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Spin-Orbital Entanglement and Violation of the Kanamori-Goodenough Rules

Andrzej M. Oleś Max-Planck-Institut f ü r Festk ö rperforschung, Stuttgart M. Smoluchowski Institute of Physics, Jagellonian University , Kraków Self-organized Strongly Correlated Electron Systems Seillac, 31 May 2006. Spin-Orbital Entanglement and Violation of the Kanamori-Goodenough Rules.

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Spin-Orbital Entanglement and Violation of the Kanamori-Goodenough Rules

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  1. 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 Spin-Orbital Entanglement and Violation of the Kanamori-Goodenough Rules • 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. Outline • Spin-orbital superexchange models • Goodenough-Kanamori rules in transition metal oxides • Example: magnetic and optical properties of LaMnO3 • Violation of Goodenough-Kanamori rules in t2g systems due to spin-orbital entanglement • Continuous orbital transition • Spin-orbital fluctuations in LaVO3

  3. Orbital physics in transition metal oxides Goodenough-Kanamori rules: AO order supports FM spin order FO order supports AF spin order C-AF A-AF Current status: Focus on Orbital Physics New Journal of Physics 2004-2005 http://www.njp.org LaVO3 t2g orbitals LaMnO3 eg orbitals

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

  5. Multipletstructureof transition metal ions Follows from three Racah parameters (Griffith, 1971): single parameter: η=JH /U [AMO et al., PRB 72, 214431 (2005)]

  6. Magnetic and optical properties of Mottinsulators (t<<U) Spin-orbital superexchange model for a perovskite, γ=a,b,c(J=4t2/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. spectral weights for increasing T AF FM Exchange constants and optical spectral weights in LaMnO3 Jc and Jab for varying orbital angle  A-AF phase orbital order: exp: F. Moussa et al., PRB 54, 15149 (1996) exp: N.N. Kovaleva et al., PRL 93, 147204 (2004) S=2 spins and egorbitals are disentangled (MF can be used) [ AMO, G. Khaliullin, P. Horsch, and L.F. Feiner, PRB 72, 214431 (2005) ]

  8. Spin waves in La1-x SrxMnO3 and in bilayer manganites Isotropic spin waves inLa1-xSrxMnO3 Anisotropic spin waves in La2-2xSr1+2xMn2O7 FM phase x=0.30 x=0.35 [ T.G. Perring et al., PRL 87, 217201 (2001) ] [ T.G. Perring et al., PRB 77, 711 (1996) ] Double exchange and superexchange explain Jab and Jc [ AMO and L.F. Feiner, PRB 65, 052414 (2002); 67, 092407 (2003)]

  9. Charge transfer insulator: KCuF3 One of the best examples of a 1D AF Heisenberg model Parameters: J =33 meV, η=0.12, R=2U/( 2Δ+Up ) =1.2 Jc and Jab for varying orbital angle  spectral weights for increasing T optical properties would help to fix the parameters Valid if S=1/2 spins and egorbitals disentangle (MF can be used) [ AMO et al., PRB 72, 214431 (2005)]

  10. Spin-orbital models with entanglement • d1 model – titanates (LaTiO3, YTiO3), S=1/2, t2gorbitals; • d2 model – vanadates (LaVO3, YVO3), S=1, t2g orbitals, (xy)1(yz/zx)1 configuration; • d9 model – KCuF3, S=1/2, eg orbitals. Spin-orbital models were derived in: d1 model [G. Khaliullin and S. Maekawa, PRL 85, 3950 (2000)] d2 model [G. Khaliullin, P. Horsch, and AMO, PRL 86, 3879 (2001)] d9model [L.F. Feiner, AMO, and J. Zaanen, PRL 78, 2799 (1997)]

  11. eg orbitals t2g orbitals Orbital degrees of freedom In t2g systems (d1,d2) two flavors are active, e.g. yz and zx along c axis – described by pseudospin operators: At finite η the orbital operators contain: GdFeO3-type distortions induce orbital interactions leading to FO order: Pseudospin operators for eg systems (d9) with 3z2-r2and x2-y2: Jahn-Teller ligand distortions favor AO order:

  12. Spin-orbital superexchange at JH=0 => chain along c axis => 2D model in ab planes

  13. Intersite spin, orbital and spin-orbital correlations Spin correlations: Orbital and spin-orbital correlations for t2g (d1 and d2)systems: Orbital and spin-orbital correlations for eg (d9) model: • Definitions follow from the structure of the spin-orbital SE at JH0; • Method: exact diagonalization of four-site systems.

  14. Intersite correlations for increasing Hund’s exchange η V=0 V=J d1 Sij – spin correlations Tij –orbital correlations Cij– spin-orbital correlations d2 • all correlations identical in d1at η=0: Sij =Tij =Cij =  0.25 [SU(4)]; • regions of Sij<0 and Tij<0 both at V=0 and V=J in d1(2) models; • Cij<0 in low-spin (S=0) states; • different signs of Sijand Tij in d9 d9 GK rules violated in d1, d2 [AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)]

  15. Spin exchange constants Jij for increasing Hund’s exchangeη V=0 V=J d1 In the shadded areas Jij is negative FM Sij is negative AF for d1 and d2t2gmodels => GK rules are violated d2 In d9eg model spin correlations Sij follow the sign of Jij => GK rules are obeyed d9 [AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)]

  16. Dynamical exchange constants due to entanglement Fluctuations of Jijare measured by Fluctuations dominate the behavior of t2g systems at η=0, V=0: for a bond <ij> fluctuations: ( S=0 / T=1 )  ( S=1 / T=0 ) d1 model: [ SU(4) symmetry ] d2 model: Fluctuations large but do not dominate for eg system at η=0, V=0: d9 model: ,i.e.,

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

  18. when only Ising term: sharp transition S=0 S=4 Continuous orbital phase transition in d2 model with full t2gorbital dynamics: V=J continuous transition quantum numbers T and Tznonconserved orbital transitions are continuous T and Tz conserved

  19. Optical spectral weights for the C-AF phase of LaVO3 mean-field approach orbital and spin-orbital dynamics orbital disorder unlike in LaMnO3 Data: S. Miyasaka et al., [ JPSJ 71, 2086 (2002) ] spin-orbital fluctuations important at T>0! [G. Khaliullin, P. Horsch, and AMO, PRB 70, 195103 (2004)]

  20. 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 LaVO3 magnetic and optical properties [G. Khaliullin, P. Horsch, and AMO, PRL 86, 3879 (2001); PRB 70, 195103 (2004)] Conclusions • Spins and orbitals disentangle in eg systems ( LaMnO3 ) • [AMO, G. Khaliullin, P.Horsch, and L.F. Feiner, PRB 72, 214431 (2005)] • 2. In systems with t2gdegrees of freedom • 3. Dynamic spin and orbital fluctuations in t2g systems: spins and orbitals are entangled static Goodenough-Kanamori rules are violated Any other experimental manifestations of entanglement?

  21. Thank you for your attention!

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