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).

1

Thanks to: I. Bloch, L. M. Duan, T. L. Ho, Z. Nussinov, S. C. Zhang.

2

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

13

• 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

6

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.

18

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

19

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

22

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.

32

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.