Exploring New States of Matter in the p -orbital Bands of Optical Lattices

1. Exploring New States of Matter in the p-orbital Bands of Optical Lattices Congjun Wu Kavli Institute for Theoretical Physics, UCSB C. Wu, D. Bergman, L. Balents, and S. Das Sarma, cond-mat/0701788. C. Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, 190406 (2006). W. V. Liu and C. Wu, PRA 74, 13607 (2006). University of Maryland, 02/05/2007

2. Collaborators L. Balents UCSB D. Bergman UCSB S. Das Sarma Univ. of Maryland W. V. Liu Univ. of Pittsburg J. Moore Berkeley Many thanks to I. Bloch, L. M. Duan, T. L. Ho, T. Mueller, Z. Nussinov for very helpful discussions.

3. Outline • Introduction. - Rapid progress of cold atom physics in optical lattices. • New direction: orbital physics in high-orbital bands; pioneering experiments. • New features of orbital physics in optical lattices. Fermions: flat bands and crystallization in honeycomb lattice. Bosons: novel superfluidity with time-reversal symmetry breaking (square, triangular lattices).

4. M. H. Anderson et al., Science 269, 198 (1995) Bose-Einstein condensation • Bosons in magnetic traps: dilute and weakly interacting systems.

5. New era: optical lattices • New opportunity to study strongly correlated systems. • Interaction effects are tunable by varying laser intensity. t : inter-site tunneling U: on-site interaction

6. t<<U t>>U Superfluid-Mott insulator transition Mott insulator Superfluid Greiner et al., Nature (2001).

7. 1st order coherence disappears in the Mott-insulating state. Noise correlation (time of flight) in Mott-insulators • Noise correlation function oscillates at reciprocal lattice vectors; bunching effect of bosons. Folling et al., Nature 434, 481 (2005); Altman et al., PRA 70, 13603 (2004).

8. Two dimensional superfluid-Mott insulator transition I. B. Spielman et al., cond-mat/0606216.

9. Low density: metal high density: band insulator Fermionic atoms in optical lattices • Observation of Fermi surface. Esslinger et al., PRL 94:80403 (2005) • Quantum simulations to the Hubbard model. e.g. can 2D Hubbard model describe high Tc cuprates?

10. New direction: orbital physics in optical lattices • Great success of cold atom physics: BEC, superfluid-Mott insulator transition, fermion superfluidity and BEC-BCS crossover … … • Next focus: resolve NEW aspects of strong correlation phenomena which are NOT well understood in usual condensed matter systems. • Orbital physics: studying new physics of fermions and bosons in high-orbital bands. Good timing: pioneering experiments; double-well lattice (NIST) and square lattice (Mainz). J. J. Sebby-Strabley, et al., PRA 73, 33605 (2006); T. Mueller and I. Bloch et al.

11. Orbital physics • Orbital: a degree of freedom independent of charge and spin. • Orbital band degeneracy and spatial anisotropy. • cf. transition metal oxides (d-orbital bands with electrons). Charge and orbital ordering in La1-xSr1+xMnO4 Tokura, et al., science 288, 462, (2000).

12. s-bond p-bond New features of orbital physics in optical lattices • px,y-orbital physics using cold atoms. • Strong anisotropy. Fermions: flat band, novel orbital ordering … … Bosons: frustrated superfluidity with translational and time-reversal symmetry breaking … … • System preparation: fermions: s-band is fully-filled; p-orbital bands are active. bosons: pumping bosons from s to p-orbital bands.

13. Double-well optical lattices J. J. Sebby-Strabley, et al., PRA 73, 33605 (2006). • Laser beams of in-plane and out-of-plane polarizations. White spots=lattice sites. Note the difference in lattice period! Combining both polarizations • The potential barrier height and the tilt of the double well can be tuned.

14. Transfer bosons to the excited band Grow the long period lattice Avoid tunneling (diabatic) Create the excited state (adiabatic) Create the short period lattice (diabatic) • Band mapping. • Phase incoherence. M. Anderlini, et al., J. Phys. B 39, S199 (2006).

15. T. Mueller, I. Bloch et al. Ongoing experiment: pumping bosons by Raman transition • Long life-time: phase coherence. • Quasi-1d feature in the square lattice.

16. Outline • Introduction. Orbital physics: good timing for studying new physics of fermions and bosons in high-orbital bands. • New features of orbital physics in optical lattices. Fermions: flat bands and crystallization in honeycomb lattice. Bosons: novel superfluidity with time-reversal symmetry breaking (square, triangular lattices).

17. cf. graphene: a surge of research interest; pz-orbital; Dirac cones. p-orbital fermions in honeycomb lattices pxy-orbital: flat bands; interaction effects dominate. C. Wu, D. Bergman, L. Balents, and S. Das Sarma, cond-mat/0701788

18. 2p 2s 1s px, py orbital physics: why optical lattices? • pz-orbital band is not a good system for orbital physics. isotropic within 2D; non-degenerate. • Interesting orbital physics in the px, py-orbital bands. 1/r-like potential • However, in graphene, 2px and 2py are close to 2s, thus strong hybridization occurs. • In optical lattices, px and py-orbital bands are well separated from s. p s

19. B B A B Artificial graphene in optical lattices • Band Hamiltonian (s-bonding) for spin- polarized fermions.

20. localized eigenstates. • Flat band + Dirac cone. p-bond Flat bands in the entire Brillouin zone! • If p-bonding is included, the flat bands acquire small width at the order of .

21. py px Enhance interactions among polarized fermions • Hubbard-type interaction: • Problem: contact interaction vanishes for spinless fermions. • Use fermions with large magnetic moments. • Under strong 2D confinement, U is repulsive and can reach the order of recoil energy.

22. Exact solution with repulsive interactions! • Crystallization with only on-site interaction! • Closest packed hexagons; avoiding repulsion. • The crystalline order is stable even with if . • The result is also good for bosons.

23. Orbital ordering with strong repulsions • Various orbital ordering insulating states at commensurate fillings. • Dimerization at <n>=1/2! Each dimer is an entangled state of empty and occupied states.

24. in unit of Experimental detection • Transport: tilt the lattice and measure the excitation gap. • Noise correlations of the time of flight image. G: reciprocal lattice vector for the enlarged unit cells; ‘+’ for bosons, ‘-’ for fermions.

25. Open problems: exotic states in flat bands • Divergence of density of states. • Interaction effects dominate due to the quenched kinetic energy; cf. fractional quantum Hall physics. • A realistic system for flat band ferromagnetism (fermions with spin). • Pairing instability in flat bands. BEC-BCS crossover? Is there the BCS limit? • Bosons in flat-bands: highly frustrated system. Where to condense? Can they condense? Possible “Bose metal” phase?

26. Outline • Introduction. • New features of orbital physics in optical lattices. Fermions: flat bands in honeycomb lattice. Bosons: novel superfluidity with time-reversal symmetry breaking. W. V. Liu and C. Wu, PRA 74, 13607 (2006); C. Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, 190406 (2006). Other’s related work: V. W. Scarola et. al, PRL, 2005; A. Isacsson et. al., PRA 2005; A. B. Kuklov, PRL 97, 2006; C. Xu et al., cond-mat/0611620 .

27. Main results: superfluidity of bosons with time reversal symmetry breaking • On-site orbital angular momentum moment (OAM). • Square lattice: staggered OAM order. • Triangular lattice: stripe OAM order.

28. (axial) are spatially more extended than (polar). On-site interaction in the p-band: orbital “Hund’s rule” • “Ferro”-orbital interaction: Lz is maximized. • cf. Hund’s rule for electrons to occupy degenerate atomic shells: total spin is maximized. • cf. p+ip pairing states of fermions: 3He-A, Sr2RuO4.

29. Band structure: 2D square lattice • Anisotropic hopping and odd parity: s-bond p-bond • Band minima: Kx=(p,0), Ky=(0,p).

30. Superfluidity with time-reversal symmetry breaking • Interaction selects condensate as • Time-reversal symmetry breaking: staggered orbital angular momentum order. • Time of flight (zero temperature): 2D coherence peaks located at

31. T. Mueller, I. Bloch et al. Quasi-1D behavior at finite temperatures • Because , px-particles can maintain phase coherence within the same row, but loose phase inter-row coherence at finite temperatures. • Similar behavior also occurs for py-particles. • The system effectively becomes 1D- like as shown in the time of flight experiment. A. Isacsson et. al., PRA 72, 53604, 2005;

32. lowest energy states Band structure: triangular lattice CW, W. V. Liu, J. Moore, and S. Das Sarma, Phys. Rev. Lett. (2006).

33. Novel quantum stripe ordering • Interactions select the condensate as (weak coupling analysis). • Time-reversal, translational, rotational symmetries are broken. • cf. Charge stripe ordering in solid state systems with long range Coulomb interactions. (e.g. high Tc cuprates, quantum Hall systems).

34. Stripe ordering throughout all the coupling regimes • Orbital configuration in each site: weak coupling strong coupling • cf. Strong coupling results also apply to the p+ip Josephson junction array systems ( e.g. Sr2RuO4).

35. Time of flight signature • Stripe ordering even persists into Mott-insulating states without phase coherence. • Predicted time of flight density distribution for the stripe-ordered superfluid. • Coherence peaks occur at non-zero wavevectors.

36. Summary • Good timing to study orbital physics in optical lattices. • New features: novel orbital ordering in flat bands; • novel superfluidity breaking time reversal symmetry.

37. hole-like vortex • particle-like vortex (t/U=0.02) Strong coupling vortex configuration of in optical lattices Ref: C. Wu et al., Phys. Rev. A 69, 43609 (2004).

38. Hidden Symmetry and Quantum Phases in Spin 3/2 Cold Atomic Systems Congjun Wu Kavli Institute for Theoretical Physics, UCSB Ref: C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 91, 186402(2003); C. Wu, Phys. Rev. Lett. 95, 266404 (2005); S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, 214428 (2005); C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602. Review paper: C. Wu, Mod. Phys. Lett. B 20, 1707 (2006).

39. Phase stability analysis

40. Px,y-band structure in triangular lattices

41. the phase of the right lobe. direction of the Lz. Strong coupling analysis • Each site is characterized by a U(1) phase , and an Ising variable . • Inter-site Josephson coupling: effective vector potential. J. Moore and D. H. Lee, PRB, 2004.

42. Bond current

43. Strong coupling analysis • The minimum of the effective flux per plaquette is . • The stripe pattern minimizes the ground state vorticity. • cf. The same analysis also applies to p+ip Josephson junction array.

44. Double well  triangular lattice frustration:

45. Condensation occurs at