Weak first-order superfluid--solid quantum phase transitions(3)

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

FIG.2:Statisticsof uctuatingSFandVBSorderparameters.Leftpanel:pointsofthe“blacksquare”areobtainedforMCcon gurationsseparatedbyequallongtimeintervalsforL=16andD=0.5727.Middlepanel:SFsti nessiscalculatedasafunctionofBfor

varioussystemsizes.Fromtoptobottomcurvesareplottedatcorresponding nite-L“transitionpoints”(seethetext),for(L=8,D=0.5742),(L=12,D=0.5738),(L=16,D=0.5727),(L=24,D=0.5715),(L=32,D=0.5712),(L=48,D=0.5705).Rightpanel:thecoarse-graineddistributionoftheaveragedensityofpointsintheBx ,By -squarealongtheB=|Bx |+|By |=constlinesforthesamesetofsystemsizesandvaluesofDasinthemiddlepanel.

itisexpectedtobemacroscopicallysmall.Itappearsasifthesuper uidorderparameterΨexperiencesanoma-louslylarge uctuationsinsmallsystems.Toexplainitwe rstspeculate(andlaterprovenumerically)thatinthevicinityofthecriticalpointthesystemisbest

,describedbythefour-dimensionalorderparameter,S

andtheO(4)-symmetryisbrokenatDc.Formally,Bx ,By andtwocomponentsofΨrepresentobservable(lin- ;correspondingly,inearlyindependent)projectionsofSthe(Bx ,By ,Ψ)-spacetheO(4)-sphereisseenasafour-dimensionalsurfacewiththespheretopology.Inthisscenario,iftheO(4)-symmetryisexactthenanyvalueof isequallyprobableatDc,i.e.solidordersalongbothS

spatialdirectionandsuper uiditymaycoexist.

Ananalogouswell-knownexampleoftheO(3)-symmetricpointisprovidedbytheSF-CBtransitioninthe2DquantumXXZ-antiferromagnetwithn.n.ex-changeinteractions.Inthiscase,theXYorderparameterSx+iSy≡ΨandtheCBorderparameterSz≡Mform

.Thecriticalpointitselfisathree-dimensionalvectorS

describedbytheSU(2)-symmetricHeisenbergHamilto-nian.SincetheO(3)-symmetryisspontaneouslybrokeninthegroundstateoftheHeisenbergmodel,atthetran-sitionbothρsandMchangediscontinuouslyinthether-modynamiclimit,butthischangeoccurswithoutenergybarriersandisprecededbyanomalouslylarge uctua-tionsandlongautocorrelationtimesin nitesystems.Iftheoutlinedpictureiscorrect,thensmallpertur-bationswhichexplicitlybreaktheO(n)symmetryatthemicroscopiclevelshouldresultingenericweak rst-ordertransitions.Indeed,inthespontaneouslyorderedstateallrenormalizationsare nite.Thussymmetrybreakingperturbations,whichcoupletothelong-rangeorder,re-sultinthenon- atmacroscopicenergy/e ectiveaction

FIG.3:Systemsizedependenceofρs(B=0)/ρs(D=0)(circles)andBmax(squares)fromFig.1.

pro lefortheorderparameter.Phasetransitionsare,then,betweenenergyminimaseparatedbymacroscopicbarrierswhichmay,however,remainrelativelyweakinsmallsystems.TheJ-currentmodelstudiedherehasnomicroscopicsymmetriesmandatingexactO(4)-orO(3)-symmetryofthecriticalpoint.WeconcludethenthatSF-VBSandSF-CBtransitionsareexpectedtobe rstorder,andintherestofthepaperwepresentevidencethatthisisindeedso.

First,wedemonstratethatrelatively“small”(hun-dredsofparticles!)systemsinthevicinityofthecriticalpointbehaveasif(Bx ,By ,Ψ) uctuationsarecon nedtosomesurface,notvolume,anditisnotpossibletohaveallthreeorderparameters uctuatingtozero.IntheleftpanelofFig.2weshowstatistical uctuationsofthe

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