We study superfluid--solid zero-temperature transitions in two-dimensional lattice boson/spin models by Worm-Algorithm Monte Carlo simulations. We observe that such transitions are typically first-order with the exception of special high-symmetry points wh
tems.Acloserlookatthetransitionpointisprovidedintheinsets.Errorbarsareshownforallpoints(insomecasesthesearesmallerthanthesymbolsize).
ciency,classical(d+1)-dimensionalanalogsofd-dimensionalquantumsystemso erasigni cantadvan-tage.Thisapproachwassuccessfullyusedpreviouslyinthestudiesofdisordered[5,7,8]andtwo-componentsystems[9].Inaddition,classicalmodelso ermorefree-domin“designing”e ectivemodelswithcomplexphasediagrams.Theso-calledJ-currentmodelofRef.[5]isobtainedbyconsideringtrajectoriesindiscreteimagi-narytime.Letn=(x,τ)denotepointsofthe(d+1)-dimensionalspace-timelattice,andintegercurrentsJν(n)withν=x ,y ,τ specifyhowmanyparticlesaregoingfromsitenindirectionν.Inthislanguage,cur-rentsinthetime-directionrepresentoccupationnum-bers,andcurrentsinthespacedirectionsrepresenthop-ping events.Thecontinuityoftrajectoriesrequiresthatν[Jν(n) Jν(n ν)]≡0.
ThesimplestJ-currentmodelathalf-integer llingfac-torisobtainedbywritingthepotentialenergytermintheparticle-holesymmetricformS2J J1
J=τ (n)
n,ν=τ
2
,(1)
andrestrictingcurrentsalongτ -bondstotakeonjusttwo
values,0and1.Thekineticenergytermissimply
SK=K
|Jν(n)|,
(2)
n,ν=τ
withtherestrictionthatallowedvaluesforspatialcur-rentsare1,0,and 1values.Toexcludesomewhat
pathologicalcaseswithtwohoppingeventshappeningat thesamespace-timepoint,wefurtherrequirethatν=τ |Jν(n)|+|Jν(n ν)|≤1.Finally,weintroduceinteractionsbetweenthespatialcurrentsonn.n.bonds
2
whichfavorVBS
SD= D |Jν(n)|
(|Jν(n+µ)|+|Jν(n µ)|).
n,ν=τ
µ=ν
(3)
Equal-timecoupling(µ=τ )favorssimultaneoushoppingeventsofparticlesonthesameplaquetteandisreminis-centoftheringexchangeterminquantummodels[2].PhononmediatedexchangeisanotherknownmechanismofdimerizationinspinPierlessystems[10],andinEq.(3)itisrepresentedbycouplingbetweenbondsconnectingthesamesitesandshiftedinthetimedirection.Withallthreetermscombined,S=SJ+SK+SD,theresultingmodelhasSF,CB,andVBSstatesinitsphasediagram.FirstwestudytheSF-VBStransitionalongtheJ=0,K=0.4line.Super uidsti nessisdeterminedbythestatisticsofwindingnumber uctuations[6]
ρs= W2 /2L,
(4)
andtheVBSorderparameterischaracterizedbythe
staggereddistributionofspatialcurrentsalongx andy directions
Bν=L (d+1)
|Jν(n)|einq,(5)
n
whereν=x andν=y forq=(π,0,0)andq=(0,π,0),
respectively.We nditconvenienttointroduceasin-gleVBSorderparameterwithpositivede niteestimatorwhichtakeson nitevalue～O(1)intheVBSphase,B=|Bx |+|By |.Forcompleteness,wealsode neheretheCBorderparameteras
Mτ(q=(π,π,0))=L (d+1)
Jτ (n)e
inq
.(6)n
andM=|Mτ|.
InFig.1weshowrescaleddatafortheSFsti nessρs/ρs(D=0)andVBSorderparameter.ThemainplotforL=16demonstratesstrongsuppressionofρsandBnearthecriticalpoint[Dc≈0.5705(2)]whichistyp-icalforcontinuousphasetransitions.Similarbehaviorwasreportedpreviouslyforthering-exchangemodelinRef.[2].However,intheinsetsweclearlyseethat nite-sizescalingisincompatiblewiththesecond-ordertransi-tionscenario—thecurvesρs(D)fordi erentsizesLinter-secteachotherwithoutanyfurtherrescalingindicatingthatlargesystemsaremoreorderedinthevicinityofthecriticalpoint.SimilarbehaviorisobservedforthecurvesB(D).Themostobviousscenarioisthena rst-ordertransitionwheretheintersectionof nite-sizecurvesatthecriticalpointisallowed.Apparently,thetransitionisweakly rst-orderbecause(i)bothorderparametersarestronglysuppressedatDc,and(ii)simulationsforL3=323systemdonotshowanyhysteresis,thoughtheautocorrelationtimeisverylongatDc.
TheothersurprisingfactisthattheregionwhereρsforL=16isbelowthecorrespondingcurvesforL=32andL=64isratherextended,whilein rst-ordertransitions