1. Frustrated Quantum Magnets in Strong Magnetic Fields F. Mila Institute of Theoretical Physics Ecole Polytechnique Fédérale de Lausanne Switzerland

2. Collaborators Theorists S. Miyahara (Lausanne  Tokyo) F. Becca (Lausanne  Trieste) B. Kumar (Lausanne) Experimentalists M. Takigawa, K. Kodama (ISSP, Tokyo) C. Berthier, M. Horvatic (Grenoble) Chemists H. Kageyama, Y. Ueda (ISSP, Tokyo)

3. Scope Introduction to Frustrated Quantum Magnets Introduction to Frustrated Quantum Magnets Dimers + Frustration  Magnetization plateaux Dimers + Frustration  Magnetization plateaux Other anomalies in SrCu 2 (BO 3 ) 2 Other anomalies in SrCu 2 (BO 3 ) 2  Dzyaloshinskii-Moriya interactions  Correlated hopping Conclusions/Perspectives Conclusions/Perspectives

4. Quantum magnets: 2 paradigms Long-range orderSpin gap Goldstone modes: Spin waves Singlet-triplet gap J’ J Δ=J’+O(J) (J«J’) H=∑ (i,j) J ij S i.S j S=1/2,1,…

5. Defining frustration Frustration = infinite degeneracy of classical ground state Shastry-Sutherland Model Effect of quantum fluctuations? J1J1 J2J2 J 2 <2J 1 J 2 >2J 1

6. Quantum approach 1) Quantum treatment of local bricks 2) Pertubative treatment of inter-brick coupling Coupled dimers Coupled triangles Frustrated motion of triplets Magnetization plateaux SrCu(BO) SrCu 2 (BO 3 ) 2 2 degrees of freedom Low-lying singlets Spin ½ kagome

7. Dimers in a magnetic field Isolated dimers Coupled dimers

8. Magnetization of spin ladders CuHpCl Chaboussant et al, EPJB ‘98 NB: CuHpCl might not be a simple ladder (Stone et al, cond-mat/0103023)

9. Frustrated ladders Metal-insulator transition for V=2t (J’=J/3)

10. Magnetization Plateau D. Cabra et al, PRL ‘97 K. Totsuka, PRB ‘98 T. Tonegawa et al, PRB ‘99 F. Mila, EPJB ‘98 Frustration Kinetic energy Repulsion Metal-insulator transition Magnetization plateau

11. Frustrated Coupled Dimers Triplet HoppingTriplet Repulsion

12. Magnetization of SrCu(BO) Magnetization of SrCu 2 (BO 3 ) 2 Kageyama et al, PRL ‘99

13. Shastry-Sutherland model Ground-state Product of singlets on J-bonds (Shastry, Sutherland, ’81) Triplets Almost immobile and repulsive (Miyahara et al, ’99) (Miyahara et al, ’00) Plateaux J’/J '.63

14. Symmetry breaking inside plateaux Miyahara et al, ’00 Hard-core bosons with repulsion

15. NMR at 1/8-plateau At least 11 different sites! K. Kodama, M. Takigawa, M. Horvatic, C. Berthier, H. Kageyama, Y. Ueda, S. Miyahara, F. Becca, F. Mila, Science ‘02

16. Magnetization profile at 1/8 Symmetry breaking 16 sites/unit cell 8-fold degenerate GS Lattice distortion Selection of one GS Magnetization in field direction Magnetization opposite to field Friedel-like oscillations NMR pattern

17. Further anomalies I 1)The magnetization raises too early 2)The gap does not close (see also ESR, Nojiri et al, 1999) 3) The magnetization jumps before plateau Kodama et al, Science ‘02 Kodama et al, unpublished Δ∕gμ B

18. Further anomalies II Tsujii et al,’03 4) No Bose condensation below plateau

19. Possible sources of deviation Correlated hopping of triplets Correlated hopping of triplets  Bound states  Bound states Spin-lattice coupling Spin-lattice coupling Dzyaloshinskii-Moriya interactions D.(SxS) Dzyaloshinskii-Moriya interactions D.(S i xS j )  Inter-dimer (D’)  Inter-dimer (D’)  Intra-dimer (D)  Intra-dimer (D)

20. Evidence of intra-dimer DM H ext =6.9 T Kodama et al, unpublished H s =H 1 -H 2 ≠0 Staggered magnetization

21. Dzyaloshinskii-Moriya interactions H D 12 D: intra-dimer DM

22. High-field effects of intra-dimer DM Kodama, Miyahara, et al, unpublished D=0.034 D’=-0.02 g s =0.023 (ED, 24 sites)

23. Properties around 1/8 plateau Exact diagonalizations? Not appropriate! Huge finite-size effects at the plateaux!  Effective hard-core boson model

24. Pertubative derivation Momoi and Totsuka, PRB’01 3rd order in J’/J

26. Mean-field phase diagram Momoi-Totsuka, ‘01 Plateaux only at 1/3 and 1/2

27. B. Kumar, F. Mila, unpublished t’ it Semi-phenomenological approach

28. Mean-field (,, ) Next step: include and

29. Conclusions/Perspectives Magnetization of SrCu 2 (BO 3 ) 2 : Remarkably rich and complex behaviour Triplet reduced mobility and repulsion  Magnetization plateaus Triplet reduced mobility and repulsion  Magnetization plateaus Intra-dimer DM interaction  Staggered magnetization  Early raise of uniform magnetization  Persistence of gap Intra-dimer DM interaction  Staggered magnetization  Early raise of uniform magnetization  Persistence of gap

30. Open issues Magnetization jump before 1/8 plateau? Magnetization jump before 1/8 plateau? Difference of behaviour below and above 1/8 plateau? Difference of behaviour below and above 1/8 plateau? Consequences of correlated hopping? Consequences of correlated hopping?  Pairing?  Pairing? Dip of the gap just before plateau? Dip of the gap just before plateau?