Part II. p-orbital physics in optical lattices

1
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,
arXiv0805.3031. 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
2
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.
3
Outline

• Introduction to orbital physics a new direction
  of orbital physics.

• Bosons unconventional BEC beyond no-node
  theorem. c.f. V. W. Lius 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. Wus talk on March 19.
4
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
Good timing pioneering experiments on
orbital-bosons. Square lattice (Mainz) double
well lattice (NIST).
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).
5
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).

LaOFeAs
La1-xSr1xMnO4
6
Advantages of optical lattice orbital systems

• 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

• Solid state orbital systems
• Jahn-Teller distortion quenches orbital degree of
  freedom
• only fermions
• correlation effects in p-orbitals are weak.

7
Novel state of orbital bosons beyond the No-node
theorem (c.f. Vincent Lius talk)

• Complex condensate wavefunctions spontaneous
  time reversal symmetry breaking.

• Orbital Hunds rule interaction ordering of
  orbital angular momentum moments.

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) C. Wu,
Mod. Phys. Lett. 23, 1 (2009).
Other groups 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 .
8
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,
arXiv0805.3031.

• 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.
9
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).
10
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.

1/r-like potential

• In optical lattices, px and py-orbital bands are
  well separated from s.

p
s
11
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).
12
Artificial graphene in optical lattices

• Band Hamiltonian (s-bonding) for spin- polarized
  fermions.

13
Flat bands in the entire Brillouin zone!

• If p-bonding is included, the flat bands acquire
  small width at the order of . Realistic band
  structures show

14
Hubbard model for spinless fermions
Exact solution Wigner crystallization
gapped state

• Close-packed hexagons avoiding repulsion.

• Particle statistics is irrelevant. The state is
  also good for bosons, and even Bose-Fermi
  mixtures.

15
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,
arXiv0805.3031.
16
Flat-band itinerant FM in p-orbitals

• Exact result in the homogenous system
  magnetization with the filling inside the flat
  band, i.e.,
• .

• More realistic system soft harmonic trap
  particle numbers of spin up and down particles
  are separately conserved.

• Self-consistent Bogoliubov-de Gennes
  calculation FM phase separation.

17
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, arXiv0805.3031.

• 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.
18
Mott-insulators with orbital degrees of freedom
orbital exchange of spinless fermion

• Pseudo-spin representation.

• No orbital-flip process. Antiferro-orbital Ising
  exchange.

19
Hexagon lattice quantum model

• For a bond along the general direction .

eigen-states of

• After a suitable transformation, the Ising
  quantization axes can be chosen just as the three
  bond orientations.

C. Wu et al, arxiv0701711v1 C. Wu, PRL 100,
200406 (2008). E. Zhao, and W. V. Liu, Phys. Rev.
Lett. 100, 160403 (2008)
20
From the Kitaev model to 120 degree model

• cf. Kitaev model Ising quantization axes form
  an orthogonal triad.

Kitaev
21
Large S picture heavy-degeneracy of classic
ground states

• Ground state constraint the two t-vectors have
  the same projection along the bond orientation.

• Ferro-orbital configurations.

• Oriented loop config t-vectors along the
  tangential directions.

22
Heavy-degeneracy of classic ground states

• General loop configurations

23
Global rotation degree of freedom

• Each loop config remains in the ground state
  manifold by a suitable arrangement of
  clockwise/anticlockwise rotation patterns.

24
Order from disorder 1/S orbital-wave correction
25
Zero energy flat band orbital fluctuations

• 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
  (s1/2)? A very promising system to arrive at
  orbital liquid state?

26
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, arXiv0805.3031.

• 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.
27
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?

28
Nontrivial orbital hybridization p-orbital
hexagonal lattice

• Along the three middle lines of Brillouin zone,
  eigen-orbitals are real.

29
Onsite Hubbard interaction for SPINLESS p-orbital
fermions

• Attraction between fermions in two orthogonal
  orbitals.

• F-wave structure intra-band pairing. Pairing
  strength vanishes along three middle lines, and
  becomes strongest at K and K.

nodal lines
30
Zero energy Andreev bound states

• If the boundary is perpendicular to the
  anti-nodal (nodal) direction, the zero energy
  Andreev bound states appear (vanish).

No Andreev Bound States
With Andreev Bound States
31
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. Wus talk on March 19.

32
Orbital ordering with strong repulsions

• Various orbital ordering insulating states at
  commensurate fillings.

• Dimerization at ltngt1/2! Each dimer is an
  entangled state of empty and occupied states.

33
Gap value and superfluid density
34
Flat-band itinerant FM in p-orbitals

• Percolation picture for flat band FM.

• Self-consistent calculation for the FM phase
  separation with a soft harmonic trap.