Supplemental Material for \Spin-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, y 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 Condensed Matter Physics, 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 electrons 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 0 1 0 1 0 1 Hel-el(i) = U niα"niα# − 2JH Siα · Siβ + 0 0 0 0 0 −i 0 −i 0 α<β lx = @0 0 iA ; ly = @0 0 0 A ; lz = @i 0 0A : 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 dy dy 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 y HSOC(i) = λ diασ(lαβ · sσσ0 )diβσ0 ; 0 O α,σ β,σ Hel (j) = "xnj;px + "ynj;py + "znj;pz ; (3) Hcf (i) = "xyni;dxy + "z 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 2 fd ; d ; d g, and dy is the creation opera- xy xz yz iασ metry allowed terms for a generic bond connecting a tor of an electron with spin σ at the site i in the or- y y 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- y 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- y orbital Coulomb interaction U and the Hund's coupling HRu-O;a^x (i) = txz;z di;xz,σpi+^ax;z,σ + h:c: + J , while (U 0−J =2) sets the strength of the inter-orbital H H y 0 electron-electron interaction, and J is the pair hopping +txy;z di;xy,σpi+^ax;y,σ + h:c: ; 2 involved. The configurations are given by (ndxy ; ndxz ; ndyz ) 2 f(1; 1; 2); (1; 2; 1); (2; 1; 1)g (4) for nd = 1 and (ndxy ; ndxz ; ndyz ) 2 f(0; 2; 2); (2; 0; 2); (2; 2; 0)g (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 jd ; 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 ^ 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 py i+^ax;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 ^x 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 2 fdyz; dxz; dxyg that is doubly occupied. We Let us now concentrate on the eigenstates of the Hamil- thus denote the basis states by jS; 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 jd ; nd = 1i and jd ; 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 j1; −1; dyzi; j1; 0; dyzi; j1; 1; dyzi; bly occupied configurations, respectively. In each of the j1; −1; dxzi; j1; 0; dxzi; j1; 1; dxzi; j1; −1; dxyi; j1; 0; dxyi; nd = 1 and nd = 2 sectors, three orbital configurations j1; 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 0 δ 0 0 −iλ 0 0 0 − pλ 0 1 2 B 0 δ 0 0 0 0 pλ 0 − pλ C B 2 2 C B 0 0 δ 0 0 iλ 0 pλ 0 C B 2 C B λ C B iλ 0 0 δ 0 0 0 i p 0 C B 2 C H = (E + u)11 + B 0 0 0 0 δ 0 i pλ 0 i pλ C (6) bloc;T 0 e 9 B 2 2 C B λ C B 0 0 −iλ 0 0 δ 0 i p 0 C B 2 C B 0 pλ 0 0 −i pλ 0 0 0 0 C B 2 2 C B− pλ 0 pλ −i pλ 0 −i pλ 0 0 0 C @ 2 2 2 2 A 0 − pλ 0 0 −i pλ 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 jd ; nd = 1i 0 0 2("xy + "z); ue = 6U − JH (assuming that U = U + 2JH) by a gap that is 2JH for the single doublon states and and δ = ("z − "xy).