Re-Entrant Quantum Phase Transitions in Antiferromagnetic Sp

1、re-entrant quantum phase transitions in antiferromagnetic sp in response to recent chemical attempts to construct higher-spin ladder materials from organic polyradicals, we study the ground-state properties of a wide class of antiferromagnetic spin-1 ladders. employing various numerical tools, we re

2、veal the rich pha re-entrantquantumphasetransitionsinantiferromagneticspin-1ladders shojiyamamoto departmentofphysics,okayamauniversity,tsushima,okayama700-8530,japan t orusakai departmentofelectronics,tokyometropolitaninstituteoftechnology,hino,tokyo191-0065,japan arxiv:cond-ma t/0 2 6553v1 cond-ma

3、t.str-el 27 jun 2021 akihisakoga departmentofappliedphysics,osakauniversity,suita,osaka565-0871,japan (received31december2021) inresponsetorecentchemicalattemptstoconstructhigher-spinladdermaterialsfromorganicpolyradicals,westudytheground-statepropertiesofawideclassofantiferromagneticspin-1lad-ders.

4、employingvariousnumericaltools,werevealtherichphasediagramandcorrectaprecedingnonlinear-sigma-modelprediction.avariationalanalysiswellinterpretsthephasecompetitionwithparticularemphasisonthere-entrantphaseboundaryasafunctionoftherunginteraction.pacsnumbers:75.10.jm,75.40.mg,75.40.cx in1983haldaneawo

5、kerenewedinterestinquan-tumspinchainspredictingastrikingcontrastbetweeninteger-andhalf-odd-integer-spinheisenbergantiferro-magnets.hisargumentwasindeedveri edinaspin-1materialni(c2h8n2)2no2(clo4)andwasgivenananalyticsupport3aswell.sincethentheenergygapsinmagneticexcitationspectra,thatis,spingaps,hav

6、ebeenacentralissueinmaterialsscience.inthelastdecademoreandmoreresearchersmadeawidevarietyofexplorationsintothespin-gapproblem,suchasthespin-peierlstransitionininorganiccompounds4,quantizedplateauxinmagnetizationcurvesandantiferromag-neticgapsintheferromagneticbackground6.amongothersdagottoetal.7poi

7、ntedoutthatanothermecha-nismofthegapformationshouldlieinaladder twocou-pledchains.aspingapwasindeedobservedinatypicaltwo-legladdermaterialsrcu2o38.moreover,super-conductivitywasbroughtaboutinitshole-dopedversion(srca)14cu24o41laddersystemscausedusfurthersurpriseexhibitingexcitationspectravaryingwith

8、thenumberoftheirlegssofarmetaloxideshavebeenrepresentativeofladdermaterials.thoughmolecule-basedoneshavebeensynthesizedinanattempttoreducethespingapsandobtainexperimentalaccesstothem,thesituationofcop-perionssupplyingtherelevantspinsremainsunchanged.thereforetheyareallspin-1 ladderan-tiferromagnetst

9、hetechniquewasfurtherdevel-opedforspatiallyinhomogeneousladdersmixed-spinladderswerealsoinvestigatedwithpartic-ularemphasisonthecompetitionbetweenmassiveandmasslessphases. incomparisonwithextensivecalculationsonspin-1 2 andthatofe ectivelyspin-1,respectively.their polyradicalstrategyhasyieldedfurthe

10、rharvestsuchasane ectivespin-1antiferromagnetonahoneycomblat-tice14andaladderferrimagnetofmixedspins1and 2 1 two-legladders.theobtainedphasediagramisreminiscentoftheprecedingsigma-modelpredictionbutcon-tainsare-entrantphaseboundary,whichcanneverbeextractedfromany eld-theoreticalargument. considering

11、thatanadvantageofassemblingorganicopen-shellmoleculesintoamagneticmaterialistheisotropicintermolecularexchangecouplings,whilethepolyradicalstrategyisaccompaniedbyspatialvariationsinmagneticinteractionwetreatawideclassofspin-1antiferromagneticladders 2 h=1 2l j=1 i=1 j i,jsi,jsi,j+1+js1,js2,j ,(1) in

12、 response to recent chemical attempts to construct higher-spin ladder materials from organic polyradicals, we study the ground-state properties of a wide class of antiferromagnetic spin-1 ladders. employing various numerical tools, we reveal the rich pha opfig.1.phasediagramsfortheantiferromagnetics

13、pin-1ladderwithtwoout-of-phaselegs.(a)a eld-theoreticalpre-diction19.thetwocriticallines(dashedlines)derivedfromthee ectivesigmamodelforladdersareinconsistentwiththesigma-modelanalysisonisolatedchains( ).theyre-mainfarapartfromeachothereveninthedecoupled-chainlimitr=0.thereforequalitativelypatched-u

14、pphasebound-aries(solidlines)werepredicted.(b)ournumerical ndings.theseries-expansionestimatesareshownby,whilethelevel-spectroscopyanalysesby2(l=6)and (l=8). -10-20-30-40-50 l= 8 -60-70 l= 8 l= 4 l= 4 l= 6 l= 6 (a)op= 0.6 (b)op= 0.8r fig.2.demonstrationofthelevelspectroscopy.thelow-est-lyingtwoeigen

15、valuesinthesubspaceofzeromagnetiza-tionasfunctionsofrcrossattransitionpointsprovidedthetwistedboundaryconditionisimposedonthehamiltonian. wherethebond-alternationparameteri,jisde nedintwowaysas 1+( 1)i+jop(out-of-phaselegs), i,j=(2) 1+( 1)jip(in-phaselegs).wecalculatetheregionof0op(ip)1andhere-after

16、setj/j tor(0).martin-delgado,shankar, andsierra19studiedthecasesofout-of-phaselegsde-rivingalow-energy-relevantsigmamodel.forthespin-sladderswithtwoout-of-phaselegs,thetopologicalangleinthee ectivesigmamodelturnsout8sop/(r+2)andreadsasthecriticallines8sop=(2n+1)(r+2) 2 (n=0,1,).however,these ndingsd

17、onotsmoothlymergewiththewell-establishedcriticalbehaviorinonedimension,2s(1 )=2n+117,asisshowninfig.1(a).thus,itisnecessarytoverifythetruescenarioallthemoreinhigherdimensions. oneofthemostreliablesolutionmaybeanumericalanalysis28onthephenomenologicalrenormalization-groupequation29.however,thescaledg

18、apsareill-naturedduetotheclosecriticalpoints,soastomakethe xedpointshardtoextractfromavailablenumer-icaldata.thenweswitchourstrategytothelevelspectroscopy30,thecoreideaofwhichissummarizedasdetectingtransitionpointsbycrossingoftworel-evantenergylevels.althoughthemethodisgeneri-callyapplicabletothegau

19、ssiancriticalpoints31,noexplicitchangeofsymmetryaccompaniesthepresentphasetransitionsandthereforeanylevelsdonotcrossnaively.inordertoovercomethedi cultyofthiskind,kitazawa32proposedtheideaofapplyingthetwistedboundarycondition,thatistosayinthepresentcase,exchangecouplingsequalto 2settingthexboundaryy

20、yxzz i=1j i,l(si,lsi,1+si,lsi,1 si,lsi,1).thentheen-ergystructureofthehamiltonianischangedandthelow-esttwolevelsareledtocrossattransitionpoints,whichisdemonstratedinfig.2.duetothelimitoftimeandmemorywellspent,wehaverestrictedourcalculationsuptol=8.weplotinfig.1(b)bare ndingsforthecross-ingpointsatl=

21、6andl=8ratherthanextrapolatethemtrickily.wearesurethatthedatauncertaintystillleftiswithinthesymbolsize.aseries-expansiontech-nique33,34guaranteesthelevelspectroscopytoworkwell.startingwithdecoupledsingletdimersonlegsorrungsandexpandingtheenergygapasapowerseriesinarelevantperturbationparameter,wecano

22、btainapartialknowledgeofphasetransitions.herewehavecalculatedthegapuptotheninthorderandfurtherap-pliedthedlogpadeapproximants35tothem.thethus-obtainedphaseboundaries,whicharealsoshowninfig.1(b),elucidatethenatureofthephasecompe-tition,thatis,thea eck-kennedy-lieb-tasaki(aklt)valence-bond-solid(vbs)3

23、onasnakelikepathversusdecoupleddimers. themostimpressive ndingsarere-entrantquantumphasetransitionswithincreasingr.theprecedingsigma-modelanalysis19isindeedenlighteningbutneverabletorevealthisnovelquantumbehavior.inordertochar-acterizeeachphase,letusconsideravariationalap-proach.weknowthatsingletdim

24、ersonrungsfig.3(h)arestabilizedforr,whereaseitherdimersonlegsfigs.3(d)and3(e)orthedoubleakltvbsfig.3(c)forr0.twomoreinterchainvbsstatesfigs.3(f)and3(g)maybeadoptedasvariationalcomponentsfortheintermediate-rregion.thusthelinearcombinationoffigs.3(c)to3(h)canbeanapproximateground-statewavefunctionfors

25、pin-1ladders.sincethepresentvari-ationalcomponentsareallasymptoticallyorthogonaltoeachother,thevariationalgroundstateturnsoutanyofthemitself36.thethus-obtainedphasediagramis e r in response to recent chemical attempts to construct higher-spin ladder materials from organic polyradicals, we study the

26、ground-state properties of a wide class of antiferromagnetic spin-1 ladders. employing various numerical tools, we reveal the rich pha presentedinfig. 4.thesigni cant stabilizationoftheintermediatephase,which isnowcharacterizedassh,andtheresultantre-entrantphaseboundaryaresuccess-fullyreproduced.con

27、sideringthatacoupleofcriticalchainsimmediatelyturnmassivewiththeirrunginter-actionswitchedon7,thepointcshouldcoincidewiththepointaundermorere ned(andthusinevitablynu-merical)variationalinvestigation. thepresentvariationalcalculationimpliespossiblephasetransitionsforin-phase-legladdersaswell,butthisi

28、stotallyduetothenaivewavefunction.numericalob-servationoftheenergystructureendsupwithnogapless pointinthisregion.thesigma-modelapproachalsoconcludesnocriticalpoint,givingthetopologicalan-gle4sindependentofbothrandp.thekeytotheground-statenatureofin-phase-legladdersisthefour-spincorrelation25.letusco

29、nsiderinteractingfourspinsofs=1 2(a =a b b);s 1 (a) plaquette singlet solid (pss) (b)double plaquettesinglet (dps)(c) double haldane (dh) (d)out-of-phase leg dimer (opld)(e)(f) in-phase legdimer (ipld)snake haldane (sh) movesfrom0to 1 r+r2.asrvariesfrom0to, (g)decoupled plaquette (dp)(h) rung dimer(

30、rd) fig.3.plaquette-singlet-solidandvalence-bond-solidstatesrelevanttothetwo-legantiferromagneticspin-1lad-ders. denotesaspin1 2 sinside. 4.0 atsite(i,j)intoaspin1.nowthat(r) isacontinuousfunctionofrandmayheredeviatefromthatineq.(3),anaiveoptimization36ofeq.(4)isnomorefeasible.however,there nedvaria

31、tionalschemeshowsusmore.forbetterunderstandingofthewavefunction(4),wevisualizeinfig.3itsspecialformsfor= 4, 2s rung dimer 3.0 and= plaquette singlet formation decoupled plaquette 2.0 r snake haldane 1.0 0.01.0 in-phase leg dimer b double haldanec leg dimer op ip 0.5a0.0a0.51.0 fig.4.variationalphase

32、diagramsforthetwo-legan-tiferromagneticspin-1ladders.thethicksolidlinesde-scribephasetransitions,whereasthethinonesrepresentthecrossoveroftheground-statenaturewithinthepresentvari-ationalscheme.thedottedlineisonlyaguideforeyes. 3 heisenberg chain31.weareallthemoreconvincedoftheimme-diategapformation

33、withrmovingawayfrom0.ontheotherhand,neitherpssnordpsincludesbothopldandshandthereforethetwocriticallinesintheout-of-phase-legregionsurviveagainsttheplaquettesingletformation. thegeneralizedstringorderparameter37o()= j 1 zzz lim|i j| sil=iexpislsj isalsousefulinchar-2 in response to recent chemical a

34、ttempts to construct higher-spin ladder materials from organic polyradicals, we study the ground-state properties of a wide class of antiferromagnetic spin-1 ladders. employing various numerical tools, we reveal the rich pha acterizingthegroundstate.o()distinguishesbetweenone-dimensionalvbsstates by

35、 itsdependence36.hence,measuringitonthelinear-chainandsnakepaths,wecandetectthetransitionsbetweendh,opld,sh,andrd,asisshowninfig.5.ifwespecifythetransitionthroughachangeofthedependenceinthevicinityof=,thatis,thechangefromthecon-vexcurvetotheconcaveone,weobtainthetransi-tionpoints(op,r)=(0.245,0)and(

36、0.6,2.125),whichareingoodagreementwiththenumerical ndingsinfig.1(b).whateverpathwetakeforo(),itspeakneversitson=inthein-phase-legregion,sug-gestingthatwecannotobservethehaldanestateofanykindthere.theplaquettesingletformationcaninsteadbevisualizedbyextendingo()toladdersas j 1 zzzzzz lim|i j| s1s,i2,i

37、l=iexpi(s1,l+s2,l)s1,js2,j 25. 0.5 2 3 456789 r= 0.00.4=op0.00.27 op= 0.6 r =2.62.4 10111213 0.250.262.152.2 (a)o ) ( (b)o ( ) fig.5.quantummontecarloestimatesofthegeneralized stringorderparameterde nedonthelinear-chain(a)andsnake(b)paths.thedashedlinesrepresenttheanalyticcal-culationsforth

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