1. Lattice instability in frustrated systems Maxim Mostovoy MPI, Stuttgart Groningen, April 22, 2004 D. Khomskii, Cologne J. Knoester, Groningen R. Moessner, Paris N.V. Prokof´ev, Amherst

2. Outline Geometrical frustration Magnetoelastic transitions in frustrated spin systems Orbital interactions Frustration of orbital ordering and its lifting

3. Geometrical frustration

4. AFM Ising on triangular lattice Ground state entropy Spin correlations at T=0 (Wannier, Houtappel, 1950) (Stephenson, 1970)

5. Ordered state

6. Hard sphere liquid Alder & Wainwright (1957) Hoover & Ree (1968) S solid > S liquid

7. Kinks Kink energy Interactions between kinks in neighboring chains

8. Kink crystal 1 kink / 3 sites M.M., N.Prokof’ev, D.Khomskii & J.Knoester, PRL 90 (2003)

9. Frustrated spins coupled to lattice Strains:

10. Free energy

11. Frustration-induced transition Free energy First order transition Ordered state Disordered state

12. Cr 3+ S = 3/2  CW = -390K T N = 12.5K first order ZnCr 2 O 4 S.-H. Lee et al., PRL 84, 3718 (2000) O. Tchernyshyov et al., PRL 88, 067203 (2001)

13. AV 2 O 4 V 3+ S = 1 first order cubic to tetragonal c < a T N (K) 40 45 35 T st (K) 50 65 97 Zn Mg Cd H. Mamiya et al., J. Appl. Phys. 81, 5289 (1997)

14. Interactions between t 2g -orbitals

15. Tetrahedron states

16. Frustration and its lifting Jahn-Teller S.-H. Lee et al., cond-mat/0312558

17. Orbital interactions Jahn-Teller interaction Coupled orbital & spin exchange (Kugel-Khomskii) `Orbital Casimir´ effect Peierls-like interaction

18. e g -orbitals

19. Jahn-Teller interaction

20. Lattice-mediated interaction

21. Kugel-Khomskii model (a): (b)(c)(d) (b)+(c): (a)

22. Orbital Casimir effect (U p,U d =  ) HolesElectrons M.M. & D. Khomskii, Phys.Rev.Lett 89, 227203 (2002), cond-mat 0304494

23. 180 0 -exchange in 2D K 2 CuF 4 1e g hole/Cu  canted antiferroorbital ferromagnetic

24. Frustrated orbital models xy yz zx xy yzzx triangular cubic pyrochlore 180 o -exchange 90 o -exchange LiNiO 2 ZnMn 2 O 4 xx yy zz KCuF 3

25. Mean field Triangular, pyrochlore ferro-obital Cubic antiferro-orbital G-type Disordered ground states arbitrary superposition

26. Order from disorder Quantum orbital fluctuations chose six unform states: G.Khaliullin, PR B 64, 212405 (2001) M.M. & D.Khomskii,PRL 89, 227203 (2002)

27. Strains

28. Orbital-strain interaction electron hole Jahn-Teller coupling Peierls coupling Strain energy

29. Ordered state layered canted antiferroorbital

30. Ordered state electrons holes LaMnO 3 KCuF 3

31. Conclusions Frustration of spin and orbital ordering makes the crystal lattice instable JT distortion occurs together with lattice distortion lifting the frustration of orbital ordering