Novel Orbital Phases of Fermions
in p-band Optical Lattices
Congjun Wu
Department of Physics, UC San Diego
Fermions:
W. C. Lee, C. Wu, S. Das Sarma, to be submitted.
C. Wu, PRL 101, 168807 (2008).
C. Wu, PRL 100, 200406 (2008).
C. Wu, and S. Das Sarma, PRB 77, 235107 (2008).
S. Z. Zhang , H. H. Hung, and C. Wu, arXiv:0805.3031.
C. Wu, D. Bergman, L. Balents, and S. Das Sarma, PRL 99,
67004(2007).
Bosons: C. Wu, Mod. Phys. Lett. 23, 1(2009).
V. M. Stojanovic, C. Wu, W. V. Liu and S. Das Sarma, PRL 101, 125301(2008).
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).
March 20, 2009, Pittsburgh
1
Collaborators:
S. Das Sarma
(UMD)
H. H. Hung, W. C. Lee
(UCSD)
L. Balents, D. Bergman
(UCSB)
Shizhong Zhang
(UIUC)
Thank W. V. Liu, J. Moore, V. Stojanovic for collaboration
on orbital physics with bosons.
Also I. Bloch, L. M. Duan, T. L. Ho, Z. Nussinov, S. C.
Zhang for helpful discussions.
Supported by NSF,
and Sloan Research
Foundation.
2
Outline
• Introduction to orbital physics: a new direction
of orbital physics.
• Bosons: unconventional BEC beyond no-node theorem.
c.f. V. W. Liu’s talk.
• Fermions in hexagonal lattice: px,y-orbital counterpart of
graphene.
1. Flat band structure and non-perturbative effects.
2. Mott insulators: orbital exchange; a new type of
frustrated magnet-like model; a cousin of the Kitaev model.
3. Novel pairing state: f-wave Cooper pairing.
4. Topological insulators – quantum anomalous Hall effect
by orbital angular momentum polarization. c.f. C. Wu’s talk
on March 19.
3
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).
4
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
5
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
6
Novel state of orbital bosons beyond the Nonode theorem (c.f. Vincent Liu’s talk)
• 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
i
i
1
1
i
1
p x  ip y
p x  ip y
i
1
i
i
i
1
1
1
i
i
i
1
p x  ip y
i
t //
1
1
1
i
i
1
i
1
i
1
1
1
i
i
1
i
1
p x  ip y
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
P-orbital fermions: 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, Hsiang-hsuan Hung, 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 et al, arxiv0701711v1;
E. Zhao, and W. V. Liu, Phys. Rev. Lett. 100, 160403 (2008)
• Novel pairing state: f-wave Cooper pairing.
W. C. Lee, C. Wu, S. Das Sarma, to be submitted.
8
p-orbital fermions in honeycomb lattices
cf. graphene: a surge
of research interest;
pz-orbital; Dirac cones.
Px, y-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 state is also good for bosons,
and even Bose-Fermi mixtures.
Spinful Fermions: flat-band itinerant ferromagnetism
• Ferromagnetism (FM) requires strong repulsive interactions ,
and thus has no well-defined weak coupling picture.
• It is accepted that it is difficult to achieve FM state conclusively
in Hubbard type modes except with flat band and Nagaoka limit.
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 formation
of dimers.
• Flat-band FM 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, Hsiang-hsuan Hung, and C. Wu, arXiv:0805.3031.
15
Flat-band itinerant FM in p-orbitals
• Exact result in the homogenous
system: magnetization with the
filling inside the flat band, i.e.,
n  0 .5 .
• More realistic system: soft
harmonic trap; particle
numbers of spin up and down
particles are separately
conserved.
  0.005ER , l  4.5a, N   N   255,
t //  U  0.24ER
• Self-consistent Bogoliubov-de
Gennes calculation: FM phase
separation.
16
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; a new
type of frustrated magnet-like model; a cousin of
Kitaev model.
C. Wu, PRL 100, 200406 (2008); C. Wu et al, arxiv0701711v1;
E. Zhao, and W. V. Liu, Phys. Rev. Lett. 100, 160403 (2008)
• Novel pairing states: f-wave pairing.
W. C. Lee, C. Wu, S. Das Sarma, to be submitted.
17
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
18
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
( 12  1  23  2 )(  12  1  23  2 )
1 1
A
(    )(    )
1
2 1
3
2 2
1
2 1
3
2 2
B
B


H ex   J ( (ri )  eˆij ) ( (rj)  eˆij )
r ,r
C. Wu et al, arxiv0701711v1; C. Wu, PRL
100, 200406 (2008). E. Zhao, and W. V.
Liu, Phys. Rev. Lett. 100, 160403 (2008)
19
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
cos
120  Kitaev
20
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.
21
Heavy-degeneracy of classic ground states
• General loop configurations
22
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 .
23
“Order from disorder”: 1/S orbital-wave correction
24
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?
25
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).
• Novel pairing state: f-wave pairing.
W. C. Lee, C. Wu, S. Das Sarma, to be submitted.
26
UNCONVENTIONAL Cooper pairing from TRIVIAL
interactions
• Most of unconventional pairing states arise from strong
correlation effects, and thus are difficult to predict and analyze.
p-wave: superfluid 3He-A and B; Sr2RuO4;
d-wave: high Tc cuprates;
Extended s-wave: Iron-based superconductors (?);
Possible f-wave: UPt3(?)
• Can we arrive at unconventional
pairing in a much easier way, say, from
trivial interactions but with nontrivial
band structures?
-

k
+
+
 k
d x2  y2
27
Nontrivial orbital hybridization: p-orbital hexagonal
lattice
p x  ip y
p x  ip y
k
K
K
k
p x  ip y
• Along the three middle
lines of Brillouin zone,
eigen-orbitals are real.
p x  ip y
K
K
28
Onsite Hubbard interaction for SPINLESS p-orbital
fermions
• Attraction between fermions in two orthogonal orbitals.
U


H int  U 
n p (r )n p (r )

r
x
y
• F-wave structure intra-band pairing. Pairing strength vanishes
along three middle lines, and becomes strongest at K and K’.
F (k )  sin 23 k x (cos 23 k x  cos 32 k y )
K
p x  ip y
nodal lines
K
p x  ip y
29
Zero energy Andreev bound states
• If the boundary is perpendicular to the anti-nodal (nodal)
direction, the zero energy Andreev bound states appear (vanish).
 
kout


With Andreev
Bound States
k in






kout


k in
No Andreev
Bound States
30
A new research direction of cold atoms
in optical lattices
• Band flatness and strong correlation: Wigner crystal and
ferromagnetism;
• Orbital exchange: a new type of frustrated magnet-like mode;
a cousin of the Kitaev model;
• f-wave Cooper pairing;
• Topological band insulator: quantum anomalous effect. c. f. C.
Wu’s talk on March 19.
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
Gap value and superfluid density
33
Flat-band itinerant FM in p-orbitals
• Percolation picture for flat band FM.




H int  U 
(
n
(
r
)
n
(
r
)

n
(
r
)
n
(
r
))
p
p ,
p ,
p ,

x ,
r
x
y
y
    1


 J  ( S p (r )  S p (r )  n p (r )n p (r ))
r
4




   ( p x, (r ) p x, (r ) p y , (r ) p y , (r )  h.c.)
x
y
x
y
r
• Self-consistent calculation
for the FM phase separation
with a soft harmonic trap.
34