due to the c-axis translational symmetry. Their couplings are

• S 1 (r = 0) couples to S 2 twice with J (r = a 1 , r = a 2 ) and twice with J 1 (r = -2a 1 , r = -2a 2 ). The same applies for S 3 (S 5 ) coupled to S 4 (S 6 ).

• S 2 (r = a 1 ) couples to S 1 twice with J (r = 0, r = a 1a 2 ) and twice with J 1 (r = 3a 1 , r = a 1 + 2a 2 ). The same applies for S 4 (S 6 ) coupled to S 3 (S 1 ).

For the couplings along [0,0,1], we have:

• S 1 (S 2 ) couples to S 3 (S 4 ) with J z .

• S 5 (S 6 ) couples to S 3 (S 4 ) with J z .

• S 3 (S 4 ) couples with J z to S 1 (S 2 ) and to S 5 (S 6 ).

With this information, we can write the torque equations for each of the six spins in the magnetic unit cell, for example:

dS x 0,1 dt = - 1 h S y 0,1 (-S) 2J + 2J 1 + J z -S J S y a1,2 + S y a2,2 + J 1 S y -2a1,2 + S y -2a2,2 + J z S y (0,0,- c 8 ),3 . (5)
Substituting Eq. 4, we can rewrite this expression as

ihω S S x 1 = -S y 1 2J + 2J 1 + J z -S y 3 J z e -i Qz 8 -S y 2 J e iQx + e iQy + J 1 e -2iQx + e -2iQy , (6)
and simplifying

ihω S S x 1 = -AS y 1 -CS y 2 -DS y 3 , (7)
where we have defined

A = 2J + 2J 1 + J z , B = A + J z , C = J e iQx + e iQy + J 1 e -2iQx + e -2iQy , D = J z e -i Qz 8 . (8)
The final torque equations for the six spins are

ihω S S x 1 = -AS y 1 -CS y 2 -DS y 3 , ihω S S y 1 = +AS x 1 + CS x 2 + DS x 3 ihω S S x 2 = +AS y 2 + C * S y 1 + DS y 4 , ihω S S y 2 = -AS x 2 -C * S x 1 -DS x 4 ihω S S x 3 = +BS y 3 + CS y 4 + DS y 5 + D * S y 1 , ihω S S y 3 = -BS x 3 -CS x 4 -DS x 5 -D * S x 1 ihω S S x 4 = -BS y 4 -C * S y 3 -DS y 6 -D * S y 2 , ihω S S y 4 = +BS x 4 + C * S x 3 + DS x 6 + D * S x 2 ihω S S x 5 = -AS y 5 -CS y 6 -D * S y 3 , ihω S S y 5 = +AS x 5 + CS x 6 + D * S x 3 ihω S S x 6 = +AS y 6 + C * S y 5 + D * S y 4 , ihω S S y 6 = -AS x 6 -C * S x 5 -D * S x 4 . (9)
This results in a 12 × 12 secular matrix (6 spins, 2 components, x, y, per spin)

M =                    0 -A 0 -C 0 -D 0 0 0 0 0 0 A 0 C 0 D 0 0 0 0 0 0 0 0 C * 0 A 0 0 0 D 0 0 0 0 -C * 0 -A 0 0 0 -D 0 0 0 0 0 0 D * 0 0 0 B 0 C 0 D 0 0 -D * 0 0 0 -B 0 -C 0 -D 0 0 0 0 0 0 -D * 0 -C * 0 -B 0 0 0 -D 0 0 D * 0 C * 0 B 0 0 0 D 0 0 0 0 0 0 -D * 0 0 0 -A 0 -C 0 0 0 0 D * 0 0 0 A 0 C 0 0 0 0 0 0 0 0 D * 0 C * 0 A 0 0 0 0 0 0 -D * 0 -C * 0 -A 0                    . ( 10
)
Diagonalizing this matrix yields the squared eigenvalues

λ 2 1 (Q) = -A 2 + C * C (11)
and

λ 2 2,3 (Q) = - A 2 2 - B 2 2 + C * C + 2J 2 z ± 1 2 J z (A + B) 2 + 32C * C -8J 2 z , (12)
each of which are two-fold degenerate because of the tetragonal symmetry of the ground state. As iω = λ, these eigenvalues correspond to the magnon branches

ω middle (Q) = A 2 -C * C (13)
and

ω acoustic,optic (Q) = A 2 2 + B 2 2 -C * C -2J 2 z ∓ 1 2 J z (A + B) 2 + 32C * C -8J 2 z . (14)
To justify these labels, and give some sense of these energies, we consider the Γ point. Substituting Q = 0 in Eq. 8 gives C = 2J + 2J 1 and D = J z , which shows that our labels were chosen in order of increasing energy:

ω acoustic (0) = 0, ω middle (0) = SJ z 1 + 2C/J z ∼ S 2J z C, ω optic (0) = SJ z 1 + 6C/J z ∼ S 6J z C, (15)
where the approximation applies for J z J. The magnon dispersion for the parameters determined by the fit to the RIXS data, J=69 meV, J 1 =17 meV (with J z =13.6 meV), can be seen in Fig. S5.

B. Calculation of the RIXS intensities

As explained in the main text and Ref. [7], the RIXS intensity for each mode, labeled n, can be written as

I n (Q) = i k in • M n,Q (r i ) 2 , (16)
with i summed over the six spins in the magnetic unit cell and

M n,Q (r i ) = c x n,Q,i S x i , c y n,Q,i S y i , 0 . (17)
As the analytic expressions for the eigenvectors are complicated, we chose to determine them by diagonalizing the secular matrix Eq. 10 numerically for each Q using SciPy [8]. As noted above, each distinct magnon branch within our model has two degenerate eigenvalues, with the two members of each pair related by a 90 degree in-plane rotation because of the tetragonal symmetry. That is

c x 1 , c y 1 , c x 2 , c y 2 , ... (18)
is degenerate with

-c y 1 , c x 1 , -c y 2 , c x 2 , ... (19)
with n, Q being implicit. It is important to enforce this symmetry when calculating the RIXS intensity.

For the fit shown in the text, we took into account the scattering geometry of the RIXS measurements, with

k in = cos θI -sin θQ, (20)
and

|Q| = 4π sin θ λ E (21)
with θ the Bragg angle and λ E the photon wavelength. Here,

I = Q × c × Q |c × Q| . (22)
In particular, as the in-plane component of Q is swept, the Q z component changes accordingly.

VI. FIRST-PRINCIPLES DETERMINATION OF EXCHANGE COUPLING CONSTANTS IN La4Ni3O8

We performed DFT calculations for La 4 Ni 3 O 8 with the all-electron full potential code WIEN2k [9,10] using the generalized gradient approximation (GGA) exchange-correlation functional [11]. In these calculations, as we did previously [2,12], we considered the influence of thee Coulomb interaction, U . The U modification of DFT is usually included to compensate for the under-localization of transition-metal 3d-electrons in DFT. Including U , conversely, tends to over-localize electrons as it "double counts" the true Coulomb and exchange interactions as explained in, for example, Refs. [13][14][15]. Because of this, it is not immediately obvious whether including U leads to a more accurate value of J. In calculations, we found that the inclusion of U simply increases the size of the gap, and also leads to a modest increase in J, as outlined in Table I. Calculations with and without U find the same insulating, charge and spin stripe-ordered ground state, which was predicted before this state was experimentally observed [5]. In our prior work, we examined the role of U at length and found that the low-spin stripe state La 4 Ni 3 O 8 was appropriately described by GGA calculations even without the inclusion of U [2,12]. We refer the reader to these papers for the reasoning behind this. Since it allows us to compute J with fewer adjustable parameters, we report values from GGA calculations in the main text, and find that this is in good accord with experiment. As is evident based on how J changes with U , the very close match between theory and experiment of 2 meV at the GGA level is likely coincidental. We note that similar values for the superexchange can be obtained from rough estimates using a sum of the Mott and charge transfer contributions, J = 2t 2 /U d + 2t 2 /[∆ + U p /2], with t = t 2 pd /∆. Using U d = 8.5 eV and U p = 7.3 eV values from a similar analysis on the cuprates [16], with a ∆ and t pd obtained from our Wannier fit for La 4 Ni 3 O 8 [17], a comparable J of 99 meV is obtained. But this, obviously, depends on the choice of U d and U p . Spin-orbit coupling is not expected to have an appreciable effect on the exchange constants for 3d transition metal ions, especially for e g states where the orbital moment is largely quenched. This has been explicitly verified in our prior calculations [5].

The exchange couplings (J, J 1 and J z ) were obtained from total energy calculations for different Ni spin configurations (labeled C1-C4 in Fig. S6) mapped to a Heisenberg model. Configuration C1 is the experimental and theoretical ground state as shown in Fig. 1 of the main text. The magnitudes of the magnetic moments of the Ni 2+ atoms were between 0.6-0.7 µ B and we confirmed that these values were similar within 0.1 µ B in the different configurations, an accuracy typical of this type of calculation, justifying the Heisenberg mapping. Different configurations C1-C4 have differing magnetic bonds:

• C1 -AFM J, AFM J 1 , AFM J z • C2 -AFM J, AFM J 1 , FM J z • C3 -FM J, FM J 1 , FM J z • C4 -FM J, AFM J 1 , FM J z

The energies per trilayer are (with each bond counted once)

E C1 = E 0 -4 × 3JS 2 -3 × 4J 1 S 2 -2 × 4J z S 2 = E 0 -3J -3J 1 -2J z E C2 = E 0 -3J -3J 1 + 2J z E C3 = E 0 + 3J + 3J 1 + 2J z E C4 = E 0 + 3J -3J 1 + 2J z (23)
where E 0 is the non-magnetic energy. Solving this set of linear equations gives

J = (E C4 -E C2 )/6 J 1 = (E C3 -E C4 )/6 J z = (E C2 -E C1 )/4. ( 24
)
whose values are listed in the main text.

VII. X-RAY ABSORPTION

Our X-ray absorption spectrum (XAS) measurements aim to compare the intensity of the O K-edge pre-peak feature as a guide to the 3d-2p orbital hybridization in nickelates and cuprates. We analyze our data on La 4 Ni 3 O 8 against literature data for La 2-x Sr x CuO 4 from Ref. [18] and Nd 1-x Sr x NiO 2 from Ref. [19]. As far as we are aware, Nd 1-x Sr x NiO 2 data are only available for in-plane polarization, so we only show this component of the polarization. The intensity of the spectra are scaled to have equivalent intensities for energies above 538 eV past the main O K-edge step. Since different measurements can have different absolute energy calibrations, we used measurements of different reference samples to put the spectra on the same energy-scale. We use Ref. [2] as our reference energy calibration for which La 4 Ni 3 O 8 was measured alongside La 2-x Sr x CuO 4 and SrTiO 3 . We then used the SrTiO 3 reference measurements in [19] to put Nd 1-x Sr x NiO 2 on the same energy scale. While in La 4 Ni 3 O 8 and La 2-x Sr x CuO 4 the pre-peak is very clear, the pre-peak in Nd 1-x Sr x NiO 2 is broader making isolating the pre-preak less immediately obvious. We took the same 'background' intensity as was used in [19], which comes from a measurement of undoped NdNiO 2 and use a lorentzian lineshape to fit. The dominant error in our analysis likely comes from inhomogeneity in the doping of the Nd 1-x Sr x NiO 2 results we compare to and some uncertainty in how to isolate all the intensity in the pre-peak. These will likely improve in the future by higher quality sample preparation. Further analysis of the ligandhole anisotropy will also be important, but also requires polarization-dependent measurements of Nd 1-x Sr x NiO 2 that are not currently available in the literature.

FIG. 1 .

 1 FIG. 1. Crystal structure and X-ray absorption spectrum (XAS). (a) Unit cell of La4Ni3O8 and Pr4Ni3O8 with Ni in purple, O in gray and La/Pr in green [39]. (b) The active trilayer nickel-oxide planes in La4Ni3O8 with an illustration of the diagonal stripe-ordered state [40]. Ni sites with extra hole character (with respect to the d 9 magnetic rows) are in purple (S = 0), whereas Ni up and down spins in the magnetic rows are colored red and blue, respectively (S = 1/2). (c)&(d) XAS data of La4Ni3O8 measured in total fluorescence yield mode with polarization perpendicular and approximately parallel to the sample c-axis for (c) the Ni L-edge and (d) the O K-edge.

FIG. 3 . 2 xFIG. 4 .

 324 FIG.3. Magnetic dispersion of La4Ni3O8. Black points are the extracted energies of the magnetic excitation. The gray line is the fit to the experimental dispersion, which is composed of the weighted sum of three dispersive magnons, called the acoustic, middle and optic modes, which are plotted as blue, orange and green lines, respectively. The thickness of all three lines represents the predicted intensity of the modes[41]. The doubling of the modes from (-1 3 , -13 ) to (0, 0) arises from magnetic twinning[41].

FIG

  FIG. S3. Magnetic dispersion of La4Ni3O8 and Pr4Ni3O8. This figure is the same as Fig.3of the main text, but with data for Pr4Ni3O8 added. Black/red points are the extracted energies of the magnetic excitation for La4Ni3O8/Pr4Ni3O8. The black line is the fit to the experimental dispersion of La4Ni3O8, which is composed of the weighted sum of three dispersive magnons, called the acoustic, middle and optic modes, which are plotted as blue, orange and green lines, respectively.

  FIG. S5. Magnon dispersion with J=69 meV, J1=17 meV and Jz=13.6 meV, as in the main text, along the (Q,Q) and (Q,0) directions (with Qz=0). The dashed curves in the left plot are for the twin domain (Q,-Q).

FIG. S6 .

 S6 FIG. S6. Magnetic supercells used for determining the magnetic exchange. Ni sites are shown as spheres with red, blue and purple denoting up-spin, down-spin and hole states, respectively, as in Fig. 1 of the main text and Fig. S4. Gray lines trace the Ni-O bond network. Each magnetic cell is a 3 √ 2a × √ 2a × c tiling of the structural cell containing 36 Ni atoms (180 atoms in total), noting that there are two trilayer units a given unit cell. Atoms that are at the edge of the unit cell are shown as full spheres rather than being cut-off.

[ 1 ]

 1 Junjie Zhang, Yu-Sheng Chen, D. Phelan, Hong Zheng, M. R. Norman, and J. F. Mitchell, "Stacked charge stripes in the quasi-2D trilayer nickelate La4Ni3O8," Proceedings of the National Academy of Sciences 113, 8945-8950 (2016).

  1-x Sr x NiO 2 is significantly less prominent than in La 4 Ni 3 O 8 and La 2-x Sr x CuO 4 , but this appears to be primarily due to a broadened pre-peak rather than a Sr x NiO 2 and the equivalent ratios for the maximum pre-peak intensities are 1.00(4):1.85(9):0.37(6). The quoted errors are the uncertainty from the least-squares fitting algorithm. The largest systematic error likely arises from the doping inhomogeneity in the data from[8]. Thus La 4 Ni 3 O 8 has somewhat less admixture than La 2-x Sr x CuO 4 , but the difference is not enough to expect qualitatively different physics. The ratio we determine is in good accord with our observed magnetic exchange in La 4 Ni 3 O 8 being around half that of the cuprates[START_REF] Dean | Persistence of magnetic excitations in La2-xSrxCuO4 from the undoped insulator to the heavily overdoped non-superconducting metal[END_REF][START_REF] Le Tacon | Intense paramagnon excitations in a large family of high-temperature superconductors[END_REF][START_REF] Peng | Influence of apical oxygen on the extent of in-plane exchange interaction in cuprate superconductors[END_REF][START_REF] Dean | Spin excitations in a single La2CuO4 layer[END_REF]. This difference likely comes from ∆ being larger in La 4 Ni 3 O 8 compared to La 2-x Sr x CuO 4 (t pd is comparable in the three materials)

lower integrated spectral weight, the broadening perhaps due to a spatially varying doping. The relative integrated weight per doped hole is 1.00(2):1.74(5):1.05

(10)

for La 4 Ni 3 O 8 :La 2-x Sr x CuO 4 :Nd 1-x

  at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences. Work at Argonne was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division (Materials Science Division) and Scientific User Facilities Division (Advanced Photon Source). X. L. and J.Q.L. were supported by the ShanghaiTech University startup fund, MOST of China under Grant No. 2016YFA0401000, NSFC under Grant No. 11934017 and the Chinese Academy of Sciences under Grant No. 112111KYSB20170059. Work at the Advanced Photon Source was supported under Contract No. DE-AC02-06CH11357. A.B. acknowledges the support from NSF DMR 2045826. We acknowledge Diamond Light Source for time on Beamline I21 under Proposal 22261 producing the data shown in Fig. 2. This research used resources at the SIX beamline of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704. We acknowledge Synchrotron SOLEIL for provision of synchrotron radiation facilities at the SEXTANTS beamline.

			(0.12, 0)	(0.17, 0)	(0.22, 0)	(0.27, 0)	(0.32, 0)	(0.37, 0)	(0.42, 0)	(0.47, 0)
		100
	Intensity (arb. units)	0 100	(0.08, 0.08)	(0.11, 0.11)	(0.14, 0.14)	(0.17, 0.17)	(0.2, 0.2)	(0.23, 0.23)	(0.26, 0.26)	(0.29, 0.29)
		0	-0.1 0 0.1 0.2	0 0.1 0.2	0 0.1 0.2	0 0.1 0.2 Energy Loss (eV) 0 0.1 0.2	0 0.1 0.2	0 0.1 0.2	0 0.1 0.2

FIG.

TABLE I .

 I Exchange constants in the GGA and GGA+U approximations (U = 4.75 eV).

	Exchange constant	GGA+U (meV)	GGA (meV)	Exchange ratio	GGA+U	GGA
	J	97.60	71.40	J/Jz	5.04	5.25
	J1	14.28	10.65	J/J1	6.83	6.7
	Jz	19.38	13.6	Jz/J1	1.36	1.28

arXiv:2008.08209v2 [cond-mat.supr-con] 16 Jan 2021

Supplemental Material: Strong Superexchange in a d 9-δ Nickelate Revealed by Resonant Inelastic X-Ray Scattering (Dated: January 19, 2021)

I. FURTHER DETAILS OF THE SINGLE-CRYSTAL SYNTHESIS

Single-crystal growth of R 4 Ni 3 O 10 (R =La, Pr) was performed as described Refs. [START_REF] Bednorz | Possible high Tc superconductivity in the Ba-La-Cu-O system[END_REF]2]. The parent Ruddlesden-Popper phases were prepared in a floating zone furnace (HKZ-1, SciDre GmbH) with 20 bar O 2 for R =La and 140 bar for R=Pr. Oxygen was flowed at a rate of 0.1 l/min during growth and the feed and seed rods were counter-rotated at 30 r.p.m. and 27 r.p.m., respectively, to improve zone homogeneity. The traveling speed of the seed was 4 mm/h and the growth time was 30 hours. 438-phase crystals were obtained by reducing the 4310-specimens in 4 mol % H 2 /Ar gas at 350 • C for five days. The resulting samples have appreciable residual strain and are very brittle, so they were mounted on copper plates for transport.

II. RESONANT BEHAVIOR

In Fig. S1 we show the resonant behavior of the magnon. The magnon peak is visible at several energies from 851.5 to 853.1 eV exhibiting a Raman-like behavior in which it appears at a constant energy loss, rather than a constant final x-ray energy. We find that the magnon is strongest at 852.7 eV (as emphasized by the dashed line at this energy). This is well above the La M 4 -edge at 849 eV, further confirming the magnetic origin of the magnon.

III. FITTING OF THE RIXS DATA

The spectra were fitted with a Gaussian function for the elastic peak, a damped harmonic oscillator model for the paramagnon and a quadratic background. The inelastic peak was convoluted with a Gaussian function to account for the energy resolution. This lineshape is described by nine parameters, but only six parameters are free to vary in the fit. For the Gaussian lineshape describing the elastic peak, the center and the width are fixed by measurements of a graphite elastic reference sample, only the amplitude is free to vary. For the inelastic mode, the temperature is fixed, and the center, width and amplitude are free. For the quadratic background, we use a function f (x) = b for x < 0 The chosen working energy that optimizes the magnon intensity, 852.7 eV, is highlighted using a dashed line and occurs above the maximum in the elastic line resonance, further confirming the magnetic nature of the observed inelastic excitations [3].


Flore Barcellini
email: flore.barcellini@lecnam.net

The Design of "Future Work" in Industrial Contexts Lessons Learned from Worker-Technology Cooperation and Work Transformation Management

Keywords: Worker-technologies cooperation, Transformation of work management, Activity-centred design, Ergonomics, Industry 4.0

This chapter proposes to discuss work transformation management as a key issue to designing working conditions to ensure safe, healthy and performant work in a context of technological transitions. Here, we view the design of future work as a transition process and as a set of projects-social, organisational, technological-that will together shape a "future of work". We consider lessons from the past regarding: (1) cooperation between workers and technology and (2) project management of work transformations, as resources to manage this transition. These lessons will be discussed with reference to ongoing research dealing with introduction of artificial intelligence (AI) or collaborative robotics at work in the French industrial context, as an illustration of the transformation of industry at a global level.

Introduction

Work situations in industry are facing a profound transformation in relation to the introduction of technology due to both an evolution of this technology (evolutions of AI algorithms and massive data analysis, collaborative robotics…) and a political invitation at an international level [START_REF] Bidet-Mayer | L'industrie du futur : une compétition mondiale[END_REF] to promote the "modernisation" of industry. Indeed, a closer look at issues raised by technological transformations reveals that some "old" ones seem to be ignored by promotors of those transformations. 1 In this context, the objective of this chapter is to recall lessons learned concerning both workers-technology cooperation and work transformation management. We discuss the relevance of these lessons learned on the basis of several ongoing collaborative research projects dealing with introduction of collaborative robotics in industry line [START_REF] Barcellini | Conception of human-robot collaborative cooperation. A case of participation in a collaborative robot design process[END_REF][START_REF] Bounouar | On the role of human operators in the design process of cobotic systems[END_REF][START_REF] Lafeuillade | Integrating collaborative robotics into work situations: the intentions of SME managers in the digital transformations of their companies[END_REF][START_REF] Saraceno | L'homme «au coeur». Du robot au cobot, le mythe du «bon» automate dans la médiatisation de «l'industrie du futur[END_REF] or artificial intelligence in various work situations (radiology, online counselling, legal activities, design and engineering) [START_REF] Barcellini | Conception of human-robot collaborative cooperation. A case of participation in a collaborative robot design process[END_REF][START_REF] Gamkrelidze | The "old" issues of the "new" artificial intelligence systems in professional activities[END_REF][START_REF] Gamkrelidze | Working with machine learning/artificial intelligence systems: workers' viewpoints and experiences[END_REF]. 2In this chapter, we propose to discuss work transformation management as a key issue to designing "a future work" to ensure safe, healthy and performant work in a context of technological transitions. Here, we view the design of future work as a transition process and as a set of projects-social, organisational, technologicalthat will together shape a "future work" in a given situation. We focus here on lessons learned from the past regarding two key issues for managing this transition:

(1) cooperation between workers and technology and (2) project management of work transformation, i.e. design project management.

Lessons Learned for Worker-Technology Cooperation

Research and Project Design Management

Is Worker-Technology Cooperation a Myth or a Possible Reality?

Worker-technology cooperation has been studied at least since massive introduction of automation at work, as well as the first surge of artificial intelligence or expert systems in the late 80s, and in particular in relation to safety. Previous works on automation, AI and cooperation between human and machine sought to qualify the feasibility of human-machine cooperation and the risks implied for human health and organisational safety [START_REF] Hoc | Towards a cognitive approach to human-machine cooperation in dynamic situations[END_REF][START_REF] Hoc | Expertise and Technology: Cognition and Human-Computer Cooperation[END_REF][START_REF] Rabardel | Human and Technologies: Cognitive Approaches of Contemporary Instruments[END_REF][START_REF] Zouinar | Évolutions de l'Intelligence Artificielle: quels enjeux pour l'activité humaine et la relation Humain Machine au travail?[END_REF]. These earlier works outlined limits such as impossibilities for an "intelligent or cooperative machine" to access and interpret context of action or to build a shared understanding of a worker situation, a poor relation between workers and technology that may be unidirectional and nonadaptative, i.e. the "machine" is not able to display dynamic behaviour in the course of the joint action with workers. These limits implied that the "machine" was not actually cooperative in a strong sense, as cooperation is anchored in the capacity of sharing common goals and of regulation of interdependent situations on the basis of understanding of a given situation of action [START_REF] Hoc | Expertise and Technology: Cognition and Human-Computer Cooperation[END_REF][START_REF] Schmidt | Remarks on the complexity of cooperative work[END_REF]. Technologies and workers were, thus, mainly in interdependence more than in cooperation [START_REF] Dekker | MABA-MABA or abracadabra? Progress on human-automation co-ordination[END_REF][START_REF] Hoc | Towards a cognitive approach to human-machine cooperation in dynamic situations[END_REF][START_REF] Schmidt | Remarks on the complexity of cooperative work[END_REF][START_REF] Zouinar | Évolutions de l'Intelligence Artificielle: quels enjeux pour l'activité humaine et la relation Humain Machine au travail?[END_REF]. Regarding safety issues, these limits in the worker-technology relationship may imply [START_REF] Hoc | Towards a cognitive approach to human-machine cooperation in dynamic situations[END_REF]: loss of human expertise and lack of mutual control over a given situation, limited situation awareness construction and thus reliability of the worker-technology systems; complacency of workers regarding technology proposals and lack of self-confidence;

and finally a lack of adaptivity of the worker-technology system due to a lack of anticipation and feedback [START_REF] Reason | Cognitive aids in process environments: prostheses or tools?[END_REF]. Moreover, these limits are reviewed as being one cause of accidents [START_REF] Funk | Flight deck automation issues[END_REF][START_REF] Sarter | How in the world did we get into that mode?" Mode error and awareness in supervisory control[END_REF]. They can be explained by: (1) the limitations of the technology, the lack of in-depth modelling of human socio-cognitive activities [START_REF] Amalberti | La conduite des systèmes à risque (PUF, collection[END_REF][START_REF] Daniellou | l'écran: l'ergonomie des salles de contrôles[END_REF][START_REF] Rabardel | Human and Technologies: Cognitive Approaches of Contemporary Instruments[END_REF]; but also (2) by the way, the introduction of technology is managed in particular with relation to socio-organisational consequences of technological changes, as outlined by the participatory approach and activity-centred ergonomics approach since the 70s [START_REF] Garrigou | Activity analysis in participatory design and analysis of participatory design activity[END_REF][START_REF] Simonsen | Routledge International Handbook of Participatory Design[END_REF].

A Lack of a Participative and Work-Centred Project Management Approach in Introducing Cooperative Technology at Work

Indeed, several studies reveal that a majority of projects to introduce new technologies "fail" (60-80% according to studies [START_REF] Combes | Comment prédire et expliquer l'échec des changements organisationnels[END_REF]) with regards to delays in actual usage of new systems, budget overrun, goal achievement, safety and health issues [START_REF] Wisner | Operation rate of robotized systems: the contribution of ergonomic work analysis[END_REF]. Different models may be considered to explain these failures. There is extensive literature about a so-called resistance to change in organisations or an organisational inertia [START_REF] Ford | Resistance to change: the rest of the story[END_REF], the necessity of a cultural change or the role of leaders and managers [START_REF] Autissier | Vandangeon-Derumez, Pas de changement sans adhésion des managers[END_REF]. However, there are several strong empirical and theoretical limitations to these models addressed by alternative research in ergonomics, sociology, organisation and management sciences (see [START_REF] Akrich | Beyond social construction of technology: the shaping of people and things in the innovation process[END_REF][START_REF] Beck | P for political: participation is not enough[END_REF][START_REF] Bernoux | Sociologie du changement dans les entreprises et les organisations[END_REF][START_REF] Garrigou | Activity analysis in participatory design and analysis of participatory design activity[END_REF][START_REF] Lapeyrière | Les aventures de Substance et Cohérence au pays des projets[END_REF][START_REF] Nascimento | Organizational change, cultural change? Insights for ergonomic intervention[END_REF] for critics of previous models) revealing alternative causes of project failure: (1) the lack of political management and of actual cooperation in project management (weakness in the political management of the project and in the definition of project goals; poor collaboration of leaders; technically driven projects; absence of true participatory design approaches) and ( 2) the work that takes place in the organisations is approached only as a set of theoretical tasks in project management. The social and organisational dimensions of work are overlooked at the beginning of the projects and addressed only at a later stage, as consequences of technological choices. The constraints and leeway related to work activity, the consequences on health and on the quality of production or safety are poorly addressed. These may lead to some hazardous consequences for performance and health of workers, including safety and reliability. In order to overcome these pitfalls, some participative and work-centred project management approaches have been proposed for more than 40 years by participatory design or activity-centred ergonomics [START_REF] Garrigou | Activity analysis in participatory design and analysis of participatory design activity[END_REF][START_REF] Simonsen | Routledge International Handbook of Participatory Design[END_REF]-sometimes supported by legislation successfully framing industrial relations as in Scandinavia [START_REF] Kanstrup | D for democracy: on political ideals[END_REF], or not as the failed "Auroux" laws in France in the early 80s [START_REF] Howell | The contradictions of French industrial relations reform[END_REF], and with more or rather less actual adoption in organisations.

So, Are Ongoing Transformations of Work in Relation to Technology Neglecting Lessons Learned from the Past?

As said in the introduction, industrial work situations are experiencing ongoing transformations in relation to introduction of technology (big data, RFID, virtual reality, AI, collaborative robotics…), these transformations are being encouraged by various national political programmes [START_REF] Bidet-Mayer | L'industrie du futur : une compétition mondiale[END_REF], such as "Industry 4.0" in Germany, "Industrie du futur" in France, "National Network for Manufacturing Innovation" in USA, "Manufacturing Industry Innovation 3.0 Strategy" in the UK, Made in China 2025.

Here, the statement is that the globalisation of the financial market, coupled with the ageing of the working population and industrial facilities and ecological issues, may make it necessary to improve production plant and competitiveness. Moreover, the COVID-19 crisis may increase the wish to preserve and to develop a strong and innovative national industrial activity. So, what can be said about this "new era" of workers-technologies cooperation and the way introduction of these technologies is managed with regards to lessons learned from past research? To address these questions, we focus on two flagship "modernisation" technologies: AI and collaborative robotics, and we analyse proposals made by the French programme "Industrie du futur" to organise transformations, as it is claimed that this is one of the only national programmes which puts "Humans at the heart" of transformation.

A Strong Techno-Determinism, a Lack of Explainability and an Under-Estimation of the Socio-organisational Impacts of Technologies

Ongoing studies [START_REF] Barcellini | Conception of human-robot collaborative cooperation. A case of participation in a collaborative robot design process[END_REF][START_REF] Bounouar | On the role of human operators in the design process of cobotic systems[END_REF][START_REF] Gamkrelidze | The "old" issues of the "new" artificial intelligence systems in professional activities[END_REF][START_REF] Gamkrelidze | Working with machine learning/artificial intelligence systems: workers' viewpoints and experiences[END_REF] and analysis of the French "Industrie du Futur" programme using concepts and framework developed by activity-centred ergonomics and sociology [START_REF] Barcellini | Conception of human-robot collaborative cooperation. A case of participation in a collaborative robot design process[END_REF][START_REF] Lafeuillade | Integrating collaborative robotics into work situations: the intentions of SME managers in the digital transformations of their companies[END_REF][START_REF] Saraceno | L'homme «au coeur». Du robot au cobot, le mythe du «bon» automate dans la médiatisation de «l'industrie du futur[END_REF] reveal a strong techno-determinism in the way technologies are still being conceived/thought of. Technologies are still seen as "remedies" to "problems" (e.g. competitiveness; safety…) with a lack of systematic analysis of those "problems". In addition, the possible consequences on company performance and worker health of the introduction of heterogeneous technologies "there and everywhere", without imagining their joint integration and the contradictions in everyday work that they may generate for workers, are not always considered. It is, however, necessary to assume there is a duality of "cooperation between workers and technology vs. subordination of workers to technology" and evaluate its risks for health, performance and safety, considering organisational and collective issues associated with the introduction of technologies. One can argue that some "old issues" are re-emerging nowadays even if they may be renewed by the enhanced performance of a "machine" [START_REF] Gamkrelidze | The "old" issues of the "new" artificial intelligence systems in professional activities[END_REF][START_REF] Saraceno | L'homme «au coeur». Du robot au cobot, le mythe du «bon» automate dans la médiatisation de «l'industrie du futur[END_REF].

Concerning AI, Gamkrelidze et al. [START_REF] Gamkrelidze | The "old" issues of the "new" artificial intelligence systems in professional activities[END_REF] pointed out that issues are tackled from an economical and technical angle, mainly to promote new AI systems. The same is observed for collaborative robotics3 [34,[START_REF] Saraceno | L'homme «au coeur». Du robot au cobot, le mythe du «bon» automate dans la médiatisation de «l'industrie du futur[END_REF]. Promotors of those technologies advocate for an augmented or collaborative approach between workers and technologies and not a substitutive approach. Beyond those promises, we observed in three studies [START_REF] Barcellini | Industry of the future: which place for work and its transformation? in Moving Work[END_REF][START_REF] Bounouar | On the role of human operators in the design process of cobotic systems[END_REF][START_REF] Gamkrelidze | Working with machine learning/artificial intelligence systems: workers' viewpoints and experiences[END_REF] that collaborative robotics or AI still raise the question of actual cooperation between workers and technologies as well as questions about transformations of professional gestures, division of labour between workers and technologies, autonomy and responsibility of workers dependent of a machine behaviours, organisational issues or more societal job issues.

For instance, actual cooperative AI or systems should be able to understand the goals and actions of workers, and workers must be able to understand the functioning and decisions of AI or cooperative systems. This goal has not been reached yet as AI is still not really "explainable" [START_REF] Gamkrelidze | Working with machine learning/artificial intelligence systems: workers' viewpoints and experiences[END_REF]. In the same direction, collaborative robotics exemplified the promises-and pitfalls-of articulation between so-called cooperative technologies and evolution of work situations. On the "promises" side, this technology is presented as "easy to implement and to maintain"; "favouring productivity gains"; virtuous by making certain workstations more attractive for young workers or by contributing to the prevention of musculoskeletal disorders (MSD)cobots taking over repetitive or strenuous tasks. This illustrates well the "technologies seen as remedies" perspective referred to above. On a previous project, 4 we tried to articulate technical and work-related issues in the design of a collaborative robot demonstrator intended to equip automobile assembly lines in order to prevent musculoskeletal disorders [START_REF] Barcellini | Conception of human-robot collaborative cooperation. A case of participation in a collaborative robot design process[END_REF]. Although the demonstrator was not industrialised, 5its design helped to understand technological locks related to dynamic regulation of interdependence between workers and cobots (e.g. dynamic re-planification of robot trajectories related to its analysis of the worker's position in space, gestures or direct communication). However, it was impossible to evaluate its potential role in MSD prevention. Indeed, the demonstrator was designed to reduce some of the biomechanical constraints incurred by workers, but it was only a necessary condition to prevent MSD and not a sufficient one. Indeed, design of the technology alone will not solve prevention and health issues; psychosocial dimensions of work, work organisation, interdependence between introduction of technology projects and other organisational projects related to the work situation [START_REF] Lafeuillade | Integrating collaborative robotics into work situations: the intentions of SME managers in the digital transformations of their companies[END_REF] are as just as necessary-or even more so-than design of technological artefacts. Yet, in the former project, neither the evolution of the professional gestures nor collective or organisational issues related to the future work organisation were addressed. This advocates for strong proposals of work transformation project management approaches, in order to jointly design technologies and future work situations, including social and organisational issues.

A Claim for More Participative and Collaborative Project Management with a Lack of Operational Proposals

As said previously, there is a claim for the centrality of humans and a "vital" need to renew design and change management, but we outlined that those discourses are still dominated by technical rationality and political communication with a poor representation of human and labour sciences (ergonomics, occupational psychology, sociology of work, management and organisational sciences, adult education). Indeed, there is a lack of understanding and proposals regarding transformation of work situations and introduction of collaborative organisations. This is obvious, for example, in the "Industrie du Futur" programme and in the operational means that are proposed to companies to achieve their transformation. On the contrary, we observe a promotion of the Lean Management model as "the" model for so-called operational excellence [START_REF] Fim | Guide pratique de l'usine du futur: enjeux et panorama de solutions[END_REF] even though the limits of this model have been outlined in the French context [START_REF] Benchekroun | Vitalité des activités et rationalité du Lean: deux études de cas[END_REF]. Moreover, despite the wish for more "collaborative innovation" processes in industries, only technical or financial tools are proposed to help industries in coping with transformations and not really alternative project management approaches supporting actual collaboration of workers in designing their future work and developing the potentially relevant uses and usefulness of technologies as one of the resources of work [START_REF] Bourmaud | From computer to instrument system: a developmental perspective[END_REF]. Finally, the place left to "Humans" is reduced to training, with training programmes that mainly ignore the importance of "past" experience and the presumed outdated competences of workers to ensure performance and safety at work while assuming that training may compensate for a lack of anticipation of organisational evolutions of work situation questions regarding organisational issues.

If humans and work are taken into account, it is often with an individual perspective and situated only at the workstation level, thus neglecting the collective dimension of work and its role in-once again-preserving performance, safety and health.

What Recommendations to Foster the Success of Projects in Terms of Health, Safety and Performance?

Here, we advocate for widespread promotion and experience of participatory or activity-centred ergonomics project design management approaches [START_REF] Barcellini | Design projects as opportunities for the development of activites[END_REF][START_REF] Béguin | Design as a mutual learning process between user and designers[END_REF][START_REF] Daniellou | Activity-oriented approaches to ergonomics: some traditions and communities[END_REF][START_REF] Garrigou | Activity analysis in participatory design and analysis of participatory design activity[END_REF] as good candidates for one of the "collaborative innovation" approaches. These approaches are designed to address political, social, organisational and technical issues related to work transformation. Indeed, the finality in work transformation related to introduction of technologies has less to do with the characteristics of the technologies themselves (products, tools, working spaces, workstations, organisation, technologies, etc.) as with the work situations-articulating individual, social, organisational and technological dimensions-in which these technologies are present. It assumes that work transformation projects are decision-making and design processes [START_REF] Bucciarelli | An ethnographic perspective on engineering design[END_REF], socially determining work situations (i.e. structured by a set of decisions made by CEO, designers or prescribers of work in general) and defining a set of prescriptions (tasks to be carried out, technologies to work with organisational structure staff allocation, contracts, timetables, rules, etc.) that workers will have to cope with [START_REF] Guerin | Understanding and Transforming Work: The Practices of Ergonomics[END_REF]. In order to be successful, these approaches argue that: (1) an effective political management of work transformation articulating and organising debates and negotiation within the different logics at stake in the project (technology, safety, health, production, human resources, etc.) in a systemic way is needed; [START_REF] Amalberti | La conduite des systèmes à risque (PUF, collection[END_REF] and that the decision-making process must be grounded in an understanding and a modelling of actual and future work situations based on an in situ activities analysis.

ANR project HECTTOR ANR-17-CE10-0011; ANR project ICARO ANR-10-CORD-0025 on collaborative robotics. Collaboration with Orange Labs or car manufacturer on introduction of AI in various work situations (radiology, online counselling, legal activities, design and engineering).

There is no consensus on the definition of cobot[START_REF] Moulières-Seban | Conception de systèmes cobotiques industriels: approche cognitique : application à la production pyrotechnique au sein d'Ariane Group[END_REF]. We will consider that cobots are robots assisting workers but remaining dependent on worker objectives and gestures. Cobots are thus seen as partners of workers with a direct physical interaction between them.

ANR funded project ICARO ANR-10-CORD-0025.

Because the demonstrator was too slow to cope with the pace of the industrial line.

The Design of "Future Work" in Industrial Contexts

Acknowledgements This chapter is based on past and current collaborations during the ANR ICARO (Corinne Grosse, Adolfo Suarez, Antoine Lasnier, Nahid Armande, colleagues from LAAS and Lirmm) and HECTTOR (Caroline Moricot, Anne-Cécile Lafeuillade, Willy Buchmann, Tahar-Hakim Benchekroun, Marco Saraceno, Celine Rosselin, Thierry Pillon, Gérard Dubey, Ali Siadat, Mouad Bounouar, Richard Béarée) projects and a collaboration with Orange Labs (Moustafa Zouinar, Tamari Gamkrelidze).

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