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Correlated States in Optical Lattices
Fei Zhou (PITP,UBC) Feb. 1, 2004
At Asian Center, UBC
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Outline
S=0 bosons (Recent experiments)
S=1 bosons1) Polar condensates2) Mott states (Spin singlet and nematic Mott states)3) Low dimensional Mott states (Dimerized valence bond crystals)4) Spin singlet condensates or superfluid
Approach: Constrained quantum rotor model ( O(2), O(3) rotors and a constraint on an inversion or an Ising constraint)
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Optical lattices
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S=1/2 Fermions in optical lattices
(small hopping limit)
Neel OrderedGapless Spin liquid HTcS made of cold atoms?
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S=0 bosons in lattices
In (a) and (b), one boson per site. t is the hopping and can be varied by tuning laser intensities of optical lattices; U is an intra-site interaction energy. In a Mott state, all bosons are localized.
M. P. A. Fisher et al., PRB 40, 546 (1989);On Mott states in a finite trap, seeJaksch et al., PRL. 81, 3108-3111(1998).
U
Mott states ( t << U)
Condensates (t >>U)
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3D real space imagine of a condensate released from a lattice (TOF)
Greiner et al., Nature 415, 41(2002)
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Absorption images of interference patterns
as the laser intensity is increased (from a to h). (a-d) BECs and (g-h) Mott insulating states.
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Probing excitationsIn (c-f), width of interference peaks versus perturbations.
(c-d) BECs and (e-f) Mott insulators
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.
/
/
),π
R:(,,φB:(
.n|α
|Sn,n|)(n:|
|iφθe
θ||iφθe
φ)(θn|
21
0
21
02
0
1
0
0
01
12
012
)
sincos
sin,
S=1 bosons with Anti-ferromagnetic interactions
.2,0,,4
)()(
02
,2121
FggM
ag
grrrrU
FF
FF
Ho, 98; Ohmi & Machida, 98; Law,98.
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|1,0> state as a triplet cooper pair
)|d|d|u(|ux
en
);|u|d|d(|uZ
en
)n,/|Sn,/|Sn,/|Sn,/(|Sn,|S
;|diφeθ
|uiφeθ
n,/|S
|diφeθ
|uiφeθ
n,/|S
21212
1
21212
1
221
121
221
121
2
11
22
22
21
22
22
21
/cos/sin
/sin/cos
(This is the Anderson-Morel state!)
Pairs given here are not polarized, i.e. .0|| nSn
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Condensates of spin one bosons (d>1)
kzyx
kC
TV
C
TVN
TVN
nQ
C
pBEC.,,
,1
,0;
)!(
),0
(~
N(Q)
Q
xy
z
Snap shots
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Half vortices
In a half vortex, each atom makes a spin rotation; a half vortex carries one half circulation of an integer vortex. A half vortex ring is also a hedgehog.
circulation
y
spin rotation
Z
x
y
x
The vortex is orientated along the z-direction; the spin rotation and circulating current occur in an x-y plane.
Z
ring
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S=1 bosons with anti-ferromagnetic interactions
in optical lattices (3D and 2D, N=2k)
Polar BEC (a)
Nematic MI (b)
Spin Singlet MI (c)
t: Hopping
).()
;)()
;)()
1
1
optVzt
SE
CE
Cc
SE
CE
CoptVzt
CEb
CE
optVzta
he critical value of is determined numerically.
.
03
)02
(4
03
)02
2(4
MN
aa
SE
MN
aa
CE
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Each site is characterized by two unit vectors, blue and red ones. a) Polar BECs (pBEC); b) Nematic mott insulators (NMI); c) Spin singlet mott insulators (SSMI).
Ordering in states present in 3d ,2d lattices
.02:);31
(2:
.312
OSSMInnNONMI
CCCCO
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Constrained quantum rotor model valid as the hopping is much less than “U”.
).()(,1)1)(2
.,2
)1
.'
]'
,[
,2)(2
kn
MIH
kn
MIHk
Nk
S
sE
cE
zt
cE
ztex
J
nkk
ik
nk
S
ln
kn
klexJ
kS
ksE
MIH
.,1;:Re SzSl
Sk
Skl
JHf exHAF
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Calculated Spectra for given N
In a),b), (n)* (n ) (ground states) as a function of n.
)(,7 nY m
0S
2
4
6
)(0,0 nY
)(,2 nY m
)(,4 nY m
)(,6 nY m
)(,3 nY m
)(,1 nY m
)(,5 nY m
1S
7
5
3
vacN
NnCn
ndn |
!
)()(
4)(
N=2kN=2k N=2k+1N=2k+1
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Microscopic wave functionsa) N=2k; b) N=2k+1
Each pair of blue and red dots is a spin singlet. S=2,4 ,6….(in a) or S=3,5,… (in b) spin excitations can be created by breaking singlets.
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Numerics I: Large N=2k limit
]0exp[~)( Qnnn
SSMI
NMI
vs. (proportional to hopping) is plotted here. Blue and Green lines represent metal stable states close to the critical point.
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The energy Vs.
SSMI
NMI
10.85
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Phase diagrams in high dimensions
t
NMIn=1
SSI NMI
n=3 NMI
SSI NMI
BEC E(k,x)
n=3
n=2
n=1
x
n
x
2
1
Mott states in a trap
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Spin singlet quantum “condensates” in 1D optical lattices (SSQC)
tDVBC SSQC(“e”)
t
SSMI SSQC(“2e”)
(a)
(b)
]).[exp(ˆ,ˆ
2
;2)ˆ(.).(0.
2.
xklkb
Nik
Ck
C
xklklaZ
H
Nkb
NkC
Echl
bk
bzklkl
tm
H
ZH
mH
fqcH
S=1, “Q=e” bosons with AF interactions ===>S=0 , “Q=e” bosons interacting via Ising gauge fieldsN=2k+1
N=2k
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A projected spin singlet Hilbert space
.,
.1,01,
;2
1,1
1,
,...1
,1,
,...|
evenlk
dk
N
kkd
xkk
kkd
kN
kkd
kN
In a projected spin singlet Hilbert space (states in a), b), d) and e)):i) An atom forms a singlet pair with another atom either at the same site or at a nearst neighboring site;ii) Each link is occupied by either one singlet pair of atoms or zero.
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An effective Hamiltonian
.,
ˆ
2
;2)ˆ(.).(,0.
;2
.
lkd
klaZH
Nkb
NkC
Echl
bk
bzlkkl
tm
H
ZH
mH
fqcH
.1,
~1,
;...11
,1,
~,1...|
...1
,1,
,...|
kkd
kkd
kN
kkd
kN
kN
kkd
kN
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Spin one bosons in optical lattices
Work in progress
• Towards fault tolerant quantum information storage
We have found
1) Half vortices in condensates
2) Nematic Mott insulators and spin singlet Mott insulators
3) Valence bond crystals (N=2k+1,1D)
4) Spin singlet condensates (1D)
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References
Zhou, Phys. Rev. Lett. 87, 080401(2001).Zhou, Int. Jour. Mod. Phys. B 17, 2643(2003).
Demler and Zhou, Phys. Rev. Lett. 88,163001(2002).Snoek and Zhou, cond-mat/0306198 (to appear in PRB).Imambekov et al., PRA. 68, 063602 (2003).
Zhou, Euro. Phys. Lett. 63 (4), 505(2003).Zhou and Snoek, Ann. Phys. 309(2), 692(2003). Constrained quantum rotor models 1) Quantum dimers, or compact gauge theories, 2) The bilinear-biqudartic spin model (S=1),3) and fractionalization of atoms etc.
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Hopping of S=1 Bosons
a) and d) SSMI states for even and
odd numbers of atoms.b) A kink-like S=0, Q=1 excitation
in an “odd” lattice.e) A string-like Q=1 excitation in
an “even” lattice.c) Hopping in an “odd” lattice
leads to kink-anti kink excitations.
f) In an “even” lattice hopping is suppressed because of a string of valence bonds between particle and hole excitations. Red dots are “charged”.
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Excitations in 1D Mott insulators
a)S=1, Q=0; b) S=0, Q=-1 for N=2k+1; c) S=0, Q=-2; d) Q=-1 for N=2k.