Spin-orbital entanglement and violation of the Goodenough-Kanamori rules

1. Spin-orbital entanglement and violation of the Goodenough-Kanamori rules L.F. Feiner A.M. Oleś, P. Horsch, G. Khaliullin Spin-orbital entanglement and GK-rules: Kraków 2008

2. Outline • Violation of GK-rules: when? • t2g systems (as compared to eg systems) • Nature of spin-orbital superexchange • Correlation functions: spin, orbital, composite • Origin of GK-violation • Entanglement • Between sites vs. between spins and orbitals • Role of SU(4) symmetry !? • Meaningful estimators? Spin-orbital entanglement and GK-rules: Kraków 2008

3. Violation of GK-rules: when? • GK-rules: FO  AF AO  FM when Torb>> Tspin ! • electron-lattice interaction (e.g. JT) • Hund’s rule coupling (JH) • What if Torb Tspin ? • fluctuating orbitals • requires: weak el-latt interaction  t2g JH = 0 (?!) Spin-orbital entanglement and GK-rules: Kraków 2008

4. t2g systems (as compared to eg) • t2g (d1: LaTiO3; d2: LaVO3); eg (d9: KCuF3) • 2 “active” orbitals: orbital pseudospin T=1/2 [t2g : along c-axis: xz, yz; eg : 3z2-r2, x2-y2] • generic Hamiltonian: with: , dependent upon : electron-lattice-coupling induced interaction (strength V) Spin-orbital entanglement and GK-rules: Kraków 2008

5. d1 (LaTiO3): S=1/2, T=1/2 • without Hund’s rule coupling (η=0): • SU(4)-invariant • SU(2)-invariant in orbital space • contains • 1st order in η contains • d9 (KCuF3): S=1/2, T=1/2 • different active orbital along each cubic axis: • not SU(2)-invariant in orbital space: no singlet tendency } • tendency towards singlets Spin-orbital entanglement and GK-rules: Kraków 2008

6. Exact numerical solution • d1, d2: 4-site chain • d9: 4-site plaquette • V=0 (pure spin-orbital SE) vs finite V • ηS as variable (strength Hund’s rule coupl.) • Correlation functions:  GK-rules: and have opposite sign Spin-orbital entanglement and GK-rules: Kraków 2008

7. [SU(4)] : d1: : • violation of GK-rules • spin-orbital entanglement V=0 : conventional d2: similar to d1 d9: conventional throughout • GK-rules obeyed Spin-orbital entanglement and GK-rules: Kraków 2008

8. V finite • d1: • small η: conventional • behavior restored • intermediate η: • fluctuation regime persists d2: similar to d1 d9: no qualitative change induced by V Spin-orbital entanglement and GK-rules: Kraków 2008

9. Origin of GK-violation • t2g as compared to eg: • intrinsic: nature of spin-orbital superexchange • not due to difference in electron-lattice coupling • SE interaction in orbital space • SU(2)-invariant for t2g: strong fluctuations • Ising-like for eg: weak fluctuations Spin-orbital entanglement and GK-rules: Kraków 2008

10. Qi+ = Ri+ = Si+ Si+ Ti_ Ti+ Qi_ = Si_ Ri_ = Ti+ Si_ Ti_ Ri·Rj + Qi·Qj + ····· + 2 Siz Sjz Tiz Tjz +1/8 Qiz = ½ ( Siz  Tiz ) Riz = ½ ( Siz +Tiz ) Entanglement • Analysis of wavefunction (at η=0): • -- alternative form of Hamiltonian • with: • -- singlets with respect to these su(2) operators  S  = ( u 1  d 2   d 1 u 2 ) / 2  S”  = ( d 1  u 2   u 1 d 2 ) / 2 Spin-orbital entanglement and GK-rules: Kraków 2008

11. -- correlation functions: Rij = - 3/8 ( 1 + cos 2α) Sij = - ¼ - ½ sin 2α Qij = - 3/8 ( 1 - cos 2α) Tij = - ¼ + ½ sin 2α Cij = - ¼ cos22α no decoupling of spins and orbitals where • apparently: d1 is close to α = 0 or α = π/2 • pure singlet with respect to either R or Q -- ansatz for wavefunction on a bond: ψ = cos α  S  + sin α  S”  -- energy ψ minimizes the energy on a bond for arbitrary α Spin-orbital entanglement and GK-rules: Kraków 2008

12. -- equivalent form of wavefunction • ψ = cos β  s; t  + sin β  t; s  ; β = π/4 – α • s; t  = (    ) ( ud + du ) / 2 • t; s  = (  +  ) ( ud  du ) / 2 -- correlation functions Rij = - 3/8 ( 1 + sin 2β) Sij = - ¼ - ½ cos 2β Qij = - 3/8 ( 1 - sin 2β) Tij = - ¼ + ½ cos 2β Cij = - ¼ sin22β no decoupling of spins and orbitals • apparently: d1 is close to β = -π/4 or β = π/4 • maximum fluctuations between spin-singlet/orbital-triplet and vice versa ! Spin-orbital entanglement and GK-rules: Kraków 2008

13. -- entanglement as measured by Von Neumann entropy • between sites: Ssite = cos2α log(1/cos2α ) + sin2α log(1/sin2α ) • maximum [equal to log(2)] at α = π/4, 3π/4, … (equal bonding contributions from R and Q) •  minimum [equal to 0] at α = 0, π/2, π, … • between orbitals and spins: (partial tracing over spin or orbital variables) Sspin/orbital = cos2β log(1/cos2β ) + sin2β log(1/sin2β ) NOTE: Cij = - ¼ sin22β gives equivalent measure • maximum [equal to log(2)] at β = π/4, 3π/4, … (equal bonding contributions from  s; t  and  t; s ) •  minimum [equal to 0] at β = 0, π/2, π, … Spin-orbital entanglement and GK-rules: Kraków 2008

14. Role of SU(4) symmetry • Bonding can occur in two (nearly) equivalent channels: R and Q • Spins and orbitals are not the natural operators to describe bonding between sites • VN entropy minimal for single-channel bonding  is this physically satisfactory? • Site-entanglement and spin/orbital entanglement are complementary • in d1 and d2 single-channel bonding occurs: maximum spin-orbital entanglement Spin-orbital entanglement and GK-rules: Kraków 2008

15. Meaningful estimators? • Which entanglement estimators are available that work for SU(4) [or any other group/algebra different from SU(2)]? • Which estimators are (especially) suitable to describe entanglement between subspaces of the Hilbert space instead of real space? • Are existing estimators physically equivalent (or: in which situations)? Spin-orbital entanglement and GK-rules: Kraków 2008

16. Spin-orbital entanglement and GK-rules: Kraków 2008

17. Spin-orbital entanglement and GK-rules: Kraków 2008