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Novel Orbital Phases of Fermions in p-band Optical Lattices Congjun Wu Department of Physics, UC San Diego Fermions: W. C. Lee, C. Wu, S. Das Sarma, to be submitted. C. Wu, PRL 101, 168807 (2008). C. Wu, PRL 100, 200406 (2008). C. Wu, and S. Das Sarma, PRB 77, 235107 (2008). S. Z. Zhang , H. H. Hung, and C. Wu, arXiv:0805.3031. C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 67004(2007). Bosons: C. Wu, Mod. Phys. Lett. 23, 1(2009). V. M. Stojanovic, C. Wu, W. V. Liu and S. Das Sarma, PRL 101, 125301(2008). 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). March 20, 2009, Pittsburgh 1 Collaborators: S. Das Sarma (UMD) H. H. Hung, W. C. Lee (UCSD) L. Balents, D. Bergman (UCSB) Shizhong Zhang (UIUC) Thank W. V. Liu, J. Moore, V. Stojanovic for collaboration on orbital physics with bosons. Also I. Bloch, L. M. Duan, T. L. Ho, Z. Nussinov, S. C. Zhang for helpful discussions. Supported by NSF, and Sloan Research Foundation. 2 Outline • Introduction to orbital physics: a new direction of orbital physics. • Bosons: unconventional BEC beyond no-node theorem. c.f. V. W. Liu’s talk. • Fermions in hexagonal lattice: px,y-orbital counterpart of graphene. 1. Flat band structure and non-perturbative effects. 2. Mott insulators: orbital exchange; a new type of frustrated magnet-like model; a cousin of the Kitaev model. 3. Novel pairing state: f-wave Cooper pairing. 4. Topological insulators – quantum anomalous Hall effect by orbital angular momentum polarization. c.f. C. Wu’s talk on March 19. 3 Research focuses of cold atom physics • Great success of cold atom physics in the past decade: BEC; superfluid-Mott insulator transition; Multi-component bosons and fermions; fermion superfluidity and BEC-BCS crossover; polar molecules … … • Orbital Physics: new physics of bosons and fermions in high-orbital bands. Good timing: pioneering experiments on orbital-bosons. Square lattice (Mainz); double well lattice (NIST). 4 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 degeneracy and spatial anisotropy. • cf. transition metal oxides (d-orbital bands with electrons). La1-xSr1+xMnO4 LaOFeAs 5 Advantages of optical lattice orbital systems • Solid state orbital systems: • Optical lattices orbital systems: Jahn-Teller distortion quenches orbital degree of freedom; rigid lattice free of distortion; only fermions; both bosons (meta-stable excited states with long life time) and fermions; correlation effects in p-orbitals are weak. strongly correlated px,y-orbitals: stronger anisotropy t // t s-bond p-bond 6 Novel state of orbital bosons beyond the Nonode theorem (c.f. Vincent Liu’s talk) • Complex condensate wavefunctions; spontaneous time reversal symmetry breaking. • Orbital Hund’s rule interaction; ordering of orbital angular momentum moments. W. V. Liu and C. Wu, PRA 74, 13607 (2006); C. Wu, Mod. Phys. Lett. 23, 1 (2009). C. Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, 190406 (2006). t p x ip y 1 i i 1 1 i p x ip y i 1 i i 1 1 i 1 p x ip y p x ip y i 1 i i i 1 1 1 i i i 1 p x ip y i t // 1 1 1 i i 1 i 1 i 1 1 1 i i 1 i 1 p x ip y p x ip y Other group’s related work: Ofir Alon et al, PRL 95 2005. 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 . 7 P-orbital fermions: px,y-orbital counterpart of graphene • Band flatness and strong correlation effect. (e.g. Wigner crystal, and flat band ferromagnetism.) C. Wu, and S. Das Sarma, PRB 77, 235107(2008); C. Wu et al, PRL 99, 67004(2007). Shizhong zhang, Hsiang-hsuan Hung, and C. Wu, arXiv:0805.3031. • P-orbital Mott insulators: orbital exchange; from Kitaev to quantum 120 degree model. C. Wu, PRL 100, 200406 (2008); C. Wu et al, arxiv0701711v1; E. Zhao, and W. V. Liu, Phys. Rev. Lett. 100, 160403 (2008) • Novel pairing state: f-wave Cooper pairing. W. C. Lee, C. Wu, S. Das Sarma, to be submitted. 8 p-orbital fermions in honeycomb lattices cf. graphene: a surge of research interest; pz-orbital; Dirac cones. Px, y-orbital: flat bands; interaction effects dominate. C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 70401 (2007). 9 What is the fundamental difference from graphene? • pz-orbital band is not a good system for orbital physics. • It is the other two px and py orbitals that exhibit anisotropy and degeneracy. • 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 10 Honeycomb optical lattice with phase stability • Three coherent laser beams polarizing in the z-direction. • Laser phase drift only results an overall lattice translation without distorting the internal lattice structure. G. Grynberg et al., Phys. Rev. Lett. 70, 2249 (1993). 11 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 12 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 13 Hubbard model for spinless fermions: Exact solution: Wigner crystallization 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 state is also good for bosons, and even Bose-Fermi mixtures. Spinful Fermions: flat-band itinerant ferromagnetism • Ferromagnetism (FM) requires strong repulsive interactions , and thus has no well-defined weak coupling picture. • It is accepted that it is difficult to achieve FM state conclusively in Hubbard type modes except with flat band and Nagaoka limit. A. Mielke and H. Tasaki, Comm. Math. Phys 158, 341 (1993). • In spite of its importance, FM has not been paid much attention in the cold atom community because strong repulsive interaction renders system unstable to the formation of dimers. • Flat-band FM in the p-orbital honeycomb lattices. • Interaction amplified by the divergence of DOS. Realization of FM with weak repulsive interactions in cold atom systems. Shizhong Zhang, Hsiang-hsuan Hung, and C. Wu, arXiv:0805.3031. 15 Flat-band itinerant FM in p-orbitals • Exact result in the homogenous system: magnetization with the filling inside the flat band, i.e., n 0 .5 . • More realistic system: soft harmonic trap; particle numbers of spin up and down particles are separately conserved. 0.005ER , l 4.5a, N N 255, t // U 0.24ER • Self-consistent Bogoliubov-de Gennes calculation: FM phase separation. 16 Px,y-orbital counterpart of graphene • Band flatness and strong correlation effect. (e.g. Wigner crystal, and flat band ferromagnetism.) C. Wu, and S. Das Sarma, PRB 77, 235107(2008); C. Wu et al, PRL 99, 67004(2007). Shizhong zhang and C. Wu, arXiv:0805.3031. • P-orbital Mott insulators: orbital exchange; a new type of frustrated magnet-like model; a cousin of Kitaev model. C. Wu, PRL 100, 200406 (2008); C. Wu et al, arxiv0701711v1; E. Zhao, and W. V. Liu, Phys. Rev. Lett. 100, 160403 (2008) • Novel pairing states: f-wave pairing. W. C. Lee, C. Wu, S. Das Sarma, to be submitted. 17 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. Antiferro-orbital Ising exchange. J 0 J 0 H ex J 1 (r ) 1 (r xˆ ) J 2t 2 / U J 2t 2 / U 18 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 ê2 px H ex J ( (r ) eˆ2 )( (r eˆ ) eˆ2 ) py • After a suitable transformation, the Ising quantization axes can be chosen just as the three bond orientations. B ( 12 1 23 2 )( 12 1 23 2 ) 1 1 A ( )( ) 1 2 1 3 2 2 1 2 1 3 2 2 B B H ex J ( (ri ) eˆij ) ( (rj) eˆij ) r ,r C. Wu et al, arxiv0701711v1; C. Wu, PRL 100, 200406 (2008). E. Zhao, and W. V. Liu, Phys. Rev. Lett. 100, 160403 (2008) 19 From the Kitaev model to 120 degree model • cf. Kitaev model: Ising quantization axes form an orthogonal triad. B s ys y H kitaev J (s x (r )s x (r e1 ) s y (r )s y (r e2 ) sx sx rA s z (r )s z (r e3 )) s zs z A 1 2 B B 0 cos 120 Kitaev 20 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 )] eˆrr }2 J z2 (r ) r ,r • Ferro-orbital configurations. or r • Oriented loop config: -vectors along the tangential directions. 21 Heavy-degeneracy of classic ground states • General loop configurations 22 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 . 23 “Order from disorder”: 1/S orbital-wave correction 24 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? 25 Px,y-orbital counterpart of graphene • Band flatness and strong correlation effect. (e.g. Wigner crystal, and flat band ferromagnetism.) C. Wu, and S. Das Sarma, PRB 77, 235107(2008); C. Wu et al, PRL 99, 67004(2007). Shizhong zhang and C. Wu, arXiv:0805.3031. • P-orbital Mott insulators: orbital exchange; from Kitaev to quantum 120 degree model. C. Wu, PRL 100, 200406 (2008). • Novel pairing state: f-wave pairing. W. C. Lee, C. Wu, S. Das Sarma, to be submitted. 26 UNCONVENTIONAL Cooper pairing from TRIVIAL interactions • Most of unconventional pairing states arise from strong correlation effects, and thus are difficult to predict and analyze. p-wave: superfluid 3He-A and B; Sr2RuO4; d-wave: high Tc cuprates; Extended s-wave: Iron-based superconductors (?); Possible f-wave: UPt3(?) • Can we arrive at unconventional pairing in a much easier way, say, from trivial interactions but with nontrivial band structures? - k + + k d x2 y2 27 Nontrivial orbital hybridization: p-orbital hexagonal lattice p x ip y p x ip y k K K k p x ip y • Along the three middle lines of Brillouin zone, eigen-orbitals are real. p x ip y K K 28 Onsite Hubbard interaction for SPINLESS p-orbital fermions • Attraction between fermions in two orthogonal orbitals. U H int U n p (r )n p (r ) r x y • F-wave structure intra-band pairing. Pairing strength vanishes along three middle lines, and becomes strongest at K and K’. F (k ) sin 23 k x (cos 23 k x cos 32 k y ) K p x ip y nodal lines K p x ip y 29 Zero energy Andreev bound states • If the boundary is perpendicular to the anti-nodal (nodal) direction, the zero energy Andreev bound states appear (vanish). kout With Andreev Bound States k in kout k in No Andreev Bound States 30 A new research direction of cold atoms in optical lattices • Band flatness and strong correlation: Wigner crystal and ferromagnetism; • Orbital exchange: a new type of frustrated magnet-like mode; a cousin of the Kitaev model; • f-wave Cooper pairing; • Topological band insulator: quantum anomalous effect. c. f. C. Wu’s talk on March 19. 31 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. 32 Gap value and superfluid density 33 Flat-band itinerant FM in p-orbitals • Percolation picture for flat band FM. H int U ( n ( r ) n ( r ) n ( r ) n ( r )) p p , p , p , x , r x y y 1 J ( S p (r ) S p (r ) n p (r )n p (r )) r 4 ( p x, (r ) p x, (r ) p y , (r ) p y , (r ) h.c.) x y x y r • Self-consistent calculation for the FM phase separation with a soft harmonic trap. 34