1. GNSF: 06-81-4-100 KITP: PHY05-51164 Krakow, 18-22 June 2008 George Jackeli Max-Planck Institute for Solid State Research, Stuttgart In collaboration with: Daniel Khomskii Univ. of Cologne Classical Dimers in Orbitally Degenerate Honeycomb Antiferromagnet Phys. Rev. Lett. 08

2. Miura et. al., JPSJ’07 Honeycomb compound: Li 2 RuO 3

3. Miura et. al., JPSJ’07 Honeycomb compound: Li 2 RuO 3

4. Ru 4+ (d 4 ), S=1 & 3-fold orbital XY XZ YZ directional hopping Honeycomb compound: Li 2 RuO 3

5. Exchange couplings: Orbital degrees are static Pott’s-like! AFM coupling ~ FM coupling ~ Coupled spin-orbital model

6. Strong AFM ~1 Weak FM ~  <<1 Free Example of an orbital and spin-coupling pattern

7. Dimensionality reduction due to orbitals: Factorization in clusters of open/closed 1D AFM chains Limit of zero Hund’s coupling All Neel type states ( S i S j +S 2 =0 ) have zero classical energy Shortest chains(=dimers) are favorable since they gain more quantum energy

8. The ground state manifold Ground state manifold is spanned by hard-core dimer coverings Spins are bound into spin-singlet on dimer bonds: Spin gaped phase = Extensive orientational degeneracy infinitely many ways of covering

9. Lifting the orientational degeneracy of dimers Possible Mechanisms: By Weak Interdimer Coupling Order-out-of-disorder by Triplet Fluctuations By Magnetoelastic Coupling “dimer Jahn-Teller” mechanism

10. Order-out-of-disorder by triplet fluctuations Fluctuations disfavor hexagonal loops, but do not fully lift the degeneracy t x,y,z S=1 S=0 s D Interdimer coupling (~  ) creates virtual triplet fluctuations (TFS) Different zero point energy of triplets may lift degeneracy h2h2 2 h1h1 2

11. Degeneracy breaking by magnetoelastic coupling Magnetoelastic coupling Elastic energy Shortening of the strong bond =Gain of magnetic energy Different distortion pattern Costs different elastic energy Selection of ground state

12. “Dimer Jahn-Teller” mechanism Li Ru Ru- displacements due to the contraction of spin-singlet bonds Li- displacements induced by contracted Ru-Ru bond Unfavorable orientation of neighboring dimers destructive interference of the distortions

13. Dimerized pattern and orbital order

14. Summary Orbital induced dimensionality reduction Spin gap formation and spontaneous dimerization in D>1 Exact realization of classical dimer problem Degeneracy lifting by “dimer Jahn-Teller” effect Formation of a peculiar VBC

15. t 2g -systems with dimer ground states (Theory) Summary S=1/2 (d 1, d 5 ) Triangular, Kagome, Honeycomb, Pyrochlore Square (elongated octahedra) S=1 (d 2, d 4 ) Honeycomb