Hubbard model calculations of phase separation in optical lattices

0.18 MB·English

by  H. Heiselberg

Hubbard model
al
ulations of phase separation in opti
al latti
es
∗
H. Heiselberg
Applied Resear
h, DALO, Lautrupbjerg 1-5, DK-2750 Ballerup, Denmark
Antiferromagneti
, Mott insulator, d-wave and gossamer super(cid:29)uid phases are
al
ulated for 2D
square latti
es from the extended Hubbard (t-J-U) model using the Gutzwiller proje
tion method
andrenormalizedmean(cid:28)eldtheory. PhaseseparationbetweenantiferromUa∼>g7n.e3tti
andd-wavesuper-
(cid:29)uidphasesisfoundnearhalf(cid:28)llingwhentheon-siterepulsionex
eeds ,and
oin
ides with
9
a (cid:28)rst order transition in the double o

upan
y. Phase separation is thus predi
ted for 2D opti
al
0
latti
eswithultra
oldFermiatomswhereasitisinhibitedin
upratesbyCoulombfrustrationwhi
h
0
insteadmayleadtostripes. Ina
on(cid:28)nedopti
allatti
etheresultingdensitydistributionisdis
on-
2
tinuousanwith extendedMottplateauwhi
henhan
estheantiferromagneti
phasebutsuppresses

n thesuper(cid:29)uidphase. ObservationofMottinsulator,antiferromagneti
,stripeandsuper(cid:29)uidphases
a in densityand momentumdistributions and
orrelations is dis
ussed.
J

9 PACSnumbers: 03.75.Ss,03.75.Lm,05.30.Fk,74.25.Ha,74.72.-h
2

] Ultra-
old atomi
Fermi-gasespresent a new opportu- niσ =aˆ†iσaˆiσ the density, Si = σσ′aˆ†iσ~σσσ′aˆiσ′ and hiji
n UP
nity to studystrongly
orrelatedquantummany-parti
le denotes nearest neighbours. is the on-site repulsive
o t
c systems and to emulate high temperature super
ondu
- interJa
tion, the nearest-neighbour hopping parameter
- tors (HT
). Opti
al latti
es realize the Hubbard model, and the spin-spin or super-ex
hange
oupling.
r
p when the periodi
al latti
e potential is strong enough so The t-J-U model allows for doubly o

upied sites and
u thatonlythelowestenergybandispopulated,andinter- thereby alsoMI transitions. As both areobserved in op-
s a
tions, densities, temperatures, et
.,
an be tuned. Re- ti
al latti
es the t-J-U model is more useful as opposed
. U/t
t
entexperimentswithopti
allatti
eshavemeasuredmo- to the t-J model whi
h allows neither. For large the
a
m mentum distributions and
orrelations, and have found t-J-U and HubbardJm=od4te2ls/Uredu
e to the t-J model with
super(cid:29)uidphasesofBoseandFermiatoms[1,2,3℄,Mott spin-spin
oupling dueto virtualhopping. At
- J U
d insulators (MI) [4, 5℄, and band insulators [6, 7℄. (cid:28)nite and the t-J-U model is to some extent dou-
n Long range Coulomb repulsion between ele
trons in ble
ounting with respe
t to the Hubbard model with
J =0
o
uprates prohibits phase separation (PS), whi
h in turn . However,whenRMFTisappliedthevirtualhop-
c
may lead to stripes, whereas PS is allowed for atoms in ping is suppressed in the Hubbard model whi
h justi(cid:28)es
[
opti
al latti
es. The various
ompeting phases
an be the expli
it in
lusion of the spin Hamiltonian as done

2
al
ulated in the 2D t-J-U model within the Gutzwiller in the t-J-U model. Also, opti
al (cid:16)superlatti
es(cid:17) provide
v proje
tion method and renormalized mean (cid:28)eld theory additional spin-spin intera
tion [10℄ and thus realize the
6
(RMFT). This method approximates the strong
orrela- un
onstrained t-J-U model.
9
tionsandgenerallyagreeswellwithfullvariationalMonte The t-J-U model on a 2D square latti
e at zero tem-
6
4 Carlo
al
ulations [8, 9℄. As will be shown below RMFT peraturewasstudiedforvariousonsite
oJuplin0g.3and0d.e5nt-
. predi
ts PS at low doping between an antiferromagneti
sities in Refs. [9, 11℄ but mainly for ≃ − ,
1
1 (AF)Neelorderandad-wavesuperU(cid:29)>ui7d.3(tdSF) phasefor whi
h is the relevant
ase for
uprates and organi
su-
8 su(cid:30)
iently strong onsite repulsion ∼ . The amount per
ondu
tors. In opti
al latti
es the Hubbard model
0 of MI, AF and dSF phases, the density distribution and
anberealizedwithanyon-siterepulsionbytuningnear
v: momentum
orrelations in opti
al latti
es are quite dif- a Feshba
h resonaJn
=e.4Ft2o/rUstrong onsite repulsion spin-
ferent from what would be expe
ted from HT

uprates. ex
hangerequires andwewill thereforestudy
i
X The t-J-U model was employed by Laughlin, Zhang the t-J-U model with this parameter
onstraint. How-
U
r and
oworkers [9℄ to study AF, HT
and (cid:16)gossamer su- ever, keeping in mind that for more moderate dou-
a
per
ondu
tivity(cid:17) in
uprates and organi
super
ondu
- ble
ounting may o

ur leading to an e(cid:27)e
tive value for
J
tors. Both the Hubbard and t-J models are in
luded in . We in
lude nearest-neighbour hopping and intera
-
H =H +H +H
U t s
the t-J-U Hamiltonian or tions only and thereforethe phase diagramis symmetri
n=1 n=N/M
around half (cid:28)lling , where is the density
H = UXi nˆi↑nˆi↓−t Xij σaˆ†iσaˆjσ + JXij SiSj, (1) orM(cid:28)lling fra
tion (N is thenumber ofele
tronsoratoms
h i h i in latti
es sites).
In the Gutzwiller approximation the trial wave fun
-
w, here aˆiσ is the usual Fermi
reation o9p,era7tor, σ40= (↑ tion|ψi=Πi[1−(1−g)nˆi↑nˆψi↓0]|ψ0iisaproje
tionofthe
↓) is the two hyper(cid:28)ne states (e.g. (−2 −2) for Na), Hartrgee-Fo
k wave fun
tion . The variational param-
eter suppresses double o

upan
y and varies between
U
0
orresponding to no doubly o

upied sites ( → ∞)
(U = 0)
∗ and 1
orresponding to no
orrelations . In the
Ele
troni
address: heiselbergmil.dk Gutzwillerproje
tionmethodthespatial
orrelationsare
2
g
t
in
luded onlythroughtherenormalizationfa
tors and UU==4400tt U=12t
gs de(cid:28)nedsu
hthathHsi=gshHsi0 andhHti=gthHti0, 0.2 0.2
ψ
∆ ∆
wψ0here the expe
tation values are with respe
t to and ∆SC ∆SC
respe
tively. The renormalization fa
tors are
al
u- m m
d d
lated by
lassi
al statisti
s [12, 13℄ 0.1 0.1
2
n 2d
g = − ,
s
(cid:18)n−2n+n−(cid:19) (2) 0 0
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
for the spin term and for the hopping term UUUUUUUUUUUUU=============8888888888888ttttttttttttt UUUUUU======666666tttttt
x n 0.2 0.2
gt = √gs −(x+d)+ −d ∆ ∆
(cid:20)rx+ rn+ (cid:21) ∆SC ∆SC
m m
x+(x+d)+ n+d . 0.1 d 0.1 d
(cid:20)rx rn (cid:21) (3)
− −
d n =n/2 m
Here isthe doubleo

upan
y, ± ± the meamn 0 0 0.1 0.2 0.3 0.4 0 0 0.1 0.2 0.3 0.4
upanddowno

upationnumbers foramagnetization
x = 1 n x = 1 n
at ea
h site, − the doping, and ± −g2±=. x = 1−n J/t =
The doubleo

upan
y proje
tionfa
torisgiven by 4Fti/gUure=10:.1O,1rd/3er,0p.5a,ra2m/3eters vs. doping for
d(x+d)/(x+x n+n gs) in RMFT for the2D t-J-Umodel.
− − .
The resulting energy is by Gutzwiller proje
tion
E =MUd+gt Ht 0+gs Hs 0. for the pairing gap, hopping averageand density
h i h i (4)
1 1 1
∆ = η2∆ + ,
The RMFT equations for the t-J-U model in the 8M Xk k d(cid:18)Ek+ Ek−(cid:19) (9)
Gutzwiller proje
tion method have been derived [11℄ in
1 ǫ ξ µ˜ ξ +µ˜
terms of the order parameters for d-wave pairing, hop- χ = γ k k− + k ,
ping average and staggerqed=m(aπg,nπe)tization (with
om- −8M Xk kξk (cid:18) Ek+ Ek− (cid:19) (10)
mensurate nesting ve
tor , in units where the 1 ξ µ˜ ξ +µ˜
k k
latti
e
on∆stainjt i=s unaˆitiyaˆ)jde(cid:28)naˆeidaˆajs 0 = ∆, x = 2M Xk (cid:18) E−k+ − Ek− (cid:19) . (11)
h i h ↓ ↑− ↑ ↓i ± (5)
m
χ = haˆ†i↑aˆj↓+aˆ†i↓aˆj↑i0, (6) The gap equationsthatddetermines the magnetization
m = (−1)ihaˆ†i↑aˆi↓+aˆ†i↓aˆi↑i0/2, (7) athnedrddeonuobrmleaoli
z
autpioann
fya
to[r1s1℄gsaraenmdogrtea
losmo pdleip
aetneddosnin
me
(+/ ) and .
respe
tively. The − in (5)ij
orresponds to a di(cid:27)er- The resulting order parameters for the t-JJ-U=m4to2/dUel
en
e(xb/eytw)een neighbour sites h i of one latti
e unit in
onstrainedtotheHubbardoUr/tt-J=modellimit
the dire
tion respe
tively. are presented in Fig. 1 for 40, 12, 8 and 6
orre-
J/t=
The resulting variational energy is [11℄ sponding to 0.1, 1/3, 0.5 and 2/3 respe
tively. At
1 half (cid:28)lling (x=0) we (cid:28)nd a(cid:28)rst orderphase transitionin
E/M = Ud−µ˜x− 2M Ek++Ek− whi
h theJdoub0l.e55ot

uUpan
7y.3jtumps from zero to a (cid:28)nite
Xk (cid:0) (cid:1) value for ≥ ( ≤ ) similar to Refs. [9, 11℄.
3
+ g J(∆2+χ2)+2g Jm2. The ground state remains an AF for smaller U without
s s
4 (8) anydSF.Thisisinagreementwithexpe
tationforthe2D
Hubbard model but in
ontrast to the results for a (cid:28)xed
J =t/3 U 7t
Here, the AF and dSF
ouples four band energies Ek± = [11℄,Uwhe5re.3ttheAFvanishesfor ≥ andadSF
pǫ (=ξk∓(gµ˜)t2++3g(∆Jdχη/k8))2γanγd −=E2k±(,cowsikth+ξkco=skq)ǫ2k+∆η 2af=, 5aa.pt3pheaalrUfs/(cid:28)ftollring7≥(a (cid:16)gosssua
mhetrhastupae
ro(cid:29)euxidis(cid:17)t)inegxiAstFs abnedtwdeSeFn
k t s k k x y k
2(cos−k cosk ) ∆ ,= 3g J∆/8 ∆ =an2dg Jm ≤ ≤ . The reJas=on4fto2r/Uthis di(cid:27)eren
e tra
esba
k
x y d s af s
− . and tothelargervaluefor inour
al
ulationwhi
h
are the dSF and AF order parameters. The Lagrange stabilizestheAFphaseanddestabilizesdSFbe
ausethe
µ˜
multiplier di(cid:27)ers from the
hemi
al potential be
ause
orresponding two order parameters
ompete. The (cid:28)rst
the renormalization fa
tors depend on density as will be ordertransitionin the doubleo

upan
y is, however,ro-
dis
ussed later. bustinthesensethatitisdrivenbytheon-siterepulsion
F(∆,χ,m,d,µ˜)=E µ˜N U 7 9t
Varyingthefreeenergy − with Jand remains at a
riti
al value ∼ − Jev=en4wt2h/eUn
respe
ttoits(cid:28)veparametersleadstothe(cid:16)gap(cid:17) equations is allowed to vary independently from the
3
Near half (cid:28)lling the
hemi
al potential de
reases with
0.6 ∆ =.02t in
reasing density in the AF phase i.e. the energy de-
SC
AF penden
e on density is
on
ave. This is unphysi
al and
PS signals PS to an AF phasexat<h0a.1lf4(cid:28)lling
oexisting with
0.4 a dSC phase at a density | |∼ shown in Fig. 2 that
/ t is determined by the Maxwell
oJnstru0
t.5io5nt. UThe P7S.3tis
J found to terminate at
oupling ≃ ( ≃ )
0.2 wherethedoubleo

upan
yundergoesa(cid:28)rstoJrd=er4ttr2a/nU-
sion from zero to a (cid:28)nite value. Relaxing the

onstraintdoesnot
hangethephasesorPSqualitatively
ex
ept for gossamer dSF.
0
Whereas PS is prohibited in
uprates by long range
0 0.1 0.2 0.3
Coulomb repulsion between ele
trons, PS is permitted
x=1-n
fortheneutral atoms trapped in opti
allatti
es where it
n(r)
leads to dis
ontinuities in the density distribution
r
Figure 2J:=Ph4at2s/eUdiagram for the t-J-U m|oxd|el under the
on- vs. trap radius . For a large number of trapped atoms
straint . AF and PS o

ur for below∆SreCsp≥e
0t.i2vte in a shallow
on(cid:28)ning potential, the Thomas-Fermi ap-

urves. Abovedotted
urvethedSFgapex
eeds . proximation applies (see, e.g. [14℄). The total
hemi
al
At halfJ(cid:28)≥lli0n.g55tthe (cid:28)rst order transition to double o

upan
y potential is then given by the lo
al
hemi
al potential
above is
omplementaryto PS(see text). µ(n) = dE/dN
V2r2 and the trap potential, whi
h is on the
form in most experiments,

onstraint. The
ase J = 0.5t (U = 8t) isd
lose to the µ(n)+V2r2 =µ(n=0)+V2R2. (14)

riti
al
ouplingandwe(cid:28)ndatransitionin atlowdop-
ing as shownmin Fig∆. 1 that also a(cid:27)e
ts the other order Thesummustbe
onstantoverthelatti
eandR
anthere-
parameters and SC. fore be set to its value at the edge or radius of o

u-
The gap depends sensitively on
oupling, magnetiza- pied latti
e sites, whi
h gives the r.h.s. in (14). The
m = d= 0
tion and do∆ub=leχo
=
upan
y.coIfs2k +cos2kth/e(2g√ap2Meq)ua-
hemi
al potential µfo(rnt=he0d)i=lute4lattti
e gas in the 2D
0ti.o3n39gives Pkp x y ≃ Hubbard model is µ(n =− 1)w=her0eas justUbe=low0
at half (cid:28)lling. At low densities th∆e gap=eq0uation half µ(cid:28)(llnin=g i1t)l=ies4btetweenU = when
simpli(cid:28)es be
ause there is no AF order af . The and when ∞. Correspondingly the
ge8nagteptr/ge3yqgusraJat)tioio≃n 23d∆Jep/d4e/tnǫkdi=ss0tohn=e teh(cid:27)Vee∆
pt,iavierwinphgaeriretionVgqu
oa≡usip-p1lia/nr(gtχid
+lie-
tarlnoedu,dltihesesiznbeu,emtwNwbheeeer=nnoPRπfSRctr2a≤oprpRaedM≤pIa2prRthica
,sleewshaNeprpe=eaR2rcπs =Rin0R2tnhp(ert)/
reVdn2r-,
vided by the bandwidth. Also the Fekrm=i s√ur2faπ
ne be- is of order c c. When more parti
les are added
F
µ
˜o=me(skF2
i−r
u4l)a(rgtw+it3hgsFJerχm/8i)m. oTmheenrt.uhm.s. of the gap eqaunad- athMe I
hpelmatie
aaul fpoortmenstiianlthheas
eantgraep(s∆eeµFiagt. h3a)lfbe(cid:28)
llaiunsge.
tion (9)
an then be
al
ulated analyti
ally to leading This gap and the
hemi
al potential above half (cid:28)lling
logarithmi
orderinthe e(cid:27)e
tive
ouplingandtoleading
an be found from parti
le-hole symmetry whi
h implies
E( x)=E(x)+MUx µ( x)=U µ(x)
orders in the density. The resulting gap be
omes − ,su
hthat − − . We
U ∆µ
1 4 1
∆= V√nexp(cid:20)−πn2(V −c0−c1n−c2n2)(cid:21) . (12) e(cid:28)vnedntthhaotuagsh thdee
droeuabselessoo

duopeasn
y ubnudteirtgroeemsaain(cid:28)srs(cid:28)tnoitre-
dertransitionto a(cid:28)nite value, i.e. theMIJrem=a4int2s/aUlong
Thehigherorcd0er
0o.r2r7e
tci1onsi0n.5d7ensityc2
an0b.e09
al
ulated with the AF phase. The approUximation is,
numeri
ally: ≃ , ≃ and ≃ . however, only a valid for large and the
onstrained t-
The AF order parameter is de(cid:28)ned as the eψxpe
- J-U mJodel maynot des
ribethe Hubbardmodel well for
tation valψue0as in Eq. (5) but for the w.f. | i in lNar>geNc∆. µT/hte MIplateau remainsin the trapR
>ent∆reµu/nVt2il
stead of | i . It therefore also attains a renormaliza- ∼
orresponding to a trap size ∼ .
tion fa
tor mAF = √gsm [11℄. Likewise the dSF or- In
reasing the number of trapped parti
les fuprther en-
d√egrs(p[araxm−e(xte+r ids)r+enornm−adli]z2e+d[∆SxC+(=x+g∆d∆)+withn+gd∆]2=) for
es doubnly=o2

upied sites and eventually a band insu-
2 qx+ qn+ qx− qn− , lator with in the
entre as seen in Fig. 3.
The density distributions and MI plateaus have been
and is shown in Fig. 1. We expe
t that the dSF
riti
al
measured experimentally for Bose atoms in opti
al lat-
temperature is similar.
ti
es by, e.g., di(cid:27)erentiating between singly and doubly
Sin
e the renormalization fa
tors depend on density
µ = dE/dN o

upied sites [2℄. A similar te
hnique applied to Fermi
d U 7.3t
the
hemi
al potential di(cid:27)ers from the La-
µ˜ atoms might observe the transition in at ≃ .
grange multiplier by
Re
entlyS
hneideret al. [5℄havemeasured
olumnden-
∂g ∂g
µ=µ˜+ Ht 0 t + Hs 0 s . sitiesforFermiatomsinopti
allatti
esand(cid:28)ndeviden
e
h i ∂n h i ∂n (13) forin
ompressibleMottandbandinsulatorphases. Even
4
oddintegers. JustasfortheAFphasewe
aninastripe
2 U=12t
phase expe
t
harge and spin
orrelations as in low en-
ergymagneti
neutrons
attering[16℄atin
ommensurate
q = 2π(0, 2x) q = π(1,1 2x)
± and ± respe
tively, ob-
1 5 10 servedasdips at these wave-numbers. However, be
ause
x
n 1 the doping varies in the trap, the Bxragg dips are dis-
tributed over the range of values for and are therefore
0.5 harder to distinguish from the ba
kg1r/o8und.xIf, 1h/o4wever,
thefourstripeperiodi
ityo

ursfor ≤ ≤ asin

uprates, we
an expe
t novelBragg dips for
harge
or-
q=(0, π/2) q=( π/2,0)
0 relationsat ± and ± andforspin
0 1 2 3 q = π(1,1 1/4) q = π(1 1/4,1)

orrelations at ± and ± .
r/R
c Pairing leads to bun
hing between opposite momenta
k = k
′
− and s-wave pairing has been observed near
Figure 3: Density distributions of FermiUato=ms12int aJn =optti/
3al the BCS-BEC
ross-over[3℄. If dSF is enhan
ed or sup-
latti
e
on(cid:28)ned by a harmoni
traNp/fNorc = ( ) pressed by stripes, the bun
hing due to dSF should be
(cid:28)lled with the numberof parti
lesn ≃ 1.14 0.5, 1n, 5≃, a0n.8d610.
orrelated
orrespondinglywith stripe anti-bun
hing.
The densitny=dis1
ontinuities from and to
the MI at due to PS between dSF and AF are absent 3D opti
al latti
es do not have a van Hove singu-
for stripe phases (see text).
larity at the Fermi surfa
e at half (cid:28)lling as the 2D
∆
k
Hubbard model, and the 2D d-wave symmetry ∝
(cosk cosk )
x y
− does not generalize to 3D. Thus we do
lowertemperaturesarerequiredforobservingtheAFand
not expe
t any signi(cid:28)
ant dSF but by generalizing the
dSF phases and the density dis
ontinuities due to PS.
RMFT equations to 3D we (cid:28)nd MI, AF and PS for the
Spin and
harge density waves in form of stripes are
same reasonsthat they appear in 2D.
not in
luded in the above RMFT
al
ulations. Stripes
are observed in several
uprates [16℄ whereas numeri
al In
on
lusion,t-J-UmodelRMFT
al
ulationspredi
t

al
ulations are model dependent. Long range Coulomb phaseseparationanddensitydis
ontinuUit>ie7s.3ntearhalf(cid:28)ll-
frustration
an explain why PS is repla
ed by stripes ingin2Dand3Dopti
allatti
eswhen ∼
oin
iding
and an AF phase at very low doping [14℄ as observed with a (cid:28)rst order transition in the double o

upan
y at
in
uprates. Cold atoms in opti
al latti
es
an dis
rim- half (cid:28)lling. Observation of phase separation would indi-
inate between the PS and stripe phases sin
e the latter
ate that long range Coulomb frustration is most likely
does not have density dis
ontinuities. the
auseforspinand
hargedensitywavesandstripesin
Bragg peaks have been observed for bosons in 3D [2℄
uprates. Contrarily, if no PS is observed but instead a
and 2D [4℄ opti
al latti
es, and dips for 3D fermions in stripe phase near half (cid:28)lling in 2D and probablyalso3D
[7℄. The Bragg peaks and dips o

ur in momentum
or- opti
al latti
es, then Coulomb frustration is not respon-
C(k,k)
′
relation fun
tions when the relative momentum sibleforstripesin
uprates. Ineither
aseopti
allatti
es
q k k = π(n ,n ) n ,n
′ x y x y
is ≡ − , where are even in- not only emulate strongly
orrelated systems, Hubbard
tegers [15℄. In an AF phase the periodi
ity of a given type models and
an determine the ground state phases
spin is two latti
e distan
es and dips also appears for su
h as MI, AF, PS, gossamer and dSF but
an even
determine more subtle e(cid:27)e
ts from Coulombfrustration.
[1℄ T. Stöferle et al., Phys. Rev. Lett. 96, 030401 (2006). [10℄ A.KleinandD.Jaks
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