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Frustrated Bosecondensates inopticallattices T. Ðuric´ and D.K.K. Lee BlackettLaboratory, ImperialCollegeLondon, PrinceConsortRoad, LondonSW72AZ,UnitedKingdom (Dated:January13,2010) WestudytheBose-condensed ground statesof bosons inatwo-dimensional optical latticeinthepresence offrustrationduetoaneffectivevectorpotential,forexample, duetolatticerotation. Weuseamappingtoa large-S frustratedmagnettostudyquantumfluctuationsinthecondensedstate.Quantumeffectsareintroduced 0 by considering a 1/S expansion around the classical ground state. The large-S regime should be relevant 1 tosystemswithmanyparticlespersite. Asthesystemapproaches theMottinsulatingstate,theholedensity 0 becomessmall.Ourlarge-S resultsshowthat,evenwhenthesystemisverydilute,theholesremaina(partially) 2 condensedsystem. Moreover, thesuperfluiddensityiscomparabletothecondensatedensity. Inotherwords, the large-S regime does not display an instability to noncondensed phases. However, for cases with fewer n than 1/3 flux quantum per latticeplaquette, we find that the fractional condensate depletion increases as the a J systemapproachestheMottphase,givingrisetothepossibilityofanoncondensedstatebeforetheMottphase isreachedforsystemswithsmallerS. 3 1 PACSnumbers:03.75.Lm,03.75.Mn,75.10.Jm,75.10.-b,75.45.+j ] s a I. INTRODUCTION ationsaroundtheclassicalstate. Inotherwords,weworkun- g derthe assumptionthat quantumeffectsdo notchangequal- - t itatively the nature of the ordering obtained for the classical n Bosonicatomsinopticallatticescandisplaysuperfluidand groundstates. Mathematically,thismeansthatwewillwork a Mott insulating phases. If the system is rotated, then, in the u corotatingframe,thisisequivalenttointroducinganeffective inalarge-S generalizationofthespinmodelandperforman q magneticfieldproportionaltotherotationfrequency1,2. This expansionin1/S toobtainthequantumeffects. Althoughour . originalmodelcorrespondsto smallS, the large-S approach t isnottheonlymeanstointroduceavectorpotentialtoasys- a canbejustifiediftheperturbativeseriesin1/S converges13–17. tem of neutral atoms. This can also be achieved3–6 through m Inthosecases, aspinwavecalculationmaygiveaccuratere- theinteractionofatomicelectricandmagneticmomentswith - sults. d an externalelectromagnetic field (Aharonov-Casherand dif- n ferential Aharonov-Bohm effects). For atoms trapped in an o opticallatticeintwo distinctinternalstates, a scheme7 using We will study how quantum fluctuations affect the order c twoadditionalRamanlaserscombinedwiththelatticeaccel- parameter, off-diagonal long-range order (ODLRO) and the [ eration or inhomogeneous static electric field has also been superfluidfractionfor differentdegreesof frustrationfor the 2 proposed. whole range of incommensurate filling. In the spin analog, v Bosonic atoms in an optical lattice can be modeled by theincommensuratefillingcorrespondstoarangeofZeeman 0 a Bose-Hubbard model. A vector potential introduces an field h up to some frustration-dependentcritical field h (α). 0 c Aharonov-Bohmphaseforthebosonhoppingfromsitetosite. Ourcalculationsweremadeforα=0,1/4,1/3,and1/2. 6 The wave function is “frustrated” if the phase twists around 1 . each plaquette add up to 2πα for some non-integerα. For a 6 OurresultsshowthatthedegreeofBosecondensationde- Bose condensateata low effectivemagneticfield, thisintro- 0 creasesashincreasestowardh . However,itdoesnotvanish ducesvorticesintothecondensate.Thepresenceoftheoptical c 9 at the limit of h = h (α). This applies to several quantities 0 lattice2,8interfereswiththeformationofanAbrikosovvortex c that we have calculated: the reduction in the order parame- : lattice1,9andquantumfluctuationsmaybeenhanced.Further, v ter,thereductioninthelargesteigenvalueofthedensityma- if the number of vortices becomes comparable to the num- i trix, and the sum of the non-macroscopiceigenvaluesof the X berofbosons,thesystemmayenterintoafractionalquantum density matrix. We also find similar conclusions for the su- r Hallstate1,2,8,10–12.However,thisrequiresaveryhighrotation a frequency or a low atomic density which is hard to achieve perfluidfraction—frustrationreducesthesuperfluidfraction in the comparison with the unfrustrated case but there is no experimentally. vanishingofthesuperfluidfractionatanyh≤h . In thiswork, we will focuson theexperimentallyaccessi- c bleregimewhereacondensatestillexiststoexaminewhether there are any precursors to such states in a frustrated Bose The paper is organized as follows. We will outline the condensate.Westudyatwo-dimensional(2D)Bose-Hubbard model and the mapping to the quantum spin model in Sec. modelonasquarelatticeforarangeofincommensuratefill- II. We describe the classical ground states (S → ∞) of the ing. In the regimeof strongon-site interaction, the modelis spin analog in Sec. III. We introduce the excitations above analogoustoaquantumeasy-planeferromagnetandthefrus- the groundstate in a 1/S expansionin Sec. IV. In Secs. V tration encouragesspin twists, i.e., the formationof vortices andVI,wecalculatethedegreeofcondensationandsuperflu- inthegroundstate. Wefindtheclassicalgroundstatesusing idityinthesystem. We makeconclusionsaboutourstudyin MonteCarlomethodsandthenwestudythequantumfluctu- thefinalsection. 2 II. MODELHAMILTONIAN The spin raising and loweringoperatorscorrespondto the creation and annihilation of hard-core bosons, respectively. Foratomstrappedinatwo-dimensionalopticallattice, we This mappingis possible because hard-corebosonshave the canfocusonasingle-bandlatticemodelifthetunnelingtbe- samecommutationrelationsasS = 1/2operators: operators tween wells within the lattice is weak comparedto the level ondifferentsitescommutebutoperatorsonthesamesitean- spacings in each well. If the tunneling is also weak com- ticommute. Themotionoftheatomstranslatestopseudospin pared to the repulsive energy U for two atoms in one well, exchange.TheeffectiveHamiltonianis thenstronglycorrelatedgroundstates,suchastheMottinsu- J latoMr,aanpypdeiaffrearsewntemlleatshaodsusphearvfleuibdesetnatpe.roposed to introduce Heff =−2Xhiji (cid:16)eiφijSî+Sˆ−j +H.c.(cid:17)−hXj Sˆzj (4) frustration in the atomic motion. This can be done through rotating the system1 or through the interaction of the atoms where J = 2t, Sˆ± = Sˆx ±Sˆy are thespin-1/2operators,and withanexternalelectromagneticfield3–6. Ifthereisonlyone h= µrepresentsianeffeictiveiZeemanfield. Notethatthisisa species of bosonic atoms, then the system is described by a ferromagnetintheabsenceoffrustration(φ =0). ij Bose-Hubbardmodelonasquarelatticewithacomplexhop- Itisnotsimpletoattacktheinfinite-U limitoftheproblem pingmatrixelement: HHubbard = H(0)+V with of hard-coreboson directly. Instead, we will relax the hard- coreconditionandallowformorethanonebosononeachsite. U H(0) = aˆ†aˆ (aˆ†aˆ −1)− µaˆ†aˆ , Wewillallow2S atomsoneachsitesothateachsitehas2S+1 2 X i i i i X i i possiblestates. Thiscorrespondstoa spin-S modelwith the i i T = −tXhiji (cid:16)eiφijaˆ†jaî+H.c.(cid:17), (1) Horaigminilatolnbioasnognisv,eaˆn,iannEdqt.hi(s4s).piTn-hSe mreoladtieolnissheipstabbeltiwsheeedn vthiae theHolstein-Primakoffrepresentation: where µ is the chemical potential and hiji denotes nearest- neighborsites iand j. The complextunnelingcouplingsap- Sî+ =cˆ†i(2S −cˆ†icî)1/2, Sîz =cˆ†icî−S. (5) pearintheHubbardHamiltonianduetothepresenceoftheef- fectivevectorpotentialA~.Whenanatommovesfromalattice wherecîareoperatorswithbosoniccommutationsandarees- siteatR~ toaneighboringsiteatR~ ,itwillgainanAharonov- sentially the original bosons aî of the Bose-Hubbard model. i j ThelimitofS → ∞ correspondstotheclassicallimitofthe Bohmphase model.Morespecifically,weneedS →∞whileJS andhre- R~j mainconstantsothatexchangeandZeemanenergiesremain φ = A~·d~r, (2) comparable. ij Z R~i Mathematically, the large-S limit provides a systematic way to control the quantum fluctuations in this problem. For neutral atoms with electric moments d~ and a magnetic e Quantum fluctuations can be introduced (see later) in a 1/S momentsd~m in an externalelectromagneticfield (E~,B~), A~ = expansionundertheassumptionthatthoseeffectsdonotalter (d~m×E~+d~e×B~)/~c3–6. Forarotatinglattice,A~=mΩ~ ×~r/~, significantlythenatureoftheorderingobtainedfortheclassi- where Ω~ is the rotation frequency and m is the mass of the calgroundstates. We willpresentresultstoleadingorderin atom. Inthiswork,westudythecaseoftheuniformeffective 1/S (i.e., we do not set S = 1/2 afterward). Physically, the magneticfield B~ = ∇~ ×A~ = Bzˆ. Results will dependon the leading-orderresultsinS shouldberelevanttoopticallattices frustration parameter α, defined as the flux per plaquette in withmanyatomspersiteonaverage. unitsof2π, Therelaxationofthemaximumsiteoccupancyto2S from a model of hard-core bosons is not the only way to control α= 1 B~·dS~ = 1 φ (3) correlationsintheBose-Hubbardmodelatweaktunneling.A 2πZ plaq 2πX ij similarmethodologyis toconsidera densebutweaklyinter- plaq actinglimitoftheBose-Hubbardmodel.Withn¯ beingtheav- wheretheintegrationisoverthesurfaceofalatticeplaquette eragebosondensitypersite,thislimitisgivenbyU →0and andthesumisperformedanticlockwiseovertheedgesofthe n¯ →∞whileUn¯remainsconstant18. Then,onecandevelopa squareplaquette. Thisparameterisonlymeaningfulbetween theoryasanexpansionin1/n¯. Thisapproachproducesresults 0and1becauseafluxof2πthroughaplaquettehasnoeffect veryclosetothe1/S expansionconsideredhere. onthesystem. Frustrationismaximalatα=1/2. NotethatourHamiltonianhaslocalgaugeinvariance.Ifwe Inthispaper,wewilluseamagneticanalogyastheframe- changethe gauge, A~ → A~+∇~χ, then the Hamiltonian stays worktostudytheBose-Hubbardproblem.Thisismosteasily unchanged if the boson and spin operators pick up a phase motivatedinthelimitofU/t→∞,eventhoughwewillnotbe change. workingdirectlyinthislimit. Insuchalimit,thesiteoccupa- tioncanberestrictedtozeroandoneboson.Then,theHilbert φij →ei(χj−χi)φij, aî →eiχiaî, Sî− →eiχiSî−. (6) space of possible states can be mapped onto a spin-half XY model. The two S states of the pseudospin correspond to Inthespinlanguage,thiscorrespondstoarotationofχ inthe z i whetheralatticecontainsabosonornot. xyplaneinspinspace. 3 Beforeproceedingtodiscussthepropertiesofthissystem, ism = [1−(h/h )2]1/2. This xymagnetizationcorresponds xy c wepointthatwemaygeneralizethistoanopticallatticecon- tosuperfluidityintheoriginalsingle-speciesHubbardmodel. taining two species of bosonic atoms, such as two hyperfine ThezmagnetizationintheSz directionM = NhSzi= Nh/h z i c states. Letusdenotethetwospeciesbyσ =↑,↓. Thisallows correspondstothenumberofatomsintheopticallatticemea- formoredegreesoffreedominthemodelHamiltonian. Two sured from half filling. For higher Zeeman fields (h > h ), c atomicspeciesmay,ingeneral,seedifferentlatticepotentials M becomessaturated and there is no xy magnetization: the z sothatthetunnelingmatrixelementsandchemicalpotentials lattice is a Mott insulator at one atom per site (or empty for couldbedifferentforthetwospecies. TheHubbardmodelfor h<−h ). c thetwospecieswouldbeoftheformH = H(0)+T with Inthepresenceofthevectorpotential,theorderingpattern Hubbard oftheclassicalgroundstatedependsontheeffectivemagnetic H(0) = 1 U aˆ† aˆ† aˆ aˆ − µ aˆ† aˆ , fluxthrougheachplaquette. Thisintroducesvorticesintothe 2iX,σ,σ′ σσ′ iσ iσ′ iσ′ iσ Xi,σ σ iσ iσ spinpattern. Italso reducesthecriticalfield hc belowwhich the xymagnetizationisnonzero. AsshownbyPázmándiand T = − tσ eiφσijaˆ†jσaîσ+H.c. , (7) Domanski19, hc is givenbyis the maximaleigenvalueof the σXhiji (cid:16) (cid:17) matrix JSeiφij. Thisis shownin Fig. 1. Note that this result for h is not restricted to the classical limit but applies for where the on-site interaction U , the exchange interaction c σσ′ allvaluesof thespin S. The spectrumof allthe eigenvalues t ,thetunnelingphaseφ ,andthechemicalpotentialµ have σ ij σ of this matrix as a function of the frustration parameter α is allacquiredadependenceontheinternalstatesofthebosons. theHofstadterspectrum20 asdiscussedoriginallyintermsof If we specialize to the case of one atom per site with strong two-dimensionaltight-bindingelectronsin the quantumHall on-site interactions, we can rule out zero or double occupa- regime. tion of each lattice site. In other words, the system should be a Mott insulator but the atom occupyingeach site can be of either internal state. Thus, each site has a spin-half de- 4.0æ æ gree of freedom: Sˆ+ = aˆ†aˆ would create a ↑ state and æ æ i i↑ i↓ æ æ Sî− = aˆ†i↓aî↑ would create a ↓ state. In this phase, the rela- 3.5 ææ ææ tive motion of the two species of atoms is still possible: the h æ æ motionofonespeciesinonedirectionmustbeaccompanied €€€€c€€ æ æ by the motion of the other species in the opposite direction. JS 3.0 ææ ææ Tsihtei.sIcnotuhneteprsfleouwdoskpeienplsanthgeuaogcec,utphaistiiosnsiamtpolnyespaitnomexcahtaenagceh. æææææææææææææææææææææææ 2.5 Therefore,inthisMottphasefortheoveralldensity,wehave again an easy-plane magnet. If we tune the interactions so that U = U = 2U , then a perturbation theory in t/U 2.0 ↑↑ ↓↓ ↑↓ 0.0 0.2 0.4 0.6 0.8 1.0 bringsustotheeffectivepseudospinHamiltonian4,6described Α byEq. (4)with J = 4t t /U,h = 2 µ −µ +8(t2−t2)/U, ↑ ↓ ↑ ↓ ↑ ↓ andφij =φ↓ij−φ↑ij. (cid:0) (cid:1) FIG.1:CriticalvalueoftheeffectiveZeemanfield,hc(α),asafunc- We can translate the phasesofthe single-speciesHubbard tionof the parameter αbeing theflux per plaquette inunits of 2π. model to this two-species system at unit filling. Superfluid- Forh>h (α)thelatticeisaMottinsulatoratoneatompersite. c ity in the single-species Hamiltonian at an incommensurate fillingcorrespondstosuperfluidityforcounterflowinthetwo- Let us now turn to the classical ground states for h < h . c speciesproblematthecommensuratefillingofoneatomper Writing the local magnetization in spherical polars, hS~ i = i site but with different relative densities of the two species. S(sinθ cosφ,sinθ sinφ,cosθ),theclassicalenergyisgiven i i i i i The advantage of considering this two-species Mott insula- by: toristhattheremaybemoredegreesoffreedomintuningthe parametersofpseudospinHamiltonian,includingtheexplicit Eclass ≃−JS2 sinθisinθjcos φi−φj+φij −hS cosθi. breakingofSz →−Szspinsymmetry. Xhiji (cid:16) (cid:17) Xi (8) Minimizingthis energy, we find that the ground-statevalues forφ andθ,Φ andΘ,mustsatisfy,foreachsitei, III. CLASSICALGROUNDSTATES i i i i JS sinΘ sinΘ sin Φ −Φ +φ = 0 i j i j ij TodeterminethegroundstatesofthepseudospinHamilto- jX=i+δ (cid:16) (cid:17) nian(4),weconsiderfirsttheS → ∞classicalgroundstates JS cosΘ sinΘ cos Φ −Φ +φ = hsinΘ (9) for the spin system. We assume that h > 0 without loss of i j i j ij i generality. In theabsenceofthe vectorpotential, the system jX=i+δ (cid:16) (cid:17) isaneasy-planeferromagnet. Forh < h = 4JS,theground wherethesummationistakenoverthefourneighboringsites c state has a uniform magnetization in the xy plane in spin of i: j = i + δ. The first equation conserves the spin cur- space. The xy component of the magnetization at each site rent(oratomiccurrentintheoriginalHubbardmodel)ateach 4 node. ThesecondspecifiesthatthereisnoneteffectiveZee- manfieldcausingprecessionaroundthezaxisinspinspace. In the original boson language, this ensures a uniform local chemicalpotentialthroughoutthe system (in the Hartreeap- proximation).Thesystemhasalocalgaugeinvarianceandwe needtofixagaugetoperformournumericalcalculations.We choosetheLandaugaugeA~= B(0,x,0)sothattheAharonov- Bohmphaseφ iszeroonallhorizontalbondsofthelattice. ij TheclassicalgroundstatesareobtainedbyusingtheMetropo- -1.0 æ æ FIG.3: Vortexpatternsfor(a)α = 1/3and(b)α = 1/2(chequer- ass2(cid:144)HLNJS--21..05ìæà ìæàòôìæàòôìæàòô ìæàòô ìæàòô ìàòô ììææààòòôô ìàòô ìæàòô ìæàòô ìæàòôìæàòôìæàòô ìæà oαo2bntπoh=a.etrhFrd1rode/crie2oaαngdtfiohi=ffnegraeu1elrr./eaa3ntriteoth3nte)w×r,eow3adirtesehug2beαqnlabet=retaiicnt6eegsdstetahagnetedenflseauwrlxaoitntpehgesrvtbaopotrletathsiqcu(devesiaottagrettoicnioneansleuscn)o.airtnsFthbooeefr cl ò ò 0 E -2.5ô ççç ç ç ç çç ç ç ç ççç ô ç ç -3.0 0.0 0.2 0.4 0.6 0.8 1.0 Α FIG.2:Groundstateenergyoftheclassicalspinsystemasafunction ofthefrustrationparameterα(fluxperplaquettedividedby2π)for differentZeemanfieldsh/JS = 0,0.5,1,1.5,2and2.5(fromtopto bottom).Theenergyissymmetricaroundthepointα=1/2. lisalgorithm. Forrationalvaluesofthefrustrationparameter FIG.4: Vortexpatternsfortwogroundstatesatα=1/4andh=0. α = p/q, the Monte Carlo simulations are done on nq×nq (a)Currentpatternperiodicon4×4square, phasepatternperiodic latticeswithperiodicboundaryconditions. Inmostcases,we on 8×8 square. (b) Current and phase patterns periodic on 4×4 find that the periodicity of the ground state is q ×q. How- squares. ever, we also find groundstates with the periodicity2q×2q in somecases. Theground-stateenergiesasfunctionsofthe fluxthroughaplaquetteareshowninFig.2. TheHalseyanalysisdoesnotcovercaseswhenα<1/3.At We can also examine the vortex pattern in these ground α = 1/4andh = 0 wefindtwo distinctgroundstate config- states. The currenton the bondjoiningsites i and j is given urations(Fig.4)withthesameenergyintheagreementwith by: I = (JS2/~)sinΘ sinΘ sin Φ −Φ +φ . Thecircu- previousresults22–24. Forbothconfigurations,thecurrentpat- ij i j i j ij lationofthesecurrentsaroundeach(cid:16) plaquettegiv(cid:17)esthevortex ternsareperiodicon4×4square.However,thephasepatterns patterns. Theseareshownforα = 1/2,1/3,and1/4inFigs. donothavethesameperiodicity:itis8×8periodicinthecon- 3and4. figurationshowninFig.4(a)but4×4inFig.4(b). Wefind IncaseofazeroZeemanfieldh=0,theclassicalHamilto- statesoftheform[Fig.4(b)]forgeneralhwhensimulations nian(8)hasbeenstudiedextensivelyinthecontextofJoseph- aredoneon4×4latticeswithperiodicboundaryconditions. son junction arrays in the presence of a perpendicular mag- Simulationsdoneonlarger4n×4nlatticesatnonzerohgive neticfield21–23. Halsey21 showedthat,forsimplefractionsin states that contain elements of both structures separated by the range 1/3 ≤ α ≤ 1/2 (e.g., α = 1/2,1/3,2/5,3/7,3/8), domainwalls. SimilarresultswerefoundbyKasamatsu24. thegroundstateshaveaconstantcurrentalongdiagonalstair- cases. Our results for h = 0 agree with these previousstud- ies. ForageneralnonzeroZeemanfield,thegroundstateswe IV. EXCITATIONSPECTRUM foundforα=1/2and1/3alsohavecurrentsindiagonalstair- cases. WecannotobtainanalyticgeneralizationoftheHalsey In this section, we compute the excitations of the system solutionforthecaseoffiniteh. Wefindthegroundstatesby usingthespin-wavetheory.Quantumeffectsareincorporated usingtheMetropolisalgorithm.Atfiniteh,thephasepatterns intheproblembyconsideringfinitevaluesofS. Wewillper- forα = 1/2andα = 1/3aresimilartothephasepatternsfor formanexpansionin powersof the parameter1/S andkeep theHalseystatesath = 0butSz hasspatialvariationaround onlythetermsofthelowestorderin1/S intheHamiltonian. afiniteaverage. Even though we are interested in S ∼ O(1), the large-S ap- 5 proachisinsomecasesjustifiedduetothegoodconvergence ThisisdiagonalizedbytheHofstadtersolution20. Theexcita- of the perturbative series13–17. Spin-wave approximationre- tionspectrumhasanenergygapofh−h andthegroundstate c lies on an assumption that the introduction of the quantum correspondstoavacuumoftheseexcitations,i.e.,thereareno fluctuationsdoesnotqualitativelychangethenatureoftheor- zero-pointfluctuationsinthegroundstate. dering obtained for classical ground state. We use this ap- For lower Zeeman fields (h < h ), Hamiltonian (12) con- c proach to investigate whether the Bose condensate becomes tainingtheanomaloustermswillhavezero-pointfluctuations unstableinanyparameterregime. which reduce the magnetizationfrom the classical value. In Starting from the classical ordered state, we use the the language of the original bosons, the fluctuations would Holstein-Primakofftransformationto representthe spin flips depletethecondensate.TheHamiltoniancanbediagonalized away fromthe classical groundstate in termsof the bosonic byageneralizedBogoliubovtransformation, operators. We willkeeponlythequadratictermsinthe final bosonic Hamiltonian. It is convenientto introducethe oper- bˆ = u αˆ +v∗ αˆ† , bˆ† = v αˆ +u∗ αˆ† i im m im m i im m im m ators~Sˆ suchthatSˆx directionisparallelto theclassicalspin Xm (cid:16) (cid:17) Xm (cid:16) (cid:17) i i (15) directionateachsite form = 1,...,I foralatticeofI sites. Toensurethatthenew Sˆx sinΘ cosΦ sinΘ sinΦ cosΘ Sˆx operatorsαˆm obeybosoniccommutationrelations,werequire i i i i i i i thematricesuandvtoobey:uu†−vv† =1anduvT−vuT =  SSˆˆyizi =−co−ssΘinicΦoisΦi −cocsoΘsiΦsiinΦi sin0Θi  SSˆîiyz(1,0) 0nq.euwadToroaptoeicrbatitanoinrtsh,aewbdeoiascgoaonnnicwalorizipteeedrtahHteoarpmsaairlsttooHnˆfita=hnecîH†nMatmecˆr,imlwtoshneoirafentMh(1es2ies) and use the Holstein-Primakoff representation of these new a 2I ×2I matrix andcˆ = (b,b†) with bˆ = (bˆ ,bˆ ,...). Then, spinoperatorsintermsofthebosonicoperators,bˆ , 1 2 i itcanbeshownthatHamiltonian(12)isdiagonalizedintothe form Sˆ+ ≡Sˆy+iSˆz =(2S −bˆ†bˆ )1/2bˆ , Sˆx =S −bˆ†bˆ . (11) i i i i i i i i i Hˆ = E + ǫ αˆ†αˆ (16) Notethatagaugetransformationcorrespondstoarotationof 0 X m m m m the spin S~ aroundthe z axis. Since these new spin variables arealignedwiththeclassicalspinconfiguration(whateverthe witheigenenergiesǫm ifwesolvetheeigenvalueproblem, choiceofgauge),thenewspin~Sisinvariantundersuchrota- ǫ tion.Therefore,thebosonicoperators,bî,aregaugeinvariant. (cid:18)M− 2Σz(cid:19)q=0. (17) Underassumptionthatthezero-pointfluctuationsaresmall so that the average numberof spin flips at each site is small where qm = (u1m,...,uNm,v∗1m,...,v∗Nm) contains the co- comparedtoS,wecanapproximate[1−bˆ†ibî/(2S)]1/2asunity. efficients of the Bogoliubov transformation and Σz = TheresultingHamiltonian,toorderO(S0),is {{1,0},{0,−1}}. We computed the spectrum for 60 × 60, 120 × 120 and Hˆ ≃ Eclass+ A−bˆ bˆ −A+bˆ bˆ†+H.c. + Cbˆ†bˆ , (12) 240×240 lattices with periodic boundary conditions, using 0 Xhiji (cid:16) ij i j ij i j (cid:17) Xi i i i theclassicalgroundstatesfromourMonteCarlosimulations discussedintheprevioussection. Ourresultsfor60×60lat- with tices and the frustration parameters α = 0, 1/2, 1/3, and 1/4 areshowninFig.5. Ourresultforα=1/4iscalculatedusing A± = JS (cosΘ cosΘ ±1)c ±i(cosΘ ±cosΘ )s , the4×4periodicclassicalgroundstatepresentedinFig.4(b). ij 2 i j ij i j ij As can be seen in Fig. 5 at h < h (α), the spectrum is h i c C = JS sinΘ sinΘ c +hcosΘ (13) gapless. Thelow-energyexcitationsaretheGoldstonemodes i i j ij i X relatedtothespontaneoussymmetrybreakingoftheglobalro- j=i+δ tationsymmetryinthexy-planeinspinspace.Inotherwords, wherec =cos(Φ−Φ +φ ),s =sin(Φ −Φ +φ )andEclass thespinsystemhaslong-rangemagnetizationinthe xyplane is the girjound-statie valjue iojf thiej classicail enjergyij[Eq. (08)]. inspinspace. WecanusehS+iastheorderparameter. Inthe i Note that all the coefficients in this Hamiltonian are gauge languageofthe originalbosonicmodel, thiscorrespondsthe invariant, confirming our above conclusion that the bosonic breakingofU(1)symmetryduetoBosecondensation.Above operators,bî,aregaugeinvariant. hc, thereis nosymmetrybreakingandwe see an energygap ThisHamiltonianalsoreducescorrectlytothecaseofh > inthesystemproportionaltoh−hcasdiscussedabove. hc(i.e.,Θi =0)whenthereisnoneedforrealigningtheaxisof The ground-state energy E0 [Eq. (16)] can be written as quantization[Eq. (10)]. Inthatcase, the“anomalous”terms E0class+∆E0,where∆E0 =∆+ mǫm/2isaquantumcorrec- bˆbˆ and bˆ†bˆ† in the Hamiltonianvanish. Then, the spin exci- tion to the classical ground-statPe energy [Eq. (8)] with ∆ = tationsaredescribedbyatight-bindingmodelwithmagnetic −JS hijicos(Φi−Φj+φij)forh = 0and−h i1/(2cosΘi) fluxthroughtheplaquettes: for hP, 0. This quantumcorrectionis of ordePrS0 while the classicalenergyis oforderS andso the fractionalchangeis Hˆh≥hc ≃−hNS −JS Xhiji (cid:16)eiφijbîbˆ†j +H.c.(cid:17)+hXi bˆ†ibî. (14) stimonasll∆inE0th/Ee0cllaasrsgfeo-Srsleivmeirta.llWateticcealsciuzleaste(6t0he×r6e0la,t1iv2e0c×or1r2ec0-, 6 ourbosonproblem.Sinceweareconsideringalatticesystem abovehalffilling, itismoremeaningfultoconsiderthe condensation of vacancies because this is the most appropriate description as h approaches h . (For the two-species model c with counterflow superfluidity, we are considering the con- densationoftheminorityspecies.) Theholedensitymatrixis definedasρh =haa†i. Theexistenceofamacroscopiceigen- ji i j value, N , corresponds to Bose-Einstein condensation. The 0 sumofallnon-macroscopiceigenvaluesgivesthenumberof holesnotincondensateandwecandefinethefractionalcon- densatedepletionastheratioofthenon-macroscopicsumto thetotalnumberofholesN whichisthetraceofthedensity h matrix. Intheanalogoftheeasy-planemagnet,weshouldstudythe spin-spincorrelationfunctionforthespincomponentsin the xy plane: ρ = hSˆ−Sˆ+i. ODLRO correspondsto a non-zero ji i j xy magnetization which is the analog of Bose condensation. Inthe large-S limit, ρ /2S is the analogofthe bosonichole FIG.5:LowenergyexcitationspectrumasafunctionoftheZeeman ji field h for 60× 60 lattices with periodic boundary conditions for densitymatrixρhjiforhclosetohc. frustrationα = 0,1/4,1/3and 1/2. Criticalvaluesh are: h (α = The macroscopic eigenvalue for our spin-spin correlation c c 0) = 4, h (α = 1/4) = 2.828, h (α = 1/3) = 2.732 and h (α = function is, to the leading order in S, given by the classical c c c 1/2) = 2.828. Abovehc,thespectrumhasafiniteenergygap. The valueN0class = i(mixy)2,wherem~ixyistheclassicalvalueofthe spectrum is gapless for h < hc indicating long-range order in the magnetizationPatsitei. Wepresentbelowourresultsforcon- system. densate and the depletion of the condensate, i.e., zero-point fluctuations which decrease the magnetization in the ground state. 240×240)andextrapolateresultstothethermodynamiclimit Theabovediscussionneedstobemodifiedinthepresence showninFig. 6. Ascanbeseen,thequantumcorrectionde- of a vector potential because the density matrices, ρ and ρh, creases to zero as the Zeeman field h approachesthe critical value h . Above h , the ground state is the classical ground arenotgauge-invariantquantities: ρji → ei(χi−χj)ρji underthe c c gauge transformation [Eq. (6)]. However, we can construct statecontainingnozero-pointfluctuations. gauge-invariantanalogs. Moreover,the eigenvaluesofρand ρh are gaugeinvarianteven thoughthe correspondingeigen- 0.06 vectorsarenot. Considerfirstthespin-spincorrelationfunc- tioninthegroundstate 0.05 ss 0.04 ρji = hSî−Sˆ+ji=ρcjilass+δρji a cl00.03 ρcjilass = ψ∗iψj with ψi =SeiΦisinΘi (18) E (cid:144) Α=0 E D 0.02 Α=1(cid:144)4 whereρclass istheclassicalvalueofthedensitymatrix(ofor- S ji Α=1(cid:144)3 der S2) and ψ is the classical value of the order parameter 0.01 i Α=1(cid:144)2 (of order S) hSˆ+i. The order parameter itself is reduced by i 0 quantumfluctuations, 0 0.2 0.4 0.6 0.8 1 h(cid:144)hcHΑL 1 hSˆ+i=ψ(1−∆), ∆ = |v |2. (19) i i i i S X im m FIG.6:Quantumcorrectiontotheground-stateenergyasafunction ofh/h (α)forα = 0,1/2,1/3and1/4. h (α)isthecriticalvalueof Thecorrectionδρtothedensitymatrixisgivenby: c c theZeemanfieldhforagivenfrustrationparameterα. S δρji ≃−ρcjilass(∆i+∆j)+ 2ei(Φj−Φi)Xq∗jnqin (20) n V. DENSITYMATRIX whereq =u +v +cosΘ(v −u ),withu andv being in in in i in in in in thecoefficientsfortheBogoliubovtransformation[Eq. (15)]. In this section, we will examine ODLRO in the density Thisdensitymatrixisnotinvariantunderagaugetransforma- matrix25,26. Consider first the case without a vector poten- tion. We obtaina gauge-invariantversionofthe densityma- tial. A macroscopically large eigenvalue of the density matrix by expressing it with respect to a gauge-covariantbasis. trixρ signalstheexistenceofBose-Einsteincondensationfor Themostnaturalbasisisthebasisformedbytheeigenvectors ji 7 of the classical density matrix ρclass. The eigenvector corre- correctionandpresentresultsintermsofthecorrectiontothe spondingtothelargesteigenvalueissimplyψ, classicallimitsasfractionsoftheclassicalsolution. i Wecanexploitthelarge-S expansiontocomputetheeigen- ρclassψ = Nclassψ with values of the density matrix. We start with calculating the ji i 0 j Xi quantumcorrectiontothenon-degeneratemacroscopiceigen- N0class =X|ψ∗iψi|2 = S2Xsin2Θi (21) vwaelucea,nNc0a.lcuSliantceethρecjileasisgeisnvlaalrugeesrothfaρnbδyρtjriebatyinagnδoρridnerpeinrtuSr-, i i bation theory. The first-order correction to N is then given 0 where Nclass is simply the classical value of the sum of the by 0 squareofthe xymagnetization(m2 )oneachsite. Itisonthe 1 xy ∆N = ψ∗δρ ψ =δρ˜ (23) orderofNS2ath=0andtendstozeroashreacheshc.Allthe 0 Nclass X i ij j 11 othereigevectorsofρclass haveeigenvaluesofzero. Usingan 0 ij orthonormalsetoftheseeigenvectorsascolumnsforaunitary if the first basis vector for δρ˜ is chosen to be the one corre- matrix U, we can construct a unitary transformation for the spondingtotheclassicalsolutionψ.Thiscorrectionisoforder densitymatrix(ρ→ρ˜,etc.), S,asopposedtoorderS2 fortheclassicalvalue. Ourresults for∆N asafractionofNclassareshowninFig.8. Weseethat 0 0 ρ˜ =U†ρU =ρ˜class+δρ˜. (22) thereductioninN islargestath=0anddecreasestozeroat 0 thecriticalfieldsh (α).Thevanishingofquantumcorrections c whereρ˜class =diag(Nclass,0,...,0).Underthegaugetransfor- ash→h (Θ →0)canbeseendirectlyfromthecoefficients 0 c i mation [Eq. (6)], all the eigenvectorsof ρji pick up a phase A− oftheanomaloustermsinHamiltonian(12)whicharere- change,e.g.,ψi →e−iχiψi sothatUij →e−iχiUij. Itiseasyto sponsible for the zero-point fluctuations in the ground state. checkthatthiscompensatesforthephasechangeinρ sothat ji ρ˜ → ρ˜ . Consequently,allthequantitiesobtainedfromthe ij ij matrix ρ˜ are gauge-invariantand therefore physically mean- 0 ingful.Inthissection,wecalculatetheeffectofquantumfluc- -0.025 Α=0 tuationsonthe density matrix. Thisrequiresonly the eigen- Α=1(cid:144)4 values of ρ˜. They are in fact the same as the eigenvaluesof ss -0.05 Α=1(cid:144)3 ρtrabnescfaourmseattihoen.two density matrices are related by a unitary Ncla0-0.075 Α=1(cid:144)2 (cid:144) Wewillnowpresentournumericalresults. Firstofall,we N0 -0.1 presentthe classical solution for the number of atoms in the D S condensate, Nclass, as given by Eq. (21). This is shown in -0.125 0 Fig.7. We seethatthisdecreasestozeroashisincreasedto -0.15 h (α). c 0 0.2 0.4 0.6 0.8 1 h(cid:144)hcHΑL 1 FIG.8: Quantumcorrection∆N tothethemacroscopiceigenvalue 2L 0.8 of the density matrix as a funct0ion of h/hc(α) for α = 0,1/4,1/3 S I and1/2. Resultshavebeenextrapolatedtothethermodynamiclimit (cid:144)H 0.6 (L→∞).hc(α)isthecriticalvalueoftheZeemanfieldhforagiven lass 0.4 Α=0 frustrationparameterα. c0 Α=1(cid:144)4 N 0.2 Α=1(cid:144)3 We can also calculate the sum of the non-macroscopic Α=1(cid:144)2 eigenvalues, Nout. Thiscorrespondsto thecondensatedeple- tion in the original boson problem. In the S → ∞ limit for 0.2 0.4 0.6 0.8 1 a lattice with I sites, the I −1 non-macroscopiceigenvalues h(cid:144)hcHΑL are all zero. The first-order quantum correctionscan be ob- tainedusingdegenerateperturbationtheory—wecanobtain theeigenvaluesastheeigenvaluesofthe(I −1)-dimensional FIG.7: Thecondensate densityper site, Nclassical/I,intheclassical 0 submatrix δρ˜ for i, j = 2,...,I which excludes the macro- limitforα = 0,1/4,1/3,1/2. (I =numberoflatticesites.) h (α)is ji c scopically occupied state. The sum of these eigenvalues is thecriticalvalueoftheZeemanfieldhforagivenfrustrationparam- eterα. simplythetraceofthesubmatrix: N = δρ˜ , (24) Next, we compute the quantum corrections to the classi- out ii Xi,1 cal solution. In the large-S expansion, these corrections are smallandtheleadingcorrectionsareoforder1/S compared Again,Nout ∝S isoneordersmallerinS thanN0class. Wefind to the classical limit. We have computed this leading-order that,justasclassicalcondensatedensity(Nclass/I)vanishesas 0 8 h→h (α),theout-of-condensatenumber,N ,alsovanishes large-S generalization of the model. In the above, we have c out ash→h (α).However,theratioofthetwoquantitiesremains comparedN with themacroscopiceigenvalueN ≃ Nclass. c out 0 0 finite. Thisratio,N /Nclass,isthefractionaldepletionofthe Strictlyspeaking,inordertodiscussthedepletionofthehole out 0 condensate. This quantity is one of interest in experiments condensate in the original boson model, we should use the which measure the degree of Bose-Einstein condensationby analoguefortheholedensitymatrixandthendividethenum- observing the time of flight of expanding condensates. Our berofholesinthesystem. Asdiscussedabove,thecorrespon- resultsforthisfractionaldepletionN /Nclass,rescaledbyS, dence is simple near h : we should consider N /2S com- out 0 c out areshowninFig.9.Theoccupationofthesenon-macroscopic paredto (S −hSˆzi)=S (1−cosΘ). Thisisqualitatively i i i i similartoPtheresultsplottePdinFig.9. 0.8 VI. SUPERFLUIDDENSITY 0.6 s s a l Bose-Einsteincondensationcanbedefinedinequilibrium. c N00.4 On the other hand, superfluidity is related to the transport (cid:144) t Α=0 properties of the system. Those two phenomena are related u o N Α=1(cid:144)4 through the phase of the macroscopic wave function (order S 0.2 Α=1(cid:144)3 parameter).Thesuperflowoccurswhenthephaseofthewave Α=1(cid:144)2 functionvariesinspace. Inthissection,we calculatethesu- perfluid density for our system as a response to an external 0 0.2 0.4 0.6 0.8 1 phase twist. The superfluiddensity, a characteristic quantity h(cid:144)hcHΑL thatdescribesthesuperfluid,measuresthephasestiffnessun- deran imposedphasevariationanddiffersfromzeroonlyin FIG.9: Fractionaldepletion N /Nclass forα = 0,1/4,1/3and1/2 the presence of the phase ordering. We find the superfluid andasafunctionofh/h (α). hout(α)0isthecriticalvalueoftheZee- fractionfollowingthecalculationsofRothandBurnett28 and c c manfieldhfor agiven frustrationparameter α. Resultshavebeen Reyetal.29 wherethesuperfluiddensityiscalculatedforthe extrapolatedtothethermodynamiclimit(L→∞). Bose-Hubbard model with real couplings. Our results show that the superfluid fraction is reduced in the presence of the modesisalsoduetotheanomaloustermsintheHamiltonian. frustration. Thisagainshouldvanishash → h . However,Fig.9 shows Thesuperfluiddensityintroducedbyconsideringachange c that the occupationremains a finite fractionof Nclass even at in the free energy of the system under imposed phase 0 the critical field h . In terms of the original boson model, variations28–30 is equivalent to the helicity modulus30 which c thisresultsuggeststhatcondensatedepletionremainsafinite differs from zero only for ordered-phase configurations and fractionofthetotalnumberofholesevenastheholedensity is consequently an indicator of the long-range phase coher- decreasestozeroath . Ourresultsatzerofrustrationagrees ence of the system. The definition is also equivalent to the c withpreviouswork17,27. definition of the superfluid density in terms of the wind- Weobservethatthisfractionaldepletiondecreasesmonot- ing numberswhich is used in the path-integralMonte Carlo ically as we increase the Zeeman field h from zero to h for methods31–33 and to Drude weight or charge stiffness which c α = 0and1/2. Forα = 1/3,thefractionaldepletionappears describesd.c.conductivity34–38. to have zero slope as a function of h near h . Interestingly, Letusconsiderasystemofsize L inthe xdirection. One c x forα = 1/4,therelativedepletionbecomesanon-monotonic waytoachievethephasetwististoimposethetwistedbound- function of the Zeeman field — the fractional depletion in- aryconditionsonthewavefunctiondescribingthesystem. If creases when h is approached. In fact, if we formally set weassumethatthephasetwistisimposedalongthe x direc- c S = 1/2,thecondensatedepletionevenreachesunitybefore tionthetwistedboundaryconditionsare h reachesh . Aswe will see inthe nextsection, thischange c inbehaviorforα=1/4isalsoseeninthesuperfluidfraction. ΨΦ¯ ~r ,...,~r +L xˆ,... =eiΦ¯ΨΦ¯ ~r ,...,~r,... (25) 1 i x 1 i Wediscussthisfurtherinourconcludingremarks. (cid:0) (cid:1) (cid:0) (cid:1) We note that N , −∆N . In other words, the trace of with respect to all coordinates of the wave function. Let us out 0 thedensitymatrixchangesduetoquantumfluctuations. This introduceaunitarytransformation means that, in the quantum magnet, there is more than one possiblemeasureof“condensation”inthegroundstate. The UΦ¯ =ePiiχ(~ri) with Φ¯ =χ ~r+Lxxˆ −χ ~r . (26) discrepancycanbe tracedto the quantumfluctuationsforSz (cid:0) (cid:1) (cid:0) (cid:1) ateachsite:Trρ= hSˆ+Sˆ−i= [S(S+1)−h(Sˆz)2i+hSˆzi]. The untwisted wave function which satisfies the periodic i i i i i i ForS = 1/2,thisisPsimply (1P/2+hSˆzi),correspondingto boundary conditions Ψ(~r1,...,~ri + Lxxˆ,...) = Ψ(~r1,...,~ri,...) i i is related to the twisted wave function via the unitary trans- thetotalbosonnumberinthPeoriginalmodelwhichisacon- servedquantity. However,foranyS > 1/2,themean-square formation UΦ¯ as |ΨΦ¯i = UΦ¯|Ψi. The Schrödinger equation fluctuation in the local z-componentwill alter the total trace for the system with twisted boundary conditions, Hˆ|ΨΦ¯i = ofthedensitymatrix. Inotherwords,thisisanartifactofour EΦ¯|ΨΦ¯i, can then be rewritten as Hˆ |Ψi = EΦ¯|Ψi where the Ψ¯ 9 twistedHamiltonianis VΦ¯ = Φ¯Jˆ/I −Φ¯2Tˆ /2I2. Calculating the ground-state en- x x x x ergyforthesystem withimposedsmalltwist withinthesec- HˆΦ¯ =UΦ†¯HÛΦ¯. (27) ondorderperturbationtheoryandusing Eq. (28), we obtain the followingexpressionforthe superfluiddensity as a frac- In other words, the eigenvalues of the twisted Hamiltonian tionofthecondensatedensity, f = I I n /N : withperiodicboundaryconditionsarethesameaseigenvalues s x y s 0 oftheoriginalHamiltonianwithtwistedboundaryconditions. 1 |hψ |Jˆ|ψ i|2 1 fs =−N0J hψ0|Tˆx|ψ0i+2Xν,0 Eνν−x E00  , Φ¯ ≪π(,30) whereN ≃ Nclassinthelarge-S limitandψ areeigenstatesof 0.8 0 0 ν originaluntwistedHamiltonianwithν=0labelingtheground state. In terms of the original boson model, N corresponds 0.6 0 s to the number of condensed particles or holes (for h < 0 or f 0.4 Α=0 h>0).Thefirsttermcorrespondstothediamagneticresponse Α=1(cid:144)4 of the condensate while the second term corresponds to the 0.2 Α=1(cid:144)3 paramagneticresponseinvolvingexcitedstates. Α=1(cid:144)2 The results obtained for the superfluid fraction within the Bogoliubov approximation are shown in Fig. 10. The lead- 0 0.2 0.4 0.6 0.8 1 ingtermduetoquantumeffectscomesfromtheparamagnetic h(cid:144)hcHΑL terminEq.(30).ThisisoforderS0. Intheabsenceoffrustra- tion(α = 0), thesystemishomogenousandthesystem con- FIG.10: Superfluiddensityasafractionoftheclassicalcondensate servesmomentum. ThismeansthattheeigenstatesareBloch density Nclass/I as a function h/h (α) for the frustration parameter states corresponding to different momenta. As a result, the 0 c α = 0,1/4,1/3 and 1/2. h (α) is thecritical value of the Zeeman currentmatrixelementinEq. (30),whichcannotcoupledif- c fieldhforagivenfrustrationparameterα. ferentmomenta,vanishes.Moreover,thekineticenergyinthe groundstateisinitselfproportionaltoN .Inthebosonmodel, 0 The superfluid velocity is proportional to the order- thismeansthatthe superfluidfractioncorrespondssimply to parameter phase gradient and an additional phase variation the kinetic energy per hole. This is a quantity which is in- χ(~r) will change the superfluid velocity by ∆~v = ~∇~χ(~r)/m dependent of h and so the superfluid density is the same as s in the continuous system. When the imposed phase gradi- thecondensatedensityinthelarge-S limitatzerofrustration. ent is small so that other excitations except increase in the (However,1/S correctionswillchangetheresult,givingasu- velocity of the superflow can be neglected the change in the perfluiddensitylargerthanthecondensatedensityforgeneral ground-stateenergycanbeapproximatedby∆E =−P~·∆~v + h, but fs → 1 as h → hc.) Similarly, the currentmatrix ele- g s M (∆~v )2/2,withM =mN beingthetotalmassofthesuper- mentvanishesforthefullyfrustratedcase (α = 1/2). Inthis s s s s case, frustration reduces the superfluid fraction in α = 1/2 fluidpartofthesystem. Herewe choosealinearphasevari- ationalongthe xˆ direction, χ(~r) = Φ¯x/L . Replacing~2/2m case to around 70%. For α = 1/3 and 1/4, an increase in x theZeemanfieldhresultsinalargerreductioninthefraction forthecontinuoussystembyJ/2forour2Ddiscretelatticewe obtainthefollowingexpressionforthesuperfluiddensity28 fs atvaluesofh closerto hc(α). Thatcanbeseen in Fig.10 fortheinhomogeneouscasesofα = 1/3and1/4. Asforthe I ∂2E (Φ¯) condensatedepletion,wenotethatthesuperfluiddensityasa ns = IxJ ∂Φ¯g2 Φ¯=0 , (28) fractionofthecondensatedensitydoesnotvanishash→hc. y (cid:12) (cid:12) Wealsonotethatthesuperfluiddensitybehavesdifferently (cid:12) where Ix,y = Lx,y/a with a being the lattice spacing. The forα = 1/3and1/4comparedtoα = 0and1/2. The same twistedHamiltonianisofthesameformastheuntwistedone qualitative change in behavior was observedfor the conden- onlywithφij replacedbyφij−Φ¯. Underassumptionthatthe satedepletioncalculatedinSec. V. phasetwist Φ¯ ≪ π we can calculatethe groundstate energy of the twisted Hamiltonian perturbatively. Expanding eiΦ¯/Lx uptothesecondorderinΦ¯ thetwistedspinHamiltonianbe- VII. CONCLUSION comes HΦ¯ = H+ Φ¯ Jˆ − Φ¯2Tˆ , (29) We have studied the ground state for bosonic atoms in a I x 2I2 x frustrated optical lattice by mapping the problem to a frus- x x trated easy-plane magnet. Using a large-S approach, we where Jˆx = iJ i(eiφii+xSî+Sî−+x −H.c.)/2 is the paramagnetic further introduce quantum effects under the assumption that currentoperatoPrand Tˆx = −J i(eiφii+xSî+Sî−+x +H.c.)/2 cor- thoseeffectsdonotchangequalitativelythenatureoftheor- respondstothekinetic-energyoPperatorforthehoppinginthe deringobtainedfortheclassicalgroundstates. Weexamined x direction. The terms in the Hamiltonian above that con- our results for any precursorto the non-superfluidor uncon- tain the twist angle can be treated as a small perturbation densedstates. 10 Wehavefoundthatfrustrationcandecreasethedepletionof intriguingto note that this case does not have a Halsey-type thecondensateandthesuperfluidfraction.However,thefrac- classical groundstate and in fact has two degenerateground tionaldepletionofthecondensateandthesuperfluidfraction stateswithdifferentphasepatterns.Onecanspeculatethatthe remainfiniteforallincommensuratefilling[h < h (α)]. The motionofdomainwallsbetweenthetwodifferentphasepat- c behaviorofthefractionalcondensatedepletionandsuperfluid ternsmaycontributetoaroutetodecondensationand/orloss fraction as a functionof filling has interesting behavior. We ofsuperfluidity. find that the cases of α = 0 and 1/2 behave differently from the cases of α = 1/3 and 1/4. Surprisingly, for the cases of Finally,wenotethatfractionalquantumHallstatesareex- smallerα,thefractionalcondensatedepletionbecomesanon- pected when the numberof vortices becomes comparable to monotonic function of the filling, decreasing as we increase thenumberofatomsorholesintheBose-Hubbardmodel. In h from zero but eventually increases as h → h . In fact, if ourlarge-S theory,thebosonnumberisproportionaltoS and c weformallysetS =1/2,thenthecomputedfractionaldeple- sothequantumHallregime,ifitexistsinsuchatheory,exists tion exceeds100%forthe α = 1/4case as h approachesh . only when h − h ∼ 1/S. Therefore, one might expect the c c We also have some evidence that the same behavior occurs condensate depletion or the reduction in the superfluid frac- in the α = 1/6 case for small system sizes. In other words, tiontobelargeash → h . Wedonotfindthisdirectlyinour c ourresultsraisethepossibility,forα<1/4,ofasecond-order perturbativetheoryin1/S. 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