Abstract
The unconventional superconductor Sr_{2}RuO_{4} has long served as a benchmark for theories of correlatedelectron materials. The determination of the superconducting pairing mechanism requires detailed experimental information on collective bosonic excitations as potential mediators of Cooper pairing. We have used Ru L_{3}edge resonant inelastic xray scattering to obtain comprehensive maps of the electronic excitations of Sr_{2}RuO_{4} over the entire Brillouin zone. We observe multiple branches of dispersive spin and orbital excitations associated with distinctly different energy scales. The spin and orbital dynamical response functions calculated within the dynamical meanfield theory are in excellent agreement with the experimental data. Our results highlight the Hund metal nature of Sr_{2}RuO_{4} and provide key information for the understanding of its unconventional superconductivity.
Similar content being viewed by others
Introduction
Conduction electrons in quantum materials form itinerant quasiparticles that propagate coherently over mesoscopic length scales, while being renormalized by local interactions akin to those in atomic physics. This dichotomy spawns a large variety of collective quantum phenomena and remains one of the major challenges of modern condensed matter physics, as epitomized by the Hubbard model describing electrons on a lattice with a single orbital per site, which has defied a complete solution until today—60 years after it was first introduced. Coulomb repulsion of oppositespin electrons residing on the same site drives the electron system toward a Mott insulating state and induces antiferromagnetic spin correlations, which have been invoked as a mediator of Cooper pairing in unconventional superconductors such as cuprates^{1} and nickelates^{2}. Whereas bonafide realizations of the Hubbard model are rare, studies on transition metal oxides^{3} and ironbased superconductors^{4} have led to the realization that atomic Hund’s rule interactions among conduction electrons with multiple active dorbitals are a source of strong electron correlations. In principle, treating spin and orbital correlations on an equal footing further increases the challenge in the theoretical description of the interacting electron system. However, recent dynamical meanfield theory (DMFT) studies of the broad family of “Hund metals"^{5} have suggested that the Hund’s rule interactions drive a largescale differentiation of spin and orbital screening energies^{6,7}. Indirect manifestations of this “spinorbital separation" include the formation of local magnetic moments and the anomalously low onset temperature of the coherent Fermiliquid state in ironbased superconductors.
Here we report a direct spectroscopic fingerprint of spinorbital separation in the archetypical Hund metal Sr_{2}RuO_{4}^{8}, which has been the subject of many years of study in view of the textbook Fermiliquid transport properties^{9} and unconventional superconducting state^{10} that develop upon cooling below the coherence temperature T_{coh} ~ 25 K and critical temperature T_{c} = 1.5 K, respectively. Recent precision experiments in the superconducting state^{11,12,13} have cast doubt on the previously advocated spintriplet pairing scenario^{14,15}, thus revitalizing the order parameter debate^{16} and the search for an indepth understanding of the Fermiliquid normal state and collective bosonic fluctuations relevant to superconductivity. The lattice structure of Sr_{2}RuO_{4} is built up of RuO_{6} octahedra in a squareplanar arrangement, and its Fermi surface comprises three bands originating from the dorbital manifold of Ru^{4+} ions in the octahedral crystal field. Owing to the availability of exceptionally clean single crystals, the electronic quasiparticle properties in these bands are very well known and clearly indicate strong electronic correlations. The temperature dependence of the Seebeck coefficient^{17,18} suggests a separation between energy scales associated with the onset of coherence of spin and orbital degrees of freedom, consistent with the notion that these correlations are governed by Hund’s rules^{19,20}. Inelastic neutron scattering (INS) studies of Sr_{2}RuO_{4}^{21,22,23,24} have revealed lowenergy incommensurate spin fluctuations (ISFs) at the inplane wavevectors q_{ISF} = (±0.3, ±0.3) and along a squareshaped ridge connecting them (Fig. 1a, inset). However, the INS spectra are limited to energies below ~50 meV and do not yield separate information on orbital excitations.
We have used Ru L_{3}edge (2838 eV) resonant inelastic xray scattering (RIXS) to obtain spectroscopic maps of spin and orbital fluctuations over a wide range of energy and momenta. Whereas the spin excitations are almost completely confined to energies below ~200 meV, significant orbital fluctuations only appear at higher energies and extend up to ~ 1 eV, thus directly confirming the theoretically predicted spinorbital separation. The RIXS spectra disagree starkly with predictions based on the standard random phase approximation (RPA), which do not capture the distinct energy scales of spin and orbital correlations, but are in excellent agreement with calculations in the framework of DMFT, which takes into account vertex corrections. Our results thus demonstrate the key role of Hund’srule interactions in inducing electron correlations in multiband metals, and highlight the capability of current manybody theory to accurately compute twoparticle correlation functions. They also shed light on the nature of potential pairing bosons for unconventional superconductivity in Sr_{2}RuO_{4}.
Results and discussion
Figure 1a shows the crystal structure of Sr_{2}RuO_{4} and the scattering geometry for the RIXS experiment. The incident xray photons were πpolarized, and the scattered photons with both σ and π polarizations were collected at the scattering angle of 90 degrees. In this geometry, the polarizations of the incident and outgoing photons are always perpendicular, selectively enhancing magnetic responses from the spin and orbital excitations while suppressing the charge response. Given the layered crystal structure of Sr_{2}RuO_{4}, we express the momentum transfer using the inplane component q, which is scanned by changing the sample angle θ. We studied two paths in the reciprocal space, q = (H, 0) and (H, H), by fixing the azimuthal angle ϕ at 0^{∘} and −45^{∘}, respectively. These paths cross the ridge and the peak of the lowenergy ISFs (inset). The measurements were performed at T = 25 K, in the FL regime of the normal state.
In Fig. 1b, we show the Ru L_{3} RIXS spectra along the two directions. Multiple peak structures are readily identified. The main feature A is composed of multiple peaks which extend up to ~ 1 eV. These peaks are assigned to spin and orbital excitations within the t_{2g} orbitals. In addition, a weaklydispersive feature B is identified at ~3 eV (blue circles). As this energy corresponds to the splitting of the transitions to the unoccupied 4d t_{2g} and e_{g} orbitals in the Ru L_{3} xray absorption spectrum (Supplementary Fig. 1a), the feature B is readily assigned to the crystal field transitions to the \({t}_{2g}^{3}{e}_{g}^{1}\) electron configurations. We note here that its intensity is maximal close to the q = (0, 0) point along the two directions.
To visualize the characteristics of the RIXS spectra, we show in Fig. 1c a colormap of the RIXS intensity. The main feature A is composed of multiple dispersions. Its lowenergy tail exhibits downward dispersion toward its local minima at q = (−0.3, 0) and (−0.7, 0) along the (H, 0) direction and at q_{ISF} = (−0.3, − 0.3) along the (H, H) direction (white triangles). These q vectors are in excellent agreement with those of the ridges and ISFs identified in the previous INS studies^{23,24}. However, the information from the INS data is limited to the lowenergy region below ~0.1 eV, whereas the full access to a large energy window in the present RIXS experiment provides comprehensive information on the ISFs. The relative intensity of spin excitations and the location of the ISF are determined by the nesting conditions between the multiple Fermi surface sheets of Sr_{2}RuO_{4}. It is well established that the nesting between the α and β sheets with q_{ISF}, as demonstrated by angleresolved photoemission measurements^{25}, drives the incommensurate spin fluctuations with q_{ISF}^{26}. In contrast, the nesting is only partial along the (H, 0) direction, resulting in the reduced intensity of the spin fluctuations along this direction.
The colormap also reveals an additional broad dispersive feature C at high energy (≳1 eV). It emanates from the top of the feature A at q = (−0.5, 0) and (−0.5, −0.5) and merges with the feature B at the (0, 0) point, generating the intensity maximum. At the highsymmetry (0, 0) point, the dd excitations to the \({d}_{{x}^{2}{y}^{2}}\) and \({d}_{3{z}^{2}{r}^{2}}\) orbitals remain localized and are almost degenerate in energy under the small tetragonal distortion of the RuO_{6} octahedra. At finite inplane q’s, the excitations to the planar \({d}_{{x}^{2}{y}^{2}}\) orbitals show energy dispersion due to large overlap integrals with the planar O 2p orbitals, while those to the outofplane \({d}_{3{z}^{2}{r}^{2}}\) orbitals have little inplane dispersion due to the small overlap integrals. The nondispersive feature B and dispersive feature C are thus primarily ascribed to the transitions to the \({d}_{3{z}^{2}{r}^{2}}\) and \({d}_{{x}^{2}{y}^{2}}\) orbitals, respectively. The dispersion of the orbital fluctuations originates from the large bandwidth of Sr_{2}RuO_{4} with a tetragonal crystal structure. In contrast, the nondispersive orbital excitations observed in orthorhombic Ca_{2}RuO_{4}^{27} and Ca_{3}Ru_{2}O_{7}^{28} indicate that the local dd excitations cannot freely move to the neighboring sites, as the rotation of the RuO_{6} octahedra significantly reduces the hopping integrals.
Having identified multiple branches of spinorbital excitations in Sr_{2}RuO_{4}, we now scrutinize the lowenergy excitations within the t_{2g} orbitals (feature A). Figure 2a shows an expanded plot of the RIXS spectra below 0.8 eV. The broad global peak maxima (red circles) disperse from 0.2 eV at the zone center q = (0, 0), where they are most sharply peaked, to the maximal energy at the zone boundary, q = (0,−0.5) and (−0.5,−0.5). We ascribe this dispersion to orbital fluctuations as we will see below. The energy scale of orbital fluctuations near the zone center agrees with that of the O Kedge RIXS data^{29}. Along the (H, H) direction, the lowenergy region contains prominent peaks due to the ISFs around q_{ISF} = (−0.3, −0.3) and subsequent shoulder structures connected to the (0, 0) point, as indicated with black circles (Supplementary Note 2). The quasielastic intensity at the (0, 0) point is significantly weaker than at q_{ISF}, consistent with polarized INS data^{22}. On the other hand, the spin fluctuation intensity is weaker along the (H, 0) direction, except for the small increase of the quasielastic intensity of the ridge scattering around (−0.3, 0) and (−0.7, 0)^{23,24}. In addition, the spectra close to the (0, 0) point contain a broad highenergy tail peaked around ~0.5 eV (Supplementary Note 5).
Figure 2b summarizes the q dispersions of the observed RIXS features. Here, the dispersion of the orbital fluctuations is defined as the global peak maxima of the RIXS spectra, and that of the spin fluctuations as the local maxima of spectral curvature deduced from the second derivative analysis (Supplementary Fig. 2). Along the (H, 0) direction, the orbital fluctuations disperse from ~0.2 eV at the (0, 0) point and reach the maximum of ~0.5 eV at the (−0.5, 0) point. Along the (H, H) direction, the dispersion is initially steeper and becomes almost flat in the region H ≤ − 0.25 at a higher energy ~0.55 eV. The spin excitations have a local minimum of 0.06 eV at q_{ISF} = (−0.3,−0.3) and approach zero energy close to the (0, 0) point. Note here that the spin and orbital fluctuations have distinct energy scales in the entire q space without a clear signature of mutual crossing. This observation is of crucial importance in testing the validity of different theoretical approaches.
To facilitate a direct connection to the INS results, we show in Fig. 2c an expanded colormap of the RIXS intensity around q_{ISF} and corresponding energy distribution curves with a step size of 0.02 eV. The RIXS intensity around q_{ISF} shows a conical shape with an isolated intensity maximum at 0.06 eV. The intensity remains centered at q_{ISF} up to 0.25 eV, consistent with the vertical INS intensity profile at q_{ISF} observed below ~0.06 eV^{30}.
We now interpret the lowenergy RIXS data in terms of theoretical spin and orbital susceptibilities \({\chi }_{{S}_{\mu }{S}_{\mu }}\) and \({\chi }_{{L}_{\mu }{L}_{\mu }}\) (μ = x, y, z), which we computed in DMFT by solving the BetheSalpeter equation using the local DMFT particlehole irreducible vertex within the Ru 4dt_{2g} subspace (Supplementary Note 3). We employed the same effective model and interaction parameters that have been established in several previous studies^{31,32,33}. Theoretical RIXS spectra are constructed by combining different components of the spin and orbital susceptibilities with matrix elements for the RIXS cross section (Supplementary Note 4). Figure 3 shows the comparison of the experimental (Fig. 3a) and theoretical DMFT (Fig. 3b) RIXS spectra along the previously defined highsymmetry momentum paths in the Brillouin zone. To highlight the importance of the DMFT dynamical vertex corrections, we also show the perturbative RPA spectra (without dynamical vertex) in Fig. 3c. It is evident that the DMFT spectra excellently capture the overall dispersion and the distribution of momentum and energy dependent maxima of the RIXS data. Specifically, the lowenergy intensity is peaked at q_{ISF} and also extrapolates continuously to the corresponding quasistatic intensity close to zero energy^{32}. Moreover, the broader maximum emanates from 0.2 eV at q = (0, 0) and disperses more steeply along the (H, H) direction. In contrast, the spectral weight distribution in RPA fails to capture the lowenergy intensity maximum at q_{ISF} and yields a spuriously sharp feature that extends from high energy to zero energy around the Γ point.
This difference between DMFT and RPA originates from the distinct behavior of the spin and orbital dynamical responses. Figure 4 shows the intensity plots of theoretical orbital (LL) and spin (SS) susceptibilities along the q = (H, 0) and (H, H) directions, obtained by averaging all components \(\frac{1}{3}{\sum }_{\mu }\,{\chi }_{{L}_{\mu }{L}_{\mu }}\) and \(\frac{1}{3}{\sum }_{\mu }\,{\chi }_{{S}_{\mu }{S}_{\mu }}\). The vertex corrections in DMFT lead to the clear energy separation of spin and orbital contributions predicted for the Hund metals (left panels). The spin response accounts for almost all the spectral weight at low energies up to ~0.2 eV, and becomes negligible above this scale. The concentrated spectral weight around q_{ISF} and weak ridge scattering around q = (−0.3, 0) and (−0.7, 0) excellently reproduce the experimental observations. The orbital response sets in at higher energies > 0.2 eV and shows broad maxima centered around commensurate momenta q = (−0.5, 0) and (−0.5, −0.5). In RPA (right panels), on the other hand, both the spin and orbital responses disperse and have spectral weight over the entire energy range. The RIXS data thus provide direct and quantitative evidence for the spinorbital separation in correlated Hund metals as captured by DMFT. Further improvement between the experiment and theory could be obtained by a more rigorous treatment of the resonance effect in the RIXS cross section, in particular on the spectral weight maximum of the orbital excitations around q = (−0.5, −0.5).
It should also be noted that the spinorbit coupling (SOC) effects have been considered only outside the vertex in our DMFT calculations. Comparison to angleresolved photoemission experiments^{31} and inelastic neutronscattering^{32} have justified this procedure on the level of the singleparticle spectra and static magnetic susceptibility. The excellent agreement we find in the present work supports the strategy also for the dynamic susceptibility. This situation is contrasted to the spinorbital J physics in the Mott insulating counterpart Ca_{2}RuO_{4}^{27} and the cubic K_{2}RuCl_{6}^{34}, whose magnetic ground states are determined by the interplay between the ionic J multiplets and the strength of intersite exchange interactions. While the t_{2g} electrons of Sr_{2}RuO_{4} carry orbital angular momentum, the finite bandwidth of the itinerant electrons partially quenches the orbital momentum. Nonetheless, the SOC brings about significant modification of the singleparticle band structure at certain highsymmetry momenta in the Brillouin zone, when multiple bands are degenerate in energy. It is well known that in Sr_{2}RuO_{4} this degeneracy occurs in the diagonal direction in the reciprocal space, which leads to the separation of the Fermisurface sheets^{31,35}. Correspondingly, the effect of SOC on the dynamical susceptibilities is most pronounced in the lowenergy spin fluctuations at q = (0, 0) and at q_{ISF}, while the effect on the orbital fluctuations remains minor (Supplementary Fig. 8).
The current findings also have implications for the microscopic mechanisms of the superconductivity in Sr_{2}RuO_{4}. As primary candidates of bosonic fluctuations mediating the Cooper pairing, the spin and orbital dynamical susceptibilities enter the Eliashberg equations, which in turn determine the SC order parameter. Our combined RIXS and DMFT+SOC results provide a comprehensive description of the momentum distribution, dispersion relation, and spinorbit composition of lowenergy magnetic excitations, which can serve as crucial input for approximate solutions of the Eliashberg equations. Recent theoretical studies suggest that static (RPA) and dynamic (DMFT) vertex approximations lead to qualitatively different SC ground states^{33,36,37,38}. Although computational challenges prohibit rigorous extrapolation of our theoretical results to low temperatures near T_{c} = 1.5 K, the RIXS data point to the critical role of dynamical vertex corrections also for the microscopic description of the superconducting order parameter.
In conclusion, we have presented Ru L_{3} RIXS measurements of the dynamical response functions in the unconventional superconductor Sr_{2}RuO_{4} over a broad range of energy and momentum. We have identified several branches of spin and orbital excitations and revealed the separation of energy scales associated with these two sets of degrees of freedom, a predicted hallmark of Hund metals which had not yet received direct experimental confirmation. The measured spectra are in excellent agreement with theoretical calculations based on DMFT including vertex corrections, while significant discrepancies with the perturbative RPA approximation are found. Our results thus epitomize the power of stateoftheart manybody theories to yield a detailed, quantitative understanding of complex electronic correlation functions in real materials. By establishing the properties of key collective modes, they also provide a solid baseline for the future identification of the nature and symmetry of SC order in this prominent model compound.
Methods
Sample growth and characterization
The Sr_{2}RuO_{4} single crystals with superconducting T_{c} ~ 1.5 K were grown by the floatingzone method^{39} and prealigned using an inhouse Laue diffractometer. Sr_{2}RuO_{4} has the tetragonal space group I4/mmm with the lattice constants of a = b = 3.903 and c = 12.901 Å. The inplane momentum transfers are expressed in the reciprocal lattice units (r.l.u.).
IRIXS spectrometer
The RIXS experiments were performed using the intermediateenergy RIXS (IRIXS) spectrometer at the P01 beamline of PETRA III at DESY^{40}. The incident xray energy was tuned to the Ru L_{3} absorption edge (2838 eV) and incoming photons were monochromatized using a highresolution monochromator composed of four asymmetricallycut Si(111) crystals. The polarization of the incident xray photons was in the horizontal scattering plane (π polarization). The polarizations of the scattered photons were not analyzed. The xrays were focused to a beam spot of 20 × 160 μm^{2} (H × V). Scattered photons from the sample were collected at the scattering angle of 90^{∘} (horizontal scattering geometry) using a SiO_{2} (10\(\bar{2}\)) (ΔE = 60 meV) diced spherical analyzer with a 1 m arm, equipped with a rectangular [100 (H) × 36 (V) mm^{2}] mask and a CCD camera, both placed in the Rowland geometry. Collected raw CCD images were transformed into RIXS spectra by summing over the vertical axis of the detector and by binning with 12.5 meV steps along the horizontal axis. To account for the xray selfabsorption effect, the RIXS intensity was normalized to the total fluorescent intensity collected with an energyresolved photon detector placed at the scattering angle of 110^{∘}. The exact position of the zero energy loss line was determined by measuring nonresonant spectra from silver paint deposited next to the sample. The overall energy resolution of the IRIXS spectrometer at the Ru L_{3}edge, defined as the full width half maximum of the nonresonant spectrum from silver, was ~80 meV. All the measurements were performed at 25 K (normal state), well above the superconducting T_{c}.
Data availability
The raw RIXS data generated in this study are available at desycloud: https://desycloud.desy.de/index.php/s/LPt7RJTHqGWLNBD.
Code availability
The numerical codes used to generate the results in this work are available at desycloud: https://desycloud.desy.de/index.php/s/LPt7RJTHqGWLNBD.
References
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to hightemperature superconductivity in copper oxides. Nature 518, 179–186 (2015).
Li, D. et al. Superconductivity in an infinitelayer nickelate. Nature 572, 624–627 (2019).
Imada, M., Fujimori, A. & Tokura, Y. Metalinsulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).
Fernandes, R. M. et al. Iron pnictides and chalcogenides: a new paradigm for superconductivity. Nature 601, 35–44 (2022).
Georges, A., Medici, L. D. & Mravlje, J. Strong correlations from Hund’s coupling. Annu. Rev. Condens. Matter Phys. 4, 137–178 (2013).
Stadler, K. M., Yin, Z. P., von Delft, J., Kotliar, G. & Weichselbaum, A. Dynamical meanfield theory plus numerical renormalizationgroup study of spinorbital separation in a threeband Hund metal. Phys. Rev. Lett. 115, 136401 (2015).
Horvat, A., Žitko, R. & Mravlje, J. Lowenergy physics of threeorbital impurity model with Kanamori interaction. Phys. Rev. B 94, 165140 (2016).
Mackenzie, A. P. & Maeno, Y. The superconductivity of Sr_{2}RuO_{4} and the physics of spintriplet pairing. Rev. Mod. Phys. 75, 657–712 (2003).
Hussey, N. E. et al. Normalstate magnetoresistance of Sr_{2}RuO_{4}. Phys. Rev. B 57, 5505–5511 (1998).
Maeno, Y. et al. Superconductivity in a layered perovskite without copper. Nature 372, 532–534 (1994).
Pustogow, A. et al. Constraints on the superconducting order parameter in Sr_{2}RuO_{4} from oxygen17 nuclear magnetic resonance. Nature 574, 72–75 (2019).
Ishida, K., Manago, M., Kinjo, K. & Maeno, Y. Reduction of the ^{17}O Knight shift in the superconducting state and the heatup effect by NMR pulses on Sr_{2}RuO_{4}. J. Phys. Soc. Jpn. 89, 034712 (2020).
Petsch, A. N. et al. Reduction of the spin susceptibility in the superconducting state of Sr_{2}RuO_{4} observed by polarized neutron scattering. Phys. Rev. Lett. 125, 217004 (2020).
Rice, T. M. & Sigrist, M. Sr_{2}RuO_{4}: an electronic analogue of ^{3}He? J. Phys. Condens. Matter 7, L643–L648 (1995).
Ishida, K. et al. Spintriplet superconductivity in Sr_{2}RuO_{4} identified by ^{17}O Knight shift. Nature 396, 658–660 (1998).
Kivelson, S. A., Yuan, A. C., Ramshaw, B. & Thomale, R. A proposal for reconciling diverse experiments on the superconducting state in Sr_{2}RuO_{4}. npj Quantum Mater. 5, 43 (2020).
Klein, Y. et al. Thermoelectric power in ruthenates: dominant role of the spin degeneracy term. MRS Online Proceedings Library 988, 9880706 (2006).
Mravlje, J. & Georges, A. Thermopower and entropy: Lessons from Sr_{2}RuO_{4}. Phys. Rev. Lett. 117, 036401 (2016).
Mravlje, J. et al. Coherenceincoherence crossover and the massrenormalization puzzles in Sr_{2}RuO_{4}. Phys. Rev. Lett. 106, 096401 (2011).
Kugler, F. B. et al. Strongly correlated materials from a numerical renormalization group perspective: How the Fermiliquid state of Sr_{2}RuO_{4} emerges. Phys. Rev. Lett. 124, 016401 (2020).
Sidis, Y. et al. Evidence for incommensurate spin fluctuations in Sr_{2}RuO_{4}. Phys. Rev. Lett. 83, 3320–3323 (1999).
Steffens, P. et al. Spin fluctuations in Sr_{2}RuO_{4} from polarized neutron scattering: Implications for superconductivity. Phys. Rev. Lett. 122, 047004 (2019).
Iida, K. et al. Horizontal line nodes in Sr_{2}RuO_{4} proved by spin resonance. J. Phys. Soc. Jpn. 89, 053702 (2020).
Jenni, K. et al. Neutron scattering studies on spin fluctuations in Sr_{2}RuO_{4}. Phys. Rev. B 103, 104511 (2021).
Damascelli, A. et al. Fermi surface, surface states, and surface reconstruction in Sr_{2}RuO_{4}. Phys. Rev. Lett. 85, 5194–5197 (2000).
Mazin, I. I. & Singh, D. J. Competitions in layered ruthenates: Ferromagnetism versus antiferromagnetism and triplet versus singlet pairing. Phys. Rev. Lett. 82, 4324–4327 (1999).
Gretarsson, H. et al. Observation of spinorbit excitations and Hund’s multiplets in Ca_{2}RuO_{4}. Phys. Rev. B 100, 045123 (2019).
Bertinshaw, J. et al. Spin and charge excitations in the correlated multiband metal Ca_{3}Ru_{2}O_{7}. Phys. Rev. B 103, 085108 (2021).
Fatuzzo, C. G. et al. Spinorbitinduced orbital excitations in Sr_{2}RuO_{4} and Ca_{2}RuO_{4}: A resonant inelastic xray scattering study. Phys. Rev. B 91, 155104 (2015).
Iida, K. et al. Inelastic neutron scattering study of the magnetic fluctuations in Sr_{2}RuO_{4}. Phys. Rev. B 84, 060402 (2011).
Tamai, A. et al. Highresolution photoemission on Sr_{2}RuO_{4} reveals correlationenhanced effective spinorbit coupling and dominantly local selfenergies. Phys. Rev. X 9, 021048 (2019).
Strand, H. U. R., Zingl, M., Wentzell, N., Parcollet, O. & Georges, A. Magnetic response of Sr_{2}RuO_{4}: Quasilocal spin fluctuations due to Hund’s coupling. Phys. Rev. B 100, 125120 (2019).
Käser, S. et al. Interorbital singlet pairing in Sr_{2}RuO_{4}: A Hund’s superconductor. Phys. Rev. B 105, 155101 (2022).
Takahashi, H. et al. Nonmagnetic J = 0 state and spinorbit excitations in K_{2}RuCl_{6}. Phys. Rev. Lett. 127, 227201 (2021).
Haverkort, M. W., Elfimov, I. S., Tjeng, L. H., Sawatzky, G. A. & Damascelli, A. Strong spinorbit coupling effects on the Fermi surface of Sr_{2}RuO_{4} and Sr_{2}RhO_{4}. Phys. Rev. Lett. 101, 026406 (2008).
Acharya, S. et al. Evening out the spin and charge parity to increase T_{c} in Sr_{2}RuO_{4}. Commun. Phys. 2, 163 (2019).
Gingras, O., Nourafkan, R., Tremblay, A.M. S. & Côté, M. Superconducting symmetries of Sr_{2}RuO_{4} from firstprinciples electronic structure. Phys. Rev. Lett. 123, 217005 (2019).
Gingras, O., Allaglo, N., Nourafkan, R., Côté, M. & Tremblay, A.M. S. Superconductivity in correlated multiorbital systems with spinorbit coupling: Coexistence of even and oddfrequency pairing, and the case of Sr_{2}RuO_{4}. Phys. Rev. B 106, 064513 (2022).
Bobowski, J. S. et al. Improved singlecrystal growth of Sr_{2}RuO_{4}. Condens. Matter 4, 6 (2019).
Gretarsson, H. et al. IRIXS: a resonant inelastic Xray scattering instrument dedicated to Xrays in the intermediate energy range. J. Synchrotron Rad. 27, 538–544 (2020).
Acknowledgements
We thank A. Damascelli, G. Khaliullin, A. Yaresko, and D. Kukusta for enlightening discussions. We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. The RIXS experiments were carried out at the beamline P01 of PETRA III at DESY. The project was supported by the European Research Council under Advanced Grant No. 669550 (Com4Com) awarded to B.K.. H.S. acknowledges financial support from the JSPS Research Fellowship for Research Abroad and GrantsinAid for Scientific Research from JSPS (KAKENHI) (number 22K13994). H.S. and L.W. acknowledge financial support from the Alexander von Humboldt Foundation. S.K. acknowledges financial support by the DFG project HA7277/31. H.U.R.S. acknowledges financial support from the ERC synergy grant (854843FASTCORR). N.K. is supported by KAKENHI (Grant Nos. 18K04715, 21H01033, and 22K19093), CoretoCore Program (No. JPJSCCA20170002) from JSPS, and a JSTMirai Program (Grant No. JPMJMI18A3). Research in Dresden benefits from the environment provided by the DFG Cluster of Excellence ct.qmat (EXC 2147, project ID 390858940) awarded to A.P.M.. The Flatiron Institute is a division of the Simons Foundation.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Contributions
H.S., L.W., J.B., Z.Y., and H.G. performed the RIXS experiments. M.K., F.J., N.K., and A.P.M. grew the Sr_{2}RuO_{4} single crystals. H.S., L.W., and M.K. performed the sample characterizations. H.G. designed the beamline and IRIXS spectrometer. H.S. analyzed the experimental data. H.U.R.S., S.K., N.W., and O.P. developed the computational framework used in the theoretical calculations. S.K., H.U.R.S., A.G., and P.H. carried out the theoretical calculations of the dynamical response functions. H.S. and P.H. constructed the theoretical RIXS intensity from the response functions. H.S., S.K., H.U.R.S., A.G., P.H., and B.K. wrote the manuscript with input from all the coauthors. B.K. initiated and supervised the project.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks the anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Suzuki, H., Wang, L., Bertinshaw, J. et al. Distinct spin and orbital dynamics in Sr_{2}RuO_{4}. Nat Commun 14, 7042 (2023). https://doi.org/10.1038/s41467023428043
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467023428043
This article is cited by

Coherent propagation of spinorbit excitons in a correlated metal
npj Quantum Materials (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.