Cold Attractive Spin Polarized Fermi Lattice Gases and the Doped Positive U Hubbard Model

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by  Adriana Moreo

Cold Attractive Spin Polarized Fermi Lattice Gases and the Doped Positive U
Hubbard Model
Adriana Moreo1 and D. J. Scalapino2
1Department of Physics and Astronomy,University of Tennessee, Knoxville, TN 37966-1200 and
Oak Ridge National Laboratory,Oak Ridge, TN 37831-6032,USA, and
2Department of Physics, University of California, Santa Barbara, CA 93106-9530 USA
(Dated: February 6, 2008)
7
0
Experimentsonpolarizedfermiongasesperformedbytrappingultracoldatomsinopticallattices,
0
allow the study of an attractive Hubbard model for which the strength of the on site interaction
2
is tuned by means of a Feshbach resonance. Using a well-known particle-hole transformation we

n discusshowresultsobtainedforthissystemcanbereinterpretedinthecontextofadopedrepulsive
a Hubbard model. In particular we show that the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state
J correspondstothestripedstateofthetwo-dimensionaldopedpositiveUHubbardmodel. Wethen

6 use the results of numerical studies of the striped state to relate theperiodicity of the FFLO state
2 tothespinpolarization. Wealsocommentontherelationship ofthedx2−y2 superconductingphase
of the doped 2D repulsive Hubbard model to a d-wave spin density wave state for the attractive

] case.
l
e
- PACSnumbers: 71.10.Fd,03.75.Ss,74.25.Ha
r
t
s
. Using standing wave laser light fields and optical Fes- a square lattice and n =c† c is the number operator.
t is is is
a hbach resonances, ultracold atomic gases can be used to In the case of the ultracold gases, a spin polarization
m
realizeavarietyofHubbardlikemodels.1,2 Technicallyif p = (N −N )/N is achieved by loading more atoms in
↑ ↓
- one wants a strong interaction, it appears easier to cre- the“up”pseudo-spinhyperfinestatethaninthe “down”
d
ate an ultracold fermi gas (e.g., 6Li) with an attractive state.
n
o interaction and experiments on polarized fermion gases In the mean-field FFLO state, the Cooper pairs have
c withattractiveinteractionshavebeencarriedouttolook a finite center of mass momentum which, for a two di-
[ for exotic superfluid states.3,4,5,6 mensional tight binding band structure, lays along the

One state of particular interest is the FFLO state (1,0) or the (1,1) direction depending upon the ratio of
1
v which arises from pairing across the spin-split Fermi |U|/t. For the case in which it lays along the (1,0) di-
9 surface of a spin polarized system.7,8 In this state the rection, the order parameter exhibits a one-dimensional
4 Cooper pairs have a finite center of mass momentum oscillation along the x direction
6 leading to spatial oscillations of the pair field order
1
parameter and the spin polarization. Here, making
0
7 use of density-matrix-renormalization-group (DMRG) ∆(ℓx)=Rehcℓ↑cℓ↓i=−∆0cos(qxℓx). (2)
results9,10forthedoped,positiveUHubbardmodelanda
0
/ well-knownparticle-holetransformation,11,12,13,14wedis- Here we have chosen the phase so that ∆(0) is negative.
at cuss the properties of the FFLO state for the case of a The spin polarization also varies in space with
m half-filled spin polarized negative U Hubbard model on
a two-dimensional lattice. We also comment on the re-
-
d lationship of this to the question of whether the doped ns(ℓx)=hnℓ↑−nℓ↓i=p−m0cos(2qxℓx). (3)
n
o positive U Hubbard model may have a dx2−y2 supercon- Herecos(q ℓ )andcos(2q ℓ )givetheleadingweakcou-
ducting ground state. x x x x
:c The Hamiltonian for a half-filled negative U Hubbard pling harmonic behavior. The spatial variation of ∆(ℓx)
v and n (ℓ ) are schematically illustrated in Fig. 1. At
model in an external Zeeman field, H, can be written as s x
i stronger coupling higher harmonics can enter giving a
X
more localized behavior.
r U
a H=−tX(c†iscjs+c†jscis)+ 2 X(ni↑−ni↓)2 proSpinocseedt,h7,e8pthoessriebhilaitsyboefenangrFeaFtLiOntesrteastteinwadsetoerrimgiinnainllgy
hijis i
whether it exists and exploring its properties. In par-
ticular, with the possibility of observing such a state in
−hX(ni↑−ni↓). (1) ultracoldatomicgases,therehavebeenanumberofthe-
oretical calculations.15,16,17,18,19,20,21 These calculations
i
have been based upon a mean field description. Here we
Here t is a nearest neighbor one-electron hopping, U is are interested in the case of a two-dimensionalhalf-filled
positive so the onsite interaction is −Uni↑ni↓ and h = lattice gaswhere it is importantto treatstrong coupling
gµ H/2. The operator c† creates a fermion with spin s effects. One would like to know if the FFLO state sur-
0 is
on site i, the sum hiji is over nearest neighbor sites of vives beyond the mean field approximation,whether the
2
stripes run vertically (or horizontally)as in Eqs.(2) and Mean field studies13,23,24,25,26,27 of the doped, two-
(3)ordiagonally,andwhethertheperiodicityofthespa- dimensional positive U Hubbard model with a nearest-
tial variation which determines q remains at its mean neighbor hopping find that the domain-walls run along
x
field value. the x or y-directions for |U|/t <∼ 3.6. For values of
The Hamiltonian for the negative U Hubbard model 3.6 <∼ |U|/t <∼ 8 the domain-walls run diagonally and
in a Zeeman field can be transformed under a unitary for |U|/t greater than 8 the domain-walls are found to
transformation to a doped positive U Hubbard model. be unstable with respect to the formation of magnetic
This transformation,introduced by Emery,11 has been a polarons.28 The mean field domain walls are found to
stapleinquantumMonteCarloworkwhereitwasusedto contain one hole per site along the wall and one says
showthatthefermionsignproblemisabsentforthehalf- that the domain wall is filled with holes. This means
filled Hubbard model with a near-neighbor hopping.12 that the spacing of the vertical (or horizontal) charged
It has also been used, as we will review, to provide a domainwallsisequaltop−1,givingthemeanfieldresult
map between the half-filled negative U Hubbard model
q =πp. (9)
in a Zeeman field and the doped positive U Hubbard x
model.12,13,14,22 Under this transformation we will see
for suchwalls. A schematic illustrationof the meanfield
thattheFFLOstatebecomesthefamiliarstriped,charge
results for p=0.125 is shown in Fig. 1a.
density, and π-phase shifted antiferromagnetic state of
the doped positive U Hubbard model. 0.2
For a unitary transformation in which
0.1
n
s
0
cℓ↑ →(−1)ℓx+ℓyd†ℓ↑ cℓ↓ →dℓ↓ (4) -0.1 ∆
(a)
the Hamiltonian given in Eq. (1) becomes -0.2
0 5 10
0.2
H=−tX(d†isdjs+d†jsdis)+ U2 X(ni↑+ni↓−1)2 0.1 ns
0
hijis
∆
-0.1
(b)
-0.2
−µX(ni↑+ni↓) (5) 0 5 10
i l
x
withµ=−h. Thatis,thehalf-fillednegativeUHubbard
model with spin polarizationp is mapped into a positive
FIG.1: (Coloronline)Schematicresultsforthespatialvaria-
U Hubbard model with a site filling hni = 1−p. Under
tionofthes-waveorderparameter∆(ℓx)(solid)andthespin
this transformation Eq. (2) becomes polarizationns(ℓx)(dashed)inanFFLOstatewithp=0.125.
(a)Themeanfieldresultwithaperiodfor∆(ℓx)of2/p=16
and (b) with a period 1/p = 8 expected from DMRG calcu-
(−1)ℓx+ℓyhMx(ℓx,ℓy)i=−∆0cosqxℓx (6) lations on long stripes.
with M (ℓ) = (d† d +d† d )/2. The system has ro-
x ℓ↑ ℓ↓ ℓ↓ ℓ↑
Nowthesearemeanfieldresultsandatlargervaluesof
tational symmetry so that Eq. (6) implies that the stag-
Uoneis dealingwithastronglycorrelatedsystem. Thus
gered spin order is modulated by cosq ℓ .
x x
one might wonder whether the FFLO state remains the
ground state. This is a question that has also played
(−1)ℓx+ℓy a central role in the high Tc cuprate problem. There
m (ℓ )= hn (ℓ ,ℓ )−n (ℓ ,ℓ )i
stag x ↑ x y ↓ x y one would like to know if the ground state of the doped
2
=−∆ cosq ℓ . (7) Hubbard model is striped or possibly a dx2−y2 super-
0 x x
conductor, and what is the nature of the interplay be-
Similarly, the spin polarization, Eq. (3), transforms to tweenstripes andpairing.29 PresentDMRG calculations
the hole density giving on6-legHubbardladdersfindastripedgroundstate.9,10,
while dynamic cluster Monte Carlo calculations30 onpe-
h(ℓx)=1−hn(ℓx,ℓy)i=p−m0cos(2qxℓx). (8) riodic lattices find evidence of a dx2−y2 superconducting
state. Whether the difference oflatticeaspectratiosand
Thus the FFLO state maps to the striped state of boundary conditions or subtle numerical biases are re-
the doped positive U Hubbard model in which charged sponsible for this disagreement is not known. It does,
stripes separate π-phase shifted antiferromagnetic re- however,appearthatthesetwostatesareverycloseinen-
gions. ergy. Here we will use the DMRG results for the striped
3
state to discuss the nature of the FFLO state and then associated with two sites connected by a hopping per-
conclude by noting what would happen for the dx2−y2 pendicular to the stripe is opposite to that ofthe case in
state. whichthehoppingisparalleltothestripe. Thesed-wave
The DMRG calculations have been carried out on 6- like pairing correlationsof the holes in the stripe are not
leg doped positive U Hubbard ladders. Periodic bound- taken into account in the mean field calculations which
ary conditions have been imposed in the 6-leg direction find filled stripes. For half-filled stripes, the spacing be-
and open boundary conditions are used in the long x- tween the charged stripes is 1/2p so that the numerical
direction. In Ref. 10, the results have been extrapo- calculations on the longer stripes imply that q for the
x
lated to infinite length ladders. These numerical results FFLO state will be twice that found in the mean field
find that the ground state exhibits the striped structure calculations.
shown in Fig. 2 for a 6×21 ladder with U/t = 12 and
12 holes corresponding to a doping p=2/21. The figure
shows the hole density qx =2πp. (12)
6 In this case, the FFLO state will have half the spa-
1
h(ℓx)= X(1−hn(ℓx,ℓy)i), (10) tial period of the mean field solution, as illustration in
6
ℓy=1 Fig. 1b.
Here we have made use of DMRG calculations which
and the staggered spin density werecarriedoutonsystemswithperiodicboundarycon-
ditions in one direction and open boundary conditions
6
mstag(ℓx)= 1 X(−1)ℓx+ℓyhn↑(ℓx,ℓy)−n↓(ℓx,ℓy)i, in the other. In cold atoms experiments, one is gener-
6 allydealingwithanunderlyingopticallatticeinaslowly
ℓy=1 varying trap potential which acts as a spatially depen-
(11)
dent chemical potential. This leads to soft boundaries
versusℓ . Forthisdoping,extrapolatedDMRGresults10
x and the simultaneous coexistence of spatially separated
show that the ground state is striped for U/t >∼ 3. For
phases. The details of this depend upon the total filling
a 6-leg tube, the stripes are found to contain 4 holes
aswellasthecurvatureoftheconfiningtrappotential.33
(2/3 filled stripes) so that the charge stripe spacing is
Inprinciple,theDMRGmethodiswellsuitedtotreating
2/(3p)=7 with hni=1−p.
such soft boundaries but here we have focused our dis-
cussiononthe idealizedcaseofatwo-dimensionallattice
1
which has one fermion per site with an attractive Fes-
0.8
hbach tuned interaction and a spin polarization p. We
)x0.6 have shown that the FFLO state of a 2D half-filled at-
h(l 0.4 tractivespinpolarizedHubbardmodelisaunitarytrans-
0.2 formofthe stripedstateofthe dopedrepulsiveHubbard
model. Thus an experimental observation of the FFLO
0
state in the above mentioned cold atom set-up would
2 imply that the 2D doped repulsive Hubbard model is
)
x
(lg 0 striped. Note that we have not shown that the striped
msta state is the ground state of the two-dimensional doped
-2 repulsiveUHubbardmodel. Thisremainsanopenques-
tion. Wehavesimplyshownthatifthestripedstatewere
-4
0 5 10 15 20 the ground state then this would imply that the FFLO
l
x state would be found in the optical lattice experiments
and that its periodicity would be given by Eq. (12). We
believe that the factor of two increase in q represents
x
FIG. 2: (Color online). This figure shows DMRG results
an important change from the mean field result and re-
s(tGa.ggHearegderseptinalm.10st)agfo(ℓrxt)h(esohloidlebhlu(ℓex))d(ednassithieeds arleodn)gatnhdetlhege flects the tendency towards dx2−y2 pairing.32 Recent ex-
directionfora21×6Hubbardladderwith12holesandU/t= periments alsosuggestthat localpairingcorrelationsare
present on the cuprate stripes.34
12. As discussed in the text, h(ℓx) corresponds to the spin
polarization ns(ℓx) and mstag(ℓx) corresponds to the s-wave FindingevidenceforanFFLOstateinacoldattractive
pairfield order parameter ∆(ℓx) of the FFLO state. spin polarized fermi lattice gas would provide evidence
that the low temperature phase of the doped positive
U Hubbard model is striped. If on the other hand, the
Calculations on an 8-leg t − J ladder find that the dx2−y2 superconducting state describes the low temper-
stripes are half-filled and there is evidence that this is ature phase of the repulsive Hubbard model, then the
the preferred filling.31 These calculations show that the transformation, Eq. (4), tells us that the cold atom sys-
holesthatmakeupthestripe exhibitshortrangepairing tem will exhibit a “d-spin density wave” ground state
correlations32 in which the sign of the singlet pair field characterizedby an order parameter
4
experimentalworkoncoldfermiongasesandL.Balents,
S.A. Kivelson, and I. Affleck for useful discussions re-
X(coskx−cosky)hc†k+Q↑ck↓i (13) garding the FFLO state. We would also like to thank
k G. Hager, G. Wellein, E. Jeckelmann and H. Fehske for
providing the data for Figure 2. A.M. is supported by
with Q = (π,π). Thus experiments on half-filled, spin
NSFundergrantsDMR-0443144andDMR-0454504,and
polarized cold fermi lattice gases with attractive on site
partial support under Contract DE-AC05-00OR22725
interactions and the search for the FFLO state have im-
with UT-Battelle, LLC. D.J.S. would like to acknowl-
portant implications for the properties of the doped re-
edge the Center for Nanophase Material Science at Oak
pulsive U Hubbard model. Likewise, our present under-
Ridge National Laboratory for support.
standing ofthe repulsiveHubbardmodel canprovidein-
formation relevant to experiments on the spin polarized
attractive fermi gas.
Acknowledgements
We would like to thank T. Esslinger, E. Mueller, and
M. Holland for helpful discussions regarding the recent
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