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
rameterforvarioussystemsizes;fromrighttoleft:(L=8,J=0.450),(L=16,J=0.4205),(L=24,J=0.4189),(L=32,J=0.4186).
VBSorderparametersforL=16andD=0.5727.ThedensityofpointsintheBx ,By “blacksquare”isnearlyhomogeneous,i.e.thesystemisequallylikelytobefoundwithsmallorrelativelylargeVBSorderorientedatanyanglerelativetox ,y -directions.Moreover,theboundaryshapesuggeststhatBx andBy uctuationshappenontheB=f(|Ψ|)surface.ThemiddlepanelofFig.2showsthatindeedthevaluesofBandρsarestronglycorrelated[ρs(B)isde nedastheaverage,seeEq.(4),overcon g-urationswiththeVBSorderparameter∈(B δ,B+δ)].Weseethat,ρsislargestwhenB→0,i.e.despitelarge uctuationsofallorderparameters,theynevervanishsimultaneously.
Finally,intherightpanelweplottheaverageden-sityofpointsintheBx ,By -planealongtheB=constcutsforvarioussystemsizes.Ineachcase,weadjustedthecouplingconstantDsothatthedistributionfunctionP(B)is“maximally at”,orthetwomaximaareatequalheights.Normalizationwassettohavethelarge-Bmax-imumequalunity.Thedensitypro lesarenearly atforL=6,12,16thoughwithatinyminimumbetween
theVBSandSFregions.Itappearsasifphysicalorderparametersbelongtosomefourdimensionalsurfaceandmaydi useoveritwithoutlargee ectiveactionbarriers.TheminimumgetsmorepronouncedforL=24,32,andbreaksthedistributionintowellseparatedpeaksforL>48,i.e.theenergybarrier nallygetslargeenoughtolocalizetheorderparameterinoneofthephases.Thetwo-peakstructureisasmokinggunevi-dencethatthetransitionisultimatelyweak rst-order.Furtherevidenceisprovidedby nite-sizescalingofthecritical-pointdistributionfunctions,i.ethedependenceofρs(B=0,L)(normalizedtoρs(D=0)asinFig.1)andthepositionoftheP(B)distributionmaximum,Bmax(L).ThedatainFig.3suggestthatbothquantitiessaturateto nitevaluesinthethermodynamiclimit.ThestudyoftheSF-CBtransitionalongtheK=0.7,D=0linerevealsaremarkablesimilaritywiththeSF-VBSpoint.AtJc=0.4184(2)thesuper uidandsolidordersexchangeplaceswithpronounced uctua-tionsofbothorderparametersinsmallsystems.These uctuationsarealmostidenticaltowhatisexpectedfortheO(3)-symmetricHeisenbergpointwherethedis-tributionfunctionforthestaggeredorderparameter,P(M<Mmax)=P(Sz)isastepfunction.However,inlargersystemsaminimuminP(M)isdeveloped.Thedouble-peakstructureofP(M)forL=32leavesnodoubtthatweareactuallydealingwiththeweak rst-ordertransition.Apparently,systemsizesinthestudyoftheSF-CBpointinRef.[1]weretoosmalltoseethe rst-ordertransition.
Atcertainconditions(whichwithJ-currentmodelcanbeeasilyachievedbyaddingappropriateterms)thesu-persolid(SS)phasemayintervenebetweenSFandSphases.Inthiscase,thevicinitytoa(quasi-)O(3)/O(4)symmetricpointwithbrokensymmetrymayrendertheSF-SSandSS-Stransitionsalso rstorder,whilenor-mallyonemightexpectthemtobeofthesecondorder.Inconclusion,wenotethatweak rst-ordertransitionsdiscussedherecanhardlybeanartifactoftheJ-currentmodelsincetheyrevealthemselvesonlargespace-timescales.
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