Novel Orbital Phases of Fermions
in p-band Optical Lattices
Congjun Wu
Department of Physics, UC San Diego
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).
Collaborators: L. Balents, D. Bergman, S. Das Sarma, H. H.
Hung, W. C. Lee, S. Z. Zhang; W. V. Liu, V. Stojanovic.
Thanks to: I. Bloch, L. M. Duan, T. L. Ho, Z. Nussinov,
S. C. Zhang.
Feb. 23, 2009, KITP
1
Outline
• Introduction to orbital physics.
New directions of cold atoms: bosons and fermions in highorbital bands; pioneering experiments.
• Bosons: meta-stable states with long life time; complexcondensations beyond the no-node theorem.
• Fermions: px,y-orbital counterpart of graphene.
1. Flat band structure and non-perturbative effects.
2. p-orbital band insulators (topological) – Haldane’s model
induced by orbital angular momentum polarization.
3. p-orbital Mott insulators: orbital exchange; a new type of
frustrated magnet-like model; a cousin of the Kitaev model.
2
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).
Orbital physics
• Orbital: a degree of freedom
independent of charge and spin.
Tokura, et al., science 288, 462, (2000).
• Orbital degeneracy and spatial
anisotropy.
• cf. transition metal oxides (d-orbital bands with electrons).
La1-xSr1+xMnO4
LaOFeAs
4
Advantages of optical lattice orbital systems
• Solid state orbital systems:
• Optical lattices orbital systems:
Jahn-Teller distortion quenches
orbital degree of freedom;
rigid lattice free of distortion;
only fermions;
both bosons (meta-stable excited
states with long life time) and
fermions;
correlation effects in p-orbitals
are weak.
strongly correlated px,y-orbitals:
stronger anisotropy
t //  t
s-bond
p-bond
5
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.
 (r1 , r2 ,...rn ) | (r1 , r2 ,...rn ) |
 (r1 , r2 ,...rn )  0
2 n
H   dr1...drn
|  i (r1 ,...rn ) |2  |  (r1 ,..rn ) |2

2m i 1
|  (r1 ,..rn ) |2
V
i j
int
n
U
i 1
ex
(ri ) 
(r i rj )
• Strong constraint: complex-valued WF  positive definite WF;
time-reversal symmetry cannot be spontanously broken.
• Feynman’s statement applies to all of superfluid, Mottinsulating, super-solid, density-wave ground states, etc.
No-node theorem doesn’t apply to orbital bosons
• Complex condensate wavefunctions; spontaneous time reversal
symmetry breaking.
• Orbital Hund’s rule interaction; ordering of orbital angular
momentum moments.
W. V. Liu and C. Wu, PRA 74, 13607 (2006);
C. Wu, Mod. Phys. Lett. 23, 1 (2009).
C. Wu, W. V. Liu, J. Moore and S. Das
Sarma, PRL 97, 190406 (2006).
t
p x  ip y
1
i
i
1
1
i
p x  ip y
i
1
1
i
i
1
i
1
p x  ip y
i
1
i
i
i
1
1
i
i
i
1
p x  ip y
1
i
i
1
i
t //
1
1
1
p x  ip y
1
1
i
p x  ip y
1
i
1
i
i
1
i
1
p x  ip y
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 .
7
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 band insulators (topological) – Haldane’s model
induced by orbital angular momentum polarization.
C. Wu, PRL 101, 186807 (2008).
• P-orbital Mott insulators: orbital exchange; from Kitaev to
quantum 120 degree model.
C. Wu, PRL 100, 200406 (2008.
8
p-orbital fermions in honeycomb lattices
cf. graphene: a surge
of research interest;
pz-orbital; Dirac cones.
pxy-orbital: flat bands;
interaction effects
dominate.
C. Wu, D. Bergman, L. Balents,
and S. Das Sarma, PRL 99,
70401 (2007).
9
What is the fundamental difference from graphene?
• pz-orbital band is not a good
system for orbital physics.
• It is the other two px and py orbitals
that exhibit anisotropy and degeneracy.
• 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
10
Honeycomb optical lattice with phase stability
• Three coherent laser beams polarizing in the z-direction.
• Laser phase drift only results an overall lattice translation
without distorting the internal lattice structure.
G. Grynberg et al., Phys. Rev. Lett. 70, 2249 (1993).
11
Artificial graphene in optical lattices
• Band Hamiltonian (s-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
12
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  . Realistic band
structures show t / t //  1%
t //  t 
p-bond
13
Hubbard model for spinless fermions:
Exact solution: Wigner crystallization
H int


 U  n p x (r )n p y (r )

r  A, B
gapped state
• Close-packed hexagons;
avoiding repulsion.
• The crystalline ordered state is
stable even with small t  .
n 
1
6
• Particle statistics is irrelevant.
The result is also good for bosons,
and Bose-Fermi mixtures.
Flat-band itinerant ferromagnetism (FM)
• FM requires strong enough repulsion and thus FM has no welldefined weak coupling picture.
• It is commonly accepted that Hubbard-type models cannot
give FM unless with flat band structure.
A. Mielke and H. Tasaki, Comm. Math. Phys 158, 341 (1993).
• 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 dimermolecule formation.
• 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.
Shizhong Zhang and C. Wu, arXiv:0805.3031.
15
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 band insulators (topological) – Haldane’s
model induced by orbital angular momentum
polarization.
C. Wu, PRL 101, 186807 (2008).
• Orbital exchange; from Kitaev to quantum 120 degree model.
C. Wu, PRL 100, 200406 (2008.
16
Topological insulators: Haldane’s QHE Model
without Landau level
• Honeycomb lattice with complex-valued next-nearest neighbor
hopping.
H NN
A


 t {c (rA )c(rB )  h.c.


r A
B




H NNN   t {ei c  (rA )c(rA )  ei c  (rB )c(rB )

r
 h.c.}
• Topological insulator at   0, p . Mass
changes sign at K1,2.




H (k )  a (k ) 1  b(k ) 3  m(k ) 2

 c(k ) I
K1
K2
17
Rotate each site around its own center
N. Gemelke, Ph. D.
thesis, Stanford
University, 2007.
• Orbital Zeeman term.
• Phase modulation on laser
beams: a fast overall oscillation
of the lattice. Atoms cannot
follow and feel a slightly
distorted averaged potential.

H zmn   Lz (r )
rA


 
 
 i 
{
p
(
r
)
p
(
r
)

p
(
r
)
p
(
r
)
x
y
y
x

• The oscillation axis slowly
precesses at the angular
frequency of  .
r A
18
Large rotation angular velocity
• Second order perturbation process generates the complex
NNN hopping.
e
e
i
i
5p
6
p  ip
t  (tei 2 / 3p )2 / 2
p
6
p  ip
  t //
p  ip
p  ip

p  ip
19
Small rotation angular velocity
0
  0.3t//
Berry curvature.
C 0
C  1
0
0
C 1
1
20
Large rotation angular velocity
  32 t//
  3t //
Berry curvature.
C  1
21
C  1
1
1
C 1
1
21
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 band insulators (topological) – Haldane’s model
induced by orbital angular momentum polarization.
C. Wu, PRL 101, 186807 (2008).
• P-orbital Mott insulators: orbital exchange; from Kitaev
to quantum 120 degree model.
C. Wu, PRL 100, 200406 (2008.
22
Mott-insulators with orbital degrees of freedom:
orbital exchange of spinless fermion
• Pseudo-spin representation.
1  12 ( px px  p y p y )  2  12 ( px p y  p y px )  3  2i ( px p y  p y px )
• No orbital-flip process. Antiferro-orbital Ising exchange.
J 0
J 0
H ex  J 1 (r ) 1 (r  xˆ )
J  2t 2 / U
J  2t 2 / U
23
Hexagon lattice: quantum 120  model
• For a bond along the general direction ê .

px , py : eigen-states of   eˆ2  cos 2  x  sin 2  y



  ê2
px
H ex  J ( (r )  eˆ2 )( (r  eˆ )  eˆ2 )
py

• After a suitable transformation, the Ising quantization axes can
be chosen just as the three bond orientations.
B
(    )(     )
1
2 1
3
2 2
1
2 1
3
2 2
1 1
A
( 12  1  23  2 )( 12  1  23  2 )
B
B


H ex   J ( (ri )  eˆij ) ( (rj)  eˆij )
r ,r
From the Kitaev model to 120 degree model
• cf. Kitaev model: Ising quantization axes form an orthogonal
triad.
B
s ys y
H kitaev   J  (s x (r )s x (r  e1 )  s y (r )s y (r  e2 )
sx sx
rA
 s z (r )s z (r  e3 ))
s zs z


A
1
2
B
B
0
120  Kitaev
cos
Large S picture: heavy-degeneracy of classic ground states
• Ground state constraint: the two -vectors have the same
projection along the bond orientation.


H ex   J {[( (r )   (r )]  eˆrr }2  J  z2 (r )
r ,r 
• Ferro-orbital configurations.
or
r
• Oriented loop config: -vectors
along the tangential directions.
26
Heavy-degeneracy of classic ground states
• General loop configurations
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  30 ,  90 ,  150 .
28
“Order from disorder”: 1/S orbital-wave correction
29
Zero energy flat band orbital fluctuations
• Each un-oriented loop has a local
zero energy model up to the
quadratic level.

E  6 JS 2 ( ) 4
• 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?
30
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.
31
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.
32