Probing the physics of strong correlations with cold fermions in

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C
T
py
% The Abdus Salam
"
International Centre forTheoretical Physics
09
SMR 1666-32
SCHOOL ON QUANTUM PHASE TRANSITIONS
AND
NON-EQUILIBRIUM PHENOMENA IN COLD ATOMIC GASES
11-22 July 2005
Probing the physics of strong correlations with cold fermions in
optical lattices: adiabatic cooling and quantum magnetism
Presented by:
Antoine Georges
Centre de Physique Theorique,
Ecole Polytechnique, France
1 I. 34014
II
-
I. *39 040 2240 11 I. F
*39 040 22463 -....
Atoms in an optical lattice:
when does the Hubbard model apply?
(?) = T'
(k
1
L x.)
A wavelength of Laser
A/2 lattice spacing
V
ER
7j2k2
Typical orders of magnitude (6Li)
.
f_A.)
I
I
LI
-1__ d
-f -
2m
.
Pioneering article: Jaksch et al, PRL 81(1998)_3108
Free-particle bands: from Bloch
waves to Wannier functions
H1 I i5k) = E,ik
k.,
-I-Vsin2(ktx)1
=
1t2
-2771
31
02
.,
dx-
"
2-
e
(x
Wannier function localised
on lattice site R:
TV-) =
LU
=
vN
:
-
'3 -it
k
st
VO=ER: 1 and 2' band (1 D)
Contour plot of 2D Wannier function V0= 10 ER
I
Li
-K..
,1
-1
'3
yf
1
2
Hoppings and couplings...
10
0.1
0.01
0001
Fic. 13
10
VO Er
20
30
- Coefficient R du Harniltonien drns hi base de Wannier, en fonction de
lit prefondeur du potentiel Vu/Er. Les coefficients tt' et A (traits pointiflé.s)
a,
sont expmies en unites de
tancks que k's coefficients d.c l'interaction UU2
etc. (traits pleins) sont en unites de E a5/a.
Large U/to. Mott insulator
.k-t-t-t + +-+
1
Intersite hopping is
blocked if:
tunneling amplitude (t)
is small enough
compared to
U = on-site repulsive
interaction
Real-space picture is simple mostly singly-occupied sites
NOTE: At large U, Mott localisation has nothing to do
with spin ordering. Gap for charge motion U.
However, at low-T, long-range spin order will (in most
cases...) set in, at a critical temperature T,
For T<T<U: random mixture of spins (paramagnet)
The phase diagram at 'h-filling (3D case)
1.5
I
I
Coherence scale (effective Fermi energy)
I
It
I
ITINERANT
(CORRE-+-+-O-t-t-ff4
"t-t-t-t-t-'t-t
FERMI
imd
015
o
'
0
I
AN
4-t-4-t-t-t-4
i
LGNET
I
.
&
"
s
I
A
I
I
20
Critical boundary calculated for a 3D cubic lattice using:
-Quantum Monte Carlo (Staiidt et i1 Piir Phvc T P17 (7fl1)Ifl 4111
-
Dynamical Mean-Field Theory approximation
Fermionse. Dynamical Mean-Field Theory
cf Rev. Mod. Phys. 68 (1996) 13
& Physics Today, 2003
'N
BATH
EITECTIVF
Effectit-ebith
-
- -
IMPTRITY PROBLEM
--
-_fl
DNIFT
LOOP
SE Lf CO'S SISTE"C% CONDITION
-/ dr/ dT'
C+ (T)G~)'(T
-r')c(Y)+U
/
/3
dT n. t (T) nj(T)
LaTiO3
VIZ
-EM,
i Strongly correlated materials
who are the suspects?
Localized orbitals (close enough to nuclei): 3d, 4f
[Materials with transition metals or rare-earth ions]
>> Strong screening: on-site matrix element
of Coulomb interaction plays the dominant role
U
- r')IFVj(r')2
Vnit
(r
'.(r)12
screene d
/drdr'W
&1
>> Narrow bandwidths (small kinetic energy)
.
-4-Cr
What do real
(strongly correlated)
solids do ?
A material poised
close to the Mott
instability:
V203
Note: slope of Tc vs.p
again the
Pomeranchuk effect!
Recent experiments
(Limelette et al, Science
2003) show that critical
Endpoint is indeed
Ising-like
Vi_Mx)aO3
500
CRITICAL MN
400
-
OO
P aramagn
iIN
nsMAUr
4
C
Paramagnetic
metal
METAL
I
PRESSURE EXP.
4-
144
444-44+1--
w
° T
100
0
4-14-44-44-0
-t ric
4-1-fkntil'ferromagne
insu1ator
*
INCREASING PRESSURE i4kbOr/cXVISIQN
ZERO PRESSURE POINT MOVES WITH TOP SCAt.E
FIG. 70. Phase
CrO" and
(V1
1973.
for doped V2O- systems,
rTj)2Q. From McWhan et a! 1971,
diagram
(V1
Example of RVB state:
The Kagome quantum
antiferromagnet
I
I ______
I
I
I
I
I
--
"
___
(I
0.41
:!.
0 2E- i-
'
TA I I
i
a
Sb--I)
1<1 I
a
4
S(S
1)
spec-trum
FIG. 2. The [ow-lying energy levels of the TAIl and 1CM-I
of the N = 27 sample The levels which possess the symmetry
expected for an ordered solution are denoted by a star. The Pisa
tower of the TAH is easily seen, quite distinct from the
first-mattnon excitations- In the ECAH1 on the contrary, the levels candidate for the building of a tower of states are mixed with other
representations in a eontinuLLm of excitations.
NO GAP IN 5=0 SECTOR!
NO mD AMCT A "T'TCIM CmDV
BREAKING
Lecheminant et a!.- C.Lhuillier's group
GAP TO S=1 SECTOR
See also: recent work on Moessner and Sondhi
on quantum dimer models.
INkJ
11\±%INO.LiflhlkJIN
LJIJXI
lattice
Kagome
optical
et al.
Rev Lett 93
[Santos
Phys
(2004) 030601]
15
IS
III
ç
Ii
>r
- II
- J5
Ir!'!t!!W
-15-li' -5
I)
x
5
l'J
IS
x
:
II
IS
(C)
sYfl-
7OS
/
/
(b) TKL using
= n/4. This lattice can be generated using/2.three
SWs with a
FIG. 1.
(a) Ideal Kagonié lattice for
=
/3 angle between themselves. Each SW is generated by three
lasers with a configuration shown in (c). (d) Enumeration of
spins in a turner and of neighboring trilners.