1. Houches/06//2006 From Néel ordered AFMs to Quantum Spin Liquids C. Lhuillier Université Paris VI & IUF

2. Houches/06//2006 B. Bernu (Paris VI) J.C. Domenge (Rutgers) H.U. Everts (Hannover) J.B. Fouet (Lausanne) A. Laeuchli (Lausanne) P. Lecheminant (Cergy-Pontoise) W. Liming (Canton) F. Mila (Lausanne) G. Misguich (Saclay) L. Pierre (Paris X) P. Sindzingre (Paris VI) C. Waldtmann (Hannover)

3. Houches/06//2006 Introduction: –Miscellaneous questions around R.V.B. w.-f. –From Néel states to purely Quantum phases Valence Bond Crystals and Valence Bond Solids True Spin Liquid phases

4. Houches/06//2006 g.s. wave-functions and spectral properties of quantum magnets RVB w.f. (Anderson 72, Liang, Douçot, Anderson 88) huge dimension of the Hilbert space of V.B. (singlets).. BUT what about spectra of excitations? In most of 2- and 3-d cases because of spontaneous symmetry breaking, these singlets do not dominate the low energy physics, and semi-classical pictures are usually OK! What can favor exotic quantum phases in 2d?

5. Houches/06//2006 Bipartite latt. Frustrated latt. Néel order versus quantum pairing : coordination number and geometry Reduction of Néel order parameter by quantum fluct. Coordination number H =   i,j  2 S i. S j

6. Houches/06//2006 Center of mass kinetics –Eigenstates: |0, k > –Scaling as 1/N G.S. + 1 phonon Q –Eigenstates: |1, k + Q> –E |1, k + q> - E |0, k > scales as 1/L Translation symmetry breakingin 2 and 3D Broken symmetry in mesoscopic solids

7. Houches/06//2006 Euler triedra kinetics –Eigenstates: |0, S, k 0 > –Scaling as 1/N |0, S, k 0 > g.s. + 1 mag. Q –Eigenstates: |1, S+1, k+ Q> –  E scales as 1/L SU(2) symmetry breaking in 2 and 3D Broken Symmetry in Néel A.F.

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11. A simple exactly solvable model and adiabatic continuation from the Ising antiferromagnet to the Heisenberg Néel antiferromagnet

12. Houches/06//2006 Partial restoration of symmetry through quantum fluctuations: “order by disorder phenomenon”

13. Houches/06//2006 J 1 -J 2 model on the triangular lattice

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16. When the adiabatic scenario fails more severely…

17. Houches/06//2006 Semi-classical Néel AFM H =  2 S i. S j Uniaxial magnet A soft mode at ( ,  )

18. Houches/06//2006 Néel ordererd AFMs H =  S i. S j H = J  S i. S j + K  (P ijkl + P -1 ijkl ) Uniaxial magnet A soft mode at ( ,  ) P ijkl i l k j P ij = 1/2 + 2 S i. S j = 1/2 + 2 (i,j) P ijkl + P -1 ijkl =  i,k  j,l  ] +2  i,j  j,k  k,l  l,i) ]  i,j  k,l  i,l  j,k  i,k) (j,l) ]

19. Houches/06//2006 A.Laeuchli, J.C. Domenge, P.Sindzingre, C.L., M.Troyer (PRL 05..) J4J4 J2J2

20. Houches/06//2006 H = J  S i. S j + K  (P ijkl + P -1 ijkl ) When K=1, J=-2: H  -   C 2 C : vector chirality C= S i x S j + S j x S k + S k x S l + S l x S i Both spin and chirality are classically ordered

21. Houches/06//2006 Low energy effective dynamics of a quantum top, scaling as S(S+1)/N biaxial magnet 3 soft modes at ( ,  ), (0,  ), ( , 0) 4-sublattice symmetry

22. Houches/06//2006 K=1, J=-2 H  -   C 2 C= S i x S j + S j x S k + S k x S l + S l x S i J Frustration ot the 4- sublattice order increases

23. Houches/06//2006 Low energy effective dynamics of a quantum top biaxial magnet 3 soft modes at ( ,  ), (0,  ), ( , 0) Low energy effective dynamics of a rigid rotator: uniaxial magnet 1 soft mode at ( ,  ),

24. Houches/06//2006 biaxial magnet Partial restoration of SU(2) symmetry by quantum fluctuations: uniaxial magnet Spin Nematic SBSB SDSD SCSC SASA C C

25. Houches/06//2006 Overcoming a further increase in frustration: Valence Bond Crystals Singlet: 1 fully optimized bond and absence of m-f interaction with other spins Staggered VBC Columnar VBC

26. Houches/06//2006 Overcoming a further increase in frustration: Valence Bond Crystals Singlet: 1 fully optimized bond and absence of m-f interaction with other spins Staggered VBC Columnar VBC

27. Houches/06//2006 More exotic phases? ? Critical phase with deconfined spinons excitations: Senthil et al Science 2004 ? n-nematic state Shannon et al. PRL 2006

28. Houches/06//2006 Valence Bond Crystals and Valence Bond Solids

29. Houches/06//2006 Valence-Bond Crystals Short range order in spin-spin correlations & L.R.O in dimer singlets or larger S=0 plaquettes!  J1-J4 model on the square lattice  J1-J2 model on the hexagonal lattice  Shastry-Sutherland model (SrCuBO)  Heisenberg on the 2D pyrochlore lattice

30. Houches/06//2006 Heisenberg on the 2D pyrochlore lattice Palmer et Chalker 02, Fouet et al. 03, Berg et al. 03 Brenig et al. The classical problem has a residual entropy per spin at T=0

31. Houches/06//2006 Heisenberg on the 2D pyrochlore lattice Gap to the first triplet Gaps in the singlet subspace Fouet et al. 2003

32. Houches/06//2006 Heisenberg on the 2D pyrochlore lattice

33. Houches/06//2006 Heisenberg on the 2D pyrochlore lattice

34. Houches/06//2006 Heisenberg on the 2D pyrochlore lattice

35. Houches/06//2006 Heisenberg on the 2D pyrochlore lattice Dispersion curves in the singlet subspace Dispersion curves in the triplet subspace

36. Houches/06//2006 Summary of the properties of VBC A spin gap and short range spin-spin correlations LRO order in singlet –singlet correlations Gapped excitations: single modes and continuums… all these excitations have integer spins (confinement of spinons) Up to now all known examples with spins are on bipartite lattices

37. Houches/06//2006 Valence-Bond Solids (A.K.L.T. 87) When individual spins obey: 2S=0 mod z A unique singlet g.s. w.-f. No lattice symmetry breaking A gap, short range spin-spin correlations but order in a non local order parameter? Integer spin excitations in the bulk Fractionalized degrees of freedom at the hedge of the sample A plausible 2D candidate with S=1

38. Houches/06//2006 The spin-1 Heisenberg antiferromagnet Hida J. Phys. Soc. Jpn 69, 4003, 2000 A Valence Bond Solid: 6 spin-1/2 components of the original spin-1 build a singlet state on each hexagon. A large spin gap & a large gap in the singlet sector Observation on m-MPYNN.BF4 N. Wada et al. J. Phys. Soc. Jpn 66, 1997

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