1. Correlated States in Optical Lattices Fei Zhou (PITP,UBC) Feb. 1, 2004 At Asian Center, UBC

2. Outline S=0 bosons (Recent experiments) S=1 bosons 1)Polar condensates 2)Mott states (Spin singlet and nematic Mott states) 3)Low dimensional Mott states (Dimerized valence bond crystals) 4)Spin singlet condensates or superfluid Approach: Constrained quantum rotor model ( O(2), O(3) rotors and a constraint on an inversion or an Ising constraint)

3. Optical lattices

4. S=1/2 Fermions in optical lattices (small hopping limit) Neel Ordered Gapless Spin liquid HTcS made of cold atoms?

5. S=0 bosons in lattices In (a) and (b), one boson per site. t is the hopping and can be varied by tuning laser intensities of optical lattices; U is an intra-site interaction energy. In a Mott state, all bosons are localized. M. P. A. Fisher et al., PRB 40, 546 (1989); On Mott states in a finite trap, see Jaksch et al., PRL. 81, 3108-3111(1998). U Mott states ( t << U) Condensates (t >>U)

6. 3D real space imagine of a condensate released from a lattice (TOF) Greiner et al., Nature 415, 41(2002)

7. Absorption images of interference patterns as the laser intensity is increased (from a to h). (a-d) BECs and (g-h) Mott insulating states.

8. Probing excitations In (c-f), w idth of interference peaks versus perturbations. (c-d) BECs and (e-f) Mott insulators

9.   S=1 bosons with Anti-ferromagnetic interactions Ho, 98; Ohmi & Machida, 98; Law,98.

10. |1,0> state as a triplet cooper pair (This is the Anderson-Morel state!) Pairs given here are not polarized, i.e.

11. Condensates of spin one bosons (d>1) N(Q) Q x y z   Snap shots

12. Half vortices In a half vortex, each atom makes a  spin rotation; a half vortex carries one half circulation of an integer vortex. A half vortex ring is also a hedgehog.  circulation y  spin rotation Z x y x The vortex is orientated along the z-direction; the spin rotation and circulating current occur in an x-y plane. Z ring

13. S=1 bosons with anti-ferromagnetic interactions in optical lattices ( 3D and 2D, N=2k) Polar BEC (a) Nematic MI (b) Spin Singlet MI (c) t: Hopping  he critical value of  is determined numerically.

14. Each site is characterized by two unit vectors, blue and red ones. a) Polar BECs (pBEC); b) Nematic mott insulators (NMI); c) Spin singlet mott insulators (SSMI). Ordering in states present in 3d,2d lattices

15. Schematic of microscopic wave functions a) NMI; b) SSMI (N=2k); c) SSMI (N=2k+1 in 1d). Each pair of blue and red dots with a ring is a spin singlet.

16. Constrained quantum rotor model valid as the hopping is much less than “U”.

17. Calculated Spectra for given N In a),b),  (n)*  (n ) (ground states) as a function of n. N=2k N=2k+1

18. Microscopic wave functions a) N=2k; b) N=2k+1 Each pair of blue and red dots is a spin singlet. S=2,4,6….(in a) or S=3,5,… (in b) spin excitations can be created by breaking singlets.

19. Numerics I: Large N=2k limit  SSMI NMI  vs.  (proportional to hopping) is plotted here. Blue and Green lines represent metal stable states close to the critical point.

20. The energy Vs.   SSMI NMI   10.85   

21. Phase diagrams in high dimensions  t NMI n=1 SSINMI n=3 NMI SSINMI BEC E(k,x) n=3 n=2 n=1 x n x 2 1 Mott states in a trap

22. Spin singlet quantum “condensates” in 1D optical lattices (SSQC) t DVBC SSQC(“e”) t SSMISSQC(“2e”) (a) (b) S=1, “Q=e” bosons with AF interactions ===> S=0, “Q=e” bosons interacting via Ising gauge fields N=2k+1 N=2k

23. A projected spin singlet Hilbert space In a projected spin singlet Hilbert space (states in a), b), d) and e)): i) An atom forms a singlet pair with another atom either at the same site or at a nearst neighboring site; ii) Each link is occupied by either one singlet pair of atoms or zero.

24. An effective Hamiltonian

25. Spin one bosons in optical lattices Work in progress Towards fault tolerant quantum information storage We have found 1) Half vortices in condensates 2)Nematic Mott insulators and spin singlet Mott insulators 3) Valence bond crystals (N=2k+1,1D) 4) Spin singlet condensates (1D)

26. References Zhou, Phys. Rev. Lett. 87, 080401(2001). Zhou, Int. Jour. Mod. Phys. B 17, 2643(2003). Demler and Zhou, Phys. Rev. Lett. 88,163001(2002). Snoek and Zhou, cond-mat/0306198 (to appear in PRB). Imambekov et al., PRA. 68, 063602 (2003). Zhou, Euro. Phys. Lett. 63 (4), 505(2003). Zhou and Snoek, Ann. Phys. 309(2), 692(2003). Constrained quantum rotor models  1) Quantum dimers, or compact gauge theories, 2) The bilinear-biqudartic spin model (S=1), 3) and fractionalization of atoms etc.

27. Hopping of S=1 Bosons a) and d) SSMI states for even and odd numbers of atoms. b) A kink-like S=0, Q=1 excitation in an “odd” lattice. e) A string-like Q=1 excitation in an “even” lattice. c) Hopping in an “odd” lattice leads to kink-anti kink excitations. f) In an “even” lattice hopping is suppressed because of a string of valence bonds between particle and hole excitations. Red dots are “charged”.

28. Excitations in 1D Mott insulators a)S=1, Q=0; b) S=0, Q=-1 for N=2k+1; c) S=0, Q=-2; d) Q=-1 for N=2k.