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A Study of Quantum Phase Transitions in Quantum Impurity Systems

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Title:
A Study of Quantum Phase Transitions in Quantum Impurity Systems
Creator:
Chowdhury, Tathagata
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (116 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
INGERSENT,J KEVIN
Committee Co-Chair:
BISWAS,AMLAN
Committee Members:
HERSHFIELD,SELMAN PHILIP
HIRSCHFELD,PETER J
HENNIG,RICHARD
Graduation Date:
12/18/2015

Subjects

Subjects / Keywords:
Conduction bands ( jstor )
Critical points ( jstor )
Electrons ( jstor )
Entropy ( jstor )
Impurities ( jstor )
Magnetic fields ( jstor )
Magnetism ( jstor )
Phase transitions ( jstor )
Quantum dots ( jstor )
Quantum entanglement ( jstor )
Physics -- Dissertations, Academic -- UF
anderson-model -- condensed-matter -- entanglement -- graphene -- kondo -- quantum-criticality -- quantum-phase-transition -- strongly-correlated
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
Kondo and Anderson impurity models with a pseudogap density of states that vanishes at the Fermi energy, feature a continuous quantum phase transition (QPT) that separates a local-moment phase from strong-coupling phases. The quantum critical points (QCPs) exhibit critical Kondo destruction, which is of current interest in connection with heavy-fermion quantum criticality. Motivated by recent experimental and theoretical developments, the numerical renormalization-group technique is used to study the quantum criticality in several pertinent models that demonstrate a QPT. The two-channel pseudogap Anderson and Kondo impurity models are of potential relevance to the Kondo effect in graphene. In the vicinity of the QCPs separating local-moment and non-Fermi liquid phases in the two-channel pseudogap Kondo model, the static local spin susceptibility can be characterized by a set of critical exponents that satisfy the scaling relations expected of an interacting system below its upper critical dimension. The dynamical local susceptibility exhibit forms consistent with frequency-over-temperature scaling, another feature associated with an interacting QCP. The observation of mixed valency in quantum critical beta-YbAlB4 has prompted study of the pseudogap Anderson model away from particle-hole symmetry. The charge response at the QCP of this model can be characterized using five critical exponents, which are found to assume nontrivial values only for 0.55< r< 1. Both the charge critical exponents and the spin critical exponents satisfy a set of scaling relations derived from an ansatz for the free energy near the QCP and can all be expressed in terms of just two underlying exponents: the correlation-length exponent and the gap exponent. The impurity entanglement entropy Se has also been studied in several variants of the Kondo model that feature critical Kondo destruction. In all cases, on the local-moment side of the QCP, the entanglement entropy contains a critical component that can be related to the order parameter characterizing the QPT. Se reaches its maximal value of ln(2Simp+1), Simp being the impurity spin, at the QCP and throughout the Kondo phase, independent of the properties of the host. Implications of these results for quantum critical systems and quantum dots are discussed. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: INGERSENT,J KEVIN.
Local:
Co-adviser: BISWAS,AMLAN.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-12-31
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by Tathagata Chowdhury.

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Embargo Date:
12/31/2016
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ASTUDYOFQUANTUMPHASETRANSITIONSINQUANTUMIMPURITYSYSTEMSByTATHAGATACHOWDHURYADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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2015TathagataChowdhury

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Idedicatethistomyparents

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ACKNOWLEDGMENTSFirstandforemost,Iwouldliketothankmyadviser,Prof.KevinIngersent,forhisguidance,encouragement,insight,understanding.Iwouldalsoliketothankhimforhisimmensehelpwiththepreparationofthisdissertation.Iwouldliketoexpressmygratitudetothemembersofmysupervisorycommittee,Prof.PeterHirschfeld,Prof.SelmanHerseld,Prof.AmlanBiswas,Prof.SusanSinnottandProf.RichardHennigfortheirtime,expertiseandsupport.ManythankstoProf.MuttalibKhandkar,Prof.DmitriMaslov,Prof.PeterHirschfeld,Prof.SelmanHerseld,Prof.KevinIngersent,Prof.Hai-PingCheng,Prof.SergeiShabanov,andProf.ChristopherStantonfortheirclassroominstructions.IamthankfultoDavidHansen,BrentNelsonandClintCollinsfortheirhelpwithvariousaspectsofcomputingandothertechnicalsupport.Iwouldalsolikethankthepastandpresentmembersofmygroup,Dr.MenxingCheng,Dr.LiliDengandChristopherWagnerfortheirhelp,suggestions,insightandconsultations.IwouldliketoacknowledgethecontributionsandinsightprovidedbyDr.JedediahPixley,Prof.QimiaoSi,Prof.StefanKirchnerandFarzanehZamani,withwhomIhavecollaboratedondierentprojects.IwouldalsoliketoacknowledgethehelpandassistanceofallmyfriendsattheUniversityofFlorida,especiallyAvinashRustagi,PeayushChoubey,NaweenAnand,MarcusPeprah,BobbyBond,HridisPalandSiddharthaGhosh.Finallyandmostimportantly,Iwouldliketothankallmyfamilymembers,speciallymyparents,fortheirhelp,understandingandconstantsupport.Ithankmywife,Amrita,forhersacricesandforbeingacontinuoussourceofencouragementandsupporttomeforthelastfewyears. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1ABriefHistoryoftheKondoEect ...................... 12 1.2PoorMan'sScaling ............................... 16 1.3VariantsoftheKondoModel ......................... 19 1.3.1PseudogapKondoModel ........................ 20 1.3.2Two-ChannelKondoModel ...................... 21 1.3.3Two-ChannelPseudogapKondoModel ................ 23 1.4KondoEectinNanostructures ........................ 24 1.5QuantumCriticalityinHeavy-FermionSystems ............... 31 1.6EntanglementEntropy ............................. 38 2NUMERICALSOLUTIONTECHNIQUES .................... 41 2.1LogarithmicDiscretizationoftheConductionBand ............. 43 2.2TridiagonalizationoftheConduction-BandHamiltonian ........... 45 2.3IterativeSolutionoftheDiscretizedProblem ................. 46 2.4NRGTreatmentoftheAndersonModel ................... 47 2.5NRGTreatementofOtherModels ....................... 48 3QUANTUMCRITICALITYINTHETWO-CHANNELPSEUDOGAPKONDOMODEL ........................................ 50 3.1Background ................................... 50 3.2ResultsforTwo-ChannelPseudogapKondoModel ............. 53 3.2.1ThermodynamicProperties ....................... 54 3.2.2StaticCriticalProperties ........................ 55 3.2.3CriticalExponentsandScalingRelations ............... 56 3.2.4CriticalBehavior ............................ 57 3.2.5DynamicalCriticalProperties ..................... 63 3.3QuantumCriticalityforDivergingDensityofStates ............. 65 3.4Conclusion .................................... 66 5

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4CRITICALCHARGEFLUCTUATIONSINAPSEUDOGAPANDERSONMODEL ........................................ 68 4.1Background ................................... 69 4.1.1PhaseDiagram ............................. 70 4.1.2CriticalSpinResponse ......................... 71 4.1.3CriticalChargeResponse ........................ 73 4.2ResultsandInterpretation ........................... 74 4.3Discussion .................................... 82 4.4Conclusion .................................... 83 5ENTANGLEMENTENTROPYNEARKONDO-DESTRUCTIONQUANTUMCRITICALPOINTS ................................. 85 5.1GeneralConsiderations ............................. 86 5.2KondoModels .................................. 90 5.2.1Simp=1 2KondoModels ......................... 92 5.2.2Simp=1Single-ChannelPseudogapKondoModel .......... 97 5.3AndersonandBose-FermiModels ....................... 101 5.4Discussion .................................... 102 5.5Conclusion .................................... 103 6CONCLUSIONANDFUTUREDIRECTIONS .................. 104 6.1Conclusion .................................... 104 6.2DirectionsforFutureWork ........................... 105 REFERENCES ....................................... 110 BIOGRAPHICALSKETCH ................................ 116 6

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LISTOFTABLES Table page 3-1CriticalexponentsattheSCRcriticalpointofthetwo-channelpseudogapKondomodel. ......................................... 61 3-2CriticalexponentsattheACRcriticalpointofthetwo-channelpseudogapKondomodel. ......................................... 61 3-3ExponentsatthestableNFLxedpointofthetwo-channelpseudogapKondomodel. ......................................... 61 4-1Chargecriticalexponents~,~,and~x,pluscorrelation-lengthexponent,attheparticle-hole-asymmetricQCPsofthepseudogapAndersonmodel. ..... 76 4-2Correlation-lengthexponentattheparticle-hole-asymmetricQCPsofthepseudogapAndersonmodel. ................................... 80 4-3Exponents~and~determineddirectlyandinferredfromscalingequations. .. 80 5-1Exponents,,andefortheSimp=1 2,one-channelKondomodel. ....... 95 5-2Exponents,1=,andefortheSimp=1 2,two-channelKondomodel. ...... 95 5-3Exponents,n,andefortheSimp=1,one-channelKondomodel ....... 97 7

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LISTOFFIGURES Figure page 1-1ResistanceminimumobservedforFeimpuritiesinaseriesofMo-Nballoys. ... 13 1-2Theparticleandholestateswhichareremovedfromtheconductionband. ... 17 1-3SchematicRGowdiagramsforvariousmodels. .................. 19 1-4Kondoeectinasingleelectrontransistor. ..................... 25 1-5Kondoeectinadoublequantumdot. ....................... 27 1-6Scaledvariationoftheconductancewithsource-drainbiasvoltageforthedoublequantumdot. ..................................... 28 1-7HoneycomblatticeofgrapheneanditscorrespondingBrillouinzone. ....... 30 1-8Electronicdispersioningrapheneascalculatedusingatight-bindingHamiltonian. 30 1-9Resistivitymeasurementsofheavy-fermionCeCu6)]TJ /F5 7.97 Tf 6.59 0 Td[(xAux. ............. 34 1-10Specicheatmeasurementsofheavy-fermionCeCu6)]TJ /F5 7.97 Tf 6.58 0 Td[(xAux. ............ 35 1-11Imaginarypartofthedynamicalsusceptibilityofheavy-fermionCeCu6)]TJ /F5 7.97 Tf 6.59 0 Td[(xAux. . 36 2-1Discretizationoftheconductionband. ....................... 42 2-2ConductionbandoftheKondomodelmappedtoasemi-innitetight-bindingchain. ......................................... 45 2-3Mappingofthediscretizedproblemtotwosemi-innitetightbindingchainsfortwo-channel. ...................................... 48 3-1SchematicRGowdiagramsofthetwo-channelpseudogapKondomodel. .... 51 3-2Temperaturedependenceoftheimpuritycontributiontothermodynamicsforthetwo-channelpseudogapKondomodel. ..................... 53 3-3SchematicplotofcrossovertemperatureTvsdistancefromthecriticalpoint. 55 3-4ExponentxfortheSCR,ACRandNFLxedpointsofthetwo-channelpseudogapKondomodel. ..................................... 58 3-5Exponent1=fortheSCR,ACRandNFLxedpointsofthetwo-channelpseudogapKondomodel. ..................................... 59 3-6Order-parameterexponentfortheSCRandACRcriticalpointsofthetwo-channelpseudogapKondomodel. ............................... 60 3-7Dynamicalsusceptibilitylocofthetwo-channelpseudogapKondomodel. ... 63 8

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3-8SchematicRGowdiagramofthetwo-channelKondomodelfordivergingdensityofstates. ........................................ 64 4-1SchematicphasediagramofthepseudogapAndersonmodel. ........... 69 4-2jQlocjvsdistancefromthephaseboundaryalongthe)-326(and"daxesforr=0:6. 75 4-3ChargecriticalexponentsofthepseudogapAndersonmodelplottedvsbandexponentr. ...................................... 77 4-4Temperature-dependentpartc)]TJ /F3 11.955 Tf 11.1 0 Td[(regcofthelocalchargesusceptibilityforbandexponentr=0:5. ................................... 82 5-1SchematicrepresentationofthemodelHamiltonians. ............... 86 5-2Entanglemententropyintheone-channel,Simp=1 2pseudogapKondomodel. .. 91 5-3Spontaneous-symmetry-breakingpartoftheentanglemententropySeandorder-parameterMlocvsdistancejj. ................................. 92 5-4TheratioSe=M2locvsjjforthetwo-channelpseudogapKondomodel. ..... 96 5-5Log-logplotofMloc,n0)]TJ /F1 11.955 Tf 9.3 0 Td[(2=3,andSevsjjforone-channelSimp=1pseudogapKondomodel. ..................................... 98 5-6RatioSe=M2locvsjjforone-channelSimp=1pseudogapKondomodel. .... 100 6-1SeandSefortheone-channelKondomodel. ................... 107 6-2Power-lawvariationofSewithrespecttodistancefromthecriticalpointandappliedeldhloc. ................................. 108 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyASTUDYOFQUANTUMPHASETRANSITIONSINQUANTUMIMPURITYSYSTEMSByTathagataChowdhuryDecember2015Chair:KevinIngersentMajor:PhysicsKondoandAndersonimpuritymodelswithapseudogapdensityofstates()/jjr,thatvanishesattheFermienergy(=0),featureacontinuousquantumphasetransition(QPT)thatseparatesalocal-momentphasefromstrong-couplingphases.Thequantumcriticalpoints(QCPs)exhibitcriticalKondodestruction,whichisofcurrentinterestinconnectionwithheavy-fermionquantumcriticality.Motivatedbyrecentexperimentalandtheoreticaldevelopments,thenumericalrenormalization-grouptechniqueisusedtostudythequantumcriticalityinseveralpertinentmodelsthatdemonstrateaQPT.Thetwo-channelpseudogapAndersonandKondoimpuritymodelsareofpotentialrelevancetotheKondoeectingraphene.InthevicinityoftheQCPsseparatinglocal-momentandnon-Fermiliquidphasesinthetwo-channelpseudogapKondomodel,thestaticlocalspinsusceptibilitycanbecharacterizedbyasetofcriticalexponentsthatsatisfythescalingrelationsexpectedofaninteractingsystembelowitsuppercriticaldimension.Thedynamicallocalsusceptibilityexhibitformsconsistentwithfrequency-over-temperaturescaling,anotherfeatureassociatedwithaninteractingQCP.Theobservationofmixedvalencyinquantumcritical-YbAlB4haspromptedstudyofthepseudogapAndersonmodelawayfromparticle-holesymmetry.ThechargeresponseattheQCPofthismodelcanbecharacterizedusingvecriticalexponents,whicharefoundtoassumenontrivialvaluesonlyfor0:55.r<1.Boththechargecriticalexponentsandthespincriticalexponentssatisfyasetofscalingrelationsderivedfromanansatzfor 10

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thefreeenergyneartheQCPandcanallbeexpressedintermsofjusttwounderlyingexponents:thecorrelation-lengthexponent(r)andthegapexponent(r).TheimpurityentanglemententropySehasalsobeenstudiedinseveralvariantsoftheKondomodelthatfeaturecriticalKondodestruction.Inallcases,onthelocal-momentsideoftheQCP,theentanglemententropycontainsacriticalcomponentthatcanberelatedtotheorderparametercharacterizingtheQPT.Sereachesitsmaximalvalueofln(2Simp+1),Simpbeingtheimpurityspin,attheQCPandthroughouttheKondophase,independentofthepropertiesofthehost.Implicationsoftheseresultsforquantumcriticalsystemsandquantumdotsarediscussed. 11

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CHAPTER1INTRODUCTIONInthischapter,IintroducetheKondoproblemandtherenormalization-groupanalysisusedtosolvethisproblem.AfterdiscussingtheKondomodel,Iturntointerestingvariantsthatexhibitmoreexoticphysics.Then,recentexperimentaldevelopmentsarereviewedwheretheKondoeecthasbeenrealizedinarticiallyengineerednanostructures.Ialsoreviewbrieytheexistenceofquantumcriticalityinheavy-fermionmaterialsanddiscussrecentexperimentsconcerningtheobservationofsuperconductivityandquantumcriticalityinamixed-valenceheavy-fermionsystem.Finallythereaderisintroducedtovariousproblemsofcurrentinterestthatformthemainfocusofthisstudy. 1.1ABriefHistoryoftheKondoEectMostmetalsshowamonotonicincreaseinresistivitywithincreasingtemperatureTduetotheenhancementofelectron-electronandelectron-phononscattering.Howeverexperimentsasearlyas1934[ 1 , 2 ]showedthatcertainmetalsexhibitaminimumin(T)[seeFig. 1-1 ],astrikingphenomenonthatremainedunexplainedforalongtime.Itwaseventuallyrealizedthatthiseectwasduetothepresenceof3dimpuritieslikeFeinthehostmetalandthetemperatureminimumwasafunctionoftheimpurityconcentrationcimp.AsatisfactoryexplanationoftheresistanceminimumwasgivenbyJ.Kondoin1964[ 3 ],whichprovidedabreakthroughinunderstandingquantumimpuritiesinmetals.Aquantumimpurity,inthiscontext,referstoanisolatedmagneticmomentarisingfromunpaired3dor4felectronspresentintransitionmetalatomssuchasFe,CeorYb[ 4 ].SuchamagneticmomentconsistsofalocalizedspindegreeoffreedomthatoversomerangeoftemperatureshasaCurie-Weisscontributiontothespinsusceptibilitygivenby[ 5 ]=C T+: (1{1) 12

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Figure1-1. ResistanceminimumobservedforFeinaseriesofMo-Nballoys[ 1 ].Reprintedgurewithpermissionfrom M.P.Sarachik,E.Corenzwit,andL.D.Longinotti,Phys.Rev.135,A1041(1964) .Copyright1964bytheAmericanPhysicalSociety. ForafreespinS,=0andC=42BS(S+1)=3kB,whereBistheBohrmagnetonandkBistheBoltzmanconstant.Positiveandnegativevaluesofindicateantiferromagneticandferromagneticspin-spincorrelations,respectively.MotivatedbythecorrelationbetweentheexistenceofaCurie-Weisstermintheimpuritysusceptibilityandtheoccurrenceoftheresistivityminimum,Kondofocusedonthemagneticnatureoftheimpurities.Heusedthes-dmodel[ 6 ]tosuccessfullyexplaintheminimuminresistivityatlowtemperatures.Thes-dmodel,nowmorecommonlyknownastheKondomodel,isdescribedbytheHamiltonian Hs)]TJ /F5 7.97 Tf 6.59 0 Td[(d=Xk;"kcyk;ck;+1 2NkXk;k0Jkk0[S+cyk#ck0"+S)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(cyk"ck0#+Sz(cyk"ck0")]TJ /F3 11.955 Tf 11.95 0 Td[(cyk#ck0#)];(1{2) 13

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wherecykcreatesaconduction-bandelectronwithspinzcomponentandwavevectork,Jkk0istheKondocouplingbetweenthelocalmagneticmomentandconductionelectrons,Nkisthenumberofunitcellsinthehost,Szisthezcomponentoftheimpurityspin,andS+andS)]TJ /F1 11.955 Tf 10.99 -4.34 Td[(representtheimpurityspinraisingandloweringoperatorsrespectively.1ThesecondsummationinEq.( 1{2 )describesexchangescatteringofconductionelectronsotheimpurity.InadditiontoprocessesinvolvingSzinwhichboththeelectronandimpurityspinsareconservedduringscattering,Hs)]TJ /F5 7.97 Tf 6.59 0 Td[(dalsocontains\spin-ip"termsinvolvingS+andS)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(,inwhichboththeconductionelectronandtheimpurityspinreversetheirspinzcomponentwhileconservingtheircombinedspinz.Kondosetouttocalculatetheconductivityinthes-dmodelforanimpurityspinS=1=2andassumedalocalk-independentinteractionJ=Jkk0.Hewentbeyondprevioustreatmentsofthemodelbykeepingperturbativecorrectionsuptothirdorderinthesmallparameter0Jandarrivedatanimpuritycontributiontotheresistancegivenby Rimp=cimp3mJ2S(S+1) 2e2~"F1)]TJ /F3 11.955 Tf 11.95 0 Td[(J0lnkBT D;(1{3)wheremistheelectronmass,Disthehalf-bandwidthoftheconductionbandand0(0)istheconduction-electrondensityofstates(")=(1=Nk)Pk(")]TJ /F3 11.955 Tf 12.79 0 Td[("k)attheFermienergy.Combiningthistermwithatypicalelectron-phononcontribution[ 5 ][therstterminEq.( 1{4 )]andatemperature-independentcontributionduetoimpurityscattering[thesecondterminEq.( 1{4 )],theresistancecanbewritteninthefollowingform[ 4 ]:Rimp(T)=aT5+c0impR0)]TJ /F3 11.955 Tf 11.96 0 Td[(cimpR1lnkBT D; (1{4) 1Eq.( 1{2 )usesamodernnotationwherethecouplingJkk0inKondo'soriginaltreatmentoftheHs)]TJ /F5 7.97 Tf 6.59 0 Td[(dmodelisreplacedbyJkk0=2. 14

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wherecimpisthedimensionlessconcentrationofmagneticimpuritiesandc0impisthetotaldimensionlessconcentrationofallimpurities,bothmagneticandnon-magnetic.Eq.( 1{4 )leadstoaresistanceminimumatatemperatureTmingivenbyTmin=R1 5a1.5c1=5imp; (1{5)thatagreeswithexperimentalresults[ 3 ].However,theexistenceofalogarithmictermln(kBT=D)intheresultindicatesthebreakdownofperturbationtheoryasT!0.Subsequentcalculationskeepinghigherorderperturbativecorrectionshavepredictedunphysicaldivergencesinpropertiessuchastheimpuritycontributiontothesusceptibility,entropyandspecicheatatacharacteristicKondotemperatureTK=(D=kB)exp()]TJ /F1 11.955 Tf 9.3 0 Td[(1=0J).ThelogarithmictermsintheperturbativeanalysisimplythattheKondoeectdependscruciallyonconduction-bandstatesatallenergyscales.BelowtheKondotemperatureTK,repeatedspin-ipscatteringbetweenthemagneticimpurityandtheconductionelectronsleadstoacomplexmany-bodyproblem.Severaladvancedmany-bodytechniqueshavebeendevelopedtondasolutiontothisproblem[ 4 ].Thes-dmodelusedbyKondotoexplaintheproblemoftheresistanceminimumisalimitingcaseofthephysicallymorerealisticAndersonmodel[ 7 ]describedbytheHamiltonian HA=Xk;"kcykck+Xnd"d+Und"nd#+1 p NkXk;(Vkdyck+Vkcykd);(1{6)wheredycreatesanelectronattheimpuritysitewithenergy"dandspinzcomponent,cykcreatesaconduction-bandelectronwithspinzcomponentandwavevectork,nd=dyd,nd=Pndisthenumberofelectronsattheimpuritysite,UistheCoulombrepulsionbetweentwoelectronsattheimpuritysite,andVkisthematrixelementfortunnelingbetweentheimpurityandtheconductionband.Thismodelwasproposedtodescribetheinteractionbetweenaspatiallycompact3dorbitalonatransitionmetalion 15

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anddelocalizedelectronsinextendedbandstatesofsand/orpcharacter.Thismodelneglectstheorbitaldegreesoffreedom.AlargevalueoftheCoulombinteractionUisessentialtodescribealocalmomentintheAndersonmodel.ThiscanbeillustratedbyconsideringtheatomiclimitofEq.( 1{6 )whereVk=0suchthatimpurityisdecoupledfromtheconductionelectrons.Inthistrivialcase,theimpurityhasthreeenergycongurations:(i)zerooccupancywithzerototalenergy,(ii)singleoccupancywitheitheraspin"oraspin#electronwithtotalenergy"d,and(iii)doubleoccupancywithspin"and#electronswithtotalenergyU+2"d.Ofthesecongurations,only(ii)hasamagneticmomentassociatedwithit.Solocal-momentformationisfavoredinthismodelundertheconditions"d)]TJ /F3 11.955 Tf 26.55 0 Td[(kBTand"d+UkBT.Eveninthepresenceofanon-zerohybridization(Vk6=0)betweenthelocalizedorbitalanddelocalizedbandelectrons,theAndersonmodelreducestothesimplieds-dmodelatenergyscalesmuchsmallerthanj"djandUprovided)]TJ /F3 11.955 Tf 9.3 0 Td[("d;U+"dmax(0V2=D;kBT),Dbeingthebandwidth,inwhichlimitonlythesubspacend=1isoccupied[ 4 ].TheexchangeinteractionoftheeectiveKondomodelisgivenbytheSchreier-Woltransformation[ 8 ]: Jkk0=VkVk01 U+"d)]TJ /F3 11.955 Tf 11.96 0 Td[("k0+1 "k)]TJ /F3 11.955 Tf 11.95 0 Td[("d:(1{7) 1.2PoorMan'sScalingIn1970,P.W.Andersondevisedaperturbativerenormalizationgroup(RG)approachknownaspoorman'sscalingtounderstandthes-dimpurityproblem[ 9 ].Theobjectiveofthisprocedureistosuccessivelyreducethewidthoftheconductionbandandeliminatehigh-energyexcitationsnearthebandedge[seeFig. 1-2 ]throughrenormalizationofthecouplings.Thisprocedurecancontinueuntilthereducedbandwidth~DkBTortherenormalizedcoupling~J1,atwhichpointperturbationtheorybreaksdown.Treatingvirtualscatteringofelectronsto/fromthebandedgesinperturbationtheoryresultsinaHamiltonianofthesameformasHs)]TJ /F5 7.97 Tf 6.59 0 Td[(dbutwithmodiedcouplings.Retaining 16

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Figure1-2. Theparticleandholestateswhichareremovedfromtheconductionband.(Adaptedfrom[ 4 ].) perturbativetermsuptosecondorderinthedimensionlessexchangecoupling~J=0J,wecanarriveatascalingequationoftheform d~J dl=~J2+O(~J3);(1{8)wherel=ln(D=~D).TheRG\xedpoints"ofthisequation,denedbytheconditiond~J=dl=0arelocatedat~J=0and~J=1.Fortheantiferromagnetic(~J>0)caseofphysicalinterest,Eq.( 1{8 )canbeintegratedtoyield~De)]TJ /F4 7.97 Tf 6.59 0 Td[(1=~J=De)]TJ /F4 7.97 Tf 6.58 0 Td[(1=0JkBTK: (1{9)ThesetoftrajectoriesdescribedbyEq.( 1{9 )ischaracterizedsolelybytheKondotemperatureTKthatisafunctionofthemodelparameters.Eq.( 1{9 )impliesthattheconductionbandwidthcanbereducedto~DkBTK,belowwhich~Jincreasestotheorderofunityandperturbationtheoryfails.2 2Fortheferromagneticside(~J<0),perturbativescalingcanbecontinueddowntoarbitraryvaluesof~DatT=0.Inthiscase,~J!0as~D!0implyingthattheimpurityspiniseectivelydecoupledfromtheconductionbandatlowtemperatures[ 9 ]. 17

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Anon-perturbativenumericalapproachtodealwithquantumimpuritymodelswaslaterdevelopedbyK.G.Wilson[ 10 ]andwassuccessfulinsolvingtheKondoproblem.ItshowedthatbelowthecharacteristicKondotemperatureTK,thelocalmagneticmomentiscollectivelyandcompletelyscreenedoutbytheconductionelectronsleadingtotheformationofastronglycoupledKondosingletatlowtemperatures.ThismethodwillbediscussedindetailinChapter 2 .Althoughthepoorman'sscalingapproachbreaksdownatlowtemperatures,itappliestheconceptofrenormalizationoftheparameterstounderstandtheKondoproblem.InthelanguageoftheRG,theKondocouplingrenormalizestowardsstrongcouplingastheenergyisloweredandtheeectiveKondocouplingisinniteattheRGxedpoint,asshowninFig. 1-3 A).Theinnitecouplingatthexedpointimpliesthattheimpurityspindegreesoffreedomarecompletelyquenchedbythedelocalizedconductionelectronstoformasinglet: jKondosingleti1 2j"iimpj#ic)-221(j#iimpj"ic;(1{10)wherejicisalinearsuperpositionoftheconductionelectronstatesofallenergyscales.BelowtheKondotemperatureTK,thesystemisinanentangledstatebetweenthelocalizedmomentandtheconductionelectrons.Thelow-energyexcitationscanbedescribedbyLandauquasiparticleexcitationshavingthesamequantumnumbersasthoseofabareelectronorhole,thatis,spin1=2andchargee,andthesystembehavesasaFermiliquid[ 4 , 11 ].Duetorepeatedspin-ipscatteringfromstatesneartheFermilevel,thequasiparticledensityofstateshasapeakattheFermienergyofwidthkBTK[ 4 ].ThisisknownastheKondoresonance.TheKondoeectalsooperatesinheavy-fermionmetalslikeCeAl3thatcontainrare-earthoractinideselements(e.g.,Ce,Yb,U,orNp)withpartiallylled4for5forbitals[ 11 { 15 ].Themainpropertiesofheavyfermionscanbemodeledasalatticeoflocalizedf-electronscoupledtoaconductionband[ 16 ].Atlowtemperatures,the 18

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Figure1-3. SchematicRGowdiagramfortheA)one-channel,B)one-channelpseudogap,C)two-channelandD)two-channelpseudogapKondomodels.Solid(open)dotssignifystable(unstable)xedpoints. resistivityandspecicheatCofthesesystemsvariesas(T)=0+AT2andC=T+T3aspredictedbyFermiliquidtheory[ 4 ].Onecharacteristicofallthesematerialsisthatatlowtemperatures,theelectroniceectivemassm(denedexperimentallyasm/0,wheremisthebareelectronmassand0isthefree-electronspecicheatcoecient)ishundredsoftimesthebaremassduetostrongcorrelationoftheelectronsthroughrepeatedscatteringofconductionelectronsfromforbitals[ 17 ].Thisismanifestedinextremelylargevaluesofthethespecicheatcoecient(oftheorderofJ/molK2comparedtomJ/molK2inordinarymetals)andthecoecientAthatenterstheresistivity,aswellasinanenhancementofthePaulisusceptibilitys[ 18 ]. 1.3VariantsoftheKondoModelThissectionintroducesseveralvariantsofthestandardKondomodelanddiscussestheirmainfeatureswithinpoorman'sscalingapproach.Someresultsfromnon-perturbativeanalysisofthesemodelsarepresentedinlatersections. 19

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1.3.1PseudogapKondoModelTheKondoeectdependsonthepresenceoffermionicexcitationsdowntoarbitrarilysmallscalesofenergyabovetheFermisurface.ThisraisesthequestionwhethertheKondoeectsurvivesinsystemswherethedensityofstatesiszeroattheFermisurface.ThepresenceofanitegapinthedensityofstatesaroundtheFermienergyturnsouttoreplacethemany-bodyKondoeectbyarenormalizedlevelcrossingbetweenthefree-impurityandquenched-impuritygroundstates[ 19 ].However,aninterestingsituationariseswhenthedensityofstatesis\pseudogapped",i.e.,itvanishesexactlyattheFermienergybutisnonzeroateveryotherenergyinthevicinityoftheFermienergy.Thissituationisfoundinavarietyofsystems.Forexample,thequasiparticledensityofstatesnearthenodesintheenergygapinunconventionalhigh-Tcd-waveandp-wavesuperconductorsvariesasj"jandj"j2respectively[ 20 ].Thedensityofstatesincertainzero-gapsemiconductorsvariesasj"jd)]TJ /F4 7.97 Tf 6.59 0 Td[(1nearthebandgapforsmallvaluesofj"j(wheredisthedimensionofthesystem)[ 21 , 22 ].Varioustwo-dimensionalsystemslikegraphenesheetshavealineardensityofstatesnearthebandgap[ 23 ].Thedensityofstatesinallthesesystemshasapower-lawvariationneartheFermienergyandforthepurposesoffundamentaltheoreticalstudy,canbefurthersimpliedbyassumingthatthepower-lawformextendsacrosstheentirebandwidth.Withthisassumption,thedensityofstatescanbedescribedby (")=0j"=Djr(D)-222(j"j);(1{11)whereDisthehalf-bandwidth,0isthedensityofstatesatthebandedge,andr>0isthepseudogapexponentthatdependsonthespecicsystemunderinvestigation.ThersttheoreticalstudyofimpuritiesinapseudogapdensityofstatesgivenbyEq.( 1{11 )wascarriedoutbyWithoandFradkin[ 24 ].Theyusedthepoorman'sscalingapproachdescribedinSec. 1.2 tostudythepseudogapKondomodelandarrivedatthe 20

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followingRGscalingequation: d~J dl=)]TJ /F3 11.955 Tf 9.3 0 Td[(r~J+~J2+O(~J3):(1{12)Inadditiontotheweak-andstrong-couplingxedpointsgivenby~J=0and~J=1,respectively,Eq.( 1{12 )alsohasanunstablexedpointat~Jc=0Jc=r.ThismeansthatforanyvalueofthecouplingJJc),thesystemowstowardstheweak-coupling(strong-coupling)xedpointgovernedby~J=0(~J!1)underparameterrenormalizationasshowninFig. 1-3 B).NRGstudiesofthepseudogapKondomodelalsorevealedazero-temperaturephasetransition(quantumphasetransition)forr<1=2,suchthattheimpurityisKondoscreenedonlywhenthecouplingislargerthanacriticalcouplingJc[ 25 , 26 ].ForJ1=2andtheimpurityformsafreelocalmomentforallvaluesofthecouplingJ>0. 1.3.2Two-ChannelKondoModelAnotherintriguingmodicationofthestandardKondomodelcanbemadebyincludingasecondindependentbandofconductionelectronsthatalsointeractswiththeimpurity.Thetwo-channelKondomodelwasintroducedbyNozieresandBlandin[ 27 ]andhasbeenproposedtounderlienon-Fermiliquidbehaviorincertainheavy-fermion 21

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compoundsandhigh-Tccupratesuperconductors[ 28 { 31 ].Thistwo-channelKondomodelisdescribedbytheHamiltonian H2CK=Hbands+J1s1S+J2s2S;(1{14)wheres1ands2arethespinsofconductionelectronchannels1and2attheimpuritysiteandSistheimpurityspin.EachJisiSterminEq.( 1{14 )canbeexpandedintotermsconsistingoftheimpurityspinoperatorsS+,S)]TJ /F1 11.955 Tf 10.99 -4.33 Td[(andSzlikeintheoriginals-dmodeldenedinEq.( 1{2 ).Forthesymmetriccase,whereJ1=J2,thetwochannelscompetewitheachothertoscreenoutthemagneticimpuritygivingrisetofrustrationinthesystem.Sincethechannelsareunabletoscreenouttheimpurityspinindividually,theresultinggroundstatefeaturesapartialscreeningoftheimpurityspin.Unlikethesituationfortheone-channelKondomodel,forthetwo-channelcase,thelow-energyexcitationscannotbeexpressedintermsofquasiparticlesandthesystemexhibitsnon-Fermiliquidbehavior[ 32 ].Howeverinthepresenceofanyasymmetrybetweenthetwochannels(J16=J2),theimpurityspinisscreenedoutbythemorestronglycoupledchannel,thusdestroyingthenon-Fermiliquidgroundstate.Infact,thereisaquantumphasetransitionthattakesplaceatJ=J1)]TJ /F3 11.955 Tf 12.49 0 Td[(J2=0.ThegroundstateoneithersideofthephasetransitionconsistsofaKondosingletbetweentheimpurityspinandtheconductionelectronsofthemorestronglycoupledchannel.Thenon-Fermiliquidbehaviorexistsinthequantumcriticalregime(J0;T>0)nearthequantumcriticalpointseparatingthetwogroundstatesandtheKondoeectisrecoveredbelowacharacteristiccrossovertemperature.Henceinordertoobservethetwo-channelKondoeectinanexperimentalsetup,itisextremelyimportanttotunethesystemtohaveidenticalKondocouplingstothetwoindependentchannels.ThebehaviordescribedabovecanbeunderstoodviaaperturbativescalingequationfortheNc-channelKondomodel[ 27 ] d~J dl=~J2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(Nc 2~J3+O(~J4):(1{15) 22

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Asusual,thexedpointsoftheproblemsatisfyd~J=dl=0.Therearetwounstablesolutions:oneat~J=0,whichisthedecoupledimpuritylimit,andoneat~J=1,whichisthestrong-couplinglimit.Thereisalsoastablesolutionat ~J=2 Nc:(1{16)Thexedpointsofthetwo-channelKondomodelareshownschematicallyinFig. 1-3 C).ForNc2,~J1,sothestablesolutionlieswithintherangeofvalidityofperturbativescaling.ForNc=2,~J=1,whichplacesthexedpointontheboundaryofvalidity.However,nonperturbativemethods[ 33 { 36 ]showthatthisxedpointdoesexistforthetwo-channelcaseanditcorrespondstoanintermediatecouplingnon-Fermiliquidgroundstate. 1.3.3Two-ChannelPseudogapKondoModelItisnaturaltowonderaboutthefateofthenon-Fermiliquidxedpointofthetwo-channelKondomodelandthecriticalpointofthepseudogapmodelifweconsideratwo-channelKondomodelwherebothconductionchannelshavethesamepseudogapdensityofstates.Thereisspeculationabouttherealizationofthetwo-channelpseudogapKondophysicsingraphene[seethediscussioninSec. 1.4 ].Thepoorman'sscalingequationforthismodelisgivenby[ 37 ] d~J dl=)]TJ /F3 11.955 Tf 9.3 0 Td[(r~J+~J2)]TJ /F1 11.955 Tf 14.77 3.02 Td[(~J3+O(~J4):(1{17)Thisshowsthatinadditiontothe~J=0and~J=1xedpoints,forr<1=4,therearetwomoreintermediate-couplingxedpointsgivenby ~J=1 2(1p 1)]TJ /F1 11.955 Tf 11.96 0 Td[(4r):(1{18)ThexedpointsofthismodelareshownschematicallyinFig. 1-3 D).Forr1=4,theserootscanbeapproximatedby(a)anunstablexedpointat~Jcr,thathasthesamer-dependenceasthecriticalpointfortheone-channelpseudogapcase,and(b)astable 23

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xedpointatJ1)]TJ /F3 11.955 Tf 12.44 0 Td[(r,thatevolvesfromtheintermediatecouplingnon-FermiliquidxedpointobtainedinSec. 1.3.2 forr=0. 1.4KondoEectinNanostructuresRecentdevelopmentsinnanofabricationallowthedesignandinvestigationofarticialnanostructuresinwhichpropertiessuchastheenergyspectrum,magneticmomentandcouplingtotheenvironmentaretunableviaexternalgatevoltagesandmagneticelds[ 38 ].TheoccurrenceoftheKondoeectinnanodeviceswaspredictedin1988[ 39 , 40 ]andrstrealizedin1998[ 41 ].Intherstsuccessfulexperiment,thequantumdotwasfabricatedbydepositingmultiplegatesonaGaAs/AlGaAsheterostructure[asshowninFig. 1-4 A)]containingatwo-dimensionalelectrongas.Atlowenoughtemperatures,thenumberofelectronspresentinthedotiswelldened(50in[ 41 ])andcanbecontrolledbytuningthegatevoltage.Forsmalldots,thechargingenergyrequiredtoaddanextraelectrontothedot(whichisrelatedtotheCoulombrepulsionenergyUintheAndersonmodel)islarge,asshowninFig. 1-4 B).However,whenthedotcontainsanunpairedspinthenvirtualtunnelingofelectronsto/fromtheleadsresultsintheformationofaKondoresonance(acollectivestateinvolvingthedotandtheleads)belowacharacteristictemperatureTK.AnumberofsubsequentexperimentshavealsoconrmedandfurtherprobedtheexistenceofKondophysicsinquantumdots[ 42 { 44 ].Electricalconductivitymeasuredthroughaquantumdotin[ 44 ]showedthattheconductancedecreasesrapidlytowardszeroasthetemperatureapproachesabsolutezerowhenthedotcontainsanevennumberofelectronsmanifestingthephenomenonofCoulombblockade.ThisfeaturecanbeunderstoodintermsoftheAndersonmodeldenedinEq.( 1{6 ).Coulombblockadesetsinthequantumdotwhentheenergycostfortheaddition/removalofanelectronto/fromthedotfarexceedsboththethermalenergyandtheappliedbias()]TJ /F3 11.955 Tf 9.3 0 Td[("d;U+"dkBT;eVbias)sothatelectronscannottunnelintooroutofthequantumdot.Here, 24

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Figure1-4. Kondoeectinasingleelectrontransistor[ 41 ].A)AquantumdotfabricatedbydepositingthreegateelectrodesonaGaAsheterostructure.Themiddleelectrodeontheleftisusedasagatetochangetheenergyofthedropletrelativetothetwo-dimensionalelectrongas.B)Schematicenergydiagramofthequantumdot,showingasituationwheretheFermienergiesofthesourceanddrainleadsarenearlyequalandfarfromthehighestoccupiedandlowestunoccupieddotstates.ThereisanenergycostUtoaddorremoveanelectronfromthequantumdot.Adaptedfrom[ 41 ]andreprintedbypermissionfromMacmillanPublishersLtd: D.Goldhaber-Gordon,HadasShtrikman,D.Mahalu,DavidAbusch-Magder,U.MeiravandM.A.Kastner,Nature391,156(1998) ,copyright1998. "distheelectronoccupationenergyatthequantumdot,Ue2=CistheCoulombrepulsionenergy,andCisthecapacitanceofthedot.However,forodd-occupancyofthequantumdot,theconductanceincreasesatlowtemperaturessignalingtheKondoeect.Inthiscase,thescreeningofthequantumdotspincreatesasingle,extendedmany-bodystatethatproducesaconductanceashighas2e2=h,whichistheunitarylimitforasingleconductancechannel[ 44 ].Itwasalsodemonstratedin[ 44 ]thatthenormalizedconductancefordierentgatevoltagesandtemperaturescollapsestoauniversalcurvewhenplottedagainstT=TK,whichisasignatureoftheKondoeect[ 4 ]. 25

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BycontrasttotheconventionalKondoeect,two-channelKondophysicshasprovedverychallengingtorealizeexperimentallyduetodicultiesin(a)fabricatingtwoelectronchannelsthataretrulyindependentand(b)tuningthetwochannelstocoupletothedotwithequalexchangecoupling.Therstsetupforobservationofthetwo-channelKondoeectinaquantum-dotsystemwasproposedbyOregandGoldhaber-Gordon[ 45 ],andlaterrealizedbyR.M.Potoket.al.[ 46 ].TheexperimentalsetupconsistsofaGaAsdouble-quantum-dotsystemasshowninFig. 1-5 D).Gatesallowprecisecontroloftheelectrostaticpotentialofthedotsandtuningofthetunnelingbarriersbetweenthereservoirsorchannels.The\magneticimpurity"correspondstothesmallerquantumdotcontaininganoddnumberofelectrons(25inanareaof0:04m2).Theconductionelectronsthatscreenoutthelocalmomentresideintworeservoirs.Onereservoirconsistsofthesourceandthedrainleads,whichalthoughphysicallyseparated,formasingleeectivereservoir(calledtheinnitereservoirir).Thesecondreservoirconsistsofthesecondquantumdotwhichhasanareaof'3m2,muchlargerthantherstdot.Theseconddothasachargingenergy(100eV)muchhigherthanthesingletformationenergyoftherstdot(13eV).ThispreventstheexchangeofelectronsbetweenthetworeservoirswhenthelargerdotistunedtoaCoulombblockadevalleywithawell-denedchargestate.Thelargedoteectivelyhasacontinuumofsingleparticlestatesandactsasthesecondindependentreservoir(calledthereservoirfr).Thesymmetricaltwo-channelKondoandnon-FermiliquidbehaviorisachievedonlywhenthegatevoltagethatcontrolthecouplingbetweenthequantumdotandthenitereservoiristunedsuchthatthecouplingsJir=Jfr.Fortheasymmetriccase,one-channelKondobehaviorisexpectedasthequantumdotshouldformaKondosingletwiththemorestronglycoupledreservoir.Figure 1-6 showsconductivitymeasurementsforthreedierentregimes.Whentheinnitereservoirismorestronglycoupledtothequantumdotthantheniteone[Fig. 1-6 A)],thezero-biasconductanceisenhanced.Thedierentialconductanceg(Vds;T)asafunctionofthesource-drainvoltageVdsandtemperatureT 26

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Figure1-5. Kondoeectinadoublequantumdot[ 46 ].A)One-channelKondoeectwherethemagneticimpurityiscoupledtoasinglereservoirofelectrons.B)Two-channelKondoeectwheretheelectroniscoupledtotwoindependentreservoirs(blueandred).C)Physicallyseparatingthetworeservoirsdoesnotsucetomakethemindependent.Ifalocalizedelectroncanhopothedottotherightreservoirandanewelectroncanhopontothedotfromtheleft,thetworeservoirswillcooperateinscreeningthelocalizedspin.D)Experimentalrealizationofthetwo-channelKondoeectusinganitereservoir(red)connectedtoconventionalleads(blue).E)Coulombblockadesuppressesexchangeofelectronsbetweenthenitereservoirandthenormalleadsatlowtemperatures.Hencethetworeservoirsactastwoindependentscreeningchannels.Adaptedfrom[ 46 ]andreprintedbypermissionfromMacmillanPublishersLtd: R.M.Potok,I.G.Rau,HadasShtrikman,YuvalOregandD.Goldhaber-Gordon,Nature446,167(2007) ,copyright2007. 27

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Figure1-6. Scaledvariationoftheconductancewithsource-drainbiasvoltageforthedoublequantumdotshowninFig. 1-5 .Resultsareshownforthreeregimesachievedbytuningthegatevoltagethatcontrolsthecouplingbetweenthequantumdotandnitereservoir.A)ForJirJfr,theconductivityshowsazero-biaspeakandwhenplottedagainstascaledtemperature,thedatacollapseontoasingleV-shapedcurve,consistentwiththescalingequationforonechannelbehavior.B)ForJirJfr,thedierentialconductanceshowsascalingthatisconsistentwiththetwo-channelbehavior.C)ForJifJfr,thereisazero-biassuppressionbecausethelocalspinformsaKondostatewiththenitereservoir.AdaptedfromFigs.3and4of[ 46 ]andreprintedbypermissionfromMacmillanPublishersLtd: R.M.Potok,I.G.Rau,HadasShtrikman,YuvalOregandD.Goldhaber-Gordon,Nature446,167(2007) ,copyright2007. obeysaformcharacteristicofthesinglechannelKondoeectgivenby[ 47 ]g(0;T))]TJ /F3 11.955 Tf 11.95 0 Td[(g(Vds;T) T2=eVds kBT2; (1{19)whereisaconstant.Whenthenitereservoiriscoupledmorestrongly[Fig. 1-6 B)],thereisasuppressionoftheconductancebecausethequantumdotformsasingletwiththenitereservoirandtheconductanceobeysaninvertedformofEq.( 1{19 ).However,inpassingfromtherstlimittothesecond,aregimeisreachedwherethescalingformoftheconductionischaracteristicofthetwo-channelKondoeectgivenby[ 48 , 49 ]g(0;T))]TJ /F3 11.955 Tf 11.96 0 Td[(g(Vds;T) T2=2YeVds kBT2; (1{20) 28

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wherethefunctionY(x)isdenedbyY(x)=8><>:3 p x)]TJ /F1 11.955 Tf 11.96 0 Td[(1forx1;cx2forx1: (1{21)ThisisdemonstratedinFig. 1-6 B)wherethedatafordierentbiasvoltagesandvarioustemperaturescollapseontoasinglecurvedescribedbyEq.( 1{20 )whenplottedagainstascaledsource-drainvoltage.Recently,theKondoeectcausedbymagneticimpuritiesingraphenehasgeneratedalotofinterest[ 50 { 57 ]duetothepossiblerealizationoftwo-channelphenomena.Grapheneconsistsofamonolayerofcarbonatomsarrangedinahoneycomblattice(leftpanelofFig. 1-7 )andthevalenceandconductionbandstoucheachotherattwoinequivalentDiracpointsKandK0atthecornersoftherstBrillouinzone[Figs. 1-7 (rightpanel)and 1-8 ].InundopedgraphenetheFermienergyliesatthistouchingpoint,andthelineardispersionofthebandsneartheDiracpointgivesrisetoalow-energydensityofstates[ 58 ] (")=2Ac j"j v2F;(1{22)whereAcistheareaofaunitcellandvFistheFermivelocity.SincethedensityofstatesvanisheslinearlyattheFermisurface,thisformsastrongcontenderfortheoccurrenceofthepseudogapKondophysicswiththeexponentr=1.Alocalmagneticmomentcanbeintroducedintographeneduetoapointdefectorvacancyinducedbyirradiationorbyplacingamagneticadatomonthegraphenesheetusingascanningtunnelingmicroscope[ 59 ].Whethertwo-channelKondophysicscanbeobservedingrapheneremainsopentodebateandseveralargumentshavebeenoeredthateithersupport[ 53 , 54 , 60 ]orcontradict[ 51 ]theproposition.Ithasbeenclaimedthatthepositionoftheadatomonthegraphenesurfaceisthekeyfactorindeterminingthenumberofeectivechannelsthatinteractwiththeimpurity[ 51 ].Partialwaveanalysisusingatight-bindingHamiltonian[ 50 ]showsthatwhentheadatomoccupiesthecenter 29

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Figure1-7. A)Honeycomblatticeofgraphene[ 58 ].B)ThecorrespondingBrillouinzone.TheDiracconesarelocatedattheKandK0points.Adaptedfrom[ 58 ]andreprintedgurewithpermissionfrom A.H.CastroNeto,F.Guinea,N.M.R.Peres,K.S.Novoselov,andA.K.Geim,Rev.Mod.Phys.81,109(2009) .Copyright2009bytheAmericanPhysicalSociety. Figure1-8. Electronicdispersioningrapheneusingatight-bindingHamiltonian.AzoominoftheenergybandsclosetooneoftheDiracpointsshowsalineardispersion,isotropicabouttheDiracpointthatholdsasymptoticallyclosetotheDiracpoints.Reprintedgurewithpermissionfrom A.H.CastroNeto,F.Guinea,N.M.R.Peres,K.S.Novoselov,andA.K.Geim,Rev.Mod.Phys.81,109(2009) .Copyright2009bytheAmericanPhysicalSociety. 30

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ofahexagoninthegraphenelattice,itcouplessymmetricallytotheKandK0valleys.Thisgivesrisetotwoindependentscreeningchannelsforthelocalizedspinresultingintwo-channelbehavior.However,iftheadatomislocatedontopofacarbonatom,itbreaksthesymmetryintheKandK0valleysandhencecouplestoaneectivesinglechannel[ 50 ].Basedonageneralgroup-theoreticalstudyoftheKondoproblem,theauthorsin[ 54 ]foundsixpossibleclassesofanisotropicfour-channelKondomodelandhavearguedseveralpossibilitiesfortherealizationofthesymmetrictwo-channelKondomodelingraphenesheets.Thusitisimperativetounderstandthetwo-channelKondomodelinthepresenceofapseudogapdensitystates.Thescalingequation[Eq.( 1{14 )]forthetwo-channelpseudogapKondomodelpredictsarichphasediagramwithatleasttwointermediatecouplingxedpoints.Althoughthequantumcriticalpropertiesoftheone-channelpseudogapproblemhavebeenstudiedextensively,notmuchisknownaboutthenatureofthequantumphasetransitionsthatariseinitstwo-channelcounterpart.Chapter 3 reportsacomprehensivestudyofthephasediagramandquantumphasetransitionthatarisesinthetwo-channelpseudogapKondomodel,aswellasthetwo-channelmodelwithapower-law-divergentdensityofstates[correspondingtonegativevaluesofrinEq.( 1{11 )]. 1.5QuantumCriticalityinHeavy-FermionSystemsAquantumphasetransition(QPT)isacontinuouszerotemperaturetransitionthatarisesinthepresenceofcompetinginteractions.Thephasetransitioncanbeaccessedbyvaryinganon-thermalparametersuchasappliedmagneticeld,externalpressureorchemicalcomposition[ 61 , 62 ].Phasetransitionsarebroadlydividedintotwocategories.Arst-orderphasetransitiondescribesasimplelevelcrossingofthegroundstateofthesystemwheretheorderparametervariesdiscontinuouslyatthephaseboundary.Themoreinterestingsecond-orderorcontinuousphasetransitioncanbecharacterizedbyanorderparameterthatiszerothroughoutonephase(the\disordered"phase),non-zerointheotherphase(the\ordered"phase)andvanishescontinuouslyonapproachtothephase 31

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boundaryfromtheorderedside.Boththecorrelationlengthandcorrelationtimedivergeonapproachtoacontinuousquantumphasetransition,whichtakesplaceataquantumcriticalpoint(QCP).Orderparameteructuationscanoccuratalllengthscalesandtimescalesandthesystembecomesscale-independent.NearaQCP,thephysicalpropertiesobeypowerlawscharacterizedbyasetofcriticalexponents.Thesecriticalexponentsarenotindependentofeachother,rathertheyobeycertainscalingrelationsthatdependontheclassofthephasetransitionundergonebythesystem.Thermaluctuationsplayacriticalroleinaclassicalphasetransitionthattakesplaceatanitetemperature.However,sinceaQPTtakesplaceatabsolutezerotemperature,allthermaluctuationsvanishandthequantumuctuationsallowedbytheHeisenberguncertaintyprincipleareresponsiblefordestructionoflongrangeorderattheQPT.Thequantumphasetransitioncanbemappedtotheclassicalphasetransitionusingthefollowinganalogy.Theclassicaldensityoperatorexp()]TJ /F3 11.955 Tf 9.29 0 Td[(H=kBT)thatentersthepartitionfunctionhasthesameformasthequantummechanicaltimeevolutionoperatorexp()]TJ /F3 11.955 Tf 9.29 0 Td[(iHt=~)foranimaginarytime=)]TJ /F3 11.955 Tf 9.3 0 Td[(i~=kBT.UsingFeynman'spathintegralformalism[ 63 ],theimaginary-timeaxiscanbeintroducedasanadditionaldimensionofthesystemspanning0<<1=T(setting~=kB=1).Theimaginary-timedimensionbehavesasaspacedimensionatzerotemperaturesinceitextendstoinnity.IfthecorrelationlengthandcorrelationtimedivergeattheQCPas/jj)]TJ /F5 7.97 Tf 6.59 0 Td[(andc/)]TJ /F5 7.97 Tf 6.59 0 Td[(zrespectively,whereisthedistancefromtheQCPalongthenon-thermalparameter,thentheQPTinddimensionscorrespondstoaclassicalphasetransitionind+zdimensions[ 61 , 64 , 65 ].Hereisknownasthecorrelationlengthexponentandzisknownasthedynamicalcriticalexponent.Forexample,z=2foraspin-densitywaveantiferromagneticQPT[ 61 , 64 ],z=2dforaMottinsulatortransition[ 66 ]andz=1forQCPsincertaininsulators[ 61 ].Quantumphasetransitionsareknowntoplayacrucialroleinavarietyofstronglycorrelatedsytsemsincludingheavyfermions[ 67 , 68 ],high-Tcsuperconductors[ 69 , 70 ] 32

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andmetal-insulatortransitions[ 71 ].Aheavy-fermionmetallatticeusuallyconsistsofelectronsfromtheconductionbandaswellaslocalizedmagneticmomentsduetopartiallylled4for5forbitals.AsubsetofQPTsthatoccurinthesematerialscanbeexplainedbytakingintoaccounttwocompetinginteractions:(i)themagneticmomentinteractingwiththeconductionbandelectronsthroughaKondo-likecouplingand(ii)antiferromagneticinteractionsbetweenthelocalmomentsthatareprimarilyoftheRuderman-Kittel-Kasuya-Yoshida(RKKY)type[ 16 ].ContinuousQPTsinitinerantelectronsystemsareconventionallydescribedwithinaGinzburg-Landau-Wilson(GLW)pictureofcriticaluctuationsofanorderparametercharacterizingaspontaneouslybrokensymmetry[ 61 , 64 , 65 ].Inthiscase,theonlyimportantcriticaluctuationsarethoseoftheorderparameter.FortheantiferromagneticQCP,theorderedphaseinthevicinityoftheQCPcanbedescribedbycollectivemodescalledspindensitywaves.However,recentexperimentshaverevealedaclassofheavy-fermionQCPsthatcannotbeexplainedusingthefamiliarGLWframeworkofcontinuousphasetransition[ 17 , 72 , 74 ].Letusconsidertheexampleoftheheavy-fermioncompoundCeCu6)]TJ /F5 7.97 Tf 6.58 0 Td[(xAux,wherexistheconcentrationofAu.ItundergoesaQPTfromaparamagneticphasetoantiferromagneticorderingwiththeincreaseinxandtheQCPissituatedatx0:1[ 75 ].InthevicinityoftheQCP,thetemperaturedependenceoftheresistivityhasaformgivenby(T)0+ATasshowninFig. 1-9 B)thatischaracteristicofnon-Fermiliquidbehavior[ 67 ].ForaFermiliquid,theresistivityisexpectedtohaveaformgivenby(T)0+AT2[ 4 ],thatisdemonstratedawayfromtheQCPasshowninFigs. 1-9 A)andC).Fig. 1-10 showsthatthespecicheatCbehavesasC=T/ln(T0=T)attheQCP(T0isattingparameter),thatisdierentfromthatpredictedforaFermi-liquid(C=T=+T2[ 4 ]).Furthermore,theimaginarypartofthedynamicalsusceptibility,asfoundusingneutronscatteringexperiments[ 73 ],hasascalingform00(!;T)=T)]TJ /F5 7.97 Tf 6.58 0 Td[(g(!=kBT): (1{23) 33

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Figure1-9. Temperaturedependenceoftheresistivityofheavy-fermionCeCu6)]TJ /F5 7.97 Tf 6.59 0 Td[(xAux[ 67 ]forA)x=0:5,B)x=xc=0:1andC)x=0.InA)andC),theresistivityisplottedversusT2.AstraightlinettothesecurvesindicateaFermiliquidbehavior.InB),theresistivityvarieslinearlywithtemperatureindicatingnon-Fermiliquidbehavior.Adaptedfrom[ 67 ]andreproducedwithpermissionfrom HilbertvonLohneysen ,J.Phys.:Condens.Matt.8,9689-9706(1996). CCBY .IOPPublishing.Allrightsreserved. 34

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Figure1-10. ThespecicheatCofheavy-fermionCeCu6)]TJ /F5 7.97 Tf 6.59 0 Td[(xAuxplottedasC=TvslogTforvariousvaluesofx[ 67 ].Atthequantumcriticalpointxc=0:1,thespecicheatexhibitsaformC=T/log(T)characteristicofanon-Fermiliquid.Reproducedwithpermissionfrom HilbertvonLohneysen ,J.Phys.:Condens.Matt.8,9689-9706(1996). CCBY .IOPPublishing.Allrightsreserved. 00dependsonboththetemperatureandfrequencyviaafractionalanomalousexponent0:74asshowninFig. 1-11 .Thedynamicsusceptibilityalsoobeysafrequency-over-temperature(!=T)scaling.TheseobservationscannotbeexplainedusingtheGLWtheorythatdescribesastandard4theorywithorderparameteructuations(knownasspindensitywaves)ind+zdimensions,wheredisthespatialdimensionandz=2isthedynamicalexponent[ 64 , 65 ].FortheantiferromagneticQPT,thedynamicalexponentz=2,suchthatd+z=5isabovetheuppercriticaldimension(dc=4)forsecond-order 35

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Figure1-11. Imaginarypartofthedynamicalsusceptibility00plottedas00kBT0:75vs!=kBTforheavy-fermioncompoundCeCu5:8Au0:2atwavevectorQ=(0:8;0;0)andforvarioustemperatures.Thesolidlinecorrespondstoaform00(!;T)=T)]TJ /F5 7.97 Tf 6.59 0 Td[(g(!=kBT)[ 72 ]withttingparameter=0:74.Insetshowsthequalityofthescalingcollapsewithttingparameter.Adaptedfrom[ 73 ]andreprintedgurewithpermissionfrom A.Schroder,G.Aeppli,E.Bucher,R.Ramazashvili,andP.Coleman,Phys.Rev.Lett.80,5623(1998) .Copyright1998bytheAmericanPhysicalSociety. phasetransitions[ 61 ].Soweexpecta\gaussian"ornon-interactingcriticalpointwheretheorderparameteructuationsareirrelevantandallthecriticalexponentsaregivenbytheirmean-eldvalues.However,theanomalousexponent0:74suggeststhatthecriticaluctuationsarerelevantandcontradictsthemean-eldvalueof=1.The!=Tscalingfurthersuggeststhatasuitablydenedrelaxationtimeduetospindampingislinearwithtemperatureaswouldbeexpectedifthecriticalpointisinteracting[ 26 ].TheseantiferromagneticQCPscanbeunderstoodonlybypostulatingadditionalcriticalmodesbeyondorder-parameteructuations[ 76 ].Ithasbeenproposed[ 77 , 78 ]thattheadditionalmodesarisefromthecriticaldestructionoftheKondoeectdueto 36

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magneticuctuations,associatedwithajumpintheFermisurfacevolume[ 79 { 81 ]fromlargeintheparamagneticphase(whereunpairedfelectronsareabsorbedintoKondoresonances)tosmallintheantiferromagneticphase(wheretheKondoresonancesaredestroyedandthefelectronsarelocalized).ThepictureofcriticalKondodestructionwasoriginallydevelopedintheKondolimitofintegerfoccupancy.Morerecently,thediscoveryofunconventionalquantumcriticality[ 82 , 83 ]intheytterbium-basedheavy-fermion-YbAlB4haspromptedinterestincriticalKondodestructionatmixedvalence[ 84 ].Thequantumcriticalpointin-YbAlB4featuresadivergenceinthemagneticsusceptibilityasT!0[ 82 ].ThemagnetizationalsoobeysaprominentH=Tscalingupontheapplicationofaexternalmagneticeld[ 83 ].Thisisconsistentwitha!=Tscalingofthesusceptibilitythatisasignatureoftheunconventionalquantumcriticalpointsinotherantiferromagneticheavy-fermionmaterials[ 77 ].Allquantumcriticalheavy-fermionsystemswerepreviouslybelievedtohaveintegervalencestabilizingthelocalmomentsinthelattice.Sothepresenceofquantumcriticalityin-YbAlB4issurprisingconsideringthemixedvalenceofthematerialwiththevalencyofYb+2:75beingsignicantlyawayfromanintegervalue[ 83 ].Thisraisesanimportantquestion:whetherKondodestructioncanoccuratamixed-valenceQCPandtheroleofvalenceuctuationsattheQCP.Atoymodelforthisphenomenonistheparticle-hole-asymmetricAndersonimpuritymodelwithadensityofstates(")/j"jrthatvanishesinpower-lawfashiononapproachtotheFermienergy"=0[ 85 ].AsdescribedinSec. 1.3.1 ,thismodelfeaturesaKondo-destructionQCPseparatingastrong-coupling(Kondo-screened)phasefromalocal-moment(Kondo-destroyed)phase[ 25 , 86 { 88 ].Astudyconductedusingacombinationofcontinuous-timequantumMonteCarloandthenumericalrenormalizationgroup(NRG)showedfortheparticularcaseofr=0:6thatKondo-destructionwasaccompaniedbydivergenceofalocalchargesusceptibilityonapproachtotheQCPfromeitherphase[ 85 ].Inthiscase,bothspinandchargeresponsesdemonstratethe 37

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frequency-over-temperatureandmagneticeld-over-temperaturescalingcharacteristicofaninteractingQCP.Ihavemadeathoroughinvestigationtounderstandthecriticalchargeuctuationsofthemodelanddeterminetherelationbetweenthecriticalspinandchargeresponsesofthesystem.TheresultsofthisinvestigationarereportedinChapter 4 ofthisdissertation. 1.6EntanglementEntropyEntanglemententropyisameasureoftheentanglementbetweensub-systemsofaquantummechanicalsystemandcapturesthedegreeofquantumnonlocalityintheground-statewavefunction.Letusconsiderapartitionthatdividesthesystemintotworegions:regionAconsistingoftheregionofinterestandregionBconsistingoftherestofthesystem.Thedensityoperatorofthesystemisgivenby^=j ih j,wherej irepresentsthegroundstatewave-functionofthesystem.Thereduceddensityoperator^AinregionAcanbeconstructedbytracingoverthesystem'sdensityoperator^overregionB,i.e.,^A=TrB^.SimilarlyonecantraceoverregionAtoobtainreduceddensityoperator^B=TrA^forregionB.TheentanglementbetweenregionsAandBisdescribedbythevonNeumannentropy Se(AjB)=)]TJ /F1 11.955 Tf 9.3 0 Td[(TrA(^Aln^A))]TJ /F1 11.955 Tf 21.92 0 Td[(TrB(^Bln^B):(1{24)Theentanglemententropyiszeroiftheregionsareunentangledandmaximumforamaximallyentangledconguration.Theentanglemententropycanquantifythequantummechanicalcorrelationsintheground-statewavefunctionofasystem.Theunderstandingoftheentanglemententropyisbelievedtoplayakeyroleinthedevelopmentofnewapplicationsinquantumcomputationandquantuminformation[ 89 ].Theentanglemententropycanalsobeusedtodescribenontrivialtopologicalorderinsituationswherealocalorderparameterisinadequate[ 90 { 94 ].Anumberofstudieshaveestablishedthattheentanglemententropyexhibitsnontrivialscalinginthevicinityofquantumphasetransitions[ 95 { 97 ]. 38

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Inthisregard,quantumimpurityproblemspresentawidearrayofsystemswheretheentanglemententropycanbeaccuratelydeterminedusinganumberofwell-establishedtechniques,anabilitythatcanhelpusunderstandtheinterplaybetweenquantumcriticalityandtheentanglemententropy.Forexample,considertheexampleofthespin-bosonmodelthatdescribesatwolevelsystem,suchasaspin-1=2impurity,coupledtoadissipativebathofharmonicoscillators[ 98 ].Theentanglemententropyinthespin-bosonmodelhasbeenstudiedusingconformaleldtheory[ 99 ],thedensity-matrixrenormalizationgroup[ 99 ],andthenumericalrenormalization-group(NRG)[ 100 ].NRGstudiesofthesub-ohmicspin-bosonmodelhaveestablishedthattheentanglemententropySeisenhancedandexhibitsacusp-likepeakatthesecond-orderphasetransitionthatseparatesthelocalizedanddelocalizedphasesofthemodel[ 100 { 102 ].Astudyoftheone-dimensionalXYmodelalsoshowsthattheentanglementbetweentwoadjacentspinsdisplaysapeakatthethecriticalpoint[ 103 , 104 ].However,duetothelimitednumberofmodelswheretheentanglementhasbeeninvestigated,whetherornotthecusp-likepeakintheSeattheQCPisagenericfeatureofsecond-orderquantumphasetransitionsremainsanopenandintriguingquestion.AsdiscussedinSec. 1.1 ,theKondoandAndersonmodelswereoriginallyusedtounderstandthemany-bodyscreeningoflocalizedmagneticmomentsduetothepresenceofdilutemagneticimpuritiesinnonmagneticmetals.Recently,theKondoeecthasbeeninvokedtounderstandavarietyoftopicsincludingthephysicsofheavy-fermionssystems,transportthroughquantumdotdevices.ThepseudogapvariantofKondoandAndersonmodelshavebeenusedtostudyquantumcriticalityduetothepresenceofmagneticimpuritiesinavarietyofsubstanceslikeunconventionalhigh-Tcsuperconductors[ 105 ]andingraphene[ 105 ].Recentstudieshavealsoshownthattuningadouble-quantum-dotsystemcanproduceapseudogapintheeectivedensityofstates[ 106 , 107 ].Althoughthesemodelshavebeenstudiedextensivelyinthepast,therehasbeennoinvestigationontheentanglementpropertiesofthesemodels.IaimtollthisgapinChapter 5 of 39

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thisdissertation.IhavestudiedtheentanglemententropyofthreevariantsoftheKondomodelthatexhibitcriticaldestructionoftheKondoeect,whereKondoscreeningissuppressedatasecond-orderquantumphasearisingduethepresenceofapseudogapintheconduction-banddensityofstatesaroundtheFermienergy[ 24 { 26 , 85 , 87 , 88 , 108 { 110 ]. 40

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CHAPTER2NUMERICALSOLUTIONTECHNIQUESThemainchallengeinsolvingquantumimpurityproblemssuchastheKondoandAndersonmodelsisthewiderangeofenergiesfromthehigh-frequencycutoscale(whichcanbeoftheorderofseveraleV)toscalesarbitrarilyclosetozero.Thereisnopredominantenergyscaleintheproblemandthecontributionfromallenergyscalesgivesrisetologarithmicdivergences.ThatiswhyanyperurbativetreatmentoftheKondoproblembreaksdownatacharacteristictemperatureknownastheKondotemperatureTK.InspiredbythescalingequationsdiscussedinSection 1.2 ,wheretheparametersoftheproblemarerenormalized,K.G.WilsonrstintroducedanonperturbativenumericalapproachtosolvetheKondoproblemessentiallyexactly[ 10 ].Thenumericalrenormalization-group(NRG)techniquecanbypassthelogarithmicdivergencesthatarisefromaperturbativeapproachandreliablydescribestheentirecrossoverfromthefreemagneticmomentregimeathightemperaturestothestronglycoupledKondosingletformationbelowTK.Inthischapter,thebasicprinciplesoftheNRGareillustratedusingasanexamplethestandardone-channelKondomodeldescribedbytheHamiltonian HK=Xk;"kcykck+J 2NkSXk;k0;;0cyk0ck00:(2{1)Forsimplicity,theconductionbandistakentobeisotropicink,i.e."k="jkj.Asaresult,theimpuritycouplesonlytos-wavestates(onesthataresphericallysymmetricabouttheimpuritysite).Inthes-wavesector,onecanconvertthesumoverkintoanintegralovertheenergies.Itisconvenienttoworkwithadimensionlessenergy="=DwhereDisthehalf-bandwidthoftheconductionband.TheHamiltonianisthustransformedintothefollowingone-dimensionalform: HK=DXZ1)]TJ /F4 7.97 Tf 6.59 0 Td[(1dcyc+F2JSX0fy0;0f00;(2{2) 41

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Figure2-1. Discretizationoftheconductionbandintologarithmicbinsfortheidealizedcaseofadensityofstatesthatisconstantovertherescaledenergiesjj<1. wheretheoperatorscsatisfytheanticommutationrelations fc;;cy0;0g=()]TJ /F3 11.955 Tf 11.95 0 Td[(0);0:(2{3)Theimpuritycouplestoonlyauniquecombinationoftheconductionstatesgivenby f0=1 FZ1)]TJ /F4 7.97 Tf 6.59 0 Td[(1dw()c;(2{4)where F2=Z1)]TJ /F4 7.97 Tf 6.58 0 Td[(1dw2():(2{5)Here,theweightingfunctionw()/p (D)where(")=N)]TJ /F4 7.97 Tf 6.59 0 Td[(1kPk(")]TJ /F3 11.955 Tf 12.84 0 Td[("k)istheconductionbanddensityofstates.TheNRGmethodwasrstdevelopedforthecaseofaatdensityofstates,forwhichonechoosesw()=(1)-249(jj).However,themethodwassubsequentlygeneralized[ 25 , 87 , 111 ]toanarbitrarydensityofstates.Forthepseudogapdensityofstates()/jjr,onecanchoosetheweightingfunctiontobe w()=jjr=2(1)-222(jj):(2{6)ThebasicstepsoftheNRGprocedureareasfollows[ 10 , 112 { 114 ]: 42

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(a)Dividetheconductionbandenergiesintoasetoflogarithmicintervals.(b)Replacethecontinuousspectrumbyadiscreetsetofstates.(c)MaptheconductionbandpartoftheHamiltonianontoatight-bindingsemi-innitechain.(d)IterativelydiagonalizetheresultingapproximationtothefullKondomodel.Thesestepswillbediscussedinturnbelow. 2.1LogarithmicDiscretizationoftheConductionBandInSec. 1.1 ,wesawthatperturbationtheoryfailstoreliablytreattheT!0groundstateofthethes-d(orKondo)modelduetothepresenceoflogarithmicdivergences.Totacklethisproblem,Wilsonproposedadivisionoftheconductionbandintologarithmicintervalsorbinssuchthatscatteringfromeachsubintervalcontributesequallytothelogarithmicdivergencesfoundusingperturbationtheory.Fig. 2-1 showsthedivisionoftheconductionbandintologarithicbinsforbothpositiveandnegativevaluesoftheenergy.Themthpositivebincoverstheenergies+m+1<<+mwhere0=1and+m=)]TJ /F5 7.97 Tf 6.59 0 Td[(mform>0.Thecorrespondingnegativebincoverstheenergies)]TJ /F5 7.97 Tf 0 -7.3 Td[(m<<)]TJ /F5 7.97 Tf 0 -7.88 Td[(m+1where)]TJ /F4 7.97 Tf 0 -7.88 Td[(0=)]TJ /F1 11.955 Tf 9.3 0 Td[(1and)]TJ /F5 7.97 Tf 0 -7.29 Td[(m=)]TJ /F1 11.955 Tf 9.3 0 Td[()]TJ /F5 7.97 Tf 6.58 0 Td[(mform>0.Here>1istheWilsondiscretizationparameterandthecontinuumlimitisobtainedifonetakes!1.Next,withineachpositive[negative]bin,onecandeneacompletesetofannihilationoperatorsa(q)m[b(q)m]andacorrespondingsetoforthonormalfunctions (q)am()[ (q)bm()]sothatwecanwrite c=1Xm=01Xq=[ (q)am()a(q)m+ (q)bm()b(q)m](2{7)andtheconductionbandpartoftheHamiltoniancanbewrittenas Hc=DXm;q;q0XZ1)]TJ /F4 7.97 Tf 6.59 0 Td[(1d[ (q)am() (q0)am()a(q)yma(q0)m+ (q)bm() (q0)bm()b(q)ymb(q0)m]:(2{8) 43

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Onechoosestheq=0functionwithineachbintobethenormalizedlinearcombinationofbinstatesthatcoupletotheimpurity: (0)am()=8><>:w()=Famfor+m+1<<+m0otherwise; (2{9) (0)bm()=8><>:w()=Fbmfor)]TJ /F5 7.97 Tf 0 -7.3 Td[(m<<)]TJ /F5 7.97 Tf 0 -7.88 Td[(m+10otherwise; (2{10)whereF2am=Z+m+m+1dw2()andF2bm=Z)]TJ /F16 5.978 Tf 0 -6.12 Td[(m+1)]TJ /F16 5.978 Tf 0 -4.84 Td[(mdw2(): (2{11)Theq6=0functionsneednotbeexplicitlyspecied.Wilsonshowedthatthesefunctionsdecoupleentirelyfromtheq=0modesinthecontinuumlimit!1.Forvaluesof>1,neglectoftheq6=0modesprovidesacontrolledapproximationtothefullproblem.HenceforthIconsideronlyq=0anddropthesuperscriptq.TheconductionbandHamiltonianbecomes: HcD1Xm=0X(amaymam+bmbymbm);(2{12)where am=1 F2amZ+m+m+1dw2()andbm=1 F2bmZ)]TJ /F16 5.978 Tf 0 -6.12 Td[(m+1)]TJ /F16 5.978 Tf 0 -4.84 Td[(mdw2():(2{13)Thuswehavereducedthecontinuumofconductionstatesintheenergyrange)]TJ /F1 11.955 Tf 9.3 0 Td[(1<<1toadiscretesetofstatesofenergies)]TJ /F5 7.97 Tf 6.58 0 Td[(n.Theimpuritycouplesonlyto f0=1Xm=0(u0mam+v0mbm);(2{14)where u0m=Fam Fandv0m=Fbm F:(2{15) 44

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Figure2-2. ConductionbandoftheKondomodelmappedtoasemi-innitetight-bindingchain.Theimpuritycouplesonlytotheendsite0ofthechain. 2.2TridiagonalizationoftheConduction-BandHamiltonianGivenEqs.( 2{12 )and( 2{14 ),onecanusetheLanczosprocedure[ 115 ]todeneasetofannihilationoperators fn=1Xm=0X(unmam+vnmbm)forn=1;2;3;:::::(2{16)wherefnannihilatesanelectronwithspinzcomponentatsitenofasemi-innitechain[showninFig. 2-2 ],suchthattheconductionbandHamiltonianisrepresentedasatight-bindingHamiltonianwithonlynearest-neighborhopping:Hc=D1Xn=0X)]TJ /F5 7.97 Tf 6.59 0 Td[(n=2[enfynfn+tn(fynfn)]TJ /F4 7.97 Tf 6.58 0 Td[(1;+H.c)] (2{17)where ()=1 2(1+)]TJ /F4 7.97 Tf 6.59 0 Td[(1)3=2:(2{18)Thecoecientsenandtncanbefoundrecursivelystartingwitht0=0.Forasymmetricdensityofstateswherew(")=w()]TJ /F3 11.955 Tf 9.3 0 Td[("),en=0forallvaluesofn.Forn1,tnapproaches1,meaningthatthehoppingcoecient)]TJ /F5 7.97 Tf 6.59 0 Td[(n=2tndecaysexponentiallyalongthechain. 45

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2.3IterativeSolutionoftheDiscretizedProblemTheKondoHamiltoniancanbewrittenasthelimitofaseriesofHamiltoniansgivenby HK=limN!1)]TJ /F5 7.97 Tf 6.58 0 Td[(N=2DHN;(2{19)whereHN,describingtheimpuritycoupledtoachainofnearestneighborsites,canbedenedforallN>0viatherecursionrelation HN=1=2HN)]TJ /F4 7.97 Tf 6.59 0 Td[(1+X[eNfyNfN+tN(fyNfN)]TJ /F4 7.97 Tf 6.59 0 Td[(1;+H.c)]:(2{20)H0describestheatomiclimitoftheHamiltoniangivenby H0=Xe0fy0f0+X;0eJSfy01 20f00;(2{21)where eJ=F20J=(2{22)isadimensionlessscaledexchangecoupling.TheseriesofHamiltoniansHNcannowbediagonalizediterativelystartingwithH0.Ateachsubsequentiteration,themany-bodybasisstatescanbecreatedfromtheeigenstatesofthepreviousiterationbytakingadirectproductwiththedegreesoffreedomoftheaddedsite.Afterafewiterations,thenumberofbasisstatesintheHilbertspaceforHNbecomestoolargeandnumericallycumbersome,sotheHilbertspaceistruncatedtoaxednumberoflow-energystatesaftereachiteration.Oncetheeigenvaluesandeigenvectorsareknownforeachiteration,otherphysicalquantitiessuchasthemagneticsusceptibilityandentropycanbecalculated.Sincetnrapidlyapproaches1withincreasingn,theHamiltoniantermsinvolvingtheendsiteofthechainalwaysdescriberescaledenergiesoforder1.ForlargeN,theeigensolutionofHNapproachesalimitwheretheeigenvaluesbecomeequivalenttothoseofiterationN)]TJ /F1 11.955 Tf 13.12 0 Td[(2.Thislimit,inwhich 46

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\HN=HN)]TJ /F4 7.97 Tf 6.58 0 Td[(2"denesanNRGxedpointanalogoustothescalingxedpointsdescribedbyd~J=dl=0.Theprocedureofdiagonalizationcanbemademoreecientbytakingadvantageofsymmetriespresentinthemodel.FortheKondomodel,theHamiltoniancommuteswiththetotalspinoperator SN=1 2NXn=0X;0fyn0fn0+S(2{23)andthetotalchargeoperatormeasuredfromhalflling QN=NXn=0Xfynfn)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2:(2{24)Atparticle-holesymmetry,HNalsocommuteswithallthethreecomponentsofthe\isospin"or\axialcharge"operatordenedas IzN=1 2QN;I+N=NXn=0()]TJ /F1 11.955 Tf 9.3 0 Td[(1)nfyn"fyn#;andI)]TJ /F5 7.97 Tf -.94 -8.19 Td[(N=(I+N)y:(2{25)HNcanbediagonalizedindependentlyinsubspaceslabeledbydierentvaluesofthequantumnumberssuchasSz,S,Q,and(incaseofparticle-holesymmetry)IsuchthateachstateisaneigenstateofI2z+(I+I)]TJ /F1 11.955 Tf 10.14 -4.33 Td[(+I)]TJ /F3 11.955 Tf 7.08 -4.33 Td[(I+)=2.Moreover,sincetheeigenvaluesareindependentofSzandQ,onecanworkwithareducedbasisconsistingonlyofstateswithSz=SandQ=2I[ 25 ]. 2.4NRGTreatmentoftheAndersonModelTheNRGcanalsobeusedtonumericallysolvetheAndersonmodelfollowingthesamebasicprocedurediscussedinthischapter.TheonlydierencebetweentheKondoandtheAndersonmodelsliesintheatomiclimitH0oftheHamiltonian.FortheAndersonmodel[refertoEq.( 1{6 )],H0isdescribedby: H0=Xe0fy0f0+e"dnd+eUnd"nd#+Xe)]TJ /F4 7.97 Tf 7.32 4.94 Td[(1=2(fy0d+H.c.)(2{26) 47

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Figure2-3. Forthetwo-channelKondomodel,thediscretizedproblemcanbemappedtotwosemi-innitetightbindingchainswithnearest-neighborhopping,eachcoupledatitsendsitetothemagneticimpurity. wheree"d,eUande)-326(aredimensionlessscaledvaluesofthemodelHamiltonianparametersgivenby e"d="d D;eU=U D;ande)-277(=F20jVj2 2D:(2{27) 2.5NRGTreatementofOtherModelsApotentialscatteringtermthataccountsfornon-magneticscatteringoftheconductionelectronsfromtheimpurity[describedbythesecondtermoftheHamiltonianinEq.( 1{13 )]canalsobeincludedeasilyintheNRGscheme.Sincetheimpurityinteractswiththerstsiteofthetight-bindingchain,wejustneedtomodifytheatomiclimit[Eq.( 2{21 )]oftheHamiltoniantoincludeatermeVPfy0f0witheV=F20V=.TheNRGcanalsobeextendedtostudythetwo-channelKondomodel.Inthiscase,thetwochannelscanbediscretizedindependentlyandmappedontotwotight-bindingchainswithnearest-neighborhoppingwithsiteoperatorsfnwhere=1;2and=";#.Theimpurityiscoupledtotheendsite(n=0)ofeachchainasshowninFig. 2-3 .AseriesofHamiltoniansHNcanbediagonalizediterativelyfollowingthesameprocedureasdiscussedinSec. 2.3 withthemodicationthatateachsuccessiveiteration,wehavetoaddthedegreesoffreedomoftwonewsites,correspondingtothetwochannels.This 48

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meansthateacheigenstateretainedfromiterationNcreates16basisstatesatiterationN+1(comparedto4suchstatesintheone-channelcase).Thisimpliesthataftertherstfewiterations,theNRGkeepsonly1=16oftheeigenstatesattheendofeachiterationaftertruncatingthehigh-energystates.Thusoneneedstoretainmoreeigenstatesaftereachiterationtoachievenumericalaccuracycomparabletothatofthesingle-channelcase.Asaresult,theNRGcalculationsarecomputationallymuchmorecostlythaninthesingle-channelcase. 49

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CHAPTER3QUANTUMCRITICALITYINTHETWO-CHANNELPSEUDOGAPKONDOMODELThischapterfocusesonthequantumphasetransitionsinthetwo-channelpseudogapKondomodelthatwasintroducedinSec. 1.3.3 .ThemotivationforthisstudyhasbeendiscussedinSec. 1.4 .Thechapterisarrangedasfollows.InSec. 3.1 ,Ireviewthetwo-channelpseudogapKondoHamiltonianandthephasediagramofthemodel.NewresultsobtainedforthemodelarediscussedinSecs. 3.2 and 3.3 .Someoftheseresultshavebeenpublishedin[ 110 ]. 3.1BackgroundIhavemainlyusedthetwo-channelpseudogapKondomodelgeneralizedtoincludepotentialscattering.TheHamiltoniancanbewrittenasHK=Hc+J 2NkSXk;k0;;;0cyk0ck00+V NkXk;k0;;cykck0; (3{1)whereHc=Xk;;"kcykck: (3{2)Here,ckdestroysanelectroninchannel(=1or2)withspin(="or#)andNkisthenumberofunitcellsinthehost.InadditiontotheantiferromagneticexchangecouplingofstrengthJbetweentheconductionbandelectronsandanimpurityspinS=1 2,themodelincludesforapotentialscatteringofelectronsatthesiteoftheimpuritywithscatteringstrengthV.Thephasediagramforthetwo-channelpseudogapKondomodelhasbeenwellestablishedbypreviousstudiesofthemodels[ 25 , 108 ].InadditiontootherparametersintheHamiltonian,thephasediagramandcriticalbehaviormodeldependcruciallyonthebandexponentr.Theexponentsr=0,r=rmax0:23andr=1playtheroleof\criticaldimensions"andthephasediagramchangesqualitativelyatthesevalues[ 25 ].Figure 3.1 showsschematicRGowdiagramsforthetwo-channelpseudogapKondomodelonthe 50

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Figure3-1. SchematicRGowdiagramsofthetwo-channelpseudogapKondomodelontheJ)]TJ /F3 11.955 Tf 11.96 0 Td[(VplaneforA)r=0,B)0
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J)]TJ /F3 11.955 Tf 12.08 0 Td[(Vplane.Sub-guresA),B),andC)areforsituationswherer=0,00,thedepletionofthedensityofstatesneartheFermienergycausestheimpuritytobeeectivelydecoupledfromtheconductionbandsasT!0forsmallvaluesoftheKondocouplingJ,resultinginastablelocal-moment(LM)xedpoint.ThelineofNFLxedpointsfoundforr=0collapsestoasingleparticle-holesymmetricxedpointlabeledNFL.TheNFLandLMxedpointsareseparatedbyaparticle-holesymmetricquantumcriticalpoint(SCR)[ 25 ].Asecondnon-Fermiliquidxedpoint(NFL')canbeaccessedforlargevaluesofJandV.NFL'correspondstoaoverscreenedavordegreeoffreedomoftheimpurity[ 108 ].FlavorisanSU(2)quantitywiththefollowinggenerators:Jzf=1 2Xk;(cyk1ck1)]TJ /F3 11.955 Tf 11.96 0 Td[(cyk2ck2); (3{3)J+f=Xk;cyk1ck2; (3{4)J)]TJ /F5 7.97 Tf -1.11 -8.28 Td[(f=(f+)y: (3{5)ThezcomponentofavorissimplyJzf=(n1)]TJ /F3 11.955 Tf 12.37 0 Td[(n2)=2,wheren1andn2aretheelectronoccupanciesinconductionchannels1and2respectively.TheNFLandNFL'phasesareseparatedbyasecondparticle-holeasymmetricquantumcriticalpoint(ACR).Itshouldbenotedthatasr!0(r!rmax),SCR!LM(SCR!NFL).Ifrisfurtherincreasedbeyondrmax0:23,boththeNFLandNFL'xedpointsvanishintherangermax
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Figure3-2. TemperaturedependenceoftheimpuritycontributiontoA)theentropy(Simp)andB)temperaturetimesthemagneticsusceptibility(Timp)forthetwo-channelpseudogapKondomodelthatconverges,asT!0,tothefollowingxedpoints:LM,NFL,SCRandACRfortheexponentr=0:2,andLM'forr=0:4.NotethattheLM'phasecannotbefoundforr=0:2asitexistsintheparameterrangermax
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3.2.1ThermodynamicPropertiesUsingtheNRG,Ihavereproducedallthepreviouslyreported[ 25 , 108 ]phasesofthetwo-channelpseudogapKondomodel.Thesephasescanbedistinguished,forexamplebytheirthermodynamicproperties.Atanytemperature,theimpuritycontributiontoathermodynamicquantitycanbefoundfromitstotalthermalexpectationvaluebysubtractingothecontributionfromtheconductionbandsonly,thatis,intheabsenceoftheimpurity.Theimpuritycontributiontotheentropy(Simp)andmagneticsusceptibility(imp)attheSCR,ACR,andNFLxedpointsforthetwo-channelKondomodelwereinvestigatedfortherange0
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Figure3-3. SchematicplotofTvsdistancefromthecriticalpoint.Here,Tisthecharacteristictemperatureforthecrossoverfromthequantumcriticalregime(T&T)tooneorotherofthelow-temperatureregimes(T.T). 3.2.2StaticCriticalPropertiesPreviousstudieshaveshownthatthecriticalbehaviorofpseudogapmagneticimpuritymodelsisrevealednotintheresponsetoanuniformmagneticeld,butinsteadbyasmallmagneticeldhthatactsattheimpuritysiteonly[ 26 , 88 ].Theresponsetothislocalmagneticeldisdemonstratedbythelocalmagnetization Mloc=hSzimpi=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(@Fimp @h(3{7)andthelocalsusceptibility loc=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(@2Fimp @h2h=0:(3{8)Mloc(T=0)vanishesthroughoutthestrong-couplingphaseandisnonzerointheweak-couplingphase,sothisquantitycanactasanorderparameterforthequantumphasetransition. 55

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3.2.3CriticalExponentsandScalingRelationsThephasetransitionsinthepseudogaptwo-channelKondomodels1canbecharacterizedbyasetofcriticalexponents,,x,andasdenedbelow:Mloc(<0;T=0;h=0+)/()]TJ /F1 11.955 Tf 9.3 0 Td[(); (3{9)Mloc(=0;T=0)/jhj1=; (3{10)loc(=0)/T)]TJ /F5 7.97 Tf 6.59 0 Td[(x; (3{11)T/jj: (3{12)Hereisthedistanceofthecouplingfromthephaseboundarywith<0(>0)denotingtheorderedphase(disorderedphase).Tisthetemperatureatwhichthesystemcrossesoverfromthehigh-temperaturequantum-criticalregimetooneofthe(stable)low-temperatureregimes,asshownschematicallyinFig. 3-3 .Inpractice,TcanbeevaluatedbyexaminingthetemperaturedependenceofathermodynamicquantitysuchasTimp.Thiscanbeillustratedusing,asanexample,theSCRcriticalpointatcriticalcouplingJ=JcseparatingtheLMphase(forJJc)intherange0Jc.Tcanbefoundbyextractingthetemperature(usinglinearinterpolation)atwhichhTimpi 1Thequantumphasetransitionsintheone-channelpseudogapKondomodelcanalsobecharacterizedbythesamesetofexponents,,x,and.However,sinceeachQPTseparatesaLM(ordered)phasefromastrong-coupling(disordered)phase,thelocalsusceptibilitydivergesontheapproachtothequantumcriticalpointfromthedisorderedsideasloc(>0)/)]TJ /F5 7.97 Tf 6.59 0 Td[([ 26 ]. 56

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crossesasuitablyselectedcutovaluechosenoneithersideofthevalueofhTimpiattheSCRcriticalpoint.Continuousphasetransitionsaredescribedbymean-eldexponentsaboveanuppercriticaldimension,butbyanomalousexponentsreectinginterparticleinteractionsifthesystemisbelowtheuppercriticaldimension.Inthelattercase,thecriticaluctuationsarerelevantintheRGsense,thecriticalpointisinteractinginnatureandoneexpectsthesingularcomponentoftheimpurityfreeenergy(Fcritimp)toexhibitcertainscalingproperties[ 61 ]nearthequantumcriticalpoint.InthevicinityoftheSCRandACRcriticalpoints,therelevantscalesaretheappliedmagneticeldhandthedistancefromthecriticalpoint.Fcritimpisthereforeexpectedtobeoftheform[ 26 ] Fcritimp=Tf Ta;jhj Tb:(3{13)UsingEqs.( 3{9 )-( 3{12 )and( 3{13 ),itcanbereadilyshownthat=1)]TJ /F3 11.955 Tf 11.96 0 Td[(b a;=b 1)]TJ /F3 11.955 Tf 11.96 0 Td[(b;x=2b)]TJ /F1 11.955 Tf 11.95 0 Td[(1;and=1 a: (3{14)SincetherewerefourdenedcriticalexponentsbutonlytwoindependentexponentsinEq.( 3{13 ),theexponentsarepredictedtoobeyapairofscalingrelations:=1+x 1)]TJ /F3 11.955 Tf 11.95 0 Td[(xand=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x) 2: (3{15) 3.2.4CriticalBehaviorSCR(0
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Figure3-4. ExponentxfortheSCR,ACRandNFLxedpointsofthetwo-channelpseudogapKondomodel.Asr!0,theresultsforxattheSCRxedpointbecomeconsistentwiththepredictionsofaeld-theoreticalleading-orderexpansion(dashedline)in[ 108 ].r=rmaxisshownusingaverticaldottedline. ACR(rmax
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Figure3-5. Exponent1=fortheSCR,ACRandNFLxedpointsofthetwo-channelpseudogapKondomodelplottedagainstthebandexponentr.TheshadedgrayregionindicatestherangewithinwhichtheexponentispredictedtoliewhenvaluesofxfromTables 3-1 , 3-2 ,and 3-3 areinsertedintoEq.( 3{15 ).r=rmaxisshownusingaverticaldottedline.Theestimatednon-systematicerrorfortheexponentsfoundusingNRGarenegligiblewithrespecttothesizeofthesymbols. electronscattered,onecandenealocal\avormagnetization"asfloc=n0=n0;1)]TJ /F3 11.955 Tf 11.99 0 Td[(n0;2,wheren0;1andn0;2aretheelectronoccupanciesofthezerothWilsonsiteofchannels1and2,respectively.ThisquantityisexpectedtobezerointheLMphaseandnon-zerointheLM'phaseduetothepresenceofafreeavormoment.Inanalogywiththelocalmagneticeld,onecanalsodeneaavoreldgivenbyhf=V=V1)]TJ /F3 11.955 Tf 12.02 0 Td[(V2,whereV1andV2arethepotentialscatteringstrengthoftheelectronsfromchannel1and2,respectively.IproposethatinadditiontothecriticalexponentsdenedinEqs.( 3{9 ),thecriticalavor 59

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Figure3-6. Order-parameterexponentfortheSCRandACRcriticalpointsofthetwo-channelKondomodelplottedagainstthebandexponentr.TheshadedgrayregionindicatestherangewithinwhichtheexponentispredictedtoliewhenvaluesofxandfromTables 3-1 and 3-2 areinsertedintoEq.( 3{15 ).r=rmaxisshownusingaverticaldottedline.Theestimatednon-systematicerrorfortheexponentsfoundusingNRGarenegligiblewithrespecttothesizeofthesymbols. responsecanbecharacterizedbyadierentsetofcriticalexponentsdenedbyfloc(>0;T=0;hf=0+)/()f; (3{16)floc(=0)/T)]TJ /F5 7.97 Tf 6.58 0 Td[(xf; (3{17)where>0representsthemagneticallydisorderedphase.flocisthestaticavorsusceptibilitygivenbyfloc=@floc=@hfjhf=0. 60

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Table3-1. CriticalexponentsattheSCRcriticalpointofthetwo-channelpseudogapKondomodel.Numbersinparenthesesindicateestimatederrorsinthelastdigit. rxy -0.2010.85(3)0.8315(3)0.09204(4)0.905(6)--0.1011.90(5)0.9740(5)0.013078(7)0.148(3)-0.0521.39(3)0.9943(6)0.002815(2)0.050(1)0.9943858(2)0.1011.90(4)0.9742(6)0.013072(8)0.150(2)0.9741906(6)0.159.50(5)0.9301(4)0.03624(2)0.330(1)0.930048(2)0.2010.85(4)0.8315(4)0.09203(4)0.908(3)0.831440(6) Table3-2. CriticalexponentsattheACRcriticalpointofthetwo-channelpseudogapKondomodel.Numbersinparenthesesindicateestimatederrorsinthelastdigit. rxxffy 0.10-0.0376(4)0.0372(5)0.912(7)--0.036885(2)0.15-0.0791(3)0.0788(2)0.849(2)--0.07887(1)0.20-0.1319(2)0.1316(2)0.767(1)--0.13177(7)0.404.14(2)0.3815(2)0.3815(4)0.4477(4)1.280(2)1.281(3)0.38145(8)0.601.88(2)0.6070(7)0.6070(6)0.2443(1)0.369(5)0.372(1)-0.801.26(4)0.799(2)0.801(2)0.1100(3)0.127(5)0.138(1)Itisknownfrompreviousstudiesofsimilarproblems[ 26 ]thattheresultscalculatedwithNRGdiscretizationparameteraslargeas9provideagoodapproximationtoexponentsgoverningthecontinuumlimitof!1.Fornumericaleciency,allthecalculationsweredoneusing=9andretaining2,000many-bodystatesaftereachNRGiterationunlessstatedotherwise.Thecriticalpropertiesforxandshowpower-lawbehavioroveratleastsixdecadesoftemperatureorlocaleld.However,itisnumericallymorechallengingtoextracttheexponentsandf,whichrequirethe Table3-3. ExponentsatthestableNFLxedpointofthetwo-channelpseudogapKondomodel.Numbersinparenthesesindicateestimatederrorsinthelastdigit. rxy -0.200.3911(2)0.4376(4)--0.100.09387(5)0.827(3)-0.050.026(1)0.932(10)0.024689(1)0.100.09385(5)0.826(2)0.093854(2)0.150.2086(2)0.6546(3)0.20858(3)0.200.3910(2)0.4380(3)0.39089(4) 61

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applicationofan\innitesimal"localmagneticeld.Thesecalculationswereperformedinquadruple-precisionoating-pointarithmeticusinglocaleldsassmallash=10)]TJ /F4 7.97 Tf 6.59 0 Td[(30.DuetotheincreasedCPUtimeforquadrupleprecision,only1,000many-bodystateswerekeptaftereachiteration.Thecriticalexponents,x,andfortheSCRandACRcriticalpointsarelistedinTables 3-1 and 3-2 ,respectively,fordierentvaluesofthebandexponentr.FortheACRcriticalpoint,theexponentsxfandfarealsolistedinTable 3-2 forrmax
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Figure3-7. Zero-temperaturedynamicalsusceptibilityloc(!)(solidlines)andstaticlocalsusceptibilityloc(T)(dashedlines)atther=0:2SCRandr=0:4ACRcriticalpointsofthetwo-channelpseudogapKondomodel. 3.2.5DynamicalCriticalPropertiesIhavealsocalculatedtheimaginarypartofthedynamicallocalspinsusceptibility00loc(=0;T=0;!)fortheSCR,ACRandNFLxedpointsofthetwo-channelKondomodel.Atlowfrequencies,thedynamicalsusceptibilitywasfoundtoobeyaform 00loc(=0;T=0;!)/sgn(!)j!j)]TJ /F5 7.97 Tf 6.59 0 Td[(y:(3{18)TheexponentyfortheSCR,ACRandNFLxedpointsislistedinTables 3-1 , 3-2 ,and 3-3 ,respectively,forvariousvaluesofthebandexponentr.Inallthecasesconsidered,y=xissatisedwithinestimatednon-systematicerror(exceptforsmallvaluesofr,wheretheRGowisextremelyslowfromACRtoNFLorNFL'xedpoints 63

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Figure3-8. SchematicRGowdiagramofthetwo-channelKondomodelinthe~J-rplanewithapower-lawdensityofstates/j"jr.Forr<0,thesolidcurverepresentsastablexedpoint~J,thedashedcurverepresentsanintermediatenon-Fermiliquidstablexedpoint~J,andthedottedcurvesrepresentunstablexedpoints~J0c(for~J<0)and~Jc(~J>0). duetoverylargevaluesofexponent,givingrisetocomputationaldiculties).ThisisillustratedinFig. 3-7 ,whichplots00loc(!;T=0)vs!(solidline)andloc(T;!=0)vsT(dashedline)atther=0:2SCRandr=0:4ACRcriticalpoints.Theequalityy=xisconsistentwithan!=Tscalingformofthedynamicspinsusceptibilitygivenby[ 26 ] 00loc(=0;!;T)=sgn(!)j!j)]TJ /F5 7.97 Tf 6.58 0 Td[(xX(!=T):(3{19)Thisshowstheabsenceofanyotherenergyscaleintheproblematcriticality,asignatureofaninteractingcriticalpoint[refertothediscussioninSec. 1.5 ][ 61 ]. 64

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3.3QuantumCriticalityforDivergingDensityofStatesSofar,IhavestudiedsituationswherethedensityofstatesvanishesexactlyattheFermienergy.AninterestingscenarioarisesifthedensityofstatesofthehostinsteadhasavanHovesingularity((")/1=p j")]TJ /F3 11.955 Tf 11.96 0 Td[("Fj)anddivergesattheFermienergy[ 117 ].AgeneralizedvanHovesingularityfeaturingadivergingdensityofstatescanbedescribedusingapower-lawdensityofstatessimilartothatusedforthepseudogapcase,givenby(")=0j"jr,butwiththeexponentr<0.OnecanusetheformalismdiscussedinChapter 2 andapplytheNRGtostudytheeectofmagneticimpuritiescoupledtofermionicbandswithdivergingdensityofstates.SincethenumberofavailableconductionelectronstatesdivergesattheFermienergy,onemightnaivelyexpectastrong-couplingbehaviorwheretheimpurityisscreenedoutforanyvalueofthecouplingJ.However,thesituationturnsouttobemoresubtle.Forr<0,theRGscalingequation[Eq.( 1{17 )]fortheNc-channelKondomodelisgivenby d~J dl=jrj~J+~J2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(Nc 2~J3:(3{20)ThexedpointsofEq.( 3{20 )(denedbyd~J=dl=0)includetrivialsolutionsat~J=0and~J=,andapairofstableintermediatecouplingsolutionsgivenby ~J=1 Nc(1p 1+2Ncjrj):(3{21)For2Ncjrj1,thesexedpointscanbeapproximatedby~Jjrjand~J2=Nc+jrj.Forasinglechannel(Nc=1),NRGcalculationshaveconrmedtheexistenceofacriticalpointat~J[ 118 ].Figure 3-8 showsaschematicRGowdiagramforthetwo-channelKondomodelwithapower-lawdensityofstatesonthe~J-rplaneasinferredfromNRGcalculations.For)]TJ /F1 11.955 Tf 9.3 0 Td[(1=4.r<0,Ihaveindeedfoundthepredicted~J(shownbysolidline)and~J(shownbydashedline)intermediatecouplingxedpoints,whichareacontinuationtor<0ofther>0SCRandNFLxedpoints,respectively.However,theNRGalsoshowsthe 65

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existenceofunstablecriticalpoints(shownbydottedlines)forlargevaluesofthecoupling~J,bothontheantiferromagnetic(~J=~Jc>0)aswellastheferromagnetic(~J=~J0c<0)side.Thesecriticalpointsarisefromstrong-couplingeectsandhencearenotpredictedbytheperturbativescalinganalysis.Theintermediate-couplingxedpointsvanishforr.)]TJ /F1 11.955 Tf 9.3 0 Td[(1=4andtheimpurityendsupbeingstrongly-coupledforanynonzerobarevalueofthecoupling(forboth~J<0and~J>0).TheexistenceofstableRGxedpointsatnegative(ferromagnetic)exchangeisadirectconsequenceofthedivergingdensityofstates.Myinvestigationindicatesthateveninthecaseofadivergingdensityofstates,thecriticalbehaviorinthevicinityoftheunstablexedpointat~J=~Jc(henceforthlabeledasSCR)canbedescribedbythesamesetofcriticalexponents,x,anddenedinEqs.( 3{9 ).Atthe~Jstablexedpoint(henceforthlabeledasNFL),thelocalmagnetizationandsusceptibilitydemonstratepower-lawdependenceonlocalmagneticeldandtemperature,respectively,justlikeinther>0case.ThecriticalexponentsfortheSCRandNFLxedpointsforr=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:1andr=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:2arelistedinTables 3-1 and 3-3 respectively.ForboththeSCRandNFLxedpoints,theexponents,x,andforr<0arefoundtobeidenticaltothosefortheSCRandNFLxedpointsofthetwo-channelpseudogapKondomodelwithbandexponentjrj>0.Thissuggeststheexistenceofamappingthattakesthenegative-rmodelwithadivergingdensityofstatestoaneectivemodelwithapositivebandexponentr.Inthepresenceofpotentialscatteringfromtheimpuritysite,thatiswhenV6=0,themodelexhibitsacomplexphasediagramconsistingofseveralstableandunstablexedpointsthatcanbemappedtotheSCRandNFLxedpointsviaspin$avortransformationand/orashiftinthetotalchargeofthemanybodygroundstate. 3.4ConclusionInthischapter,Ihavemadeathoroughinvestigationofthecriticalpropertiesofthetwo-channelKondomodelinthepresenceofapseudogapdensityofstatesthatvanishes 66

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attheFermienergy.Thecriticalspinresponseatallthequantumcriticalpointsandstablenon-Fermiliquidxedpointsofthemodelcanbecharacterizedbyasetofcriticalexponentsthatobeycertainscalingrelationsexpectedofainteractingcriticalpointbelowitsuppercriticaldimension.ThedynamicalsusceptibilitywasfoundtodivergeattheSCRandACRcriticalpointswithacriticalexponentythatisconsistentwithan!=Tscalingformofthedynamicsusceptibility.Ihavealsodenedexponentsfandxfthatcharacterizethecriticalavorresponseofthemodel.AttheACRcriticalpoint,theexponentsfandxfwerefoundtobeidenticaltoandxrespectively,conrmingthesymmetricnatureoftheACRcriticalpointunderaspin$avortransformation.IhavealsostudiedtheRGxedpointsofthetwo-channelKondomodelinthepresenceofadensityofstatesfeaturingapower-lawdivergenceattheFermienergy.Thecriticalexponentsforadivergingdensityofstateswithr<0arefoundtobeequaltothoseforthepseudogapcasewithbandexponentjrj>0.Thisworkcanbeofpotentialrelevancetomagneticimpuritiesingraphene. 67

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CHAPTER4CRITICALCHARGEFLUCTUATIONSINAPSEUDOGAPANDERSONMODELThischapterisbasedonapublishedpaper[ 119 ].AllthepublishedcontentsarereprintedwithpermissiongrantedunderthecopyrightpolicyoftheAmericanPhysicalSocietyfrom TathagataChowdhuryandKevinIngersent,Phys.Rev.B91,035118(2015) .Copyright(2015)bytheAmericanPhysicalSociety.Throughoutthischapter,\we"and\our"referstotheauthorsofRef.[ 119 ].AsdiscussedinSec. 1.5 ,therecentdiscoveryofunconventionalquantumcriticalityinmixed-valenceheavy-fermionYbAlB4haspromptedsignicantinterestintheroleofvalenceuctuationsatKondo-destructionquantumcriticalpoints(QCPs).Inthischapter,westudythecriticalchargeresponseofaparticle-holeasymmetricAndersonmodelwithapseudogapdensityofstates((")/j"jr).Thismodelfeaturesaquantumphasetransition(QPT)separatingalocal-moment(LM)phasefromastrong-coupling(SC)phase,andhasbeenusedasatoymodeltounderstandtheroleofvalenceuctuationsinaKondodestructionQCP[ 85 ].WehaveextendedthenumericalresultsprovidedinRef.[ 85 ]bydeterminingacompletesetofstaticchargecriticalexponentsfordierentvaluesofthebandexponentrontherange3=8.r<1overwhichtheasymmetricpseudogapAndersonmodelhasaninteractingQCPthatisdistinctfromthatofitssymmetriccounterpart[ 25 , 88 ].WeprovideaunieddescriptionofboththespinandchargecriticalbehaviorsintermsofanansatzfortheformofthefreeenergyneartheQCP,expressingallcriticalexponentsintermsofjusttwounderlyingexponents[ 26 , 120 ],whichcanbetermed(inthenomenclatureofclassicalphasetransitions)the\correlation-length"exponent(r)andthe\gap"exponent(r).1Theansatzleadstoscalingequationsthatareobeyedtohighaccuracybynumericallydeterminedvaluesofthechargeexponents. 1Intheliteratureonimpurityquantumphasetransitions,=isusuallywrittenas(1+x)=2or1)]TJ /F3 11.955 Tf 11.95 0 Td[(=2,where=1)]TJ /F3 11.955 Tf 11.95 0 Td[(x0istheanomalousmagneticexponent. 68

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Figure4-1. SchematicphasediagramofthepseudogapAndersonmodelonthe{"dplaneforxedUandforbandexponentsA)02.Theoutlineoftherestofthechapterisasfollows:InSec. 4.1 ,wereviewthephasediagramofthepseudogapAndersonmodelanddiscussthecriticalspinandchargeresponseofthesystemattheKondodestructionQCP.OurnumericalresultsarepresentedandinterpretedinSec. 4.2 .ImplicationsoftheseresultsforabroaderclassofquantumimpuritymodelsarediscussedinSec. 4.3 . 4.1BackgroundThisworkaddressesanAndersonmodeldescribedbyEq.( 1{6 )withapseudogapdensityofstatesdescribedbyEq.( 1{11 ).Thevaluesof0inEq.( 1{11 )andVinEq.( 1{6 )aecttheimpuritypropertiesonlyinasinglecombination,thehybridizationwidth)-277(=0V20.WehaveworkedwithunitswheregB=kB=~=1. 69

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4.1.1PhaseDiagramThephasediagramofthepseudogapAndersonmodelhasbeenwellestablishedbypreviouswork[ 25 , 87 ].Acutofthephasediagramonthe|"dplaneforaxedvalueofU>0isshownschematicallyfor00,theimpuritydegreeoffreedomiscompletelyquenchedinthelimitofabsolutetemperatureT!0,andasinglestrong-coupling(SC)phaseoccupiestheentirehalf-space(U;"d;\apartfromitsboundaryplane)-341(=0.Throughoutthisphase,theimpuritycontributionstothestaticspinsusceptibilityandtheentropysatisfylimT!0Timp=0andSimp(T=0)=0,respectively.Theground-state\charge"Q,denedtobetheexpectationvalueofthetotalelectronoccupancyofthebandandtheimpuritylevelmeasuredwithrespecttohalflling,evolvessmoothlyfrom1to)]TJ /F1 11.955 Tf 9.3 0 Td[(1as"disraisedfromto1.Forr>0,bycontrast,thereislocal-moment(LM)phasespanning)]TJ /F3 11.955 Tf 9.3 0 Td[(U<"d<0,)]TJ /F3 11.955 Tf 11.84 0 Td[(<)]TJ /F5 7.97 Tf 7.32 -1.79 Td[(c(U;"d))]TJ /F5 7.97 Tf 7.31 -1.79 Td[(c(U;)]TJ /F3 11.955 Tf 9.3 0 Td[(U)]TJ /F3 11.955 Tf 12.43 0 Td[("d)withinwhichthegroundstatecontainsanunquenchedspindegreeoffreedomcharacterizedbylimT!0Timp=1=4andSimp(T=0)=ln2.For)]TJ /F3 11.955 Tf 11.61 0 Td[(>)]TJ /F5 7.97 Tf 7.31 -1.79 Td[(c(U;"d),thesystemliesinoneofthreeSCphases.Thesymmetricstrong-coupling(SSC)phase,reachedonlyfor"d=)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2[seesolidhorizontallineinFig. 4-1 A)],haslimT!0Timp=r=8andSimp(T=0)=2rln2,suggestiveofpartialquenchingoftheimpurityspin.Theasymmetricstrong-couplingphasesASC)]TJ /F1 11.955 Tf 10.98 1.79 Td[(andASC+,reachedfor"d>)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2and"d<)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2,respectively,sharethepropertieslimT!0Timp=0andSimp(T=0)=0,indicatingcompletequenchingoftheimpuritydegreeoffreedom.Forr>0,theground-statechargetakesonlyintegervalues(incontrasttothecaser=0):Q=0intheLMandSSCphases,Q=1intheASCphase.ItshouldbenotedthattheSSCphasecanbereachedonlyfor0
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4.1.2CriticalSpinResponseOntheboundarybetweentheLMandSCphases,thethermodynamicpropertiestakevaluesdistinctfromthoseineitherphase.Forexample,limT!0Timp(T)=X(r),wherer=8)]TJ /F5 7.97 Tf 7.31 -1.79 Td[(c),andthereforeservesasanorderparameterfortheLM-SCQPT,whilethezero-temperaturelimitofsdivergesonapproachtotheQPTfromtheSCsideandisinnitethroughouttheLMphase.Forr>1,Mlocisdiscontinuousacrossthephaseboundaries,meaningthattheQPTsarerst-order.For00)/g)]TJ /F5 7.97 Tf 6.59 0 Td[(; (4{3c)s(g=0)/T)]TJ /F5 7.97 Tf 6.58 0 Td[(x; (4{3d) 71

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whereonecandenethenonmagneticdistancetocriticalitytobeg=)]TJ /F7 11.955 Tf 26.99 0 Td[()]TJ /F1 11.955 Tf 13.27 0 Td[()]TJ /F4 7.97 Tf 7.32 -1.79 Td[(0atxedU0and"d0or,alternatively,g=U0)]TJ /F3 11.955 Tf 12.6 0 Td[(Uatxed)]TJ /F4 7.97 Tf 49.91 -1.8 Td[(0and"d0,whereineithercase)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(0=)]TJ /F5 7.97 Tf 19.74 -1.79 Td[(c(U0;"d0).Onecanalsodeneacorrelation-lengthexponentviatherelation T/jgj;(4{4)whereTisatemperaturecharacterizingthecrossoverfromthequantum-criticalregime(s/T)]TJ /F5 7.97 Tf 6.59 0 Td[(xforTT)toeithertheLMphase(s/T)]TJ /F4 7.97 Tf 6.59 0 Td[(1forg<0,TT)oroneoftheSCphases(s'constforg>0,TT).Eachofthecriticalexponents,,,x,andhasanontrivialdependence[ 26 ]onthebandexponentr.For0
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belowitsuppercriticaldimension,impliesthat =)]TJ /F1 11.955 Tf 11.96 0 Td[(; (4{7a)=2)]TJ /F3 11.955 Tf 11.96 0 Td[(; (4{7b)1===)]TJ /F1 11.955 Tf 11.96 0 Td[(1; (4{7c)x=2=)]TJ /F1 11.955 Tf 11.96 0 Td[(1: (4{7d)EliminationoffromEqs.( 4{7 )yieldsEqs.( 4{5 ). 4.1.3CriticalChargeResponseReference[ 85 ]investigatedlocalchargeuctuationsinthevicinityoftheKondo-destructionQPTsinthepseudogapAndersonmodel.Thelocalchargeresponseisthevariationoftheimpuritycharge^nd,whichenterstheHamiltonianwithcoupling"d,soitisnaturaltodenealocalchargesusceptibility c=)]TJ /F3 11.955 Tf 9.29 0 Td[(@h^ndi=@"dj"d="d0;(4{8)nearapoint(U0;"d0;)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(0)onthephaseboundary.ItwasreportedinRef.[ 85 ]thatcremainsniteonpassagethroughtheparticle-hole-symmetricQCPsthatoccurfor00. 73

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4.2ResultsandInterpretationWehavesystematicallyextendedtheresultsofRef.[ 85 ]throughstudyoftheparticle-hole-asymmetricpseudogapAndersonmodelwithdierentvaluesofthebandexponentrwithintheranger
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Figure4-2. jQlocjvsdistancefromthephaseboundaryalongthe)-327(and"daxesforr=0:6.Filled(hollow)symbolsrepresentpointsintheLM(ASC)]TJ /F1 11.955 Tf 7.08 1.8 Td[()phase.Thelinearvariationsonthislog-logplotindicatepower-lawbehaviorinaccordancewithEqs.( 4{11 ). benotedthatthesepowerlawsrevealthemselvesinbothphases(unlikethepower-lawvariationofMloc,whichoccursonlyontheLMsideoftheQCP).Theparalleltrendsofthedataonthislog-logplotsuggeststhatvariationofQlocwithrespectto)-326(andwithrespectto"disgovernedbyacommoncriticalexponent~,i.e., jQloc("d="d0)j/jgj~; (4{11a)jQloc(g=0)j/j"d)]TJ /F3 11.955 Tf 11.96 0 Td[("d0j~: (4{11b) 75

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Table4-1. Chargecriticalexponents~,~,and~x,pluscorrelation-lengthexponent,attheparticle-hole-asymmetricQCPsofthepseudogapAndersonmodelforbandexponentsrbetween0:4and0:9.Exponent~wasobtainedindependentlyfromtstoEqs.( 4{11a )and( 4{11b ).Parenthesesenclosetheestimatednonsystematicerrorinthelastdigit.EachchargecriticalexponentagreestowithinitsestimatederrorwiththevalueobtainedbysubstitutingintotheappropriatescalingrelationinEqs.( 4{15 ). r~( 4{11a )~( 4{11b )~~x 0.401.000(2)1.000(1)0.0000(1)4.24(4)0.501.000(3)1.000(2)0.0000(2)2.36(4)0.521.000(4)0.0000(2)2.22(3)0.540.997(4)0.000(1)2.08(3)0.560.965(6)0.021(1)1.98(4)0.580.876(5)0.0658(6)1.88(4)0.600.7910(6)0.7913(5)0.210(2)0.1164(1)1.77(4)0.700.472(4)0.474(4)0.524(4)0.3569(4)1.45(3)0.800.263(2)0.265(2)0.728(8)0.582(2)1.29(4)0.900.109(4)0.105(5)0.872(2)0.790(2)1.13(6) ThissuppositionisconrmedinFig. 4-3 A),whichplotsvaluesof~obtainedfromEqs.( 4{11a )and( 4{11b )fordierentbandexponentsovertherange0:4r0:9.Forr<0:4,itprovesverydiculttodistinguishthesymmetricandasymmetricQCPs(whichmergeatr=r'0:375),whileforr0:9powerlawstendtobecomeill-denedasthesystemnearsitsuppercriticaldimension[ 88 ]atr=1.TheotherstrikingfeatureofFig. 4-3 A)isthesharpbreakaroundr=0:55betweenthepinnedvalue~=1forr.0:55andthemonotonicdecreaseof~overtherange0:55.r<1.Thisdecreaseof~pointstoavariationoftheimpurityvalencearoundtheQCPthatbecomesmorerapidwithincreasingrandpresumablybecomesdiscontinuousforr>1.Therdependencesofthecriticalexponents~xand~characterizingthelocalchargesusceptibilityareplottedinFigs. 4-3 B)and 4-3 C),respectively.Eachoftheseexponentsispositiveforr&0:55,whileitappearstovanishforr.0:55.Table 4-1 summarizesthenumericalvaluesofthethreechargecriticalexponentsdenedinEqs.( 4{9 )and( 4{11 )andof,thecorrelation-lengthexponent.Alsolistedisanestimateofthenon-systematicerrorinthelastdecimalplaceofeachexponent. 76

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Figure4-3. ChargecriticalexponentsofthepseudogapAndersonmodelplottedvsbandexponentr:A)~obtainedindependentlyfromthevariationofQlocwithrespectto"dandwithrespectto,B)~x,andC)~.Inallcases,theestimatednonsystematicerrorissmallerthanthedatasymbol.Shadingindicatestherangewithinwhicheachexponentispredictedtoliewhenthevaluesofthecorrelation-lengthexponentfromTable 4-1 areinsertedintoEqs.( 4{15 ). 77

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Exponent~xfromEq.( 4{9b )generallyhasthesmallesterrorbecauseitcanbeobtainedfromtsoflocovermanydecadesofT.Thereisconsiderableuncertaintyinthevaluesof,whichwereobtainedbyinterpolatingfromdataatdiscretetemperaturesTthevalueTatwhichTimp(T)passesoutsideanarrowwindowsurroundingitscriticalvalueX(r).Whenallowanceismadefortheseuncertainties,Table 4-1 suggestsaninterestingrelationamongthechargecriticalexponents,namely, ~=1)]TJ /F1 11.955 Tf 12.39 0 Td[(~:(4{12)Finally,wenotethatthethresholdvalueofthebandexponentr'0:55seemstocoincidewiththepointwherethecorrelation-lengthexponentpassesthrough=2.Manyoftheempiricalobservationsnotedintheprecedingparagraphscanbeunderstoodthroughanextensionofthescalingansatzusedpreviously[ 26 ]toexplainthecriticalspinresponse.WepostulatethatthesingularcomponentoftheimpurityfreeenergytakestheformgiveninEq.( 4{6 )withageneralizeddenitionofthenonmagneticdistancefromcriticality,namely, g=(p)]TJ /F9 11.955 Tf 11.95 0 Td[(p0)^n0:(4{13)Here,^n0isthelocalunitnormaltothephaseboundaryatp0=(U0;"d0;)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(0),h=0inathree-dimensionalEuclideanspaceofnonmagneticcouplingsp=(U;"d;\thedirectionof^n0ischosensothatitpointsintotheSCphase.Thisformisassumedtoholdforjp)]TJ /F9 11.955 Tf 10.73 0 Td[(p0jmuchsmallerthantheradiiofcurvatureofthephaseboundaryatp0,inwhichcasejgjisjusttheperpendiculardistancefromptothephaseboundary.TheextendedansatzreproducesthecriticalspinresponseinEqs.( 4{3 )withexponentssatisfyingEqs.( 4{5 ),irrespectiveofwhethertheapproachtothephaseboundaryisalongtheU,"d,or)-327(axis(oralonganydirectioninbetween).Theansatz 78

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alsorecoversthecriticalchargeresponse2inEqs.( 4{9 )and( 4{11 ),aswellastwofurtherpower-lawbehaviors, Qloc(g=0)/jhj1=~; (4{14a)c(T=g=0)/jhj)]TJ /F4 7.97 Tf 7.7 2.11 Td[(~; (4{14b)withallcriticalexponentsexpressedasfunctionsofand: ~=)]TJ /F1 11.955 Tf 11.96 0 Td[(1; (4{15a)~=2)]TJ /F3 11.955 Tf 11.95 0 Td[(; (4{15b)~x=2=)]TJ /F1 11.955 Tf 11.95 0 Td[(1; (4{15c)1=~=()]TJ /F1 11.955 Tf 11.95 0 Td[(1)=; (4{15d)~=(2)]TJ /F3 11.955 Tf 11.96 0 Td[()=: (4{15e)Equations( 4{15a )and( 4{15b )notonlyconrmEqs.( 4{12 ),butalsoshowthatsincethelocal\eld""dconjugatetothelocalchargeentersthefreeenergyinthesamemannerasdoUand,thechargecriticalexponents~,~,and~xarefunctionssolelyof,unliketheirspincounterparts,,andx,whichalsodependon.Forallcasesstudiedontherange0:55.r0:9,thedirectlydeterminedexponents~,~x,and~liewithinthebounds(representedbyshadedregionsinFig. 4-3 )obtainedbyinsertingnumericalestimatesofintoEqs.( 4{15 ).Giventheratherlargeuncertaintiesin,amorerigoroustestofthescalingrelationsisprovidedbyTable 4-2 ,whichcomparesthedirectlydeterminedvalueoffor0:6r0:9withonesinferredthroughthescalingrelationsfromtheNRGvaluesof~,~x,and~.Foreachbandexponentr0:8,allvaluesofagreetowithintheirestimatednonsystematicerrors,providingstrongnumericalsupportforthevalidityofEqs.( 4{15 ).Thediscrepanciesbetweenthevariousestimates 2WithgasdenedinEq.( 4{13 ),Eq.( 4{11b )becomesaspecialcaseofEq.( 4{11a ). 79

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Table4-2. Correlation-lengthexponentattheparticle-hole-asymmetricQCPsofthepseudogapAndersonmodelforbandexponentsrbetween0:6and0:9,asobtaineddirectlyandviathescalingequations( 4{15 )fromthechargecriticalexponentslistedinTable 4-1 .Exceptforr=0:9,thevariousestimatesofforagivenrallagreetowithintheestimatednonsystematicerrorinthelastdigitofeachvalue(enclosedinparentheses). rfoundfrom direct~( 4{11a )~( 4{11b )~~x 0.61.77(4)1.7910(6)1.7913(5)1.790(2)1.7915(6)0.71.45(3)1.472(4)1.474(2)1.476(4)1.474(1)0.81.29(4)1.263(2)1.265(2)1.272(8)1.264(2)0.91.13(6)1.109(4)1.105(5)1.128(2)1.117(2) Table4-3. Exponents~and~asdetermineddirectly(\dir.")fromEqs.( 4{14 )forbandexponentsrbetween0:6and0:9.AlsolistedarevaluesofthesameexponentsinferredfromscalingequationsEqs.( 4{15d )and( 4{15e ),respectively,usingthebestestimateoffromTable 4-2 andavalueoffoundviaEq.( 4{7d )fromthetabulatedvalueofthemagneticexponentx.Exceptforr=0:9,thedirectlydeterminedandinferredexponentsagreetowithintheestimatednonsystematicerrorinthelastdigitofeachvalue(enclosedinparentheses). r1=~(dir.)1=~( 4{15d )~(dir.)~( 4{15e )x 0.60.4941(5)0.4934(2)0.1306(6)0.1300(2)0.79057(6)0.70.3517(3)0.3512(6)0.393(3)0.390(1)0.8315(1)0.80.2240(7)0.222(2)0.620(6)0.619(3)0.88021(7)0.90.130(6)0.109(3)0.80(2)0.820(4)0.928(2) offorr=0:9canbeattributedtothedicultymentionedaboveinidentifyingclearpower-lawbehaviorsforbathexponentsapproaching1.Table 4-3 lists,forbandexponents0:6r0:9,directlycomputedvaluesoftheexponents1=~and~denedinEqs.( 4{14 )aswellthevaluesofthesameexponentspredictedfromscalingEqs.( 4{15d )and( 4{15e ),respectively.Forr=0:9,itproveddiculttoobtainarobustpower-lawvariationofQlocwithh,sonodirectlycomputedvalueisrecordedfor1=~.Theinputstothescalingequationsare(i)thevalueofthecorrelation-lengthexponentfoundfrom~xusingEq.( 4{15c )[seerightmostcolumnofTable 4-2 ],and(ii)avalueofthegapexponentfoundviaEq.( 4{7d )fromthemagneticexponentx.Thevaluesofx[alsolistedinTable 4-3 ]areeitherdirectlycomputedin 80

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theAndersonmodel(forr=0:7)orobtainedbyreningpreviousresults[ 26 ]forthepseudogapKondomodel.Thatthedirectlycomputedvaluesinallcasesbutone(1=~forr=0:9)agreewiththeirscalingpredictionstowithintheestimatednonsystematicerrorsfurthersupportsthevalidityoftheextendedscalingansatzcontainedinEqs.( 4{6 )and( 4{13 ).Theextendedscalingansatzhasimplicationsnotonlyforrelationsamongcriticalexponentsbutalsofortherelativemagnitudeofresponsesatdierentpointspnearp0.NRGrunsperformedforxedjp)]TJ /F9 11.955 Tf 11.96 0 Td[(p0jbutforvariousanglesbetweenp)]TJ /F9 11.955 Tf 11.97 0 Td[(p0andthelocalnormal^n0areconsistentwiththehypothesisthatlocalspinandchargepropertiesdependonlyongasdenedinEq.( 4{13 ).Forr.0:55(whichistherangeinwhich>2),thescalingrelationsinEqs.( 4{15 )predictthat~>1andand~x;~<0.Incontrast,wendnumericallythat~,~x,and~arepinnedattrivialvaluesof1,0and0respectively.Inordertoexplainthestrongdeviationsfromscalingoverthisrangeofbandexponents,itturnsouttobeessentialtoconsiderthehithertoneglectedregular(analytic)parts, Fregimp=)]TJ /F1 11.955 Tf 10.5 8.09 Td[(1 2regcg2)]TJ /F1 11.955 Tf 13.16 8.09 Td[(1 2regsh2+:::;(4{16)ofthetotalimpurityfreeenergyFimp=Fcritimp+Fregimp.TheregulartermsimpartapiecetoQlocvaryinglinearlywithg(i.e.,~=1)andaconstantlocalchargesusceptibility(formallycorrespondingto~x=~=0).Forany>2,thesecontributionsdominatethechargeresponsesdescribedbyEqs.( 4{15 )thatarisefromthecriticalpartofthefreeenergy.Thecondition>2doesnotprecludeadivergentlocalspinsusceptibility,whichdependsnotonlyonbutalsothemagneticexponentx.Indeed,nontrivialcriticalbehaviorinthespinsectorpersistsforr!0+,inwhichlimitthereisadivergentcorrelation-lengthexponent'1=r(Refs.[ 88 ]and[ 121 ]).Itshouldbepointedoutthatanonanalyticchargeresponse,albeitsub-leading,isstillpresentintherangeofbandexponentswhere>2.ThisisillustratedinFig. 4-4 , 81

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Figure4-4. Temperature-dependentpartc)]TJ /F3 11.955 Tf 11.96 0 Td[(regcofthelocalchargesusceptibilityforbandexponentr=0:5.ThelinettedthroughtheNRGdatapointscorrespondstoc)]TJ /F3 11.955 Tf 11.95 0 Td[(regc/T0:16. alog-logplotofc(T))]TJ /F3 11.955 Tf 12.95 0 Td[(regc[whereregcc(T!0)]versustemperaturefortherepresentativecaser=0:5.Anempiricaltc)]TJ /F3 11.955 Tf 12.31 0 Td[(regc/T0:16isincloseagreementwiththeexpectationbasedonEq.( 4{15c )ofatemperatureexponent1)]TJ /F1 11.955 Tf 11.95 0 Td[(2==0:15(3). 4.3DiscussionThisworkhasshedlightonthecriticallocalchargeresponsefoundpreviouslyneartheKondo-destructionquantumcriticalpoint(QCP)inthepseudogapAndersonimpuritymodelawayfromparticle-holesymmetry[ 85 ].The\eld"conjugatetothelocalcharge(i.e.,theimpurityoccupancy)istheimpuritylevelenergy"d.Changing"ddoesnotdestroyorrestoretheSU(2)spin-rotationinvariancethatdistinguishesthemodel'sstrong-couplingphasefromitsbroken-symmetrylocal-momentphase.Forthisreason,"djoinsothermodelcouplings,suchastheinteractionstrengthUandthe 82

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hybridizationwidth,whosecollectivedeviationgfromthephaseboundaryentersanansatz[Eq.( 4{6 )]forthefreeenergyinthescalingcombinationg=T1=,distinctfromthejhj=T=scalingofthelocalmagneticeld.3Firstandsecondpartialderivativesofthefree-energywithrespectto"dexhibitpower-lawvariationswithexponents~,~,~,~,and~xthatdependon,but(apartfrom~and~)areindependentof.Presumably,thecorrespondingpartialderivativesofthefreeenergywithrespecttoUand)-327(wouldbedescribedbythesamesetofexponents.Inallcasesstudiednumericallyinthiswork,thelocalchargeresponseattheQCPhasprovedtobelesssingularthanthelocalspinresponse.Itseemsintuitivelyreasonablethattheresponsetotheorder-parametereldismoresingularthanthattootherperturbationsofthesystem.Indeed,onecanargueonthatthisshouldbetrueatanyinteractingQCPdescribedbythescalingansatzEq.( 4{6 ),examplesofwhichhavebeenidentiedinanumberofotherquantumimpuritymodels[ 120 , 122 { 126 ].AtsuchaQCP,theresponsetotheorder-parametereldwillbethemostsingularresponseprovidedthatthegapexponentsatises>1,aconditionthatcanbeshownusingEqs.( 4{5 )and( 4{7d )[allderivedfromEq.( 4{6 )]tobeequivalentto+>1.SinceanyinteractingQCPisexpectedtosatisfy>0(describingacontinuouspower-lawriseoftheorderparameter)and1(=1beingthemean-eldvalue),+>1shouldbesatisedquitegenerally. 4.4ConclusionInsummary,inthischapterwehaveprovidedauniedpictureofcriticalspinandchargeresponsesatquantumcriticalpointsintheparticle-hole-asymmetricpseudogap 3Inthelimitofunitimpurityoccupancy,thepseudogapAndersonmodelcanbemappedatlowenergiestoapseudogapKondomodelwithanexchangecouplingJandapotentialscatteringV.InthisKondomodel,weexpectthatthecriticalpartoftheimpurityfreeenergysatisesthescalingansatzinEq. 4{6 withjgjbeingthelengthoftheperpendiculartothephaseboundary[parameterizedasJ0=Jc(V0)]thatpassesthroughthepoint(J;V)representingthebaremodelparameters. 83

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AndersonHamiltonian,atoymodelforinvestigatingcriticalKondodestructionatmixedvalence.Allcriticalexponentshavebeenrelatedtojusttwounderlyingexponents:thecorrelation-lengthexponentandthegapexponent.Thechargesusceptibilitydivergesatthetransitionprovided<2,whilefor>2thelocalchargeresponseisregularwithnonanalyticcorrections.Wehavearguedthatnonanalyticresponsestonon-symmetry-breakingeldsareagenericfeatureofinteractingQCPsinquantumimpuritymodels,althoughsuchresponsesshouldbelesssingularthanthosetoaeldbreakingthesymmetrythatdistinguishesthephasesoneithersideoftheQCP. 84

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CHAPTER5ENTANGLEMENTENTROPYNEARKONDO-DESTRUCTIONQUANTUMCRITICALPOINTSThischapterisbasedonapublishedpaper[ 127 ].AllthepublishedcontentsarereprintedwithpermissiongrantedunderthecopyrightpolicyoftheAmericanPhysicalSocietyfrom J.H.Pixley,TathagataChowdhury,M.T.Miecnikowski,JaimieStephens,ChristopherWagner,andKevinIngersent,Phys.Rev.B91,245122(2015) .Copyright(2015)bytheAmericanPhysicalSociety.Throughoutthischapter,\we"and\ours"refertotheauthorsofRef.[ 127 ].AsdiscussedinSec. 1.6 ,thereisconsiderablecurrentinterestinquantifyingthequantummechanicalentanglementwithinstronglycorrelatedsystems,particularlyinthevicinityofquantumphasetransitions.OneofthemostcommonlystudiedmeasuresofentanglementistheentanglemententropydenedinEq.( 1{24 ).Inthischapter,wepresentthepropertiesexhibitedbytheentanglemententropyinthreevariantsoftheKondomodel,namely(i)thepseudogapone-channelspin-1=2,(ii)thepseudogaptwo-channelspin-1=2,and(iii)thepseudogapone-channelspin-1Kondomodels.AllthesemodelsfeatureaKondodestructionquantumcriticalpoint(QCP)separatingalocal-moment(LM)phasefromastrong-couplingphase.BycombininganalyticandNRGcalculations,weestablishthattheentanglemententropySebetweenamagneticimpurityanditsenvironmentcontainsacriticalcomponentinthevicinityofthequantumcriticalpointsinthesemodels.Foraspin-Simpimpuritymoment,weshowthatentanglemententropySetakesitsmaximalvalueofln(2Simp+1)attheQCPandthroughouttheKondophase,anddecreasesinapower-lawfashiononentryintotheKondo-destroyedorlocal-momentphase.ThesebehaviorshighlightsomedierencesbetweentheKondoandspin-bosonmodelsstudiedpreviouslyin[ 100 { 102 ].Theremainderofthischapterisorganizedasfollows:Section 5.1 reviewsgeneralcharacteristicsoftheentanglemententropyinquantumimpuritysystemsandsummarizestheuniversalbehaviorsthatwendatKondo-destructionQCPs.Detailedanalysisof 85

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Figure5-1. SchematicrepresentationofthemodelHamiltoniansconsideredinthischapter.TheimpuritydegreesoffreedomdescribedbyHimparecoupledviaHhost-imp(wavyline)tothehostdegreesoffreedomdescribedbyHhost.WecalculatetheimpurityentanglemententropyusingEq.( 1{24 )withregionAdenedtocontainjusttheimpurityandregionBcontainingallhostdegreesoffreedom. variousKondomodelsispresentedinSecs. 5.2 .WediscussourresultsinSec. 5.4 andconcludeinSec. 5.5 . 5.1GeneralConsiderationsTheentanglemententropymeasurestheextenttowhichtworegions(labeledasAandB)ofaquantummechanicalsystemseparatedbyaboundaryareentangledwitheachother.Inquantumimpurityproblems,theentanglemententropybetweentheimpurityandtherestofthesystemisdenedbytakingregionAtocontainsolelytheimpuritydegreesoffreedom,whileregionBdescribesthehost(i.e.,therestofthesystem),asshownschematicallyinFig. 5-1 .Upontracingoutthehost,weobtaintheimpurityreduceddensityoperator^impactinginavectorspaceofdimensiondimp.Equation( 1{24 )thengivestheimpurityentanglemententropy[ 100 , 101 ] Se=)]TJ /F5 7.97 Tf 11.43 16.34 Td[(dimpXi=1pilnpi;(5{1) 86

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wherefpigisthesetofeigenvaluesof^imp.Thecorrespondingeigenstatesfjiigmustrespectthesystem'ssymmetries,aconstraintthatallowstheeigenvaluespitobeexpressedintermsofexpectationvaluesofimpurityoperatorsthatcanreadilybecalculatedusingtheNRG.Sincethehostdegreesoffreedomhavebeencompletelytracedout,theimpurityentanglemententropymeasuresonlytheentanglementbetweentheimpurityandthehostasawhole.Detailsofthehost|suchasthenumber,dispersion,andanyinternalinteractionsoftheconductionbands|inuenceSeonlyinsofarastheyaecttheimpuritymatrixelementsthatdeterminetheeigenvaluesof^imp.Foragroundstateofproductformj i=jiimpjihost,onecanchoosep1=1andpi=0foralli>1,implyingthatSe=0.Attheotherextreme,astateofmaximalentanglementbetweentheimpurityanditshostisdescribedbypi=1=dimpforalli,leadingtoSe=lndimp.Acomplicationarisesifthesystemisnotinapurestate,asislikelytobethecasewhenthereisground-statedegeneracy.Forexample,inthetriviallimitwheretheimpurityandthehostaredecoupled,n-folddegeneracyoftheimpuritygroundstateresultsin^imphavingnvaluespi=1=nanddimp)]TJ /F3 11.955 Tf 12.56 0 Td[(nvaluespi=0,implyingthatSe=lnn.Inordertoavoidsuchmisleadingindicationsofentanglement,itisnecessarytobreaktheground-statedegeneracyoftheimpuritytoobtainapurestate.Inthepresentchapter,wherewehavetreatedmagneticimpurities,theground-statedegeneracycanbeliftedbytheapplicationofaninnitesimallocalmagneticeldhlocthatcouplessolelytotheimpuritythroughaHamiltoniantermhlocSzimp,whereSimpistheimpurityspinoperatorandtheLandegfactorandtheBohrmagnetonhavebothbeensettounity.Forthisreason,wehaveconsideredatwo-parameterfunctionSe(x;hloc),wherexisanonthermal,nonmagneticparameterthattunesthesystemthroughaQCPatx=xc.Inmanycases,wehaveemployedareducedvariable=(x)]TJ /F3 11.955 Tf 11.57 0 Td[(xc)=xcsuchthattheQCPislocatedat=0.Italsoprovesconvenienttodenethelocal-eld-dependentpartoftheentanglemententropy(notethesign) Se(x;hloc)=Se(x;0))]TJ /F3 11.955 Tf 11.96 0 Td[(Se(x;hloc);(5{2) 87

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andtointroducetheshorthandnotations S+e(x)=Se(x;hloc=0+); (5{3a)Se(x)=Se(x;hloc=0+) (5{3b)representing,respectively,thedegeneracy-liftedentanglemententropyandthereductioninentanglemententropyduetospontaneoussymmetrybreaking.InSecs. 5.2 ,wereportresultsfortheentanglemententropyinseveralKondoimpurityHamiltoniansofthegeneralform H=Hhost+Himp+Hhost-imp(5{4)whereHhostdescribesoneormorefermionicbandsaswellas,possibly,abosonicbath.ThetermHimpdescribestheisolatedimpurity,andHhost-impaccountsforthecouplingbetweenthehostandtheimpurity.Thefermionicbandsareassumedtohaveadispersion"kgivingrisetoanidealizeddensityofstatesgivenbyEq.( 1{11 )thatvanishesattheFermienergyinapower-lawfashioncharacterizedbyapseudogapbandexponentr.QCPsariseintheKondomodelsincases00)fromalocal-momentphase(spanning<0),inwhichtheKondoeectisdestroyed.Inthemodelsconsideredinthisstudy,theKondophaseexhibitsexactscreening,over-screening,orunder-screeningoftheimpurityspin.Ineachcaseanappropriateorderparameterforthequantumphasetransitionisthehloc!0+limitofthelocalmagnetizationMloc(;hloc)=)]TJ /F1 11.955 Tf 11.3 0 Td[(limT!0hSzimpi,whichvanishesthroughouttheKondophase,andinthelocal-momentphaseclosetotheQCPobeys Mloc(;hloc=0+)/()]TJ /F1 11.955 Tf 9.3 0 Td[();(5{5) 88

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whereistheorder-parameterexponent.Atthecriticalvalueofthetuningparameter,thelocalmagnetizationsatises Mloc(=0;hloc)/jhlocj1=;(5{6)whereisanothercriticalexponent.Weshowverygenerally|independentoffeaturessuchasparticle-holesymmetryorasymmetry,thedegreeofimpuritychargeuctuations,whethertheKondophaseinvolvesexact,over-,orunder-screening|thatuponapproachtotheQCPfromthelocal-momentside,theentanglemententropysatises Se(;hloc)=aM2loc;(5{7)whereaisaconstantthatdependsondetailsofthemodel.WhencombinedwithEqs.( 5{5 )and( 5{6 ),Eq.( 5{7 )impliesthat Se(<0;hloc=0+)/()]TJ /F1 11.955 Tf 9.3 0 Td[()e; (5{8a)Se(=0;hloc)/jhlocj1=e; (5{8b)with e=2; (5{9a)1=e=2=; (5{9b)scalingrelationsthataredemonstratedexplicitlyinthenumericalresultspresentedbelow.WehavesolvedthequantumimpurityproblemsintroducedaboveusingtheNRG[ 10 , 114 ]asadaptedtotreatquantumimpurityproblemsinvolvingapseudogappedfermionicdensityofstates[ 25 , 87 ]asexplainedinChapter 2 .TheimpurityentanglemententropywasfoundviaEq.( 5{1 )usingreduceddensitymatrixeigenvaluespiexpressedintermsofexpectationvalues(convergedforlargeNRGiterationnumberscorrespondingtoasymptoticallylowtemperatures)ofcertainimpurityoperatorsspeciedinthesections 89

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thatfollow.WehaveusedaWilsondiscretizationparameter39,arangethathasbeenshownpreviouslytoprovideanaccurateaccountofthecriticalexponents[ 26 , 123 , 126 , 128 ].ClosetotheQCP,wefounditnecessarytoemployquadruple-precisionoating-pointcalculationsinordertoaccuratelyresolvetheentanglemententropy,andinparticular,thevalueofSe()denedinEq.( 5{3b ). 5.2KondoModelsTheKondomodelsstudiedaspartofthisdissertationaredescribedbyKondoHamiltoniansoftheformofEq.( 5{4 )with Hhost=Xk;;"kcykck; (5{10a)Himp=hlocSzimp; (5{10b)Hhost-imp=JSimpX;0;cy01 20c00+VX;cy0c0; (5{10c)whereckdestroysaconductionelectronofwavevectork,spinzcomponent=1 2";#,channelindex2f1;:::;Kg,andenergy"k;c0=1=p NkPkckdestroysanelectronofspinzcomponentandchannelindexattheimpuritysite;Nkisthenumberofunitcellsinthehost;andSimpisthespinoperatorforaspin-Simpimpurity.Thehost-impuritycouplingischaracterizedbyanantiferromagneticexchangeJ>0andapotentialscatteringV.InSec. 5.2.1 ,weconsiderpseudogapKondomodelsdescribedbyEqs.( 5{4 )and( 5{10 )withanimpurityspinSimp=1 2andchannelnumbersK=1(exactlyscreenedimpurity)andK=2(overscreenedimpurity).Section 5.2.2 treatsaspin-oneimpurity,focusingontheunderscreenedK=1pseudogapmodel.Ineachofthesecases,wehaveconsideredVtobeheldxedanddenethedistancefromcriticalitytobe=(J)]TJ /F3 11.955 Tf 11.95 0 Td[(Jc)=Jc. 90

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Figure5-2. Degeneracy-liftedentanglemententropyS+evsdimensionlessKondocoupling0Jfortheone-channel,Simp=1 2pseudogapKondomodelforvariouscombinationsofthebandexponentrandthedimensionlesspotentialscattering0V.Resultsforr=0:2andr=0:3areatparticle-holesymmetry(V=0).Particle-hole-asymmetricresultsareshownforr=0:4(0V=0:109),r=0:6(0V=0:54),andr=0:8(0V=6:2,plotting0J=6onthehorizontalaxis).Ineachcase,S+etakesitsmaximumvalueofln2'0:693forallJJc. Evenwiththeadditionofadegeneracy-liftingeldthatcouplestoSzimp,theKondoHamiltonians[Eqs.( 5{10 )]exhibitspin-rotationsymmetryaboutthezaxisandhenceconservethezcomponentoftotalspin.Thisensuresthattheeigenstatesof^impcanbechosentobetheconventionalbasisstatesjmisuchthatSzimpjmi=mjmi.Then, Se=)]TJ /F5 7.97 Tf 21.62 14.94 Td[(SXm=)]TJ /F5 7.97 Tf 6.59 0 Td[(Spmlnpm:(5{11) 91

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Figure5-3. Spontaneous-symmetry-breakingpartoftheentanglemententropySe()andorder-parameterMloc(plottedas2M2loc)vsdistancejj=(Jc)]TJ /F3 11.955 Tf 11.76 0 Td[(J)=JcfromtheQCPinthelocal-momentphaseoftheone-channelspin-1=2Kondomodel.Resultsforr=0:2andr=0:3areatparticle-holesymmetry(V=0).Resultsforr=0:4,r=0:6andr=0:8areatparticle-holeasymmetry(V6=0).ThecoincidenceofthetwodatasetsforeachrconrmsEq.( 5{16 )andstraight-linetsyieldexponentslistedinTable 5-1 . 5.2.1Simp=1 2KondoModelsForaspin-1=2Kondoimpurityandinthepresenceofspin-rotationsymmetryaboutthezaxis,theeigenvaluesof^imparejusttheimpurityspin-upandspin-downoccupationprobabilities p1=2=1 2Mloc;(5{12)andEq.( 5{1 )reducesto Se=S2(1 2+Mloc);(5{13) 92

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where S2(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(xlnx)]TJ /F1 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)ln(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)(5{14)isthebinaryentropyfunction.ExpandingEq.( 5{13 )forjMlocj1 2gives Se'ln2)]TJ /F1 11.955 Tf 11.95 0 Td[(2M2loc)]TJ /F1 11.955 Tf 11.95 0 Td[((4=3)M4loc;(5{15)aresultthatholdsforanyS=1 2Kondomodel,irrespectiveofthenumberanddispersionoftheconductionbands.Equation( 5{15 )impliesthateveninthepresenceofaninnitesimalmagneticeld,theentanglemententropytakesitsmaximumpossiblevalueofln2atanymagneticQCPandthroughouttheKondophase.ThisistruebothwhentheimpuritymomentisexactlyscreenedwithFermi-liquidexcitations(asisthecaseforK=1)andwhenitisover-screenedwithanon-Fermiliquidmany-bodyspectrum(asforK2).TakingintoaccountalsoEq.( 5{5 ),onendsthatonapproachtotheQCPfromthelocal-momentside(!0)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(), Se()'2M2loc/()]TJ /F1 11.955 Tf 9.3 0 Td[()2;(5{16)realizingEq.( 5{7 )witha=2aswellasexemplifyingEqs.( 5{8a )and( 5{9a ).AnotherimportantconclusionthatcanbedrawnfromEqs.( 5{1 )and( 5{12 )isthatS+evanishesonlyforjMlocj!1 2,correspondingtoavanishingKondocouplingJ.EventhoughtheKondoeectisdestroyedfor0
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ofparticle-hole-asymmetricQCPsatJ=Jc;a(r),V=Vc(r)foranyrsatisfyingr'3=81,leadingtoamuchweakerfeatureinS+evsatthelocationofthequantumphasetransition.WehavealsoperformedNRGcalculationsforthetwo-channelpseudogapKondomodel[thecaseofEq.( 5{10 )withK=2].ThephasediagramandcriticalpropertiesofthismodelarediscussedindetailinChapter 3 .Thismodelfeatures,for0
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r0V1=e 0.200.16025(1)0.02645(4)0.324(2)0.300.35499(2)0.07398(2)0.71007(4)0.40.1090.57553(2)0.15606(3)1.15106(2)0.60.540.18759(2)0.11696(5)0.3754(3)0.86.20.07578(2)0.06373(3)0.156(2) Table5-1. Exponents,,andedenedinEqs.( 5{5 ),( 5{6 ),and( 5{8a ),respectively,fortheSimp=1 2,one-channelKondomodelwiththevecombinationsofthebandexponentranddimensionlesspotentialscattering0VillustratedinFig. 5-3 .Parenthesesenclosetheestimatednonsystematicerrorinthelastdigit.TheexponentsobeyEq.( 5{9a )withintheerrors,apartfromweakviolationsforr=0:2andr=0:8,wherethesmallvaluesof1=impedeaccurateevaluationofande. r0V1=e 0.0500.050(1)0.001815(2)0.140(5)0.1000.150(2)0.013075(2)0.327(4)0.1500.330(1)0.036233(8)0.667(3)0.2000.908(3)0.09205(2)1.815(9)0.400.791.280(6)0.4477(2)2.560(7)0.601.110.3720(6)0.2453(5)0.744(2)0.801.490.1465(9)0.112(2)0.294(2) Table5-2. Exponents,1=,andedenedinEqs.( 5{5 ),( 5{6 ),and( 5{8a ),respectively,fortheSimp=1 2,two-channelKondomodelatparticle-symmetry(V=0)aswellasparticle-holeasymmetry(V6=0)andsevendierentvaluesofthebandexponentr.Parenthesesenclosetheestimatednonsystematicerrorinthelastdigit.TheexponentsobeyEq.( 5{9a )towithinamarginthatiswithinnonsystematicerrorsforr0:2,butbecomeslargerforlowervaluesofr,likelyduetothesmallvaluesofexponent1=. 3.1 [ 25 , 108 ].Forrmax
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Figure5-4. TheratioSe=M2locvsjjforthetwo-channelpseudogapKondomodelforseveralvaluesofrattheparticle-holesymmetricQCP(r=0:05,0:10,0:15,and0:20)aswellasattheparticle-holeasymmetricQCP(r=0:40and0:80).Theratiosconvergeto2asjj!0conrmingthevalidityofEq.( 5{16 ). Theexponentefoundusingpower-lawtsofEq.( 5{16 )arelistedinTable 5-2 forthesymmetricQCP(r=0:05,0:10,0:15,and0:20)aswellasfortheasymmetricQCP(r=0:40,0:60,and0:80).ThecriticalexponentsandforthismodelarealsolistedfromTables 3-1 and 3-2 .TheexponentsandeobeyEq.( 5{16 )towithinestimatednon-systemicerrors(showninparenthesis)forr0:20.Forr<0:2,thedeviationsfromEq.( 5{16 )increasesasrisdecreased,likelyduetothesmallvaluesofexponent1=preventingMlocfromattainingsmallenoughvaluesevenatthecriticalpoint.InFig. 5-4 ,wehaveplottedtheratioofSe=M2locforvariousvaluesofthebandexponentr(hollow 96

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r0Vne 0.1000.062(5)0.19(1)0.13(1)0.1500.116(1)0.386(5)0.24(1)0.2000.207(1)0.787(4)0.423(4)0.2500.51(1)2.7(1)1.00(5)0.400.50.08960(5)0.4509(3)0.180(1)0.600.80.04300(5)0.2128(2)0.089(1) Table5-3. Exponents,n,andedenedinEqs.( 5{5 ),( 5{18 ),and( 5{8a ),respectively,fortheSimp=1,one-channelKondomodelwithsixcombinationsofthebandexponentrandthedimensionlesspotentialscattering0V.Parenthesesenclosetheestimatednonsystematicerrorinthelastdigit.Ineachcase,n3,implyingthatvariationsofn0arenegligiblecomparedtothoseofMlocinthevicinityofthequantumcriticalpoint.TheexponentsobeyEq.( 5{9a )withintheerrors,apartfromaweakviolationforr=0:2andastrongeroneforr=0:6,wherethesmallvalue1==0:02589(2)impedesaccurateevaluationofande. symbols).Forallvaluesofther,theratioconvergeto2,aspredictedbyEq.( 5{16 ).ThedierentconvergenceratesoftheratiocanbeexplainedbyconsideringcorrectionsoftheorderM4locgivenbyEq.( 5{15 )totheleadingorderterminEq.( 5{16 )andthefactthattheexponentdecreasesasriseitherincreasedordecreasedfromitsvalueatr=rmax.Apower-lawtoftheratio(showninFig. 5-4 bydashedlines)veriesthattheratioSe=M2loc!2(withinestimatederrorbars)as!0forallvaluesofrexceptr=0:05. 5.2.2Simp=1Single-ChannelPseudogapKondoModelForaspin-1Kondoimpurityandinthepresenceofspin-rotationsymmetryaboutthezaxis,theeigenvaluesof^impcanbeparameterized p1=1 2(n0Mloc); (5{17a)p0=1)]TJ /F3 11.955 Tf 11.95 0 Td[(n0; (5{17b)wherewehaveintroducedn0=h(Szimp)2i.Forsuchanimpurity,wehavefocusedexclusivelyontheone-channelpseudogapKondomodel.Atparticle-holesymmetry(V=0),thismodelhasaQCPforanybandexponentrontherange0
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Figure5-5. Log-logplotofMloc,n0)]TJ /F1 11.955 Tf 11.96 0 Td[(2=3,andSevsjjasthecriticalpointisapproachedfromthelocal-momentsidefortheone-channelSimp=1pseudogapKondomodel.Theplotsarefortwocombinationsofthepseudogapexponentandthedimensionlesspotentialscattering:A)(r;0V)=(0:4;0:5),andB)(r;0V)=(0:6;0:8).ThelinearvariationofMloc,n0)]TJ /F1 11.955 Tf 11.95 0 Td[(2=3,andSewithrespecttojjexempliesEqs.( 5{5 ),( 5{18 ),and( 5{8a ),respectively. 98

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anyr'0:245.SubstitutingEqs.( 5{17 )intoEq.( 5{1 ),settingn0=2=3andkeepingonlyleadingtermsinMloc,onearrivesattheresult Se()3 4M2loc/()]TJ /F1 11.955 Tf 9.29 0 Td[()2;(5{19)providingarealizationofEq.( 5{7 )witha=3=4.AswasthecaseforSimp=1 2,thepredictedbehaviorisconsistentwithEqs.( 5{8a )and( 5{9a ).Figures 5-5 A)andB)presentNRGresultsfortwocombinationsofthepseudogapexponentandthedimensionlesspotentialscattering:(r;0V)=(0:4;0:5)and(0:6;0:8),respectively.Here,wehavecomputedMloc=hSzimpiandn0=h(Szimp)2i,thenusedEqs.( 5{1 )and( 5{17 )tondSe.Figure 5-5 showsthat,forbothvaluesofr,onapproachto 99

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Figure5-6. RatioSe=M2locvsjjfortheone-channelspin-1pseudogapKondomodelandforfourvaluesofthepseudogapexponentr.Theconvergenceoftheratioto0:75as!0)]TJ /F1 11.955 Tf 10.98 -4.34 Td[(conrmsthevalidityofEq.( 5{19 ).Therateofconvergencedependsonrthroughtheorderparameterexponent(r). eachQCPfromthelocal-momentside,Mloc,n0)]TJ /F1 11.955 Tf 11.5 0 Td[(2=3,andSevanishinpower-lawfashionaccordingtoEqs.( 5{5 ),( 5{18 ),and( 5{8a ),respectively.TheexponentsextractedfromsuchplotsaresummarizedinTable 5-3 forthetwoexamplesshowninthegure,plustheparticle-hole-symmetricQCPsforr=0:10,0:15,0:20,and0:25.Ineachcase,theexponentssatisfyn>3,conrmingtheconjecturethatdeviationsofn0from2=3canbeneglectedinthevicinityoftheQCP.Equation( 5{9a )isalsosatisedwithoneminorviolation(forr=0:25)andonemoresignicantoneforr=0:6thatcanbeattributed 100

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tothesamecause(asmallexponent1=)asintheone-channelandtwo-channelSimp=1 2Kondomodels.Figure 5-6 plotstheratioSe=M2loconthesamelogarithmicjjscaleasusedforFigure 5-5 forfourcombinationsofthepseudogapexponentr.Forr=0:25;V=0,theratioconvergesrapidlyto0:75,aspredictedinEq.( 5{19 ).Theconvergenceisprogressivelyslowerforr=0:2;0V=0,r=0:4;0V=0,andr=0:6;0V=0:8.Asforthetwo-channelcase,thistrendcanbeexplainedbythetakingintoaccounttheleadingcorrectiontoEq.( 5{19 )insituationswheren>2,namely, Se()3 4M2loc+9 32M4loc;(5{20)andnotinginTable 5-3 thedecreaseinthevalueofasriseitherincreasestoward1ordecreasedtoward0fromr=rmax.WecanextrapolatetheratioSe=M2locfor!0usingapower-lawtoftheratiovstoobtain0:7499(3)and0:746(6)forr=0:4andr=0:6,respectively,inexcellentagreementwiththepredictedvaluea=3=4.WethereforehavetakenFigure 5-6 asprovidingconrmationoftherelationEq.( 5{19 ). 5.3AndersonandBose-FermiModelsOurcollaboratorshavealsoinvestigatedtheentanglementpropertiesin(i)anAndersonmodelwithapseudogapdensityofstatesthatfeaturesaKondo-destructionQCPasdiscussedinSec. 4.1.1 ,and(ii)aBose-FermiKondomodelwherecompetitionbetweenthefermionicbandsandbosonicbathgivesrisetoaQPT[ 60 , 121 , 123 , 126 , 128 { 130 ].Theresultsofthesestudiesarepublishedin[ 127 ].ThebehavioroftheentanglemententropyintheBose-FermiKondomodelissimilartothatintheKondomodelsconsideredinthischapter,i.e.,Sehasamaximumvalueofln(2Simp+1)throughouttheKondophaseandvanishesasSe2M2loconapproachtotheQCPfromthelocal-momentside.InthepseudogapAndersonmodel,chargeuctuationsproduceanonanalyticleadingvariationofSeneartheQCPwithacriticalexponentthatdependsonlyonthebandexponentrintherange0:55.r<1.Awayfromparticle-holesymmetry,Semayriseonapproach 101

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totheQCPfromtheKondoside,producingacusppeakinSepreciselyatthequantumphasetransition.However,undercertainsituations,theentanglemententropydecreasescontinuously,albeitnonanalytically,onpassingfromtheKondophasetothelocal-momentphase. 5.4DiscussionOneuniversalfeatureofourresultsisthepresenceofanonzeroentanglementonentrytothelocal-moment(Kondo-destroyed)phase.Sucharesidualentanglementimpliesthatthegroundstateisnotasimpleproductofanimpuritystateandanenvironmentalstate.ThisresulthassignicantimplicationsfortheoreticalandnumericaldescriptionsoftheKondo-destroyedphase.Forexample,withinalarge-NmeaneldtheoryofthepseudogapKondomodel[ 24 ],thelocalmomentisrepresentedwithfermionicspinonsfandtheeectiveHamiltonianisaresonant-levelmodelwithahybridizationb=h^biMF=hfyc0iMF(where^bisabosonicoperator).Atthislevel,Kondodestructioncorrespondstob!0,implyingthatthelocalmomentiscompletelyfreeandnolongerentangledwiththeconductionband.Thus,suchastaticmean-eldtheorycannotreproducethenonzeroentanglemententropythatwendintheKondo-destroyedphase.Ourresultscanbeunderstood,however,intermsofabosonicoperator^b(!)thathasavanishingstaticcomponentandgiverisestoadynamicalKondoeect.OurresultsalsoimplythattheKondo-destroyedphasecannotbecapturedinvariationalquantumMonteCarlostudiesoftheKondolatticethattreatbasastaticvariationalparameter.ItwillbeinterestingtotryandconsidermoregeneralvariationalwavefunctionsthatcantreattheKondo-destroyedphasemoreaccurately.Kondo-destroyedquantumcriticalpointshavebeeninvokedtounderstandtheunconventionalquantumcriticalityobservedinexperimentsonheavy-fermionmetals[ 131 ].AsaresultofthefailureoftheHertz-Millis-Moriyatheory[ 64 , 65 , 132 ]ofthespin-density-wavetransitiontodescribetheexperimentaldata[ 76 ],theconceptoflocalquantumcriticality[ 77 ]hasbeenusedtounderstandtheenergy-over-temperaturescaling 102

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inthedynamicspinsusceptibility,thepresenceofanadditionalenergyscale,andajumpintheFermi-surfacevolume.Thetheoryoflocalquantumcriticalityisbasedontheextended-dynamicalmean-eldtheoryoftheKondolattice[ 77 ],whichndsthatforsucientlystrongquantumuctuationstheKondoeectisindeeddestroyedattheantiferromagneticQCP.TheresultsofthepresentstudyimplythatacontinuouslossofentanglementisexpectedattheKondo-breakdownQCPsbelievedtooccurincertainheavy-fermionsystems[ 133 ]. 5.5ConclusionWehavestudiedthequantummechanicalentanglementbetweenamagneticimpurityanditsenvironmentinseveralmodelsthatfeaturecriticaldestructionoftheKondoeect.IntheKondo-destroyedphaseofeachmodelstudied,wehaveidentiedatermintheentanglemententropyvaryingwithacriticalexponente=2,whereisthecriticalexponentgoverningtheorderparametercharacterizingthequantumphasetransition.Inaddition,wehaveestablishedthattheresponseofSetoalocalmagneticeldgivesrisetoapartoftheentanglemententropythatvarieswithacriticalexponent1=e=2=,whereisthecriticalexponentgoverningtheresponseoftheorderparameterattheQCPtoalocalmagneticeld.WehaveestablishedverygenerallythatinKondomodels,theratioofthecriticalpartoftheentanglemententropytothesquareoftheorderparameterdependsonlyonthemagnitudeoftheimpurityspin,andnotonthenumberofconductionchannels.InallvariantsoftheKondomodelthatwehaveconsidered,Seremainspinnedatitsmaximalvalueofln(2Simp+1)throughouttheKondophase. 103

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CHAPTER6CONCLUSIONANDFUTUREDIRECTIONS 6.1ConclusionInthisdissertation,Ihaveusedthenumericalrenormalization-grouptechniquetostudyseveralvariantsoftheKondomodelwithapseudogapdensityofstatesthatvanishesattheFermienergy("F=0)as(")/j"jr.IhavealsoinvestigatedthepseudogapAndersonmodel.AllthemodelsconsideredinthisstudyexhibitcriticaldestructionoftheKondoeectataquantumphasetransition(QPT),thatseparatesthelocal-momentorKondo-destroyedphasefromstrong-couplingphases.InChapter 3 ,Ihavestudiedthepseudogaptwo-channelKondomodelthatisofcurrentinterestinconnectionwithKondoeectingrapheneduetomagneticadatoms.Itwasfoundthatinthevicinityofthequantumcriticalpoints(QCPs)andstablenon-Fermiliquidgroundstatesofthemodel,thelocalcriticalspinresponseischaracterizedbyasetofcriticalexponents.Intherange0
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describethecriticalspinresponseofthemodeltoshowthatallcriticalexponentsofthemodel,includingthosethatcharacterizethecriticalspinandchargeresponseofthesystem,arerelatedtojusttwounderlyingexponents.Thesearethecorrelationlengthexponentandthegapexponent.ThechargesusceptibilitydivergesattheQCPprovided<2.For>2,thechargesusceptibilitywasfoundtoberegularwithnonanalyticcorrections.IhavearguedthatininteractingQCPsingeneral,theresponsestonon-symmetrybreakingeldswouldbelesssingularthanthosetoasymmetrybreakingeld.InChapter 5 ,Ihavestudiedtheentanglemententropy,amanifestlynon-localquantity,thatmeasuresthequantumcorrelationsbetweentheimpurityandhost.TheentanglementpropertieswerestudiedforthreevariantsofthepseudogapKondomodel:thespin-1=2one-channel,spin-1=2two-channel,andspin-1one-channelKondomodels.Inallthemodelsconsidered,thecriticalpartoftheentanglemententropySevanishesonapproachtotheQCPfromthelocal-momentsidewithacriticalexponente=2,whereistheorderparameterexponent.ThisimpliesthatSevariesasSe/M2loc,whereMlocistheorderparameteroftheQPT.Theresponseofthecriticalpartoftheentanglemententropytoalocalmagneticeldcanalsobecharacterizedbyanexponent1=e=2=,wherethecriticalexponentgovernsthespinresponsetoalocalmagneticeldattheQCP.InalltheKondomodels,theentanglemententropySeisfoundtobepinnedatitsmaximumvalueofln(2Simp+1)throughouttheKondophase. 6.2DirectionsforFutureWorkInallcasesstudiednumericallyinChapter 4 ofthisdissertation,thelocalchargeresponseatthequantumcriticalpointwasfoundtobelesssingularthanthelocalspinresponse.However,itisstraightforwardtocomeupwithexamplewherethereverseorderingholds.InterchangeofspinandchargedegreesoffreedommapstheU>0AndersonmodelinzeromagneticeldtoaU<0Andersonmodelatparticle-holesymmetry.Inthepresenceofapseudogappeddensityofstatesdescribedbyexponent 105

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0
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Figure6-1. Entanglemententropyintheone-channelpseudogapKondomodelatthesymmetriccriticalpoint.Resultsareshownforbandexponentr=0:2.A)ThedegeneracyliftedentanglemententropyS+e(solidsymbols)andtheentanglemententropyatzeroeldSeplottedagainstthedimensionlessKondocoupling0J.ThecriticalcouplingatJ=Jcisplottedusingdottedlines.S+eandSeareidenticaltoeachotherinthestrong-couplingphase(J>Jc)anddiersfromeachotherinthelocal-momentphase(J
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Figure6-2. PlotsofSeagainstA)distancejj=j(J)]TJ /F3 11.955 Tf 11.96 0 Td[(Jc)=Jcjfromthecriticalpoint,andB)appliedeldhlocatthecriticalpointfortheone-channelpseudogapKondomodelforbandexponentr=0:2.ThelinearvariationofSeonthelog-logplotsindicatesapower-lawvariationwithrespecttobothandhloc.Ineachcase,apower-lawtofthenumericaldataisshownusingdashedlines. againstdimensionlessKondocoupling0J.ThecriticalcouplingatJ=Jcisshownusingaverticaldashedline.SeandS+ewerefoundtobeidenticalthroughouttheKondophase(J>Jc)anddierinthelocal-momentphase(JJcandapproacheszeroasJ!1.Thisisbecauseatthestrong-couplingxedpointat0J=1,theimpurityandtheelectronsatthef0siteformaspinsingletandhenceareeectivelydecoupledfromtherestofthesystematlowtemperatures.Figure 6-1 showsthatthecriticalpartoftheentanglemententropySe=Se)]TJ /F3 11.955 Tf 12.66 0 Td[(S+eiszerointhestrongcouplingphase(J>Jc)andvanishescontinuouslyonapproachtotheKondo-destructionQCPfromthelocal-momentside.WehavealsoveriedthatinthevicinityoftheQCP,Sehasanontrivialpower-lawdependenceonboththeappliedlocaleldhlocandthedistancefromthecriticalpoint=(J)]TJ /F3 11.955 Tf 12.25 0 Td[(Jc)=JcandobeysEqs.( 5{8a )and( 5{8b ).This 108

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isillustratedinFig. 6-2 whereIhaveplottedSeagainstA)distancefromthecriticalpoint,andB)appliedlocaleldhlocatthecriticalpoint.Theexponentse0:64and1=e0:206wereobtainedviapower-lawtsofthedata.Myimmediategoalistoextendthisstudytounderstandtheentanglementpropertiesofthemodelatbothparticle-holesymmetricandasymmetriccriticalpointsforvariousvaluesofrintherange0
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BIOGRAPHICALSKETCHTathagataChowdhurywasborninSerampore,asuburbofKolkata,India.AftercompletinghisschoolingatDonBoscoSchool,Bandel,hejoinedJadavpurUniversityfromwherehereceivedBachelorofSciencedegreewithphysicshonors.SubsequentlyhegraduatedwithaMasterofSciencefromtheIndianInstituteofTechnologyatKanpur.HejoinedtheUniversityofFloridainthefallof2009andhasbeenworkingwithProf.KevinIngersentsince2010.Tathagata'sresearchfocusesonquantumphasetransitionsandtheassociatedquantumcriticalityinarticialnano-structuresandheavyfermionssystems.HereceivedhisPh.D.fromtheUniversityofFloridainthefallof2015.Heisalsointerestedinhiking,photography,wildlifeandpainting. 116