Novel orbital physics with cold atoms in optical lattices Congjun Wu Department of Physics, UC San Diego Wu, arXiv:0805.3525, to appear in PRL. izhong Zhang and C. Wu, arXiv:0805.3031. M. Stojanovic, C. Wu, W. V. Liu and S. Das Sarma, PRL 101, 125301(2008). Wu, PRL 100, 200406 (2008). Wu, and S. Das Sarma, PRB 77, 235107 (2008). Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 67004(2007). Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, 190406 (2006). V. Liu and C. Wu, PRA 74, 13607 (2006). Oct 22, 2008, UCLA.
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Novel orbital physics with cold atoms in optical lattices
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). - PowerPoint PPT Presentation
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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
Many thanks to I. Bloch, L. M. Duan, T. L. Ho, Z. Nussinov, S. C. Zhang for helpful discussions.
New directions of cold atoms: orbital physics in high-orbital bands; pioneering experiments.
• Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbative effects
• Orbital exchange, frustrations, order from disorder, orbital liquid?.
4
M. H. Anderson et al., Science 269, 198 (1995)
Bose-Einstein condensation
314cm10~1~ nKTBEC
• Bosons in magnetic traps: dilute and weakly interacting systems.
5
New era of cold atom physics: optical lattices
• Strongly correlated systems.
• Interaction effects are tunable by varying laser intensity.
U
tt : inter-site tunnelingU: on-site interaction
6
Superfluid-Mott insulator transition
Superfluid
Greiner et al., Nature (2001).
Rb87
Mott insulator
t<<Ut>>U
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).
8
Orbital physics
• Orbital band degeneracy and spatial anisotropy.
• cf. transition metal oxides (d-orbital bands with electrons).
La1-xSr1+xMnO4
• Orbital: a degree of freedom independent of charge and spin.Tokura, et al., science 288, 462,
(2000).
LaOFeAs
9
Advantages of optical lattice orbital system
tt//
• Optical lattices orbital systems:
rigid lattice free of distortion;
both bosons (meta-stable excited states with long life time) and fermions;
Jahn-Teller distortion quenches orbital degree of freedom;
only fermions;
correlation effects in p-orbitals are weak.
10
Pumping bosons by Raman transition
T. Mueller, I. Bloch et al., Phys. Rev. Lett. 99, 200405 (2007).
• Quasi-1d feature in the square lattice.
• Long life-time: phase coherence.
xpyp
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=) states.
12
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?.
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.
• 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?.
C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 70401 (2007).
cf. graphene: a surge of research interest; pz-orbital; Dirac cones.
25
px, py orbital physics: why optical lattices?
• pz-orbital band is not a good system for orbital physics.
• 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.
• Interesting orbital physics in the px, py-orbital bands.
isotropic within 2D; non-degenerate.
1s
2s
2p1/r-like potential
s
p
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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).
27
Artificial graphene in optical lattices
].)ˆ()([
].)ˆ()([
.].)ˆ()([
333
212
111//
cherprp
cherprp
cherprptHAr
t
• Band Hamiltonian (-bonding) for spin- polarized fermions.
1p2p
3p
yx ppp2
1
2
31
yx ppp2
1
2
32
ypp 3
1e2e
3e
A
B B
B
28
• If -bonding is included, the flat bands acquire small width at the order of . Realistic band structures show
Flat bands in the entire Brillouin zone!
• Flat band + Dirac cone.
• localized eigenstates.
t
-bond
tt//
%1/ // tt
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Realistic Band structure with the sinusoidal optical potential
3~1
22
)cos(2 i
i rpVm
H
s-orbital bands
px,y-orbital bands
• Excellent band flatness.
Hubbard model for spinless fermions: Exact solution: Wigner crystallization in
Hubbard
6
1n • Particle statistics is irrelevant. The result is also good for bosons.
• Close-packed hexagons; avoiding repulsion.
• The crystalline ordered state is stable even with small . t
gapped state
)()(,
int rnrnUHyx p
BArp
31
Orbital ordering with strong repulsions
10/ // tU
2
1n
• 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
Flat-band ferromagnetism (FM) • FM requires strong repulsion to overcome kinetic energy cost, and thus has no well-defined weak coupling picture.
• We propose the realistic flat-band ferromagnetism in the p-orbital honeycomb lattices.
• Interaction amplified by the divergence of DOS. Realization of FM with weak repulsive interactions in cold atom systems.
• 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.
• 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.
Shizhong Zhang and C. Wu, arXiv:0805.3031; .
A. Mielke and H. Tasaki, Comm. Mat. Phys 158, 341 (1993).
33
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).
34
Mott-insulators with orbital degrees of freedom: orbital exchange of spinless fermion
• Pseudo-spin representation.
)(2
11 yyxx pppp )(
2
12 xyyx pppp )(
23 xyyxi pppp
• No orbital-flip process. Exchange is antiferro-orbital Ising.
UtJ /2 2
)ˆ()( 11 xrrJH ex 0J
0J
UtJ /2 2
• For a bond along the general direction .
xp : eigen-states of yxe 2sin2cosˆ2
e
2e
)ˆ)ˆ()(ˆ)(( 22 eererJH ex
Hexagon lattice: quantum model 120
)ˆ)(()ˆ)((,
ijjijirr
ex ererJH
• After a suitable transformation, the Ising quantization axes can be chosen just as the three bond orientation.
A B
B
B
11 ))(( 22
312
122
312
1
))(( 22
312
122
312
1
yp
yx pp ,
• cf. Kitaev model.
))()()()(
)()((
32
1
errerr
errJH
zzyy
xxr
kitaev
36
Large S picture: heavy-degeneracy of classic ground states
• Ground state constraint: the two -vectors have the same projection along the bond orientation.
or r
zirr
ex rJerrJH )(}ˆ)]()({[( 22
,
• Ferro-orbital configurations.
• Oriented loop config: -vectors along the tangential directions.
37
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 .
150,90,30
38
“Order from disorder”: 1/S orbital-wave correction
39
Zero energy flat band orbital fluctuations
42 )(6 JSE
• Each un-oriented loop has a local zero energy model up to the quadratic level.
• 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?
40
Summary
1
1ii
1
1
ii
11
i
i
1
1ii
1
1
ii
11
i
i1 1
i
i
Orbital Hund’s rule and complex superfludity of spinless bosons
px,y-orbital counterpart of graphene: flat band and Wigner crystalization.
Orbital frustration.
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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 double-well lattice of NIST.V. M. Stojanovic, C. Wu, W. V. Liu and S. Das Sarma, PRL 101, 125301 (2008).