Exploring New States of Matter in the p-orbital
Bands 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
1
Collaborators
L. Balents
UCSB
D. Bergman
UCSB
S. Das Sarma
Univ. of Maryland
W. V. Liu
Univ. of Pittsburg
J. Moore
Berkeley
Many thanks to I. Bloch, L. M. Duan, T. L. Ho, T. Mueller, Z. Nussinov
for very helpful discussions.
2
Outline
• Introduction.
- Rapid progress of cold atom physics in optical lattices.
- New direction: orbital physics in high-orbital bands;
pioneering experiments.
• New features of orbital physics in optical lattices.
Fermions: flat bands and crystallization in honeycomb lattice.
Bosons: novel superfluidity with time-reversal symmetry
breaking (square, triangular lattices).
3
Bose-Einstein condensation
• Bosons in magnetic traps: dilute and weakly interacting
systems.
M. H. Anderson et al., Science 269, 198 (1995)
TBEC ~ 1K
n ~ 1014 cm 3
4
New era: optical lattices
• New opportunity to study strongly correlated systems.
• Interaction effects are tunable by varying laser intensity.
t
U
t : inter-site tunneling
U: on-site interaction
5
Superfluid-Mott insulator transition
Superfluid
Mott insulator
t<<U
t>>U
87
Rb
Greiner et al., Nature (2001).
6
Noise correlation (time of flight) in Mott-insulators

k1

k2

• 1st order coherence n(k ) disappears
in the Mott-insulating state.
• Noise correlation function oscillates
at reciprocal lattice vectors; bunching
effect of bosons.
 


  
n(k1 )n(k2 )  n(k1 ) n(k2 )   (k1  k2  G)
G
Folling et al., Nature 434, 481 (2005); Altman et al.,
PRA 70, 13603 (2004).
7
Two dimensional superfluid-Mott insulator transition
V / ER  12
V / ER  20
I. B. Spielman et al., cond-mat/0606216.
V / ER  21
8
Fermionic atoms in optical lattices
• Observation of Fermi surface.
40
K : Fm 
99
22
,
97
22
Esslinger et al.,
PRL 94:80403
(2005)
Low density: metal
high density: band insulator
H  t  ci c j  U  ni ni
i , j ,
t
i
   ni
U
i
• Quantum simulations to the Hubbard model.
e.g. can 2D Hubbard model describe high Tc cuprates?
9
New direction: orbital physics in optical lattices
• 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.
• Orbital physics: studying new physics of fermions and
bosons in high-orbital bands.
Good timing: pioneering experiments; double-well lattice
(NIST) and square lattice (Mainz).
J. J. Sebby-Strabley, et al., PRA 73, 33605 (2006); T. Mueller and I. Bloch et al.
10
Orbital physics
• Orbital: a degree of freedom
independent of charge and spin.
• Orbital band degeneracy and
spatial anisotropy.
• cf. transition metal oxides (dorbital bands with electrons).
Charge and orbital ordering in
La1-xSr1+xMnO4
Tokura, et al., science 288, 462, (2000).
11
New features of orbital physics in optical lattices
• px,y-orbital physics using cold
atoms.
• Strong anisotropy.
Fermions: flat band, novel orbital
ordering … …
Bosons: frustrated superfluidity
with translational and time-reversal
symmetry breaking … …
• System preparation:
-bond
p-bond
t //  t
fermions: s-band is fully-filled; p-orbital bands are active.
bosons: pumping bosons from s to p-orbital bands.
12
Double-well optical lattices
J. J. Sebby-Strabley, et al., PRA 73, 33605 (2006).
• Laser beams of in-plane and out-of-plane polarizations.
I xy
Iz
White spots=lattice sites.
Note the difference in
lattice period!
Combining both
polarizations
• The potential barrier height and the
tilt of the double well can be tuned.
13
Transfer bosons to the excited band
Grow the long period lattice
Create the excited state (adiabatic)
Avoid tunneling (diabatic)
Create the short period lattice (diabatic)
• Band mapping.
• Phase incoherence.
M. Anderlini, et al., J. Phys. B 39, S199 (2006).
14
Ongoing experiment: pumping bosons by Raman transition
• Long life-time: phase coherence.
• Quasi-1d feature in the square
lattice.
py
px
T. Mueller, I. Bloch et al.
15
Outline
• Introduction.
Orbital physics: good timing for studying new physics of
fermions and bosons in high-orbital bands.
• New features of orbital physics in optical lattices.
Fermions: flat bands and crystallization in honeycomb lattice.
Bosons: novel superfluidity with time-reversal symmetry
breaking (square, triangular lattices).
16
p-orbital fermions in honeycomb lattices
pxy-orbital: flat bands;
interaction effects
dominate.
C. Wu, D. Bergman, L. Balents,
and S. Das Sarma, condmat/0701788
cf. graphene: a surge
of research interest;
pz-orbital; Dirac cones.
17
px, py orbital physics: why optical lattices?
• pz-orbital band is not a good system for orbital physics.
isotropic within 2D; non-degenerate.
• Interesting orbital physics in the px,
py-orbital bands.
• 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.
1/r-like
potential
2p
2s
1s
p
s
18
Artificial graphene in optical lattices
• Band Hamiltonian (-bonding) for spinpolarized fermions.
H t  t // 

 
[
p
(
r
)
p
(
r
 1 1  eˆ1 )  h.c.]
ê2
ê1
B
B
A

r A


 [ p (r ) p1 (r  eˆ2 )  h.c]


 [ p3 (r ) p3 (r  eˆ3 )  h.c]

2
p1 
p2 
3
2
3
2
p x  12 p y
ê3
B

p2
p1
p x  12 p y
p3   p y
p3
19
Flat bands in the entire Brillouin zone!
• Flat band + Dirac cone.
• localized eigenstates.
• If p-bonding is included, the flat
bands acquire small width at the
order of t  .
t //  t 
p-bond
20
Enhance interactions among polarized fermions


H

U
n
(
r
)
n
(
r
 px p y )
int
• Hubbard-type interaction:

r  A, B
• Problem: contact interaction vanishes for spinless
fermions.
• Use fermions with large magnetic
moments.
• Under strong 2D confinement,
U is repulsive and can reach
the order of recoil energy.

 
S1, 2  ( x1  x2 )
53
Cr (S  3,   6 B )

x2

B

x1
py
px
21
Exact solution with repulsive interactions!
• Crystallization with only on-site
interaction!
• Closest packed hexagons;
avoiding repulsion.
• The crystalline order is stable
even with t  if U  t .
n 
1
6
• The result is also good for
bosons.
22
Orbital ordering with strong repulsions
n  12
U / t//  10
• Various orbital ordering
insulating states at
commensurate fillings.
• Dimerization at <n>=1/2! Each dimer is an entangled state
of empty and occupied states.
23
Experimental detection
• Transport: tilt the lattice and measure the excitation gap.
• Noise correlations of the time of
flight image.
 




C (k1 , k2 )  n(k1 )n(k2 )  n(k1 ) n(k2 )


C ( q )   dk
 q  q
 
C (k  2 , k  2 )
 q
 q     (d  G )
n( k  2 ) n( k  2 )
G
G: reciprocal lattice vector for the
enlarged unit cells; ‘+’ for bosons,
‘-’ for fermions.
in unit of 2p / 3a
24
Open problems: exotic states in flat bands
• Divergence of density of states.
• Interaction effects dominate due to the quenched
kinetic energy; cf. fractional quantum Hall physics.
• 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?
25
Outline
• Introduction.
• New features of orbital physics in optical lattices.
Fermions: flat bands in honeycomb lattice.
Bosons: novel superfluidity with time-reversal
symmetry breaking.
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).
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 .
26
Main results: superfluidity of bosons with time
reversal symmetry breaking
• On-site orbital angular
momentum moment (OAM).
• Square lattice:
staggered OAM order.
1
1
i
i
1
1
i
1
p x  ip y
i
i
1
1
i
1
i
1
1
i
i
1
i
i
i
i
i
i
1
1
i
i
1
p x  ip y
i
1
1
i
1
• Triangular lattice:
stripe OAM order.
i
i
1
1
1
1
i
1
1
i
1
i
i
1
i
1
27
On-site interaction in the p-band: orbital “Hund’s rule”
H int
U

2
1 z 2
{
n

( Lr ) }
r
3
1
1
2
r
n  p x p x  p y p y , Lz  i ( p x p y  p y p x )
• “Ferro”-orbital interaction: Lz is maximized.
1
i
1
i
p x  i p y (axial) are spatially more extended than p x , y (polar).
• 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.
28
Band structure: 2D square lattice
• Anisotropic hopping and odd parity:
t //  t


H t  t // { px (r ) px (r  eˆx )  h.c.  x  y}

r


 t { p (r ) p y (r  eˆx )  h.c.  x  y}

r

y
-bond
p-bond
• Band minima: Kx=(p,0), Ky=(0,p).
 x (k x , k y )  t // cos k x  t  cos k y
 y (k x , k y )  t cos k x  t // cos k y
29
Superfluidity with time-reversal symmetry breaking
• Interaction selects condensate as

G
1
1
i
1
1



{ ( Kx
 i Ky
)}N0 0
N 0! 2
• Time-reversal symmetry breaking:
staggered orbital angular momentum
order.
i
i
1
1
i
i
i
i
1
1
1
1
i
• Time of flight (zero temperature):
2D coherence peaks located at
p
(( m  ) , 0)
a
1
2
p
(0, (n  ) )
a
1
2
30
Quasi-1D behavior at finite temperatures
• Because t  t// , px-particles can
maintain phase coherence within the
same row, but loose phase inter-row
coherence at finite temperatures.
px
py
• Similar behavior also occurs for pyparticles.
• The system effectively becomes
1D- like as shown in the time of
flight experiment.
A. Isacsson et. al., PRA 72, 53604, 2005;
31
T. Mueller, I. Bloch et al.
Band structure: triangular lattice


H t  t // { p (r ) pi (r  eˆi )  h.c.}

r

i
CW, W. V. Liu, J. Moore, and S. Das
Sarma, Phys. Rev. Lett. (2006).

K1
a0 ê2
a0 ê1
a0ê3
 K 1

K3

K2
(
4p
,0)
3a0

K1, 2,3 lowest energy states
 K 2
 K 3
32
Novel quantum stripe ordering
• Interactions select the condensate as (weak coupling analysis).
 K 2
 K 3
i
1
1
1
i
i
i
1
i
i
1
i
1
1
{
( K 2  i K 3 )}N 0 0
N 0!
2
1
1
1
i
1
i
i
i
i
1
1
i
i
1
1
1
i
• Time-reversal, translational, rotational symmetries are broken.
• cf. Charge stripe ordering in solid state systems with long
range Coulomb interactions. (e.g. high Tc cuprates, quantum
Hall systems).
33
Stripe ordering throughout all the coupling regimes
• Orbital configuration in each site: eir (cos  p x  i r sin  p y )
i
i
1
i
1
1
1
i
i
1
1
1
i
1
i
i
p

6
1
i
i
i
1
1
i
i
1
i
p

4
weak coupling
i
i
i
1
i
i
1
1
1
i
1
i
i
i
1
1
1
1
1
i
1
i
1
1
1
i
i
1
i
1
strong coupling
• cf. Strong coupling results also apply to the p+ip Josephson
junction array systems ( e.g. Sr2RuO4).
34
Time of flight signature
• Stripe ordering even
persists into Mott-insulating
states without phase
coherence.
• Predicted time of flight
density distribution for the
stripe-ordered superfluid.
• Coherence peaks occur
at non-zero wavevectors.
35
Summary
• Good timing to study orbital physics in optical lattices.
• New features: novel orbital ordering in flat bands;
novel superfluidity breaking time reversal symmetry.
i
1
1
i
i
1
1
i
i
i
1
1
1
i
i
i
1
i
1
i
1
1
1
i
i
1
i
1
36
Strong coupling vortex configuration of in optical
lattices
• hole-like vortex
à
®
• particle-like vortex
®
d l A = 2 p,
W= h

IM
2 m L2
W= h

IM
2 m L2
(t/U=0.02)
Ref: C. Wu et al., Phys. Rev. A 69, 43609 (2004).
37
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).
38
Phase stability analysis
39
Px,y-band structure in triangular lattices
40
Strong coupling analysis
• Each site is characterized by a U(1) phase  , and an
Ising variable  .
 : the phase of the right lobe.
1
i
1
i
 : direction of the Lz.
• Inter-site Josephson coupling: effective vector potential.
1
H eff   nt //
2
 cos[
r1
 r 2  Ar1,r 2 ( r1 ,  r 2 )]
 r1
r1
r1 , r2
Ar1,r 2 ( r1 ,  r 2 )   r1 r1   r 2 r 2
J. Moore and D. H. Lee, PRB, 2004.
r 1
r 2
r2
r 2
41
Bond current
p
 
6
42
Strong coupling analysis

r3
• The minimum of the effective
flux per plaquette is  1 / 6 .
1
i
i
1

r1
i
1

1
1
6
1
i 
i
1
2p
1
A

'
 r ,r 6 ( r1   r 2   r 3 )
r ,r '
1
i
i
1
1
1
6
1

r2
1
6
1
• The stripe pattern minimizes
the ground state vorticity.
1

6
1

1
1
6
1
• cf. The same analysis also applies to
p+ip Josephson junction array.
43
Double well  triangular lattice
e
i
p
3
e
i
2p
3
e
i
p
3
e
i
2p
3
frustration:
e ip
ei 0
44
Condensation occurs at

2p
K  ( ,0)
3
45
46