Table Of ContentPairingstates ofa polarizedFermi gastrapped ina one-dimensional optical lattice
A. E. Feiguin1 and F. Heidrich-Meisner2
1Microsoft Station Q, University of California, Santa Barbara, California 93106
2Materials Science and Technology Division, Oak Ridge National Laboratory,
Oak Ridge, Tennessee, 37831, and Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA
(Dated:February1,2008)
Westudythepropertiesofaone-dimensional(1D)gasoffermionstrappedinalatticebymeansofthedensity
matrixrenormalizationgroupmethod, focusingonthecaseofunequalspinpopulations, andstrongattractive
interaction. In the low density regime, the system phase-separates into a well defined superconducting core
8
andafullypolarized metalliccloud surrounding it. Wearguethat thesuperconducting phase corresponds to
0
a1DanalogoftheFulde-Ferrell-Larkin-Ovchinnikov(FFLO)state,withaquasi-condensateoftightlybound
0
bosonicpairswithafinitecenter-of-massmomentumthatscaleslinearlywiththemagnetization. Inthelarge
2
densitylimit,thesystemallowsforfour phases: inthecore, weeitherfindaFockstateoflocalizedpairsor
n ametallicshellwithfreespin-downfermionsmovinginafullyfilledbackgroundofspin-upfermions. Asthe
a magnetizationincreases,theFockstatedisappearstogiveroomforametallicphase,withapartiallypolarized
J
superconductingFFLOshellandafullypolarizedmetalliccloudsurroundingthecore.
2
PACSnumbers:03.75.Ss,03.75.Mn,03.75.Hh,71.10.Pm,71.10.Fd
]
l
e
- Ultracold atoms in optical lattices can be used to study larized) Fermi surface.7 However, the instability against an
r
t modelsofstronglycorrelatedfermionsincleanandcontrolled FFLO state may be enhanced in low dimensions, rendering
s
experimentalconditions. Inparticular,coldgasesprovidean thisstatemorerobust.
.
at optimalplaygroundtostudythecrossoverbetweenaBardeen- The uniform 1D polarized Fermi gas has been stud-
m Cooper-Schrieffer (BCS) superfluid, with extended Cooper ied by means of bosonization and renormalization group
pairs, and a Bose-Einstein condensate (BEC), composed of techniques,13providingevidencefortheexistenceofanFFLO
-
d molecules of tightly bound pairs.1 As cold atom gases can state. The case of fermions confined to a 1D trap has
n also be realizedin opticallattices, dimensionalcrossoveref- been analyzed using modified versions of the Gaudin-Yang
o fects can be accessed.2 In particular, 1D optical lattices can Hamiltonian,14,15 a minimal integrable model of fermions in
c
be prepared by strongly ramping up the amplitudes of two the continuum interacting via a contact potential. A two-
[
outofthreecounterpropagatinglightwaves.2This,combined shellstructurehas beenpredicted,16,17 with a partially polar-
4 with the possibility of tuning the interactions, allows for the ized phase of the FFLO type in the center of the trap, and
v
realization of the fermionic 1D Hubbard model, with, in eitherfullypairedorfullypolarizedwings,dependingonthe
2
experiments, two hyperfine states interacting via an onsite strengthofthemagneticfieldor, equivalently,thetotalmag-
7
potential.3 In the case of a spin imbalanced fermion mix- netization.
1
4 ture, magnetized superconducting states are expected, such In this work we investigate the FFLO state in a Hubbard
. as,e.g.,asuperfluid-normalmixture4 ortheonepredictedby chain,thusaccoutingfortheopticallattice:
7
FuldeandFerrell5andLarkinandOvchinnikov6(FFLOstate)
0
7 over four decadesago. The FFLO state7 is characterizedby H = − t L−1 c† c +h.c. +U L n n
0 pairing across a spin-split Fermi surface, with the resulting X (cid:16) iσ i+1σ (cid:17) X i↑ i↓
: Cooperpairshavinga finitecenter-of-massmomentum,pro- i=1,σ i=1
v
portionaltothespinpolarization,andconsequently,anoscil- L
Xi latory phase in the superconductingcorrelationfunction. Its + V X(x−L/2)2ni, (1)
r experimentalobservationhaseludedcondensedmatterphysi- i=1
a
cistuntilveryrecently,whenitwasdetectedinheavy-fermion where c† creates a fermion with spin σ =↑,↓ at site l,
systems.8 Its realization in cold atom systems has acquired ℓσ
n = c† c , n = n +n is the local density, t is the
particular relevance for the field of high-T superconductiv- ℓσ ℓσ ℓσ ℓ ℓ↑ ℓ↓
c
ity. MoreoandScalapino9 haverecentlypointedoutthat,by hopping parameter, and U is the onsite interaction energy,
which in this work is negative. We define x = ia, where a
exploitingaparticle-holetransformation,10itspresenceinthe
is the lattice spacing, set to unity. We add a harmonic con-
2D attractive Hubbard model may imply the existence of a
fining potential parameterized by a constant V. The Hub-
stripedphaseinitsrepulsivecounterpart.
bardmodelwithV = 0hasbeenextensivelystudied,andits
Experiments on 3D traps11,12 suggest the existence of properties are well documented in the literature.18 The low-
phase-separated shells, with a superfluid core, and a par- energy properties of the Hamiltonian (1) with U < 0 are
tiallypolarizednormalcloud,andafuturecharacterizationof thoseofaLuther-Emeryliquid.19 Atsmallattractiveinterac-
thesuperconductingphasesmayestablishwhethertheFFLO tions,fermionsformCooper-pair-likeboundstateswithaspin
phase is present or not. Theoretical work indicates that the gap, reminiscentof the superconductinggap in conventional
FFLO phaseisstablein anarrowwindowaroundthe(unpo- BSC superconductors. In the case of stronginteractions, the
2
1.5 U(a=)-8t, L=80, N=40, V=0.002t (b) 0.03 0.08 0.1 (b) N=80, V=0.0S00z=58t (c) 1.5
<n>i 1 0.02zz>/S<Si 0.06 pair|n|lm1100--23 2 π Sz /L 01.5kmax
0.5 0.01 Npair 10-41 10 0 0.05 0.1 0.15eff0.2eff 0
1.20 (c) (d) 0 pairn/k0.04 |l-m| Szeff/Leff
Sz=0 0.06
2δ<(n)>i00..48 SSSSSzzzzz=====246810 00..0024z2δ)><(Si 0.02 (a) U=-8t, L=80, N=40, V=0.002tincreasing Sz=0,2,4,6,8,10
00 π/2 π
0 0 momentum k
0 20 40 60 0 20 40 60 80
site i site i
FIG.2: (coloronline)(a)Pairmomentumdistributionfunctionfor
FIG.1: (coloronline)Resultsforthe1DattractiveHubbardmodel N = 40fermionswithNpair = P npair. Thedottedlinesshow
k k
(U = 8t,L=80),withN =40fermionsconfinedbyaparabolic results for the sum in Eq. (2) restricted to the center of the trap.
−
potential with strength V = 0.002t, for different magnetizations: (b)Spatialdecayofpaircorrelations(squares). Dashedline: Fitto
(a)Densityprofile,(b)magnetizationprofile,(c)chargefluctuations, npair cos(k l m)/l mα (compareRef.13). Datashown
and(d)spinfluctuations. folrmL=∝100,N|=−80,|V|=−0.00|05t,Sz =8,correspondingtothe
sameeffectivedensityρeff =N√V.(c)Momentumkmax,atwhich
thedistributionshownin(a)ispeaked,vs.magnetizationofthecore
pairs becometightly boundwith their extensionof the order Sezff/Leff (Leff ≈40,seetext).
of the lattice spacing only, effectivelybehavingas hard-core
bosons,whilethespindegreeoffreedommovestohighener-
Forthispurpose,itisillustratingtoanalyzethenaturalpair
gies. Superconductingcorrelationsthen decay algebraically.
excitations of the system in terms of a bosonic description
The parabolic trap adds an extra ingredient that leads to the
withcreationandannihilationoperatorsb† =c† c† andb =
emergenceof non-uniformstates, and the inclusion of a lat- i i↑ i↓ i
c c . In the low-density limit, indeed, the pairs behave in
ticemayproducearicherphasediagram. Ithasbeenshown, i↓ i↑
for instance, that this allows for the possibility of engineer- goodapproximationlikebosonssince[b†i,bi]=1−2ni ≈1,
butwe will use it generically, as it providesan intuitive pic-
ing states of hard-core bosons such as pure Fock states that
cannot be found in the continuum.20 At the same time, the ture. We further define the pair one-particle density matrix
physics of the system without a lattice can be recovered in (OPDM)asρij =hb†ibji.
thelowdensitylimit. Whilerecentstudieshaveaddressedthe InFig.2,weshowthepairmomentumdistributionfunction
unpolarized situation21, this article’s chief case is the polar- (MDF)
izedone. Notethatthecorrespondingsituationwithrepulsive
interactionshasbeenstudiedin,e.g.,Ref.22. npkair =(1/L)Xexp[ik(l−m)]ρlm. (2)
Weusethedensitymatrixrenormalizationgroup(DMRG) lm
method23toobtainthegroundstatepropertiesofthismodelin Intheunpolarizedcase, weseeasharppeakcenteredatmo-
finite systems, fordifferentnumbersofparticlesN = N↑ + mentum k = 0, as observed for quasi-condensates of hard-
N↓,andvaluesofthetotalmagnetizationSz =(N↑−N↓)/2. core bosons.20 As the magnetization increases, the distribu-
In our calculationswe choose L = 80, U = −8t, and V = tionexhibitstwomaxima,centeredatmomenta±k . Note
max
0.002t,unlessotherwisestated. Inthefollowingwedescribe that by restricting the summation over l,m to the core part
two typical situations in the low and large density regimes, ofthesystems, thefeaturesinnpair becomemoreprominent
k
whichillustratethemainfeaturesoftheproblem.
(dotted lines in Fig.2). From Fig. 2 (c), we see that mo-
We start by looking at the low density case with N = 40 mentumk isproportionaltothemagnetizationinthecore
max
particles.InFig.1weshowthelocaldensityhniiandthespin Seff, that is obtained by integrating hSizi over the effective
projectionhSzialongwiththefluctuationsofbothquantities. sizeoftheregionoccupiedbytheFFLOstate,fromFig.1(a).
i
Wedefinethechargefluctuationsash(δni)2i=hn2ii−hnii2, Ourresultsreproducepreciselythebehaviorexpectedforthe
andsimilarforthespinfluctuationsh(δSz)2i. FFLOstate,inwhichthepairspossessafinitecenter-of-mass
i
The system exhibits a nearly half-filled density profile in momentum k = kF↑ − kF↓ (where kFσ is the Fermi vec-
the center with a sharp edge, and oscillations accompanied tor of the spin-σ fermions), which is predicted to grow as
bylargefluctuations.Asthemagnetizationgrows,thecoreof k = π(n↑ −n↓)/L = 2πSz/L.7 In Fig.2 (b) we show the
thesystembecomespartiallypolarized,displayingchargeand spatialdecayofthepaircorrelations,consistentwithapower-
spin oscillations, surrounded by fully polarized clouds with lawdecayoftheformnpair ∝cos(k |l−m|)/|l−m|α,as
lm max
spin ↑-fermions only. We can clearly identify two well de- predictedinRef.13.
finedphases: thefullypolarizedmetallicwings,surrounding The bosonic quasi-condensate can be studied by means
a polarized state in the core that we wish to characterize in of Penrose and Onsager’s description of the superfluid or-
moredetailinthefollowing. der parameter.24 The naturalorbitals (NO) ψ of the system
α
3
0.4
U=-8t, L=80, N=40, V=0.002t 0.12 2.5 U=-8t, L=80, N=80, V=0.002t (b)
(a)
0.4 SSS<SSS<zzzzzzSS======zz024024>> ffoorr SSzz==22 pairλα00..0048 0.4 <n>i1.251 00..23z><Si
ψ||α=0 ii 00 10ind2e0x α30 z>4*<Si 0.05 00.1
0.2 0.2 1.2 (c) Sz=0 (d)
2δ<(n)>i00..48 SSSSzzzz====481260 00..0046z2δ)><(Si
0 0 0.02
00 1100 2200 3300 4400 5500 6600 7700 8800
ssiittee ii
0 0
0 20 40 60 0 20 40 60 80
site i site i
FIG.3: (coloronline)Naturalorbitalψ0ofthepairOPDMforN =
40confinedfermionsanddifferentmagnetizations. Theinsetshows
FIG. 4: (color online) Sameparameters as in Fig.1, but N = 80
theOPDM’seigenvaluesλα fermions. (a) Densityprofile, (b) magnetization profile, (c) charge
fluctuations,and(d)spinfluctuations.
willsimplybethesingleparticleeigenstates–inthebosonic
sense – of the pair OPDM, and the corresponding eigenval- the band insulating behavior in the center of the trap. The
uesλ representtheir occupations. The NO with the largest FockstatesurvivessmallvaluesofSz butdisappearsrapidly,
α
eigenvalue, ψ , is the single-particle state in which quasi- andthelocalmagnetizationbecomesalmostuniform,roughly
0
condensationtakesplace. ThelowestNOandtheeigenvalues equaltotheaveragetotalmagnetization,hSzi ≈ Sz/N. The
i
of the OPDM are presented in the main panel and the inset fluctuationsaremostprominentintheintermediateregionbe-
of Fig. 3, respectively. The distribution of eigenvaluesqual- tweenthewingsandthecore,andafterreachingamaximum
itatively resembles the results for a trapped gas of hard-core there,theydecreaseagaintovaluessimilartothoseobserved
bosons.20 In the unpolarizedcase we observea macroscopic inthecenter.
occupationofthe lowesteigenstate, correspondingto the or- More details are revealed in Figs. 5(a) and (b), where we
derparameter.Theeffectofincreasingthemagnetizationisto plotthe densitieshn i for Sz = 2 and 8, respectively. Ob-
iσ
decreasetheoccupationofthebosoniccondensate,rendering viously,thewingsarefullypolarized,andthe↑-spinsfillthe
theprofileofλα vsαlesspronounced. ThelowestNO,rep- coreofthetrapwithaplateauofdensityhni↑i=1.TheFock
resentingthe bosonicorderparameter,resemblesthe density statesurvivesatSz = 2andwefindadistributionhn ithat
i↓
profile in the unpolarized case, see Fig. 1(a). As the mag- resembles the one of trapped spinless fermions.20 Fig. 5(c)
netization increases, it develops sharp oscillations, in agree- showstheMDFcalculatedinshellscenteredinthemiddleof
mentwiththeresultsfortheMDF.AsobservedinRef.9,we the trap, by restricting the sum in Eq. (2). As we cross the
seethattheunpairedfermionsaccumulateinthenodesofthe differentphasessurroundingthecore,thedistributionevolves
orderparameter,effectivelyformingmagneticdomainwalls. from the bulk result with two peaks to a featureless uniform
Thesingle-particlewavefunctioniswellconfinedtothecore profiledescribinglocalizedpairs. Thisshowsthatinthecen-
of the system, an indicationof the phase separationbetween tralregion,onlythe↓-fermionsexhibitanydynamics.
the FFLO state in the center and the fully polarizedmetallic These results establish the existence of three distinct
wings. Thisobservation,i.e.,theconfinementoftheNOtoa phases: first, there is a metallic, fully polarized phase at
windowoflengthLeff ≈ L/2furthercorroboratestheuseof the wings, behaving like a partially-filled band of spinless
theeffectivequantitiesLeff andSezff intheinsetofFig.2(b). fermionsandsecond,apartiallypolarizedphaseinthecenter
We now turn our focusto the dense limit, by lookingat a thatisametaloffreelymoving↓-fermionsinauniformback-
systemwithN = 80particles. Theresultsforthelocalden- groundof↑-fermions,withaneffectivesiteenergyof−U.An
sity and magnetization are shown in Fig. 4, along with their intermediate,third shellseparatesthe two, with featuresthat
fluctuations.IntheunpolarizedcaseSz =0,aplateauatden- resemble those observed in the FFLO state in the low den-
sityhn i=2appearsinthecenter,surroundedbyanextended sity regime. Notice thata fourthphase, theFockstate in the
i
regionwithoscillationsaroundhn i=1.Thestateinthecen- center, survives at small values of the magnetization, before
i
terisapureFockstateoflocalizedpairs,withoutcoherence, givingwaytothemetalliccore.
anddecoupledfromtherestofthesystem. Asaconsequence, In order to shed light on the nature of this intermediate
the charge and spin fluctuations in the center of the trap are phase, we analyze the pair OPDM’s spectrum and NOs. In
totally suppressed, which are signatures of a band insulator Fig.5(d),weplotthefirstthreeNOsthatdonotcorrespondto
with localized bound pairs, and both finite charge and spin localizedboundstates. TheseNOsarepreciselysituatedinthe
gaps. Thefluctuationsincreaseabruptlyattheboundarywith regionwhere the intermediate phase is found. We have also
thepartially-filled,unpolarizedregion. Asthetotalmagneti- calculatedtheMDFbyrestrictingthesummationinEq.(2)to
zationincreases, pairsarebrokenandthepolarizedfermions onlythisregionofthelattice (andthereflectedsectoronthe
movetowardthe edges, effectivelysuppressingthe gapsand oppositeside). ThisrestrictedMDFexhibitsfinite center-of-
4
1.2 U=-8t, L=80, N=80, V=0.002t (b) Sz=8 1.2 ofdensityandmagnetizationconsidered(notshownhere),we
(a) Sz=2 haveonlyseenafullypolarizedphasesurroundingthecore.
1 1
> 0.8 0.8> In conclusion, we have numerically studied the attractive
σ σ
<ni0.6 0.6<ni Hubbardmodelina parabolictrappingpotential,indifferent
0.4 <di> 0.4 densityandmagnetizationregimes. Intheunpolarizedlimit,
spin up
0.2 spin down 0.2 the system tends to be superconducting, but when the den-
0 0 sityincreases,aninsulatingFockstateoflocalizedpairsforms
0 20 40 60 0 20 40 60 80
site i site i inthecenterofthetrap,displacingthesuperconductingstate
0.014 (c) Sz=8 (d) pairα00..0023 1 taonwceadrd, ttwheobdoifufenrdeanrtielsim. iWtshaerenrtehaeliszpeidn:pAotpluolwatidoennissitiimesbwale-
λ 0.01
pairn/Nkpair0.012 f1u9l<l ln,mk<60 00 2in0αd=ex24 05α,2660 000...468ψ||α hpthaeavrtceoeffnfodeuucntcidtvienalgysitbnreunhceatruvcreeoarwse,aitah1nDdtwfgoualslryeogfpinooonlansr,-iizanetdwermealclettidanelglfiiscnpewidnilnsegusss-
2259<<ll,,mm<<5550 αα==2279,,2380 0.2 fermions. As the density increases, four phases emergeas a
0.01-π -π/2 0 π/2 π 50 60 70 800 functionofSz: aFockstate inthecenterofthetrapisgrad-
momentum k site i ually replaced by free ↓-fermions moving in a fully filled ↑
background. A superconducting shell separates it from the
FIG. 5: (color online) (a),(b) Particle density for niσ , σ = , , fully polarized wings. At finite magnetizations, the super-
for N = 80 fermions, and Sz = 2,8, respectivehly. iThe do↑ub↓le conductingstate becomespartiallypolarized,and canbe de-
occupancy hdii = hni↑ni↓i is also included. (c) Pair momentum scribed as a FFLO state with an oscillating order parameter
distributionfunctionmeasuredinconcentricshellsofdifferentradii,
forSz = 8. (d)FirstthreeNOsofthepairOPDMthatarenotjust and tightly boundpairs with a finite center-of-massmomen-
tum. Inthemean-fieldtheory,theFFLOphaseisregardedto
localizedstates.TheinsetshowstheeigenvaluesofthepairOPDM.
bestableinaverynarrowwindowaroundtheFermisurface,7
butwefindthatthepresenceofthetraphelpstostabilizeitby
phase-separatingthesystem.
massmomentathatcoincidewiththoseobservedinFig.5(c)
inthefullnpair,evidencingtheFFLOphase. We thank A. Moreo, D.J. Scalapino, and M. Troyer for
k
Ourresultsinthedilutelimitareingeneralagreementwith helpfuldiscussions,andS.Trebstforacriticalreadingofthe
Refs.16,17. However,wehavenotfoundevidenceforafully manuscript. We are grateful to the Kavli Institute for The-
pairedstateinthewings.AccordingtotheanalysisofRef.16, oretical Physics at UCSB, where the idea for this work was
17,thisphaseshouldbestableifthemagneticfieldissmaller conceived, and The Aspen Center for Physics for their hos-
thanhalfthebindingenergyofthebosonicpairs. Thisshould pitality. F.H.-M.’s work is supported in part by NSF grant
occurbelow a criticalpolarizationof P = 2Sz/N = 0.2 in DMR-0706020,andby contractDE-AC05-00OR22725with
the strong coupling limit. In our calculations, for all values UT-Battelle,LLC.
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