Novel orbital physics with cold atoms in optical lattices Congjun Wu Department of Physics, UC San Diego C. Wu, arXiv:0805.3525, to appear in PRL. Shizhong Zhang and C. Wu, arXiv:0805.3031. V. M. Stojanovic, C. Wu, W. V. Liu and S. Das Sarma, PRL 101, 125301(2008). C. Wu, PRL 100, 200406 (2008). C. Wu, and S. Das Sarma, PRB 77, 235107 (2008). C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 67004(2007). 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). Oct 22, 2008, UCLA. Collaborators L. Balents UCSB D. Bergman Yale S. Das Sarma Univ. of Maryland W. V. Liu Univ. of Pittsburg W. C. Lee, H. H. Hung UCSD I. Mondragon Shem J. Moore Berkeley Shi-zhong Zhang UIUC Many thanks to I. Bloch, L. M. Duan, T. L. Ho, Z. Nussinov, S. C. Zhang for helpful discussions. Outline • Introduction to orbital physics. New directions of cold atoms: orbital physics in high-orbital bands; pioneering experiments. • Bosons: exotic condensate, complex-superfluidity breaking time-reversal symmetry. • Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbative effects • Orbital exchange, frustrations, order from disorder, orbital liquid?. 3 Bose-Einstein condensation • Bosons in magnetic traps: dilute and weakly interacting systems. M. H. Anderson et al., Science 269, 198 (1995) TBEC ~ 1K n ~ 1014 cm 3 4 New era of cold atom physics: optical lattices • Strongly correlated systems. • Interaction effects are tunable by varying laser intensity. t U t : inter-site tunneling U: on-site interaction 5 Superfluid-Mott insulator transition Superfluid Mott insulator t<<U t>>U 87 Rb Greiner et al., Nature (2001). 6 Research focuses of cold atom physics • Great success of cold atom physics in the past decade: BEC, superfluid-Mott insulator transition, fermion superfluidity and BEC-BCS crossover … … • New focus: novel strong correlation phenomena which are NOT easily accessible in solid state systems. • New physics of bosons and fermions in high-orbital bands. Good timing: pioneering experiments; square lattice (Mainz); double well lattice (NIST); quasi 1D polariton lattice (stanford). J. J. Sebby-Strabley, et al., PRA 73, 33605 (2006); T. Mueller et al., Phys. Rev. Lett. 99, 200405 (2007); C. W. Lai et al., Nature 450, 529 (2007). Orbital physics • Orbital: a degree of freedom independent of charge and spin. Tokura, et al., science 288, 462, (2000). • Orbital band degeneracy and spatial anisotropy. • cf. transition metal oxides (d-orbital bands with electrons). La1-xSr1+xMnO4 LaOFeAs 8 Advantages of optical lattice orbital system • Solid state orbital systems: Jahn-Teller distortion quenches orbital degree of freedom; only fermions; correlation effects in p-orbitals are weak. • Optical lattices orbital systems: rigid lattice free of distortion; both bosons (meta-stable excited states with long life time) and fermions; strongly correlated px,y-orbitals: stronger anisotropy s-bond p-bond t // t 9 Pumping bosons by Raman transition • Long life-time: phase coherence. • Quasi-1d feature in the square lattice. py px T. Mueller, I. Bloch et al., Phys. Rev. Lett. 99, 200405 (2007). 10 Lattice polariton condensation in the p-orbital C. W. Lai et al, Nature 450, 529 (2007) • Quasi 1D polariton lattice by deposing metallic strips. • Condensates at both s-orbital (k=0) and p-orbital (k=p) states. Outline • Introduction. • Bosons: complex-superfluidity breaking time-reversal symmetry. New condensates different from Feynman’s argument of the positive-definitiveness of ground state wavefunctions. 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). Other group’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 . • Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbative effects • Orbital exchange, frustrations, order from disorder, orbital liquid?. 12 Feynman’s celebrated argument • The many-body ground state wavefunctions (WF) of boson systems in the coordinate-representation are positive-definite in the absence of rotation. (r1 , r2 ,...rn ) | (r1 , r2 ,...rn ) | (r1 , r2 ,...rn ) 0 2 n 2 2 H dr1...drn | ( r ,... r ) | | ( r ,.. r ) | i 1 n 1 n 2m i 1 | (r1 ,..rn ) |2 V i j int n U i 1 ex (ri ) (r i rj ) • Strong constraint: complex-valued WF positive definite WF; time-reversal symmetry cannot be broken. • Feynman’s statement applies to all of superfluid, Mottinsulating, super-solid, density-wave ground states, etc. New states of bosons beyond Feynman’s argument • How to bypass this argument to obtain complex-valued many-body wavefunctions of bosons and spontaneous breaking of time-reversal symmetry? Solution 1: Excited (meta-stable) states of bosons: bosons in higher orbital bands ---- orbital physics of bosons. Solution 2: Feynman’s argument does not apply to Hamiltonians linearly dependent on momentum. Ground states of spinful bosons with spin-orbit coupling (e.g. excitons). C. Wu and I. Mondragon-Shem, ariXiv:0809.3532. Ferro-orbital interaction for spinless p-orbital bosons • A single site problem: two orbitals px and py with two spinless bosons. V (r1 r2 ) g (r1 r2 ) x2 y 2 Lz 0 : U g dr | x (r ) |4 polar 1 ( px px p y p y ) 0 2 1 2 ( x iy ) 2 Lz 2 : { ( p x p x p y p y ) 1 ( px ip y ) 2 | 0 2 E0 43 U i px p y } 0 2 1 axial i 1 i E2 23 U E 2 E0 15 Orbital Hund’s rule for p-orbital spinless bosons H int U 2 1 z 2 2 { n r r 3 ( Lr ) } n p x p x p y p y , Lz i ( p x p y p y p x ) • Bosons go into one axial state to save repulsive interaction energy, thus maximizing orbital angular momentum. • Axial states (e.g. p+ip) are spatially more extended than polar states (e.g. px or py). • cf. Hund’s for electrons; p+ip superconductors. 1 1 1 polar i 1 axial i 16 “Complex” superfluidity with time-reversal symmetry breaking • Staggered ordering of orbital angular momentum moment in the square lattice. t 1 i i 1 1 i 1 i t // 1 i i i 1 1 1 i • Time of flight (zero temperature): Bragg peaks located at fractional values of reciprocal lattice vectors. p p W. V. Liu and C. Wu, PRA 74, (( m 12 ) , 0) (0, (n 12 ) ) 13607 (2006). a a 17 “Complex” superfluidity with time-reversal symmetry breaking • Stripe ordering of orbital angular momentum moment in the triangular lattice. i 1 1 i i 1 i 1 i 1 1 1 i i 1 1 i i i 1 i 1 i 1 i 1 i 1 • Each site behaves like a vortex with long range interaction in the superfluid state. Stripe ordering to minimize the global vorticity. C. Wu, W. V. Liu, J. Moore and S. Das Sarma, Phy. Rev. Lett. 97, 190406 (2006). 18 Strong coupling analysis r3 1 i • Ising variable for vortex vorticity: i 1 r1 i 1 r2 16 1 s 1 1 i 1 • The minimum of the effective flux per plaquette is 1 / 6 . i i i 1 1 2p 1 A ' r ,r 6 (s r1 s r 2 s r 3 ) r ,r ' 1 1 3 1 • The stripe pattern minimizes the ground state vorticity. 1 6 1 1 6 1 1 1 1 • cf. The same analysis also applies to p+ip Josephson junction array. 19 Staggered plaquette orbital moment 2p p 6 1 3 cf. d-density-wave state in high Tc cuprate: Sudip Chakravarty, R. B. Laughlin, Dirk K. Morr, and Chetan Nayak, Phys. Rev. B 63, 094503 (2001) 20 Digression: Exciton with Rashba SO coupling in 2D quantum well • Excitons: bound states of electrons and holes (heavy). Effective bosons at low densities below the Mott-limit. 1 Sz 2 jhh, z 3 2 • The SO coupling still survives in the center-of-mass motion. Let us consider the sector with fixed heavy hole spin. Kramer doublet: (e , h 2 ), (e , h 2 ) 3 R R me* / M , M me* mhh* 3 2 2 2 H EX d r { Vex (r )} 2M g R (i ys x i xs y ) 2 Ground state condensates: Kramer doublet • A harmonic trap: Vex (r ) 12 MT2 r 2 • SO length scale: 1 / k0 /( MR ) l /( MT ) • Half-quantum vortex and spin skyrmion texture configuration in the strong SO coupling limit. 1/ 2 g (r )ei f (r ) , 1/ 2 , i g ( r )e f (r ) jz L S 1 2 lk 0 4 Outline • Introduction. • Bosons: Complex-superfluidity beyond Feynman’s argument. • Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbative effects. Shizhong Zhang and C. Wu, arXiv:0805.3031. C. Wu, and S. Das Sarma, PRB 77, 235107 (2008). C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 67004(2007). • Orbital exchange, frustrations, order from disorder, orbital liquid?. 23 p-orbital fermions in honeycomb lattices cf. graphene: a surge of research interest; pz-orbital; Dirac cones. pxy-orbital: flat bands; interaction effects dominate. C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 70401 (2007). 24 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. • 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. 1/r-like potential 2p 2s 1s p s 25 Honeycomb optical lattice with phase stability • Three coherent laser beams polarizing in the z-direction. G. Grynberg et al., Phys. Rev. Lett. 70, 2249 (1993). 26 Artificial graphene in optical lattices • Band Hamiltonian (s-bonding) for spinpolarized fermions. H t t // [ p ( r ) p ( r 1 1 eˆ1 ) h.c.] ê2 ê1 B B A r A [ p (r ) p1 (r eˆ2 ) h.c] [ p3 (r ) p3 (r eˆ3 ) h.c] 2 p1 p2 3 2 3 2 p x 12 p y ê3 B p2 p1 p x 12 p y p3 p y p3 27 Flat bands in the entire Brillouin zone! • Flat band + Dirac cone. • localized eigenstates. • If p-bonding is included, the flat bands acquire small width at the order of t . Realistic band structures show t / t // 1% t // t p-bond 28 Realistic Band structure with the sinusoidal optical potential • Excellent band flatness. px,y-orbital bands s-orbital bands 2 2 H V cos( pi r ) 2m i 1~ 3 29 Hubbard model for spinless fermions: Exact solution: Wigner crystallization in Hubbard H int U n p x (r )n p y (r ) r A, B gapped state • Close-packed hexagons; avoiding repulsion. • The crystalline ordered state is stable even with small t . n 1 6 • Particle statistics is irrelevant. The result is also good for bosons. Orbital ordering with strong repulsions n 12 U / t// 10 • Various orbital ordering insulating states at commensurate fillings. • Dimerization at <n>=1/2! Each dimer is an entangled state of empty and occupied states. 31 Flat-band ferromagnetism (FM) • FM requires strong repulsion to overcome kinetic energy cost, and thus has no well-defined weak coupling picture. • Hubbard-type models cannot give FM unless with the flat band structure. However, flat band models suffer the stringent condition of fine-tuned long range hopping, thus are difficult to realize. A. Mielke and H. Tasaki, Comm. Mat. Phys 158, 341 (1993). • In spite of its importance, FM has not been paid much attention in cold atom community because strong repulsive interaction renders system unstable to the dimer-molecule formation. • We propose the realistic flat-band ferromagnetism in the porbital honeycomb lattices. • Interaction amplified by the divergence of DOS. Realization of FM with weak repulsive interactions in cold atom systems. Shizhong Zhang and C. Wu, arXiv:0805.3031; . 32 Outline • Introduction. • Bosons: new states of matter beyond Feynman’s argument of the positive-definitiveness of ground state wavefunctions. Complex-superfluidity breaking time-reversal symmetry. • Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbative effects • Orbital exchange and frustrations, order from disorder; orbital liquid? C. Wu, PRL 100, 200406 (2008). 33 Mott-insulators with orbital degrees of freedom: orbital exchange of spinless fermion • Pseudo-spin representation. 1 12 ( px px p y p y ) 2 12 ( px p y p y px ) 3 2i ( px p y p y px ) • No orbital-flip process. Exchange is antiferro-orbital Ising. J 0 J 0 H ex J 1 (r ) 1 (r xˆ ) J 2t 2 / U J 2t 2 / U 34 Hexagon lattice: quantum 120 model • For a bond along the general direction ê . px , py : eigen-states of eˆ2 cos 2 x sin 2 y H ex J ( (r ) eˆ2 )( (r eˆ ) eˆ2 ) ê2 px py • After a suitable transformation, the Ising quantization axes can be chosen just as the three bond orientation. B 1 3 1 3 ( 2 1 2 2 )( 2 1 2 2 ) ˆ H ex J ( (ri ) eij ) ( (rj) eˆij ) 1 1 r ,r A ( 12 1 23 2 )( 12 1 23 2 ) • cf. Kitaev model. B B H kitaev J ( x (r ) x (r e1 ) r y (r ) y (r e2 ) z (r ) z (r e3 )) Large S picture: heavy-degeneracy of classic ground states • Ground state constraint: the two -vectors have the same projection along the bond orientation. H ex J {[( (r ) (r )] eˆi }2 J z2 (r ) r ,r • Ferro-orbital configurations. or r • Oriented loop config: -vectors along the tangential directions. 36 Global rotation degree of freedom • Each loop config remains in the ground state manifold by a suitable arrangement of clockwise/anticlockwise rotation patterns. • Starting from an oriented loop config with fixed loop locations but an arbitrary chirality distribution, we arrive at the same unoriented loop config by performing rotations with angles of 30 , 90 , 150 . 37 “Order from disorder”: 1/S orbital-wave correction 38 Zero energy flat band orbital fluctuations • Each un-oriented loop has a local zero energy model up to the quadratic level. E 6 JS 2 ( ) 4 • The above config. contains the maximal number of loops, thus is selected by quantum fluctuations at the 1/S level. • Project under investigation: the quantum limit (s=1/2)? A very promising system to arrive at orbital liquid state? 39 Summary Orbital Hund’s rule and complex superfludity of spinless bosons px,y-orbital counterpart of graphene: flat band and Wigner crystalization. Orbital frustration. i 1 1 i i 1 1 i i i 1 1 1 i i i 1 i 1 i 1 1 1 i i 1 i 1 40 More work • Orbital analogue of anomalous quantum Hall effect – topological insulators in the p- band. C. Wu, arXiv:0805.3525, to appear in PRL. • Incommensurate superfluidity in the doublewell lattice of NIST. V. M. Stojanovic, C. Wu, W. V. Liu and S. Das Sarma, PRL 101, 125301 (2008). 41

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