Supplemental Material for “-Orbiton in Ca2RuO4 Revealed by Resonant Inelastic X-ray Scattering”

L. Das,1 F. Forte,2, 3 R. Fittipaldi,2, 3 C. G. Fatuzzo,4 V. Granata,2, 3 O. Ivashko,1 M. Horio,1 F. Schindler,1 M. Dantz,5 H. M. Rønnow,6 W. Wan,7 N. B. Christensen,7 J. Pelliciari,5, ∗ P. Olalde-Velasco,5, † N. Kikugawa,8, 9 T. Neupert,1 A. Vecchione,2, 3 T. Schmitt,5 M. Cuoco,2, 3 and J. Chang1 1Physik-Institut, Universit¨atZ¨urich,Winterthurerstrasse 190, CH-8057 Z¨urich,Switzerland 2CNR-SPIN, I-84084 Fisciano, Salerno, Italy 3Dipartimento di Fisica “E.R. Caianiello”, Universit`adi Salerno, I-84084 Fisciano, Salerno, Italy 4Institute of Physics, Ecole´ Polytechnique Fed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland 5Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 6Institute for , Ecole´ Polytechnique Fed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland 7Department of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 8National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, 305-0047 Japan 9National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA We present the model Hamiltonian employed for the analysis of the RIXS cross section. In order to extract the character of the RIXS spectra, we follow their evolution in terms of the Hund’s coupling JH and the Coulomb interaction U. We provide a detailed description of the electronic configurations at the ruthenium site and we determine the superexchange amplitude allowing for the propagation of local spin-orbiton excitations.

I. MODEL HAMILTONIAN term. We assume that these parameters obey relations that follow from considering the Coulomb interaction in 0 The model Hamiltonian of the propagating a rotationally symmetric system, namely, U = U + 2JH 0 within the ruthenium-oxygen plane for the bands close to and J = JH. 1 the Fermi level can be built by considering the interaction We denote by the 3-vector of 2 × 2 matrices s = 2 σ terms at the ruthenium and oxygen sites and the kinetic is the spin operator expressed through the vector of the term for the ruthenium-oxygen connectivity. Concerning three Pauli matrices σ. Furthermore the 3-vector of 3×3 1,2 the local ruthenium Hamiltonian Hloc, we consider the matrices l is the projection of the orbital angular momen- complete Coulomb interaction for the t2g electrons, the tum operator to the t2g subspace. It has components spin-orbit coupling, and the tetragonal crystal field po- (lk)αβ = ikαβ, with k = x, y, z, such that l × l = −i l. tential. The on-site Coulomb, spin-orbit and crystal field Explicitly, in the basis (dyz, dxz, dxy), the matrices for contributions are given by: the orbital operators are X X       Hel-el(i) = U niα↑niα↓ − 2JH Siα · Siβ + 0 0 0 0 0 −i 0 −i 0 α<β lx = 0 0 i , ly = 0 0 0  , lz = i 0 0 .   0 −i 0 i 0 0 0 0 0 0 JH X + U − niαniβ + (2) 2 α<β The local Hamiltonian at the oxygen site only includes X +J 0 d† d† d d , the on-site energy term, which is introduced to take into iα↑ iα↓ iβ↑ iβ↓ account the energy difference between the occupied or- α<β bitals of oxygen and ruthenium. It reads X X † HSOC(i) = λ diασ(lαβ · sσσ0 )diβσ0 , 0 O α,σ β,σ Hel (j) = εxnj,px + εynj,py + εznj,pz , (3)  Hcf (i) = εxyni,dxy + εz ni,dxz + ni,dyz , where j labels a given oxygen site and nj,p are the H (i) = H (i) + H (i) + H (i) , k loc Hel-el SOC cf occupation number operators of the pk orbital, with where i labels the ruthenium site, α, β are indices run- k = x, y, z, at that site. ning over the three orbitals in the t sector, i.e., Lastly, we introduce the ruthenium-oxygen hopping 2g term, which we assume includes only the tetragonal sym- α, β ∈ {d , d , d }, and d† is the creation opera- xy xz yz iασ metry allowed terms for a generic bond connecting a tor of an with spin σ at the site i in the or- † † ruthenium to an oxygen atom along the x-direction, for bital α. Furthermore ni,α ≡ di,α,↑di,α,↑ + di,α,↓di,α,↓ example. This ruthenium-oxygen hopping along the x- † and ni,α,σ ≡ di,α,σdi,α,σ are the respective density op- direction then reads erators. The interaction is parametrized by the intra-  †  orbital Coulomb interaction U and the Hund’s coupling HRu-O,aˆx (i) = txz,z di,xz,σpi+ˆax,z,σ + h.c. + J , while (U 0−J /2) sets the strength of the inter-orbital H H  †  0 electron-electron interaction, and J is the pair hopping +txy,z di,xy,σpi+ˆax,y,σ + h.c. , 2

involved. The configurations are given by

(ndxy , ndxz , ndyz ) ∈ {(1, 1, 2), (1, 2, 1), (2, 1, 1)} (4)

for nd = 1 and

(ndxy , ndxz , ndyz ) ∈ {(0, 2, 2), (2, 0, 2), (2, 2, 0)} (5)

for nd = 2. It is worth pointing out that the presence of the ligand oxygens allows to have charge transfer between the 2p and d orbitals which makes configurations of the type d5-2p1 directly relevant for the processes involved in the RIXS probe and for the construction of a low FIG. 1. (color online). Energy spectrum of the S = 1 and 4 energy effective Hamiltonian. When considering a one-doublon |d ; nd = 1i block Hbloc,T of the Hamiltonian at ruthenium-oxygen-ruthenium bond, the configurations the ruthenium site, for a representative value of the tetragonal d5-2p2-d3 are the lowest-energy intermediate states that crystal field splitting δ = 0.3 eV. enter in the spin and orbital exchange processes. In contrast, configurations of the type d5-2p1 as well as d6-2p0 represent higher energy states, due to the energy where j = i+a ˆ is the index of the oxygen site neighbor- x cost related to the charge transfer from p to d orbitals ing the ruthenium site i in the x-direction and p† i+ˆax,z,σ and the local Coulomb interactions at the ruthenium site. is the operator creating an electron in the oxygen pz or- bital at site i +a ˆx with spin σ. The hopping terms along In this section, we will focus on the d4 electronic config- the ruthenium-oxygen bonds in y-direction are defined urations. As we will show later, they form the set of the similarly. lowest energy levels of the Hamiltonian at the Ru site. In the following, we aim to represent Hloc in the spin/orbital sectors of the d4 subspace. We characterize the basis II. ELECTRONIC STATES AT RUTHENIUM states by using the quantum numbers S of the total spin, SITE mS of the total spin projection in z direction and the orbital α ∈ {dyz, dxz, dxy} that is doubly occupied. We Let us now concentrate on the eigenstates of the Hamil- thus denote the basis states by |S, mS, αi. In the nd = 2 tonian Hloc for the single ruthenium atom (in which subspace, only mS = 0 appears, as all the electrons in case we can drop the site index). It is useful to intro- the doubly occupied sites need to form spin singlet con- duce a notation for the local ruthenium configurations figurations. In the nd = 1 subspace, only mS = −1, 0, 1 that takes into account the number nd of doubly occu- appear, as the electrons in the doubly occupied site need pied orbitals (so-called doublons). We recall that we have to form a singlet. The remaining two electrons can then n = 4 electrons in the stochiometric Ru4+ state and that be in a spin-singlet or spin-triplet state. 4 they fill only the t2g subspace. Then, we indicate with The spin-triplet sector of the d subspace 4 4 |d ; nd = 1i and |d ; nd = 2i the states with n = 4 is 9-dimensional. A basis for this subspace  electrons in the d-shell and nd = 1 and nd = 2 dou- is given by |1, −1, dyzi, |1, 0, dyzi, |1, 1, dyzi, bly occupied configurations, respectively. In each of the |1, −1, dxzi, |1, 0, dxzi, |1, 1, dxzi, |1, −1, dxyi, |1, 0, dxyi, nd = 1 and nd = 2 sectors, three orbital configurations |1, 1, dxyi . In this basis, the Hamiltonian Hloc, pro- are possible, which are distinguished by the orbital occu- jected on the spin-triplet subspace, has the following pation numbers (ndxy , ndxz , ndyz ) of the three d-orbitals matrix representation

 δ 0 0 −iλ 0 0 0 − √λ 0  2  0 δ 0 0 0 0 √λ 0 − √λ   2 2   0 0 δ 0 0 iλ 0 √λ 0   2   λ   iλ 0 0 δ 0 0 0 i √ 0   2  H = (E + u)11 +  0 0 0 0 δ 0 i √λ 0 i √λ  (6) bloc,T 0 e 9  2 2   λ   0 0 −iλ 0 0 δ 0 i √ 0   2   0 √λ 0 0 −i √λ 0 0 0 0   2 2  − √λ 0 √λ −i √λ 0 −i √λ 0 0 0   2 2 2 2  0 − √λ 0 0 −i √λ 0 0 0 0 2 2 3

4 with 119 the identity matrix in the 9 × 9 block, E0 = arated in energy from the spin-triplet sector |d ; nd = 1i 0 0 2(εxy + εz), ue = 6U − JH (assuming that U = U + 2JH) by a gap that is 2JH for the single doublon states and and δ = (εz − εxy). Had we diagonalized only the in- 2JH as well as ∼ 5JH for the two-doublon sectors, with teraction part of the Hamiltonian in this subspace, we small corrections set by the spin-orbit parameter λ and would have found the eigenvalue ue. The eigenvalues of the crystal field energy δ. the matrix (6) should be compared to the energies in the spin-singlet subspaces, where the spin-orbit coupling is ineffective. We will only compare the contributions from We now consider the eigenstates of the block Hamilto- the interaction terms in the Hamiltonian as they repre- nian Hbloc,T as they represent the low energy excitations 4 sent the dominant energy scales in the problem. In the within the |d ; nd = 1i sector. For the case at hand the 4 singlet sector of |d ; nd = 1i, the interacting part of the crystal field potential corresponds to a compression of 0 Hamiltonian has the eigenvalue 6U +JH (in place of theu ˜ the octahedra, i.e., εxy < εz, and thus the splitting δ is above). Diagonalizing the interaction in the singlet sector positive. From the lowest to the highest energy config- 4 0 of the |d ; nd = 2i states yields two eigenvalues 6U + JH urations, the eigenvalues and associated eigenvectors in 0 and 6U + 4JH. Hence, all these configurations are sep- the same basis as the Hamiltonian are given by

√ 1 2 2 1 T A1 : EA = δ − λ − δ − 2δλ + 9λ ,A1 = √ (a1, 0, −a1, −ia1, 0, −ia1, 0, 1, 0) , 1 2 c1 √ a = √ 2λ ; c = 1 + 4a2, √ 1 δ−λ+ δ2−2δλ+9λ2 1 1 1 2 2 1 T A2 : EA = δ − δ + 4λ ,A2 = √ (0, a2, 0, 0, −ia2, 0, 0, 0, 1) , 2 2 c2 √ 2 2 a = −δ+ √δ +4λ ; c = 2 , 2 2 2λ 2 1+ √ δ √ δ2+4λ2 1 2 2 1 T A3 : EA = δ − δ + 4λ ,A3 = √ (0, a3, 0, 0, ia3, 0, 1, 0, 0) , 3 2 c3 a = −a ; c = c = 2 , 3 2 3 2 1+ √ δ δ2+4λ2 B : E = δ − λ, B = 1 (−i, 0, −i, −1, 0, 1, 0, 0, 0)T , 1 B1 √ 1 2 1 2 2 1 T B2 : EB = δ − λ + δ − 2δλ + 9λ ,B2 = √ (−a5, 0, a5, ia5, 0, ia5, 0, 1, 0) , 2 2 c5 √ (7) a = √ 2λ ; c = 1 + 4a2, √ 5 −δ+λ+ δ2−2δλ+9λ2 5 5 1 2 2 1 T B3 : EB = δ + δ + 4λ ,B3 = √ (0, −a6, 0, 0, ia6, 0, 0, 0, 1) , 3 2 c6√ 2 2 a = δ+ √δ +4λ ; c = 1 , 6 2 2λ 6 1 − √ δ √ 2 2 δ2+4λ2 1 2 2 1 T B4 : EB = δ + δ + 4λ ,B4 = √ (0, a7, 0, 0, ia7, 0, 1, 0, 0) , 4 2 √ c7 2 2 a = δ+ √δ +4λ ; c = c = 1 , 7 2 2λ 7 6 1 − √ δ 2 2 δ2+4λ2 T B : E = δ + λ, B = √1 (0, 0, i, 0, 0, 1, 0, 0, 0) , 5 B5 5 2 T B : E = δ + λ, B = √1 (−i, 0, 0, 1, 0, 0, 0, 0, 0) . 6 B6 6 2

In the Fig. 1, we show the evolution of the energy eigen- and A3 = |Tz = −1i are the only states that enter in the values for a representative value of the tetragonal crystal spin-orbit entangled ground-state. field splitting δ as a function of the strength of spin-orbit coupling. As one can see, the sector of those lowest en- ergy levels is made up by 9 spin/orbital entangled states III. RIXS CROSS SECTION AT THE OXYGEN that can be grouped in 2 distinct blocks: A and B. For K-EDGE vanisihing spin-orbit coupling λ = 0 the states in each of the two blocks are degenerate. The A block, consist- To calculate the polarization dependent RIXS inten- ing of the three states A1, A2, A3, is lowest in energy, sity at the oxygen K-edge, we consider the cross section with an energy bandwidth of ∼ 50 meV, while B spans associated to the transition from the oxygen core 1s to an energy range between 300 meV and 500 meV, for the the 2p level within the fast collision (FC) approximation. characteristic value of λ =75 meV. As already pointed One can then consider the case of electric field polar- out in Ref. 3 in the theoretical description of the neutron ized along the z and y directions at the planar oxygens, scattering experiments, the configurations A1,A2,A3 are corresponding to the horizontal and vertical experimen- the lowest three projected states building up an effective tal configuration, respectively, and similarly for the other pseudospin T = 1. Thus, A1 = |Tz = 0i, A2 = |Tz = 1i oxygens. The generic amplitude of the cross section can 4

FIG. 2. RIXS cross sections for planar oxygens for vertical [(a)–(c)] and horizontal polarization [(d)–(f)]. The panels are for different values of JH. All the energy scales are in eV. The amplitude I is in arbitrary units. The Coulomb interaction strengths are U 0 = 1.4 eV and U = 2.2 eV. be expressed as tained from first-principle calculations for the class of ruthenates. In particular, we have εx = εy = εz ≡ εp, X X † 2 A(pk) = |hm|pj,k,σpj,k,σ|0i| δ [ω − (Em − E0)] , with εp − εxy = −1.0 eV, δ = 0.3 eV, txz,z = 1.2 eV j∈Op m,σ and txy,z = 0.7 txz,z in order to include the effect of ro- (8) tations of the octahedra and orthorhombic distortions. † For the present calculations, variations in the d-p energy where pj,k,σ creates an electron with spin σ in the 2pk orbital, k = x, y, z, at the oxygen site j and Em are the offset (εp − εxy) and in the ruthenium-oxygen hopping energies of excited states |mi and E0 is the energy of the integrals do not qualitatively change the results. ground state |0i. The results presented in the main text refer to the In Fig. 2 we report two representative trends for the calculated RIXS spectra corresponding to i) fixed values computation of the RIXS cross-section of a ruthenium- 0 oxygen-ruthenium cluster for which we have determined of U = 2.2 eV, U = 1.4 eV, while varying JH, and ii) 0 the spectra by exact diagonalization of the full Hamilto- fixed values of U = 1.4 eV, JH = 0.5 eV, while varying nian as introduced in the Sec. I. We have also analyzed U. The aim is to understand the nature of the interme- the case of a single octahedron RuO6 as a guide for the diate and high-energy excitations occurring in the RIXS analysis of the spectra when comparing clusters with a spectra with respect to the role played by the Hund’s single or two ruthenium sites. In this Section we focus coupling and the Coulomb interaction. As one can see on the evolution of the intensity of the RIXS spectra as from the inspection of Fig 2, the spectra are highly asym- a function of the Hund’s coupling JH and the Coulomb metric when comparing the intensity associated with the interaction U at the ruthenium site. We consider a rep- transitions in the pz and px(y) channels, i.e., for the hor- resentative set of electronic parameters for the Ru-O- izontal and vertical electric field polarizations, respec- Ru cluster that is consistent with typical amplitudes ob- tively. Furthermore, the spectra are marked by several 5

FIG. 3. RIXS cross sections for planar oxygens for vertical [(a)-(c)] and horizontal polarization [(d)-(f)]. The panels are for different values of U. All the energy scales are in eV. The amplitude I is in arbitrary units. The Coulomb and Hund interaction 0 strengths are: U =1.4 eV and JH =0.5 eV.

excitations in different energy sectors. In the range of ruthenium sites. The observation of a shift in energy by a energy below 100 meV, we observe distinct excitations change in the amplitude of U confirms this interpretation with a significant intensity only for the z polarization (see Fig. 3). case. Such features are related to the transitions within the A manifold at the ruthenium site. Then, we have a set of excitations in the range ∼ 0.3 eV to 0.6 eV which IV. DISPERSION OF THE LOW ENERGY arise from the transitions between the A and B blocks EXCITATIONS at the ruthenium site, only depending on the crystal field potential and the spin-orbit coupling. Above 0.8 eV As discussed above, the nature of the low energy the spectra exhibit clear features developing in the range excitations in the spectra is related to the ruthenium ∼ [2JH, 2JH + δ] and ∼ [4JH, 4JH + 2δ] that correspond states in the block {A1,A2,A3,B1,B2,B3,B4,B5,B6}. to singlet-to-triplet transitions occurring respectively at We introduce the notation Aa(i), a = 1, 2, 3, and Bb(i), a single ruthenium site or at two neighboring ruthenium b = 1, ··· , 6, to indicate the triplet sector states and one sites, with the possibility of having also an orbital flip doublon nd = 1 for the ruthenium at the site i. We of the doublon configuration. That these excitations are already demonstrated by explicit calculation the RIXS related by the Hund’s coupling is highlighted by the fact cross section at the oxygen K-edge that the RIXS pro- that a variation of JH mainly leads to a shift of the peaks cess allows to have an excitation of B character at the only in that window of energies (see Fig. 2). Finally, there ruthenium site. In this Section we aim to demonstrate is a block of excitations that occurs at energies of the or- that such an excitation can propagate due to second or- der of U and depends on the number of doublons at the der exchange processes. 6

0.08 ∆=0.26 eV ∆=0.28 eV 0.06 ∆=0.3 eV L eV

H 0.04 AB J 0.02

0.00 0.00 0.05 0.10 0.15 ΛHeVL

FIG. 4. (color online). Evolution of the exchange constant

JA1B2 , as a function of λ for different crystal field amplitudes.

0 0 Indeed, for any configuration in the blocks Aa(i) and Bb(i ), at two neighboring ruthenium sites i and i , the second 0 order exchange amplitude JAaBb , for the case i = i + 2ˆax, is given by

X hBb(i)Aa(i + 2ˆax)|HRu-Ru,2ˆa (i)|mihm|HRu-Ru,−2ˆa (i + 2ˆax)|Aa(i)Bb(i + 2ˆax)i J = x x . (9) AaBb [E − (E + E )] m m Aa Bb

Here, HRu-Ru,2ˆax (i) describes the ruthenium-ruthenium hopping process via the oxygen site through successive appli- cations of HRu-O,aˆx (i) and HRu-O,−aˆx (i + 2ˆax) and is expressed as

X † X † HRu-Ru,2ˆax (i) = tz di,xz,σdi+2ˆax,xz,σ + txy di,xy,σdi+2ˆax,xy,σ + h.c.. (10) σ σ

The intermediate states |mi are eigenstates with en- tent with the amplitude estimated from first-principle 0 ergy Em of the local Hamiltonian with configurations calculations in Ref. 4), while U = 1.5, JH = 0.4 eV and 5 3 3 5 0 d -d or d -d of the ruthenium d-orbitals on the sites U = U + 2JH. Considering the two-dimensional char- i and i + 2ˆa . As done for the d4 configurations, one x acter of the system, the bandwidth (i.e., 4JAaBb ) would can construct all the d3 and d5 states and obtain the be of the order of 35–40 meV at the characteristic spin- corresponding eigenvalues and eigenvectors in each given orbital coupling λ = 75 meV. We observe in Fig. 4 that spin sector. With these one can determine all the ex- JA1B2 → 0 as λ → 0. This behavior is specific to the change amplitudes of the type J , for a = 1, 2, 3, and AaBb particular excitation B2. Other JAaBb remain finite even b = 1, ··· , 6. at vanishing spin-orbit coupling. It is thus the creation For instance, let us analyze a representative case of of B-sector excitations that is spin-orbit activated, not 0 the configuration A1(i) ⊗ B2(i ). The resulting JA1B2 the fact that these excitations propagate. is reported in Fig. 4 assuming tz = txy = 0.3 eV for the ruthenium-ruthenium hopping parameters (consis-

∗ Present address: Department of Physics, Massachusetts In- (2006). stitute of Technology, Cambridge,MA 02139, USA 3 A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen, † Present address: Instituto de Fisica, Benemerita Univer- D. L. Abernathy, J. T. Park, A. Ivanov, J. Chaloupka, G. sidad Autonoma de Puebla, Apdo. Postal J-48, Puebla, Khaliullin, et al., Nat. Phys. advance online publication, Puebla 72570, Mexico (2017), URL http://dx.doi.org/10.1038/ nphys4077. 1 M. Cuoco, F. Forte, and C. Noce, Phys. Rev. B 74, 195124 4 E. Gorelov, M. Karolak, T. O. Wehling, F. Lechermann, (2006). A. I. Lichtenstein, and E. Pavarini, Phys. Rev. Lett. 104, 2 M. Cuoco, F. Forte, and C. Noce, Phys. Rev. B 73, 094428 226401 (2010).