spin correlated states in optical lattices fei zhou ( itp, utrecht ) april 15, 2003 pims, banff...
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Spin Correlated States in Optical Lattices
Fei Zhou (ITP, Utrecht) April 15, 2003
PIMS, Banff National Park, Canada
Acknowledgement:
E.Demler (Harvard), F. D. Haldane (Princeton), P. W. Wiegmann (Chicago) G. Barkema, M. Snoek, J. Wiemer (Utrecht)
Funded by the FOM, the Netherlands
A lab for baby universe ?
n(2))
Creation of vortices, monopoles and half vortices.
(1), n(1))
2
21
ZSS
R
Atoms in optical lattices Vs. electrons in Cuprates
a) Free from imperfections
b) Known interactions
c) Tunable coupling constants
d) ld, 2d and 3d lattices
e) S=0, 1/2, 1, 3/2 atoms
a) Defects or disorder
b) Material dependent
c) Barely changeable
d) layered structures
e) S=1/2 electrons
S=1/2 Fermions in optical lattices
(small hopping limit)
Neel OrderedGapless Spin liquid HTcS made of cold atoms?
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)
Absorption images of interference patterns as the laser intensity is increased (from a to h).
(a-d) BECs and (g-h) Mott insulating statesGreiner et al., Nature 415, 39( 02)
.
2/1
0
2/1
)0,2
(:,
0
1
0
),0(:
.0||,|)1(:|
1|2
sin0|cos1|
2sin
),(|
RB
nSnnn
ie
ie
n
S=1 bosons with Anti-ferromagnetic interactions
.2,0
,,4
)()(
02
,2121
F
ggM
ag
grrrrU
FF
FF
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
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
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
Schematic of microscopic wave functions
a) NMI; b) SSMI (N=2k); c) SSMI (N=2k+1 in 1d).Each pair of blue and red dots with a ring is a spin singlet.
.|!
2/)(~,0;|
!
)(~,1;
vac
k N
Nk
Ck
C
SSMIvac
k N
Nnk
C
NMIEEtz
CS
.02:
);31
(2:
.312
OSSMI
nnNONMI
CCCCO
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.
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