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Novel orbital physics with cold atoms in optical lattices

Congjun Wu

Department of Physics, UCSD

C. Wu, arxiv:0801.8888. C. Wu, and S. Das Sarma, arxiv:0712.4284.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).

April 17, UC Davis

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Stop

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Media

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Disto

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Shame

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On CNN!

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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 Yale

W. V. Liu Univ. of Pittsburg

S. Das Sarma Univ. of Maryland

J. Moore Berkeley

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Outline

• Introduction.

• Bosons: complex-superfluidity breaking time-reversal symmetry.

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.

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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

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Good timing: pioneering experiments; double-well lattice (NIST) and square lattice (Mainz).

Orbital physics with cold atoms

• New physics of bosons and fermions in high-orbital bands.

J. J. Sebby-Strabley, et al., PRA 73, 33605 (2006); T. Mueller and I. Bloch et al., PRL (2007).

• Great success of cold atom physics:

BEC, superfluid-Mott insulator transition, fermion superfluidity and BEC-BCS crossover … …

• New focus: novel strong correlation phenomena which are NOT accessible in usual solid state systems.

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Orbital physics

• Orbital band degeneracy and spatial anisotropy.

• cf. transition metal oxides (d-orbital bands with electrons).

Charge and orbital ordering in La1-xSr1+xMnO4

• Orbital: a degree of freedom independent of charge and spin.

Tokura, et al., science 288, 462, (2000).

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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;

strongly correlated px,y-orbitals: stronger anisotropy

-bond -bond

• 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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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, 33605 (2006).

xyI zI

• The potential barrier height and the tilt of the double well can be tuned.

• Laser beams of in-plane and out-of-plane polarizations.

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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)

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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

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Outline

• Introduction.

• Bosons: complex-superfluidity breaking time-reversal symmetry. New states of matter beyond 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, frustration, order from disorder.

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Feynman’s argument

• The many-body ground state wavefunctions of boson systems with short range interactions are positive-definite in the absence of rotation.

• It does NOT apply to excited states! Possibility of complex-valued many-body wavefunctions.

• Strong constraint: complex-valued WF positive definite WF; time-reversal cannot be broken.• It applies to all of superfluid, Mott-insulating, super-solid, density-wave states, etc.

),...,( 21 nrrrψ

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Ferro-orbital interaction for spinless p-orbital bosons

4|)(| rdrgU xφ∫=

• Two bosons on the same site: three states.

⟩±=

±−±=

++

++++++

0|)(2

1

0}2

)(2

1{:2

2yx

yxyyxxz

pp

ppi

ppppL

0)(2

1:0 ++++ += yyxxz ppppL

polar

+ UE3

40 =

11−i

i−

axial

UE3

22 =±

)()( 2121 rrgrrV −=− δ

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Orbital Hund’s rule for p-orbital bosons (spinless)

11−i

i−

11− polar

axial

• 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.

• Bosons go into one axial state to save repulsive interaction energy, thus maximizing orbital angular momentum.

)(,

})(3

1{

222

int

xyyxzyyxx

zr

rr

ppppiLppppn

LnU

H

++++ −−=+=

−= ∑

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Band structure: 2D square lattice• Anisotropic hopping and odd parity:

}..)ˆ()({

}..)ˆ()({//

yxcherprpt

yxcherprptH

xyr

y

xxr

xt

↔+++−

→+++=

∑

∑+

⊥

+

rr

rr

r

r

⊥>> tt//

-bond -bond

• Band minima: Kx=(,0), Ky=(0,).

yxyxx ktktkk coscos),( // ⊥−=ε

yxyxy ktktkk coscos),( //+−= ⊥ε

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“Complex” superfluidity with time-reversal symmetry breaking

• Interaction selects condensate as

0)}(2

1{

!

10

0

NKyKxG

iN

++ += ψψψ

i

i−11−

i1−1

i−

1−1i−

i11−

i−

i

• Time of flight (zero temperature): 2D coherence peaks located at )0,)((

2

1

am

+ ))(,0(

2

1

an

+

//t

⊥t

• Strong coupling analysis: inter-site Josephson coupling fixes the AF- orbital angular momentum order.

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Quasi-1D behavior at finite temperatures

• Because , px-particles can maintain phase coherence within the same row, but loose inter-row phase coherence at finite temperatures.

T. Mueller, I. Bloch et al.

//tt <<⊥

A. Isacsson et. al., PRA 72, 53604 (2005).

• Similar behavior also occurs for py-particles.

• The system effectively becomes 1D-like as shown in the time of flight experiment.

xp

yp

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10ea

1Kr

2Kr

3Kr )0,

3

4(

0a

20ea

30ea

CW, W. V. Liu, J. Moore, and S. Das Sarma, Phys. Rev. Lett. (2006)..}.)ˆ()({// cherprptH ii

rit ++= ∑ + rr

r

Band structure: triangular lattice

lowest energy states

3,2,1Kr

1Krψ 2K

rψ3Krψ

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Weak coupling: quantum stripe ordering

1

1−ii−

1

1−ii−

1−

1i−i

1−

1i−i

1−1i−

i1−1

i−

i1− 1

i−i

• Interactions select the condensate as (weak coupling analysis).

2Krψ

3Krψ

i+

0)}(2

1{

!

10

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0

NKK i

N++ + ψψ

• cf. Charge stripe ordering in solid state systems with long range Coulomb interactions. (e.g. high Tc cuprates, quantum Hall systems).

• Time-reversal, translational, rotational symmetries are broken.

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Stripe ordering throughout all the coupling regimes

)sin(cos yrxi pipe r ααφ +

1

1−ii−

1−

1

i−i

1−1

i−

i

1

1−ii−

1−

1

i−i

1−1

i−

i1− 1

i−

i

• Strong coupling analysis: p-wave Josephson junction array. Stripe ordering to minimize the global vorticity.

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α =

• Intermediate coupling: Gutzwiller mean-field analysis also shows stripe ordering.

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Staggered plaquette orbital moment

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θ =Δ

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)  

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• 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.

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Outline

• Introduction.

• Bosons: Complex-superfluidity beyond Feynman’s argument.

• Fermions: px,y-orbital counterpart of graphene, flat bands and non-perturbation effects.

C. Wu, arxiv:0810.8888; C. Wu, and S. Das Sarma, arxiv:0712.4284.C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99, 67004(2007).

• Orbital exchange and frustration.

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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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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).

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Artificial graphene in optical lattices

{

}].)ˆ()([

].)ˆ()([

.].)ˆ()([

333

212

111//

cherprp

cherprp

cherprptHAr

t

+++

+++

++=

+

+

∈

+∑

rr

rr

rrr

• 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 .

Flat bands in the entire Brillouin zone!

• Flat band + Dirac cone.

• localized eigenstates.

⊥t-bond

⊥>> tt//

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Realistic Band structure with the sinusoidal optical potential

∑=

⋅+∇

−=3~1

22

)cos(2 i

i rpVm

Hrrh

s-orbital bands

px,y-orbital bands

• Excellent band flatness.

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Enhance interactions among polarized fermions

)()(,

int rnrnUHyx p

BArp

rrr∑∈

=• Hubbard-type interaction:

• Problem: contact interaction vanishes for spinless fermions.

)63,(SCr 53Bμμ ==• Use fermions with large

magnetic moments.

• Under strong 2D confinement, U is repulsive and can reach the order of recoil energy. )( 212,1 xxS

rrr−⊥

Br

px

py

1xr

2xr

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Exact solution: Wigner crystallization

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1=n • The result is also good for bosons.

• Close-packed hexagons; avoiding repulsion.

• The crystalline ordered state is stable even with small . ⊥t

gapped state

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Orbital ordering with strong repulsions

10/ // =tU

2

1=n

• Various orbital ordering insulating states at commensurate fillings.

• Dimerization at <n>=1/2! Each dimer is an entangled state of empty and occupied states.

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Experimental detection

)()()()(),( 212121 knknknknkkCrrrrrr

−=

• Noise correlations of the time of flight image.

G: reciprocal lattice vector for the enlarged unit cells; ‘+’ for bosons, ‘-’ for fermions.

∑∫ −±∝−+

−+=

Gqq

qq

Gdknkn

kkCkdqC )(

)()(

),()(

22

22rr

rr

rrrr

rr

rr

δ

in unit of a3/2

• Work in progress: flat-band ferromagnetism in p-orbital honeycomb lattices. A realistic system to investigate itinerant ferromagnetism with cold atoms.

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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, frustration, order from disorder; orbital liquid?

C. Wu, arxiv:0801.8888.

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Mott-insulators with orbital degree of freedom: orbital exchange (spinless fermion)

• Pseudo-spin representation.

)(2

11 yyxx pppp ++ −=τ )(

2

12 xyyx pppp ++ +=τ )(

23 xyyxi pppp ++ −=τ

• No orbital-flip process. Exchange is AF-Ising-like. UtJ /2 2=

• For a bond along the general direction .

yxyyxx pppppp ϕϕϕϕ cossin,sincos +−=′+=′

)ˆ()( 11 xrrJH ex += ττ

ϕeigen-state of

yxe τϕτϕτ ϕ 2sin2cosˆ2 +=⋅r

ϕe

ϕτ 2e⋅r

)ˆ)ˆ()(ˆ)(( 22 ϕϕϕ ττ eererJHex ⋅+⋅=rr

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Hexagon lattice: model

2

,,

}ˆ)]()({[()ˆ)()(ˆ)(( irr

iirr

exex errJererJHH ⋅′−−=⋅′⋅−=⇒ ∑∑′′

ττττ rrrr

• Honeycomb lattice: at sub-lattice A or B, rotate p-orbitals 180 degree around x-axis or xy-diagonal, respectively.

o120

• Non-frustrated lattice: square, triangular, and Kagome lattices.

• cf. Kitaev model, .

zyxei ˆ,ˆ,ˆˆ =

3232

2323

1111

ˆ)(ˆ)(,ˆ)(ˆ)(

ˆ)(ˆ)(,ˆ)(ˆ)(

ˆ)(ˆ)(,ˆ)(ˆ)(

erererer

erererer

erererer

BBAA

BBAA

BBAA

⋅−→⋅⋅→⋅⋅−→⋅⋅→⋅⋅−→⋅⋅→⋅

ττττττττττττ

rrrr

rrrr

rrrr

)ˆ)ˆ()(ˆ)(( 22 ϕϕϕ ττ eererJHex ⋅+⋅=rr

1e

3e

2e

A B

B

B

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Frustration in the honeycomb lattice: model and loop rep.

• Large S picture: for each bond, the two τ-vector projections along that bond are the same.

o120

or∑∑ +⋅′−−=′ r

zirr

ex rJerrJH )(}ˆ)]()({[( 22

,

τττrr

• Ferromagnetic configurations.

• Oriented loop config: τ-vectors along the tangential directions.

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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 .

ooo 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 θΔ=Δ 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.

• Open questions: the quantum limit (s=1/2)? A very promising system to arrive at orbital liquid state?

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Summary

1

1−ii−

1−

1

i−i

1−1

i−

i

1

1−ii−

1−

1

i−i

1−1

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 and loop representation.

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• 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 band. Where to condense?

frustrated superfluidity (c.f. frustrated magnets).

On-going projects: exotic states in flat bands

• Divergence of density of states. Interaction effects dominate due to the quenched kinetic energy; cf. fractional quantum Hall physics.

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Orbital exchange (spinless fermion)• Pseudo-spin representation.

)(2

11 yyxx pppp ++ −=τ )(

2

12 xyyx pppp ++ +=τ )(

23 xyyxi pppp ++ −=τ

• No orbital-flip process. Exchange is AF-Ising-like.

UtJ /2 2=

• For a bond along the general direction .

yxyyxx pppppp ϕϕϕϕ cossin,sincos +−=′+=′

)ˆ()( 11 xrrJH ex += ττ

ϕeigen-state of

yxe τϕτϕτ ϕ 2sin2cosˆ2 +=⋅r

ϕe

• Ising-quantization axis is correlated with bond orientation.

)ˆ)ˆ()(ˆ)(( 22 ϕϕϕ ττ eererJHex ⋅+⋅=rr

ϕτ 2e⋅r

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Orbital-ordering in square, triangular, and Kagome lattices

• Square lattice: staggered px, and py configuration.

)ˆ)ˆ()(ˆ)(( ϕϕϕ ττ eererJHr

ex ⋅+⋅=∑ rr3,23,211 , ττττ −→→

• Triangular and Kagome lattices: rotate p-orbitals at each site at 180 degree around x-axis.

• For each bond, the projections of two tau-vectors sum to zero .

ϕϕ2−

τr