Anomalous critical behavior near the quantum critical point

The Landau-Ginzburg-Wilson paradigm for critical phenomena is spectacularly successful whenever the critical temperature is finite and all fluctuation modes, with characteristic energies much smaller than the thermal energy, obey classical statistics. In z

Anomalouscriticalbehaviornearthequantumcriticalpointofahole-dopedLa2CuO4

Y.Chen,1WeiBao,1J.E.Lorenzo,2A.Stunault,3J.L.Sarrao,1S.Park,4,5,6andY.Qiu4,5

LosAlamosNationalLaboratory,LosAlamos,NM87545

2

CNRS,BP166X,F-38043,Grenoble,France

3

InstitutLaue-Langevin,BP156,F-38042,Grenoble,France

4

NISTCenterforNeutronResearch,NationalInstituteofStandardsandTechnology,Gaithersburg,MD20899

5

Dept.ofMaterialsScienceandEngineering,UniversityofMaryland,CollegePark,MD20742

6

HANAROCenter,KoreaAtomicEnergyResearchInstitute,Daejeon305-600,Korea

(Dated:February2,2008)

TheLandau-Ginzburg-Wilsonparadigmforcriticalphenomenaisspectacularlysuccessfulwhen-everthecriticaltemperatureis niteandall uctuationmodes,withcharacteristicenergiesmuchsmallerthankBTC,obeyclassicalstatistics.Inzero-temperaturequantumcriticalphenomena,clas-sicalthermal uctuationsarereplacedbyzero-pointquantum uctuationsandquantum-mechanicalgeneralizationoftheLandau-Ginzburg-Wilsonparadigmhasbeenacentraltopicincondensed-matterphysics.Inthisneutron-scatteringstudyonspin uctuationsnearthequantumcriticalpointinducedbyhole-dopinginLa2Cu1 xLixO4(0.04≤x≤0.1),thephaseboundaryforquantumcrossoverexpectedfromthegeneralizedquantumtheoryforcriticalphenomenawasobservedforthe rsttime.Furthermore,criticalexponentandscalingfunctionbecomeanomalousnearthequantumcriticalpoint,whichhasnotbeenexpectedincurrenttheories.

1

arXiv:cond-mat/0408547v1 [cond-mat.str-el] 25 Aug 2004

Interestingphenomena,includinghigh-TCsupercon-ductivity,occurinlaminarcupratesbelowatempera-tureT J/kB≈1500K,whereJisthemagneticex-changeenergybetweenthenearest-neighborspinsintheCuO2plane.Inthistemperaturerange,quantum uc-tuationsbecomedominantandaddoneextradimension,ξT≡¯hc/kBT,wherecisthespin-wavevelocity,tothe2-dimensional(2D)CuO2squarelattice[1,2](Fig.Atthezerotemperature,aquantumphasetransitionfromtheantiferromagnetic(AF)phasetoaparamagneticoneoccursatacriticalholedoping,a.k.a.quantumcriti-calpoint(QCP),xc,inthecuprates.Thus,fordopingx>xcontheparamagneticsideoftheQCP,magneticcorrelationlengthatT=0,ξ0,is nite.GeneralizingtheLandau-Ginzburg-Wilsonparadigmforcriticalphenom-enainclassicalsystemstothequantumspinsysteminthe(2+1)-dimensionalspace,theshorterofξTandξ0setsthelong-wavelengthcuto forspin uctuationsandisex-pectedtodeterminetheuniversalmagneticphenomenawithuniversalcriticalTheE/Tscalinginspindynamics,observedinpersesamplesofhole-dopedLa2CuO4usingBa,SrorLidopant,withsuperconduct-ingorinsulatinggroundstate,andwithcommensurateorincommensuratedynamicspincorrelations[4,5,6,7],canbeunderstoodasthephysicalconsequenceofξT<ξ0atT>TX≡¯hc/kBξ0.Inthisneutronscatteringstudyonquantumspindy-namicsinhole-dopedLa2CuO4,wedeterminethedop-ingdependenceofthecrossovertemperatureTX,belowwhichtheE/Tscalingbreaksdownandcrossesovertoaconstant-energyscaling,consistentwithξ0<ξT.Themostinterestingresultofthisstudy,however,isthatthecriticalexponentaintheE/Tchangesfromanexpecteda≈1toananomalousa≈0.65whendopingisreducedtowardtheAFQCPatxc.Thechangeofthe

FIG.1:(color)PhasediagramsforSrandLi-dopedLa2CuO4whereincreasingholeconcentrationsuppressestheantiferro-magnetic(AF)phaseatthequantumcriticalpointxcanddrivestheSrdopedmaterialsuperconducting(SC).Theline lledareadenotesspin-glassphase.WeobservethecrossoveratTX(redsymbols)betweentheE/Tscaling(QC)andtheE/Γ0scaling(QD)regimesinLi-dopedLa2CuO4.Theinsetillustratesthe(2+1)-dimensionalspaceformedbytheCuO2plane(Cu:redcircles,O:yellowcircles)andthequantumξT.Thebluehemisphereindicatestheextentofthemagneticcorrelationlengthatzerotemperature,ξ0.

criticalexponentisinconsistentwiththeclassicaluniver-salityofcriticalexponents.Inanalogtoarecentexampleinheavyfermionthesubstantialreductionoftheaexponentinthe2DspinsystemsuggestsextraphysicsinquantumcriticalphenomenawhichinvalidatestheclassicalLandau-Ginzburg-WilsonWechooseLi-dopedLa2CuO4singlecrystalsforthisstudy(TableI).Isotopicallyenriched7Li(98.4%)wasusedtoreduceneutronabsorptionofnaturalLi.ThecrystalshaveorthorhombicCmcasymmetryinthetem-

The Landau-Ginzburg-Wilson paradigm for critical phenomena is spectacularly successful whenever the critical temperature is finite and all fluctuation modes, with characteristic energies much smaller than the thermal energy, obey classical statistics. In z

2

TABLEI:DopingxofLa2Cu1 xLixO4andmainexperimen-talquantitiesdeterminedinthisstudyxTX(K)

0.65±0.06

b

≈1

≈1

The Landau-Ginzburg-Wilson paradigm for critical phenomena is spectacularly successful whenever the critical temperature is finite and all fluctuation modes, with characteristic energies much smaller than the thermal energy, obey classical statistics. In z

FIG.3:(color)Scalingplotsofthemagneticresponsefunc-tion.a,b,TheE/Tscalingathightemperatureforx=0.1and0.04,respectively.Insetinbshowsthegoodness-of- tforthecriticalexponenta=0.65forx=0.04.c,d,DeparturefromtheE/Tscalingatlowtemperature.ThesolidlineinaandcisEq.(1)usingEq.(2)asthescalingfunction,inbanddusingEq.(3)asthescalingfunction.ColorofsymbolsdenotestemperatureasinFig.2.

Forx=0.1,identicaltothecaseforx=0.06[7],a≈1andthescalingfunctionis

f1(y)=y/(1+y2).

(2)

ThepeakpositionEp/kBT=b=0.28±0.01inFig.3aislargerthanb=0.18forx=0.06.ForBaorSr-dopedLa2CuO4,theexponentahasbeendeterminedonlyforx=0.14anditisalsoa=0.94±0.06≈1[6].Thusforx≥0.06,hole-dopedLa2CuO4hasanormalcriticalexpo-nenta≈1,consistentwithaCurie’slawforthestaggeredmagneticsusceptibility.

ForLidopingx=0.04whichisclosetotheAFQCPofxc≈0.03[11],afractionala=0.65±0.06isrequiredforthedatatocollapseontoasinglescalingcurve(Fig.3b).Previously,inheavyfermionalloyCeCu5.9Au0.1,whichisalsonearanAFQCP,single-crystalneutronscat-teringdataarecollapsedontotheE/Tscalingwitha=0.74±0.1[8].ThesameprocedureasinRef.[8],whichisunbiasedwithrespecttotheselectionoff(y),isusedtooptimizeaforx=0.04,andthegoodness-of- trepre-sentedbythechi-squarestatisticaltestisshownintheinsettoFig.3b.ThespectruminFig.3bisbroaderthanthatinFig.3aandcannotbedescribedbyEq.(2).Weadoptempiricallythescalingfunctionforx=0.04,

f2(y)=sgn(y)yu1/(1+yv

1),

(3)

wherey1≡|y|[(v u)/u]1/vwithu=0.77±0.02and

v=1.33±0.02asthebest ts.Theparameterb=0.21±0.01.Notethatfor|y| 1,f2(y)～sgn(y)|y|uisalsoanomalousanddoesnotconformtotheusualanalyticformχ′′(E)～EatsmallE[3].

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TheheavyfermionmetalCeCu5.9Au0.1(z=2anddi-mensiond>2)andourx=0.04cupratesample(z=1andd=2)belongtodi erentuniversalityclasseswiththeef-fectivedimension(d+z)aboveandbelowtheuppercrit-icaldimension,respectively[1,21].However,theyarebothclosetoanAFQCPandsharewithinerrorbarsananomalousaexponentsubstantiallysmallerthanunity.Inourstudyofdopingdependence,furthermore,hole-dopedLa2CuO4recoversthenormala≈1andanalyticχ′′(E)forx≥0.06.Thesechangesincriticalbehaviorareveryunusual.AnomalousaexponentfortheheavyfermionmetalCeCu5.9Au0.1isrecentlyexplainedbyin-cludinglocaldegreesoffreedomintheKondoscreen-ing,inadditiontotraditionallong-wavelengthmagnetic uctuations,intheantiferromagneticquantumphasetransition[9].ClosertoourcaseofdopedcupratesneartheAFQCP,currently,“decon ned”degreesoffreedomassociatedwithfractionalizationofmagneticorderpa-rameterinpure2Dantiferromagnetarebeingexploredtheoretically[10].LikeRef.[9],thistheorygoesbeyondtheLandau-Ginzburg-Wilsonparadigm,whichfocusesonlong-wavelength uctuationsoftheorderparameterincriticalphenomena.Whetherincludingtheseadditional“decon ned”degreesoffreedominthecriticaltheoryfortheQCPwould,similartothecaseforheavyfermionmetals[9],producetheanomalousaexponentandf2(y)reportedhereforcupratesisunknown.Nevertheless,ourobservationofanomalouscriticalphenomenaneartheQCPinanotherclassofcorrelatedelectronicmaterialsuggeststhegeneralityofthephenomena,andwouldcer-tainlyencouragethebuddingresearchinterestonsubtlequantuminterferencee ectsnearaQCP.

InFig.3candd,lowtemperaturedataareaddedtothescalingplotsforx=0.1and0.04,respectively.Sincethepeakpositionshiftstotherightwithdecreasingtem-perature,theE/Tscalingcannotbesatis ed.Instead,asshowninFig.4aandb,belowTX,thelocalmagneticresponsefunctionχ′′(E)isnearlytemperatureindepen-dent.Thecurvesinthe guresare

χ′′(E)=Bf(E/Γ0),

(4)

whereBisaconstant,andthesamepairofscalingfunctions,f1(y)andf2(y),areusedinEq.(4)withΓ0=0.66±0.01and1.11±0.03meVforx=0.1and0.04,respectively.Notethatthereisnodetectablegapinχ′′(E)inFig.4forx=0.1and0.04,asforx=0.06re-portedpreviously[7].Insometheoreticalmodelsofnon-randomquantumantiferromagnets,aquantumspinliq-uidhasagap, ～¯hc/ξ0,nearE=0inχ′′(E)belowTX[21].Thegaplessχ′′(E)observedinoursamplesisob-tainedinthesetheorieswithstrongdampingofthespinliquidbydopedholes,Γ≈ [22,23].Partial llingoftheHaldanegapin1Dquantumspinliquidbydopedholeshasbeenreportedlately[24].Inothertheoreticalmod-elsofquantumantiferromagnets,severaltypesofgaplessspinliquidsexistwithoutimpurityscattering[25].Clas-

The Landau-Ginzburg-Wilson paradigm for critical phenomena is spectacularly successful whenever the critical temperature is finite and all fluctuation modes, with characteristic energies much smaller than the thermal energy, obey classical statistics. In z

FIG.4:(color)Temperatureinvarianceofthemagneticre-sponsefunctionatT<TX.a,Thedopingx=0.1.ThesolidlineisEq.(4)usingEq.(2)asthescalingfunction.b,x=0.04.ThesolidlineisEq.(4)usingEq.(3)asthescalingfunction.ColorofsymbolsdenotestemperatureasinFig.2.

si cationofourobserved2Dgaplessspinliquidinhole-dopedcuprateswouldbeaninterestingtheoreticaltask.Thecrossovertemperature,TX,fromtheE/Tscal-ingtotheE/Γ0scalingregimecanbeconvenientlylocatedbythepeakintemperaturedependenceofS(Q,E)attheE→0limit[7],sinceathightemperature,S(Q,E→0)∝T afromEq.(1)-(3),andatlowtemper-atureS(Q,E→0)∝TfromEq.(2)-(4).ThebluecurvesatE=0inFig.2aandbareS(Q,E→0)forx=0.1and0.04,respectively.TheextractedTXfromFig.2,to-getherwithpreviouslydeterminedvalueforx=0.06[7],areshowninthephasediagraminFig.1andarelistedinTableI.Underthedome-shapedphaseboundary,spin uctuationenergy,whichwouldapproachzeroaccord-ingtotheE/Tscaling,saturatesata niteΓ0.ThusthecrossoveratTXkeepsspin uctuationsenergeticatlowtemperature,whichmightbeessentialforthehighvalueofTCifsuperconductivityincupratesismediatedmagneticallybyspin uctuations.

ExperimentalsearchfortheE/TscalingwasinitiallystimulatedbythemarginalFermiliquidphenomenology[4,5,26].Thesubsequentobservationofthescalingininsulatingsample[7]supportstheQCPasthecommonmechanism.ThebreakdownoftheE/Tscalingbelowadome-shapedcrossoverboundaryreportedhereexcludestheexistenceofadditionalmag-neticQCPsintheinvestigateddopingrange.ThegappedquantumspinliquidpredictedbelowTXfornon-random2Dantiferromagnetsdoesnotsurviveinourdopedcuprates.Thisobservationdisfavorsthosetheo-rieswhichrelyontherobustnessofthegappedquantumspinliquidagainstdopingforhigh-TCsuperconductiv-ityincuprates[27].Whiletheapplicationoftheclassi-calLandau-Ginzburg-Wilsonparadigmforcriticalphe-

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nomenatoquantumcriticalphenomenaincupratesnat-urallyexplainstheuniversaloccurrenceoftheE/Tscal-inginhole-dopedLa2CuO4andthecrossovertothe -nitemagnetic-energyregimeatlowtemperature[2,3],theanomalouscriticalexponentandscalingfunctionneartheQCPreportedherehaveyettobeunderstood,probablybygoingbeyondtheLandau-Ginzburg-Wilsonparadigmandtakingintoaccountadditionalquantuminterferencee ectsexistingnearthezerotemperature[9,10].WethankQ.M.Si,A.Zheludev,S.-H.Lee,C.Bro-holm,S.Sachdev,A.V.Chubukov,F.C.Zhang,C.M.Varma,X.G.Wen,A.Balatsky,A.Abanov,L.Yu,Z.Y.WengandY.Bangforusefuldiscussions.SPINSatNISTissupportedpartiallybyNSF.WorkatLANLissup-portedbyU.S.DepartmentofEnergy.

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