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/ C. Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, (2006). W. V. Liu and C. Wu, PRA 74, (2006). University of Maryland, 02/05/2007

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

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

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

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

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

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

10 Good timing: pioneering experiments; double-well lattice (NIST) and square lattice (Mainz). New direction: orbital physics in optical lattices Orbital physics: studying new physics of fermions and bosons in high-orbital bands. J. J. Sebby-Strabley, et al., PRA 73, (2006); T. Mueller and I. Bloch et al. 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.

11 Orbital physics Orbital band degeneracy and spatial anisotropy. cf. transition metal oxides (d- orbital bands with electrons). Charge and orbital ordering in La 1-x Sr 1+x MnO 4 Orbital: a degree of freedom independent of charge and spin. Tokura, et al., science 288, 462, (2000).

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

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

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

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

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

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

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

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

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

21 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. pxpx pypy

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

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

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

25 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? Open problems: exotic states in flat bands Interaction effects dominate due to the quenched kinetic energy; cf. fractional quantum Hall physics. Divergence of density of states.

26 Outline Bosons: novel superfluidity with time-reversal symmetry breaking. 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/ W. V. Liu and C. Wu, PRA 74, (2006); C. Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, (2006). New features of orbital physics in optical lattices. Introduction. Fermions: flat bands in honeycomb lattice.

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 On-site interaction in the p-band: orbital “Hund’s rule” (axial) are spatially more extended than (polar). “Ferro”-orbital interaction: L z is maximized. cf. Hund’s rule for electrons to occupy degenerate atomic shells: total spin is maximized. cf. p+ip pairing states of fermions: 3 He-A, Sr 2 RuO 4.

29 Band structure: 2D square lattice Anisotropic hopping and odd parity:  -bond  -bond Band minima: K x =( ,0), K y =(0,  ).

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

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

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

34 Stripe ordering throughout all the coupling regimes weak coupling Orbital configuration in each site: strong coupling cf. Strong coupling results also apply to the p+ip Josephson junction array systems ( e.g. Sr 2 RuO 4 ).

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

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

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, (2003); C. Wu, Phys. Rev. Lett. 95, (2005); S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, (2005); C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/ Review paper: C. Wu, Mod. Phys. Lett. B 20, 1707 (2006).

39 Phase stability analysis

40 P x,y -band structure in triangular lattices

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

42 Bond current

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

44 Double well  triangular lattice frustration:

45 Condensation occurs at

46