novel orbital phases of fermions in p-band optical lattices congjun wu department of physics, uc san...
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Novel Orbital Phases of Fermionsin p-band Optical Lattices
Congjun Wu Department of Physics, UC San Diego
Feb. 23, 2009, KITP
Collaborators: L. Balents, D. Bergman, S. Das Sarma, H. H. Hung, W. C. Lee, S. Z. Zhang; W. V. Liu, V. Stojanovic.
W. C. Lee, C. Wu, S. Das Sarma, in preparation.C. Wu, PRL 101, 168807 (2008).C. Wu, PRL 100, 200406 (2008). C. Wu, and S. Das Sarma, PRB 77, 235107 (2008).S. Zhang , H. H. Hung, and C. Wu, arXiv:0805.3031. C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 67004(2007).
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Thanks to: I. Bloch, L. M. Duan, T. L. Ho, Z. Nussinov, S. C. Zhang.
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Outline
• Introduction to orbital physics.
• Bosons: meta-stable states with long life time; complex-condensations beyond the no-node theorem.
New directions of cold atoms: bosons and fermions in high-orbital bands; pioneering experiments.
• Fermions: px,y-orbital counterpart of graphene.
1. Flat band structure and non-perturbative effects.
3. p-orbital Mott insulators: orbital exchange; a new type of frustrated magnet-like model; a cousin of the Kitaev model.
2. p-orbital band insulators (topological) – Haldane’s model induced by orbital angular momentum polarization.
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); 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).
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Orbital physics
• Orbital 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
• Solid state orbital systems:
Jahn-Teller distortion quenches orbital degree of freedom;
only fermions;
correlation effects in p-orbitals are weak.
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Advantages of optical lattice orbital systems
tt//
• 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
-bond -bond
Bosons: Feynman’s no-node theorem • The many-body ground state wavefunctions (WF) of bosons in the coordinate-representation are positive-definite in the absence of rotation.
0),...,( 21 nrrr),...,( 21 nrrr |),...,(| 21 nrrr
• Strong constraint: complex-valued WF positive definite WF; time-reversal symmetry cannot be spontanously broken.
)(|),..(|
)(|),..(||),...(|2
...
int2
1
1
21
21
1
2
1
jiji
n
n
iiexnni
n
in
rrVrr
rUrrrrm
drdrH
• Feynman’s statement applies to all of superfluid, Mott-insulating, super-solid, density-wave ground states, etc.
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No-node theorem doesn’t apply to orbital bosons
W. V. Liu and C. Wu, PRA 74, 13607 (2006);C. Wu, Mod. Phys. Lett. 23, 1 (2009).
• Complex condensate wavefunctions; spontaneous time reversal symmetry breaking.
1
1ii
1
1
ii
11
i
i
1
1ii
1
1
ii
11
i
i1 1
i
i
i
i11
i11
i
11i
i11
i
i
//t
t
yx ipp yx ipp
yx ipp yx ipp
C. Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, 190406 (2006).
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 .
yx ipp
yx ipp
yx ipp
• Orbital Hund’s rule interaction; ordering of orbital angular momentum moments.
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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.
C. Wu, PRL 101, 186807 (2008).
• P-orbital band insulators (topological) – Haldane’s model induced by orbital angular momentum polarization.
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pxy-orbital: flat bands; interaction effects dominate.
p-orbital fermions in honeycomb lattices
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.
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What is the fundamental difference from graphene?
• 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.
• It is the other two px and py orbitals that exhibit anisotropy and degeneracy.
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).
• Laser phase drift only results an overall lattice translation without distorting the internal lattice structure.
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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
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• 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
Hubbard model for spinless fermions: Exact solution: Wigner crystallization
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1n • Particle statistics is irrelevant. The result is also good for bosons, and Bose-Fermi mixtures.
• Close-packed hexagons; avoiding repulsion.
• The crystalline ordered state is stable even with small . t
gapped state
)()(,
int rnrnUHyx p
BArp
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Flat-band itinerant ferromagnetism (FM)
• FM requires strong enough repulsion and thus FM has no well-defined weak coupling picture.
• 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.
• It is commonly accepted that Hubbard-type models cannot give FM unless with flat band structure.
• 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 dimer-molecule formation.
Shizhong Zhang and C. Wu, arXiv:0805.3031.
A. Mielke and H. Tasaki, Comm. Math. Phys 158, 341 (1993).
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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.
• Orbital exchange; from Kitaev to quantum 120 degree model.
C. Wu, PRL 100, 200406 (2008.
C. Wu, PRL 101, 186807 (2008).
• P-orbital band insulators (topological) – Haldane’s model induced by orbital angular momentum polarization.
Topological insulators: Haldane’s QHE Model without Landau level
..)()({ chrcrctH BAr
ANN
• Honeycomb lattice with complex-valued next-nearest neighbor hopping.
A
B
.}.
)()()()({
ch
rcrcercrcetH BBi
rAA
iNNN
• Topological insulator at . Mass changes sign at K1,2.
,0
Ikc
kmkbkakH
)(
)()()()( 231
1K 2K
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Rotate each site around its own center
N. Gemelke, Ph. D. thesis, Stanford
University, 2007.
• Phase modulation on laser beams: a fast overall oscillation of the lattice. Atoms cannot follow and feel a slightly distorted averaged potential.• The oscillation axis slowly precesses at the angular frequency of .)()()()({
)(
rprprprpi
rLH
xyyAr
x
Arzzmn
• Orbital Zeeman term.
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6
i
e
6
5i
e
Large rotation angular velocity
2/)( 23/2 itet
• Second order perturbation process generates the complex NNN hopping.
//t
ipp
ipp
ipp
ipp
ipp
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Small rotation angular velocity 0
//3.0 t
1C
0C
Berry curvature.
1
0
0
1C
20
21
Large rotation angular velocity
//2
3 t//3t
1C
1CBerry curvature.
1C
11
121
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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.
C. Wu, PRL 101, 186807 (2008).
• P-orbital band insulators (topological) – Haldane’s model induced by orbital angular momentum polarization.
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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. Antiferro-orbital Ising exchange.
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 orientations.
A B
B
B
11 ))(( 22
312
122
312
1
))(( 22
312
122
312
1
ypyx pp ,
From the Kitaev model to 120 degree model
A B
B
B
xx yy
zz
• cf. Kitaev model: Ising quantization axes form an orthogonal triad.
))()(
)()()()((
3
21
err
errerrJH
zz
yyxxAr
kitaev
Kitaev
cos02
1
120
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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
zrrrr
ex rJerrJH )(}ˆ)]()({[( 22
,
• Ferro-orbital configurations.
• Oriented loop config: -vectors along the tangential directions.
Heavy-degeneracy of classic ground states
• General loop configurations
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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
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“Order from disorder”: 1/S orbital-wave correction
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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?
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Summary
px,y-orbital counterpart of graphene: strong correlation from band flatness.
Topological insulator: quantum anomalous Hall effect.
orbital exchange: frustration, quantum 120 degree model.
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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.