Re-Entrant Quantum Phase Transitions in Antiferromagnetic Spin-1 Ladders

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 reveal 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-mat.str-el] 27 Jun 2002

AkihisaKoga

DepartmentofAppliedPhysics,OsakaUniversity,Suita,Osaka565-0871,Japan

(Received31December2001)

Inresponsetorecentchemicalattemptstoconstructhigher-spinladdermaterialsfromorganicpolyradicals,westudytheground-statepropertiesofawideclassofantiferromagneticspin-1lad-ders.Employingvariousnumericaltools,werevealtherichphasediagramandcorrectaprecedingnonlinear-sigma-modelprediction.Avariationalanalysiswellinterpretsthephasecompetitionwithparticularemphasisonthere-entrantphaseboundaryasafunctionoftherunginteraction.PACSnumbers:75.10.Jm,75.40.Mg,75.40.Cx

In1983Haldaneawokerenewedinterestinquan-tumspinchainspredictingastrikingcontrastbetweeninteger-andhalf-odd-integer-spinHeisenbergantiferro-magnets.Hisargumentwasindeedveri edinaspin-1materialNi(C2H8N2)2NO2(ClO4)andwasgivenananalyticsupport[3]aswell.Sincethentheenergygapsinmagneticexcitationspectra,thatis,spingaps,havebeenacentralissueinmaterialsscience.Inthelastdecademoreandmoreresearchersmadeawidevarietyofexplorationsintothespin-gapproblem,suchasthespin-Peierlstransitionininorganiccompounds[4],quantizedplateauxinmagnetizationcurvesandantiferromag-neticgapsintheferromagneticbackground[6].AmongothersDagottoetal.[7]pointedoutthatanothermecha-nismofthegapformationshouldlieinaladder twocou-pledchains.Aspingapwasindeedobservedinatypicaltwo-legladdermaterialSrCu2O3[8].Moreover,super-conductivitywasbroughtaboutinitshole-dopedversion(SrCa)14Cu24O41LaddersystemscausedusfurthersurpriseexhibitingexcitationspectravaryingwiththenumberoftheirlegsSofarmetaloxideshavebeenrepresentativeofladdermaterials.Thoughmolecule-basedoneshavebeensynthesizedinanattempttoreducethespingapsandobtainexperimentalaccesstothem,thesituationofcop-perionssupplyingtherelevantspinsremainsunchanged.Thereforetheyareallspin-1

ladderan-tiferromagnetsThetechniquewasfurtherdevel-opedforspatiallyinhomogeneousladdersMixed-spinladderswerealsoinvestigatedwithpartic-ularemphasisonthecompetitionbetweenmassiveandmasslessphases.

Incomparisonwithextensivecalculationsonspin-1

2

andthatofe ectivelyspin-1,respectively.Their

polyradicalstrategyhasyieldedfurtherharvestsuchasane ectivespin-1antiferromagnetonahoneycomblat-tice[14]andaladderferrimagnetofmixedspins1and

2

1

two-legladders.Theobtainedphasediagramisreminiscentoftheprecedingsigma-modelpredictionbutcon-tainsare-entrantphaseboundary,whichcanneverbeextractedfromany eld-theoreticalargument.

Consideringthatanadvantageofassemblingorganicopen-shellmoleculesintoamagneticmaterialistheisotropicintermolecularexchangecouplings,whilethepolyradicalstrategyisaccompaniedbyspatialvariationsinmagneticinteractionwetreatawideclassofspin-1antiferromagneticladders

2

H=1

2L

j=1

i=1

J γi,jSi,j·Si,j+1+J⊥S1,j·S2,j

,(1)

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 reveal the rich pha

OPFIG.1.Phasediagramsfortheantiferromagneticspin-1ladderwithtwoout-of-phaselegs.(a)A eld-theoreticalpre-diction[19].Thetwocriticallines(dashedlines)derivedfromthee ectivesigmamodelforladdersareinconsistentwiththesigma-modelanalysisonisolatedchains( ).Theyre-mainfarapartfromeachothereveninthedecoupled-chainlimitr=0.Thereforequalitativelypatched-upphasebound-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)δOP= 0.6

(b)δOP= 0.8r

FIG.2.Demonstrationofthelevelspectroscopy.Thelow-est-lyingtwoeigenvaluesinthesubspaceofzeromagnetiza-tionasfunctionsofrcrossattransitionpointsprovidedthetwistedboundaryconditionisimposedontheHamiltonian.

wherethebond-alternationparameterγi,jisde nedintwowaysas

1+( 1)i+jδOP(out-of-phaselegs),

γi,j=(2)

1+( 1)jδIP(in-phaselegs).Wecalculatetheregionof0≤δOP(δIP)≤1andhere-aftersetJ⊥/J tor(≥0).Martin-Delgado,Shankar,

andSierra[19]studiedthecasesofout-of-phaselegsde-rivingalow-energy-relevantsigmamodel.Forthespin-Sladderswithtwoout-of-phaselegs,thetopologicalangleinthee ectivesigmamodelturnsout8πSδOP/(r+2)andreadsasthecriticallines8SδOP=(2n+1)(r+2)

2

(n=0,±1,···).However,these ndingsdonotsmoothlymergewiththewell-establishedcriticalbehaviorinonedimension,2S(1 δ)=2n+1[17],asisshowninFig.1(a).Thus,itisnecessarytoverifythetruescenarioallthemoreinhigherdimensions.

Oneofthemostreliablesolutionmaybeanumericalanalysis[28]onthephenomenologicalrenormalization-groupequation[29].However,thescaledgapsareill-naturedduetotheclosecriticalpoints,soastomakethe xedpointshardtoextractfromavailablenumer-icaldata.Thenweswitchourstrategytothelevelspectroscopy[30],thecoreideaofwhichissummarizedasdetectingtransitionpointsbycrossingoftworel-evantenergylevels.Althoughthemethodisgeneri-callyapplicabletotheGaussiancriticalpoints[31],noexplicitchangeofsymmetryaccompaniesthepresentphasetransitionsandthereforeanylevelsdonotcrossnaively.Inordertoovercomethedi cultyofthiskind,Kitazawa[32]proposedtheideaofapplyingthetwistedboundarycondition,thatistosayinthepresentcase,exchangecouplingsequalto 2settingthexboundaryyyxzz i=1J γi,L(Si,LSi,1+Si,LSi,1 Si,LSi,1).Thentheen-ergystructureoftheHamiltonianischangedandthelow-esttwolevelsareledtocrossattransitionpoints,whichisdemonstratedinFig.2.Duetothelimitoftimeandmemorywellspent,wehaverestrictedourcalculationsuptoL=8.WeplotinFig.1(b)bare ndingsforthecross-ingpointsatL=6andL=8ratherthanextrapolatethemtrickily.Wearesurethatthedatauncertaintystillleftiswithinthesymbolsize.Aseries-expansiontech-nique[33,34]guaranteesthelevelspectroscopytoworkwell.Startingwithdecoupledsingletdimersonlegsorrungsandexpandingtheenergygapasapowerseriesinarelevantperturbationparameter,wecanobtainapartialknowledgeofphasetransitions.Herewehavecalculatedthegapuptotheninthorderandfurtherap-pliedtheDlogPad´eapproximants[35]tothem.Thethus-obtainedphaseboundaries,whicharealsoshowninFig.1(b),elucidatethenatureofthephasecompe-tition,thatis,theA eck-Kennedy-Lieb-Tasaki(AKLT)valence-bond-solid(VBS)[3]onasnakelikepathversusdecoupleddimers.

Themostimpressive ndingsarere-entrantquantumphasetransitionswithincreasingr.Theprecedingsigma-modelanalysis[19]isindeedenlighteningbutneverabletorevealthisnovelquantumbehavior.Inordertochar-acterizeeachphase,letusconsideravariationalap-proach.Weknowthatsingletdimersonrungs[Fig.3(h)]arestabilizedforr→∞,whereaseitherdimersonlegs[Figs.3(d)and3(e)]orthedoubleAKLTVBS[Fig.3(c)]forr→0.TwomoreinterchainVBSstates[Figs.3(f)and3(g)]maybeadoptedasvariationalcomponentsfortheintermediate-rregion.ThusthelinearcombinationofFigs.3(c)to3(h)canbeanapproximateground-statewavefunctionforspin-1ladders.Sincethepresentvari-ationalcomponentsareallasymptoticallyorthogonaltoeachother,thevariationalgroundstateturnsoutanyofthemitself[36].Thethus-obtainedphasediagramis

E

r

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 reveal the rich pha

presentedinFig.

4.Thesigni cant

stabilizationoftheintermediatephase,which

isnowcharacterizedasSH,andtheresultantre-entrantphaseboundaryaresuccess-fullyreproduced.Consideringthatacoupleofcriticalchainsimmediatelyturnmassivewiththeirrunginter-actionswitchedon[7],thepointCshouldcoincidewiththepointAundermorere ned(andthusinevitablynu-merical)variationalinvestigation.

Thepresentvariationalcalculationimpliespossiblephasetransitionsforin-phase-legladdersaswell,butthisistotallyduetothenaivewavefunction.Numericalob-servationoftheenergystructureendsupwithnogapless

pointinthisregion.Thesigma-modelapproachalsoconcludesnocriticalpoint,givingthetopologicalan-gle4πSindependentofbothrandδP.Thekeytotheground-statenatureofin-phase-legladdersisthefour-spincorrelation[25].LetusconsiderinteractingfourspinsofS=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)

θmovesfrom0to

π

1 r+r2.Asrvariesfrom0to∞,

(g)Decoupled Plaquette

(DP)(h)

Rung Dimer(RD)

FIG.3.Plaquette-singlet-solidandvalence-bond-solidstatesrelevanttothetwo-legantiferromagneticspin-1lad-ders. denotesaspin1

2

’sinside.

4.0

atsite(i,j)intoaspin1.Nowthatθ(r)

isacontinuousfunctionofrandmayheredeviatefromthatinEq.(3),anaiveoptimization[36]ofEq.(4)isnomorefeasible.However,there nedvariationalschemeshowsusmore.Forbetterunderstandingofthewavefunction(4),wevisualizeinFig.3itsspecialformsforφ=π

4,

2’s

Rung Dimer

3.0

andθ=

π

Plaquette Singlet Formation

Decoupled Plaquette

2.0

r

Snake Haldane

1.0

0.01.0

In-Phase Leg Dimer

B

Double HaldaneC

Leg Dimer

δOP

δIP

0.5A0.0A0.51.0

FIG.4.Variationalphasediagramsforthetwo-legan-tiferromagneticspin-1ladders.Thethicksolidlinesde-scribephasetransitions,whereasthethinonesrepresentthecrossoveroftheground-statenaturewithinthepresentvari-ationalscheme.Thedottedlineisonlyaguideforeyes.

3

Heisenberg

chain[31].Weareallthemoreconvincedoftheimme-diategapformationwithrmovingawayfrom0.Ontheotherhand,neitherPSSnorDPSincludesbothOPLDandSHandthereforethetwocriticallinesintheout-of-phase-legregionsurviveagainsttheplaquettesingletformation.

Thegeneralizedstringorderparameter[37]O(θ)=

j 1

zzz

lim|i j|→∞ Sil=iexp[iθSl]Sj isalsousefulinchar-2

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 reveal the rich pha

acterizingthegroundstate.O(θ)distinguishesbetweenone-dimensionalVBSstates

by

itsθdependence[36].Hence,measuringitonthelinear-chainandsnakepaths,wecandetectthetransitionsbetweenDH,OPLD,SH,andRD,asisshowninFig.5.Ifwespecifythetransitionthroughachangeoftheθdependenceinthevicinityofθ=π,thatis,thechangefromthecon-vexcurvetotheconcaveone,weobtainthetransi-tionpoints(δOP,r)=(0.245,0)and(0.6,2.125),whichareingoodagreementwiththenumerical ndingsinFig.1(b).WhateverpathwetakeforO(θ),itspeakneversitsonθ=πinthein-phase-legregion,sug-gestingthatwecannotobservetheHaldanestateofanykindthere.TheplaquettesingletformationcaninsteadbevisualizedbyextendingO(θ)toladdersas

j 1

zzzzzz

lim|i j|→∞ S1S,i2,il=iexp[iθ(S1,l+S2,l)]S1,jS2,j [25].

0.5

[2]

[3]

[4][5][6][7][8][9]

r= 0.00.4δ=OP0.30.20.10.00.27

δOP= 0.6

r =2.62.4

[10][11][12][13]

0.250.262.152.2

(a)O θ ) ( (b)O ( ) θ

FIG.5.QuantumMonteCarloestimatesofthegeneralized

stringorderparameterde nedonthelinear-chain(a)andsnake(b)paths.Thedashedlinesrepresenttheanalyticcal-culationsfortheAKLTVBS(4)anddecoupleddimers24(

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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 reveal the rich pha

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