1 / 17

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

Spin-orbital entanglement and violation of the Goodenough-Kanamori rules. L.F. Feiner A.M. Ole ś , P. Horsch, G. Khaliullin. Outline. Violation of GK-rules: when? t 2g systems (as compared to e g systems) Nature of spin-orbital superexchange

bruis
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related