Theory of probing orbitons with RIXS Luuk Ament Lorentz Institute, Leiden, the Netherlands Giniyat Khaliullin Max-Planck-Institute FKF Stuttgart, Germany Fiona Forte Salerno University Salerno, Italy Jeroen van den Brink Lorentz Institute Leiden, the Netherlands

Orbital ordering Why do orbitals order? LaMnO3 Orbital order in plane Goodenough (1963) Orbital order in plane Why do orbitals order? Lattice distortion (Jahn-Teller) 2. Orbital and spin dependent superexchange

Kugel-Khomskii model Superexchange interaction involving spins and orbitals. Orbitals are degenerate, no coupling to the lattice. Orbitals determine overlap t  J ~ t2/U x2-y2 3d e2g x2-y2 3z2-r2 3d e2g 3z2-r2

Jahn-Teller Vs. Superexchange Both lead to orbital order, so why is it interesting? Excitations are very different! Local crystal field excitations Vs. dispersing orbitons Superexchange: spins and orbitals entangle. Jahn-Teller: spins and orbitals decouple, orbitals frozen out at low T.

YTiO3 A good candidate for orbitons. Why? t2g orbitals: directed away from oxygen ions. No cooperative JT phase transition seen. TiO6 octahedra are tilted, but only slightly deformed. Spin wave spectrum is isotropic. Raman data: temperature dependence. C. Ulrich et al., PRL 97, 157401 (2006) LA & G. Khaliullin, to be published

YTiO3 Ti has 3d t2g1 configuration Ferromagnetic Mott insulator at low temperature: spin and charge degrees of freedom frozen out Ti O Y Two scenario’s: Lattice distortions split t2g orbitals. Orbital fluctuations dominate over Jahn-Teller distortions. Degenerate t2g orbitals with superexchange interactions. Both models lead to orbital order, but with very different orbital excitations.

YTiO3 - superexchange What are the possible hopping processes via oxygen? ‘Out-of-plane’ hopping is symmetry forbidden. ‘In-plane’ hopping: only via one of the two 2p’s allowed. Result: t2g orbitals are conserved and confined to their plane. x z Ti x z Ti Ti O Ti O Y O y O y Expand in t/U: Superexchange interaction, dependent on bond direction.

YTiO3 - superexchange Superexchange interaction dependent on bond direction. y-direction xz xy 3d t2g Ti yz Ti

YTiO3 - superexchange Superexchange Hamiltonian has an orbitally ordered ground state with 4 sublattices: Condense: In analogy to magnons: collective excitations (orbitons) on top of the ordered ground state. Orbiton gap induced by octahedron tilting: t_2g to e_g hopping -> nonconservation of orbital type. Pictures from E. Saitoh et al., Nature 410, 180 (2001) and Khaliullin et al., Phys. Rev. B68, 205109 (2003).

Indirect RIXS off YTiO3 YTiO3 wres (~460 eV) Measure energy and momentum transfer YTiO3 Ti 3d eg level Say: electron into e_g, not t_2g wres (~460 eV) Ti 2p level Core hole couples to valence electrons via core hole potential

RIXS data on YTiO3 Low energy part for 3 momentum transfers q along [001]-direction: C. Ulrich, et al., to be published Spectral weight increases with larger q. Maximum of 250 meV peak shows little dispersion. Multi-phonons? Multi-magnons? Orbital excitations? C. Ulrich, G. Ghiringhelli, L. Braicovich et al., PRB 77, 113102 (2008)

RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons. 3d eg If core hole potential is not of A1g symmetry: 3d t2g 2p Core hole Mechanism 1: core hole potential shakes up t2g electrons S. Ishihara et al., PRB 62, 2338 (2000)

RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons: U 3d t2g 3d eg U Mechanism 2: superexchange bond is modified

RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons: 3d t2g 3d eg Core hole potential effectively lowers Hubbard U: U-Uc Core hole Mechanism 2: superexchange bond is modified F. Forte et al., PRL 101, 106406 (2008) S. Ishihara et al., PRB 62, 2338 (2000) Magnons: J. Hill et al., PRL 100, 097001 (2008) J. Van den Brink, EPL 80, 47003 (2007)

Results Calculate effective scattering operator (UCL): J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006) L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007) Two RIXS mechanisms: 1. Coulomb-induced shakeup Polarization Multiplet structure for example if  = t2g yz: Transferred momentum H from three-band Hubbard model can be obtained by cluster calculation. We take all equal. Mechanism applicable to both J-T and superexchange models.

Results Calculate effective scattering operator (UCL): J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006) L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007) 2. Superexchange bond modification Two RIXS mechanisms: Hamiltonian, two-orbiton only Enhanced fluctuations, create one- and two-orbitons Applies only to superexchange model of YTiO3.

? Results RIXS Mechanism Superexchange modification Local orbital flip C. Ulrich et al., to be published Lattice distortions: (local dd-excitations) E. Pavarini et al., New J. Phys. 7, 188 (2005) Orbiton physics: RIXS Mechanism Physics of YTiO3 Lattice distortions Super- exchange Superexchange modification Local orbital flip ? 2-orbiton continuum 2-orbiton continuum 1-orbiton shoulder

RIXS data on YTiO3 Temperature dependence Say: 250 meV peak strongly enhanced at low T - orbital dynamics strongly influenced by magnetic correlations, natural in Kugel-Khomskii model. Local dd-excitations are not expected to show strong T-dependence. Low-energy peak is magnon peak (corresponds to 16 meV magnons) Large increase of spectral weight in low-T ferromagnetic state Peaks sharpen at low temperature C. Ulrich et al., to be published

LaMnO3 Mn 3d4, high-spin configuration: eg t2g Mott insulator, A-type AFM at low temperature (FM layers). Kugel-Khomskii model without Hund’s rule coupling: To first order, orbitals of different layers decouple!

LaMnO3 - Superexchange eg orbitals order ‘antiferro-orbitally’: eg t2g Excitations: eg orbital waves (orbitons) E. Saitoh et al., Nature 410, 180 (2001) J. van den Brink, F. Mack, P. Horsch and A. Oles, Phys. Rev. B. 59, 6795 (1999).

LaMnO3 - Single orbitons Initial Intermediate Final eg Stress contrast with indirect magnetic RIXS: no change in S^z. Looks like Heisenberg, but no conservation of Tz. This leads to single orbiton excitations. J. van den Brink, P. Horsch, F. Mack & A. M. Oles, PRB 59, 6795 (1999)

Orbitons in indirect RIXS Orbital Hamiltonian: J. van den Brink, F. Mack, P. Horsch and A. Oles, Phys. Rev. B. 59, 6795 (1999). Intermediate state Hamiltonian for superexchange modification: q=0 gives 0 intensity to first order in UCL, but not to second order. with F. Forte, LA and J. van den Brink, Phys. Rev. Lett. 101, 106406 (2008). S. Ishihara and S. Maekawa, PRB 62, 2338 (2000)

Results Orbiton RIXS spectrum for LaMnO3 Two-orbiton continuum Experiments: see nothing at low energy (down to ~100 meV). Look at dispersion of dd-excitations? One-orbiton peak F. Forte, L. Ament and J. van den Brink, Phys. Rev. Lett. 101, 106406 (2008).

Conclusion RIXS is an excellent probe of orbital excitations, discrimination between Jahn-Teller and superexchange driven order is possible. RIXS data for YTiO3 best explained with orbitons. Lattice distortion scenario doesn’t work.

LaMnO3 Probably Jahn-Teller dominated eg orbitals: directed towards oxygen ions leads to higher Jahn-Teller coupling than t2g orbitals. Cooperative JT phase transition around T = 800 K. 2-sublattice orbital order below 800 K. Magnetic order sets in only below TN = 140 K. JT splitting EJT = 0.7 eV. Classical orbitals describe experimental data well.

Vs. 2 competing scenario’s Jahn-Teller Superexchange 3d t2g Vs. Jahn-Teller Superexchange Kugel-Khomskii: strong coupling between spins and orbitals, leads to sensitivity of orbital sector to changes in magnetic sector. Local excitations: No dispersion Collective excitations: Strong dispersion