14non-Local Quantum Criticality in Ce(Ru_{1-x} Fe_x)_2 Ge_2 (x = x_c = 0.76)-第2页
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14non-Local Quantum Criticality in Ce(Ru_{1-x} Fe_x)_2 Ge_2 (x = x_c = 0.76)-2
teringstudiesonKondolatt；Theobservedlineshapeχ(q,；Arandomphaseapproximatio；smallq.U(q=0.55?A?1)isal；Themodi?edLorentzianintr；Ourneutronscatteringmeas；Weacknowledgestimulating；ackno
teringstudiesonKondolatticesystems[13,18,19,20,21,22],thedynamicresponseinCe(Ru0.24Fe0.76)2Ge2isbroadandquasi-elastic.However,Fig.2bshowsthattheLorentzianlineshape[13,18,19,20,21,22]commontoKondolatticesagreesverypoorlywithourmeasuredχ(q,E),bothforasmallwavenumber,0.35?A?1wherethemomentsareinter-acting,andatalargewavenumber1.15?A?1wheretheresponseispurelylocal.We?ndinsteadthatatallE,forallTfrom1.9K-200K,andforthecompleteq-rangeprobedinourexperimentthatourdatacanbesatisfactorilydescribedbyasimplephenomenologicalexpression:χ(q,E)=χq(T)/[1?iE/Γq(T)]β.Theobservedlineshapeχ(q,E)iscontrolledbyanenergyscaleΓq(T),andbyadynamicalexponentβ=0.15±0.05.ThetemperaturedependenceofΓq(T)whichweextractfromthese?tsisplottedfortwovaluesofqinFig.2c.Forq=1.15?A?1,Γ(q)hasthefamiliarT-dependenceofaKondoimpuritysystem,initiallydecreasingwithTbeforesaturatingandincreasingweaklybelowTK,whichweidentifyas～5K.Dynamicsonthislocallengthscaleareconsequentlynotcritical.Incontrast,Γqdeterminedforq=0.35?A?1approacheszerowithdecreasingT,asexpectedforcriticalslowing-downassociatedwiththeT=0phasetransition.WiththeexceptionofthelowestTatthelargestq,ΓqisapproximatelylinearinT,Γq=θq+aqT.Theq-dependenceofθqisshowninFig.2d,demonstratingthatdynamiccriticality,i.e.Γq→0,canonlybeachievedifθq→0.ForCe(Ru0.24Fe0.76)2Ge2thisoccursasqapproachesthepropagationwavevectorofincipientAForder,0.2±0.1?A?1.Weconcludethatatalltemperaturesχ(q,E,T)isdominatedatshortlengthscalesbytheexcitationsofindividualKondomoments,whilethelong-wavelength?uctuationsbecomeincreasinglylong-livedandultimatelycriticalasT→0.Arandomphaseapproximation(RPA)analysisofχq(T)showsthatthesingularbehav-iorfoundneartheQCPinCe(Ru0.24Fe0.76)2Ge2requiresthecollaborationofbothlocalandlong-rangedynamicalcorrelations.Thelong-rangecorrelationsaregraphicallydemon-stratedinFig.3a,wherewehaveplottedχloc(T)/χq(T)atseveraltemperatures.Anincreasingsuppressionofχloc/χqisobservedatsmallqasthetemperatureislowered,sig-nallingthegrowthoflongrangeAFcoupling.Anestimateofthiscoupling,Uq(T)canbeobtainedintheRPAapproximationbynotingthatχloc(T)/χq(T)=1?Uq(T)χloc(T).Uqdeducedfromthisanalysisat4.4KisplottedintheinsetofFig.3a,demonstratingthattheinteractionsarelongranged,rapidlyvanishingforwavevectorslargerthan～0.6?A?1.AsshowninFig.3b,thetemperaturedependencesofUqareverydi?erentforlargeandsmallq.U(q=0.55?A?1)isalmosttemperatureindependent,whileU(q=0.275?A?1)increasesalmostafactoroffourbetween20Kand1.5K.Incontrast,χloc(T)initiallyincreaseswithdecreasingtemperature,butultimatelysaturatesbelow～5K.ThisRPAanalysisrevealsthatlocal?uctuationsinitiallyprovideabiastowardscriticalityinCe(Ru0.24Fe0.76)2Ge2,butultimatelyitistheintermomentcouplingUq(T)whichactuallydrivescriticality.AsimilarconclusionwasreachedforU2Zn17[21],aheavyfermionantiferromagnetwithaN′eeltemperatureTN=9.7K.However,neitheradeparturefromLorentzianlineshape,noranyothernFle?ectswereobservedinthissystem,whichisnotataQCP.Themodi?edLorentzianintroducedaboveandtheobservedT-linearityofΓq(T)impliesthatourdatashouldalsodisplaynFlE/(T+θq)scaling.Tocon?rmthis,wehaveplotted′′χ(q,E)(T+θW)0.51forq=0.35,0.55,1.35and1.75?A?1asfunctionsofE/kB(T+θq)inFig.4.Anexcellentcollapseofthedatatakenatdi?erenttemperaturesisobservedateachq,spanningthreeordersofmagnitudeinthescalingvariableE/kB(T+θq).Theex-ponent0.51istakenfromthetemperaturedependenceofχ0(T),aswasalsofoundinthelocallycriticalsystemsUCu4Pd[8,9]andCeCu5.9Au0.1[6].Unlikethosesystems,we?ndinCe(Ru0.24Fe0.76)2Ge2thatthedynamicalscalingfunctionitselfrequiresasecondexponentβ=0.15.OurneutronscatteringmeasurementshaveestablishedthattherearelocalmomentspresentinCe(Ru0.24Fe0.76)2Ge2whichexperienceincreasingAFcouplingasT→0.Although?uctu-ationsofthesemomentsareobservedoneverylengthscale,χqonlydivergesattheresidualorderingwavevectorofthenearbyAFphase.Correspondingly,weobserveasubstantialKondosuppressionofthelocal?uctuationsbelow～20K.Thedominanceofthelongwavelengthcorrelationsatthelowesttemperatures,andthe?niteKondotemperatureattheQCPtogetherimplythattheT=0antiferromagnetictransitioninCe(Ru0.24Fe0.76)2Ge2isacollectiveinstabilityofthestronglyinteractingquasiparticles,andisnotlocallycritical.Themean?eldviewofsuchaphasetransitionrequiresasinglediverginglengthscalewithanaccompanyingdivergingtimescale,leadingtoLorentzianlineshapesforχinenergyandwavevector.We?ndinsteadthatthesusceptibilityχ(q,E,T)iswelldescribedbyamodi-?edLorentzianexpression,encompassingtheE/Tscalingwhichweobserveateverywavevector.WeacknowledgestimulatingdiscussionswithP.Coleman,A.J.Millis,andQ.M.Si.MCAthanksT.GortenmulderandR.Hendrikxforinvaluabletechnicalassistance,andacknowledgesthehospitalityoftheMSMgroupatLeidenduringtheearlystagesofthisproject.WethankI.P.Swainsonforcarryingouttheneutrondi?ractionmeasurements.WorkattheUniversityofMichiganwassupportedbyNSF-DMR-997300.[1]G.R.Stewart,Rev.ofMod.Physics73,797(2001).[2]S.Sachdev,QuantumPhaseTransitions(CambridgeUniversityPress,Cambridge,England,1999).[3]J.A.Hertz,Phys.Rev.B14,).[4]A.J.Millis,Phys.Rev.B48,);privatecommunication.[5]P.Coleman,PhysicaB259-261,353(1999).[6]A.Schr¨oderetal.,Nature407,351(2000).[7]QimiaoSietal.,Nature413,804(2001).[8]M.C.Aronsonetal.,Phys.Rev.Lett.75,725(1995).[9]M.C.Aronsonetal.,Phys.Rev.Lett.87,1).[10]S.S¨ullowetal.,Phys.Rev.Lett.82,).[11]H.Rietscheletal.,J.Magn.Magn.Mater.76-77,105(1988);A.Boehmetal.,J.Magn.Magn.Mater.76-77,150(1988)[12]M.B.Fontesetal.,Phys.Rev.B53,).[13]B.D.RainfordandS.J.Dakin,Phil.Mag.B65,).[14]ThisvalueisbelowtheestimateofFontesetal.[12],however,theirresultsarenotinconsistentwithxc=0.76.[15]W.Montfrooijetal.,unpublished.[16]χoinCe(Ru0.24Fe0.76)2Ge2hassubstantialanisotropy,re?ectingpreferredorientationand/orIsing-likebehavior.However,weonlyusetheT-dependenceofχ0inouranalysis,notitsabsolutevalue.[17]N.E.Bickers,D.L.FoxandJ.W.Wilkins,Phys.Rev.B36,).[18]M.Loewenhauptetal.,JournaldePhysique40,C4-142(1979).[19]G.Aeppli,E.BucherandG.Shirane,Phys.Rev.B32,).[20]A.I.Goldmanetal.,Phys.Rev.B33,).[21]C.Broholmetal.,Phys.Rev.Lett.58,917(1987).[22]U.Walter,M.Loewenhaupt,E.Holland-MoritzandW.Schlabitz,Phys.Rev.B36,1981(1987).FIG.1:(a):ThemagneticphasediagramforCeRu2Ge2asfunctionsofpressure[10](dashedlines)andFedoping(?:FMphaseboundary[12],?:AFphaseboundary,presentwork).FilledcirclesrepresentthepressuredependentKondotemperature,whichis?niteattheQCP[10].Thesolidlinesareguidestotheeye.Theelectricalresistivityisquadraticintemperature[10]intheshadedpartofthephasediagram(FL).(b):ThefullycorrectedS(q,E)forCe(Ru0.24Fe0.76)2Ge2asafunctionofneutronmomentumtransfer??qandenergytransferE,forT=7.5K.(c):ThestaticsusceptibilityχqforT=2.9K(?),7.5K(?)and15.2K(△).NotetheincipientAForderaround??1)at2.9K.the(101)Braggpeak(q=1.64AFIG.3:(a):χloc(T)/χq(T)asafunctionofqforT=1.9K(?),4.4K(△),7.5K(??),and15.2K(?).Asexplainedinthetext,thisquantityisdirectlyrelatedtotheinteractionUq,whichisplottedintheinsetforT=4.4K.(b):Whilethetemperaturedivergenceofχloc(?)iscuto?below～5KbytheKondoe?ect,U(q=0.275?A?1)(?)increasesmonotonicallytothelowestT.Forq=0.55?A?1(?),UqisT-independent.FIG.4:Scalingofthedynamicresponseforvariousq-values.Theneutronscatteringdataχ”(q,E)havebeenmultipliedby(T+θW)α(α=0.51,seetext),anddisplayedversusthereducedvariableE/(T+θq),withθqasinFig.2d.Notethatthevariousq-valuesareo?setbyhalfadecadealongtheverticalaxis.Thetemperaturesrangefrom1.9K(darkestsymbols)to200K(lightestsymbols),andthereissubstantialoverlapinE/(T+θq)amongthe11temperaturesdisplayedinthis?gure.Everyscalingcurverepresentsabout3000independentdatapoints.FIG.2:(a):χq(T)forq=0.35?A?1(?),forq=0.45?A?1(△)andintegratedoverallq,χloc(T)(?).Thesolidlineistheq=0,dcsusceptibilityχ0.Alsoshown(?,inμ2B)istheaverageofSq(T)over1&q&1.8?A?1,demonstratingtheonsetofKondo-screeningatT≈20K.(b):χ”(q,E)/Eforq=0.35?A?1(?)andq=1.15?A?1(?)atT=4.4K.Thedataatq=1.15?A?1havebeendividedby2forthesakeofplottingclarity.Thesolidlinesarethebest?tstothemodi?edLorentzianlineshapedescribedinthetext,withβ=0.15.Thedashedcurveisthebest?tLorentzianlineshape(β=1).(c):TheenergylinewidthΓqofχ”(q,E)/Eforq=0.35?A?1(?)andq=1.15?A?1(?).(d):q-dependenceoftheresiduallinewidthθq,explainedinthetext.AlsoshownisθWatq=0.包含各类专业文献、中学教育、各类资格考试、幼儿教育、小学教育、外语学习资料、文学作品欣赏、生活休闲娱乐、14non-Local Quantum Criticality in Ce(Ru_{1-x} Fe_x)_2 Ge_2 (x = x_c = 0.76)等内容。