1

Exploring New States of Matter in the p-orbitalBands of Optical Lattices

Congjun Wu

Kavli Institute for Theoretical Physics, UCSB

C. Wu, D. Bergman, L. Balents, and S. Das Sarma, cond-mat/0701788.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).

University of Maryland, 02/05/2007

2

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

W. V. Liu Univ. of Pittsburg

S. Das Sarma Univ. of Maryland

J. Moore Berkeley

3

Outline

• Introduction.

• New features of orbital physics in optical lattices.

- Rapid progress of cold atom physics in optical lattices.

Bosons: novel superfluidity with time-reversal symmetry breaking (square, triangular lattices).

- New direction: orbital physics in high-orbital bands; pioneering experiments.

Fermions: flat bands and crystallization in honeycomb lattice.

4

M. H. Anderson et al., Science 269, 198 (1995)

Bose-Einstein condensation

314 cm10~1~ nKTBEC

• Bosons in magnetic traps: dilute and weakly interacting systems.

5

New era: optical lattices• New opportunity to study strongly correlated systems.

• Interaction effects are tunable by varying laser intensity.

U

tt : inter-site tunnelingU: on-site interaction

6

Superfluid-Mott insulator transitionSuperfluid

Greiner et al., Nature (2001).

Rb87

Mott insulator

t<<Ut>>U

7

)()()()()( 212121 GkkknknknknG

• 1st order coherence disappears in the Mott-insulating state.

)(kn

• Noise correlation function oscillates at reciprocal lattice vectors; bunching effect of bosons.

Noise correlation (time of flight) in Mott-insulators

Folling et al., Nature 434, 481 (2005); Altman et al., PRA 70, 13603 (2004).

1k

2k

8

Two dimensional superfluid-Mott insulator transition

I. B. Spielman et al., cond-mat/0606216.

12/ REV 20/ REV 21/ REV

9

Fermionic atoms in optical lattices

U

t

ii

iji i

iji

n

nnUcctH

,,

• Observation of Fermi surface.

Low density: metal high density: band insulator

2

7

2

9

2

9

2

940 ,: FmK

Esslinger et al., PRL 94:80403 (2005)

• Quantum simulations to the Hubbard model.e.g. can 2D Hubbard model describe high Tc cuprates?

10

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

New direction: orbital physics in optical lattices

• Orbital physics: studying new physics of fermions and bosons in high-orbital bands.

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

• Great success of cold atom physics:

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

• Next focus: resolve NEW aspects of strong correlation phenomena which are NOT well understood in usual condensed matter 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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New features of orbital physics in optical lattices

tt//

Fermions: flat band, novel orbital ordering … …

Bosons: frustrated superfluidity with translational and time-reversal symmetry breaking … …

• px,y-orbital physics using cold atoms.

fermions: s-band is fully-filled; p-orbital bands are active.bosons: pumping bosons from s to p-orbital bands.

-bond -bond

• Strong anisotropy.

• System preparation:

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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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Ongoing experiment: pumping bosons by Raman transition

T. Mueller, I. Bloch et al.

• Quasi-1d feature in the square lattice.

• Long life-time: phase coherence.

xpyp

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Outline

• Introduction.

• New features of orbital physics in optical lattices.

Bosons: novel superfluidity with time-reversal symmetry breaking (square, triangular lattices).

Fermions: flat bands and crystallization in honeycomb lattice.

Orbital physics: good timing for studying new physics of fermions and bosons in high-orbital bands.

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pxy-orbital: flat bands; interaction effects dominate.

p-orbital fermions in honeycomb lattices

cf. graphene: a surge of research interest; pz-orbital; Dirac cones.

C. Wu, D. Bergman, L. Balents, and S. Das Sarma, cond-mat/0701788

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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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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

20

• 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//

21

Enhance interactions among polarized fermions

)()(,

int rnrnUHyx p

BArp

• Hubbard-type interaction:

• Problem: contact interaction vanishes for spinless fermions.

)63,(SCr 53B • Use fermions with large

magnetic moments.

• Under strong 2D confinement, U is repulsive and can reach the order of recoil energy.

)( 212,1 xxS

B

px

py

1x

2x

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Exact solution with repulsive interactions!

6

1n

• Crystallization with only on-site interaction!

• The result is also good for bosons.

• Closest packed hexagons; avoiding repulsion.

• The crystalline order is stable even with if .t tU

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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.

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Experimental detection

)()()()(),( 212121 knknknknkkC

• 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

22

• Transport: tilt the lattice and measure the excitation gap.

in unit of a3/2

25

• 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-bands: highly frustrated system. Where to condense? Can they condense? Possible “Bose metal” phase?

Open problems: exotic states in flat bands

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

• Divergence of density of states.

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Outline

Bosons: novel superfluidity with time-reversal symmetry breaking.

Other’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 .

W. V. Liu and C. Wu, PRA 74, 13607 (2006); C. Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97, 190406 (2006).

• New features of orbital physics in optical lattices.

• Introduction.

Fermions: flat bands in honeycomb lattice.

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Main results: superfluidity of bosons with time reversal symmetry breaking

• On-site orbital angular momentum moment (OAM).

11i

i11

i

iyx ipp yx ipp

i

i11

i11

i

11i

i11

i

i

1

1ii

1

1

ii

11

i

i

1

1ii

1

1

ii

11

i

i1 1

i

i

• Square lattice: staggered OAM order.

• Triangular lattice: stripe OAM order.

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On-site interaction in the p-band: orbital “Hund’s rule”

)(,

})(3

1{

222

int

xyyxzyyxx

zr

rr

ppppiLppppn

LnU

H

11i

i

(axial) are spatially more extended than (polar).

yx pip yxp ,

• “Ferro”-orbital interaction: Lz is maximized.

• cf. Hund’s rule for electrons to occupy degenerate atomic shells: total spin is maximized.

• cf. p+ip pairing states of fermions: 3He-A, Sr2RuO4.

11

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

}..)ˆ()({

}..)ˆ()({//

yxcherprpt

yxcherprptH

xyr

y

xxr

xt

tt//

-bond -bond

• Band minima: Kx=(,0), Ky=(0,).

yxyxx ktktkk coscos),( //

yxyxy ktktkk coscos),( //

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Superfluidity with time-reversal symmetry breaking

• Interaction selects condensate as

0)}(2

1{

!

10

0

NKyKxG

iN

• Time-reversal symmetry breaking: staggered orbital angular momentum order.

i

i11

i11

i

11i

i11

i

i

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

)0,)((2

1

am

))(,0(

2

1

an

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Quasi-1D behavior at finite temperatures

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

32

10ea

1K

2K

3K )0,

3

4(

0a

20ea

30ea

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

rit

Band structure: triangular lattice

lowest energy states

3,2,1K

1K 2K

3K

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Novel quantum stripe ordering

1

1ii

1

1ii

1

1ii

1

1ii

11i

i11

i

i1 1

ii

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

2K

3K

i

0)}(2

1{

!

10

32

0

NKK i

N

• 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

weak coupling

1

1ii

1

1

ii

11

i

i

1

1ii

1

1

ii

11

i

i1 1

i

i

1

1ii

1

1ii

1

1ii

1

1ii

11i

i11

i

i1 1

ii

• Orbital configuration in each site:

strong coupling

4

6

• cf. Strong coupling results also apply to the p+ip Josephson junction array systems ( e.g. Sr2RuO4).

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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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Summary

• Good timing to study orbital physics in optical lattices.

1

1ii

1

1

ii

11

i

i

1

1ii

1

1

ii

11

i

i1 1

i

i

• New features: novel orbital ordering in flat bands;

novel superfluidity breaking time reversal symmetry.

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Strong coupStrong coupling vling vortex configuration of in ortex configuration of in optical latticesoptical lattices

Ref: C. Wu et al., Phys. Rev. A 69, 43609 (2004).

àd l® A®

=2 p,

W=hI2 m L2M

W=hI2 m L2M

(t/U=0.02)

• hole-like vortex • particle-like vortex

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Hidden Symmetry and Quantum Phases in Spin 3/2 Cold Atomic Systems

Congjun Wu

Kavli Institute for Theoretical Physics, UCSB

Ref: C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 91, 186402(2003);

C. Wu, Phys. Rev. Lett. 95, 266404 (2005);

S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, 214428 (2005);

C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.

Review paper: C. Wu, Mod. Phys. Lett. B 20, 1707 (2006).

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Phase stability analysis

40

Px,y-band structure in triangular lattices

41

Strong coupling analysis

• Each site is characterized by a U(1) phase , and an Ising variable .

i

i11: the phase of the right lobe.

: direction of the Lz.

)],(cos[2

1212,1

,21//

21

rrrrrr

rreff AntH

• Inter-site Josephson coupling: effective vector potential.

1r

2r

2r

1r

2211212,1 ),( rrrrrrrrA

J. Moore and D. H. Lee, PRB, 2004.

1r

2r

42

Bond current

6

43

Strong coupling analysis

)( 3216

1

,,2

1

'

' rrr

rrrri A 1r

2r

3r

6

1i

i11

i

i11

1

1ii

• cf. The same analysis also applies to p+ip Josephson junction array.

• The minimum of the effective flux per plaquette is .

1 1 1

1

1

6

1

6

1

1

6

1

6

1

1

• The stripe pattern minimizes the ground state vorticity.

6/1

44

Double well triangular lattice

frustration:

ie

3

ie 3

2i

e

3

2ie3

i

e

0ie

45

Condensation occurs at

)0,3

2(

K

46