Introduction

The presence of strong intra-atomic spin-orbit coupling (SOC) in quantum materials can give rise to a variety of many-body phenomena, ranging from spin liquid phases to topologically protected states1,2,3. Since the SOC constant λ scales with the atomic number, it is strong in 5d-electron materials, while it is naturally less prominent in 4d and 3d systems. Nonetheless, it can still shape the electronic and magnetic properties of the latter materials. For instance, SOC is a critical ingredient for the emergence of excitonic magnetism in the 4d antiferromagnetic (AFM) Mott insulator Ca2RuO44,5,6,7,8,9, and it can lead to prevalent Kitaev exchange interactions in other 4d ruthenates10 as well as in 3d cobaltates11.

Collective excitations (quasiparticles) that possess a spin-orbit-entangled character are hallmarks of the role of SOC in shaping the electronic properties of quantum materials. Notably, the dispersion relation of such spin-orbit exciton (SOE) modes encodes the relevant microscopic interaction parameters, from which model Hamiltonians can be constructed, as demonstrated for several insulating 3d materials12,13,14,15. Another example is the 5d AFM Mott insulator Sr2IrO4, where strong SOC splits the t2g-orbital manifold into separated bands with total angular momentum j = 1/2 and 3/216,17,18,19,20,21,22, and a SOE mode arises from transitions of holes across the split manifold23,24,25,26,27,28. Resonant inelastic x-ray scattering (RIXS) experiments at the Ir L3-edge23,24 and at the O K-edge26,27,28 found that the SOE dispersion exhibits a maximum at the Γ-point and a minimum at the AFM Brillouin zone (BZ) boundary. The corresponding propagation of the SOE can be described by theories that include the interaction with magnons in the exciton hopping process23,24. In hole- and electron-doped Sr2IrO4 without long-range AFM order, the SOE retains its typical dispersion as long as short-lived magnetic excitations (paramagnons) exist29,30.

Recently, the influence of the intra-atomic SOC on the electronic structure and macroscopic properties of correlated metals has moved into the center of attention. For instance, the debate about the role of SOC for the unconventional superconductivity in the correlated 4d metal Sr2RuO431,32,33,34,35,36 has been revived37,38,39,40. Yet, whereas collective excitations of conduction electrons in metallic systems, such as plasmons41,42 and paramagnons43,44, have been extensively studied, the propagation of spin-orbit entangled modes in such systems has remained elusive. One notable exception are heavy-fermion metals45 where spin-orbit excitations can be mediated by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between local f-electron moments and conduction electrons46, which, however, is quite different from Sr2RuO4 where only d-electrons are situated at the Fermi level.

Here we use RIXS at the O K-edge to examine the low-energy excitation spectrum of the correlated 4d-electron material Sr2RhO4. Similarly to Sr2RuO4, the material is a paramagnetic (PM) metal with a sharply defined Fermi surface47,48,49, but with one additional valence electron its 4d5 configuration is isoelectronic to Sr2IrO4 (5d5). The distinct insulating AFM ground state of the latter material is usually rationalized by its narrow, half-filled j = 1/2 band that is further split into upper and lower Hubbard bands due to on-site Coulomb repulsion U16,17. In the case of Sr2RhO4, it was suggested that the moderate SOC is incapable of fully splitting the j = 1/2 and 3/2 bands, and thus even an increased U cannot open a Mott gap, leaving the material metallic17. Specifically, extensive studies on the electronic structure of Sr2RhO4 pointed out that the metallic ground state is actually the result of a concerted interplay between SOC, Coulomb interaction, and structural distortions49,50,51,52, although the definite role of SOC and whether it acts in an effective, ‘Coulomb-enhanced’53,54 fashion is still under debate. Most importantly, however, it is not clear to what extent the ionic j = 1/2 and 3/2 picture, which is natural for insulating materials, is applicable in a correlated metal.

Our RIXS experiment reveals a well-defined, dispersive SOE mode, whose emergence is an explicit manifestation of the splitting of the t2g manifold into bands carrying total angular momentum j = 1/2 and 3/2, in close analogy to the spin-orbit split electronic structure of Sr2IrO4. We observe, however, that the SOE in Sr2RhO4 exhibits a markedly distinct dispersion relation, which we model and discuss in the context of the different ground states of the two materials. Our results provide insights into the collective dynamics of spin-orbit entangled quasiparticles, which had previously remained elusive in correlated metallic systems.

Results

Identification of orbital states in XAS

The experiment was performed on Sr2RhO4 single-crystals grown by the optical floating zone method (see Materials and Methods). Figure 1a shows the unit cell of Sr2RhO4 in the tetragonal space group I41/acd, which is closely related to Sr2IrO455. The lattice parameters a, b = 5.45 Å and c = 25.75 Å were determined from a Rietveld refinement of powder x-ray diffraction data of a pulverized Sr2RhO4 crystal (see Supplementary Note 1). The scattering geometry used for the XAS and RIXS measurements is sketched in Fig. 1b. While the scattering angle between the incoming (k) and outgoing photon beam (k’) was fixed to 130∘, the projected momentum transfer within the RhO2 planes could be varied by changing the incident angle θ. Furthermore, an azimuthal rotation of the sample by 45∘ (not shown here) allowed us to orient the in-plane momentum transfer either along the Rh-O-Rh bond direction [(0,0)-(π,0) direction], or along the diagonal direction [(0,0)-(π,π) direction]. Note that the former direction coincides (approximately) with the x/y direction as defined within the RhO6 octahedral network (Fig. 1a), while the latter direction coincides with the crystallographic a/b direction.

Fig. 1: Experimental scattering geometry and orbital states probed in XAS and RIXS.
figure 1

a Crystal structure of Sr2RhO4, with the solid black lines corresponding to the tetragonal unit cell. The crystallographic a, b, and c-axis are indicated, as well the x, y, and z-axis of the RhO6 octahedral reference frame. The labels Oa and Op mark the apical and in-plane oxygen ions, respectively. b Schematic of the scattering geometry of the O K-edge RIXS experiment. The square planar lattice of a RhO2 plane is illustrated, where the RhO6 octahedral tilts are omitted for simplicity. The scattering plane (gray) is spanned by the z-direction and either the x- or the diagonal x + y direction (not shown here). The former case corresponds to transfer of the projected in-plane momentum along the (0,0)-(π,0) direction in reciprocal space, and the latter case along the (0,0)-(π,π) direction. The incident photons are linearly π-polarized, and k (k') corresponds to the incoming (outgoing) photon beam. The angle between k and the sample surface is denoted by θ. c Splitting of the t2g hole level into the j = 1/2 and 3/2 spin-orbit multiplets. d XAS across the O K-edge measured in total electron yield for θ = 6∘ (top panel) and θ = 80∘ (bottom panel). The vertical dashed line marks the incident photon energy (528.1 eV) used for the RIXS measurements. The labels refer to the Rh t2g orbitals (dxz, dyz, dxy) that are hybridized with 2p states either from the Oa or Op ions. e Schematics of the orbital hybridizations that are relevant in d. Blue (gray) boxes indicate hybridizations with in-plane (apical) oxygen p orbitals.

Full size image

Figure 1d shows the XAS pre-edge structure of the O K-edge of Sr2RhO4, which corresponds to transitions into unoccupied states of hybridized O 2p and Rh 4d (t2g) orbitals. Clear differences are visible between spectra taken for θ close to grazing incidence (upper panel) and close to normal incidence (lower panel). The angular dependence of the XAS features is due to the spatial extensions of the lobes of the three active Rh t2g-orbitals (dxz, dyz, dxy), as well as crystallographically distinct oxygen sites of the RhO6 octahedra, differentiating the apical (Oa) and the in-plane oxygen sites (Op). Depending on the chosen scattering geometry, the XAS signal contains contributions from specific combinations of the Rh t2g orbitals with either the Oa or the Op ions, each of which possesses three different p orbitals (px, py, pz). Schematics of the Rh-O hybridized orbital states that are relevant for our study are displayed in Fig. 1e. The assignment of the XAS features in Fig. 1d to these hybridized states is consistent with a previous O K-edge XAS study on Sr2RhO456.

Spin-orbit entangled excitations in RIXS

Figure 2 shows RIXS spectra measured with an incident photon energy of 528.1 eV. For small θ, this energy coincides with the XAS resonance energy of Rh dyz and dxz states hybridized with pz orbitals of the planar Op ions (Fig. 1d). For larger values of θ, the Rh dxy states hybridized with px and py orbitals of Op become the dominant feature in the XAS (Fig. 1d), although we note that the maximum of the latter XAS resonance is situated at slightly higher energies. A comprehensive survey of the dependence of the inelastic features in the RIXS spectra as a function of the incident energy and the incident angle θ is given in Supplementary Fig. 3. Notably, we find that the low-energy features in the RIXS spectra (below 1 eV energy loss), on which we will focus in the following, do not depend significantly on the incident energy. In fact, the peak around 300 meV energy loss in Fig. 2 is present across the entire Rh dyz/dxz resonance for θ = 6∘, as well as the dxy resonance for θ = 80∘, although the peak intensity decreases when moving away from the XAS maximum energy (see Supplementary Fig. 3). However, the~300 meV peak is absent (or below the detection limit) when the incident photon energy is tuned to 527.6 eV, which corresponds to the resonance of the Rh dyz/dxz states hybridized with py/px orbitals of the Oa ions (Fig. 1d and Supplementary Fig. 3). As a consequence of these observations, the key RIXS measurements of this study (Fig. 2) were carried out exclusively at 528.1 eV.

Fig. 2: Spin-orbit exciton in Sr2RhO4.
figure 2

O K-edge RIXS spectra taken at different incident angles θ, which varies the projected in-plane momentum transfer along (a), the (0,0)-(π,0) direction and (b), the (0,0)-(π,π) direction, as illustrated in the insets. Spectra are offset in vertical direction for clarity. c Exemplary fit (solid blue line) of the RIXS spectrum taken at θ = 65∘. The fitted peak profiles of the quasi-elastic line (gray dotted line) and the SOE (gray area) are indicated. d Exemplary fit (solid red line) of the RIXS spectrum taken at θ = 9∘ for the (0,0)-(π,π) direction.

Full size image

As a first step, we inspect the RIXS spectra in Fig. 2a. Each spectrum in the panel was collected at a different in-plane momentum transfer, which was achieved by varying the incident angle θ. The corresponding coverage of the in-plane BZ along the (0,0)-(π,0) direction is illustrated in the inset in Fig. 2a. As the central result, we find that the peak centered around ~300 meV for θ = 65∘ disperses to higher energies for increasing momentum transfer (decreasing θ), and broadens in its linewidth. A similar behavior is observed when the momentum transfer is oriented along the (0,0)-(π,π) direction (Fig. 2b). For a detailed analysis, we fit the RIXS spectra by the sum of the quasi-elastic peak situated around zero-energy loss and a damped harmonic oscillator (DHO) function57 that accounts for the dispersive peak around ~300 meV. Two exemplary fit results are shown in Fig. 2c, d, while the complete set of fitted spectra and additional information about the fitting procedure are given in the Methods section and Supplementary Fig. 4. Note that the DHO model is used due to the broad linewidth and asymmetric lineshape of the peak, which is present even at (0,0) momentum transfer, i.e., at the 2D BZ center. The application of the DHO model is further justified when considering it is strongly damped, in particular for large momentum transfers where the damping parameter extracted in the fits becomes comparable to ω0 (Fig. 3a, b). Nevertheless, while the DHO fit captures the rising edge of the dispersive peak accurately (Fig. 2c, d), a single DHO function does not account for additional spectral weight that emerges at higher energy losses. This spectral weight is essentially featureless and reminiscent of a broad background contribution, which will be discussed in more detail below.

Fig. 3: Exciton dispersion and propagation in Sr2RhO4 and Sr2IrO4.
figure 3

a Energy ω0 of the SOE (filled blue symbols) extracted from fits of the RIXS spectra with a DHO model (see text and Fig. 2) plotted as a function of the in-plane momentum transfer in units of \(1/{a}^{{\prime} }\), where \({a}^{{\prime} }=a/\sqrt{2}\) is the Rh-Rh bond length. The positions of the maximum spectral intensity (open blue symbols) are superimposed for comparison. Filled gray symbols correspond to the energy of the SOE in Sr2IrO4, extracted by Lorentzian fits of Ir L3-edge RIXS data in ref. 24. The gray (blue) line corresponds to a tight-binding model for the exciton hopping in Sr2IrO4 (Sr2RhO4). b Damping parameter γ from the DHO fits. The procedure for determining the error bars is detailed in the Methods sections. c Schematic of the incoherent nearest-neighbor hopping of the SOE in an AFM background. We use the hole language which is convenient to describe the d5 configuration containing a single hole in the t2g manifold. Lattice site A (B) represents spin-up (spin-down) magnetic sublattice of the Ir-d5 ions. The orange arrow indicates the promotion of A site from the j = 1/2 ground state (black spin) to the excited j = 3/2 state (blue spin). The blue cloud represents the exciton. Hole hopping between neighboring sites is denoted by \(\tilde{t}\). In the virtual state, holes interact via the on-site Coulomb repulsion U. In the final state of the exchange process, the resulting occupation (red spin) of the A site is incompatible with the magnetic order of the sublattices (red wiggled lines) and a magnon is created. As a consequence, exciton hopping from A to B is incoherent. An additional hopping of the exciton to a next-nearest neighbor (not shown here) can create another frustrated spin that is required for a double-spin flip to reestablish the proper sublattice order. d Coherent hopping of the SOE in a paramagnetic metal. A t2g hole in the ground (excited) state of d5 ion is indicated by blue sphere. Left: the exchange process involving high-energy (∝U) virtual states as in Sr2IrO4 but without a magnon creation. Right: the exciton motion in Sr2RhO4 using a single hole hopping between Rh ions with different charge configurations, d5 and d6, which are present in the metallic ground state. Charge fluctuations between d5 and d4 states (discussed in Supplementary Note 4) similarly contribute to the exciton hopping.

Full size image

The oscillator frequencies ω0 extracted with the DHO model are shown in Fig. 3a (filled blue symbols). In addition, we display the positions of the maximum spectral intensity (open blue symbols) for comparison. Both positions are similar in proximity to the BZ center at (0,0), but deviate for large momentum transfer along the (0,0)-(π,0) and (0,0)-(π,π) directions. This deviation arises because for large damping of the DHO, the center of the peak no longer coincides with the actual oscillator frequency ω0. Since the damping parameter γ in the DHO fits becomes larger for higher momentum transfers (Fig. 3b), the discrepancy between dispersion and peak position thus increases towards the zone boundary. Because the DHO model has the spectral correspondence of a collective mode, whereas the direct physical meaning of maximum spectral intensity might be ambiguous, we focus on the ω0 positions in the following.

In any case, we note that the two differently extracted peak energies both fall into an energy interval between ≈250 and 375 meV, with the minimum energy at (0,0). This energy scale is significantly higher than the reported single- and bimagnon energies in Sr2IrO4, which revolve around 40 meV58 and 160 meV26, respectively. Typical magnon energies in 4d-electron compounds are also well below 100 meV3,5,6,7. We therefore rule out a purely magnetic origin of the dispersive Sr2RhO4 peak. Furthermore, we consider its assignment to a peak seen in optical conductivity59 as implausible, because this peak is centered only around 180 meV and reproduced by theoretical calculations both with and without SOC54. Optical transitions that involve p − d charge transfer were reported at energies higher than 2 eV56 and can also be observed in our RIXS spectra at these high energies (Supplementary Fig. 3). On the other hand, the energy regime of our observed low-energy peak is remarkably similar to \(\frac{3}{2}\lambda \approx\) 285 meV, that is, the energy difference between the j = 1/2 and 3/2 levels in the t2g manifold (Fig. 1c), according to the free-ion SOC constant λ ≈ 190 meV of Rh4+60. Moreover, quantum chemistry calculations predict a spin-orbit excitation around 250 meV in Sr2RhO461. In analogy to Sr2IrO4, we therefore assign the observed mode to excitations of a hole across the spin-orbit split t2g manifold, which is referred to as SOE23,24.

Comparison to excitons in iridates

A direct comparison between the SOE dispersions in Sr2RhO4 and Sr2IrO4 is provided in Fig. 3a. The most distinctive differences are that the energy scale in Sr2IrO4 is higher and the curvature of the dispersion curves around the Γ-point (0,0) is inverted. We note that the SOE energy in Sr2IrO4 was determined from fits with Lorentzian functions to Ir L3-edge RIXS data in ref. 24. While the differences in the data analysis and RIXS process possibly obscure details for a quantitative comparison of the two compounds, the statements about the different energy scales and curvature of the dispersions are robust.

In the 5d material Sr2IrO4, the expected SOC splitting is large, \(\frac{3}{2}\lambda \approx\) 570 meV24, which explains the distinct energy scales in Fig. 3a. Notably, for Sr2RhO4, the good agreement between the observed SOE energy and \(\frac{3}{2}\lambda\) indicates that SOC is neither quenched nor ‘Coulomb-enhanced’. We note, however, that a decisive determination of the mode energy requires future RIXS experiments with the improved resolution or the use of complementary techniques, such as Raman spectroscopy25.

The inverted dispersions in Fig. 3a imply that the exciton propagation processes in the AFM insulator and nonmagnetic metal are qualitatively different. In Sr2IrO4, exciton motion involves the exchange of the j = 1/2 and j = 3/2 states, using high-energy virtual hoppings of electrons between Ir4+ ions (Fig. 3c), similar to spin-exchange in magnets. In the AFM state, the nearest-neighbor exciton hopping creates a flipped spin, i.e., a magnon, and a coherent motion of the exciton is only possible within the same (A or B) magnetic sublattice23,24, when the pairs of flipped spins can relax into the AFM ground state. This results in a dispersion minimum at the AFM zone boundary \((\frac{\pi }{2},\frac{\pi }{2})\) and a maximum at the Γ-point24, in close analogy to the motion of a doped hole in the AFM CuO2 planes of cuprates62.

In paramagnetic Sr2RhO4, the exciton hopping does not generate magnons, and no band-folding effects, associated with the magnetic unit cell doubling, occur. The major difference from Sr2IrO4 is, however, that the metallic ground state of Sr2RhO4 comprises valence states other than the dominant Rh4+(d5) one, in particular Rh3+(d6) and Rh5+(d4). Charge fluctuations between these states can promote the SOE motion without involving any high-energy exchange processes (Fig. 3d, right panel). The overall amplitude of this contribution is proportional to the densities of the d6 and d4 charge configurations, and its sign is given by the sign of the hopping integral of the t2g electrons. The latter is negative, because this hopping is mainly via the oxygen p orbitals53, i.e., \(t=-{t}_{pd}^{2}/{{{\Delta }}}_{pd}\), where tpd is the overlap between Rh and O states and Δpd > 0 is the charge-transfer gap. Therefore, the SOE dispersion is determined by two competing mechanisms: a virtual exchange process that dominates in the Mott insulator Sr2IrO4, and a real charge transfer process that generates the dispersion minimum at the Γ-point in metallic Sr2RhO4.

On a qualitative level, the SOE hopping amplitude is \(\tau \sim 2{\tilde{t}}^{2}/U-2\tilde{t}n\), where n is the average density of the d6 charge states (which is equal to that of the d4 states), reduced from its ‘free-electron’ value by the on-site Coulomb repulsion. The parameter \(\tilde{t}\simeq \frac{2}{3}{t}_{pd}^{2}/{{{\Delta }}}_{pd}\) refers to the hoppings between the spin-orbit j levels in Fig. 3 and includes the SOC projection factor of 2/3. A quantitative theory should include mixing of the SOE modes with the underlying electronic particle-hole continuum, as the exciton motion in metals is coupled to the fermionic density fluctuations (Fig. 3d). The observed broad linewidth of the SOE and the additional spectral weight that is not captured by the DHO fits (Fig. 2c, d and Methods) might be a signature of this mixing.

Modeling of the exciton propagation

Further insights into the dispersion of the SOE can be gleaned from the application of a semi-quantitative tight-binding model on a square lattice, representing the Rh (Ir) sites in the RhO2 (IrO2) planes, with τ1, τ2, and τ3, the nearest, next-nearest, and third-nearest neighbor exciton hopping parameters, respectively. In a model capturing the dispersion of Sr2IrO4, the hopping amplitudes are positive, with τ1 possessing a small value, while τ2 and τ3 are large and of similar magnitude. The small value of τ1 is due to the AFM background, which inhibits a coherent nearest-neighbor hopping23,24 (Fig. 3c). The resulting dispersion for a parameter choice with a τ1: τ2: τ3 ratio of 1:4:4 and τ1 = 2 meV is shown as the gray line in Fig. 3a.

In contrast, the hopping amplitudes for Sr2RhO4 are negative, and the τ1 value is dominant, for instance such that the τ1: τ2: τ3 ratio is 4:1:0.7 and τ1 = − 20 meV (blue line in Fig. 3a). The leading τ1 value is rationalized by a coherent hopping of SOE in the PM background, as shown in Fig. 3d, and its negative value implies a predominance of the second term in the relation \(\tau \sim 2{\tilde{t}}^{2}/U-2\tilde{t}n\) introduced above. In fact, with \({t}_{pd}^{2}/{{{\Delta }}}_{pd}\approx 0.2\) eV typical for t2g electrons, the Coulomb interaction U ≈ 2.3 eV in Sr2RhO454, and a density n ≈ 1/8, this estimate yields a τ1 comparable to the one obtained from our modeling. It is interesting to note that in terms of the Hubbard model, the above value of n places Sr2RhO4 half-way between the free-electron and the Mott-insulating limits (n = 1/4 and n = 0, correspondingly).

Despite the significantly smaller values of τ2 and τ3 compared to τ1 in Sr2RhO4, a model that completely disregards them fails to describe the experimentally observed dispersion around (0,0) well. We interpret the necessity to include τ2 and τ3 as a consequence of the spatially extended character of the 4d wavefunctions, which may result in longer-range hoppings. Nevertheless, for the precise determination of the τ1: τ2: τ3 ratio, data of the SOE energies beyond the maximum momentum transfer of the present O K-edge RIXS experiment would be required. Future experiments that provide a wider coverage of reciprocal space, for instance Rh L3-edge RIXS or inelastic neutron scattering, are therefore highly desirable.

Discussion

In summary, our observation and description of a collective, spin-orbit entangled mode provide direct evidence for the significance of SOC in Sr2RhO4. The presence of the mode indicates that the splitting of the t2g manifold into states with total angular momentum j = 1/2 and 3/2 is an essential ingredient of the low-energy electronic structure of Sr2RhO4, which needs to be taken into account in future theoretical models of this material and other 4d-electron compounds. We find that the energy scale of the SOE mode is close to 3/2 of the SOC constant λ of Rh4+, which implies the presence of unquenched orbital moments. Our comparison to the SOE dispersion in the AFM insulating 5d analogue Sr2IrO4 reveals how excitonic quasiparticles behave when they propagate on a metallic background and magnonic dressings are stripped off.

Furthermore, our results suggest that the SOE is a property of the RhO2 planes and possesses a 2D-like character, as the mode resonates exclusively at energies that correspond to hybridized states of Rh and planar O sites (Supplementary Fig. 3). Such a character is reminiscent of the extensively studied excitons in 2D layered materials, including monolayer transition metal dichalcogenides63. Notably, the propagation of excitons in the latter materials can be controlled by external tuning parameters64, and a bosonic condensation can be achieved for instance by optical pumping65. Whether the properties of the SOE in Sr2RhO4 can be tuned in a similar fashion remains to be examined in future studies.

Extending our findings beyond Sr2RhO4, our insights into the differences of the SOE propagation in antiferromagnetic insulating versus metallic systems hold significant implications for a variety of 4d and 5d transition metal compounds, including those based on ruthenium8,66,67,68,69,70,71,72. In particular, the charge fluctuation mechanism might also be critical for the SOE propagation in the correlated metal and superconductor Sr2RuO4, where previous RIXS experiments at the O K-edge have reported a spin-orbit entangled mode with an energy of a few hundred meV67 and a study at the Ru L3-edge has identified dispersive branches of spin-orbital excitations73. Yet, the d4 ground state of the Ru4+ ions with two holes in the t2g shell suggests the emergence of a more complex SOE multiplet structure than that of Rh and Ir ions in d5 configuration, calling for complementary experimental and theoretical investigations.

Methods

Sample growth

Single crystals of Sr2RhO4 were synthesized using the optical-floating zone technique. Off-stoichiometric quantities of Rh2O3 (Alfa Aesar, 99.9%) and SrCO3 (Alfa Aesar, 99.994%) powder with the molar ratio 1.15:4 were ground together and calcinated at 1100 ° C for 36 hrs. Subsequently the powder was reground and compressed into a dense rod which was then sintered at 1400 ° C for 48 hrs. To promote the oxidization of Rh3+ to Rh4+, both calcination steps were performed under flowing O2 gas. The floating zone growth was performed using a CSC FZ-T-10000-H-II-VP furnace equipped with four 1.5 kW halogen lamps. A maximum of 7 bar oxygen partial gas pressure was applied and crystals were grown at a rate of 10 mm/hr to limit the amount of evaporation from highly volatile Rh-oxides. The largest single-crystals had dimensions of approximately 1.5 cm × 0.5 cm × 0.4 cm, and it was found that the crystals cleave easily. A characterization of the physical properties of the crystals is given in the Supplementary Note 1.

RIXS experiment

The XAS and RIXS spectra were collected at the U41-PEAXIS beamline74,75,76 of BESSY II at the HZB, Berlin. The measurements were carried out at T = 15 K with linearly π-polarized photons with energies tuned to the O K-edge (≈530 eV). The effective energy resolution of the RIXS measurements was ΔE ≈ 90 meV. The scattering geometry of the experiment is sketched in Fig. 1b.

Analysis of the RIXS data

An overview of all fitted RIXS spectra is presented in Supplementary Fig. 4. The zero-energy loss in each spectrum was determined by acquiring a reference spectrum from carbon-tape prior to each measurement at different incident angles. To that end, a thin piece of carbon tape film which covers a small portion of the sample surface was attached on the Sr2RhO4 single-crystal. The effect of height difference between sample and tape was found to be negligible by scanning the tape at multiple positions along the beam. The employed energy resolution did not allow us to resolve fine details around zero-energy loss. Therefore, the elastic peak was modeled by a Voigt function77,78, capturing both elastic and quasi-elastic contributions in a single peak. The SOE was modeled using a damped harmonic oscillator (DHO) function57, which can be expressed as

$$f(\omega )\propto \frac{\gamma \omega }{{({\omega }^{2}-{\omega }_{0}^{2})}^{2}+4{\gamma }^{2}{\omega }^{2},}$$
(1)

where ω0 denotes the oscillator frequency and γ is the damping parameter. The experimental energy resolution of ΔE = 90 meV was taken into account via Gaussian convolution of the DHO function. Note that for large damping, the oscillator frequency can deviate from the position ωp of the peak maximum, as \({\omega }_{0}^{2}={\omega }_{{{{\rm{p}}}}}^{2}+{\gamma }^{2}\) (for γ≤ω0). As a consequence, we find that the peak position and ω0 deviate in the RIXS spectra in Figs. 2 and 3 for high momentum transfers, where the SOE feature becomes increasingly broad due to larger damping79. An additional complication arises from excess spectral weight at the high-energy shoulder of the SOE peak, which cannot be captured by the DHO function employed for the SOE. The emergence of this spectral weight can be due to the particle-hole continuum or multiple-exciton processes, and is reminiscent of the asymmetry of the SOE feature in the O K-edge RIXS spectra of Sr2IrO426,27. In our approach, we do not attempt to capture this incoherent contribution by fitting additional peaks at higher energies. Instead, we only fit the ‘coherent’ part of the peak, that is, the low-energy shoulder as well as a small range of the high-energy shoulder. A possible error that can be introduced by the choice of the cut-off of the fitting range of the high-energy shoulder is reflected by the error bars for ω0 shown in Fig. 3a. Specifically, we extracted ω0 for a number of different fits with differently chosen cut-offs of the fitting range of the high-energy shoulder (see Supplementary Fig. 5). The resulting spread of ω0 is indicated by the error bars at the filled blue symbols in Fig. 3a. The average energy defines the position of the data point. The error bars of open blue symbols in Fig. 3a reflect a conservative estimate of the uncertainty in our assignment of the maximum of the intensity, in correspondence to the sampling interval of our RIXS data points, which is 16 meV.