Coupled spin-orbital fluctuations in a three orbital model for 4d and 5d oxides with electron fillings n=3,4,5 -- Application to \rm NaOsO_3, \rm Ca_2RuO_4, and \rm Sr_2IrO_4
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Coupled spin-orbital fluctuations in a three orbital model for d and d oxides with electron fillings n = 3 , , — Application toNaOsO , Ca RuO , and Sr IrO Shubhajyoti Mohapatra and Avinash Singh ∗ Department of Physics, Indian Institute of Technology, Kanpur - 208016, India (Dated: February 3, 2021)A unified approach is presented for investigating coupled spin-orbital fluctuationswithin a realistic three-orbital model for strongly spin-orbit coupled systems withelectron fillings n = 3 , , t g sector of d yz , d xz , d xy orbitals. A generalizedfluctuation propagator is constructed which is consistent with the generalized self-consistent Hartree-Fock approximation where all Coulomb interaction contributionsinvolving orbital diagonal and off-diagonal spin and charge condensates are included.Besides the low-energy magnon, intermediate-energy orbiton and spin-orbiton, andhigh-energy spin-orbit exciton modes, the generalized spectral function also showsother high-energy excitations such as the Hund’s coupling induced gapped magnonmodes. We relate the characteristic features of the coupled spin-orbital excitationsto the complex magnetic behavior resulting from the interplay between electronicbands, spin-orbit coupling, Coulomb interactions, and structural distortion effects,as realized in the compounds NaOsO , Ca RuO , and Sr IrO . I. INTRODUCTION
The 4 d and 5 d transition metal (TM) oxides exhibit an unprecedented coupling betweenspin, charge, orbital, and structural degrees of freedom. The complex interplay betweenthe different physical elements such as strong spin-orbit coupling (SOC), Coulomb interac-tions, and structural distortions results in novel magnetic states and unconventional collec-tive excitations. In particular, the cubic structured NaOsO and perovskite structuredCa RuO and Sr IrO compounds, corresponding to d n electronic configuration of the TMion with electron fillings n =3,4,5 in the t sector, respectively, are at the emerging researchfrontier as they provide versatile platform for the exploration of SOC-driven phenomenainvolving collective electronic and magnetic behavior including coupled spin-orbital excita-tions.The different physical elements give rise to a rich variety of nontrivial microscopic featureswhich contribute to the complex interplay. These include spin-orbital-entangled states,band narrowing, spin-orbit gap, and explicit spin-rotation-symmetry breaking (due to SOC),electronic band narrowing due to reduced effective hopping (octahedral tilting and rotation),crystal field induced tetragonal splitting (octahedral compression), orbital mixing (SOCand octahedral tilting, rotation) which self consistently generates induced SOC terms andorbital moment interaction from the Coulomb interaction terms, significantly weaker electroncorrelation term U compared to 3 d orbitals and therefore critical contribution of Hund;scoupling to local magnetic moment. These microscopic features contribute to the complexinterplay in different ways for electron fillings n =3,4,5, resulting in significantly differentmacroscopic properties of the three compounds, which are briefly reviewed below alongwith experimental observations about the collective and coupled spin-orbital excitations asobtained from recent resonant inelastic X-ray scattering (RIXS) studies.The nominally orbitally quenched d compound NaOsO undergoes a metal-insulatortransition (MIT) ( T MI = T N = 410 K) that is closely related to the onset of long-rangeantiferromagnetic (AFM) order. Various mechanisms, such as Slater-like, magnetic Lif-shitz transition, and AFM band insulator have been proposed to explain this unusual andintriguing nature of the MIT.
Interplay of electronic correlations, Hund’s coupling,and octahedral tilting and rotation induced band narrowing near the Fermi level in thisweakly correlated compound results in the weakly insulating state with G-type AFM or-der, with magnetic anisotropy and large magnon gap resulting from interplay of SOC, bandstructure, and the tetragonal splitting.
The Os L resonant edge RIXS measurementsat room temperature show four inelastic peak features below 1.5 eV, which have been in-terpreted to correspond to the strongly gapped ( ∼
58 meV) dispersive magnon excitationswith bandwidth ∼
100 meV, excitations (centered at ∼ t manifold, andexcitations from t to e g states and ligand-to-metal charge transfer for the remaining twohigher-energy peaks. The intensity and positions of the three high-energy peaks appearto be essentially temperature independent.The nominally spin S =1 d compound Ca RuO undergoes a MIT at T MI =357 K andmagnetic transition at T N =110 K ( ≪ T MI ) via a structural phase transition involving acompressive tetragonal distortion, tilt, and rotation of the RuO octahedra. The low-temperature AFM insulating phase is thus characterized by highly distorted octahedrawith nominally filled xy orbital and half-filled yz, xz orbitals. This transition has alsobeen identified in pressure, chemical substitution, strain, and electrical currentstudies, and highlights the complex interplay between SOC, Coulomb interactions, andstructural distortions.Inelastic neutron scattering (INS) and Raman studies on Ca RuO have revealedunconventional low-energy ( ∼
50 and 80 meV) excitations interpreted as gapped trans-verse magnon modes and possibly soft longitudinal (“Higgs-like”) or two-magnon excitationmodes. From both Ru L -edge and oxygen K -edge RIXS studies, multiple nontrivial exci-tations within the t manifold were observed recently below 1 eV. Two low-energy ( ∼
80 and 350 meV) and two high-energy ( ∼
750 meV and 1 eV) excitations were identifiedwithin the limited energy resolution of RIXS. From the incident angle and polarisation de-pendence of the RIXS spectra, the orbital character of the 80 meV peak was inferred to bemixture of xy and xz/yz states, whereas the 0.4 eV peak was linked to unoccupied xz/yz states. Guided by phenomenological spin models, the low-energy excitations (consisting ofmultiple branches) were interpreted as composite spin-orbital excitations (also termed as“spin orbitons”).Finally, SOC induced novel Mott insulating state is realized in the d compoundSr IrO , where band narrowing of the spin-orbital-entangled electronic states near theFermi level plays a critical role in the insulating behavior. The AFM insulating groundstate is characterized by the correlation induced insulating gap within the nominally J =1/2bands emerging from the Kramers doublet, which are separated from the bands of the J =3/2quartet by energy 3 λ/
2, where λ is the SOC strength. The RIXS spectra show low-energydispersive magnon excitations (up to 200 meV), further resolved into two gapped magnonmodes with energy gaps ∼
40 meV and 3 meV at the Γ point corresponding to out-of-plane and in-plane fluctuation modes, respectively.
Weak electron correlation effect andmixing between the J =1/2 and 3/2 sectors were identified as contributing significantly tothe strong zone-boundary magnon dispersion as measured in RIXS studies. In addition,high-energy dispersive spin-orbit exciton modes have also been revealed in RIXS studies inthe energy range 0.4-0.8 eV. This distinctive mode is also referred to as the spin-orbitonmode, and has been attributed to the correlated motion of electron-hole pair excitationsacross the renormalized spin-orbit gap between the J =1/2 and 3/2 bands. Most of the theoretical studies involving magnetic anisotropy effects and excitations inabove systems have mainly focused on phenomenological spin models with different exchangeinteractions obtained as fitting parameters to the experimental spectra. However, the in-terpretation of experimental data remains incomplete since the character of the effectivespins, the microscopic origin of their interactions, and the microscopic nature of the mag-netic excitations are still debated.
Realistic information about the spin-orbital characterof both low and high-energy collective excitations, as inferred from the study of coupledspin-orbital excitations, is clearly important since the spin and orbital degrees of freedomare explicitly coupled, and both are controlled by the different physical elements such asSOC, Coulomb interaction terms, tetragonal compression induced crystal-field splitting be-tween xy and yz, xz orbitals, octahedral tilting and rotation induced orbital mixing hoppingterms, and band physics.Due to the intimately intertwined roles of the different physical elements, a unified ap-proach is therefore required for the realistic modeling of these systems in which all physicalelements are treated on an equal footing. The generalized self-consistent approximation ap-plied recently to the n = 4 compound Ca RuO provides such a unified approach. Involvingthe self-consistent determination of magnetic order within a three-orbital interacting electronmodel including all orbital-diagonal and off-diagonal spin and charge condensates generatedby the different Coulomb interaction terms, this approach explicitly incorporates the com-plex interplay and accounts for the observed behavior including the tetragonal distortioninduced magnetic reorientation transition, orbital moment interaction induced orbital gap,SOC and octahedral tilting induced easy-axis anisotropy, and Coulomb interaction inducedanisotropic SOC renormalization. Extension to the n = 5 compound Sr IrO , providesconfirmation of the Hund’s coupling induced easy-plane magnetic anisotropy, which is re-ponsible for the ∼
40 meV magnon gap measured for the out-of-plane fluctuation mode. Towards a generalized non-perturbative formalism unifying the magnetic order andanisotropy effects on one hand and collective excitations on the other, the natural exten-sion of the above generalized condensate approach is therefore to consider the generalizedfluctuation propagator in terms of the generalized spin ( ψ † µ [ σ α ] ψ ν ) and charge ( ψ † µ [ ] ψ ν ) op-erators in the pure spin-orbital basis of the t orbitals µ, ν = yz, xz, xy and spin components α = x, y, z . The generalized operators include the normal ( µ = ν ) spin and charge opera-tors as well as the orbital off-diagonal ( µ = ν ) cases which are related to the generalizedspin-orbit coupling terms ( L α S β , where α, β = x, y, z ) and the orbital angular momentumoperators L α . Constructing the generalized fluctuation propagator as above will ensure thatthis scheme is fully consistent with the generalized self-consistent approach involving thegeneralized condensates.The different components of the generalized fluctuation propagator will therefore natu-rally include spin-orbitons and orbitons, corresponding to the spin-orbital ( L α S β ) and orbital( L α ) moment fluctuations, besides the normal spin and charge fluctuations. The normal spinfluctuations will include in-phase and out-of phase fluctuations with respect to different or-bitals, the latter being strongly gapped due to Hund’s coupling. The spin-orbitons willinclude the spin-orbit excitons measured in RIXS studies of Sr IrO .The structure of this paper is as below. The three-orbital model within the t sector(including SOC, hopping, Coulomb interaction, and structural distortion terms), and thegeneralized self-consistent formalism including orbital diagonal and off-diagonal condensatesare reviewed in Sec. II and III. After introducing the generalized fluctuation propagator inSec. IV, results of the calculated fluctuation spectral functions are presented for the cases n = 3 , , , Ca RuO , Sr IrO ) in SectionsV, VI, VII. Finally, conclusions are presented in Sec. VIII. The basis-resolved contributionsto the total spectral function showing the detailed spin-orbital character of the collectiveexcitations are presented in the Appendix. II. THREE ORBITAL MODEL WITH SOC AND COULOMB INTERACTIONS
In the three-orbital ( µ = yz, xz, xy ), two-spin ( σ = ↑ , ↓ ) basis defined with respect to acommon spin-orbital coordinate axes (Fig. 1), we consider the Hamiltonian H = H band + H cf + H int + H SOC within the t manifold. For the band and crystal field terms together,we consider: H band+cf = X k σs ψ † k σs ǫ yz k ′ ǫ xz k ′
00 0 ǫ xy k ′ + ǫ xy δ ss ′ + ǫ yz k ǫ yz | xz k ǫ yz | xy k − ǫ yz | xz k ǫ xz k ǫ xz | xy k − ǫ yz | xy k − ǫ xz | xy k ǫ xy k δ ¯ ss ′ ψ k σs ′ (1)in the composite three-orbital, two-sublattice ( s, s ′ = A , B) basis. Here the energy offset ǫ xy (relative to the degenerate yz/xz orbitals) represents the tetragonal distortion inducedcrystal field effect. The band dispersion terms in the two groups correspond to hoppingterms connecting the same and opposite sublattice(s), and are given by: ǫ xy k = − t (cos k x + cos k y ) ǫ xy k ′ = − t cos k x cos k y − t (cos 2 k x + cos 2 k y ) ǫ yz k = − t cos k x − t cos k y ǫ xz k = − t cos k x − t cos k y ǫ yz | xz k = − t m (cos k x + cos k y ) ǫ xz | xy k = − t m (2 cos k x + cos k y ) ǫ yz | xy k = − t m (cos k x + 2 cos k y ) . (2)Here t , t , t are respectively the first, second, and third neighbor hopping terms forthe xy orbital. For the yz ( xz ) orbital, t and t are the nearest-neighbor (NN) hoppingterms in y ( x ) and x ( y ) directions, respectively, corresponding to π and δ orbital overlaps.Octahedral rotation and tilting induced orbital mixings are represented by the NN hoppingterms t m (between yz and xz ) and t m , t m (between xy and xz, yz ). In the n = 4 case cor-responding to the Ca RuO compound, we have taken hopping parameter values: ( t , t , t , t , t )=( − . , . , , − . , . t m =0.2 and t m = t m =0.15( ≈ . / √ | t | =200meV. The choice t m = t m corresponds to the octahedral tilting axis oriented along the ± ( − ˆ x + ˆ y ) direc-tion, which is equivalent to the crystal ∓ a direction (Fig. 1). The t m and t m ,m values (a) (b) FIG. 1: (a) The common spin-orbital coordinate axes ( x − y ) along the Ru-O-Ru directions, shownalong with the crystal axes a, b . (b) Octahedral tilting about the crystal a axis is resolved alongthe x, y axes, resulting in orbital mixing hopping terms between the xy and yz, xz orbitals. taken above approximately correspond to octahedral rotation and tilting angles of about 12 ◦ ( ≈ . For the on-site Coulomb interaction terms in the t g basis ( µ, ν = yz, xz, xy ), we consider: H int = U X i,µ n iµ ↑ n iµ ↓ + U ′ X i,µ<ν,σ n iµσ n iνσ + ( U ′ − J H ) X i,µ<ν,σ n iµσ n iνσ + J H X i,µ = ν a † iµ ↑ a † iν ↓ a iµ ↓ a iν ↑ + J P X i,µ = ν a † iµ ↑ a † iµ ↓ a iν ↓ a iν ↑ = U X i,µ n iµ ↑ n iµ ↓ + U ′′ X i,µ<ν n iµ n iν − J H X i,µ<ν S iµ . S iν + J P X i,µ = ν a † iµ ↑ a † iµ ↓ a iν ↓ a iν ↑ (3)including the intra-orbital ( U ) and inter-orbital ( U ′ ) density interaction terms, the Hund’scoupling term ( J H ), and the pair hopping interaction term ( J P ), with U ′′ ≡ U ′ − J H / U − J H / U ′ = U − J H . Here a † iµσ and a iµσ arethe electron creation and annihilation operators for site i , orbital µ , spin σ = ↑ , ↓ . Thedensity operator n iµσ = a † iµσ a iµσ , total density operator n iµ = n iµ ↑ + n iµ ↓ = ψ † iµ ψ iµ , and spindensity operator S iµ = ψ † iµ σ ψ iµ in terms of the electron field operator ψ † iµ = ( a † iµ ↑ a † iµ ↓ ). Allinteraction terms above are SU(2) invariant and thus possess spin rotation symmetry.Finally, for the bare spin-orbit coupling term (for site i ), we consider the spin-spacerepresentation: H SOC ( i ) = − λ L . S = − λ ( L z S z + L x S x + L y S y )= (cid:16) ψ † yz ↑ ψ † yz ↓ (cid:17) (cid:16) iσ z λ/ (cid:17) ψ xz ↑ ψ xz ↓ + (cid:16) ψ † xz ↑ ψ † xz ↓ (cid:17) (cid:16) iσ x λ/ (cid:17) ψ xy ↑ ψ xy ↓ + (cid:16) ψ † xy ↑ ψ † xy ↓ (cid:17) (cid:16) iσ y λ/ (cid:17) ψ yz ↑ ψ yz ↓ + H . c . (4)which explicitly breaks SU(2) spin rotation symmetry and therefore generates anisotropicmagnetic interactions from its interplay with other Hamiltonian terms. Here we have usedthe matrix representation: L z = − i i , L x = − i i , L y = i − i , (5)for the orbital angular momentum operators in the three-orbital ( yz, xz, xy ) basis.As the orbital “hopping” terms in Eq. (4) have the same form as spin-dependent hoppingterms i σ.t ′ ij , carrying out the strong-coupling expansion for the − λL z S z term to secondorder in λ yields the anisotropic diagonal (AD) intra-site interactions:[ H (2)eff ] ( z )AD ( i ) = 4( λ/ U (cid:2) S zyz S zxz − ( S xyz S xxz + S yyz S yxz ) (cid:3) (6)between yz, xz moments if these orbitals are nominally half-filled, as in the case of Ca RuO .This term explicitly yields preferential x − y plane ordering (easy-plane anisotropy) forparallel yz, xz moments, as enforced by the relatively stronger Hund’s coupling.For later reference, we note here that condensates of the orbital off-diagonal (OOD) one-body operators as in Eq. (4) directly yield physical quantities such as orbital magneticmoments and spin-orbital correlations: h L α i = − i (cid:2) h ψ † µ ψ ν i − h ψ † µ ψ ν i ∗ (cid:3) = 2 Im h ψ † µ ψ ν ih L α S α i = − i (cid:2) h ψ † µ σ α ψ ν i − h ψ † µ σ α ψ ν i ∗ (cid:3) / h ψ † µ σ α ψ ν i λ int α = ( U ′′ − J H / h L α S α i = ( U ′′ − J H / h ψ † µ σ α ψ ν i (7)where the orbital pair ( µ, ν ) corresponds to the component α = x, y, z , and the last equationyields the interaction induced SOC renormalization, as discussed in the next section. III. SELF-CONSISTENT DETERMINATION OF MAGNETIC ORDER
We consider the various contributions from the Coulomb interaction terms (Eq. 3) in theHF approximation, focussing first on terms with normal (orbital diagonal) spin and chargecondensates. The resulting local spin and charge terms can be written as:[ H HFint ] normal = X iµ ψ † iµ [ − σ. ∆ iµ + E iµ ] ψ iµ (8)where the spin and charge fields are self-consistently determined from:2∆ αiµ = U h σ αiµ i + J H X ν<µ h σ αiν i ( α = x, y, z ) E iµ = U h n iµ i U ′′ X ν<µ h n iν i (9)in terms of the local charge density h n iµ i and the spin density components h σ αiµ i .There are additional contributions resulting from orbital off-diagonal (OOD) spin andcharge condensates which are finite due to orbital mixing induced by SOC and structuraldistortions (octahedral tilting and rotation). The contributions corresponding to differentCoulomb interaction terms are summarized in Appendix A, and can be grouped in analogywith Eq. (8) as:[ H HFint ] OOD = X i,µ<ν ψ † iµ [ − σ. ∆ iµν + E iµν ] ψ iν + H . c . (10)where the orbital off-diagonal spin and charge fields are self-consistently determined from: ∆ iµν = (cid:18) U ′′ J H (cid:19) h σ iνµ i + (cid:18) J P (cid:19) h σ iµν iE iµν = (cid:18) − U ′′ J H (cid:19) h n iνµ i + (cid:18) J P (cid:19) h n iµν i (11)in terms of the corresponding condensates h σ iµν i ≡ h ψ † iµ σ ψ iν i and h n iµν i ≡ h ψ † iµ ψ iν i .The spin and charge condensates in Eqs. 9 and 11 are evaluated using the eigenfunctions( φ k ) and eigenvalues ( E k ) of the full Hamiltonian in the given basis including the interactioncontributions [ H HFint ] (Eqs. 8 and 10) using: h σ αiµν i ≡ h ψ † iµ σ α ψ iν i = E k Since all generalized spin h ψ † µ σ ψ ν i and charge h ψ † µ ψ ν i condensates were included in the selfconsistent determination of magnetic order, the fluctuation propagator must also be definedin terms of the generalized operators. We therefore consider the time-ordered generalizedfluctuation propagator:[ χ ( q , ω )] = Z dt X i e iω ( t − t ′ ) e − i q . ( r i − r j ) × h Ψ | T [ σ αµν ( i, t ) σ α ′ µ ′ ν ′ ( j, t ′ )] | Ψ i (14)in the self-consistent AFM ground state | Ψ i , where the generalized spin-charge operatorsat lattice sites i, j are defined as σ αµν = ψ † µ σ α ψ ν , which include both the orbital diagonal( µ = ν ) and off-diagonal ( µ = ν ) cases, as well as the spin ( α = x, y, z ) and charge ( α = c )operators, with σ α defined as Pauli matrices for α = x, y, z and unit matrix for α = c .In the random phase approximation (RPA), the generalized fluctuation propagator is1obtained as:[ χ ( q , ω )] RPA = 2[ χ ( q , ω )] − [ U ][ χ ( q , ω )] (15)in terms of the bare particle-hole propagator [ χ ( q , ω )] which is evaluated by integrating outthe electronic degrees of freedom:[ χ ( q , ω )] µ ′ ν ′ α ′ s ′ µναs = 12 X k " h k | σ αµν | k − q i s h k | σ α ′ µ ′ ν ′ | k − q i ∗ s ′ E ⊕ k − q − E ⊖ k + ω − iη + h k | σ αµν | k − q i s h k | σ α ′ µ ′ ν ′ | k − q i ∗ s ′ E ⊕ k − E ⊖ k − q − ω − iη (16)The matrix elements in the above expression are evaluated using the eigenvectors of the HFHamiltonian in the self-consistent AFM state: h k | σ αµν | k − q i s = ( φ ∗ k µ ↑ s φ ∗ k µ ↓ s )[ σ α ] φ k − q ν ↑ s φ k − q ν ↓ s (17)and the superscripts ⊕ ( ⊖ ) refer to particle (hole) states above (below) the Fermi energy.The subscripts s, s ′ indicate the two (A/B) sublattices. In the composite spin-charge-orbital-sublattice ( µναs ) basis, the [ χ ( q , ω )] matrix is of order 72 × 72, and the form of the [ U ]matrix in the RPA expression (Eq. 15) is given in Appendix B.The spectral function of the excitations will be determined from:A q ( ω ) = 1 π Im Tr[ χ ( q , ω )] RPA (18)using the RPA expression for [ χ ( q , ω )]. When the collective excitation energies lie withinthe AFM band gap, it is convenient to consider the symmetric form of the denominator inthe RPA expression (Eq. 15):[ U ][ χ ( q , ω )][ U ] − [ U ] (19)and in terms of the real eigenvalues λ q ( ω ) of this Hermitian matrix, the magnon energies ω q for momentum q are determined by solving for the zeroes: λ q ( ω = ω q ) = 0 (20)corresponding to the poles in the propagator.Results of the calculated spectral function will be discussed in the subsequent sectionsfor different electron filling cases ( n = 3 , , 5) with applications to corresponding 4 d and25 d transition metal compounds. Broadly, our investigation of the generalized fluctuationpropagator will provide information about the dominantly spin, orbital, and spin-orbitalexcitations, as the generalized spin and charge operators ψ † µ σ α ψ ν include spin ( µ = ν , α = x, y, z ), orbital ( µ = ν , α = c ), and spin-orbital ( µ = ν , α = x, y, z ) cases. Alsoincluded will be the high-energy spin-orbit exciton modes involving particle-hole excitationsacross the renormalized spin-orbit gap between spin-orbital entangled states of different J sectors, as in the n = 5 case relevant for the Sr IrO compound. V. n = 3 — APPLICATION TO NaOsO The strongly spin-orbit coupled orthorhomic structured 5 d osmium compound NaOsO ,with nominally three electrons in the Os t g sector, exhibits several novel electronic andmagnetic properties. These include a G-type antiferromagnetic (AFM) structure with spinsoriented along the c axis, a significantly reduced magnetic moment ∼ µ B as measured fromneutron scattering, a continuous metal-insulator transition (MIT) that coincides with theAFM transition ( T N = T MIT = 410 K) as seen in neutron and X-ray scattering, and a largespin wave gap of 58 meV as seen in resonant inelastic X-ray scattering (RIXS) measurementsindicating strong magnetic anisotropy. Two different mechanisms contributing to SOC-induced easy-plane anisotropy and largemagnon gap for out-of-plane fluctuation modes were identified for the weakly correlated5 d compound NaOsO in terms of a simplified picture involving only the normal spin andcharge densities. Both essential ingredients — (i) small moment disparity between yz, xz and xy orbitals and (ii) spin-charge coupling effect in presence of tetragonal splitting — areintrinsically present in the considered three-orbital model on the square lattice. A realisticrepresentation of magnetic anisotropy in NaOsO is therefore provided by the consideredmodel, while maintaining uniformity of lattice structure across the n = 3 , , − λL α S α for α = x, y, z . Due to the small momentdisparity m yz,xz > m xy resulting from the broader xy band, the interaction term in Eq. (6)dominates over the other two terms, leading to the easy-plane anisotropy for parallel yz, xz moments enforced by the Hund’s coupling. With increasing U , this effect weakens as the3 ( π /2, π /2) ( π ,0) ( π , π ) (0,0) 0 50 100 150 200 250 300 ω ( m e V ) π /2, π /2) ( π ,0) ( π , π ) ( π /2, π /2) (b) ω q ( m e V ) FIG. 2: (a) Low energy part of the calculated spectral function from the generalized fluctuationpropagator shows the magnon excitations in the self-consistent state with planar AFM order, and(b) magnon dispersion showing the gapless and gapped modes corresponding to in-plane and out-of-plane fluctuations. moments saturate m yz,xz,xy ≈ U limit. In the second mechanism, the SOCinduced decreasing xy orbital density n xy with spin rotation from z direction to x − y planecouples to the tetragonal distortion term, and for positive ǫ xy the energy is minimized forspin orientation in the x − y plane.We will consider the parameter set values U = 4, J H = U/ U ′′ = U − J H / 2, SOC=1.0,and ǫ xy = 0 . | t | =300 meV.Thus, U = 1 . ǫ xy = 0 . 15 eV, which are realistic values for the NaOsO compound. Initially, we will also set t m ,m ,m = 0 for simplicity, and focus on the easy-planeanisotropy and large magnon gap for out-of-plane fluctuations.Self consistent determination of magnetic order using the generalized approach discussedin Sec. III confirms the easy-plane anisotropy. Starting in nearly z direction, the AFM orderdirection self consistently approaches the x − y plane in a few hundred iterations. Initially,we will discuss magnetic excitations in the self consistent state with AFM order along theˆ x or ˆ y directions. Although these orientations correspond to metastable states as discussedlater, they provide convenient test cases for explicitly confirming the gapless in-plane andgapped out-of-plane magnon modes in the generalized fluctuation propagator calculation.The low-energy part of the calculated spectral function using Eq. (18) is shown in theFig. 2(a) as an intensity plot for q along symmetry directions of the Brillouin zone. The4 (a) m agnon gap ( m e V ) SOC (eV) (b) m agnon gap ( m e V ) U (eV) (c) m agnon gap ( m e V ) ε xy (eV) FIG. 3: Variation of the calculated magnon gap showing effects of (a) SOC, (b) Hubbard U , and(c) tetragonal distortion ǫ xy , on the easy-plane magnetic anisotropy. gapless and gapped modes corresponding to in-plane and out-of-plane fluctuations reflectthe easy-plane magnetic anisotropy. The calculated gap energy 60 meV is close to themeasured spin wave gap of 58 meV in NaOsO . Also shown for comparison in Fig. 2(b)is the magnon dispersion calculated from the poles of the RPA propagator as described inSec. IV. Focussing on the magnon gap in Fig. 2(b), which provides a measure of the SOCinduced easy-plane anisotropy, effects of various physical quantities are shown in Fig. 3.The gapless Goldstone mode corresponding to in-plane rotation of AFM ordering directionin the x − y plane involves only small changes in spin densities h ψ † µ σ α ψ µ i for α = x, y and µ = yz, xz , and also in generalized spin densities h ψ † µ σ α ψ ν i for α = x, y and µ = yz, xz withfixed ν = xy . For example, the magnetization values m xyz = 0 . 82 and m xxz = 0 . 84 changeto m yyz = 0 . 84 and m yxz = 0 . 82 when the ordering is rotated from x to y direction. Thus,the Goldstone mode is nearly pure spin mode and the small orbital character reflects theeffectively suppressed spin-orbital entangement in the n = 3 AFM state. In contrast, the n = 5 case corresponding to Sr IrO shows strongly coupled spin-orbital character of theGoldstone mode (Appendix C) due to the extreme spin-orbital entanglement.We now consider the easy-axis anisotropy effects in our self consistent determination ofmagnetic order. With respect to the AFM order orientation (azimuthal angle φ ) within theeasy ( x − y ) plane, we find an easy-axis anisotropy along the diagonal orientations φ = nπ/ n = 1 , , , 7) even for no octahedral tilting. This anisotropy is due to an orientationdependent Dzyaloshinski-Moriya (DM) interaction:[ H (2)eff ] ( z )DM ( i ) = − λ / λ z / U ˆ z. ( S yz × S xz ) (21)which is generated in the usual strong-coupling expansion from the combination of the SOC5term ψ † yz ( iσ z λ z / ψ xz and the charge term ψ † yz ( − λ / ψ xz in Eq. (10) involving the real partof the off-diagonal charge condensate h ψ † xz ψ yz i . Due to the strong orientation dependence ofthis condensate, the DM interaction strength D ( φ ) = 2 λ ( φ ) λ z /U vanishes for φ along the x, y axes (hence the gapless in-plane mode in Fig. 2), and is significant near the diagonalorientations, resulting in easy-axis anisotropy and small relative canting between yz, xz moments about the z axis.This orientation dependent DM interaction highlights the important role of orbital off-diagonal condensates in the self-consistent determination of magnetic order and anisotropyin the n = 3 case. The resulting C symmetry of the easy-axis ± (ˆ x ± ˆ y ) is reduced to C symmetry ± (ˆ x − ˆ y ) in the presence of octahedral tilting. The important anisotropy effectsof the octahedral tilting induced inter-site DM interactions are discussed below. We findthat the DM axis lies along the crystal b axis, leading to easy axis direction along the crystal a axis. Both these directions are interchanged in comparison to the Ca RuO case, whichfollows from a subtle difference in the present n = 3 case as explained below.Following the analysis carried out for the Ca RuO compound, within the usual strong-coupling expansion in terms of the normal ( t ) and spin-dependent ( t ′ x , t ′ y ) hopping termsinduced by the combination of SOC and orbital mixing hopping terms t m ,m due to octa-hedral tilting, the DM interaction terms generated in the effective spin model are obtainedas: [ H (2)eff ] ( x,y )DM = 8 tt ′ x U X h i,j i x ˆ x. ( S i,xz × S j,xz ) + 8 tt ′ y U X h i,j i y ˆ y. ( S i,yz × S j,yz ) ≈ t | t ′ x | U X h i,j i (ˆ x + ˆ y ) . ( S i,yz × S j,yz ) (22)where we have taken t ′ x = − t ′ y = − ive and S xi,xz = S xi,yz (due to Hund’s coupling) as earlier,but with S zi,xz = − S zi,yz for the n = 3 case as obtained in our self consistent calculation whichis discussed below. The effective DM axis (ˆ x + ˆ y ) is thus along the crystal b axis (Fig. 1)for the yz orbital, resulting in easy-axis anisotropy along the crystal a direction, as well asspin canting about the DM axis in the z direction.Results for various physical quantities are shown in Tables I and II. Starting with initialorientation along the ˆ x or ˆ y directions, the AFM order direction self consistently approachesthe easy-axis direction in a few hundred iterations, explicitly exhibiting the strong easy-axisanisotropy within the easy ( x − y ) plane due to the octahedral tilting induced DM interaction,6 TABLE I: Self consistently determined magnetization and density values for the three orbitals ( µ )on the two sublattices ( s ), showing easy-axis anisotropy along the crystal a axis due to octahedraltilting induced DM interaction. Here t m ,m = 0 . µ (s) m xµ m yµ m zµ n µ yz (A) 0.598 − xz (A) 0.557 − − xy (A) 0.541 − µ (s) m xµ m yµ m zµ n µ yz (B) − xz (B) − − xy (B) − λ α = λ + λ int α and the orbitalmagnetic moments h L α i for α = x, y, z on the two sublattices. Bare SOC value λ =1.0. s λ x λ y λ z h L x i h L y i h L z i A 1.179 1.179 1.364 0.032 − − along with small spin canting in the z direction about the DM axis. The small momentdisparity m yz,xz > m xy and the negligible orbital moments can also be seen here explicitly.The renormalized SOC strength λ z is enhanced relative to the other two components, whichfurther reduces the SOC induced frustration in this system with nominally one electron ineach of the three orbitals.With octahedral tilting included, the orbital resolved electronic band structure in the self-consistent AFM state (Fig. 4) shows the AFM band gap between valence and conductionbands, SOC induced orbital mixing and band splittings, the fine splitting due to octahedraltilting, and the asymmetric bandwidth for xy orbital bands characteristic of the 2nd neighborhopping term t which connects the same magnetic sublattice. The calculated magnondispersion evaluated using Eq. 20 is shown in Fig. 5. As expected, both in-plane andout-of-plane magnon modes are gapped due to the easy-axis and easy-plane anisotropiesdiscussed above.The high energy part of the spectral function is shown in the series of panels in Fig. 6for different SOC strengths. The two groups of modes here correspond to: (i) the Hund’scoupling induced gapped magnon modes for out-of-phase spin fluctuations (the two disper-7 -3-2-1 0 1 2 (0,0) ( π ,0) ( π , π ) (0,0) (0, π ) ( π ,0) yz xz xy E k - E F ( e V ) FIG. 4: Calculated orbital resolved electronic band structure in the self-consistent state with AFMorder along the crystal a axis due to octahedral tilting induced DM interaction. Here t m ,m = 0 . yz ), green ( xz ), blue ( xy ). sive modes starting at energies 0.7 and 0.8 eV from the left edge in panel (a)), and (ii) thespin-orbiton modes (starting at energy below 0.6 eV) which are inter-orbital magnetic exci-tons corresponding to the lowest-energy particle-hole excitations across the AFM band gapinvolving yz/xz orbitals (particle) and xy orbital (hole) states (Fig. 4). Through the usualresonant scattering mechanism, these modes are pulled down in energy below the continuumby the U ′′ interaction term, and form well defined propagating modes. ω q ( m e V ) ( π /2, π /2) ( π ,0) ( π , π ) ( π /2, π /2) FIG. 5: Magnon dispersion for the self consistently determined magnetic order as given in TableI in the presence of octahedral tilting ( t m ,m = 0 . ( π /2, π /2) ( π ,0) ( π , π ) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 ω ( e V ) λ = 0 (a) ( π /2, π /2) ( π ,0) ( π , π ) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 ω ( e V ) (b) λ = π /2, π /2) ( π ,0) ( π , π ) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 ω ( e V ) (c) λ = π /2, π /2) ( π ,0) ( π , π ) (0,0) 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ω ( e V ) λ = ε xy = 0 (d) FIG. 6: Gapped magnon modes and dominantly magnetic exciton modes seen in the high-energypart of the spectral function calculated in the self consistent AFM state including octahedral tilting,for different SOC ( λ ) values shown in the panels. The spin-orbiton mode involving xy orbital shifts to higher energy when ǫ xy decreases tozero (panel (d)) which lowers the dominantly xy valence band (Fig. 4) and thus increasesthe particle-hole excitation energy. The splitting of the exciton modes in panel (c) is due tothe SOC induced splitting of electronic bands as seen in Fig. 4, which is then reflected inthe particle-hole excitation energies. The combination of orbitals for these exciton modesindicates that L x and L y components of the orbital angular momentum are involved inthese coupled spin-orbital fluctuations. There is an additional spin-orbiton mode involvingonly yz, xz orbitals (and L z component) which is formed at higher energy near 0.8 eV (flatband near the left edge iin panel (a)). With increasing SOC, one of the high-energy modes(involving yz, xz orbitals) acquires significant spin-orbit exciton character.9 VI. n = 5 — APPLICATION TO Sr IrO The perovskite structured 5 d compound Sr IrO exhibits an AFM insulator state dueto strong SOC induced splitting of the t states, with four electrons in the nominallyfilled and non-magnetic J =3/2 sector and one electron in the nominally half filled andmagnetically active J =1/2 sector. The SOC induced splitting of 3 λ/ J =1/2 sector, all of these play a crucial role in the stabilization of the AFMinsulator state. Both low-energy magnon excitations and high-energy spin-orbit excitonsacross the renormalized spin-orbit gap have been intensively studied using RIXS experimentsand variety of theoretical approaches. In this case, we have taken realistic parameter values U = 3, J H = U/ 7, SOC strength λ = 1 . 35, and ǫ xy = − . 5, along with hopping terms: ( t , t , t , t , t , t m )=(-1.0, 0.5, 0.25,-1.0, 0.0, 0.2), all in units of the realistic hopping energy scale | t | =290 meV. The self consis-tently determined results for various physical quantities are given in Table III for magneticorder in the x direction. All ordering directions within the x − y plane are equivalent, con-sistent with the easy-plane magnetic anisotropy due to Hund’s coupling induced anisotropicinteractions. The octahedral rotation induces small in-plane canting of spins but the cant-ing axis is free to orient in any direction. The strong Coulomb interaction induced SOCrenormalization by nearly 2/3 (Table IV) agrees with the pseudo-orbital based approach. The strong orbital moments and their correlation with the magnetic order direction (TableIV) reflect the strong SOC induced spin-orbital entanglement. TABLE III: Self consistently determined magnetization and density values, showing small spincanting about the z axis due to octahedral rotation induced DM interaction. Here t m = 0 . µ (s) m xµ m yµ m zµ n µ yz (A) 0.186 − xz (A) − xy (A) − − µ (s) m xµ m yµ m zµ n µ yz (B) − − xz (B) 0.185 0.049 0 1.654 xy (B) 0.172 − TABLE IV: Self consistently determined renormalized SOC values λ α = λ + λ int α and the orbitalmagnetic moments h L α i for α = x, y, z on the two sublattices. Bare SOC value λ =1.35. s λ x λ y λ z h L x i h L y i h L z i A 1.882 1.882 1.871 0.367 0.091 0B 1.882 1.882 1.871 − The low energy part of the spectral function along with the calculated magnon dispersionobtained using Eq. 20 are shown in Fig. 7, clearly showing the gapless and gapped modescorresponding to in-plane and out-of-plane fluctuation modes, consistent with the easy-planeanisotropy. The magnon gap ≈ 45 meV is close to the result obtained using the pseudo-orbital based approach, and in agreement with recent experiments. It should be noted that along with the full generalized spin sector, the orbital off-diagonalcharge sector ( ψ † µ ψ ν ) related to the orbital moment operators L x,y,z has also been included inthe above calculations. This allows transverse fluctuations of orbital moments to be includedwhich are important in view of their strong magnitude (Table IV). This is in contrast tothe n = 3 case where this charge sector could be neglected as orbital moments were nearlyquenched. Indeed, the exactly gapless Goldstone mode seen in Fig. 7 is obtained only if the ( π /2, π /2) ( π ,0) ( π , π ) (0,0) 0 50 100 150 200 250 300 ω ( m e V ) ω q ( m e V ) ( π /2, π /2) ( π ,0) ( π , π ) (0,0) ( π ,0) FIG. 7: (a) Low energy part of the spectral function and (b) magnon dispersion in the self-consistentstate for the n = 5 case with planar AFM order including octahedral rotation, showing the gaplessand gapped modes corresponding to in-plane and out-of-plane fluctuations. ( π /2, π /2) ( π ,0) ( π , π ) (0,0) 300 400 500 600 700 800 ω ( m e V ) π /2, π /2) ( π ,0) ( π , π ) (0,0) 0 200 400 600 800 1000 1200 ω ( m e V ) H = U/8 ( π /2, π /2) ( π ,0) ( π , π ) (0,0) 0 200 400 600 800 1000 1200 ω ( m e V ) H = 0 FIG. 8: The spin-orbit exciton modes (near 500 meV) and orbiton modes (near 300 meV) seen inthe high-energy part of the spectral function for n = 5, calculated in the self-consistent state withplanar AFM order including octahedral rotation, and with Hund’s coupling term (a) J H = U/ U/ 8, and (c) 0. ψ † µ ψ ν sector is included, indicating the coupled spin-orbital nature of the Goldstone mode.If the magnetic order direction rotates from x to y direction, the h L x i component transformsto h L y i component, as illustrated in Appendix C showing the detailed composition of thespin-orbital character of the Goldstone mode.At intermediate energy ( ∼ 300 meV), the spectral function shows (Fig. 8) the dominantlyorbital fluctuation modes (orbitons) which arise from J =1/2 sector particle-hole excitationsbetween xy, yz and yz, xz orbitals, corresponding to in-plane ( L y ) and out-of-plane ( L z )transverse fluctuations, respectively, of the orbital moment h L x i . The orbiton mode splittingseen beyond ( π, 0) is due to the Hund’s coupling induced easy-plane magnetic anisotropy, andvanishes as J H → 0, as seen in Fig. 8(c). In addition, Fig. 8 shows the high-energy ( ∼ J =1/2 and3/2 sector states, which matches closely with the earlier results obtained using the pseudo-orbital based approach. As discussed in the previous ( n = 3) case, collective modes arisefrom particle-hole excitations which are converted to well defined propagating modes splitoff from the continuum by the Coulomb interaction induced resonant scattering mechanism. VII. n = 4 — APPLICATION TO Ca RuO For moderate tetragonal distortion ( ǫ xy ≈ − xy orbital in the 4 d compoundCa RuO is nominally doubly occupied and magnetically inactive, while the nominally half-filled and magnetically active yz, xz orbitals yield an effectively two-orbital magnetic system.2Hund’s coupling between the two S = 1 / yz, xz orbital S = 1 / S = 1 spin system. However, the rich interplay be-tween SOC, Coulomb interaction, octahedral rotations, and tetragonal distortion results incomplex magnetic behaviour which crucially involves the xy orbital and is therefore beyondthe above simplistic picture.Treating all the different physical elements on the same footing within the unified frame-work of the generalized self-consistent approach explicitly shows that the rich interplaygives rise to the complex magnetic behavior in Ca RuO including SOC induced easy-planeanisotropy, octahedral tilting induced easy-axis anisotropy, spin-orbital coupling inducedorbital magnetic moments, Coulomb interaction induced strongly anisotropic SOC renor-malization, decreasing tetragonal distortion induced magnetic reorientation transition fromplanar AFM order to FM ( z ) order, and orbital moment interaction induced orbital gap. Stable FM and AFM metallic states were also obtained near the magnetic phase boundaryseparating the two magnetic orders. The self-consistent determination of magnetic orderhas also explicitly shown the coupled nature of spin and orbital fluctuations, as reflected inthe ferro and antiferro orbital fluctuations associated with in-phase and out-of-phase spintwisting modes, highlighting the strong deviation from conventional Heisenberg behaviourin effective spin models, as discussed recently to account for the magnetic excitation mea-surements in INS experiments on Ca RuO . In the following, we will take U = 8 in the energy scale unit (200 meV) and J H = U/ 5, sothat U = 1 . U ′′ = U/ . J H = 0 . 32 eV. These are comparable to reportedvalues extracted from RIXS ( J H = 0 . 34 eV) and ARPES ( J H = 0 . Thehopping parameter values considered are as given in Sec. II, and the bare SOC value λ = 1.Fig. 9 shows the generalized fluctuation spectral function calculated for the same param-eter set as considered in the self-consistent study. Several well defined propagating modesare seen here including: (i) the low-energy (below ∼ 70 meV) dominantly spin (magnon)excitations involving the magnetically active yz, xz orbitals and corresponding to in-planeand out-of-plane fluctuations which are gapped due to the magnetic anisotropies, (ii) theintermediate-energy (150 and 200 meV) dominantly orbital excitations (orbitons) involvingparticle-hole excitations between xy (hole) and yz, xz (particle) states, (iii) the intermediate-energy (400 and 450 meV) dominantly spin-orbital excitations (spin-orbitons) involving xy ( π /2, π /2) ( π ,0) ( π , π ) (0,0) 0 50 100 150 200 250 ω ( m e V ) π /2, π /2) ( π ,0) ( π , π ) (0,0) 300 350 400 450 500 550 600 650 ω ( m e V ) FIG. 9: The generalized fluctuation spectral function for the n = 4 case, showing coupled spin-orbital excitations including low-energy magnon modes, intermediate-energy orbiton and spin-orbiton modes, and high-energy spin-orbit exciton modes. (hole) and yz, xz (particle) states, and (iv) the high-energy (550 meV) dominantly spin-orbital excitations (spin-orbit excitons) involving particle-hole excitations between yz, xz (hole) and yz, xz (particle) states of nominally different J sectors. The spin-orbital ampli-tudes of the SOC-induced spin-orbital entangled J states are strongly renormalized by thetetragonal splitting and the electronic correlation induced staggered field.The spin-orbital characterization of the various collective excitations mentioned above isinferred from the basis-resolved contributions to the total spectral functions in the gener-alized spin-charge basis which explicitly show the relative spin-orbital composition of thevarious excitations (Appendix C). The spectral function for realistic SOC value λ =0.5 andwith improved ω resolution is shown in Fig. 10. The presence of sharply defined collectiveexcitations for the magnon, orbiton, and spin-orbiton modes which are clearly separatedfrom the particle-hole continuum highlights the rich spin-orbital physics in the n = 4 casecorresponding to the Ca RuO compound. Many of our calculated magnon spectra featuressuch as the magnon gaps at the (0 , 0) point for in-plane and out-of-plane modes, weak dis-persive nature along the magnetic zone boundary, as well as the overall energy scale of themagnon modes are in excellent agreement with the INS study. The orbiton mode energyscale is also qualitatively comparable to the composite excitation peaks obtained around 80meV in the Raman study. ( π /2, π /2) ( π ,0) ( π , π ) (0,0) 0 40 80 120 160 200 ω ( m e V ) π /2, π /2) ( π ,0) ( π , π ) (0,0) 300 350 400 450 500 550 600 650 ω ( m e V ) FIG. 10: The generalized fluctuation spectral function for the n = 4 case with improved ω resolutionand bare SOC value λ =0.5, showing (a) the magnon and orbiton modes, and (b) the spin-orbitonand spin-orbit exciton modes. VIII. CONCLUSIONS Following up on the generalized self-consistent approach including orbital off-diagonalspin and charge condensates, investigation of the generalized fluctuation propagator revealsthe composite spin-orbital character of the different types of collective excitations in stronglyspin-orbit coupled systems. A realistic representation of magnetic anisotropy effects due tothe interplay of SOC, Coulomb interaction, and structural distortion terms was includedin the three-orbital model, while maintaining uniformity of lattice structure in order tofocus on the coupled spin-orbital excitations. Our unified investigation of the three electronfilling cases n = 3 , , , Ca RuO , Sr IrO provides deep insight into how the spin-orbital physics in the magnetic ground state isreflected in the collective excitations. The calculated spectral functions show well definedpropagating modes corresponding to dominantly spin (magnon), orbital (orbiton), and spin-orbital (spin-orbiton) excitations, along with the spin-orbit exciton modes involving spin-orbital excitations between states of different J sectors induced by the spin-orbit coupling.5 Appendix A: Orbital off-diagonal condensates in the HF approximation The additional contributions in the HF approximation arising from the orbital off-diagonalspin and charge condensates are given below. For the density, Hund’s coupling, and pairhopping interaction terms in Eq. 3, we obtain (for site i ): U ′′ X µ<ν n µ n ν → − U ′′ X µ<ν [ n µν h n νµ i + σ µν . h σ νµ i ] + H . c . − J H X µ<ν S µ . S ν → J H X µ<ν [3 n µν h n νµ i − σ µν . h σ νµ i ] + H . c .J P X µ = ν a † µ ↑ a † µ ↓ a ν ↓ a ν ↑ → J P X µ<ν [ n µν h n µν i − σ µν . h σ µν i ] + H . c . (A1)in terms of the orbital off-diagonal spin ( σ µν = ψ † µ σ ψ ν ) and charge ( n µν = ψ † µ ψ ν ) oper-ators. The orbital off-diagonal condensates are finite due to the SOC-induced spin-orbitalcorrelations. These additional terms in the HF theory explicitly preserve the SU(2) spinrotation symmetry of the various Coulomb interaction terms.Collecting all the spin and charge terms together, we obtain the orbital off-diagonal(OOD) contributions of the Coulomb interaction terms:[ H HFint ] OOD = X µ<ν (cid:20)(cid:18) − U ′′ J H (cid:19) n µν h n νµ i + (cid:18) J P (cid:19) n µν h n µν i− (cid:18) U ′′ J H (cid:19) σ µν . h σ νµ i − (cid:18) J P (cid:19) σ µν . h σ µν i (cid:21) + H . c . (A2) Appendix B: Coulomb interaction matrix elements in the orbital-pair basis Corresponding to the above HF contributions of the Coulomb interaction terms in theorbital off-diagonal sector, we obtain (for site i ):[ H int ] OOD = X µ<ν (cid:20)(cid:18) − U ′′ J H (cid:19) n µν n † µν − (cid:18) U ′′ J H (cid:19) σ µν . σ † µν (cid:21) + X µ<ν (cid:20)(cid:18) J P (cid:19) n µν n † νµ − (cid:18) J P (cid:19) σ µν . σ † νµ + H . c . (cid:21) (B1)where n † µν = n νµ and σ † µν = σ νµ . The above form shows that only the pair-hoppinginteraction terms ( J P ) are off-diagonal in the orbital-pair ( µν ) basis. We will use the aboveCoulomb interaction terms in the orbital off-diagonal sector in the RPA series in order to6ensure consistency with the self-consistent determination of magnetic order including theorbital off-diagonal condensates.The Coulomb interaction terms in the orbital diagonal sector can be cast in a similarform:[ H int ] OD = X µ (cid:20)(cid:18) − U (cid:19) σ µ . σ µ + (cid:18) U (cid:19) n µ n µ (cid:21) + X µ<ν (cid:20)(cid:18) − J H (cid:19) σ µ . σ ν + U ′′ n µ n ν (cid:21) (B2)which include the Hubbard, Hund’s coupling, and density interaction terms.The form of the [ U ] matrix used in the RPA series Eq. (15) is now discussed below. Inthe composite spin-charge-orbital-sublattice ( µναs ) basis, the [ U ] matrix is diagonal in spin,charge, and sublattice sectors. There are two possible cases involving the orbital-pair ( µν )basis. In the case µ = ν , the [ U ] matrices in the spin ( α = x, y, z ) and charge ( α = c ) sectorsare obtained as:[ U ] µ ′ µ ′ α ′ = αµµα = x,y,z = U J H J H J H U J H J H J H U [ U ] µ ′ µ ′ α ′ = αµµα = c = − U − U ′′ − U ′′ − U ′′ − U − U ′′ − U ′′ − U ′′ − U (B3)corresponding to the interaction terms (Eq. B2) for the normal spin and charge densityoperators. Similarly, for the six orbital-pair cases ( µ, ν ) corresponding to µ = ν , the [ U ]matrix elements in the spin ( α = x, y, z ) and charge ( α = c ) sectors are obtained as:[ U ] µναµνα = x,y,z = U ′′ + J H / U ] µναµνα = c = U ′′ − J H / U ] νµαµνα = x,y,z = J P [ U ] νµαµνα = c = − J P (B4)corresponding to the interaction terms (Eq. B1) involving the orbital off-diagonal spin andcharge operators.7 FIG. 11: The basis-resolved contributions to the total spectral function for the low-energy magnon(left panel) and intermediate-energy orbiton (center and right panels) modes, showing dominantlyspin ( µ = ν , α = x, y, z ) and orbital ( µ = ν , α = c ) character of the fluctuation modes, respectively. Appendix C: Basis-resolved contributions to the total spectral function The detailed spin-orbital character of the collective excitations can be identified from thebasis-resolved contributions to the total spectral functions. This is illustrated here for theexcitations shown in Fig. 9 for the n = 4 case corresponding to the Ca RuO compound.Fig. 11 shows dominantly spin excitations involving yz, xz orbitals for the magnon modes(below 70 meV) and dominantly orbital excitations involving xy and yz, xz orbitals forthe orbiton modes (150 and 200 meV). Similarly, Fig. 12 shows dominantly spin-orbitalexcitations involving xy and yz, xz orbitals for the spin-orbiton modes (400 and 450 meV),and dominantly spin-orbital excitations involving yz, xz orbitals for the spin-orbit excitonmodes (550 meV).8 FIG. 12: The basis-resolved contributions to the total spectral function for the intermediate-energy spin-orbiton (left and center panels) and high-energy spin-orbit exciton (right panel) modes,showing dominantly spin-orbital character ( µ = ν , α = x, y, z ) involving xy and yz, xz orbitals (leftand center panels) and yz, xz orbitals (right panel). Similarly, for the n = 5 case corresponding to Sr IrO , the detailed spin-orbital characterof the Goldstone mode and gapped mode at q = (0 , 0) seen in Fig. 7 is shown in Fig.14, explicitly illustrating the effect of extreme spin-orbital entanglement and the resultingcorrespondence (Fig. 13) between magnetic ordering directions, spin moments, and orbitalmoments.9 FIG. 13: The extreme spin-orbital-entanglement induced correspondence between (a) magneticordering directions, (b) sign of magnetic moments for the three orbitals, and (c) orbital currentinduced orbital moments for the three orbitals, for the n = 5 case corresponding to Sr IrO . ω = 0 meV(a) I m [ χ ( q , ω ) ] µ ν α µ ν α µν =yz yzxz xzxy xyyz xzxz xy xy yzxz yzxy xzyz xy q =(0,0) 0 2 4 6 8 10 12 14 x y z c ω = 46 meV(b) I m [ χ ( q , ω ) ] µ ν α µ ν α q =(0,0) FIG. 14: The basis-resolved contributions to the total spectral function for the (a) gapless in-planemagnon mode and (b) gapped out-of-plane magnon mode for the n = 5 case corresponding toSr IrO with extreme spin-orbital entanglement. ∗ Electronic address: [email protected] W. 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