Universal Magnetic Properties of Frustrated Quantum Antiferromagnets in Two Dimensions

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

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aUniversalmagneticpropertiesoffrustratedquantumantiferromagnetsintwodimensionsAndreyV.Chubukov1,2,T.Senthil1andSubirSachdev11DepartmentsofPhysicsandAppliedPhysics,P.O.Box208284,YaleUniversity,NewHaven,CT06520-8284and2P.L.KapitzaInstituteforPhysicalProblems,Moscow,Russia(February1,2008)AbstractWepresentatheoryoffrustrated,two-dimensional,quantumantiferromag-netsinthevicinityofaquantumtransitionfromanon-collinear,ingasigma-modelforbosonic,spin-1/2,spinon elds,weobtainuniversalscalingformsforavarietyofobservables.Ourresultsarecomparedwithnumericaldataonthespin-1/2triangularantiferromagnet.

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We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

Ausefulclassi cationoftwo-dimensional,quantum,Heisenbergantiferromagnetsispro-videdbythestructureofthemagnetically-orderedgroundstate:thespin-condensatesonthesitescaneitherbecollinearornon-collineartoeachother.Collinearmagnetshavebeenextensivelystudiedinrecentyearsandmanyoftheirpropertiesarereasonablywellunder-stood.TheypossessanO(3)/O(2)orderparameterwhose uctuationsdescribethelowtemperature(T)propertiesofthemagnetically-orderedstate[1].Thequantum-disorderedstatehasonlyintegerspinexcitations(thespinonsarecon ned)andspin-Peierlsorderisexpectedforcertainvaluesofthesingle-sitespin[2].The nite-Tcrossoverbetweenthesetwostateshasalsobeenstudiedinsomedetail[3].

Lessisknown,however,aboutnon-collinearantiferromagnets,whicharethesubjectofthispaper.Examplesincludethetriangular,kagome,andsquare(with rst,second,andthirdneighborinteractions)lattices.Themagnetically-orderedstatecompletelybreaksthespin-rotationsymmetry,yieldinganSO(3)orderparameter[4].Spaceandtimede-pendenttwistsofthisorderparameterthende nethreeindependentspin-sti nesses,spin-susceptibilities,andassociatedspin-wavevelocities.Forsimplicity,wewillrestrictourat-tentionheretomagnetswithcoplanarspinsandaninternalsymmetry(aC3vsymmetryonthetriangularandkagomelattices,andascrewaxissymmetryfortheincommensurateplanarspiralsonthesquarelattice),whichleadstojusttwoindependentsti nesses(ρ⊥,ρ ),susceptibilities(χ⊥,χ ),andspin-wavevelocities(c⊥=(ρ⊥/χ⊥)1/2,c =(ρ /χ )1/2);morecomplicatednon-collinearmagnetswillhavesimilarproperties.Thelong-wavelengthactionfortheSO(3)orderparameterhasanSO(3)×O(2)symmetry,theO(2)beingacontinuummanifestationoftheinternalsymmetrynotedabove[4].AspacetimedimensionD=2+ studyofsmall uctuationsoftheSO(3)orderparameteraboutthemagnetically-orderedstatewasperformedbyAzariaet.al.[5];theyfoundthatthesti nessesandsuscepti-bilitiesbecameasymptoticallyequaluponapproachingthecriticalpointseparatingthemagnetically-orderedandquantum-disorderedphases,withthecriticaltheorypossessinganenlargedO(4)symmetry.AlargeNtheorybaseduponSp(N)symmetry[6]foundasim-ilarmagnetically-orderedstate,butwasalsoabletoaccessthequantum-disorderedphase.

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

Thelatterstatewaspredictedtobeafeatureless,fullygappedspin- uid,withuncon ned,bosonicspin-1/2spinonexcitations.Wealsonotethattherearealternativeapproachestothequantumdisorderedphase[7]whicharequitedisconnectedfromthestructureoftheorderedstate.

Inthispaper,weshallpresentatheoryoftheuniversal, nite-Tpropertiesofnon-collinearantiferromagnetsinthevicinityofthecriticalpoint.Wewilldescribethecrossoverfromthemagnetically-orderedstate,withitslow-lyingspin-waveexcitations,tothefullygappedquantum-disorderedstateviaanintermediatequantum-criticalregion.Ourresultsareincompleteagreementwithsomepreviousstudiesofthemagnetically-orderedstate[5]andthequantumdisorderedstate[6],andestablishafundamentalconnectionbetweentheO(4)-symmetriccriticalpointofRef.[5]andthedecon nedbosonicspinonsofRef.[6];arelatedconnectionwasnotedrecentlyinRef.[8].WewillalsoobtainnewresultsforthelowTbehaviorofthedynamicstructurefactoranduniformsusceptibilityofmagnetically-orderedantiferromagnets.

Ourmotivationforthisstudyissimilartothatfortheanolagousrecentstudyofcollinearantiferromagnets[3].AgivenS=1/2antiferromagnetmaybeeithermagnetically-ordered(asisexpectedforthetriangularlattice)orquantum-disordered(thekagomelattice)[9].AtlowT,themagneticallyorderedmagnethasthermally-excitedclassicalspin-wave uctu-ations(therenormalized-classical(RC)region),whilethequantum-disorderedmagnethasonlyactivateddeviationsfromitsground-stateproperties.AthigherThoweverboththesemagnetsareexpectedtocrossovertoaquantum-critical[1](QC)regionwhereclassicalandthermal uctuationsareequallyimportant.Manypropertiesofthisregionareuniversal,andarethusamenabletonumericalandexperimentaltests.Inparticular,therearesigni cantquantitativedi erencesbetweentheQCbehaviorofcollinearandnon-collinearmagnets,whichareadirectconsequenceofthepresenceofdecon nedspinonsinthelatter.

Webeginbypresentingoure ectiveaction.Wechoosetodescribethelocalspincon g-urationbyanSU(2)rotationaboutareferenceorderedstate.ThechoiceofSU(2)ratherthanSO(3)issigni cant,andhastheimmediateconsequenceofsuppressingthevortices[10]

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

associatedwithπ1(SO(3))=Z2forwhichtheSU(2) eldisdouble-valued.ThischoiceismotivatedpartlybytheresultsofRef.[6],wherevorticesweresuppressedinthequantum-disorderedphasebyaHiggscondensate.WeparametrizetheSU(2)matrixbytwocomplexnumbersz1,z2with|z1|2+|z2|2=1,andwritedownthemostgeneral,long-wavelengthactionwithanSU(2)×O(2)invariance:

S= d2xdτ1

4 z µz µz z 2 (1)

0000000Itiseasytoshowthatgx=1/2ρ0⊥,gτ=1/2χ⊥,γx=(ρ ρ⊥)/ρ⊥,γτ=(χ χ⊥)/χ⊥,

wherethesuperscript0denotesbarevalues;notethatiftheγµ=0,ShasanenlargedO(4)symmetry.TheactionScanbeexplicitlyderivedbyalong-wavelengthanalysisofthemodelsofRefs.[5]and[6];wehavealsolearnedofarecentstudyofSbyAzariaet.al.[11].Thestaggeredspin-structurefactor(wavevectorsmeasuredasdeviationsfromtheordering

)canbeshowntobetheFouriertransformofRez(x1,τ1)z(x2,τ2).NotewavevectorG

thatthisisquarticinthez,consistentwiththeidenti cationofthezquantaasspin-1/2bosonicspinons.

WestudiedSbygeneralizingztoanN-component,unit-length,complexvector,andperforminga1/Nexpansion;SthenhasaSU(N)×O(2)invariance,whileforγµ=0itisinvariantunderO(2N).ThismethodallowsustoworkdirectlyinD=2+1andaccessboththeQCandRCregions.NotethattheextensiontolargeNisdi erentfromthatusedinRefs.[8,12].

WeexpectthatSpossessesquantum-disorderedandmagnetically-ordered(withthezquantacondensed)asthecouplings(saygx)arevaried.AkeypropertyofthepresentlargeNexpansionisthatthelong-distancephysicsatthecriticalpointatgx=gcisO(2N)-symmetric.Thisismanifestedinthemagneticallyorderedphase(gx<gc)bythecriticalbehaviorofthesti nesses.Josephsonscalingisobeyedbythefullyrenormalizedρ ,ρ⊥,χ ,χ⊥allofwhichvanishas(gc gx)ν,whereνisthecorrelationlengthexponent(ν=1 16/3Nπ2+O(1/N2)).However,therelativedi erencesbetweenthesti nessesalsovanishatthecriticalpoint:wede ned 1=(ρ ρ⊥)/ρ⊥, 2=(χ χ⊥)/χ⊥,andfound

2

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

1=γ1(ξJ) φ1+γ2(ξJ) φ2

2=γ1(ξJ) φ1 2γ2(ξJ) φ2(2)

whereγ1=(2γx+γτ)/3;γ2=(γx γτ)/3,andξJistheJosephsonlengthmeasuredinlatticeunits.Thepositivecrossoverexponentsφ1,2measuretheirrelevancyoftheγµtermsinS;theγµareactually‘dangerously’-irrelevantas 1,2controllong-wavelengthphysicsforgx<gc.Toorder1/N,wefoundφ1=1+32/3π2N,φ2=1+112/15π2N[13].Wenowpresentourscalingresultsforthewavevector-(k)andfrequency-(ω)dependentstaggered(χs)anduniform(χu)spinsusceptibilitiesinthevicinityofgx=gc.Werestrictourselvestogx<gc,althoughmorecompleteresultshavebeenobtained[14].Wefound

χs(k,ω)=22πN0

h¯c2⊥ 2kBT 2 NkBT k, kBTΦuω,x, 1, 2(3)

whereN0istheon-sitemagnetizationatT=0,Φ1s,Φ1uareuniversalfunctionsofthedimensionalvariablesω=h¯ω/kBT,x=NkBT/4πρ⊥.Wefoundtheexponentη¯=1+32/3π2N.TheprefactorofΦsremainsnon-singularatgx=gcasN0～¯¯=(1+η(gc gx)βwith2β¯)ν.Allscalingfunctionsarede nedsuchthattheyremain niteasx→∞.Asbefore[3],theargumentxdetermineswhetherthesystemisintheQC(x 1)orRC(x 1)region.

Animportantdi erenceintheabovescalingformsfromthoseforcollinearmagnets[3]isinthevalueofη¯.Herewehaveη¯closetounity,whiletheanalogousexponentforcollinearmagnetswasclosetozero.Thisisaconsequenceofthepresencehereofdecon nedspinons:itisthezquantawhichbehavelikealmostfreeparticles(atT=0, z z ～1/k2 ηwithηcloseto0)whilethestaggeredsusceptibilityisacorrelatorofacompositeoperatoroftwo

¯spinons(χs～1/k2 ηwithη¯closeto1).

WehavecomputedΦs,Φuina1/Nexpansiontolinearorderin 1,2.Wedescribeourresultsastheyrelatetovariousobservables.

Correlationlength.Asincollinearmagnets,wede nethecorrelationlength,ξ,from

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

thelong-distancee r/ξdecayoftheequal-timespin-spincorrelationfunction.Wefoundthat,toorder1/N,thereisasimplerelationshipbetweenthevaluesofξforcollinearandnon-collinearmagnets.Forallvaluesofx,thenon-collinearξisprecisely1/2thepreviouslycomputedξ[3]fortheisotropicO(2N)sigmamodel.Thefactorof1/2isasignatureofdecon nedspinons.Thecollinearexpressionforξ[3]howevermustbeusedwiththee ectivevaluesρs=ρ⊥(1+N 1/(2N2 2)),χ=χ⊥(1+N 2/(2N2 2)andc=(ρs/χ)1/2;noticealsothefactorof4di erenceinthecoulingconstantin(1)andin[3].ForthephysicalcaseN=2,wehaveto rstorderin 1,2thatρs=(2ρ⊥+ρ )/3,c=(2c⊥+c )/3,andourresultforξisthenconsistentintheRCregionwiththatofAzariaet.al.[5].

Staticuniformsusceptibility.TheresultforχuisobtainedbyevaluatingtheresponsetoavectorpotentialcoupledtotheconservedchargeoftheSU(N)symmetry.

IntheRCregion(NkBT 4πρs)weobtained

χu= gµB

(N+1)χ⊥2χ⊥NkBT

h¯c 2kBT√4π 1 0.31

5+1)/2],x¯=NkBT/4πρsandα=0.8+O(1/N).Notethattheslope

ofthelinearinTtermisprecisely1/2ofthatintheO(2N)sigma-model[3].Thefactorof1/2isagainasignatureofspin-1/2spinonsandshouldbeamenabletoexperimentaltests.Staggereddynamicsusceptibilityandstructurefactor.IntheRCregion,thescalingform

(3)forχscollapsesintoareducedscalingforminwhichthephysicalξ,ratherthanc/kBTisthemostimportantlengthscale[1,3].Wefound

χs(k,iωn)=2N0

4πρs

ξ2f(kξ,ωnξ/c)

(N+1)/(N 1)×(6)

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

werefisascalingfunction.Notethatcomputationswereinfactdoneonlytoorder1/N-theformatarbitraryNfollowsfromareasonableguessaboutthewavefunctionrenormalizationofthecomposite eld.Theoverallfactorin(6)is

chosensuchthatf(0,0)=1+O(1/N),Thebehavioroff(x,y)atintermediatex,y=O(1)israthercomplicated,chie ybecausespin-wavevelocityalsoacquiresasubstantialdownturnrenormalizationatkξ=O(1)[1].Howeveratkξ～ωξ/c 1,velocityrenormalizationisirrelevantandweobtained

f(x,y)= N 1

x2+y2 log(x2+y2)

ρs 2 ¯k,ω)Ξ(

16(ω+¯/2iδ)2)1 η

2(9)k(1+whereAN√ω2)/12Θ2+...]where...standforhigherpowersof2Θ/√=1+O(1/N).Atsmallkandω,wehaveReΦs=(ωandforregularcorrec-

tionsin1/N.ForImΦsweobtainedthefollowingasymptoticlimitsforlargeN

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

ImΦs= ANsin(πη¯/2)(

ω

πe ω2 2¯/2k)1 η;ω 1,

ω 1and

ω.

IntheT=0quantum-disorderedphase,ImΦsshowsaclearsignatureofdecon nedspinons-thespectralweightat xedkisabroadbandcontinuumratherthanthedelta-functionpeakpresentincollinearmagnets[14].

Localsusceptibilityandspin-latticerelaxation.ThelocaldynamicstructurefactorS(ω)isgivenbyS(ω)= d2kS(k,ω)/4π2.Simpleinspectionthenshowsthatforω～cξ 1,theintegrationovermomentumisalsocon nedtok～ξ 1andthereforeS(ω)isauniversalobservable.ThesmallfrequencylimitofS(ω)isdirectlyrelatedtothespin-latticerelaxationrateofnuclearspinscoupledtoelectronicspinsintheantiferromagnet:1/T1∝S(ω→0).IntheRCregion,we ndusingourpreviousresultsforthescalingfunctionsthatforω～cξ 1

S(ω)∝2N0ξ

4πρs (3N+1)/2(N 1)(10)

ForN=2,wethenhave1/T1∝T7/2ξ.

DeepintheQCregion,wefound

S(

whereK(Nρ⊥ NkBTω)ω(11)5 1)/64πatωBNsin(πη¯/2)/32πatω)=

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

exponentiallylarge,andthereisauniversalcontributiontoS(k)fromclassical uctuationswhichscalesasξ2.IntheRCregionwethenhaveS(k)≈kBTχs(k,0),whereχs(k,0)isgivenby(6).Atk=0thisyieldsS(0)∝T2N/(N 1)ξ2.ForN=2,wethenobtainS(0)∝T4ξ2.

ApplicationtotheS=1/2triangularantiferromagnet.Weperformeda1/SexpansiononthisantiferromagnettoobtaintheT=0valuesofρ⊥,ρ ,χ⊥,χ (alltoorder1/S),andN0(toorder1/S2).ForS=1/2thisgaveusN0=0.266;χ⊥=0.07/Ja2;χ=0.077/Ja2;ρs=0.087J;c=(ρs/χ)1/2=1.06Ja.FortheuniformsusceptibilityintheRCregimewethenobtainedχu=(gµB/h¯a)2[0.08/J+0.08kBT/J2+O(T2/J3)].Ontheotherhand,intheQCregime,wehaveχu=(gµB/h¯a)2[0.13kBT/J2+0.07/J(kBT/2πρs)1 1/ν+...].Thetemperaturedependenceinthesubleadingtermislikelytobequitesmallintheregionofexperimentalinterest(kBT～2πρs),andwecanwellapproximatethistermbyaconstant.Notehowever,thatthefactor0.07isanN=∞result-the1/Ncorrectionstothisfactorhavenotbeencomputed.Further,thecorrelationlengthbehavesintheRCregimeasξ≈0.25(4πρs/kBT)1/2exp[4πρs/kBT]where4πρs≈1.09J,anddeepintheQCregionasξ=0.53Ja/kBT.

Considernowthenumericalresultsforχu.Thedataofrecentseriesexpansionstudies[15]showthatχuobeysaCurie-WeisslawathighT,passesthroughamaximumatT≈0.4J,andthenfallsdown.Theregionbelowthemaximumisquitesmall;nevertheless,we ttedthisdatabyastraightlineandfound0.13±0.03fortheslopeandabout0.06fortheintercept-theresultsareinbetteragreementwithourQCratherthantheRCresult.Finally,atverylowT,weexpectacrossovertotheRCregime,andthecorrespondingvalueofχuatT=0isalsoconsistentwiththedata.WealsocomparedthedataforthecorrelationlengthandS(0)atkBT～0.4JandfoundroughconsistencywithourexpressionsinthecrossoverregionbetweenQCandRCregimes.Notethatourinterpretationofthenumericaldataisdi erentfromthatinRef.[15].

Toconclude,wehavepresentedatheoryofthecriticalpropertiesofnon-collinearquan-tumantiferromagnetsintwodimensions.Ourkeyassumptionwasonthevalidityofa

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

continuumdescriptioninSU(2)variables,whichsuppressedvortexexcitations.However,wewerethenabletoshowthatourresultswereconsistentwithearlierlargeN[6]andD=2+ [5]studies.Thequantumdisorderingtransitionwasdescribedbyananisotropicsigma-modelforspin-1/2,bosonicspinon elds.Allphysicalobservablesinvolveacollectivemodeoftwospinons,andwecomputedexplicitscalingformsforavarietyofexperimentallymeasurablequantities.OurresultsforχuintheQCregionareroughlyconsistentwithrecentnumericaldataonthespin-1/2triangularantiferromagnet[15];thismaybeviewedasindirectevidenceforthepresenceofdecon nedspinons.However,numericalresultsalsoseemtoindicatethattheTrangewhereQCbehaviormaybeobservedisrathernarrowforthissystem.Moredetailedstudies,especiallyinquantum-disorderednon-collinearmagnetswillbequiteuseful.

TheresearchwassupportedbyNSFGrantNo.DMR-9224290.S.S.isgratefulforLPTHE,Universit´eParis7,forhospitality.WethankP.Azaria,B.Delamotte,P.Lechmen-niat,D.MouhannaandN.Readforhelpfuldiscussions.

We present a theory of frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum-disordered phase. Using a sigma-model for bosonic, spin-1/2, spinon fi

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