1. Seillac, 31 May 20061 Spin-Orbital Entanglement and Violation of the Kanamori-Goodenough Rules Andrzej M. Oleś Max-Planck-Institut für Festkörperforschung, Stuttgart M. Smoluchowski Institute of Physics, Jagellonian University, Kraków Self-organized Strongly Correlated Electron Systems Seillac, 31 May 2006 Peter Horsch, Max-Planck-Institut FKF, Stuttgart Giniyat Khaliullin, Max-Planck-Institut FKF, Stuttgart Louis-Felix Feiner, Philips Research Laboratories, Eindhoven Institute of Theoretical Physics, Utrecht University oo

2. Seillac, 31 May 20062 Outline Spin-orbital superexchange models Goodenough-Kanamori rules in transition metal oxides Example: magnetic and optical properties of LaMnO 3 Violation of Goodenough-Kanamori rules in t 2g systems due to spin-orbital entanglement Continuous orbital transition Spin-orbital fluctuations in LaVO 3

3. Seillac, 31 May 20063 Orbital physics in transition metal oxides Current status: Focus on Orbital Physics New Journal of Physics 2004-2005 http://www.njp.org LaVO 3 t 2g orbitals LaMnO 3 e g orbitals C-AFA-AF Goodenough-Kanamori rules: AO order supports FM spin order FO order supports AF spin order

4. Seillac, 31 May 20064 Electron interactions and multiplet structure [AMO and G. Stollhoff, PRB 29, 314 (1984)] Two parameters: U – intraorbital Coulomb interaction, J H – Hund’s exchange Anisotropy in Hund’s exchange:

5. Seillac, 31 May 20065 [AMO et al., PRB 72, 214431 (2005)] Multiplet structure of transition metal ions Follows from three Racah parameters (Griffith, 1971): single parameter: η=J H /U

6. Seillac, 31 May 20066 Magnetic and optical properties of Mott insulators (t<<U) Spin-orbital superexchange model for a perovskite, γ=a,b,c (J=4t 2 /U): contains orbital operators: By averaging over orbital operators one finds effective spin model: Here spin and orbital operators are disentangled. Superexchange determines partial optical sum rule for individual band n: [G. Khaliullin, P. Horsch, and AMO, PRB 70, 195103 (2004)]

7. Seillac, 31 May 20067 Exchange constants and optical spectral weights in LaMnO 3 J c and J ab for varying orbital angle  spectral weights for increasing T [ AMO, G. Khaliullin, P. Horsch, and L.F. Feiner, PRB 72, 214431 (2005) ] AF FM S=2 spins and e g orbitals are disentangled (MF can be used) A-AF phase orbital order: exp: F. Moussa et al., PRB 54, 15149 (1996)exp: N.N. Kovaleva et al., PRL 93, 147204 (2004)

8. Seillac, 31 May 20068 Spin waves in La 1-x Sr x MnO 3 and in bilayer manganites Isotropic spin waves in La 1-x Sr x MnO 3 [ AMO and L.F. Feiner, PRB 65, 052414 (2002); 67, 092407 (2003) ] Double exchange and superexchange explain J ab and J c FM phase Anisotropic spin waves in La 2-2x Sr 1+2x Mn 2 O 7 [ T.G. Perring et al., PRL 87, 217201 (2001) ] [ T.G. Perring et al., PRB 77, 711 (1996) ] x=0.30x=0.35

9. Seillac, 31 May 20069 Charge transfer insulator: KCuF 3 J c and J ab for varying orbital angle  Valid if S=1/2 spins and e g orbitals disentangle (MF can be used) spectral weights for increasing T Parameters: J =33 meV, η =0.12, R=2U/( 2Δ+U p ) =1.2 One of the best examples of a 1D AF Heisenberg model optical properties would help to fix the parameters [ AMO et al., PRB 72, 214431 (2005) ]

10. Seillac, 31 May 200610 Spin-orbital models with entanglement d 1 model – titanates (LaTiO 3, YTiO 3 ), S=1/2, t 2g orbitals; d 2 model – vanadates (LaVO 3, YVO 3 ), S=1, t 2g orbitals, (xy) 1 (yz/zx) 1 configuration; d 9 model – KCuF 3, S=1/2, e g orbitals. Spin-orbital models were derived in: d 1 model [G. Khaliullin and S. Maekawa, PRL 85, 3950 (2000)] d 2 model [G. Khaliullin, P. Horsch, and AMO, PRL 86, 3879 (2001)] d 9 model [L.F. Feiner, AMO, and J. Zaanen, PRL 78, 2799 (1997)]

11. Seillac, 31 May 200611 Orbital degrees of freedom In t 2g systems (d 1,d 2 ) two flavors are active, e.g. yz and zx along c axis – described by pseudospin operators: At finite η the orbital operators contain: Pseudospin operators for e g systems (d 9 ) with 3z 2 -r 2 and x 2 -y 2 : GdFeO 3 -type distortions induce orbital interactions leading to FO order: Jahn-Teller ligand distortions favor AO order: e g orbitalst 2g orbitals

12. Seillac, 31 May 200612 Spin-orbital superexchange at J H =0 => chain along c axis => 2D model in ab planes

13. Seillac, 31 May 200613 Intersite spin, orbital and spin-orbital correlations Spin correlations: Orbital and spin-orbital correlations for t 2g (d 1 and d 2 ) systems: Orbital and spin-orbital correlations for e g (d 9 ) model: Definitions follow from the structure of the spin-orbital SE at J H  0; Method: exact diagonalization of four-site systems.

14. Seillac, 31 May 200614 Intersite correlations for increasing Hund’s exchange η V=0V=J S ij – spin correlations T ij – orbital correlations  C ij – spin-orbital correlations [AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)] d1d1 d2d2 d9d9 all correlations identical in d 1 at η=0: S ij =T ij =C ij =  0.25 [SU(4)]; regions of S ij <0 and T ij <0 both at V=0 and V=J in d 1(2) models; C ij <0 in low-spin (S=0) states; different signs of S ij and T ij in d 9 GK rules violated in d 1, d 2

15. Seillac, 31 May 200615 [AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)] V=0V=J Spin exchange constants J ij for increasing Hund’s exchange η d1d1 d2d2 d9d9 In the shadded areas J ij is negative FM S ij is negative AF for d 1 and d 2 t 2g models => GK rules are violated In d 9 e g model spin correlations S ij follow the sign of J ij => GK rules are obeyed

16. Seillac, 31 May 200616 Dynamical exchange constants due to entanglement Fluctuations of J ij are measured by Fluctuations dominate the behavior of t 2g systems at η=0, V=0: d 1 model: d 2 model: [ SU(4) symmetry ] Fluctuations large but do not dominate for e g system at η=0, V=0: d 9 model:,i.e., for a bond fluctuations: ( S=0 / T=1 )  ( S=1 / T=0 )

17. Seillac, 31 May 200617 Quantum corrections in spin-orbital models [AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)] Large corrections beyond MF due to spin-orbital entanglement

18. Seillac, 31 May 200618 Continuous orbital phase transition in d 2 model with full t 2g orbital dynamics: V=J continuous transition when only Ising term: sharp transition orbital transitions are continuous S=0S=4 quantum numbers T and T z nonconserved T and T z conserved

19. Seillac, 31 May 200619 Optical spectral weights for the C-AF phase of LaVO 3 mean-field approach orbital and spin-orbital dynamics [G. Khaliullin, P. Horsch, and AMO, PRB 70, 195103 (2004)] spin-orbital fluctuations important at T>0! orbital disorder unlike in LaMnO 3 Data: S. Miyasaka et al., [ JPSJ 71, 2086 (2002) ]

20. Seillac, 31 May 200620 Conclusions 1.Spins and orbitals disentangle in e g systems ( LaMnO 3 ) [AMO, G. Khaliullin, P.Horsch, and L.F. Feiner, PRB 72, 214431 (2005)] 2. In systems with t 2g degrees of freedom 3. Dynamic spin and orbital fluctuations in t 2g systems: spin triplet orbital singlet spin singlet orbital triplet [AMO, P. Horsch, L.F. Feiner, and G. Khaliullin, PRL 96, 147205 (2006)] 4. Joint spin-orbital fluctuations in LaVO 3 magnetic and optical properties [G. Khaliullin, P. Horsch, and AMO, PRL 86, 3879 (2001); PRB 70, 195103 (2004)] spins and orbitals are entangled static Goodenough-Kanamori rules are violated Any other experimental manifestations of entanglement?

21. Seillac, 31 May 200621 Thank you for your attention!