Frustrated Bose condensates in optical lattices

10 Pages·0.24 MB·

by  T. Duric

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)eiφijSˆi+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†jaˆi+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- Sˆi+ =cˆ†i(2S −cˆ†icˆi)1/2, Sˆiz =cˆ†icˆi−S. (5)
pearintheHubbardHamiltonianduetothepresenceoftheef-
fectivevectorpotentialA~.Whenanatommovesfromalattice wherecˆiareoperatorswithbosoniccommutationsandarees-
siteatR~ toaneighboringsiteatR~ ,itwillgainanAharonov- sentially the original bosons aˆi 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, aˆi →eiχiaˆi, Sˆi− →eiχiSˆi−. (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†jσaˆiσ+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
Sˆi− = aˆ†i↓aˆi↑ 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  SSˆˆiiyz(1,0) 0nq.euwadToroaptoeicrbatitanoinrtsh,aewbdeoiascgoaonnnicwalorizipteeedrtahHteoarpmsaairlsttooHnˆfita=hnecˆiH†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,bˆi,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†ibˆi/(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,bˆi,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)eiφijbˆibˆ†j +H.c.(cid:17)+hXi bˆ†ibˆi. (14) stimonasll∆inE0th/Ee0cllaasrsgfeo-Srsleivmeirta.llWateticcealsciuzleaste(6t0he×r6e0la,t1iv2e0c×or1r2ec0-,
6
ourbosonproblem.Sinceweareconsideringalatticesystem
abovehalffilling, itismoremeaningfultoconsiderthe con-
densation 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 = hSˆi−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 ma- trix 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†¯HˆUΦ¯. (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+xSˆi+Sˆi−+x −H.c.)/2 is the paramagnetic further introduce quantum effects under the assumption that
currentoperatoPrand Tˆx = −J i(eiφii+xSˆi+Sˆi−+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. However,ourresultsforthefluc-
phasetransitiontoanon-condensedstatewherequantumfluc- tuationsaroundnon-Halsey-typegroundstatessuggestthatan
tuationsarelargeenoughtodestroyBosecondensation. Itis instabilitytoanon-condensedstatemaybepossible.
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