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Quantum Phase Transitions of Magnetic Impurities in Dissipative Environments

Permanent Link: http://ufdc.ufl.edu/UFE0042386/00001

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Title: Quantum Phase Transitions of Magnetic Impurities in Dissipative Environments
Physical Description: 1 online resource (166 p.)
Language: english
Creator: Cheng, Mengxing
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation presents results of theoretical research on quantum phase transitions in systems where a magnetic impurity hybridizes with a fermionic host and is also coupled, via the impurity charge, to one or more bosonic modes representing dissipative environments. Two such dissipative quantum impurity models are studied using the numerical renormalization-group technique. The charge-coupled Bose-Fermi Anderson model describes a magnetic impurity that hybridizes with conduction electrons in a metal and is also coupled to a bath of dispersive bosons. The conduction-electron density of states is constant and the bosonic bath is captured by a spectral function characterized by a bath exponent s. The following properties of the model are established: (i) As the impurity-bath coupling increases from zero at fixed impurity-band hybridization, the effective Coulomb interaction between two electrons in the impurity level is progressively renormalized from its repulsive bare value until it eventually becomes attractive. For weak hybridization, this renormalization in turn produces a crossover from a conventional, spin-sector Kondo effect to a charge Kondo effect. (ii) At particle-hole symmetry, and for sub-Ohmic bath exponents 0 < s < 1, further increase in the impurity-bath coupling results in a continuous, zero-temperature quantum phase transition to a broken-symmetry phase in which the ground-state impurity occupancy acquires an expectation value other than 1. The response of the impurity occupancy to a locally applied electric potential features the hyperscaling of critical exponents and energy-over-temperature scaling that are expected at an interacting critical point. For the Ohmic case s=1, the transition is instead of Kosterlitz-Thouless type. (iii) Away from particle-hole symmetry, the quantum phase transition is replaced by a smooth crossover, but signatures of the symmetric quantum critical point remain in the physical properties at elevated temperatures and/or frequencies. In the pseudogap Anderson-Holstein model, a magnetic impurity level hybridizes with a pseudogapped fermionic host characterized by a band exponent r and is also coupled, via the impurity charge, to a local boson mode. We find that the pseudogap Anderson-Holstein model shows distinctive low-temperature quantum fluctuations in two regimes, depending on the strength of the impurity-boson coupling. We study two cases of band exponents: 0 < r < 1 and r=2. (i) For 0 < r < 1, the pseudogap Anderson-Holstein model exhibits continuous quantum phase transitions with anomalous critical exponents. At fixed weak impurity-boson couplings, as the impurity-band hybridization increases from zero, transitions occur between a local-moment phase and two strong-coupling (Kondo) phases. However, at fixed strong impurity-boson couplings, increase in the impurity-band hybridization instead leads to continuous quantum phase transitions from a local-charge phase to another two strong-coupling phases. Particle-hole asymmetry in the model with weak impurity-boson couplings acts in a manner analogous to a local magnetic field applied to the model with strong impurity-boson couplings. (ii) For r=2, the pseudogap Anderson-Holstein model can effectively describe a particular boson-coupled two-quantum-dot setup. In this case, quantum phase transitions between local spin (charge) and strong-coupling phases are manifested by peak-and-valley features in the gate-voltage (magnetic-field) dependence of the linear electrical conductance through the device.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Mengxing Cheng.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Ingersent, J. Kevin.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-12-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042386:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042386/00001

Material Information

Title: Quantum Phase Transitions of Magnetic Impurities in Dissipative Environments
Physical Description: 1 online resource (166 p.)
Language: english
Creator: Cheng, Mengxing
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation presents results of theoretical research on quantum phase transitions in systems where a magnetic impurity hybridizes with a fermionic host and is also coupled, via the impurity charge, to one or more bosonic modes representing dissipative environments. Two such dissipative quantum impurity models are studied using the numerical renormalization-group technique. The charge-coupled Bose-Fermi Anderson model describes a magnetic impurity that hybridizes with conduction electrons in a metal and is also coupled to a bath of dispersive bosons. The conduction-electron density of states is constant and the bosonic bath is captured by a spectral function characterized by a bath exponent s. The following properties of the model are established: (i) As the impurity-bath coupling increases from zero at fixed impurity-band hybridization, the effective Coulomb interaction between two electrons in the impurity level is progressively renormalized from its repulsive bare value until it eventually becomes attractive. For weak hybridization, this renormalization in turn produces a crossover from a conventional, spin-sector Kondo effect to a charge Kondo effect. (ii) At particle-hole symmetry, and for sub-Ohmic bath exponents 0 < s < 1, further increase in the impurity-bath coupling results in a continuous, zero-temperature quantum phase transition to a broken-symmetry phase in which the ground-state impurity occupancy acquires an expectation value other than 1. The response of the impurity occupancy to a locally applied electric potential features the hyperscaling of critical exponents and energy-over-temperature scaling that are expected at an interacting critical point. For the Ohmic case s=1, the transition is instead of Kosterlitz-Thouless type. (iii) Away from particle-hole symmetry, the quantum phase transition is replaced by a smooth crossover, but signatures of the symmetric quantum critical point remain in the physical properties at elevated temperatures and/or frequencies. In the pseudogap Anderson-Holstein model, a magnetic impurity level hybridizes with a pseudogapped fermionic host characterized by a band exponent r and is also coupled, via the impurity charge, to a local boson mode. We find that the pseudogap Anderson-Holstein model shows distinctive low-temperature quantum fluctuations in two regimes, depending on the strength of the impurity-boson coupling. We study two cases of band exponents: 0 < r < 1 and r=2. (i) For 0 < r < 1, the pseudogap Anderson-Holstein model exhibits continuous quantum phase transitions with anomalous critical exponents. At fixed weak impurity-boson couplings, as the impurity-band hybridization increases from zero, transitions occur between a local-moment phase and two strong-coupling (Kondo) phases. However, at fixed strong impurity-boson couplings, increase in the impurity-band hybridization instead leads to continuous quantum phase transitions from a local-charge phase to another two strong-coupling phases. Particle-hole asymmetry in the model with weak impurity-boson couplings acts in a manner analogous to a local magnetic field applied to the model with strong impurity-boson couplings. (ii) For r=2, the pseudogap Anderson-Holstein model can effectively describe a particular boson-coupled two-quantum-dot setup. In this case, quantum phase transitions between local spin (charge) and strong-coupling phases are manifested by peak-and-valley features in the gate-voltage (magnetic-field) dependence of the linear electrical conductance through the device.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Mengxing Cheng.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Ingersent, J. Kevin.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042386:00001


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QUANTUMPHASETRANSITIONSOFMAGNETICIMPURITIESINDISSIPATIVEENVIRONMENTSByMENGXINGCHENGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010

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c2010MengxingCheng 2

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Idedicatethisdissertationtomyfamily,particularly,tomyselfforworkingonthisPh.D.programformorethanveyears;tomyparentsforraisingandsupportingme;tomywifeforherpatienceandunderstanding;tomysonforbringingmeenormouspleasureeveryday;tomyparents-in-lawfortheirhelp;andtomygrandparents,uncles,andauntsforencouragingme. 3

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ACKNOWLEDGMENTS MygreatestappreciationgoestoKevinIngersentforsupervisingmeasadviserthroughoutthetimeittookmetocompletethisresearchandwritethedissertation.Kevinhasbeenarolemodelforinquisitiveness,healthyskepticism,clearwriting,andhonestyinthescienticcommunity.Hispatienceandencouragementhashelpedmethroughseveralperiodsofdoubt.HisphysicalinsightandknowledgehaveguidedmyPh.D.studies.Themembersofmydissertationcommittee,GregoryStewart,SelmanHersheld,SergeiObukhov,andSimonPhillpot,havegenerouslygiventheirtimeandexpertisetoimprovemywork.Ithankthemfortheircontributionsandtheirgood-naturedsupport.Iamgratefultopersonswhosharedtheirmemoriesandexperiences,especiallyYinanYuandfamily,BoLiuandfamily,YuningWu,XingyuanPan,ChungweiWang,Yun-WenChen,VivekMishra,LiliDeng,TomoyukiNakayama,andmanyotherfriendsattheDepartmentofPhysics.Imustalsoacknowledgeformergroupcolleagues,MatthewGlossopandBrianLane,whogenerouslysharedtheirunderstandingofphysicsandcomputingskillsduringmyrsttwoyearsatUFasajuniorgraduatestudent.IhavetothankLuisG.G.V.DiasdaSilvaforvaluablediscussions.Ithankaswellthemanyteachers,friends,colleagues,andstaffwhoassisted,advised,andsupportedmyresearchandwritingeffortsovertheyears.Especially,IneedtoexpressdeepappreciationtoKristinNicholaforherhospitalityandassistance.MythanksmustgoalsotoDavidHansenandBrentNelsonfortheirtechnicalsupport.IalsoneedtoexpressmygratitudetothestaffoftheUFHigh-PerformanceComputingCenteratwhichmostofthecomputationalworkincludedinthisdissertationwasperformed.FinancialsupporthasbeenprovidedbytheDivisionofMaterialsResearchoftheU.S.NationalScienceFoundation. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1Outline ...................................... 13 1.2TheKondoEffect ................................ 13 1.3OverviewofQuantumPhaseTransitions ................... 16 1.4QuantumCriticalityofHeavy-FermionMaterials ............... 18 1.5QuantumPhaseTransitionsinQuantum-DotandSingle-MoleculeDevices 23 2NUMERICALRENORMALIZATION-GROUPMETHODFORQUANTUMIMPURITYMODELS ....................................... 25 2.1OverviewofNumericalRenormalization-GroupMethod .......... 25 2.2NumericalRenormalizationGroupwithaFermionicBand ......... 27 2.2.1LogarithmicDiscretization ....................... 28 2.2.2MappingontoaSemi-InniteChain .................. 30 2.2.3IterativeDiagonalization ........................ 31 2.3NumericalRenormalizationGroupwithaBosonicBath ........... 34 3RESULTSFORCHARGE-COUPLEDBOSE-FERMIANDERSONMODEL .. 38 3.1Introduction ................................... 38 3.2ModelandSolutionMethod .......................... 39 3.2.1Charge-CoupledBose-FermiAndersonHamiltonianandRelatedModels .................................. 39 3.2.2NumericalRenormalization-GroupTreatment ............ 42 3.3PreliminaryAnalysis .............................. 46 3.3.1ZeroHybridization ........................... 46 3.3.2ZeroElectron-BosonCoupling ..................... 50 3.3.3ExpectationsfortheFullModel .................... 53 3.4Results:SymmetricModelwithSub-OhmicDissipation .......... 57 3.4.1NRGFlowsandFixedPoints ..................... 58 3.4.2CriticalCoupling ............................ 62 3.4.3CrossoverScale ............................ 63 3.4.4ThermodynamicSusceptibilities .................... 65 3.4.5LocalChargeResponse ........................ 68 5

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3.4.5.1Staticlocalchargeresponse ................ 68 3.4.5.2Dynamicallocalchargesusceptibility ........... 70 3.4.6ImpuritySpectralFunction ....................... 72 3.4.7Spin-KondotoCharge-KondoCrossover ............... 75 3.5Results:SymmetricModelwithOhmicDissipation ............. 76 3.5.1FixedPointsandThermodynamicSusceptibilities .......... 76 3.5.2StaticLocalChargeSusceptibilityandCrossoverScale ...... 77 3.5.3ImpuritySpectralFunction ....................... 78 3.6Results:AsymmetricModel .......................... 78 3.7Summary .................................... 82 4RESULTSFORPSEUDOGAPANDERSON-HOLSTEINMODEL ........ 117 4.1Introduction ................................... 117 4.2ModelHamiltonianandPreliminaryAnalysis ................ 118 4.2.1PseudogapAnderson-HolsteinModel ................ 118 4.2.2ReviewofRelatedModels ....................... 120 4.2.2.1PseudogapAndersonmodel. ................ 120 4.2.2.2Anderson-Holsteinmodel .................. 121 4.3Results:SymmetricPAHModelwithBandExponents0
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LISTOFTABLES Table page 3-1Crossovere-bcouplingc0forthezero-hybridizationmodel ........... 50 3-2Correlation-lengthcriticalexponentvsbathexponents ............. 65 3-3Staticcriticalexponents,1=,x,and ...................... 71 4-1Correlation-lengthcriticalexponents1and2vsbandexponentr ....... 127 4-2ExponentsdescribingthelocalspinresponseatthecriticalpointCsoftheparticle-hole-symmetricPAHmodel ........................ 129 4-3Exponentsdescribingthelocalspinresponseoftheparticle-hole-asymmetricPAHmodel ...................................... 132 7

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LISTOFFIGURES Figure page 3-1Levelcrossingofthezero-hybridizationmodel ................... 84 3-2Dependenceofthelevel-crossingcouplingc0onthediscretizationforthezero-hybridizationmodel ............................... 85 3-3Schematicphasediagramofthesymmetriccharge-coupledBFAmodelforbathexponents0
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3-19Scalingwith!=Toftheimaginarypartofthedynamicallocalchargesusceptibility00c,loc(!,T)atthecriticale-bcouplingcfors=0.2 ............... 102 3-20ImpurityspectralfunctionA(!;T=0)vsfrequency!fors=0.8 ........ 103 3-21Variationwithe-bcouplingoftwocharacteristicenergyscales!Hand)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(Kfors=0.8 ....................................... 104 3-22DetailoftheimpurityspectralfunctionA(!;T=0)aroundfrequency!=0fors=0.8 ....................................... 105 3-23Variationwithe-bcoupling
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4-5PhaseboundariesofthesymmetricPAHmodel:Variationwithbosoniccouplingofthecriticalhybridizationwidths)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c1and)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(c2 .................. 144 4-6SymmetricPAHmodelatweakbosoniccoupling:Temperaturedependenceoftheimpuritycontribution ............................. 145 4-7SymmetricPAHmodelatstrongbosoniccoupling:Temperaturedependenceoftheimpuritycontribution ............................. 146 4-8LocalspinresponseforsymmetricPAHmodelnearthespin-sectorquantumphasetransition ................................... 147 4-9LocalchargeresponseforsymmetricPAHmodelnearthecharge-sectorquantumphasetransition ................................... 148 4-10BehaviorsoflocalpropertiesforsymmetricPAHmodelatbothweakandstrongcouplings ....................................... 149 4-11PhaseboundariesofthePAHmodelonthed-)]TJ /F1 11.955 Tf 10.1 0 Td[(planeforthreeweakbosoniccouplings ....................................... 150 4-12PhaseboundariesofthesymmetricPAHmodelontheh-)]TJ /F1 11.955 Tf 10.1 0 Td[(planeforthreestrongbosoniccouplings .................................. 151 4-13TemperaturedependenceofimpuritycontributionsforU2=0double-quantum-dotdeviceatweakbosoniccoupling .......................... 152 4-14TemperaturedependenceofimpuritycontributionsforU2=0double-quantum-dotdeviceatstrongbosoniccoupling .......................... 153 4-15PhasediagramsofU2=0double-quantum-dotdevice .............. 154 4-16LinearconductancegforU2=0double-quantum-dotdevicenearthespin-sectorquantumphasetransition .............................. 155 4-17LinearconductancegforU2=0double-quantum-dotdevicenearthecharge-sectorquantumphasetransition .............................. 156 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyQUANTUMPHASETRANSITIONSOFMAGNETICIMPURITIESINDISSIPATIVEENVIRONMENTSByMengxingChengDecember2010Chair:KevinIngersentMajor:Physics Thisdissertationpresentsresultsoftheoreticalresearchonquantumphasetransitionsinsystemswhereamagneticimpurityhybridizeswithafermionichostandisalsocoupled,viatheimpuritycharge,tooneormorebosonicmodesrepresentingdissipativeenvironments.Twosuchdissipativequantumimpuritymodelsarestudiedusingthenumericalrenormalization-grouptechnique. Thecharge-coupledBose-FermiAndersonmodeldescribesamagneticimpuritythathybridizeswithconductionelectronsinametalandisalsocoupledtoabathofdispersivebosons.Themetallichostisdescribedbyaconstantdensityofstates,whilethebathisdescribedbyaspectraldensityproportionalto!s,wherethevalueoftheexponentsdependsontheparticularrealizationofthemodel.Thefollowingpropertiesofthemodelareestablished:(i)Astheimpurity-bathcouplingincreasesfromzeroatxedimpurity-bandhybridization,theeffectiveCoulombinteractionbetweentwoelectronsintheimpuritylevelisprogressivelyrenormalizedfromitsrepulsivebarevalueuntiliteventuallybecomesattractive.Forweakhybridization,thisrenormalizationinturnproducesacrossoverfromaconventional,spin-sectorKondoeffecttoachargeKondoeffect.(ii)Atparticle-holesymmetry,andforsub-Ohmicbathexponents0
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impurityoccupancytoalocallyappliedelectricpotentialfeaturesthehyperscalingofcriticalexponentsand!=Tscalingthatareexpectedataninteractingcriticalpoint.FortheOhmiccases=1,thetransitionisinsteadofKosterlitz-Thoulesstype.(iii)Awayfromparticle-holesymmetry,thequantumphasetransitionisreplacedbyasmoothcrossover,butsignaturesofthesymmetricquantumcriticalpointremaininthephysicalpropertiesatelevatedtemperaturesand/orfrequencies. InthepseudogapAnderson-Holsteinmodel,amagneticimpuritylevelhybridizeswithafermionichostwhosedensityofstatesvanishesasjjrattheFermienergy(=0)andisalsocoupled,viatheimpuritycharge,toalocalbosonmode.WendthatthepseudogapAnderson-Holsteinmodelshowsdistinctivelow-temperaturequantumuctuationsintworegimes,dependingonthestrengthoftheimpurity-bosoncoupling.Westudytwocasesofbandexponents:0
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CHAPTER1INTRODUCTION 1.1Outline Thedissertationisorganizedasfollows.Thisrstchapterprovidesbackgroundknowledgeandmotivationfortheresearch.WerstreviewthehistoryandmodelsofKondophysics.Afterabriefoverviewofthegeneralconceptofaquantumphasetransition,weprovideadiscussionofrelevantexperimentalresultsandreviewrelatedtheory,motivatingthestudyofquantumphasetransitionsofamagneticimpurityinadissipativeenvironment.Chapter 2 describestheformalismofthenumericalrenormalizationgroup,thetechniqueweusetoinvestigatethisareaofphysics.Chapter 3 presentsresultsforacharge-coupledBose-FermiAndersonmodeldescribingamagneticimpuritythathybridizeswithconductionelectronsinahostmetalandisalsocoupledtoadispersivebosonicbath.Chapter 4 presentsresultsforapseudogapAnderson-Holsteinmodelofamagneticimpuritythathybridizeswithapseudogappedfermionichostandisalsocoupled,viaitscharge,toalocalbosonmode.Chapter 5 summarizesourresultsandindicatesfuturedirections. 1.2TheKondoEffect Thebehaviorofmagneticimpuritiesinmetalshasbeenachallengingandstimulatingeldsincethe1930s[ 1 ].Theexperimentalobservationofalow-temperatureminimumintheresistanceofnominallypuremetalssuchassilverandgoldwasnotexplaineduntil1964,whenKondo[ 2 ]calculatedtheresistivityinthesingle-impuritys-dmodel,nowmoreusuallyreferredtoastheKondomodel,describedbytheHamiltonian ^HK=Xkkcykck+J NkXk,k0,,0cyk1 20ck00S.(1) Here,ckistheannihilationoperatorforaconductionelectronofspinzcomponent=1 2andenergyk.JisthestrengthoftheexchangeinteractionbetweentheimpurityspinSandtheconductionelectronspinattheimpuritysite,writteninthetermsofthe 13

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setofPaulimatrices0.Nkisthenumberofunitcellsinthehostmetaland,hence,thenumberofinequivalentwavevectorkvalues. Kondo'sperturbativecalculationtothethirdorderinJcorrectlydescribestheobservedupturnoftheresistanceatlowtemperatures[ 2 ].However,italsoproducesanunphysicallogarithmicdivergenceoftheresistanceatT=0.Later,amoreadvancedperturbativeapproachusingmany-bodydiagrammatictechniques[ 3 ]tocarryoutinnite-ordersummationofleadingdiagramsworkedwellforferromagneticexchange(J<0)butforantiferromagneticexchange(J>0)movedthedivergencefromT=0toT=TK'(kB))]TJ /F7 7.97 Tf 6.59 0 Td[(1Dexp[)]TJ /F4 11.955 Tf 9.3 0 Td[(1=(0J)],wherekBisBoltzmann'sconstant,Dishalfofthebandwidth,and0istheFermi-energydensityofstates.Inthelate1960s,Andersonandhiscollaboratorsdevelopedanewscalingapproach[ 4 7 ]whichshedlightontheKondoproblem.ByadjustingJtocompensateforprogressivelyeliminatinghigh-energyexcitations,theyfoundascalingtrajectorysuggestingthattheantiferromagneticcouplingJapproachesinnityatverylowtemperatures.Thismeansthattheimpurityisstronglycoupledtotheconductionelectronstoformaspinsinglet,leavingacompletelycompensatedgroundstatewithnon-magneticbehavior.In1975,Wilsoncombinedtheideaofscalingwiththatofrenormalizationfromhigh-energytheorytodevelopthenumericalrenormalization-group(NRG)method,anon-perturbativetoolwhichheinitiallyappliedtotheKondoproblem[ 8 ],successfullygivingtheentirepictureofcrossoverfromanasymptoticallyfreemagneticimpurityathightemperaturestoastronglyboundKondosingletatlowtemperatures. Inordertounderstandtheformationofthelocalmomentinahostmetal,Andersonhadpreviouslyproposedamodelconsideringscatteringofconductionelectronsofftransitionmetalorrareearthimpuritieswhosedlevelsmixweaklywiththehostmetal'sconductionbandandexperienceaCoulombinteractionfordoubleoccupancy[ 9 ].The 14

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Andersonmodelis ^HA=Xkkcykck+d^nd+1 p NkXk(Vkcykd+Vkdyck)+U^nd"^nd#.(1) Here,dannihilatesanelectronofspinzcomponent=1 2(or=",#)andenergyd<0intheimpuritylevel,^nd=dyd,^nd=^nd"+^nd#,andU>0istheCoulombrepulsionbetweentwoelectronsintheimpuritylevel.NkhasthesamedenitionasinEq.( 1 ).Vkisthehybridizationbetweentheimpurityandaconduction-bandstateofenergykannihilatedbyfermionicoperatorck.Henceforth,wefollowcommonpracticeandassumealocalhybridizationVk=V. IntheatomiclimitV=0,thepossibleimpuritycongurationsare(i)anemptystatej0iwithenergy0;(ii)singlyoccupiedstatesj"iandj#iwithenergyd;and(iii)adoublyoccupiedstatej"#iwithenergy2d+U.If)]TJ /F5 11.955 Tf 9.3 0 Td[(dkBTandd+UkBT,thesinglyoccupiedstateswillbeenergeticallyfavored,leavinganetspin="or#ontheimpurity.InthenoninteractinglimitU=0,theeffectofthehybridizationVistobroadentheimpuritydleveltoaLorentzianresonanceofwidth)-359(=0jVj2,where0istheconductionbanddensityofstates.ForageneralcasewithU>0andV6=0,theconditionsforlocal-momentformationaremodiedtobe)]TJ /F5 11.955 Tf 9.3 0 Td[(dmax(,kBT)andd+Umax(,kBT).Inthislimit,theAndersonHamiltonianEq.( 1 )canbemappedontotheKondoHamiltonianEq.( 1 )throughtheSchrieffer-Wolfftransformation[ 10 ],givingtheexchangeinteractionJas J=2jVj21 )]TJ /F5 11.955 Tf 9.3 0 Td[(d+1 d+U.(1) Thus,theKondomodeldescribesalimitingsituationofthemoregeneralAndersonmodel. Sincetheendof1990s,theKondoandAndersonmodelsofmagneticimpuritieshavebeenrealizedbothinsemiconductorquantumdots[ 11 12 ]andinsingle-moleculetransistors[ 13 14 ].Atlowtemperatures,interactionbetweenalocalizedelectronwith 15

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anunpairedspininthedotormoleculeandconductionelectronsintheelectricalleadsproducesaKondo(orAbrikosov-Suhl)resonanceintheimpuritydensityofstatesattheFermienergy,whichgreatlyenhancesthezero-biasconductanceevenuptotheunitarylimit[ 15 ].Zero-biasconductancedatacollapsetoauniversalfunctionofanormalizedratioT=TK,indicatingthattheKondotemperatureTKisthecharacteristicenergyscale. 1.3OverviewofQuantumPhaseTransitions Phasetransitionsareubiquitousandfamiliar.Inourdailylife,weallhaveobservedwaterboilingandsnowmelting.Amongclassicexamplesinphysicstextbooksaremagneticandsuperconductingphasetransitionsreacheduponloweringtemperature.Theseclassicalphasetransitionsresultfromadelicatebalanceofenergyandentropyreachedbytuningthetemperature[ 16 ]. Duringrecentyears,enormousattentionhasbeenattractedtoanotherkindofphasetransition,namely,quantumphasetransitions(QPTs)occurringattheabsolutezerooftemperature(T=0)[ 17 18 ].QPTsarecloselyassociatedwithmanyfundamentalproblemsincondensedmatterphysics,includingtherichbehaviorsofheavy-fermionmetals[ 19 ],thehigh-temperaturesuperconductivityincupratecompounds[ 20 21 ],themetal-insulatortransitionindisorderedelectronicsystems[ 22 ],andthequantumHalleffectintwo-dimensionalelectronsystems[ 23 ]. AtT=0,thermaluctuationsarecompletelysuppressed.However,Heisenberg'suncertaintyprincipletellsusthatitisimpossibletosimultaneouslydetermineboththepositionandmomentumofaparticle.Thusthebasiclawofquantummechanicspredictsnon-thermalzero-pointuctuationsthatmaydestroylong-rangemacroscopicorderjustasthermaluctuationsdoatclassicalphasetransitions.Atrstglance,QPTsseemtobeoflittlepracticalinterestbecausetheabsolutezerooftemperatureisnotaccessibleinanyreal-worldexperiment.However,asfoundoverthelasttwodecades,thepresenceofaQPTatT=0hasgreatinuenceonthemeasurable 16

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physicalpropertiesatT>0,producingfascinatingphasediagramsinavarietyofmaterials[ 19 24 25 ]. TounderstandthephysicsofQPTs,itisusefultodrawanalogieswiththetheoryofclassicalphasetransitions[ 17 ].Aclassicalphasetransitiontakesplacewhenasystemiscooleddowntoatransitiontemperature,whereasaquantumphasetransitionoccursattheabsolutezerooftemperaturewhensomeexternalnon-thermalparameterKsuchaspressureormagneticeldistunedtoitscriticalvalueKc.Traditionally,phasetransitionsareclassiedarst-orderorsecond-order(continuous).Atarst-orderclassicalphasetransition(forexamples,icemeltsat0Celsius),alatentheatisinvolvedandtheorderparameterjumpsdiscontinuouslyatthetransitiontemperature,whileatarst-orderquantumphasetransition,thereisalevel-crossingofthegroundstateatK=Kc.Moreinterestingisthesecond-orderphasetransitionwherelong-rangeordering,arisingfromspontaneoussymmetrybreaking,vanishescontinuouslyasapproachingthetransition.Classically,asT!Tc,thespatialcorrelationlengthdivergesas/j(T)]TJ /F6 11.955 Tf 12.71 0 Td[(Tc)=Tcj)]TJ /F14 7.97 Tf 6.59 0 Td[(.Hereisthecorrelation-lengthcriticalexponent.Asaconsequenceofthediverging,manyphysicalpropertiesshowscalingformscontrolledbyasetofcriticalexponentscharacterizingtheuniversalityclassofthephasetransition.Incontrast,continuousquantumphasetransitionsaremorecomplicatedbecausespaceandtimeareintertwinedinthecriticalregion(asexplainedinthenextparagraph).Indeed,asK!Kc,notonlythespatialcorrelationlengthdivergesas/j(K)]TJ /F6 11.955 Tf 12.26 0 Td[(Kc)=Kcj)]TJ /F14 7.97 Tf 6.59 0 Td[(,butthecorrelationlengthinthetimedirectionalsodivergesas/z,wherezisthedynamicalcriticalexponent. Aquantumphasetransitioncanbeunderstoodinthelanguageofquantumstatisticalmechanicsasfollows[ 18 ].Theweightfunctione)]TJ /F14 7.97 Tf 6.58 0 Td[(H(=1=kBT,wherekBistheBoltzmannconstant)inthepartitionfunctionlooksexactlylikethequantum-mechanicaltime-evolutionoperatore)]TJ /F8 7.97 Tf 6.59 0 Td[(iHT=~foranimaginarytimeT=)]TJ /F6 11.955 Tf 9.29 0 Td[(i~.WithinFeynman'spath-integralformalism,theimaginarytimeactslikeanadditionaldimension.In 17

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thissense,theexpressionforthepartitionfunctionforarealquantumsysteminddimensionslookslikethepartitionfunctionforaclassicalsystemind+1dimensions,exceptthatthetimedimensionhasanitelength~.AtT=0,theextratimedimensionisextendedtoinnityandwehaveatrued+1classicalsystem.Atanon-zerotemperature,anite-sizecrossoverinthetimedirectionoccurswhenthecorrelationtimebecomeslargerthan~. Animportantpointmissingfromtheintuitivepicturepresentedinthepreviousparagraphisthatspaceandtimeneednottoenterthepathintegralonthesamefooting[ 18 ].Ingeneral,timescalesasthepowerzofaspatialdimension,wherezistheaforementioneddynamicalexponent.ThissuggeststhataquantumphasetransitioninddimensionscorrespondsmorecloselytoaclassicalphasetransitioninD=d+zdimensions.Whetherthislong-believedcorrespondencebetweenquantumandclassicaltransitionsgenericallyholdstruehasrecentlybeenthesubjectofheateddebate[ 26 27 ],asdescribedinmoredetailinthefollowingsection 1.4 1.4QuantumCriticalityofHeavy-FermionMaterials Thestudyoflow-temperatureexcitationsofinteractingfermionsisamongthemostimportantaspectsofcondensed-matterphysics.Historically,LandauproposedaFermi-liquidtheorypostulatingthatthelow-energyexcitationsofinteractingfermionscanbedescribedbylong-livedquasiparticleswhosequantumnumbers(suchaschargeandspin)arethesameasthoseofthefermionsintheabsenceofinteractions[ 28 ].TolowestorderinkBT=F,whereFistheFermienergy,Fermi-liquidtheorypredictsaquadratictemperaturedependenceoftheresistivity(T)=0+AT2,alineartemperaturedependenceofthespecicheatC(T)=T,andatemperature-independentmagneticsusceptibility=M=Bconst.Fermi-liquidtheoryhasbeensuccessfullyappliedtoavarietyofinteractingsystemsrangingfromhelium-3liquidtonormalmetalslikecopperorevencomplexcompoundslikeCeCu6inwhichtheinteractionsbetweenlocalizedf-electronsareverystrong. 18

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Sincethe1990s,aseriesofexperimentsinheavy-fermionmaterials[ 29 32 ],agroupofd)]TJ /F1 11.955 Tf 12.63 0 Td[(andf)]TJ /F1 11.955 Tf 9.3 0 Td[(electroncompoundsfamousfortheirgiganticeffectivemasses(determinedexperimentallybymeasuringtheratioofspecicheattotemperatureC=T),havechallengedthevalidityofLandauFermi-liquidtheory.Forinstance,CeCu6)]TJ /F8 7.97 Tf 6.59 0 Td[(xAuxexhibitsmarkednon-Fermi-liquidbehaviorsatx=0.1,i.e.,C=T=aln(T0=T),=M=B=0)]TJ /F5 11.955 Tf 12.27 0 Td[(p T,and(T)=0+A0T[ 33 ].Similarnon-Fermi-liquidbehaviorswerealsoobservedinYbRh2Si2anditsvariantsbytuninganexternalmagneticeld;infactthereisanupturnC=T/T)]TJ /F7 7.97 Tf 6.59 0 Td[(0.3belowaverylowtemperatureatwhichthelnTbehaviorofC=Tiscutoff[ 34 35 ].Thesenon-Fermi-liquidbehaviorsinheavy-fermionmaterialsseemalwaystobeassociatedwithquantumphasetransitionsinvolvingmagneticordering. Hertzrstputforwardatheorytodescribemagneticquantumphasetransitionsinmetals[ 36 ]andtheworkwaslaterre-examinedandcorrectedbyMillis[ 37 ].Assumingthatlow-energyfermionicexcitationscanbecompletelyintegratedoutandtheorderparameteristheonlysignicantuctuationmodenearthetransition,aQPTinddimensionscanbeconnectedtoaclassical4-eldtheoryind+zdimensions,wherezistheaforementioneddynamicalexponent.Specically,itisfoundthatz=3foraferromagneticQPTandz=2foranantiferromagneticQPT. Hertz-MillistheorypredictsC=T/p Tand/T3=2forantiferromagnetic(z=2)systemsind=3dimensions.Thesetemperaturedependencesareconsistentwithexperimentalobservationsinanumberofheavy-fermionmaterialssuchasCeNi2Ge2andCeCu2Si2[ 30 32 ].However,theycontradictthosebehaviorsmentionedaboveinCeCu6)]TJ /F8 7.97 Tf 6.58 0 Td[(xAuxandYbRh2Si2.Roschin1997pointedoutthatthelnTbehaviorofC=TandthelineardependenceofresistivityinCeCu6)]TJ /F8 7.97 Tf 6.59 0 Td[(xAuxcanbeexplainedwithinHertz-Millistheorybyassumingthatspinuctuationsexistonlyind=2dimensions[ 38 ].However,subsequentexperimentalresultshavemadethisexplanationuntenable.First,neutron-scatteringdataforCeCu6)]TJ /F8 7.97 Tf 6.58 0 Td[(xAuxexhibitE=T 19

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scalingwithananomalousfractionalexponentoveressentiallytheentireBrillouinzone[ 39 ],suggestinganinteractingtypeofcriticalpointonlyexpectedbelowtheuppercriticaldimensiondu.However,Hertz-Millistheorypredictsdu=4)]TJ /F4 11.955 Tf 10.41 0 Td[(2=2,souctuationsind=2or3dimensionswouldbeatorabovetheuppercriticaldimension,leadingtoaGaussian(mean-eld)criticalpoint.Second,experimentsontheHalleffectforYbRh2Si2haveobservedasharpchangeoftheFermisurfaceatthecriticalmagneticeld[ 40 ],whichcontradictsthesmoothevolutionoftheFermisurfaceimpliedbyHertz-Millistheoryandindicatesasuddendisappearanceofalargenumberofchargecarriersatthecriticalpoint.InordertoexplaintheexperimentalphenomenacontradictingtheconventionaltheoryofQPT,Siandcollaboratorsproposedatheoryoflocallycriticalquantumphasetransitions[ 26 27 ].ThemainideaofthetheoryisthatlocaldegreesoffreedomassociatedwiththeKondoeffectandthelong-wavelengthuctuationsoftheorderparameterbecomecriticalsimultaneouslyattheQPT.ThisisdifferentfromtheHertz-Millispicturewhereonlycriticallong-rangeorder-parameteructuationsareimportant.Inthefollowingofthissection,webrieyreviewthetheoryoflocallycriticalquantumphasetransitionsandmotivateourstudyonarelatedmodel. ItisgenerallyacceptedthatQPTsinheavy-fermionmaterialsstemfromcompetitionbetweenKondoscreeningoflocalmomentsandtheRuderman-Kittel-Kasuya-Yosida(RKKY)interactionattemptingtoalignthoselocalmoments[ 41 ].AmicroscopicmodelcapturingthisessentialphysicsistheKondolatticemodel, ^HKL=Xij,tijcyicj+XiJKSisc,i+XijIijSiSj.(1) Here,tijisthetight-bindingparameterthatdeterminestheconduction-banddispersion"kandhencethedensityofstates()=(Nk))]TJ /F7 7.97 Tf 6.59 0 Td[(1Pk()]TJ /F5 11.955 Tf 12.46 0 Td[("k).cyicreatesanelectronofspinzcomponent=1 2atsitei.JKistheKondocouplingbetweenalocalizedspinSiandtheconduction-electronspinsc,iatthesamelatticesite.ThetendencytowardmagnetismentersthroughtheRKKYinteractionIijbetweenthelocalizedspins 20

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ondifferentsitesiandj.TheFouriertransformofIijdeterminestheRKKYdensityofstatesl()=(Nk))]TJ /F7 7.97 Tf 6.59 0 Td[(1Pk()]TJ /F6 11.955 Tf 11.96 0 Td[(Ik). Sietal.havestudiedtheKondolatticemodelwithintheframeworkofextendeddynamicalmean-eldtheory(EDMFT),wherethelatticemodel( 1 )ismappedontoadissipativequantumimpuritymodeltheBose-FermiKondomodel(BFKM),describedbytheHamiltonian[ 26 27 ] ^HBFKM=Xk,kcykck+Xq!qyqq+JKSsc+XqgqS(q+y)]TJ /F13 7.97 Tf 6.59 0 Td[(q).(1) Here,asinglelocalizedspinSiscouplednotonlytotheon-siteconduction-bandspinscbutalsotothreedissipativebosonicbathsofharmonicoscillatorsdescribedbyq,whichrepresentmagneticuctuationsgeneratedbyspinsatallotherlatticesites.Inthedissipativeimpuritymodel,thecouplingoftheimpuritytotheconductionbandisfullyspeciedbytheexchangefunctionJ()=(Nk))]TJ /F7 7.97 Tf 6.59 0 Td[(1JKPk()]TJ /F5 11.955 Tf 12.4 0 Td[(k),whiletheinteractionbetweentheimpurityandthebosonicbathsiscompletelydescribedbyabathspectralfunctionB(!)(Nq))]TJ /F7 7.97 Tf 6.59 0 Td[(1Pqg2q(!)]TJ /F5 11.955 Tf 12.39 0 Td[(!q).Toreproducetheeffectofotherlatticesitesontherepresentativesite,theexchangefunctionandthebathspectralfunctionenteringtheimpuritymodelaredeterminedbyapairofself-consistencyequationsinvolvingtheaforementionedconduction-banddensityofstates()andtheRKKYdensityofstatesl()oftheKondolatticemodel. ItisbelievedthattheRKKYinteractioninCeCu6)]TJ /F8 7.97 Tf 6.59 0 Td[(xAuxishighlyanisotropicandcanbeapproximatedasPijIijSziSzj[ 42 43 ].WithinEDMFT,thissituationisdescribedbyaself-consistentIsing-anisotropicBose-FermiKondomodel,whichhasbeensolvedusingvariousmethodsincluding-expansion[ 27 ],quantumMonteCarlo[ 42 43 ],andthenumericalrenormalizationgroup[ 44 45 ].Allthemethodsagreethat:(a)Formagneticuctuationsin3DwheretheRKKYdensityofstatesl()hasasquare-rootonsetatitsloweredge,thelatticestaticsusceptibilityattheorderingmomentumdivergesatthecriticalpointwhereasthelocalstaticsusceptibilitystaysnite.(b)Bycontrast,for2D 21

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magneticuctuationswheretheRKKYdensityofstatesundergoesajumpatitsloweredge,boththelatticestaticsusceptibilityattheorderingmomentumandthelocalstaticsusceptibilitybecomesingularsimultaneouslyatthecriticalpoint.Furthermore,thecriticallocalstaticsusceptibilityshowsapower-lawdependenceontemperaturewithafractionalexponent'0.75asobservedinexperimentsinCeCu6)]TJ /F8 7.97 Tf 6.58 0 Td[(xAux[ 39 ],whilethecriticallocaldynamicalsusceptibilityexhibitsE=Tscalingwiththesamefractionalexponent. AlthoughthelocallycriticaltheorybasedontheKondolatticemodelreproducesthefeaturesofseveralkeyexperiments,theoriginalmodel( 1 )takesintoaccountlocalspinuctuationsbutignoreslocalchargeuctuations.Itisnaturalistoaskwhathappensifchargeuctuationsareincludedinatwo-bandextendedHubbardmodel[ 46 ] ^HTBHM=Xi"d^ndi+U^ndi"^ndi#+Xt(dyici+H.c.)+V^ndi^nci+JKSdisci# (1) +Xij tijXcyicj+Vij(^ndi)]TJ /F4 11.955 Tf 11.95 0 Td[(1)(^ndj)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+IijSdiSdj!. Themodelinvolvestwokindsofelectrons,namely,thestronglycorrelateddelectronsandthebandcelectrons.Here,distheenergyofthedlevelandUistheHubbardinteractionford-electrondoubleoccupancyatthesamesite.^ndi=P^ndi.tisthehybridizationbetweendelectronsandcelectronsatthesamesite.Visthelocalcharge-screeninginteractionandJKistheKondocoupling.tijistheintersitehoppingforcelectrons.VijandIijarethenon-localdensity-densityinteractionsandspin-exchangeinteractionsbetweendelectrons.WithintheframeworkofEDMFT,thetwo-bandextendedHubbardmodelismappedontoadissipativequantumimpuritymodel,whichhasbeenfoundinitsmixed-valenceregime(andatinniteU)toexhibitanovelphaseassociatedwithstandardFermi-liquidspinexcitationsandunusualnon-Fermi-liquidchargeexcitations[ 46 47 ].Thisimpuritymodeliscloselyrelatedtoacharge-coupledBose-FermiAndersonmodelthatwasrstdevelopedbyHaldanetoaddressthe 22

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mixed-valencephenomenoninrare-earthmaterials[ 48 ].Haldanesolvedthemodelwithamean-eldapproximation[ 49 ]thatleavesmanyopenquestionsaboutthestrong-correlationregime.InChapter 3 ,wereportessentiallyexactnumericalresultsforthecharge-coupledBose-FermiAndersonmodel. 1.5QuantumPhaseTransitionsinQuantum-DotandSingle-MoleculeDevices Studyinggate-tunablearticialnanoscaledevices,suchassemiconductorquantumdotsandsingle-moleculetransistors,isoneofthemostactiveareasofcurrentscienticresearchdrivenbypotentialapplicationsinnanoelectronicsandquantumcomputing.Thesedevices'outstandingfeatures,suchaspreciseexperimentalcontrolbymeansofgateelectrostaticsandlargespacingsbetweendiscreteenergylevelsduetonanoscalespatialconnementofasmallnumberofelectrons,makeitfeasibletoobservequantumphenomenaatrelativelyhightemperatures.Forinstance,theKondo(orAbrikosov-Suhl)resonancehasbeenobservedinbothsemiconductorquantumdots[ 11 12 ]andsingle-moleculetransistors[ 13 14 ]. Whileheavy-fermioncompoundsareprototypicalmaterialsforstudyingbulkquantumphasetransitionsaswehaveseeninthelastsection,semiconductorquantumdotsandsingle-moleculetransistorsprovidenewmeanstoinvestigateboundaryquantumphasetransitionsinwhichonlyazero-measuresubsetofsystemdegreesoffreedombecomecritical[ 50 ].Experimentally,boundaryQPTshavebeenrealizedinquantum-dotsetupswheretwoconductionelectronreservoirsarecompetingtolocalizeamagneticdot[ 51 ]aswellasinsingle-C60transistorsshowingamagneticeldandgate-inducedsinglet-triplettransition[ 52 ].Theoretically,certainquantum-dotsystemshavebeenproposedtorealizeanXY-anisotropicBose-FermiAndersonmodelexhibitinganinterestingboundaryQPT[ 53 ]. OneboundaryQPToftheoreticalinterestisthatoccurringintheKondoandAndersonimpuritymodelswhentheconduction-banddensityofstatesvanishesasjjrattheFermienergy(=0)[ 54 ],asituationthatcanberealizedinanumberof 23

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systemsincludingunconventionald-wavesuperconductors(r=1)[ 55 ]andcertaindouble-quantum-dotsetups(r=2)[ 56 57 ].ThepseudogapinthedensityofstatesinhibitsKondoscreeningandincursalocal-momentphaseforweakimpurity-bandhybridization,whereasthemagneticimpurityisstillscreenedbytheconductionbandthroughtheKondoeffectforstrongimpurity-bandhybridization.ThesetwostablephasesareseparatedbyaQPT,atwhichthereisadivergentresponsetoamagneticeldappliedlocallytotheimpurityspin[ 58 ]. Furthermore,inquantum-dotorsingle-moleculedevices,itisnaturalthatvibrationalmodes(localphonons)playsignicantrole.Anumberofrecentexperimentshaveinvestigatedphonon-assistedelectrontransportthroughaquantumdot[ 59 ]ormolecule[ 60 61 ]intheKondoregime.TheessentialphysicsoftheseexperimentsseemstobecapturedbytheAnderson-Holsteinmodel[ 62 ]^HAH=X"d^nd+U^nd"^nd#+Xk,"kcykck+1 p NkXk,(Vkcykd+Vkdyck)+(^nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1)(a+ay)+!0aya, (1) whichaugmentstheAndersonimpuritymodelEq.( 1 )withaHolsteincoupling[ 63 ]oftheimpurityoccupancytoalocalboson(phonon)modeofenergy!0(setting~=1). InChapter 4 ,wereportourstudyofapseudogapAnderson-Holsteinmodelincorporatingtheeffectsbothofapseudogappedconductionbandandcouplingtoalocalbosonmode.ThemodelturnsouttoexhibitinterestingQPTsthathaveremarkablesignaturesinthenite-temperatureproperties. 24

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CHAPTER2NUMERICALRENORMALIZATION-GROUPMETHODFORQUANTUMIMPURITYMODELS Inthischapter,abriefintroductiontothenumericalrenormalization-group(NRG)methodforsolvingquantumimpuritymodelsinSection 2.1 isfollowedbyamoretechnicaldescriptionofthemethod.TheconventionalNRGmethodforpure-fermionicproblemsissummarizedinSection 2.2 .Section 2.3 describestheextensionoftheNRGtotreatproblemsincludingbosonicbaths. 2.1OverviewofNumericalRenormalization-GroupMethod Aquantumimpuritymodeldescribesasmallsubsystemwithafewdegreesoffreedomcoupledtoalargesubsystemwithacontinuumofstates[ 64 ].AtypicalexampleistheKondoproblem(asdiscussedinSec. 1.2 )whereamagneticimpurityinteractswithametallicconductionband.Studyofquantumimpurityproblemsischallengingbecauseonehastodealwithawiderangeofenergiesfromanultra-violetcutoff(oftenoforderelectronvolts)downtothesmallestscale(typicallysetbykBT,whichmaybeaslowas1eVinquantum-dotexperiments).Perturbationtheoryusuallyfailsforsuchproblemsduetotheappearanceofinfrareddivergences. Inthe1970s,K.G.Wilsondevelopedanon-perturbativemethod-thenumericalrenormalizationgroup-tosolvetheKondoproblem.TheNRGrstsuccessfullygavetheentirecrossoverfromafree-magnetic-momentregimeathightemperaturestoastronglycoupledKondo-singletregimeatlowtemperatures[ 8 ].TheNRGinvolvesthreekeysteps:discretizationoftheconductionband,tridiagonalizationoftheHamiltonian,anditerativesolution. Discretization:Thefullrangeofconduction-bandenergies)]TJ /F6 11.955 Tf 9.3 0 Td[(DDisdividedintoasetoflogarithmicintervalsboundedby=D)]TJ /F8 7.97 Tf 6.59 0 Td[(kfork=0,1,2,...,where>1istheWilsondiscretizationparameter.Thecontinuumofstateswithineachintervalisreplacedbyasinglestate,namely,theparticularlinearcombinationofbandstateswithintheintervalthatdirectlycouplestotheimpuritysite. 25

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Tridiagonalization:TheLanczosprocedure[ 65 ]isusedtoconverttheconduction-bandpartoftheHamiltonian[therstterminEqs.( 1 )and( 1 )]intoatight-bandingform^Hband=DPP1n=0nfynfn+n)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(fynfn)]TJ /F7 7.97 Tf 6.59 0 Td[(1,+fyn)]TJ /F7 7.97 Tf 6.59 0 Td[(1,fn.Theoperatorf0annihilatesaspin-electroninthelinearcombinationofconduction-bandstatesthatcoupletotheimpurity.Theoperatorfnforn>0annihilatesalinearcombinationofelectronsofenergyjjD)]TJ /F8 7.97 Tf 6.59 0 Td[(n. Iterativesolution:Thefullquantum-impurityproblemcanbesolvedbyiterativelydiagonalizingasequenceofHamiltonians^HN(N=0,1,2,...)satisfyingarecursiverelation^HN+1=1=2^HN+P"N+1fyN+1,fN+1,+PN+1)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(fyN+1,fN+fyNfN+1,,whichequivalentlydescribetight-bindingchainsofincreasinglength.Many-bodyenergiesandmatrixelementsfromthediagonalizationarethenusedtocalculatestaticanddynamicproperties.Practically,itisnotfeasibletokeeptrackofalltheeigenstatesbeyondtherstfewiterationsbecausethedimensionoftheFockspaceincreasesexponentiallyasweaddsitestothechain.Instead,theNslowestlyingmany-particlestatesareretainedaftereachiteration.ItisnecessarytocheckthatNshasbeenchosensufcientlylargetoensureconvergenceofphysicalproperties. AfteritsrstapplicationtotheKondomodel[ 8 ],theNRGwasthengeneralizedtomanysituations:theAndersonmodelincludingchargeuctuationsattheimpuritysite[ 66 67 ];thetwo-channelKondomodelwherethemagneticimpurityiscoupledtotwoindependentconductionbands[ 68 ];thetwo-impurityKondomodeltreatingcompetitionbetweenKondoscreeningandcorrelationbetweenmagneticimpurities[ 69 ];themorecomplextwo-impurity,two-channelKondomodel[ 70 ];andtheKondo/Andersonmodelwithapseudogapinthedensityofstates[ 71 ].AnothergeneralizationhasbeenthedevelopmentofabosonicNRG,rstappliedtothespin-bosonmodel[ 72 ]andthentotheBose-FermiKondomodel[ 73 74 ].ManyoftheseapplicationsaredescribedinarecentreviewbyBullaetal.[ 64 ]. 26

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Sections 2.2 and 2.3 describethetechnicalformalismoftheNRGinmoredepth.Thereaderwhoisnotinterestedinthesedetailsmayskiptherestofthischapterandmovedirectlytothepresentationofresultsinchapter 3 2.2NumericalRenormalizationGroupwithaFermionicBand ThissectiondescribestheNRGtreatment[ 66 67 ]ofthesingle-impurityAndersonmodel( 1 ),whichcanbewrittenconvenientlyas ^HA=^Himp+^Hband+^Himp)]TJ /F10 7.97 Tf 6.58 0 Td[(band,(2) where^Himp=d^nd+U^nd"^nd#, (2)^Hband=Xk,kcykck, (2)^Himp-band=1 p NkXk,)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Vkcykd+Vkdyck. (2) Here,dannihilatesanelectronofspinzcomponent=1 2(or=",#)andenergyd<0intheimpuritylevel,^nd=dyd,^nd=^nd"+^nd#,andU>0istheCoulombrepulsionbetweentwoelectronsintheimpuritylevel.Vkisthehybridizationbetweentheimpurityandaconduction-bandstateofenergykannihilatedbyfermionicoperatorck.Nkisthenumberofunitcellsinthehostmetaland,hence,thenumberofinequivalentkvalues.Hereafterinthischapter,summationoverrepeatedspinindicesisassumed.Forsimplicity,weconsideronlyspatiallyisotropicproblems,whichmeansthatkandVkdependonlyonjkj,sothattheimpurityinteractsonlywiths-wavestatesabouttheimpuritysite.Then,itisconvenienttoreplacesummationoverkbyintegrationoverkandreplaceVkbyV(). Theinteractionbetweentheimpurityandtheconductionbandiscompletelydeterminedbythehybridization-widthfunction\()=()jV()j2,where()isthe 27

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densityofstatesoftheconductionband.Inthiswork,weconsiderfunctionsoftheform \()=8>><>>:)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(0j=DjrforjjD,0otherwise.(2) Thecaser=0correspondstoaregularmetallicbandwithaconstantdensityofstates,whereasr>0describesapseudogappeddensityofstates. Introducingadimensionlessscale"==Dforconvenience,theHamiltonianEq.( 2 )canbetransformedintotheone-dimensionalform:^Himp=d^nd+U^nd"^nd#, (2)^Hband=DZ1)]TJ /F7 7.97 Tf 6.58 0 Td[(1d""cy"c", (2)^Himp)]TJ /F10 7.97 Tf 6.59 0 Td[(band=r )]TJ /F7 7.97 Tf 6.78 -1.79 Td[(0D F(fy0d+dyf0), (2) where f0=F)]TJ /F7 7.97 Tf 6.59 0 Td[(1Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1d"w(")c",(2) whichincludesaweightingfunction w(")=s \("D) )]TJ /F7 7.97 Tf 6.78 -1.79 Td[(0,(2) andanormalizationfactor F2=Z1)]TJ /F7 7.97 Tf 6.58 0 Td[(1d"[w(")]2.(2) 2.2.1LogarithmicDiscretization Thecontinuousbandspectrum)]TJ /F4 11.955 Tf 9.29 0 Td[(1"1()]TJ /F6 11.955 Tf 9.3 0 Td[(DD)isdividedintoasequenceofintervals.Thenthpositive(negative)intervalextendsoverenergy(-)from)]TJ /F7 7.97 Tf 6.58 0 Td[((n+1)to)]TJ /F8 7.97 Tf 6.59 0 Td[(n(n=0,1,2,...).InhisoriginaltreatmentoftheKondoandAndersonmodelsinwhichtheconduction-banddensityofstatesisconstant,namelyw(")=1,Wilsondeneda 28

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completesetoforthonormalfunctions[ 8 ] np(")=8>><>>:1 p dnei!np"for)]TJ /F7 7.97 Tf 6.59 0 Td[((n+1)")]TJ /F8 7.97 Tf 6.59 0 Td[(n,0outsidethisinterval,(2) wheredn=)]TJ /F8 7.97 Tf 6.58 0 Td[(n(1)]TJ /F4 11.955 Tf 12.14 0 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1)isthewidthofthenthinterval,pistheFourierharmonicindexrunningfromto+1,!n=2=dnisthefundamentalfrequencyforthenthinterval,andthesuperscript+()]TJ /F1 11.955 Tf 9.3 0 Td[()standsforpositive(negative)".Theconductionelectronoperatorc"canbeexpandedinthisbasisas c"=Xnpanp +np(")+bnp )]TJ /F8 7.97 Tf -.43 -7.89 Td[(np("),(2) intermsofoperatorsanp=Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1d"[ +np(")]c", (2)bnp=Z1)]TJ /F7 7.97 Tf 6.59 0 Td[(1d"[ )]TJ /F8 7.97 Tf -.43 -7.89 Td[(np(")]c", (2) whichsatisfytheusualfermionicanticommutationrelations.Withthisexpansion,theoperatorf0denedinEq.( 2 )isexpandedas f0=F)]TJ /F7 7.97 Tf 6.59 0 Td[(1XnpanpZ)]TJ /F20 5.978 Tf 5.76 0 Td[(n)]TJ /F21 5.978 Tf 5.76 0 Td[((n+1)d" +np(")+bnpZ)]TJ /F7 7.97 Tf 6.59 0 Td[()]TJ /F21 5.978 Tf 5.75 0 Td[((n+1))]TJ /F7 7.97 Tf 6.59 0 Td[()]TJ /F20 5.978 Tf 5.76 0 Td[(nd" )]TJ /F8 7.97 Tf -.43 -7.89 Td[(np(").(2) TheintegralsinEq.( 2 )arezerounlessp=0.Thismeansthatthep=0stateistheonlyonethatcouplesdirectlytotheimpurity. Foranarbitraryw("),onecangeneralizeWilson'sapproachbydening +n0(")=8>><>>:w(")=F+nfor)]TJ /F7 7.97 Tf 6.59 0 Td[((n+1)")]TJ /F8 7.97 Tf 6.59 0 Td[(n,0otherwise, (2) )]TJ /F8 7.97 Tf -.42 -7.98 Td[(n0(")=8>><>>:w(")=F)]TJ /F8 7.97 Tf -1.59 -7.29 Td[(nfor)]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(n")]TJ /F4 11.955 Tf 21.92 0 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[((n+1),0otherwise, (2) 29

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withnormalizationfactors [F+n]2=Z)]TJ /F20 5.978 Tf 5.76 0 Td[(n)]TJ /F21 5.978 Tf 5.75 0 Td[((n+1)d"[w(")]2,[F)]TJ /F8 7.97 Tf -1.59 -7.89 Td[(n]2=Z)]TJ /F7 7.97 Tf 6.59 0 Td[()]TJ /F21 5.978 Tf 5.76 0 Td[((n+1))]TJ /F7 7.97 Tf 6.58 0 Td[()]TJ /F20 5.978 Tf 5.76 0 Td[(nd"[w(")]2.(2) Withthischoice, f0=F)]TJ /F7 7.97 Tf 6.59 0 Td[(1XnF+nan0+F)]TJ /F8 7.97 Tf -1.59 -7.89 Td[(nbn0.(2) Inthenewbasis,thebandtermEq.( 2 )istransformedinto ^Hband'DXn"+nayn0an0+")]TJ /F8 7.97 Tf 0 -7.89 Td[(nbyn0bn0,(2) where"+n=[F+n])]TJ /F7 7.97 Tf 6.58 0 Td[(2Z)]TJ /F20 5.978 Tf 5.76 0 Td[(n)]TJ /F21 5.978 Tf 5.75 0 Td[((n+1)d""[w(")]2, (2)")]TJ /F8 7.97 Tf 0 -7.89 Td[(n=[F)]TJ /F8 7.97 Tf -1.59 -7.89 Td[(n])]TJ /F7 7.97 Tf 6.58 0 Td[(2Z)]TJ /F7 7.97 Tf 6.59 0 Td[()]TJ /F21 5.978 Tf 5.75 0 Td[((n+1))]TJ /F7 7.97 Tf 6.59 0 Td[()]TJ /F20 5.978 Tf 5.76 0 Td[(nd""[w(")]2. (2) Here,followingtheargumentbyWilson,weignorethep6=0conductionbandstatesbecausethesestatesinteractonlyindirectlywiththeimpurity,andfullydecoupleinthecontinuumlimit!1. 2.2.2MappingontoaSemi-InniteChain UsingtheLanczosmethod[ 65 ],thebandHamiltonianEq.( 2 )canbemappedtotheform ^Hband=D1Xn=0"nfynfn+n)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(fynfn)]TJ /F7 7.97 Tf 6.59 0 Td[(1,+fyn)]TJ /F7 7.97 Tf 6.58 0 Td[(1,fn,(2) where00.Theoperatorfnexhibitsonlynearest-neighborcouplingtofn1,,representingtheformofasemi-innitechain.Thisnewsetofoperatorsfnisconstructedfroman0andbn0viaanorthogonaltransformation fn=Xmnmam0+nmbm0,(2) where,0m=F+m=Fand0m=F)]TJ /F8 7.97 Tf -1.59 -7.3 Td[(m=FwerealreadydenedinEq.( 2 ).Theremainingcoefcientsnmandnmaswellastheparameters"nandninthechainformEq.( 2 ) 30

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aredeterminedbyrecursiverelations"n=Xm(2nm"+m+2nm")]TJ /F8 7.97 Tf 0 -7.89 Td[(m), (2)n+1n+1,m=("+m)]TJ /F5 11.955 Tf 11.96 0 Td[("n)nm)]TJ /F5 11.955 Tf 11.96 0 Td[(nn)]TJ /F7 7.97 Tf 6.59 0 Td[(1,m, (2)n+1n+1,m=(")]TJ /F8 7.97 Tf 0 -7.89 Td[(m)]TJ /F5 11.955 Tf 11.96 0 Td[("n)nm)]TJ /F5 11.955 Tf 11.96 0 Td[(nn)]TJ /F7 7.97 Tf 6.59 0 Td[(1,m, (2)1=Xm(2n+1,m+2n+1,m), (2) with00.Theserecursiverelationshavethefeaturethat"n=0forallniftheweightingfunction(hybridizationwidth)hasthesymmetryw(")=w()]TJ /F5 11.955 Tf 9.3 0 Td[(")[\()=\()]TJ /F5 11.955 Tf 9.3 0 Td[()].Inthespecialcaseofaconstantdensityofstatesw(")=1,Wilsonderivedanexplicitexpressionforthehoppingcoefcients n=(1+)]TJ /F7 7.97 Tf 6.59 0 Td[(1)(1)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(n)]TJ /F7 7.97 Tf 6.59 0 Td[(1) 2p 1)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(2n)]TJ /F7 7.97 Tf 6.58 0 Td[(1p 1)]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(2n)]TJ /F7 7.97 Tf 6.59 0 Td[(3)]TJ /F8 7.97 Tf 6.59 0 Td[(n=2!1 2(1+)]TJ /F7 7.97 Tf 6.58 0 Td[(1))]TJ /F8 7.97 Tf 6.59 0 Td[(n=2forn1.(2) Formoregeneralweightingfunctions,theaboverecursiverelationsmustbecalculatednumerically.Itisgenerallyfoundthatthehoppingcoefcientsnandtheon-siteenergies"ndropoffas)]TJ /F8 7.97 Tf 6.59 0 Td[(n=2forlargen. 2.2.3IterativeDiagonalization Afterthelogarithmicdiscretizationthatdividesthecontinuousbandspectrumintoasequenceofintervals.andchainmappingthattransformstheoriginalbandpartoftheAndersonHamiltonianintoasemi-innitechain,thesingle-impurityAndersonHamiltonianistransformedinto ^HA=D1Xn=0"nfynfn+n)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(fynfn)]TJ /F7 7.97 Tf 6.59 0 Td[(1,+fyn)]TJ /F7 7.97 Tf 6.58 0 Td[(1,fn+d^nd+U^nd"^nd#+r )]TJ /F7 7.97 Tf 6.78 -1.8 Td[(0D F(fy0d+dyf0).(2) Becausethechainparametersnand"ndecreaseas)]TJ /F8 7.97 Tf 6.59 0 Td[(n=2,itisconvenienttointroducescaledparameters en=)]TJ /F7 7.97 Tf 6.59 0 Td[(1n=2"n,tn=)]TJ /F7 7.97 Tf 6.59 0 Td[(1n=2n,(2) 31

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where,=1 2(1+)]TJ /F7 7.97 Tf 6.59 0 Td[(1)1=2isaconventionalfactor.Then,oneseethattheHamiltonianEq.( 2 )canberecoveredasthelimitofaseriesofniteHamiltonians ^HA=limN!1)]TJ /F8 7.97 Tf 6.58 0 Td[(N=2D^HN,(2) whereHNsatisestherecursiverelation ^HN+1=1=2^HN+eN+1fyN+1,fN+1,+tN+1(fyN+1,fN,+fyN,fN+1,).(2) TheinitialHamiltonian ^H0=e0fy0f0+~"dnd+~Und"nd#+~)]TJ /F7 7.97 Tf 6.78 4.94 Td[(1=2(fy0d+dyf0),(2) whichincludesonlytheoperatorsf0andd,withthescaledcouplings ~"d="d D,~U=U D,~)-277(=)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(0F2 2D.(2) RepeateduseoftherecursiverelationEq.( 2 )allowsonetosolvetheseriesofHamiltonians^HN.Theprocedureisbasicallyasfollows.Assumethatwehavealreadysolvedtheeigenvalueequation ^HNjr,Ni=E(r,N)jr,Ni,(2) andknowalltheeigenenergiesE(r,N)andmatrixelementshr,Njfynjr0,Ni.Thebasisfor^HN+1canbeconstructedasthedirectproductoftheeigenstatesjr,Niandthenewdegreesoffreedomoftheaddedsite.Explicitly,foreachofthestatesjr,Ni,therearefourcorrespondingstatesj1,r,Ni=j0iN+1jr,Ni, (2)j2,r,Ni=j"iN+1jr,Ni, (2)j3,r,Ni=j#iN+1jr,Ni, (2)j4,r,Ni=j"#iN+1jr,Ni. (2) 32

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SandwichingEq.( 2 )betweenthesestatesgivesthematrixelementsof^HN+1,viz.,hi0,r0,Nj^HN+1ji,r,Ni=1=2E(r,N)i,i0r,r0+eN+1hi0jfyN+1,fN+1,jiir,r0+tN+1(hi0jfyN+1,jiihr0,NjfN,jr,Ni+hr0,NjfyN,jr,Nihi0jfN+1,jii). (2) DiagonalizingthismatrixgivesrisetoE(r,N+1)andhr,N+1jfynjr0,N+1i,whichwillbeusedtosolve^HN+2,andsoon.Notethattheground-stateenergyissettobezeroaftereachdiagonalization. Thewholerecursiveprocedurecanbeunderstood[ 8 ]intermsofarenormalizationgrouptransformationR: ^HN+1=R[^HN],(2) whichtransformstheHamiltonianspeciedbyasetofeigenenergiesandmatrixelementsintoanotherHamiltonianofthesameformbutwithanewsetofeigenenergiesandmatrixelements.Actually,duetotheodd-evenalternationpropertiesoffermionicchains,RitselfhasnoxedpointbutR2doeshavexedpoints. ThenumericallaborofdiagonalizingtheHamiltoniancanbesignicantlyreducedbytakingadvantageofsymmetries.TheHamiltonianEq.( 2 )commuteswiththetotalspinoperator ^S=1 2Xnfyn,,0fn,0+1 2dy,0d0,(2) where,0isthesetofPaulimatrices,andwiththetotalchargeoperator ^Q=^nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1+Xn)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(fyn"fn"+fyn#fn#)]TJ /F4 11.955 Tf 11.95 0 Td[(1.(2) Atparticle-holesymmetry(d=)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 2U)only,theHamiltonianalsocommuteswiththeSU(2)isospin(axialcharge)operators ^Iz=1 2^Q,^I+=)]TJ /F6 11.955 Tf 9.3 0 Td[(dy"dy#+Xn()]TJ /F4 11.955 Tf 9.3 0 Td[(1)nfyn"fyn#)]TJ /F4 11.955 Tf 4.73 -7.03 Td[(^I)]TJ /F12 11.955 Tf 7.09 11.48 Td[(y.(2) 33

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Consequently,theHamiltoniancanbediagonalizedinsubspaceslabeledbyconservedquantumnumbersS,Sz,Q,and(atparticle-holesymmetryonly)I. ThedimensionoftheFockspacegrowsbyafactoroffourateachsuccessiveiteration.Afterafewiterations,thisbecomestoobigtoallowallstatestobecomputed,soatruncationmustbeintroduced.TheNseigenstateswiththelowestenergiesarekeptaftereachdiagonalization.ProvidedthatNsischosenlargeenough,theNRGsolutionatiterationNallowsaccurateevaluationofstaticthermodynamicpropertiesattemperaturesT'(D=kB))]TJ /F8 7.97 Tf 6.59 0 Td[(N=2andofzero-temperaturedynamicalpropertiesatfrequenciesj!j'(D=~))]TJ /F8 7.97 Tf 6.59 0 Td[(N=2.FortheAndersonmodel,Ns'500isbigenoughtoaccuratelycalculatetheeigenenergies,whilemuchlargerNs('2000)isneededtocalculatetheimpuritycontributiontotheentropyandthespinmagneticsusceptibility.Althoughthistruncationmayseemtobequestionable,itissuccessfulinpracticebecausethediscardedhigh-energyeigenstateshaveverylittleinuenceonthelow-energyeigenstatesofthenextiteration. 2.3NumericalRenormalizationGroupwithaBosonicBath WewilldescribethebosonicNRG[ 72 ]usingtheexampleofthespin-bosonmodel,whichdescribestunnelingwithinatwo-statesystemcoupledtoabosonicbath[ 75 ].Themodelhasmanyproposedapplications,includingfrictionaleffectsonbiologicalandchemicalreactionrates[ 76 ],coldatomsinaquasi-one-dimensionalopticaltrap[ 77 ],aquantumdotcoupledtoLuttinger-liquidleads[ 78 ],andstudyofentanglementbetweenaqubitanditsenvironment[ 79 80 ].Inmanycases,thedissipativebosonicbathcanbedescribedbyaspectraldensity[formallydenedinEq.( 2 )below]thatisproportionalto!satlowfrequencies!.Thespin-bosonmodelwithanOhmic(s=1)bathhaslongbeenknown[ 75 ]toexhibitaKosterlitz-ThoulessQPTbetweendelocalizedandlocalizedphases.TheexistenceofaQPTforsub-Ohmic(0
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providedbytheNRG[ 72 ],byperturbativeexpansionin=saboutthedelocalizedxedpoint[ 82 ],andthroughexact-diagonalizationcalculations[ 83 ]. TheHamiltonianofthespin-bosonmodelreads^HSB=^Hspin+^Hbath+^Hspin)]TJ /F10 7.97 Tf 6.59 0 Td[(bath=Sx+Xq!yqyqq+1 p NqSzXqgyq(q+yq), (2) whereisthetunnelingbetweenspin-upandspin-downstates,yqandqarethecreationandannihilationoperatorsforaharmonicoscillatorofenergy!q,Nqisthenumberofoscillatorsinthebath(thenumberofdistinctvaluesofq),andgqparameterizesthecouplingbetweentheimpurityspinzcomponentandthedisplacementoftheoscillatorwithwavevectorq.Theinteractionbetweenthespinandthebathisentirelydeterminedbythespectralfunction,takentohavethepower-lawform B(!) NqXqg2q(!)]TJ /F5 11.955 Tf 11.96 0 Td[(!q)=8>><>>:B01)]TJ /F8 7.97 Tf 6.59 0 Td[(s!sfor0!,0otherwise,(2) whichischaracterizedbyanuppercutoff,anexponentsthatmustsatisfys>)]TJ /F4 11.955 Tf 9.3 0 Td[(1toensurenormalizability,andadimensionlessprefactorB0. ThebosonicNRGisverysimilartothefermionicNRGdescribedinthelastsection,exceptthatthespectralfunctionEq.( 2 )isnonzeroonlyforpositivefrequencieswhereasthehybridization-widthfunctionEq.( 2 )isdenedforbothpositiveandnegativefrequencies. Introducingadimensionlessscaley=!=,thebathandspin-bathtermsoftheHamiltoniancanbewrittenas ^Hbath=Z10dyyyyy,(2) and ^Hspin)]TJ /F10 7.97 Tf 6.59 0 Td[(bath=r B0B2 Sz(b0+by0).(2) 35

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Here,thenewoperatorb0isdenedas b0=B)]TJ /F7 7.97 Tf 6.59 0 Td[(1Z10dyW(y)y,(2) withaweightingfunction W(y)=s B(y) B0,(2) andanormalizationfactor B2=Z10dy[W(y)]2.(2) Thecontinuousbathspectrum0y1(0!)isdividedintoasequenceofintervals,thenthofwhichextendsovertheenergyrangefrom)]TJ /F7 7.97 Tf 6.58 0 Td[((n+1)to)]TJ /F8 7.97 Tf 6.59 0 Td[(n(n=0,1,2,...).Withineachinterval,acompletesetoforthonormalfunctionssimilartothatofthefermionicNRGcanbeintroduced.ThebathtermintheHamiltonianisapproximated ^Hbath'Xm!mymm,(2) with !m=B)]TJ /F7 7.97 Tf 6.58 0 Td[(2mZ)]TJ /F20 5.978 Tf 5.76 0 Td[(m)]TJ /F21 5.978 Tf 5.75 0 Td[((m)]TJ /F21 5.978 Tf 5.76 0 Td[(1)dyy[W(y)]2,(2) m=B)]TJ /F7 7.97 Tf 6.59 0 Td[(1mZ)]TJ /F20 5.978 Tf 5.76 0 Td[(m)]TJ /F21 5.978 Tf 5.75 0 Td[((m)]TJ /F21 5.978 Tf 5.76 0 Td[(1)dyW(y)y,(2) andanormalizationfactor B2m=Z)]TJ /F20 5.978 Tf 5.75 0 Td[(m)]TJ /F21 5.978 Tf 5.76 0 Td[((m)]TJ /F21 5.978 Tf 5.75 0 Td[(1)dy[W(y)]2.(2) Deninganewbasis bn=Xmunmm,(2) whereu0m=Bm=B,Eq.( 2 )canbemappedviatheLanczosprocedure[ 65 ]ontoatight-bindingHamiltonian ^Hbath=1Xn=0"Bnbynbn+Bn)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(bynbn)]TJ /F7 7.97 Tf 6.59 0 Td[(1+byn)]TJ /F7 7.97 Tf 6.59 0 Td[(1bn,(2) 36

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whereB00.Theremainingcoefcientsunm,"Bn,andBninthechainformEq.( 2 )aredeterminedbytherecursiverelations"Bn=Xmu2nm!m, (2)Bn+1un+1,m=(!m)]TJ /F5 11.955 Tf 11.95 0 Td[("Bn)unm)]TJ /F5 11.955 Tf 11.96 0 Td[(Bnun)]TJ /F7 7.97 Tf 6.59 0 Td[(1,m, (2)1=Xmu2n+1,m, (2) with!mgivenbyEq.( 2 ).Asaresultoftheone-sidedformofthebathspectralfunction,"BnandBndecreaseas)]TJ /F8 7.97 Tf 6.58 0 Td[(nforlargen. ThechainHamiltoniancanbesolvedrecursivelyasinthefermionicNRG.AnotherapproximationmustbeintroducedbecausethepresenceofbosonsaddsthefurthercomplicationthattheFockspaceisunboundedevenforasingle-sitechain,makingitnecessarytorestrictthemaximumnumberofbosonsperchainsitetoanitenumberNb.WemustensurethatNbissufcientlylargetoproducereliablephysicalproperties. Finally,inordertosolveproblemswithbothfermionicandbosonicchains,wemustkeepinmindthatthespiritoftheNRGistotreatfermionsandbosonsofthesameenergyscaleatthesameiteration.Sincethebosoniccoefcients"BnandBn(/)]TJ /F8 7.97 Tf 6.59 0 Td[(n)inEq.( 2 )decaywithsiteindexntwiceasfastasthefermioniccoefcients"nandn(/)]TJ /F8 7.97 Tf 6.59 0 Td[(n=2)inEq.( 2 ),afterafewiterationstheiterativeprocedurerequiresextensionofthebosonicchainonlyforeverysecondsiteaddedtothefermionicchain. 37

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CHAPTER3RESULTSFORCHARGE-COUPLEDBOSE-FERMIANDERSONMODEL Thischapterisbasedonapublishedpaper.AllthepublishedcontentsarereprintedwithpermissiongrantedunderthecopyrightpolicyoftheAmericanPhysicalSocietyfromMengxingCheng,MatthewT.Glossop,andKevinIngersent,Phys.Rev.B80,165113(2009).Copyright(2009)bytheAmericanPhysicalSociety. 3.1Introduction Thischapterpresentsourinvestigationofacharge-coupledBose-FermiAn-derson(BFA)modelinwhichtheimpuritynotonlyhybridizeswithconduction-bandelectronsbutalsoiscoupled,viaitselectronoccupancy,toabathrepresentingacousticphononsorotherbosonicdegreesoffreedomwhosedispersionextendstozeroenergy.Themodelwasintroducedmorethan30yearsago[ 48 49 ]inconnectionwiththemixed-valenceproblem.Aspinlessversionofthemodelwasalsodiscussedinthesamecontext[ 84 ].Morerecently,verysimilarmodelshavebeenshowntoariseaseffectiveimpurityproblemsintheextendedDMFTforone-andtwo-bandextendedHubbardmodels[ 46 47 ]. OurNRGstudyofthecharge-coupledBFAmodelwithbosonicbathscharacterizedbyexponents0
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Ohmicbaths(correspondingtos=1),theQPTisfoundtobeofKosterlitz-Thoulesstype.Particle-holeasymmetryactsinamanneranalogoustoamagneticeldataconventionalferromagneticorderingtransition,smearingthediscontinuouschangeintheground-stateasafunctionofe-bcouplingintoasmoothcrossover.Signaturesofthesymmetricquantumcriticalpointremaininthephysicalpropertiesatelevatedtemperaturesand/orfrequencies. Therestofthischapterisorganizedasfollows.Section 3.2 introducesthecharge-coupledBFAHamiltonianandsummarizestheNRGmethodusedtosolvethemodel.Section 3.3 containsapreliminaryanalysisofthemodel,focusingonthebosonicrenormalizationoftheeffectiveelectron-electroninteractionwithintheimpuritylevel.Numericalresultsforthesymmetricmodelwithsub-Ohmic(0
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^Hbath=Xq!qayqaq (3)^Himp-band=1 p NkXk,)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Vkcykd+Vkdyck, (3)^Himp-bath=1 p Nq(^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1)Xqq)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(aq+ay)]TJ /F13 7.97 Tf 6.59 0 Td[(q. (3) Here,dannihilatesanelectronofspinzcomponent=1 2(or=",#)andenergyd<0intheimpuritylevel,^nd=dyd,^nd=^nd"+^nd#,andU>0istheCoulombrepulsionbetweentwoelectronsintheimpuritylevel.Vkisthehybridizationbetweentheimpurityandaconduction-bandstateofenergykannihilatedbyfermionicoperatorck,andqcharacterizesthecouplingoftheimpurityoccupancytobosonsinanoscillatorstateofenergy!qannihilatedbyoperatoraq.Nkisthenumberofunitcellsinthehostmetaland,hence,thenumberofinequivalentkvalues.Correspondingly,Nqisthenumberofoscillatorsinthebath,andthenumberofdistinctvaluesofq.Withoutlossofgenerality,wetakeVkandqtoberealandnon-negative.Notethat,throughoutthechapterinallmathematicalexpressionsandlabels,wedropallfactorsofthereducedPlanckconstant~,Boltzmann'sconstantkB,theimpuritymagneticmomentgB,andtheelectronicchargee. Again,themodeldescribesamagneticimpuritythathybridizeswithmetallichostandiscoupled,viatheimpuritycharge,toabathofdispersivebosonsrepresentingdissipativeenvironments.Tofocusonthemostinterestingphysicsofthemodel,weassumeaconstanthybridizationVk=Vandaatconduction-banddensityofstates(perunitcell,perspin-zorientation) ()1 NkXk()]TJ /F5 11.955 Tf 11.95 0 Td[(k)=8>><>>:0=(2D))]TJ /F7 7.97 Tf 6.58 0 Td[(1forjjD,0otherwise,(3) 40

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deningthehybridizationwidth)-318(=0V2.Thebosonicbathiscompletelyspeciedbyitsspectraldensity,whichwetaketohavethepurepower-lawform B(!) NqXq2q(!)]TJ /F5 11.955 Tf 11.96 0 Td[(!q)=8>><>>:(K0)21)]TJ /F8 7.97 Tf 6.58 0 Td[(s!sfor0!,0otherwise,(3) characterizedbyanuppercutoff,anexponentsthatmustsatisfys>)]TJ /F4 11.955 Tf 9.3 0 Td[(1toensurenormalizability,andadimensionlessprefactorK0.Inthiswork,wepresentresultsonlyforthecase=Dinwhichthebathandbandshareacommoncutoff.WealsoadopttheconventionthatK0isheldconstantwhileonevaries,whichwetermtheelectron-boson(e-b)coupling.Itshouldbeemphasized,though,thatthekeyfeaturesofthemodelareanonvanishingFermi-leveldensityofstates(0)>0andtheasymptoticbehaviorB(!)/!sfor!!0.Relaxinganyoralloftheremainingassumptionslaidoutinthisparagraphwillnotaltertheessentialphysicsofthemodel,althoughitmayaffectnonuniversalproperties,suchasthelocationsofphaseboundaries. Formanypurposes,itisconvenienttorewrite[ 66 67 ]theimpuritypartoftheHamiltonian(droppingaconstanttermd) ^Himp=d(^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+U 2(^nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1)2,(3) whered=d+U=2.Mostoftheresultspresentedbelowwereobtainedforthesymmetricmodelcharacterizedbyd=)]TJ /F6 11.955 Tf 9.3 0 Td[(U=2ord=0,forwhichtheimpuritystatesnd=0andnd=2aredegenerateinenergy.Section 3.6 addressesthebehavioroftheasymmetricmodel. Inanyrealizationof^HCCBFAinvolvingcouplingofacousticphononstoamagneticimpurityoraquantumdot,thevalueofthebathexponentswilldependonthepreciseinteractionmechanism.However,phasespaceconsiderationssuggestthatanysuchsystemwilllieinthesuper-Ohmicregimes>1.Modelscloselyrelatedto^HCCBFAhavealsobeenconsideredinthecontextofextendedDMFT[ 47 ],atechniquefor 41

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systematicallyincorporatingsomeofthespatialcorrelationsthatareomittedfromtheconventionalDMFToflatticefermions[ 86 ].ExtendedDMFTmapsthelatticeproblemontoaquantumimpurityprobleminwhichacentralsiteinteractswithbothafermionicbandandoneormorebosonicbaths,thelatterrepresentinguctuatingeffectiveeldsduetointeractionsbetweendifferentlatticesites.Thecharge-coupledBFAmodelservesasthemappedimpurityproblemforvariousextendedHubbardmodelswithnonlocaldensity-densityinteractions[ 46 47 ].Inthesesettings,theeffectivebathexponentsisnotknownapriori,butisdeterminedthroughself-consistencyconditionsthatensurethatthecentralsiteisrepresentativeofthelatticeasawhole.TheextendedDMFTtreatmentofotherlatticemodels[ 26 44 45 ]givesrisetoexponents0
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fermions[ 73 74 ].Thefullrangeofconduction-bandenergies)]TJ /F6 11.955 Tf 9.29 0 Td[(DD(bosonic-bathenergies0!)isdividedintoasetoflogarithmicintervalsboundedby=D)]TJ /F8 7.97 Tf 6.59 0 Td[(k(!=)]TJ /F8 7.97 Tf 6.58 0 Td[(k)fork=0,1,2,...,where>1istheWilsondiscretizationparameter.Thecontinuumofstateswithineachintervalisreplacedbyasinglestate,namely,theparticularlinearcombinationofband(bath)stateswithintheintervalthatenters^Himp-band(^Himp-bath).Thediscretizedmodelisthentransformedintoatight-bindingforminvolvingtwosetsoforthonormalizedoperators:(i)fn(n=0,1,2,...)constructedaslinearcombinationsofallckhavingjkj
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fermionsandbosonsofthesameenergyscaleatthesameiteration.Sincethebosoniccoefcientsdecaywithsiteindextwiceasfastasthefermioniccoefcients,afterafewiterationstheiterativeprocedurerequiresextensionofthebosonicchainonlyforeverysecondsiteaddedtothefermionicchain.Inthiswork,wehavechosenforsimplicitytoworkwithasinglehigh-energycutoffscaleD.Itisthenconvenienttoaddtothebosonicchainateveryeven-numberediteration,sothatthehighest-numberedbosonicsiteisM(N)=bN=2c,wherebxcisthegreatestintegerlessthanorequaltox. TheNRGmethodreliesontwoadditionalapproximations.Evenforpure-fermionicproblems,itisnotfeasibletokeeptrackofalltheeigenstatesbecausethedimensionsoftheFockspaceincreaserapidlyasweaddsitestothechains.Therefore,onlythelowestlyingNsmany-particlestatescanberetainedaftereachiteration.ThepresenceofbosonsaddsthefurthercomplicationthattheFockspaceisinnite-dimensionalevenforasingle-sitechain,makingitnecessarytorestrictthemaximumnumberofbosonsperchainsitetoanitenumberNb.ProvidedthatNsandNbarechosentobesufcientlylarge(asdiscussedinSec. 3.4.1 ),theNRGsolutionatiterationNprovidesagoodaccountoftheimpuritycontributiontophysicalpropertiesattemperaturesTandfrequencies!oforderD)]TJ /F8 7.97 Tf 6.58 0 Td[(N=2. Hamiltonian( 3 )commuteswiththetotalspin-zoperator ^Sz=1 2(^nd")]TJ /F4 11.955 Tf 12.06 0 Td[(^nd#)+1 2Xn)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(fyn"fn")]TJ /F6 11.955 Tf 11.96 0 Td[(fyn#fn#,(3) thetotalspin-raisingoperator ^S+=dy"d#+Xnfyn"fn#)]TJ /F4 11.955 Tf 6.36 -7.03 Td[(^S)]TJ /F12 11.955 Tf 7.08 11.48 Td[(y,(3) andthetotalchargeoperator ^Q=^nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1+Xn)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(fyn"fn"+fyn#fn#)]TJ /F4 11.955 Tf 11.95 0 Td[(1,(3) 44

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whichmeasuresthedeviationfromhalf-llingofthetotalelectronnumber.Onecaninterpret ^Iz=1 2^Q,^I+=)]TJ /F6 11.955 Tf 9.3 0 Td[(dy"dy#+Xn()]TJ /F4 11.955 Tf 9.3 0 Td[(1)nfyn"fyn#)]TJ /F4 11.955 Tf 4.73 -7.02 Td[(^I)]TJ /F12 11.955 Tf 7.09 11.47 Td[(y(3) asthegeneratorsofanSU(2)isospinsymmetry(originallydubbedaxialchargeinRef.[ 69 ]).Since[^Himp-bath,^I]6=0,thecharge-coupledBFAmodeldoesnotexhibitfullisospinsymmetry.However,thissymmetryturnsouttoberecoveredintheasymptoticlow-energybehavioratcertainrenormalization-groupxedpoints. AsdescribedinRef.[ 66 ],thecomputationaleffortrequiredfortheNRGsolutionofaproblemcanbegreatlyreducedbytakingadvantageoftheseconservedquantumnumbers.Inparticular,itispossibletoobtainallphysicalquantitiesofinterestwhileworkingwithareducedbasisofsimultaneouseigenstatesof^S2,^Sz,and^QwitheigenvaluessatisfyingSz=S.WithoneexceptionnotedinSec. 3.4.7 ,anyNsvaluespeciedbelowrepresentsthenumberofretained(S,Q)multiplets,correspondingtoaconsiderablylargernumberof(S,Sz,Q)states. Evenwhenadvantageistakenofallconservedquantumnumbers,NRGtreatmentofthecharge-coupledBFAmodelremainsmuchmoredemandingthanthatoftheAndersonmodel[Eq.( 3 )]ortheAnderson-Holsteinmodel[Eq.( 3 )].Beingnondispersive,thebosonsinthelastmodelenteronlytheatomic-limitHamiltonian^H0,allowingsolutionviathestandardNRGiterationprocedure.ForBose-Fermimodelssuchas^HCCBFA,theneedtoextendabosonicchainaswellasafermioniconeateveryeven-numberediterationN>0,expandsthebasisof^HNfrom4Nsstatesto4(Nb+1)Nsstates,andmultipliestheCPUtimebyafactor(Nb+1)3.SincewetypicallyuseNb=8or12inourcalculations,theincreaseincomputationaleffortisconsiderable. ThechoiceofvaluefortheNRGdiscretizationparameterinvolvestrade-offsbetweendiscretizationerror(minimizedbytakingtobenotmuchgreaterthan1)andtruncationerror(reducedbyworkingwith1).Experiencefromsimilarkindofproblems[ 58 73 74 ]indicatesthatcriticalexponentscanbedeterminedveryaccurately 45

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usingquitealarge.Mostoftheresultspresentedintheremainingsectionsoftheworkwereobtainedfor=9,with=3beingemployedinthecalculationoftheimpurityspectralfunction.Forconvenienceindisplayingtheseresults,weset=D=1andomitallfactorsof0andK0. 3.3PreliminaryAnalysis Webeginbyexaminingthespecialcasesinwhichtheimpuritylevelisdecoupledeitherfromtheconductionbandorfromthebosonicbath.UnderstandingthesecasesallowsustoestablishsomeexpectationsforthebehaviorofthefullmodeldescribedbyEq.( 3 ). 3.3.1ZeroHybridization ThekeypointofthissectionistoprovethatthemaineffectofthebosonsistoreducetheCoulombinteractionUtoaneffectiveoneUeff,whichmightbenegativeforsufcientlylargee-bcoupling.Ifonesets)-343(=0inEq.( 3 ),thentheconductionbandcompletelydecouplesfromtheremainingdegreesoffreedomandcanbedroppedfromthemodel,leavingthezero-hybridizationmodel ^HZH=d(^nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+U 2(^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1)2+Xq!qayqaq+1 p Nq(^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1)Xqq)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(aq+ay)]TJ /F13 7.97 Tf 6.59 0 Td[(q.(3) TheFockspaceseparatesintosectorsofxedimpurityoccupancy(nd=0,1,or2),withineachofwhichtheHamiltoniancanberecast,usingdisplaced-oscillatoroperators and,q=aq+q p Nq!q(nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1),(3) inthetriviallysolvableform ^HZH(nd)=^H0imp+Xq!qaynd,qand,q,(3) where ^H0imp=d(^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+Ueff 2(^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1)2.(3) 46

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ThebosonsactontheimpuritytoreducetheCoulombinteractionfromitsbarevalueUtoaneffectivevalue Ueff=U)]TJ /F4 11.955 Tf 16.93 8.09 Td[(2 NqXq2q !q=U)]TJ /F4 11.955 Tf 13.55 8.09 Td[(2 Z10B(!) !d!.(3) ForthebathspectraldensityinEq.( 3 )with)]TJ /F4 11.955 Tf 9.3 0 Td[(10,forwhichEqs.( 3 )and( 3 )give Ueff=U)]TJ /F4 11.955 Tf 13.16 8.09 Td[(2(K0)2 s.(3) Forweake-bcouplings,Ueffispositiveandthegroundstateof^HZHliesinthesectornd=1wheretheimpurityhasaspinzcomponent1 2.However,Ueffisdrivennegativeforsufcientlylarge,placingthegroundstateinthesectornd=0ornd=2wheretheimpurityisspinlessbuthasacharge(relativetohalflling)of)]TJ /F4 11.955 Tf 9.3 0 Td[(1or+1. Figure 3-1 illustratesthisrenormalizationoftheCoulombinteractionforthesymmetricmodel(d=0),inwhichthend=0andnd=2statesalwayshavethesameenergy.Inthiscase,allfourimpuritystatesbecomedegenerateatacrossovere-bcoupling K0c0=p sU=2.(3) Theimpuritycontributionstophysicalpropertiesatthisspecialpoint,whichischaracterizedbyeffectiveparameters)-365(=U=d=0,areidenticaltothoseatthefree-orbitalxedpoint[ 66 67 ]oftheAndersonmodel. Forthegeneralcaseofanasymmetricimpurity,thesectorsnd=0and2haveaground-stateenergydifferenceE0(nd=2))]TJ /F6 11.955 Tf 12.34 0 Td[(E0(nd=0)=2dforanyvalueof.TheoverallgroundstateofEq.( 3 )isadoublet(nd=1,S=1 2)forsmalle-bcouplings,crossingovertoasinglet(nd=0ford>0,ornd=2ford<0)forlarge.AtK0c0=p s(U=2)-222(jdj)=,apointofthree-foldground-statedegeneracy,theimpurity 47

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contributionstolow-temperature(Tjdj)physicalpropertiesareidenticaltothoseatthevalence-uctuationxedpointoftheAndersonmodel[ 66 67 ]. UsingtheNRGwithonlyabosonicchain[Eq.( 3 )]coupledtotheimpuritysite,wehaveconrmedtheexistenceford=0ofasimplelevelcrossingfromaspin-doubletgroundstateforc0.Intheformerregime,thebosonscoupleonlytothehigh-energy(nd=0,2)impuritystates,sothelow-lyingspectrumisthatoffreebosonsobtainedbydiagonalizingHNRGbathgiveninEq.( 3 ).Here,NRGtruncationplaysanegligibleroleprovidedthatoneworkswithNb8(say). For>c0,thelow-lyingbosonicexcitationsshould,inprinciple,correspondtononinteractingdisplacedoscillatorshavingpreciselythesamespectrumastheoriginalbath.However,theoccupationnumberayqaqinthegroundstateofEq.( 3 )obeysaPoissondistributionwithmean2q=(Nq!2q).Thus,thetotalnumberofbosonscorrespondingtooperatorsaqsatisfying)]TJ /F7 7.97 Tf 6.58 0 Td[((k+1)>><>>>:(K0)2 lnfors=1,(K0)2 )]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F8 7.97 Tf 6.59 0 Td[(s)]TJ /F4 11.955 Tf 11.96 0 Td[(1 (1)]TJ /F6 11.955 Tf 11.95 0 Td[(s)(1)]TJ /F8 7.97 Tf 6.59 0 Td[(s)kotherwise.(3) ThebathstatesinthekthintervalarerepresentedbyNRGchainstates0mk,withthegreatestweightbeingbornebystatem=k.Thus,afaithfulrepresentationofthedisplaced-oscillatorspectrumrequiresinclusionofstateshavingbymbmuptoseveraltimesh^nmi0;basedonexperiencewiththeAnderson-Holsteinmodel[ 62 ],oneexpectsNb4h^nmi0tosufce.Giventhath^nmi0/(1)]TJ /F8 7.97 Tf 6.59 0 Td[(s)m,itisfeasibletomeetthisconditionasm!1solongasthebathexponentsatisess1.Indeed,forOhmicandsuper-Ohmicbathexponents,theNRGspectrumfornottoomuchgreaterthanc0isfoundtobenumericallyindistinguishablefromthatfor=0.Fors<1,bycontrast,therestrictionbymbmNbleads,for>c0andlargeiterationnumbers,toanarticiallytruncatedspectrumthatcannotreliablyaccessthelow-energyphysical 48

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properties.Nonetheless,observationofthislocalizedbosonicspectrumservesasausefulindicator,bothinthezero-hybridizationlimitandinthefullcharge-coupledBFAmodel,thattheeffectivee-bcouplingremainsnonzero. AnotherinterpretationofEq.( 3 )isthatattheenergyscaleE=)]TJ /F8 7.97 Tf 6.58 0 Td[(kcharacteristicofintervalk,thee-bcouplingtakesaneffectivevalue~(E)governedbytherenormalization-groupequation d~ dln(=E)=1)]TJ /F6 11.955 Tf 11.95 0 Td[(s 2~,(3) whichimpliesthatthee-bcouplingisirrelevantfors>1,marginalfors=1,andrelevantfors<1.WhiletheNRGmethodiscapableoffaithfullyreproducingthephysicsof^HCCBFAforarbitraryrenormalizationsofd,U,and)]TJ /F1 11.955 Tf 6.78 0 Td[(,itsvalidityisrestrictedtotheregion )]TJ /F6 11.955 Tf 5.48 -9.68 Td[(K0~2.NB 41)]TJ /F6 11.955 Tf 11.95 0 Td[(s 1)]TJ /F8 7.97 Tf 6.59 0 Td[(s)]TJ /F4 11.955 Tf 11.96 0 Td[(1!1)166(!NB 4ln.(3) For=9andNB=8,asusedinmostofourcalculations,theupperlimitonthesaferangeofK0~variesfrom1.7fors=1to0.9fors=0. Wenowfocusonthevalueofthecrossovere-bcouplingc0determinedusingtheNRGapproach.Figure 3-2 showsforvedifferentbosonicbathexponentssthatK0c0hasanalmostlineardependenceontheNRGdiscretizationintherange1.64.WebelievethattheriseinK0c0withreectsareductionintheeffectivevalueofK0arisingfromtheNRGdiscretization.Itisknown[ 66 67 ]thatinNRGcalculationsforfermionicproblems,theconduction-banddensityofstatesattheFermienergytakesaneffectivevalue (0)=0=0=A,(3) where A=ln 21+)]TJ /F7 7.97 Tf 6.58 0 Td[(1 1)]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(1.(3) 49

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Table3-1. Crossovercouplingc0for^HZH[Eq.( 3 )]withU=0.1,d=0,andvedifferentvaluesofthebathexponents:Comparisonbetweenc0(exact)givenbyEq.( 3 )andc0(!1),theextrapolationtothecontinuumlimitofnumericalvaluesobtainedforNs=200,Nb=16,and1.64.Parenthesessurroundtheestimatednonsystematicerrorinthelastdigit. s0.20.40.60.81.0 c0(exact)0.1770.2510.3070.3550.396c0(!1)0.188(4)0.250(2)0.307(2)0.355(2)0.397(3) ThegeneraltrendofthedatainFig. 3-2 isconsistentwiththerebeingananalogousreductionofthebosonicbathspectraldensitythatrequiresthereplacementofK0by K0=K0=A,s,(3) whenextrapolatingNRGresultstothecontinuumlimit=1.However,wehavenotobtainedaclosed-formexpressionforA,s. Table 3-1 listsvaluesc0(!1)extrapolatedfromthedataplottedinFig. 3-2 .Fors0.4,thesevaluesareingoodagreementwithEq.( 3 ).Fors=0.2,however,theextrapolatedvalueofc0liessignicantlyabovetheexactvalue,indicatingthatforgiventheNRGunderestimatesthebosonicrenormalizationofU.Thisismostlikelyanotherconsequenceoftruncatingthebasisoneachsiteofthebosonictight-bindingchain. InanalyzingourNRGresultsforthefullcharge-coupledBFAmodel,weattempttocompensatefortheeffectsofdiscretizationandtruncationbyreplacingEq.( 3 )by UNRGeff=Uh1)]TJ /F4 11.955 Tf 11.96 0 Td[((=c0)2i.(3) Here,c0isnotthetheoreticalvaluepredictedinEq.( 3 ),butratherisobtainedfromrunscarriedoutfor)-320(=0butotherwiseusingthesamemodelandNRGparametersasthedatathatarebeinginterpreted. 3.3.2ZeroElectron-BosonCoupling For=0,thebosonicbathdecouplesfromtheelectronicdegreesoffreedom,whicharethendescribedbythepureAndersonmodel.Inthissection,webrieyreview 50

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aspectsoftheAndersonmodelthatwillproveimportantininterpretingresultsforthecharge-coupledBFAmodel.ForfurtherdetailsconcerningtheAndersonmodel,seeRefs.[ 1 ]and[ 66 67 ]. Forany)]TJ /F5 11.955 Tf 11.81 0 Td[(>0,andforanyUanddd+U=2(whetherpositive,negative,orzero),thestablelow-temperatureregimeoftheAndersonmodelliesonalineofstrong-couplingxedpointscorrespondingto)-352(=1.Atanyofthesexedpoints,thesystemislockedintothegroundstateoftheatomicHamiltonian^H0,andtherearenoresidualdegreesoffreedomontheimpuritysiteoronsiten=0ofthefermionicchain;theNRGexcitationspectrumisthatoftheHamiltonian[ 66 67 ] ^HNRGSC(V1)=D1Xn=1Xn)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(fynfn)]TJ /F7 7.97 Tf 6.59 0 Td[(1,+fyn)]TJ /F7 7.97 Tf 6.59 0 Td[(1,fn+V1 Xfy1f1)]TJ /F4 11.955 Tf 11.95 0 Td[(1!.(3) Thecoefcientsnareidenticaltothoseentering^HNRGband[Eq.( 3 )],exceptthathere1=0.NotethatinEq.( 3 ),thesumovernbeginsat1ratherthan0. AsshowninRef.[ 66 67 ],thestrong-couplingxedpointsoftheAndersonmodelareequivalentapartfromashiftof1intheground-statechargeQdenedinEq.( 3 )tothelineoffrozen-impurityxedpointscorrespondingtod=1,)-324(=U=0,withNRGexcitationspectradescribedby ^HNRGFI(V0)=^HNRGband+V0 Xfy0f0)]TJ /F4 11.955 Tf 11.96 0 Td[(1!.(3) Themappingbetweenalternativespecicationsofthesamexed-pointspectrumis 0V0=)]TJ /F4 11.955 Tf 9.3 0 Td[((0V1))]TJ /F7 7.97 Tf 6.58 0 Td[(1,(3) where0[seeEq.( 3 )]istheeffectiveconduction-banddensityofstates. Thexed-pointpotentialscatteringisrelatedtotheground-stateimpuritychargeviatheFriedelsumrule, h^nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1i0=2 arccot)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(0V0=2 arctan)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(0V1.(3) 51

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Forjdj,)]TJ /F2 11.955 Tf 15.58 0 Td[(UD,onendsthat h^nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1i0=)]TJ /F4 11.955 Tf 16.07 8.09 Td[(8d)]TJ ET q .478 w 249.83 -44.83 m 284.85 -44.83 l S Q BT /F5 11.955 Tf 249.83 -56.02 Td[(AU2,(3) whereAisdenedinEq.( 3 ). EventhoughthestablexedpointoftheAndersonmodelforany)]TJ /F5 11.955 Tf 11.45 0 Td[(>0isoneofthestrong-couplingxedpointsdescribedabove,theroutebywhichsuchaxedpointisreachedcanvarywidely,dependingontherelativevaluesofU,d,and)]TJ /F1 11.955 Tf 6.77 0 Td[(.Forourimmediatepurposes,itsufcestofocusonthesymmetric(d=0)model,forwhichthereisasinglestrong-couplingxedpointcorrespondingtoV0=orV1=0.Iftheon-siteCoulombrepulsionisstrongenoughthatthesystementersthelocal-momentregime(T,)]TJ /F2 11.955 Tf 16.57 0 Td[(U),thenitispossibletoperformaSchrieffer-Wolfftransformation[ 10 ]thatrestrictsthesystemtothesectornd=1andreducestheAndersonmodeltotheKondomodeldescribedbytheHamiltonian^HK=^Hband+Jz 4Nk(^nd")]TJ /F4 11.955 Tf 12.06 0 Td[(^nd#)Xk,k0cyk"ck0")]TJ /F6 11.955 Tf 11.95 0 Td[(cyk#ck0#+J? 2NkXk,k0dy"d#cyk#ck0"+H.c., (3) where 0Jz=0J?=8)]TJ ET q .478 w 261.9 -432.37 m 278.17 -432.37 l S Q BT /F5 11.955 Tf 261.9 -443.56 Td[(U.(3) ThestablexedpointisapproachedbelowanexponentiallysmallKondotemperatureTKwhenthespin-ipprocessesassociatedwiththeJ?termin^HKcausetheeffectivevaluesof0Jzand0J?torenormalizetostrongcoupling,resultinginmany-bodyscreeningoftheimpurityspin. MotivatedbythediscussioninSec. 3.3.1 ,wealsoconsiderthecaseofstrongon-siteCoulombattraction.Inthelocal-chargeregime(T,)]TJ /F2 11.955 Tf 18.36 0 Td[()]TJ /F6 11.955 Tf 27.37 0 Td[(U),acanonicaltransformationsimilartotheSchrieffer-Wolfftransformationrestrictsthesystemtothesectorsnd=0andnd=2,andmapstheAndersonmodelontoachargeKondomodel 52

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describedbytheHamiltonian^HCK=^Hband+Wd Nk(^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1)Xk,k0cyk"ck0"+cyk#ck0#)]TJ /F5 11.955 Tf 11.96 0 Td[(k,k0+2Wp NkXk,k0dy"dy#ck#ck0"+H.c., (3) where 0Wd=0Wp=2)]TJ ET q .478 w 263.65 -145.47 m 286.57 -145.47 l S Q BT /F5 11.955 Tf 263.65 -156.66 Td[(jUj.(3) Inthiscase,thestablexedpointisapproachedbelowanexponentiallysmall(charge)KondotemperatureTKwhenthecharge-transferprocessesassociatedwiththeWptermin^HCKcausetheeffectivevaluesof0Wdand0Wptorenormalizetostrongcoupling,resultinginmany-bodyscreeningoftheimpurityisospindegreeoffreedom[associatedwiththed-operatortermsinEqs.( 3 )]. BetweentheoppositeextremesoflargepositiveUandlargenegativeUisamixed-valenceregimeT,jUj)]TJ /F1 11.955 Tf 10.09 0 Td[(inwhichinteractionsplayonlyaminorrole.Here,thestablexedpointisapproachedbelowatemperatureoforder)]TJ /F1 11.955 Tf 10.1 0 Td[(whentheeffectivevalueofp )]TJ /F5 11.955 Tf 6.77 0 Td[(=(2D)scalestostrongcoupling,signalingstrongmixingoftheimpuritylevelswiththesingle-particlestatesoftheconductionband. 3.3.3ExpectationsfortheFullModel Insightintothebehaviorofthefullcharge-coupledBFAmodeldescribedbyEqs.( 3 )( 3 )canbegainedbyperformingaLang-Firsovtransformation^HCCBFA!^H0CCBFA=^U)]TJ /F7 7.97 Tf 6.59 0 Td[(1^HCCBFA^Uwith ^U=exp"(^nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1)Xqq p Nq!q)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(aq)]TJ /F6 11.955 Tf 11.96 0 Td[(ayq#.(3) Thetransformationeliminates^Himp-bath,leaving ^H0CCBFA=^H0imp+^Hband+^Hbath+^H0imp-band,(3) 53

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where^H0impisasdenedinEqs.( 3 )and( 3 ),and ^H0imp-band=1 p NkXk,(Vkexp"Xqq)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(aq)]TJ /F6 11.955 Tf 11.96 0 Td[(ayq p Nq!q#cykd+Vkexp")]TJ /F12 11.955 Tf 11.29 11.35 Td[(Xqq)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(aq)]TJ /F6 11.955 Tf 11.95 0 Td[(ayq p Nq!q#dyck).(3) InadditiontorenormalizingtheimpurityinteractionfromUtoUeffentering^H0imp,thee-bcouplingcauseseveryhybridizationeventtobeaccompaniedbythecreationandannihilationofarbitrarilylargenumbersofbosons. InthecaseoftheAnderson-Holsteinmodel[Eq.( 3 )],variouslimitingbehaviorsareunderstood[ 87 ].Intheinstantaneouslimit!0)]TJ /F1 11.955 Tf 6.77 0 Td[(,thebosonsadjustrapidlytoanychangeintheimpurityoccupancy;for20=!0U!0,thephysicsisessentiallythatoftheAndersonmodelwithU!Ueff,whilefor20=!0D,U,)]TJ /F1 11.955 Tf 12.26 0 Td[(,thereisalsoareductionfrom)]TJ /F1 11.955 Tf 10.1 0 Td[(to)-166(exp[)]TJ /F4 11.955 Tf 9.3 0 Td[((0=!0)2]intherateofscatteringbetweenthend=0andnd=2sectors,reectingthereducedoverlapbetweenthegroundstatesinthesetwosectors.Intheadiabaticlimit!0)]TJ /F1 11.955 Tf 6.78 0 Td[(,thephononsareunabletoadjustonthetypicaltimescaleofhybridizationevents,andneitherUnor)]TJ /F1 11.955 Tf 10.1 0 Td[(undergoessignicantrenormalization. Similaranalysisforthecharge-coupledBFAmodeliscomplicatedbythepresenceofacontinuumofbosonicmodeenergies!,onlysomeofwhichfallintheinstantaneousoradiabaticlimits.Nonetheless,wecanuseresultsforthecases)-422(=0(Sec. 3.3.1 )and=0(Sec. 3.3.2 ),aswellasthosefortheAnderson-Holsteinmodel,toidentifylikelybehaviorsofthefullmodel.Specically,wefocushereontheevolutionwithdecreasingtemperatureoftheeffectiveHamiltoniandescribingtheessentialphysicsofthesymmetric(d=)]TJ /F6 11.955 Tf 9.3 0 Td[(U=2)modelatthecurrenttemperature.ThiseffectiveHamiltonianisobtainedundertheassumptionthatrealexcitationsofenergyabovethegroundstateETwhereisanumberaround5,saymakeanegligiblecontributiontotheobservableproperties,andthuscanbeintegratedfromtheproblem. Basedontheprecedingdiscussion,oneexpectsthatathightemperaturesT)]TJ /F1 11.955 Tf 6.77 0 Td[(,thephysicsofthecharge-coupledBFAmodelwillbeverysimilartothatoftheAnderson 54

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modelwithUreplacedby~U(T),where ~U(E)=U)]TJ /F4 11.955 Tf 13.55 8.09 Td[(2 Z1EB(!) !d!.(3) Notethat~U(0)isidenticaltoUeffdenedinEq.( 3 ).ForthebathspectraldensityinEq.( 3 )withs>0, ~U(E)=U)]TJ /F4 11.955 Tf 13.15 8.09 Td[(2(K0)2 s1)]TJ /F4 11.955 Tf 11.95 0 Td[((E=)s.(3) WhenanalyzingNRGdata,weinsteaduse ~UNRG(E)=Un1)]TJ /F4 11.955 Tf 11.96 0 Td[((=c0)21)]TJ /F4 11.955 Tf 11.96 0 Td[((E=)so,(3) wherec0istheempiricallydeterminedvaluediscussedinconnectionwithEq.( 3 ). If,upondecreasingthetemperaturetosomevalueTLM,thesystemcomestosatisfy~U(TLM)=max(TLM,\,thenitshouldenteralocal-momentregimedescribedbytheeffectiveHamiltonian^HLM=^HK+^Hbathwiththeexchangecouplingsin^HK[Eq.( 3 )]determinedbyEq.( 3 )withU!~U(TLM),similartowhatisfoundintheAnderson-Holsteinmodel[ 85 ].Sincetheycoupleonlytothehigh-energysectorsnd=0andnd=2thatareprojectedoutduringtheSchrieffer-Wolfftransformation,thebosonsshouldplaylittlefurtherroleindeterminingthelow-energyimpurityphysics.TheoutcomeshouldbeaconventionalKondoeffectwherethee-bcouplingcontributesonlytoarenormalizationoftheKondoscaleTK. If,instead,atsomeT=TLCthesystemsatises~U(TLC)=)]TJ /F5 11.955 Tf 9.3 0 Td[(max(TLC,\,thenitshouldenteralocal-chargeregimedescribedbytheeffectiveHamiltonian ^HLC=^HCK+^Hbath+^Himp-bath.(3) BasedonthebehavioroftheAnderson-Holsteinmodel[ 85 ],oneexpectsWdin^HCK[Eq.( 3 )]tobedeterminedbyEq.( 3 )withU!~U(TLC),butwithWpexponentiallydepressedduetotheaforementionedreductionintheoverlapbetweenthegroundstatesofthend=0andnd=2sectors.Thebosonscoupletothelow-energy 55

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sectoroftheimpurityFockspace,sotheyhavethepotentialtosignicantlyaffecttherenormalizationofWdandWpuponfurtherreductioninthetemperature.Inparticular,thetermin^HLC,whichfavorslocalizationoftheimpurityinastateofwell-denednd=0or2,directlycompeteswiththeWpdouble-chargetransfertermthatisresponsibleforthechargeKondoeffectofthenegative-UAndersonmodel.ThisnontrivialcompetitiongivesrisetothepossibilityofaQPTbetweenqualitativelydistinctgroundstatesofthecharge-coupledBFAmodel. Betweentheseextremes,thesystemcanenteramixed-valenceregimeofsmalleffectiveon-siteinteraction.Inthisregime,onemustretainalltheimpuritydegreesoffreedomofthecharged-coupledBFAmodel.Theimpurity-bandhybridizationcompeteswiththee-bcouplingforcontroloftheimpurity,againsuggestingthepossibilityofaQPT. Eachoftheregimesdiscussedabovefeaturescompetitionbetweenband-mediatedtunnelingwithinthemanifoldofimpuritystatesandthelocalizingeffectofthebosonicbath.Althoughthetunnelingisdominatedbyadifferentprocessinthethreeregimes,italwaysdrivesthesystemtowardanondegenerateimpuritygroundstate,whereasthee-bcouplingfavorsadoubly-degenerate(nd=0,2)impuritygroundstate.Inordertoprovideauniedpictureofthethreeregimes(andtheregionsoftheparameterspacethatlieinbetweenthem),wewillnditusefultointerpretourNRGresultintermsofanoveralltunnelingrate,whichhasabarevalue 'q J2?+2)]TJ /F6 11.955 Tf 25.01 0 Td[(D=+16W2p.(3) Here,Wpisassumedtobenegligiblysmallinthelocal-momentregime,andJ?tobesimilarlynegligibleinthelocal-chargeregime.Ifrenormalizestolargevalueswhilethee-bcouplingscalestoweakcoupling,thenoneexpectstorecoverthestrong-couplingphysicsoftheAndersonmodel.If,ontheotherhand,becomesstrongwhilebecomesweak,thesystemshouldenteralow-energyregimeinwhichthebathgoverns 56

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theasymptoticlow-energy,long-timeimpuritydynamics.Whetherornoteachofthesescenariosisrealizedinpractice,andwhetherornotthereareanyotherpossiblegroundstatesofthemodel,canbedeterminedonlybymoredetailedstudy.ThesequestionsareansweredbytheNRGresultsreportedinthesectionsthatfollow. 3.4Results:SymmetricModelwithSub-OhmicDissipation Thissectionpresentsresults1forHamiltonian( 3 )withU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d>0andwithsub-Ohmicdissipationcharacterizedbyanexponent0
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natureoftheQPTisqualitativelydifferentthanfor01,thee-bcouplingisirrelevant,andthesystemisintheKondophaseforall)]TJ /F5 11.955 Tf 10.09 0 Td[(>0. Theremainderofthissectionpresentstheevidenceforthepreviousstatements.Werstdiscusstherenormalization-groupowsandxedpoints.Wethenturntothebehaviorinthevicinityofthephaseboundary,focusinginparticularonthecriticalresponseoftheimpuritychargetoalocalelectricpotential.Followingthat,wepresentresultsfortheimpurityspectralfunction,andshowthatthelow-energyscaleextractedfromthisspectralfunctionsupportsthequalitativepicturelaidoutintheparagraphsaboveandsummarizedinFig. 3-3 3.4.1NRGFlowsandFixedPoints Figure 3-4 plotstheschematicrenormalization-groupowsofthecouplingsenteringEq.( 3 )anddenedinEq.( 3 )forasymmetricimpurity(U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d)coupledtobathdescribedbyanexponent0
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alone,and(ii)thestrong-couplingexcitationsoftheKondo(orsymmetricAnderson)model,correspondingtofreeelectronswithaFermi-levelphaseshiftof=2.Thisspectrum,whichexhibitsSU(2)symmetrybothinthespinandcharge(isospin)sectors,isidenticaltothatfoundthroughouttheKondophaseoftheparticle-hole-symmetricIsingBFKHamiltonian[ 73 74 ](amodelinwhichthebosonscoupletotheimpurity'sspinratherthanitscharge). TheschematicRGowdiagraminFig. 3-4 showsalocalizedxedpointcorrespondingto=1and=0.However,thisisreallyalineofxedpointsdescribedby^HLC[Eq.( 3 )]witheffectivecouplings=1,Wp=0,and0Wd<1.SinceWp=0,theimpurityoccupancytakesaxedvaluend=0or2.(Itisimportanttodistinguishnd,usedtocharacterizethexed-pointexcitations,fromthephysicalexpectationvalueof^nd.ThelatterquantityisdiscussedinSec. 3.4.5.1 .) Eachxedpointalongthelocalizedlinehasanexcitationspectrumthatdecomposesintothedirectproductof(i)bosonicexcitationsidenticaltothoseatthelocalizedxedpointofthespin-bosonmodel[ 72 ]withthesamebathexponents,and(ii)fermionicexcitationsdescribedbyaHamiltonian ^HNRGL,f=^HNRGband+Wd(nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1) Xfy0f0)]TJ /F4 11.955 Tf 11.96 0 Td[(1!,(3) whichisjustthediscretizedversionof^HCK[Eq.( 3 )]withWp=0andtheoperator^ndreplacedbytheparameternd.Thelow-lyingmany-bodyeigenstatesof^HNRGL,fappearindegeneratepairs,onememberofeachpaircorrespondingtond=0andtheothertond=2.Thexed-pointcouplingWdincreasesmonotonicallyasthebaree-bcouplingdecreasesfrominnity,anddivergesonapproachtothephaseboundary.AsillustratedinFig. 3-6 ,thisdivergencecanbettedtothepower-lawform Wd/()]TJ /F5 11.955 Tf 11.96 0 Td[(c))]TJ /F14 7.97 Tf 6.59 0 Td[(for!+c.(3) 59

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ForreasonsthatwillbeexplainedinSec. 3.4.5.1 ,thenumericalvalueofcoincides,towithinasmallerror,withthatoftheorder-parameterexponentdenedinEq.( 3 ). Thefree-orbitalxedpoint(=c0,=0)isunstablewithrespecttoabare)]TJ /F2 11.955 Tf 10.45 0 Td[(6=0oranydeviationoffromc0lim)]TJ /F9 7.97 Tf 4.82 0 Td[(!0c(\.Thelocal-momentxedpoint(==0),atwhichtheimpurityhasaspin-1 2degreeoffreedomdecoupledfromthebandandfromthebath,isreachedonlyforbarecouplings)-277(=0(hence,=0)and
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effectiveHamiltonian ^HNRGLC(Wd=1)=^HNRGSC(0)+^HNRGSBM.(3) Here,^HNRGSC(0)[Eq.( 3 )]actsonlyonfermionicchainsitesn1,andyieldstheKondo/Andersonstrong-couplingexcitationspectrum,while ^HNRGSBM=^HNRGbath+2Wp)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(j*ih+j+j+ih*j+K0 p (s+1))]TJ /F2 11.955 Tf 5.48 -9.69 Td[(j*ih*j)-222(j+ih+j)]TJ /F6 11.955 Tf 12.95 -9.69 Td[(b0+by0(3) actsontheremainingdegreesoffreedomintheprobleminasubspaceofstatesallcarryingquantumnumbersS=Sz=Q=0.^HNRGSBMispreciselythediscretizedformofthespin-bosonHamiltonianwithtunnelingrate=4Wpanddissipationstrength=2(K0)2=.Thesetwocouplingscompetewithoneanother,withthreepossibleoutcomes:(1)canscaletoinnityandtozero,resultinginowtothedelocalizedxedpoint(theKondoxedpointofthecharge-coupledBFAmodel);(2)canscaletoinnityandtozero,yieldingowtothelocalizedxedpoint;or(3)bothcouplingscanrenormalizetonitevalues=C,=Catthecriticalpoint.Thispictureimpliesthattheuniversalcriticalbehaviorofthecharge-coupledBFAmodelshouldbeidenticaltothatofthespin-bosonmodel,theconduction-bandelectronsservingonlytodressthend=0,2impuritylevelsandtorenormalizetheimpuritytunnelingrateandthedissipationstrength. GiventhattheNRGapproachnecessarilyinvolvesFock-spacetruncation,itisinstructivetoexaminethedependenceofthexed-pointspectraontheparametersNsandNbdenoting,respectively,thenumberofstatesretainedfromoneNRGiterationtothenextandthemaximumnumberofbosonsallowedpersiteofthebosonicchain.Figure 3-7 shows,forrepresentativebathexponentss=0.2ands=0.8,thattheenergyofthelowestbosonicexcitationat=cconvergesrapidlywithincreasingNsandNb.Thisbehaviorsuggeststhatfor=9,atleast,Ns=500andNb=8aresufcientforstudyingthephysicsatthecriticalpoint. 61

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Bycontrast,thelowestbosonicexcitationenergyfor=1.1c,plottedinFig. 3-8 ,convergesonlyslowlywithrespecttoNb.Thispointstothefailureofthetruncatedbosonicbasisdeepinsidethelocalizedphaseofthesub-Ohmicmodel,wherethemeanbosonnumberpersiteisexpectedtodivergeaccordingtoEq.( 3 ).Thisinterpretationisconrmedbycalculationoftheexpectationvalueofthetotalbosonnumber, ^BN=M(N)Xmbymbm,(3) whereM(N)denotesthehighestlabeledbosonicsitepresentatiterationN.Ourresultsforh^B20ivsNb(notshown)areshowninFig. 3-9 ,withconvergencebyNb=8atthecriticalpoint,butnoevidenceofsuchconvergenceforane-bcoupling10%overthecriticalvalue. Recently,Bullaetal.appliedastarreformulationoftheNRGtothespin-bosonmodel[ 88 ].Whilethisapproachprovidesagooddescriptionofthelocalizedxedpoint,itdoesnotcorrectlycapturethephysicsofthedelocalizedphase(correspondingtotheKondophaseofthepresentmodel)orofthecriticalpointthatseparatesthetwostablephases.Forthisreason,weprefertoworkwiththechainformulationsummarizedinSec. 3.2 3.4.2CriticalCoupling Figure 3-10 plotsthecriticale-bcouplingc(\forxedU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2dandfourdifferentvaluesofthebathexponents.Asexpected,withincreasing)]TJ /F1 11.955 Tf 6.78 0 Td[(,thecriticalcouplingincreasessmoothlyfromc()-362(=0)c0,reectingthefactthatentrytothelocalizedphaserequiresane-bcouplingsufcientlylargenotonlytodriveUeffnegative,butalsotoovercomethereductionintheelectronicenergythatderivesfromthehybridization.Webelievethattheverticalslopeofthes=0.2phaseboundaryasitapproachesthehorizontalaxisinFig. 3-10 isanartifactstemmingfromthesamesourceastheNRGoverestimateofc0forthesamebathexponent.(SeethediscussionofFig. 3-2 inSec. 3.3.1 .) 62

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Inthesubsectionsthatfollow,weshowthatthecriticalpropertiesofthecharge-coupledBFAmodelmap,underinterchangeofspinandchargedegreesoffreedom,ontothoseofthespin-coupledBFAmodelstudied(alongwiththecorrespondingIsingBFKmodel)inRef.[ 74 ].Thespin-coupledmodelisdescribedbyEqs.( 3 )( 3 )and( 3 )( 3 ),withEqs.( 3 )and( 3 )replacedby ^Himp-bath=1 2p Nq(^nd")]TJ /F4 11.955 Tf 12.05 0 Td[(^nd#)Xqgq)]TJ /F6 11.955 Tf 5.47 -9.68 Td[(aq+ay)]TJ /F13 7.97 Tf 6.59 0 Td[(q,(3) and ^HNRGimp-bath=K0g 2p (s+1)(^nd")]TJ /F4 11.955 Tf 12.05 0 Td[(^nd#))]TJ /F6 11.955 Tf 5.48 -9.69 Td[(b0+by0.(3) Inlightoftheparallelsbetweentheuniversalcriticalbehaviorofthetwomodels,itisofinteresttocomparetheircriticalcouplings,makingdueallowancefortheadditionalprefactorof1 2thatentersEqs.( 3 )and( 3 ). Figure 3-11 plotsthesdependenceofcandgc=2forxedvaluesofU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2dand)]TJ /F1 11.955 Tf 6.78 0 Td[(.Forall0
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xedpointtypicallygovernsthebehaviorattemperaturesmuchgreaterthantheKondotemperatureTKoftheAndersonmodelobtainedbysetting=0inEq.( 3 ).FortemperaturesbetweenoforderTKandacrossoverscaleT,thesystemexhibitsquantumcriticalbehaviorcontrolledbythermaluctuationsabouttheunstablecriticalpoint.Finally,thephysicsintheregimeT.Tisgovernedbyoneorotherofthetwostablexedpoints:Kondoorlocalized. Forxedvaluesofallotherparameters,oneexpectsTtovanishasthee-bcouplingapproachesitscriticalvalueaccordingtoapowerlaw: T/j)]TJ /F5 11.955 Tf 11.95 0 Td[(cjfor!c,(3) whereisthecorrelation-lengthexponent[ 17 ].ThecrossoverscalecanbedetermineddirectlyfromtheNRGsolutionviatheconditionT/)]TJ /F8 7.97 Tf 6.58 0 Td[(N=2,whereNisthenumberoftheiterationatwhichthemany-bodyenergylevelscrossovertothoseofastablexedpoint.Thereissomearbitrarinessastowhatpreciselyconstitutescrossoverofthelevels.DifferentcriteriawillproduceT()valuesthatdifferfromoneanotherbya-independentmultiplicativefactor.ItisoflittleimportancewhatdenitionofNoneuses,providedthatitisappliedconsistently. Figure 3-12 showstypicaldependencesofTonc)]TJ /F5 11.955 Tf 11.96 0 Td[(intheKondophase.Equation( 3 )holdsverywelloverseveraldecades,asdemonstratedbythelinearbehaviorofthedataonalog-logplot.Wendthatthenumericalvaluesof(s),someofwhicharelistedinTable 3-2 ,areidentical(withinsmallerrors),tothoseofthespin-bosonandIsingBFKmodelsforthesamebathexponents.Thissupportsthenotionthatthecriticalpointofthecharge-coupledBFAmodelbelongstothesameuniversalityclassasthecriticalpointsofthespin-bosonandIsingBFKmodels.However,toconrmthisequivalence,wemustcompareothercriticalexponents,asreportedbelow. 64

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Table3-2. Correlation-lengthcriticalexponentvsbathexponentsforthecharge-coupledBose-FermiAndersonmodel(CC-BFA,thiswork)andfortheIsing-anisotropicBose-FermiKondomodel(BFK,fromRefs.[ 73 ]and[ 74 ]).Parenthesessurroundtheestimatednonsystematicerrorinthelastdigit. s0.20.40.60.8 (CC-BFA)4.99(3)2.52(2)1.97(4)2.12(6)(BFK)4.99(5)2.50(1)1.98(3)2.11(2) 3.4.4ThermodynamicSusceptibilities Inthissubsection,weconsidertheresponseofthecharge-coupledBFAmodeltoaglobalmagneticeldHandtoaglobalelectricpotential.TheseexternalprobesentertheHamiltonianthroughanadditionalterm ^Hext=HSz+Q,(3) whereSzandQaredenedinEqs.( 3 )and( 3 ),respectively.Inparticular,wefocusonthestaticimpurityspinsusceptibilitys,imp=)]TJ /F5 11.955 Tf 9.3 0 Td[(@2Fimp=@H2andthestaticimpuritychargesusceptibilityc,imp=@2Fimp=@2.Here,Fimp=(F),where(X)isthedifferencebetween(i)thevalueofthebulkpropertyXwhentheimpurityispresentand(ii)thevalueofXwhentheimpurityisremovedfromthesystem.ItisstraightforwardtoshowthatTs,imp=)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(hh^S2zii)-222(hh^Szii2, (3)Tc,imp=)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(hh^Q2ii)-222(hh^Qii2, (3) where,foranyoperator^A, hh^Aii=Tr^Aexp()]TJ /F4 11.955 Tf 10.86 2.66 Td[(^H=T) Trexp()]TJ /F4 11.955 Tf 10.86 2.66 Td[(^H=T).(3) Notethatwiththeabovedenitions,limT!1Ts,imp=1 8butlimT!1Tc,imp=1 2,afactoroffourdifferencethatmustbetakenintoaccountwhencomparingthetwosusceptibilities.SinceeachTimpiscalculatedasthedifferenceofbulkquantities,itsevaluationusingtheNRGmethodiscomplicatedbysignicantdiscretizationand 65

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truncationerrors.Inordertoobtainreasonablywell-convergedresultsforTimp,weretainNs=2000statesaftereachNRGiteration.However,eventhisnumberisinsufcienttoallowreliableextractionofimp(Timp)=TasT!0. Figure 3-13 plotsNRGresultsforTs,imp(T)and1 4Tc,imp(T),calculatedforbathexponents=0.8anddifferentvaluesofthee-bcoupling.Forc0(seeSec. 3.3.1 ),bothimpuritysusceptibilitiesbehaveverymuchastheydointheAndersonmodel:withdecreasingtemperature,Tc,impquicklyfallstowardzero,signalingquenchingofchargeuctuationsuponentryintothelocal-momentregime,whereasTs,impinitiallyrisestowarditslocal-momentvalueof1 4,beforedroppingtozeroforTTonapproachtotheKondoxedpoint.Withincreasing,thechargeresponsegrowsandthespinresponseissuppressed.Thetwosusceptibilitiesareapproximatelyequivalentfor=c0,wheretheeffectiveCoulombinteractionUeff=0.Forstillstrongere-bcouplings,Ts,impplungesrapidlyasthetemperatureisdecreased,whereasTc,imprstrisesonentrytothelocal-chargeregimebeforedroppingtosatisfy limT!0Tc,imp(T)=0for
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ThebehaviorofthestaticimpurityspinsusceptibilityisqualitativelyunchangeduponcrossingfromtheKondophasetothelocalizedphase.However,for>c,Tc,impapproachesatlowtemperaturesanonzerovaluethatcanbeinferredfromtheeffectiveHamiltonian^HNRGL,f[Eq.( 3 )].ElectronsneartheFermilevelexperienceans-wavephaseshift (!=0)=8>><>>:0fornd=0,)]TJ /F5 11.955 Tf 11.95 0 Td[(0fornd=2,(3) wherendlabelsthetwodisconnectedsectorsof^HNRGL,f,and 0=arctan(0Wd),00=2,(3) with0beingtheeffectiveconduction-banddensityofstatesdenedinEq.( 3 ).Itisthenstraightforwardtoshowthat limT!0Tc,imp(T)=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(20=)2.(3) Equations( 3 ),( 3 ),and( 3 )togetherimplythat limT!0Tc,imp(T)/()]TJ /F5 11.955 Tf 11.96 0 Td[(c)2for!+c.(3) Asthisexampleillustrates,thethermodynamicsusceptibilitiescontainsignaturesofanevolutionfromaspin-Kondoeffecttoacharge-Kondoeffect.Furthermore,Eqs.( 3 )and( 3 )suggestthatc,impmayserveastheorder-parametersusceptibilityfortheQPT.However,neithersusceptibilitymanifeststhevanishingofthecrossoverscaleTonapproachtothetransitionfromtheKondoside.Moreover,theconservationofQpreventsc,impfromacquiringananomaloustemperaturedependenceinthequantum-criticalregime[ 90 ].Thus,oneisledtoconcludethattheresponsetoaglobalelectricpotentialdoesnotprovideaccesstothecriticaluctuationsneartheQPT. 67

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3.4.5LocalChargeResponse GiventhenatureofthecouplinginHamiltonian( 3 )betweentheimpurityandthebosonicbath,weexpecttobeabletoprobethequantumcriticalpointthroughthesystem'sresponsetoalocalelectricpotentialthatactssolelyontheimpuritycharge,enteringtheHamiltonianviaanadditionalterm ^Hc,loc=(^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1).(3) AnonzeroisequivalenttoashiftindenteringEq.( 3 )awayfromitsbarevalued+U=2=0. Inthissubsectionweshowthatforsub-Ohmicbathexponents0
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phases.Thisquantityvanishesforallc,itsonsetbeingdescribedbythepowerlaw lim!0Qloc(,;T=0)/()]TJ /F5 11.955 Tf 11.96 0 Td[(c)for!+c.(3) Inthelocalizedphase,thepresenceofaninnitesimallocalpotentialrestrictstheeffectiveHamiltonian( 3 )tojustonendsector:nd=0for>0,ornd=2for<0.ThensubstitutingEq.( 3 )intotheFriedelsumruleh^ndi0=2(0)=yields lim!0Qloc(;T=0)=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(2sgn acot)]TJ /F5 11.955 Tf 5.47 -9.69 Td[(0Wd.(3) ThelatterrelationexplainstheequalityoftheexponentsenteringEqs.( 3 )and( 3 ).ItshouldalsobenotedthatEqs.( 3 ),( 3 ),and( 3 )togetherimplythat lim!0Q2loc(;T=0)=limT!0Tc,imp(T).(3) Atthecriticalpoint,theresponsetoasmall-but-nitepotentialobeysanotherpowerlaw, Qloc(;=c,T=0)/jj1=.(3) ThisbehaviorisexempliedinFig. 3-15 forfourdifferentvaluesofs. Figure 3-16 showsalogarithmicplotofthestaticlocalchargesusceptibilityc,loc(T;!=0)vstemperatureTforbathexponents=0.4andanumberofe-bcouplingsstraddlingc.Inthequantum-criticalregime,thesusceptibilityhastheanomaloustemperaturedependence c,loc(T;!=0)/T)]TJ /F8 7.97 Tf 6.59 0 Td[(xforTTTK,(3) characterizedbyacriticalexponentx.ForTT(),thetemperaturevariationapproachesthatofoneorotherofthestablexedpoints.IntheKondophase,thesusceptibilityisessentiallytemperatureindependent,signalingcompletequenchingoftheimpurity,andthezero-temperaturevaluedivergesonapproachtothecritical 69

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couplingas c,loc(;!=T=0)/(c)]TJ /F5 11.955 Tf 11.95 0 Td[())]TJ /F14 7.97 Tf 6.59 0 Td[(for!)]TJ /F8 7.97 Tf 0 -7.89 Td[(c.(3) Inthelocalizedphase,bycontrast,c,loc(T,;!=0)=lim!0Q2loc(,;T=0) Tfor>candTT, (3) indicativeofaresidualimpuritydegreeoffreedom.Preciselyatthecriticale-bcoupling,Eq.( 3 )isobeyedallthewaydowntoT=0. Table 3-3 liststhenumericalvaluesofthecriticalexponents,1=,x,and,forfourdifferentsub-Ohmicbathexponentss.Foreachs,thesecriticalexponentsareidenticalwithinestimatederrortothoseofthespin-bosonandIsingBFKmodels.Inallcases,wendthatx=stowithinourestimatednonsystematicnumericalerror.Wealsonotethatfors1 2,thevalueofliesclosetoitsmean-eldvalueof1.Itisconceivablethatthedeviationsoffrom1areartifactsoftheNRGdiscretizationandtruncationapproximations. TheexponentsinTable 3-3 obeythehyperscalingrelations =1+x 1)]TJ /F6 11.955 Tf 11.96 0 Td[(x,2=(1)]TJ /F6 11.955 Tf 11.96 0 Td[(x),=x,(3) whichareconsistentwiththeansatz F=Tfj)]TJ /F5 11.955 Tf 11.96 0 Td[(cj T1=,jj T(1+x)=2(3) forthenonanalyticpartofthefreeenergy.Suchhyperscalingsuggeststhatthequantumcriticalpointisaninteractingone[ 17 ]. 3.4.5.2Dynamicallocalchargesusceptibility Thedynamicallocalchargesusceptibilityis c,loc(!,T)=iZ10dte)]TJ /F8 7.97 Tf 6.59 0 Td[(i!t[^nd(t))]TJ /F4 11.955 Tf 11.96 0 Td[(1,^nd(0))]TJ /F4 11.955 Tf 11.95 0 Td[(1].(3) 70

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Table3-3. Staticcriticalexponents,1=,x,anddenedinEqs.( 3 )and( 3 )( 3 ),respectively,forfourdifferentvaluesofthebosonicbathexponents.Parenthesessurroundtheestimatednonsystematicerrorinthelastdigit. s1=x 0.22.0005(3)0.6673(1)0.1997(2)0.997(4)0.40.7568(2)0.4283(2)0.4002(4)1.0117(6)0.60.3923(1)0.2501(7)0.600(2)1.1805(5)0.80.2130(1)0.1111(1)0.800(2)1.703(3) Itsimaginarypart00c,loccanbecalculatedwithintheNRGas 00c,loc(!,T)= Z(T)Xm,m0m0j^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1jm2)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F8 7.97 Tf 6.58 0 Td[(Em0=T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F8 7.97 Tf 6.59 0 Td[(Em=T(!)]TJ /F6 11.955 Tf 11.95 0 Td[(Em0+Em).(3) Here,jmiisamany-bodyeigenstatewithenergyEm,andZ(T)=Pme)]TJ /F8 7.97 Tf 6.59 0 Td[(Em=Tisthepartitionfunction.Equation( 3 )producesadiscretesetofdelta-functionpeaksthatmustbebroadenedtorecoveracontinuousspectrum.Followingstandardprocedure[ 91 ],weemployGaussianbroadeningofdeltafunctionsonalogarithmicscale: (j!j)-55(jEj)!e)]TJ /F8 7.97 Tf 6.59 0 Td[(b2=4 p bjEjexp)]TJ /F4 11.955 Tf 10.49 8.09 Td[((lnj!j)]TJ /F4 11.955 Tf 17.93 0 Td[(lnjEj)2 b2,(3) withthechoiceofthebroadeningwidthb=0.5ln. (a)Zerotemperature.Figure 3-17 plots00c,loc(!;T=0)vs!forbathexponents=0.2andaseriesofe-bcouplings
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thatwehaveobtainedareconsistentwiththerelation x=y=sfor0
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be A(!,T)=)]TJ /F5 11.955 Tf 9.3 0 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1ImGd(!,T),(3) wheretheretardedimpurityGreen'sfunctionis Gd(!,T)=)]TJ /F6 11.955 Tf 9.3 0 Td[(iZ10dtei!td(t),dy(0)+.(3) ThespectralfunctioncanbecalculatedwithintheNRGusingtheformulation A(!,T)=1 Z(T)Xm,m0m0jdyjm2)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F8 7.97 Tf 6.58 0 Td[(Em0=T+e)]TJ /F8 7.97 Tf 6.59 0 Td[(Em=T(!)]TJ /F6 11.955 Tf 11.96 0 Td[(Em0+Em),(3) wherethenotationisthesameasinEq.( 3 ).Torecoveracontinuousspectrum,wehaveagainappliedEq.( 3 )tothedelta-functionoutputofEq.( 3 ),choosingthebroadeningfactorb=0.55lnthatbestsatisestheFermi-liquidresultA(!=0,T=0)=1=)]TJ /F1 11.955 Tf 10.09 0 Td[(fortheAndersonmodel.Inordertoachievesatisfactoryresults,wenditnecessarytoworkwithasmallerdiscretizationparameter(=3insteadofthevalue=9employedforallthequantitiesreportedabove)andtoretainmorestates(Ns=1200ratherthanthe500thattypicallysufces).Sincethespectralfunctionsshownbelowareallspin-independent,wehenceforthdroptheindexonA.Fortheparticle-hole-symmetricmodelconsideredinthissection,thespectralfunctionissymmetricabout!=0. Figure 3-20 plotsA(!;T=0)vs!fors=0.8andaseriesofvalues.For=0,werecoverthespectralfunctionoftheAndersonmodel,featuringanarrowKondoresonancecenteredatzerofrequencyandbroadHubbardsatellitebandscenteredaround!=1 2U.Increasingthee-bcouplingfromzerohastwoinitialeffectsadisplacementoftheHubbardbandstosmallerfrequencies,andabroadeningofthelow-energyKondoresonancethatcanbothbeattributedtotheboson-inducedrenormalizationoftheCoulombinteractiondescribedinEq.( 3 ). WeexpecttheHubbardpeaklocationstoobey!H'1 2Uefffor0c0.However,thepeaklocationsplottedinFig. 3-21 (a)arebetterttedbyj!Hj=0.4U)]TJ ET BT /F1 11.955 Tf 227.35 -687.85 Td[(73

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2=(s),which(giventhediscretizationandtruncationeffectsdiscussedinSec. 3.3.1 )appearstorepresentastrongerbosonicrenormalizationthanthatpredictedbyj!Hj=1 2Ueff.WebelievethatthisdiscrepancyarisesprimarilyfromtherapidbroadeningoftheKondoresonancewithincreasing,whichshiftsthelocalmaximumofthecombinedspectralfunction(thesumoftheKondoresonanceplusHubbardsatellitebands)toafrequencysmallerinmagnitudethanthecentralfrequencyoftheHubbardpeakbyitself. Thewidth2)]TJ /F10 7.97 Tf 13.05 -1.8 Td[(KoftheKondoresonance,plottedinFig. 3-21 (b),provestobeequal(uptoamultiplicativeconstant)tothecrossoverscaleTdenedinSec. 3.4.3 .For.c0,thevariationinbothscalesiswelldescribedbythereplacementofUintheexpression[ 66 67 ]fortheKondotemperatureofthesymmetricAndersonmodelby~UNRG(U=2)[givenbyEq.( 3 )],theeffectiveCoulombinteractiononentrytothelocal-momentregime.ThedashedlineinFig. 3-21 (b)showsthattheresultingformula, )]TJ /F10 7.97 Tf 6.77 -1.8 Td[(K=CKs 8~UNRG)]TJ ET q .478 w 188.12 -311.62 m 228.58 -311.62 l S Q BT /F5 11.955 Tf 197.91 -322.81 Td[(Aexp )]TJ /F5 11.955 Tf 10.5 8.09 Td[(A~UNRG 8)]TJ /F12 11.955 Tf 31.87 28.64 Td[(!,(3) whereAisdenedinEq.( 3 ),providesanexcellentdescriptionof)]TJ /F10 7.97 Tf 6.78 -1.8 Td[(Koveralmosttheentirerange0
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Inthelocalizedphase(>c),thereisnovestigeoftheKondoresonance,buthigh-energyHubbard-likepeaksreappear;seethecurvesfor=0.5and0.6inFig. 3-20 .Inaddition,thereisapairoflow-energypeakscenteredat!'T,asshowninFig. 3-22 3.4.7Spin-KondotoCharge-KondoCrossover Basedontheanalysisofthezero-hybridizationlimitpresentedinSec. 3.3.1 ,oneexpectsspinuctuationstodominatetheimpuritybehaviorintheregionc0,butchargeuctuationstobedominantforc0
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eitherinversestaticsusceptibility,indicatingthattheKondoeffecthasmixedspinandchargecharacter. Figure 3-24 presentsa-)]TJ /F1 11.955 Tf 10.1 0 Td[(phasediagramfors=0.8andxedU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d,showingdatapointsalongthephaseboundary=c(\andalongthecrossoverboundary=X(\,denedasthee-bcouplingatwhichtheKondoresonancewidth2)]TJ /F10 7.97 Tf 13.05 -1.8 Td[(Kismaximalforthegiven)]TJ /F1 11.955 Tf 6.77 0 Td[(.Thefactthatthelatterlinerisesalmostverticallyfrom=c0at)-463(=0providesfurtherconrmationofthepictureofacrossoverfromaspin-Kondoeffecttoacharge-KondoeffectresultingfromthechangeinthesignofUeff,andestablishesthevalidityoftheschematicphasediagram(Fig. 3-3 )presentedintheintroductiontothissection. 3.5Results:SymmetricModelwithOhmicDissipation ThissectionpresentsresultsforHamiltonian( 3 )withU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d>0andanOhmicbath(i.e.,s=1).Werstdiscussthebehaviorofthestaticlocalchargesusceptibility.Weshowthat,incontrastwiththesub-Ohmiccase0c0,Wp=0,and0Wd<1.Anotherimportant 76

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departurefromthesub-Ohmiccaseisthatfors=1thereisnolongeradistinctcriticalpointreachedbyowalongtheseparatrixfromthefree-orbitalxedpoint;ratherthesetwoxedpointsmergeass!1)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(,leavingacriticalendpointat=c0,=0.Strictly,thisisalineofcriticalendpointsdescribedby^HLC[Eq.( 3 )]witheffectivecouplings=c0,Wp=0,and0Wd<1.Foraxedbarevalueof)]TJ /F1 11.955 Tf 6.77 0 Td[(,theendpointvalueofWdisjustthelimitofthelocalizedxed-pointvalueofWdasthebarecouplingapproachesthephaseboundaryc(\. Thebehaviorsofthestaticimpurityspinandchargesusceptibilitiesarequalitativelyverysimilartothoseforasub-Ohmicbath,asdiscussedinSec. 3.4.4 .Theonlysignicantdifferenceisthatfors=1,limT!0Tc,imp(T)undergoesadiscontinuousjumpfromitsvalueof0forctoanonzerovaluefor=+c.ThisjumpcanbeunderstoodthroughEqs.( 3 )and( 3 )asaconsequenceofthefactthatWddoesnotdivergeonapproachtothecriticalcoupling. 3.5.2StaticLocalChargeSusceptibilityandCrossoverScale Figure 3-26 isalogarithmicplotofthestaticlocalchargesusceptibilityc,loc(T;!=0)vstemperatureTfordifferente-bcouplings.OntheKondosideofthephaseboundary,c,loc(T;!=0)isproportionalto1=Tathightemperatures,butlevelsoffforT.T.Wenditconvenienttodene T=4=c,loc(!=T=0)for!)]TJ /F8 7.97 Tf 0 -7.89 Td[(c,(3) therebyremovingtheambiguityinthedenitionofthecrossoveriterationN(seeSec. 3.4.3 )ontheKondosideofthes=1quantumphasetransition. For!)]TJ /F8 7.97 Tf 0 -7.29 Td[(c,thecrossoverscalevanishesaccordingto(seeFig. 3-27 ) T/exp")]TJ /F6 11.955 Tf 40.21 8.09 Td[(C p 1)]TJ /F4 11.955 Tf 11.95 0 Td[((=c)2#.(3) Inthelocalizedphase,c,loc(T;!=0)satisesEq.( 3 )overtheentiretemperaturerangeTU.Sincethecriticalandlocalizedxedpointssharethe 77

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sametemperaturevariation,nocrossoverscalecanbeidentiedonthelocalizedsideofthephaseboundary.Moreover,theorderparameterlim!0Qloc(;T=0)doesnotvanishcontinuouslyas!+c,butratherundergoesadiscontinuousjumpatthetransition,asshowninFig. 3-27 .Themagnitudeofthisjumpisnonuniversal,beingrelatedviaEq.( 3 )tothevalueofWdatthecriticalendpoint. ThepropertiesdescribedaboveareanalogoustothoseoftheKondomodel[Eq.( 3 )]atthetransitionbetweentheKondo-screenedphase(reachedforJ?6=0andJz>jJ?j)andthelocal-momentphase(reachedforJzjJ?j).SuchbehaviorsarecharacteristicofaKosterlitz-ThoulesstypeofQPT. 3.5.3ImpuritySpectralFunction Figure 3-28 showstheimpurityspectralfunctionA(!;T=0)foranOhmicbath.ThebehaviorintheKondophaseissimilartothatinthesub-OhmiccasediscussedinSec. 3.4.6 :Asthee-bcouplingincreasesfromzero,theHubbardsatellitebandsareinitiallydisplacedtosmallerfrequenciesaccordingto!H'1 2Ueff[Fig. 3-29 (a)],whilethewidth2)]TJ /F10 7.97 Tf 13.05 -1.79 Td[(KoftheKondoresonance[Fig. 3-29 (b)]rstrisesbeforefallingsharplyonapproachto=c.Justasfor0
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tothegeneralsituationofanasymmetricimpurity,startingwiththesub-Ohmiccase00)orh^ndi'2(ford<0).ForT.TL,theimpuritydegreesoffreedomwillbefrozen,thebosonicspectrumwillrapidlyapproachstrongcoupling,andtheconductionelectronswillhaveanexcitationspectrumcorrespondingto^HNRGFIinEq.( 3 )withasmallvalueofjV0j. Giventheequivalenceof^HNRGSCand^HNRGFI,itseemslikelythatthelow-energybehavioroftheasymmetricmodelwillbethesameinthesmall-andlarge-limits.Thissuggeststhatthemany-bodyeigenstatesevolveadiabaticallyasthee-bcouplingisincreasedfrom=0+to!1,withouttheoccurrenceofaninterveningQPT. 79

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Fors=1,thee-bcouplingismarginal,ratherthanrelevant.Oneagainexpectsacontinuousevolutionofthelow-energyNRGspectrumwiththebarevalueof.However,inthisOhmiccase,thebosonicexcitationsshouldcorrespondtononinteractingdisplacedoscillatorsratherthanthe(truncated)strong-couplingspectrumfoundfor0
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jdj.Itisthenreasonabletohypothesizethathnd)]TJ /F4 11.955 Tf 11.9 0 Td[(1i0intheasymmetriccharge-coupledBFAmodelshouldbeclosetotheground-stateimpuritychargeoftheAndersonmodelwiththesame)]TJ /F1 11.955 Tf 10.1 0 Td[(andd,butwithUreplacedby~U(E)[Eq.( 3 )]evaluatedatE'Tf.Ournumericalresultssupportthisconjecture.Forexample,Fig. 3-31 showsthatclosetoparticle-holesymmetry(d=)]TJ /F6 11.955 Tf 9.3 0 Td[(U=2),theAnderson-modelchargecalculatedfor~UNRG(E)[Eq.( 3 )]withE=0.3U(solidlines)reproducesquitewellthevalueofh^nd)]TJ /F4 11.955 Tf 12.77 0 Td[(1i0(symbols)overquiteabroadrangeofe-bcouplings0.2 3c,wherec'0.29835isthecriticalcouplingofthesymmetricproblem. Inthesmall-limit,onecanalsoestimatetheboson-localizationtemperatureTLbyconsideringtheevolutionwithdecreasingToftheeffectivevalueofh^nd)]TJ /F4 11.955 Tf 12.23 0 Td[(1i0.Theimpuritychargedoesnotrenormalize,whiletolowestordertheeffectivee-bcouplingobeysEq.( 3 )[ 46 ].DeningTLbythecondition~(TL)jh^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1i0j=CL,wend TL'C)]TJ /F7 7.97 Tf 6.58 0 Td[(1Lh^nd)]TJ /F4 11.955 Tf 11.96 0 Td[(1i02=(1)]TJ /F8 7.97 Tf 6.59 0 Td[(s).(3) InFig. 3-32 ,symbolsrepresentTLvaluesextractedfromthecrossoverofbosonicexcitationsintheNRGspectrum,whilesolidlinesshowtheresultsofevaluatingEq.( 3 )usingCL=3andtheh^nd)]TJ /F4 11.955 Tf 12.16 0 Td[(1i0valuesshowninFig. 3-31 .ThealgebraicrelationbetweenthenumericalvaluesofTLandh1)]TJ /F4 11.955 Tf 12.71 0 Td[(^ndi0iswellobeyedoverarangeofe-bcouplingsthatextendsbeyondcofthesymmetricproblem. Figure 3-33 plotsthestaticlocalchargesusceptibilitycalculatedfors=0.4atthecriticale-bcouplingofthesymmetricmodel.Ford6=0,c,locfollowsthequantumcriticalbehaviorc,loc(T;!=0)/T)]TJ /F8 7.97 Tf 6.59 0 Td[(xfromahigh-temperaturecutoffoforderTKdowntoacrossovertemperatureT,belowwhichthesusceptibilitysaturates.BasedonEq.( 3 )withtheidenticationd,oneexpectsT/jdj2=(1+x)and,hence, c,loc(;=c,!=T=0)/jdj)]TJ /F7 7.97 Tf 6.58 0 Td[(2x=(1+x).(3) 81

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Thelog-logplotintheinsetofFig. 3-33 hasaslope0.57thatisfullyconsistentwithEq.( 3 ). Theresultsofthisworkshowthatgainingdirectaccesstothequantumcriticalpointofthecharge-coupledBFAmodelrequiressimultaneousnetuningoftwoparameters:thee-bcouplingasafunctionofthehybridization)]TJ /F1 11.955 Tf 10.1 0 Td[(andtheon-siteCoulombrepulsionU;andtheparticle-holeasymmetry(determinedinourcalculationssolelybyd=d+U=2,butingeneralalsoaffectedbytheshapeoftheconduction-banddensityofstates).Whileitmayproveverychallenging,orevenimpossible,toachievethisfeatinanyexperimentalrealizationofthemodel,itshouldbeamorefeasibletasktocarryoutaroughtuningofparametersthatplacesthesysteminthequantumcriticalregimeoversomewindowofelevatedtemperaturesand/orfrequencies. 3.7Summary Inthischapterwehaveconductedadetailedstudyofthecharge-coupledBose-FermiAndersonmodel,inwhichamagneticimpuritybothhybridizeswithastructurelessconductionbandandiscoupled,viaitscharge,toadissipativeenvironmentrepresentedbyabosonicbathhavingaspectralfunctionthatvanishesas!sforfrequencies!!0.Withincreasingcouplingbetweentheimpurityandthebath,wendacrossoverfromaconventionalKondoeffectinvolvingconduction-bandscreeningoftheimpurityspindegreeoffreedomtoacharge-Kondoregimeinwhichthedelocalizedelectronsquenchimpuritychargeuctuations. Underconditionsofstrictparticle-holesymmetry,furtherincreaseintheimpurity-bathcouplinggivesrisefor0
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criticalpointcharacterizedbyhyperscalingrelationsofcriticalexponentsand!=Tscalinginthedynamicallocalchargesusceptibility.Moreover,thecontinuousphasetransitionofthepresentmodelbelongstothesameuniversalityclassasthetransitionsofthespin-bosonandtheIsing-anisotropicBose-FermiKondomodels.ForanOhmic(s=1)bosonicbathspectrum,thequantumphasetransitionisofKosterlitz-Thoulesstype. Inthepresenceofparticle-holeasymmetry,thequantumphasetransitiondescribedinthepreviousparagraphisreplacedbyasmoothcrossover,butforsmall-to-moderateasymmetries,signaturesofthesymmetricquantumcriticalpointremaininthephysicalpropertiesatelevatedtemperaturesand/orfrequencies. 83

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Figure3-1. Symmetric,zero-hybridizationmodeldenedby^HZHinEq.( 3 )withd=0:evolutionwithe-bcoupling2ofthelowesteigenenergyinthespinsector(nd=1,solidline)andinthechargesector(nd=0,2,dashedline).Alevelcrossingoccursat=c0speciedinEq.( 3 ). 84

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Figure3-2. Dependenceofthelevel-crossingcouplingc0onthediscretizationfortheNRGsolutionof^HZH[Eq.( 3 )]withU=0.1,d=0,Ns=200,Nb=16,andvedifferentvaluesofthebathexponents.Dashedlinesshowlineartstothedata. 85

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Figure3-3. Schematicphasediagramofthesymmetriccharge-coupledBFAmodelforbathexponents0
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Figure3-4. Schematicrenormalization-groupowsonthe-planeforthesymmetriccharge-coupledBFAmodelwithabathexponent0
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Figure3-5. Low-lyingmany-bodyenergiesENvseveniterationnumberNfors=0.2,)-278(=0.5,andvedifferentcouplings,whosecriticalvaluec'0.530086.(a)ThespectrumoftheKondoxedpointisreachedfor=0,wherethebosonicbathisdecoupledfromtheimpurity.(b)AsapproachescfromtheKondoside,thequantumcriticalregimeisaccessed(6N26)beforetheenergylevelscrossovertotheKondoxedpoint.(c)Thespectrumofthequantumcriticalpoint.(d)Whenisclosetoitscriticalvaluebutslightinthelocalizedphase,theenergylevelscrossoverfromthequantumcriticalpoint(6N30)tothelocalizedphase.(e)Deepinthelocalizedphase,thespectrumofthelocalizedxedpointisreachedatsmallN.Thecrossoverintheenergyowsdenesalow-energyscaleTthatvanishesasg!gcinthefashionofEq.( 3 ).NotethatthelocalizedphaseisactuallynotcapturedbyauniquexedpointbutcorrespondstoalineofxedpointswithdifferentWd.Seetextfordiscussion. 88

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Figure3-6. (Color)Fixed-pointcouplingWdenteringEq.( 3 )vse-bcoupling)]TJ /F5 11.955 Tf 11.96 0 Td[(cinthelocalizedphasenearthephaseboundaryat=c.ResultsareshownforU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,=9,Ns=500,Nb=8,fourdifferentvaluesofthebathexponents,and)-277(=0.5,1.0,10,and50fors=0.2,0.4,0.6,and0.8,respectively(seefootnote 1 onpage 57 ).Thepower-lawdivergenceofWdas!+c[Eq.( 3 )]isreectedinthelinearbehaviorsofdataonalogarithmicscale.Thenumericalvaluesoftheexponentobtainedhereareidentical(towithinsmallerrors)tothoselistedinTable 3-3 89

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Figure3-7. Dependenceoftheenergyoftherstbosonicexcitationatthecriticalpoint(=c)ontheNRGtruncationparametersNbandNs.ResultsareshownforU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,)-277(=0.01,=9,andbathexponentss=0.2ands=0.8.Intheleftpanels,Ns=500,whileintherightpanelsNb=8. 90

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Figure3-8. Dependenceoftheenergyoftherstbosonicexcitationinthelocalizedphase(=1.1c)ontheNRGtruncationparametersNbandNs.AllotherparametersareasinFig. 3-7 91

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Figure3-9. Dependenceofh^B20i,denedinEq.( 3 )andevaluatedatcharacteristictemperaturescaleofiterationN=20,ontheNRGtruncationparametersNbfor(a)s=0.2and(b)s=0.8.Seetextfordiscussion. 92

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Figure3-10. (Color)Criticalcouplingcvshybridizationwidth)]TJ /F1 11.955 Tf 10.09 0 Td[(forU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,=9,Ns=500,Nb=8,andthebathexponentsslistedinthelegend. 93

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Figure3-11. Variationwithbathexponentsofthecriticalcouplingscandc0inthecharge-coupledBFAmodel(thiswork)andgc=2inthespin-coupledBFAmodel(Ref.[ 74 ]).ResultsareshownforU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,)-277(=0.01,=9,Ns=500,andNb=8. 94

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Figure3-12. (Color)CrossoverscaleTvsc)]TJ /F5 11.955 Tf 11.95 0 Td[(ontheKondosideofthecriticalpointforfourdifferentvaluesofthebathexponents,withallotherparametersasinFig. 3-6 .Theslopeofeachlineonthislog-logplotgivesthecorrelation-lengthexponent(s)denedinEq.( 3 ). 95

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Figure3-13. (Color)Temperaturedependenceoftheimpuritycontributiontothestaticspin(left)andcharge(right)susceptibilitiesfors=0.8,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,)-277(=0.01,=9,Ns=2000,Nb=8,anddifferentvaluesofthee-bcoupling.Dottedcurvescorrespondtoe-bcouplingslyingbetweenthevaluesspeciedinthelegendfortheadjacentnondottedcurves.For=c0'0.396,thespinandchargesusceptibilitiesareequivalent:s,imp(T)'1 4c,imp(T).Forc0,thechargeresponsedominates.Forc'0.5052181,limT!0Tc,imp(T)=0,whereasfor>c,thelimitingvalueisnonzeroandobeysEqs.( 3 )and( 3 ). 96

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Figure3-14. (Color)Impuritychargelim!0)]TJ /F6 11.955 Tf 8.74 -.3 Td[(Qloc(,;T=0)vse-bcoupling)]TJ /F5 11.955 Tf 11.95 0 Td[(cforfourdifferentvaluesofthebathexponents.AllotherparametersareasinFig. 3-6 .Asapproachescfromabove,lim!0)]TJ /F6 11.955 Tf 8.75 -.3 Td[(Qloc(,;T=0)vanishes(leftpanel)inapower-lawfashion(rightpanel)describedbyEq.( 3 ). 97

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Figure3-15. (Color)ImpuritychargeQloc(;=c,T=0)vslocalelectricpotentialjjforfourdifferentvaluesofthebathexponents.AllotherparametersareasinFig. 3-6 .ThedashedlinesrepresenttstotheformofEq.( 3 ). 98

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Figure3-16. (Color)Staticlocalchargesusceptibilityc,loc(T;!=0)vstemperatureTfors=0.4,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,)-277(=1.0(seefootnote 1 onpage 57 ),=9,Ns=500,Nb=8,andfordifferentvaluesofthee-bcouplingstraddlingthecriticalvaluec'1.02905. 99

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Figure3-17. (Color)Imaginarypartofthedynamicallocalchargesusceptibility00c,loc(!;T=0)vsfrequency!fors=0.2,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,)-277(=0.5(seefootnote 1 onpage 57 ),=9,Ns=500,Nb=8,anddifferente-bcouplings
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Figure3-18. (Color)Criticalstaticanddynamicalresponse:c,loc(T;=c,!=0)vsT(circles)and00c,loc(!;=c,T=0)vs!(squares)fortworepresentativebathexponentss=0.2ands=0.8.AllotherparametersareasinFig. 3-6 .Theequalityoftheslopesofthestaticanddynamicalchargesusceptibilitiesforagivenbathexponentsindicatesthatthecorrespondingcriticalexponentssatisfyx=y. 101

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Figure3-19. (Color)Scalingwith!=Toftheimaginarypartofthedynamicallocalchargesusceptibility00c,loc(!,T)atthecriticale-bcouplingc'0.53008fors=0.2,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,)-277(=0.5(seefootnote 1 onpage 57 ),=9,Ns=500,Nb=8,anddifferenttemperaturesTTK=0.425. 102

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Figure3-20. (Color)ImpurityspectralfunctionA(!;T=0)vsfrequency!fors=0.8,U=)]TJ /F4 11.955 Tf 9.29 0 Td[(2d=0.1,)-278(=0.01,=3,Ns=1200,Nb=8,anddifferentvaluesofthee-bcoupling.Fortheseparameters,UeffdenedinEq.( 3 )changessignatc0'0.369andthecriticalcouplingisc'0.474. 103

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Figure3-21. (Color)Variationwithe-bcouplingoftwocharacteristicenergyscalesextractedfromthezero-temperatureimpurityspectralfunction.AllparametersexceptarethesameasinFig. 3-20 .(a)Location!HoftheupperHubbardpeak.Thedashedlineshows!H=0.4U)]TJ /F5 11.955 Tf 11.95 0 Td[(2=(s).(b)Kondoresonancewidth(fullwidthathalfheight)2)]TJ /F10 7.97 Tf 13.05 -1.8 Td[(K.Thedashedline,representingthepredictionofEq.( 3 )withCK=0.82andwith~UNRGinEq.( 3 )evaluatedatE=U=2=jdj,tsthedataoveralmosttheentirerange0
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Figure3-22. (Color)DetailoftheimpurityspectralfunctionA(!;T=0)aroundfrequency!=0fors=0.8,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,)-278(=0.01,=3,Ns=1600,Nb=8,anddifferente-bcouplingsstraddlingthecriticalvaluec'0.47458.Forc,A(!;T=0)ispinnedtothevaluepredictedbyFermi-liquidtheory.For>c,theKondoresonancedisappears,leavingapairoflow-energypeakscenteredatj!joforderthecrossovertemperatureT('1.410)]TJ /F7 7.97 Tf 6.58 0 Td[(8for=0.475). 105

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Figure3-23. (Color)Variationwithe-bcoupling
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Figure3-24. Phaseboundaryc(\andcrossoverboundaryX(\(denedinthetext)fors=0.8,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,=3,Ns=1200,andNb=8.ThedataareconsistentwiththeschematicphasediagramshowninFig. 3-3 107

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Figure3-25. Schematicrenormalization-groupowsonthe-planeforthesymmetricmodelwithbathexponents=1.TrajectoriesrepresenttheowofthecouplingsenteringEq.( 3 )anddenedinEq.( 3 )underdecreaseinthehigh-energycutoffsontheconductionbandandthebosonicbath.Aseparatrix(dashedline)formstheboundarybetweenthebasinsofattractionoftheKondoxedpoint(K)andalineoflocalizedxedpoints(L).Flowalongtheseparatrixistowardthefree-orbitalxedpoint(FO)locatedat=c0.For=0only,thereisowawayfromFOtowardthelocal-momentxedpoint(LM)at=0. 108

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Figure3-26. (Color)Staticlocalchargesusceptibilityc,loc(T;!=0)vstemperatureTfors=1,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,)-277(=0.01,=9,Ns=800,Nb=12,anddifferente-bcouplings.OntheKondosideoftheQPT(c). 109

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Figure3-27. Variationwithe-bcouplingofthelocalchargesusceptibilityc,loc(!=T=0)intheKondophasec,fors=1,U=)]TJ /F4 11.955 Tf 9.29 0 Td[(2d=0.1,)-278(=0.01,=9,Ns=800,andNb=12.ThedottedlineshowsatofthesusceptibilitydatausingEqs.( 3 )and( 3 ). 110

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Figure3-28. (Color)ImpurityspectralfunctionA(!;T=0)vs!fors=1,U=)]TJ /F4 11.955 Tf 9.29 0 Td[(2d=0.1,)-278(=0.01,=3,Ns=1200,Nb=12,anddifferentvaluesofthee-bcoupling.Fortheseparameters,Ueff[Eq.( 3 )]changessignatc0'0.413andthecriticalcouplingisc'0.669. 111

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Figure3-29. (Color)Variationwithe-bcouplingoftwocharacteristicenergyscalesextractedfromthezero-temperatureimpurityspectralfunction.AllparametersexceptarethesameasinFig. 3-28 .(a)Location!HoftheupperHubbardpeak.Thedashedlineshows!H()=0.4U)]TJ /F5 11.955 Tf 11.96 0 Td[(2=.(b)Kondoresonancewidth(fullwidthathalfheight)2)]TJ /F10 7.97 Tf 13.05 -1.8 Td[(K.Thedashedline,representingthepredictionofEq.( 3 )withCK=0.82andwith~UNRGinEq.( 3 )evaluatedatE=U=2=jdj,tsthedataoveralmosttheentirerange0
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Figure3-30. (Color)ImpurityspectralfunctionA(!;T=0)vsfrequency!onalogarithmicscalefors=1,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.1,)-277(=0.01,=3,Ns=1200,Nb=12,anddifferente-bcouplings.For
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Figure3-31. (Color)Variationinthemagnitudeh1)]TJ /F4 11.955 Tf 12.06 0 Td[(^ndi0oftheground-stateimpuritychargewithe-bcouplingfors=0.4,U=0.1,)-277(=0.01,=9,Ns=500,andNb=8.Symbolsrepresentresultsforvevaluesoftheimpurityasymmetryd=d+U=2.Thesolidlinescorrespondingtoeachcased6=0representtheimpuritychargecalculatedbysolvingtheAndersonmodel[Eq.( 3 )]forthesamedvaluebutusinganeffectiveCoulombinteraction~UNRG(0.3U)[Eq.( 3 )].Thed=0symbolsshowvaluesoflim!0)]TJ /F6 11.955 Tf 8.75 -.3 Td[(Qloc(,;T=0),connectedbyaninterpolatingline. 114

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Figure3-32. (Color)VariationinthebosoniclocalizationtemperatureTLwithcouplingfors=0.4,U=0.1,)-277(=0.01,=9,Ns=500,Nb=8,andvariousimpurityasymmetriesd=d+U=2.ThesolidlineswereobtainedbyevaluatingEq.( 3 )withtheh1)]TJ /F4 11.955 Tf 12.06 0 Td[(^ndi0valuesshowninFig. 3-31 andwithCL=3. 115

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Figure3-33. (Color)Staticlocalchargesusceptibilityc,loc(T;!=0)vstemperatureTfors=0.4,U=0.1,)-277(=0.01,'0.29835,=9,Ns=500,Nb=8,andvariousimpurityasymmetriesd=d+U=2.Thee-bcouplingequalsthecriticalcouplingcofthesymmetriccased=0.Inset:zero-temperaturestaticlocalchargesusceptibilityc,loc(!=T=0)vsd. 116

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CHAPTER4RESULTSFORPSEUDOGAPANDERSON-HOLSTEINMODEL ThischapterisbasedonamanuscriptbyMengxingChengandKevinIngersentcurrentlyinnalpreparationforsubmissiontoPhys.Rev.B. 4.1Introduction Inthischapter,wetheoreticallyinvestigateapseudogapAnderson-Holstein(PAH)modelofamagneticimpuritylevelthathybridizeswithafermionichostwhosedensityofstatesvanishesasjjrattheFermienergy(=0)andisalsocoupled,viatheimpuritycharge,toalocal-bosonmode.Themodelisrelevanttoelectrontransportinnanoscaledevices,aswediscussedinSec. 1.5 Ournumericalrenormalization-group(NRG)studyofthepseudogapAnderson-Holsteinmodelrevealsthat:(i)For0
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forthePAHmodelwith0
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abosonmodeofenergy!0annihilatedbyoperatora.Withoutlossofgenerality,wetakeVkandtoberealandnon-negative.Forcompactnessofnotation,wedropallfactorsofthereducedPlanckconstant~,Boltzmann'sconstantkB,theimpuritymagneticmomentgB,andtheelectronicchargee. Theconduction-banddispersionkandthehybridizationVkaffecttheimpuritydegreesoffreedomonlythroughthescatteringrate ~\() NkXkjVkj2()]TJ /F5 11.955 Tf 11.95 0 Td[(k).(4) Tofocusonthemostinterestingphysicsofthemodel,weassumeasimpliedform ~\()=)]TJ /F2 11.955 Tf 29.59 0 Td[(j=Djr(D)-222(jj),(4) where(x)istheHeaviside(step)function.Inthisnotation,thecaser=0representsaconventionalmetallicscatteringrate.Thisworkfocusesoncasesr>0inwhichthescatteringexhibitsapower-lawpseudogaparoundtheFermienergy.Wewillassumehenceforththatsuchascatteringratearisesfromalocalhybridization(Vk=V)andadensityofstatesvaryingas()(Nk))]TJ /F7 7.97 Tf 6.59 0 Td[(1Pk()]TJ /F5 11.955 Tf 12.85 0 Td[(k)=0j=Djr(D)-296(jj),inwhichcase)-329(=0V2.However,theresultsbelowapplyalsotosituationsinwhichthehybridizationcontributestotheenergydependenceof~\(). Theassumptionthat~\()exhibitsapurepower-lawdependenceovertheentirewidthoftheconductionbandisaconvenientidealization.Morerealisticscatteringratesinwhichthepower-lawvariationisrestrictedtoaregionaroundtheFermienergyexhibitthesamequalitativephysics,withmodicationonlyofnonuniversalpropertiessuchascriticalcouplingsandKondotemperatures. ThepropertiesoftheHamiltonianspeciedbyEqs.( 4 )( 4 )turnouttodependonwhetherornotthesystemisinvariantundertheparticle-holetransformationck!cy)]TJ /F13 7.97 Tf 6.59 0 Td[(k,,d!)]TJ /F6 11.955 Tf 26.75 0 Td[(dy,a!)]TJ /F6 11.955 Tf 26.76 0 Td[(a,whichmapsd!)]TJ /F5 11.955 Tf 26.76 0 Td[(d,k!)]TJ /F5 11.955 Tf 26.76 0 Td[()]TJ /F13 7.97 Tf 6.59 0 Td[(k,Vk!V)]TJ /F13 7.97 Tf 6.59 0 Td[(k.Forthe 119

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symmetricscatteringrategiveninEq.( 4 ),theconditionforparticle-holesymmetryisd=0correspondingtod=)]TJ /F7 7.97 Tf 10.5 4.7 Td[(1 2U. 4.2.2ReviewofRelatedModels BeforeaddressingthefullPAHmodeldescribedbyEq.( 4 ),itisusefultoreviewtwolimitingcasesthathavebeenstudiedpreviously.OneisthepseudogapAndersonmodelinabsenceoftheimpurity-bosoncouplingandtheotheristheAnderson-Holsteinmodelwithametallichostcorrespondingtothecaser=0. 4.2.2.1PseudogapAndersonmodel. Forzerobosoniccoupling=0,thePAHmodelreducestothepseudogapAndersonmodel[ 71 92 95 ]plusfreelocalbosons.Intheconventional(r=0)Andersonimpuritymodel,thegenericlow-temperaturelimitisastrong-coupling(Kondoormixedvalence)regimeinwhichtheimpurityleveliseffectivelyabsorbedintotheconductionband[ 66 67 ].Inthepseudogapvariantofthemodel,thedepressionofthescatteringratearoundtheFermienergygivesrisetoacompetinglocal-momentphaseinwhichtheimpurityretainsanunscreenedspindegreeoffreedomallthewaydowntotheabsolutezerooftemperature.TheT=0phasediagramofthismodeldependsonthepresenceorabsenceofparticle-holesymmetry[ 71 ]. Forthesymmetriccased=0(d=)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 2U)andanybandexponent0)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c),thecorrespondingpropertiesareSimp=2rln2,Ts,imp=r=8,and0(0)=)]TJ /F4 11.955 Tf 9.3 0 Td[((1)]TJ /F6 11.955 Tf 12 0 Td[(r)(=2)sgn(allindicativeofincompletequenchingoftheimpuritydegreesoffreedom).Thequantumphasetransitiontakesplaceataninteractingquantumcriticalpoint[ 58 96 97 ].Forthesymmetriccaseand 120

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r1 2,bycontrast,theSSCxedpointisunstable[ 71 93 ]andthesystemliesintheLMphaseforallvaluesof)]TJ /F1 11.955 Tf 6.78 0 Td[(. Awayfromparticle-holesymmetry,themodelremainsintheLMphasedescribedaboveforjdj<1 2U(i.e.,)]TJ /F6 11.955 Tf 9.3 0 Td[(U)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c,themodelliesinoneoftwoasymmetricstrong-coupling(ASC)phaseshavinglow-temperaturepropertiesSimp=0andTs,imp=0.Ford>0,theFermi-energyphaseshiftis0(0)=)]TJ /F5 11.955 Tf 9.3 0 Td[(sgn,whiletheground-statecharge(totalfermionnumbermeasuredfromhalf-lling)isQ=)]TJ /F4 11.955 Tf 9.3 0 Td[(1.Ford<0,bycontrast,0(0)=+sgnandQ=+1.WelabelthesetwophasesASC)]TJ /F1 11.955 Tf 10.41 1.79 Td[(andASC+accordingtothesignofQ. Forrr,therearetwoasymmetricquantumphasetransitions(oneford>0,theotherford<0)[ 71 ].Theasymmetrictransitionstakeplaceatinteractingquantumcriticalpointsforr
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^H=^Himp+^Hband+^HbosonwithUenteringEq.( 4a )replacedby Ueff=U)]TJ /F4 11.955 Tf 11.95 0 Td[(22=!0.(4) ItcanbeseenfromEq.( 4 )thatthiseffectiveon-siteCoulombinteractionchangessignat=0,where 0=p !0U=2.(4) Forweakbosoniccouplings<0,theinteractionisrepulsive,andforjdj<1 2Uefftheimpuritygroundstateisaspindoubletwithnd=1and=1 2.For>0,bycontrast,thestrongcouplingtothebosonicbathyieldsanattractiveeffectiveon-siteinteractionandforjdj<)]TJ /F7 7.97 Tf 10.49 4.7 Td[(1 2Ueffthetwolowest-energyimpuritystatesarespinlessbuthaveacharge(relativetohalflling)Q=nd)]TJ /F4 11.955 Tf 12.37 0 Td[(1=1;notethatthesetwostatesaredegenerateonlyunderconditionsofstrictparticle-holesymmetry(d=0). ThefullAnderson-Holsteinmodelwith)]TJ /F2 11.955 Tf 10.26 0 Td[(6=0exhibitsacontinuousevolution[ 62 85 ]ofitsphysicalpropertieswithincreasingbosoniccoupling.ThepropertiesforagivenareessentiallythoseoftheconventionalAndersonimpuritymodelwithUreplacedbyUeff().Inparticular,for)]TJ /F2 11.955 Tf 10.64 0 Td[(jdjU,thereisasmoothcrossoverfromaconventionalspin-sectorKondoeffectfor0(andthusUeff)]TJ /F1 11.955 Tf 6.77 0 Td[()toacharge-sectoranalogoftheKondoeffectfor0(and)]TJ /F6 11.955 Tf 9.3 0 Td[(Ueff)]TJ /F1 11.955 Tf 6.77 0 Td[().Aprimarygoalofthepresentworkwastounderstandhowthisphysicsismodiedbythepresenceofapseudogapintheimpurityscatteringrate. 4.3Results:SymmetricPAHModelwithBandExponents0
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modelwithUreplacedbyUeffdenedinEq.( 4 ).Aquantumcriticalpointlocatedat)-324(=)]TJ /F8 7.97 Tf 30.61 -1.8 Td[(c1(r,U,<0)')]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c(r,Ueff)hasuniversalpropertiesindistinguishablefromthoseatthecriticalpointofthepseudogapAndersonmodel.Forstrongerbosoniccouplings>0,thereisinsteadaquantumphasetransitionat)-298(=)]TJ /F8 7.97 Tf 29.97 -1.79 Td[(c2(r,U,>0)betweenthesymmetricstrong-couplingphaseandalocal-chargephaseinwhichtheimpurityhasaresidualtwo-foldchargedegreeoffreedom.Thecriticalexponentsdescribingthelocalchargeresponseatthethe)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c2criticalpointareidenticaltothosecharacterizingthelocalspinresponseatthe)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c1criticalpoint. AllnumericalresultspresentedinthissectionwereobtainedforasymmetricPAHmodelwithU=)]TJ /F4 11.955 Tf 9.29 0 Td[(2d=0.5,forabosonicenergy!0=0.1,forNRGdiscretizationparameter=3,andrestrictingthenumberofbosonstonomorethanNb=40.Exceptwherenotedotherwise,Ns=500stateswereretainedaftereachNRGiteration. 4.3.1NRGSpectrumandFixedPoints TherstevidencefortheexistenceofmultiplephasesofthesymmetricPAHmodelcomesfromtheeigenspectrumof^HN.Thisspectrumcanbeusedtoidentifystableandunstablerenormalization-groupxedpoints,aswellastemperaturescalescharacterizingcrossoversbetweenthosexedpoints. (1)Weakbosoniccouplings.Figure 4-2 (a)showsforr=0.4,=0.05<0'0.158,andsevendifferentvaluesof)]TJ /F1 11.955 Tf 6.78 0 Td[(thevariationwitheveniterationnumberNoftheenergyoftherstexcitedmultiplethavingquantumnumbersS=1,Q=0.Forsmallvaluesof)]TJ /F1 11.955 Tf 6.78 0 Td[(,thisenergyENatrstriseswithincreasingN,buteventuallyfallstowardthevalueELM=0expectedatthelocal-moment(LM)xedpointcorrespondingtoeffectivemodelcouplings)-476(==0andU=1.Atthisxedpoint,theimpuritynd=1doubletasymptoticallydecouplesfromthetight-bindingchainoflengthN+1,leavingalocalizedspin-1 2degreeoffreedom.TheimpuritythermodynamiccontributionsSimp=ln2,Ts,imph^S2zi=1 4,andTc,imph(^nd)]TJ /F4 11.955 Tf 12.32 0 Td[(1)2i=0aswellastheFermi-energyphase 123

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shift0(0)=0,coincidewiththecorrespondingpropertiesattheLMxedpointofthepseudogapAndersonmodel(seeSec. 4.2.2.1 ). Forlarge)]TJ /F1 11.955 Tf 6.77 0 Td[(,ENinsteadrisesmonotonicallytoreachalimitingvalueESSC'1.119characteristicofthesymmetricstrong-coupling(SSC)xedpoint,correspondingtoeffectivecouplings)-428(=1,U==0.Here,theimpuritylevelformsaspinsingletwithanelectronontheend(n=0)siteofthetight-bindingchain.Thesingletformationfreezesouttheendsite,leavingfree-fermionicexcitationsonachainofreducedlengthN,leadingtoaFermi-energyphaseshift0(0)=)]TJ /F4 11.955 Tf 9.3 0 Td[((1)]TJ /F6 11.955 Tf 12.25 0 Td[(r)(=2)sgn.Thisphaseshift,alongwiththeimpuritycontributionsSimp=2rln2,Ts,imp=r=8,andTc,imp=r=2areidenticaltotheSSCpropertiesinthepseudogapAndersonmodel. TheLMandSSCxedpointsdescribethelarge-N(low-energyD)]TJ /F8 7.97 Tf 6.59 0 Td[(N=2)physicsforallinitialchoicesofthehybridizationexcept)-319(=)]TJ /F8 7.97 Tf 30.49 -1.79 Td[(c1'0.3166805,inwhichspecialcaseENrapidlyapproachesEc'0.6258andremainsatthatenergyuptoarbitrarilylargeN.ThisbehaviorcanbeassociatedwithanunstablepseudogapcriticalpointseparatingtheLMandSSCphases.ThecriticalpointcorrespondstothePAHmodelwith=0and)]TJ /F5 11.955 Tf 6.78 0 Td[(=Uequalinganr-dependentcriticalvalue. Thelow-energyNRGspectrumateachoftherenormalization-groupxedpointsidentiedintheprecedingparagraphsLM,SSC,andcriticalvarieswiththebandexponentrandtheNRGdiscretizationparameter.However,forgivenrand,thexed-pointspectraarefoundtobeidenticaltothoseoftheparticle-hole-symmetricpseudogapAndersonmodel(asdescribedinSec. 4.2.2 ).Eachofthesexedpointscanbeinterpretedascorrespondingtoaneffectivebosoncoupling=0andexhibitstheSU(2)isospinsymmetrythatisbrokeninthefullPAHmodel. (2)Strongbosoniccouplings.Figure 4-3 (a)plotstheenergyateveniterationsoftherstNRGexcitedstatehavingquantumnumbersS=Q=0,forr=0.4,=0.2>0'0.158,andsevendifferent)]TJ /F1 11.955 Tf 10.1 0 Td[(values.For)]TJ /F5 11.955 Tf 11.94 0 Td[(>)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(c2'0.6878956,ENeventuallyowstothevalueESSC'1.119identiedintheweak-bosonic-coupling 124

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regime,andexaminationofthefullNRGspectrumconrmsthatthelow-temperaturebehaviorisgovernedbythesameSSCxedpoint. For)]TJ /F5 11.955 Tf 11.18 0 Td[(<)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(c2,ENowstozero,thevaluefoundattheLMxedpoint.Infact,allthexed-pointmany-bodystatesobtainedfor>0turnouttohavethesameenergiesasstatesattheLMxedpoint.However,thequantumnumbersofstatesinthe>0spectrumandtheLMspectraarenotidentical,butratherarerelatedbytheinterchangesS$IandSz$Iz[SeeEq.( 2 )forthedenitionoftheisospin(axialcharge)].Wethereforeassociatethe>0spectrumwithalocal-charge(LC)xedpoint,correspondingto)-303(==0andU=,atwhichtheimpurityhasaresidualisospin-1 2degreeoffreedom.ThisxedpointshareswithitsLMcounterpartthepropertiesSimp=ln2and0(0)=0,butisdistinguishedbyhavingTs,imp=0andaCurie-likechargesusceptibilityTc,imph(^nd)]TJ /F4 11.955 Tf 12.4 0 Td[(1)2i=1.(Duetothedifferentdenitionsofthespinandchargesusceptibilities,itismostappropriatetocompare1 4c,impwiths,imp.) For)-482(=)]TJ /F8 7.97 Tf 34.39 -1.79 Td[(c2,ENrapidlyapproachesandremainsatthesamecriticalvalueEcasfoundfor<0and)-431(=)]TJ /F8 7.97 Tf 33.17 -1.8 Td[(c1.Onceagain,however,themany-bodyspectrumisrelatedtothatatthecorrespondingweak-bosonic-couplingxedpointbyinterchangeofspinandisospinquantumnumbers,leadingtotheinterpretationofthisxedpointasachargeanalogofthecriticalpointoftheconventionalpseudogapAndersonmodel. Thepropertiesdescribedabovearesummarizedintheschematicrenormalization-groupowdiagramshowninFig. 4-4 .ArrowsindicatethedirectioninwhichtheeffectivevaluesofthecouplingUeffand)]TJ /F1 11.955 Tf 10.1 0 Td[(evolvewithincreasingNRGiterationnumberN,i.e.,underprogressivereductionoftheeffectivebandcutoffD)]TJ /F8 7.97 Tf 6.59 0 Td[(N=2.Thehigh-temperaturelimitofthemodelisgovernedbythefree-orbital(FO)xedpoint,correspondingtobaremodelparametersU=)-428(==0,atwhichSimp=ln4,Ts,imp=1 8,Tc,imp=1 2,and0(0)=0.DashedlinesmarktheseparatricesbetweenthebasinsofattractionoftheLM,LC,andSSCxedpointsdescribedabove.FlowalongeachseparatrixisfromFOtowardoneorotheroftwoquantumcriticalpointseithertheconventionalspin 125

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pseudogapcriticalpointCsreachedforUeff>0,oritschargeanalogCcreachedforUeff<0. 4.3.2PhaseBoundariesandCrossoverScales Figure 4-5 showsthephaseboundaries)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(c1and)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c2vsforU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2"d=0.5,!0=0.1,andfourdifferentvaluesofthebandexponentr.Foreachr,)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c1and)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c2arebothfoundtovanishat0=0.15812(1),closetothevaluep !0U=2'0.158114predictedbyEq.( 4 ). Withdecreasingj)]TJ /F2 11.955 Tf 9.76 0 Td[()]TJ /F4 11.955 Tf 12.28 0 Td[()]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c1j,ENinFig. 4-2 (a)remainsclosetoEcoveranincreasingnumberofiterationsbeforeheadingeithertoELMortoESSC.Toquantifythiseffect,itisusefultodenethresholdenergyvaluesEwhereELM
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Table4-1. Correlation-lengthcriticalexponents1and2vsbandexponentrfortheparticle-hole-symmetricPAHmodel.Parenthesessurroundtheestimatednonsystematicerrorinthelastdigit. r12 0.26.22(1)6.22(1)0.35.14(1)5.14(1)0.45.84(1)5.84(1) abehaviorthatissketchedqualitativelyinFig. 4-3 (b)andisconrmedquantitativelyinFig. 4-3 (c).AsshowninTable 4-1 ,forallthevaluesofrthatwehavestudied,thenumericalvaluesof1and2coincidetowithinourestimatederrors. 4.3.3ThermodynamicProperties Intheweakbosoniccouplingregime,Figure 4-6 plotsthedependenceofTs,imp,1 4Tc,imp,andSimpontemperatureforr=0.4,=0.05,andvarious)]TJ /F1 11.955 Tf 10.1 0 Td[(straddlingthecriticalvalue)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c1.Thelow-temperaturebehaviorofTs,impclearlyshowsthataQCPseparatestheSSCandLMphases.Exactlyatthecriticalpoint(blueline),Ts,impexhibitsrenormalization-freebehavioratlowtemperaturesandmaintainsacriticalvalue'0.1348.When)]TJ /F1 11.955 Tf 10.09 0 Td[(deviatesslightlyfromthecriticalvalue,Ts,imptracesthecriticalbehavioruntilitcrossesbelowacertaintemperaturetoeither1=4fortheLMphase(solidsymbols)orr=8fortheSSCphase(emptysymbols).ThereisaweakerreectionoftheQCPin1 4Tc,imp,whichhasacriticalvalue'0.0158and,asthetemperaturedecreases,fallstowardzerointheLMphasebutrisestowardr=8intheSSCphase.Simphasacriticalvalue'0.6945andreacheseitherln2fortheLMphaseor2rln2fortheSSCphaseatlowtemperatures.(ThecriticalvalueofSimpisclosetobutapparentlydistinctfromln2.Thiswillbeinvestigatedfurtherinfuturework.) Inthestrongbosoniccouplingregime,Figure 4-7 plotsthedependenceofTs,imp,1 4Tc,imp,andSimpontemperatureforr=0.4,=0.2,andvarious)]TJ /F1 11.955 Tf 10.1 0 Td[(straddlingthecriticalvalue)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c2.Here,itisthelow-temperaturebehaviorofTc,impthatmostclearlyshowsthataQCPseparatestheSSCandLCphases.Exactlyatthecriticalpoint)-356(=)]TJ /F8 7.97 Tf 31.38 -1.79 Td[(c2(blueline),Tc,impexhibitsrenormalization-freebehavioratlowtemperatures. 127

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When)]TJ /F1 11.955 Tf 10.1 0 Td[(deviatesslightlyfromthecriticalvalue,1 4Tc,imptracesthecriticalbehavior'0.1348untilitcrossesoverbelowacertaintemperaturetoeither1=4fortheLCphase(solidsymbols)orr=8fortheSSCphase(emptysymbols).Inthiscase,Ts,imptracesitscriticalvalue'0.0158untilfallstozerofortheLCphaseandrisestowardr=8fortheSSCphase.ThebehaviorofSimpupondecreasingtemperaturesisverysimilartothatinFig. 4-6 4.3.4LocalResponseandUniversalityClass Inordertoinvestigateingreaterdetailthepropertiesofthespinandchargevariantsofthepseudogapcriticalpoint(CsandCcinFig. 4-4 ),itisnecessarytoidentifyanappropriateorderparameterforeachquantumphasetransition. (1)Weakbosoniccoupling.InthepseudogapKondoandAndersonmodels,thecriticalpropertiesmanifestthemselves[ 58 ]throughtheresponsetoalocalmagneticeldhthatcouplesonlytotheimpurityspin.SuchaeldenterstheAndersonmodelthroughanadditionalHamiltonianterm^H=1 2h(^nd")]TJ /F4 11.955 Tf 11.95 0 Td[(^nd#).Theorderparameterforthepseudogapquantumphasetransitionisthelimitingvalueash!0ofthelocalmoment Mloc=h1 2(^nd")]TJ /F4 11.955 Tf 12.06 0 Td[(^nd#)i,(4) andtheorder-parametersusceptibilityisthestaticlocalspinsusceptibility s,loc=)]TJ /F4 11.955 Tf 12.5 0 Td[(limh!0Mloc h.(4) BasedonthesimilaritiesnotedabovebetweenthepseudogapAndersoncriticalpointandtheCscriticalpointofthePAHmodel(i.e.,thepropertiesofthephasesoneithersideofeachtransition,theNRGspectrumatthetransition,andthevalueoftheorder-parameterexponent),weexpectthatthetwoquantumphasetransitionsalsotosharethesameorderparameter.Accordingly,thebehaviorsofMlocands,locinthevicinityofthecriticalhybridizationwidth)-339(=)]TJ /F8 7.97 Tf 30.96 -1.79 Td[(c1,shouldbedescribedbyexponents1, 128

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1,1,andx1denedinthefollowingfashion: Mloc()]TJ /F5 11.955 Tf 14.97 0 Td[(<)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c1;h!0,T=0)/()]TJ /F8 7.97 Tf 11.66 -1.79 Td[(c1)]TJ /F4 11.955 Tf 11.96 0 Td[(\1, (4a)s,loc()]TJ /F5 11.955 Tf 14.98 0 Td[(>)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(c1;T=0)/()]TJ /F2 11.955 Tf 14.32 0 Td[()]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F8 7.97 Tf 6.78 -1.8 Td[(c1))]TJ /F14 7.97 Tf 6.59 0 Td[(1, (4b)Mloc(h;)-278(=)]TJ /F8 7.97 Tf 34.97 -1.79 Td[(c1,T=0)/jhj1=1, (4c)s,loc(T;)-278(=)]TJ /F8 7.97 Tf 34.97 -1.79 Td[(c1)/T)]TJ /F8 7.97 Tf 6.59 0 Td[(x1. (4d) TheprecedingexpectationsareprovedcorrectbyNRGcalculations,asdemonstratedinFig. 4-8 forr=0.4and=0.05,thecasetreatedinFig. 4-2 .Thecriticalexponentsextractedasbest-tslopesoflog-logplotsarelistedinTable 4-2 forthreevaluesofthebandexponentr.Thevaluesofindividualexponentsvarywithr,butareindependentofotherHamiltonianparameters(U,!0,and)andwellconvergedwithrespecttotheNRGparameters(,Ns,andNb).Towithintheirestimatedaccuracy,theexponentsforagivenrobeythehyperscalingrelations 1=1+x1 1)]TJ /F6 11.955 Tf 11.96 0 Td[(x1,21=(1)]TJ /F6 11.955 Tf 11.95 0 Td[(x1),1=1x1,(4) whichareconsistentwiththescalingansatz F=Tfj)]TJ /F2 11.955 Tf 9.43 0 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c1j T1=1,jhj T(1+x1)=2(4) forthenonanalyticpartofthefreeenergyataninteractingcriticalpoint[ 17 ]. (2)Strongbosoniccoupling.WehaveseenabovethattheNRGspectrumandlow-temperaturethermodynamicsattheCcxedpointarerelatedtothoseattheCs Table4-2. ExponentsdescribingthelocalspinresponseatthecriticalpointCsoftheparticle-hole-symmetricPAHmodel,evaluatedforthreevaluesofthebandexponentr.Anumberinparenthesesindicatestheestimatedrandomerrorinthelastdigitofeachexponent. r11=1x11 0.20.15(1)0.02630(2)0.9488(2)5.85(6)0.30.34(1)0.07364(1)0.8629(3)4.41(3)0.40.90(1)0.1845(1)0.6885(2)3.95(5) 129

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xedpointbyinterchangeofspinandchargedegreesoffreedom.OnethereforeexpectstobeabletoprobethecriticalpropertiesviatheresponsetoalocalelectricpotentialthatentersthemodelthroughanadditionalHamiltonianterm^H=(^nd)]TJ /F4 11.955 Tf 11.08 0 Td[(1).Theorderparametershouldbethe!0limitingvalueofthelocalcharge Qloc=h^nd)]TJ /F4 11.955 Tf 11.95 0 Td[(1i,(4) andtheorder-parametersusceptibilityshouldbethestaticlocalchargesusceptibility c,loc=)]TJ /F4 11.955 Tf 12.74 0 Td[(lim!0Qloc .(4) Inthevicinityofthecriticalpoint)-277(=)]TJ /F8 7.97 Tf 29.49 -1.79 Td[(c2,oneexpectsthefollowingcriticalbehaviors: Qloc()]TJ /F5 11.955 Tf 14.98 0 Td[(<)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c2;!0,T=0)/()]TJ /F8 7.97 Tf 11.66 -1.8 Td[(c2)]TJ /F4 11.955 Tf 11.95 0 Td[(\2, (4a)c,loc()]TJ /F5 11.955 Tf 14.98 0 Td[(>)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c2;T=0)/()]TJ /F2 11.955 Tf 14.31 0 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c2))]TJ /F14 7.97 Tf 6.59 0 Td[(2, (4b)Qloc(;)-278(=)]TJ /F8 7.97 Tf 34.97 -1.79 Td[(c2,T=0)/jj1=2, (4c)c,loc(T;)-278(=)]TJ /F8 7.97 Tf 34.97 -1.8 Td[(c2)/T)]TJ /F8 7.97 Tf 6.58 0 Td[(x2. (4d) TheseexpectationsareborneoutbytheNRGresults.Figure 4-9 demonstratesthepredictedcriticalbehaviorsforthecaser=0.4,=0.2treatedinFig. 4-3 .Thevaluesofthecriticalexponents2,2,1=2,andx2areequaltothoseof1,1,1=1,andx1listedinTable 4-2 withsmallerrors,asdemonstratedasfollows. (3)Comparisonbetweenweakandstrongbosoniccoupling.Figure 4-10 (a)superimposesthevariationwith)]TJ /F1 11.955 Tf 10.1 0 Td[(oftheorderparameterinthevicinityofther=0.2andr=0.4criticalpoints.Theequalityoftheslopesofthelog-logplotsatthespin-andcharge-sectorquantumphasetransitionsshowsthat1=2.Similarly,Fig. 4-10 (b)showsthatthetemperaturevariationoftheorder-parametersusceptibilitiesisconsistentwithx1=x2.Indeed,foreachvalueofrthatwehaveexamined,wendthatallexponentsatthecharge-sectorcriticalpointareindistinguishable(withinourestimatederrors)fromthecorrespondingexponentsatthespin-sectorcriticalpointofthePAH 130

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modelandatthecriticalpointofthepseudogapKondomodel(asgiveninTableIofRef.[ 58 ]).Thisleadsustoconcludethatallthreecriticalpointslieinthesameuniversalityclass. 4.4Results:GeneralPAHModelwithBandExponents00.BasedonthediscussioninSec. 4.2.2.2 ,oneexpectsthePAHmodeltoexhibitthepropertiesasthepseudogapAndersonmodelwithUeffreplacingthebareCoulombinteractionU.Figure 4-11 plotsphaseboundariesofthePAHmodelonthed-)]TJ /F1 11.955 Tf 10.09 0 Td[(planeforr=0.4(left)and0.6(right),andthreeweakbosoniccouplings<0.ThemodelremainsintheLMphaseforjdj<1 2Ueffand)]TJ /F5 11.955 Tf 11.4 0 Td[(<)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c(r,U,d,))]TJ /F8 7.97 Tf 6.78 -1.8 Td[(c(r,U,)]TJ /F5 11.955 Tf 9.3 0 Td[(d,);otherwiseitliesinoneorotheroftheasymmetricstrong-couplingphasesdescribedinSec. 4.2.2.1 :ASC)]TJ /F1 11.955 Tf 10.41 1.8 Td[(ford>0orASC+ford<0.Thecriticalhybridizationwidth)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(cdecreasesmonotonicallyasincreases,suggestingacontractionoftheLMphaseuponincreasingbosoniccoupling. WecanassociatethetransitionsfromtheLMphasetotheASCphaseswithquantumcriticalpointsC.InthevicinityofC,thelocalspinresponsesexhibitpower-lawbehaviorsastheydonearthesymmetricquantumcriticalpointCs.Table 4-3 listscriticalexponentsatC.ComparisonwithTable 4-2 showsthatCsandChavethesamelow-temperaturephysicsforr=0.2and0.3butnotforr=0.4,0.6,and0.8.ThisisconsistentwiththepseudogapAndersonmodel,wheretheCsandCcriticalpointsareidenticalfor0
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Table4-3. ExponentsdescribingthelocalspinresponseatthecriticalpointsCoftheparticle-hole-asymmetricPAHmodel,evaluatedforvevaluesofthebandexponentr.Anumberinparenthesesindicatestheestimatedrandomerrorinthelastdigitofeachexponent. r11=1x11 0.20.15(1)0.02630(2)0.9488(2)5.85(6)0.30.34(1)0.07364(1)0.8629(3)4.41(3)0.40.59(1)0.1569(1)0.7275(3)3.12(2)0.60.188(1)0.1173(2)0.7896(4)1.41(1)0.80.079(1)0.0644(5)0.879(1)1.10(1) phaseboundariesofthesymmetricPAHmodelontheh-)]TJ /F1 11.955 Tf 10.1 0 Td[(planeforr=0.4(left)and0.6(right),andthreestrongbosoniccouplings>0.ThemodelremainsintheLCphaseforjhj0orASC"forh<0.Here,"or#indicatesthatthegroundstatehasaspinzcomponentSz=1 2.Thecriticalhybridizationwidth)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(cincreasemonotonicallyasincreases,indicatinganexpansionoftheLCphaseuponincreasingbosoniccoupling. InthevicinityofthecriticalpointsC",#markingthetransitionsfromtheLCphasetotheASC",#phases,thelocalchargeresponsesexhibitpower-lawbehaviorsastheydointhevicinityofCc.ThevaluesofthecriticalexponentsareequaltothoselistedinTable 4-3 withinsmallerrors. 4.5Results:DoubleQuantumDotswithU2=0 ThissectionaddressesthePAHmodelwithabandexponentr=2,acaseofparticularinterestbecauseithasapossiblerealizationindoublequantumdots.Belowwepresentresultsnotonlyfortheimpuritycontributionstothermodynamicpropertiesbutalsoforthelinearconductanceofsuchadouble-dotsysteminthevicinityofitsspin-andcharge-sectorquantumphasetransitions. 4.5.1EffectivePseudogapModelforDoubleQuantumDots Themotivationforfocusingonthecaser=2comesfromstudies[ 56 57 ]oftwoquantumdotscoupledinparalleltoleft(L)andright(R)leads,andgatedinsuch 132

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amannerthatthelow-energyphysicsisdominatedbyjustonesingle-particlestateoneachdot.Itisassumedthatoneofthedots(dot1)issmallandhencestronglyinteracting,whiletheother(dot2)islarger,hasanegligiblechargingenergy,andcanbeapproximatedasanoninteractingresonantlevel.Thissetupcanbedescribedbythetwo-impurityAndersonHamiltonian ^HDD=Xi,i^ni+U1^n1"^n1#+X`,k,`kcy`kc`k+Xi,`,k,Vi`)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(dyic`k+H.c..(4) Here,diannihilatesanelectronofspinzcomponentandenergyiinthedoti(i=1,2),^ni=dyidiisthenumberoperatorforsuchelectrons,andc`kannihilatesanelectronofspinzcomponentandenergy`kinlead`(`=L,R).Forsimplicity,theleadsareassumedtohavethesamedispersionLk=Rk,correspondingtoatop-hatdensityofstates()=0(D)-251(jj)with0=(2D))]TJ /F7 7.97 Tf 6.59 0 Td[(1,andtohybridizesymmetricallywiththedotssothatViL=ViR.Undertheseconditions,thedotscoupleonlytothesymmetriccombinationofleadelectronsannihilatedbyck=(cLk+cRk)=p 2witheffectivehybridizationsVi=p 2ViL. AkeyfeatureofEq.( 4 )isthevanishingofthedot-2CoulombinteractionU2associatedwithaHamiltoniantermU2^n2"^n2#.Thisallowsonetointegrateoutdot2toyieldaneffectiveAndersonmodelforasingleimpuritycharacterizedbyalevelenergy1,anon-siteinteractionU1,andascatteringrate[ 56 ] ~)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(1()=()]TJ /F5 11.955 Tf 11.96 0 Td[(2)2 ()]TJ /F5 11.955 Tf 11.96 0 Td[(2)2+)]TJ /F7 7.97 Tf 18.73 4.12 Td[(22)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(1(D)-222(jj),(4) where)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(i=0V2ifori=1,2.Thepresenceofdot2intheoriginalmodelmanifestsitselfhereasaLorentzianholein~)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(1()ofwidth)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(2centeredon=2.For2=0(aconditionthatmightbeachievedinpracticebytuningaplungergatevoltageondot2),~)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(1()/2inthevicinityoftheFermienergy,providingarealizationofther=2pseudogapAndersonmodel[ 56 ]. 133

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Intheremainderofthissection,weconsiderthedouble-dotdeviceintroducedinRef.[ 56 ],augmentedbyaHolsteincouplingbetweendot1andlocalbosons.Suchasystem,modeledbyaHamiltonian^HDD+!0aya+(^n1)]TJ /F4 11.955 Tf 12.31 0 Td[(1)(a+ay),canbemapped(followingRef.[ 56 ])ontotheeffectivesingle-impuritymodel^H=X1^n1+U1^n"^n#+Xk,kcykck+!0aya+Xk,V1)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(dy1ck+H.c.+(^n1)]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F6 11.955 Tf 5.48 -9.69 Td[(a+ay (4) withthescatteringrate~)]TJ /F7 7.97 Tf 6.77 -1.8 Td[(1()=(Nk))]TJ /F7 7.97 Tf 6.58 0 Td[(1PkV21()]TJ /F5 11.955 Tf 11.95 0 Td[(k)asdenedinEq.( 4 ). Allnumericalresultspresentedinthissectionwereobtainedforastronglyinteractingdot1havingU1=0.5,forabosonicenergy!0=0.1,forNRGdiscretizationparameter=2.5,andrestrictingthenumberofbosonstonomorethanNb=40.Exceptwherenotedotherwise,Ns=500stateswereretainedaftereachNRGiteration. 4.5.2ImpurityThermodynamicProperties Intheweakbosoniccouplingregime,wendaphasetransitionbetweentheASCandLMphasesbytuningthedot-1energy1.Figure 4-13 plotsthetemperaturedependenceoftheimpuritycontributiontothethermodynamicquantitiesTs,imp,1 4Tc,imp,andSimpforaweakbosoniccoupling=0.1andvarious1straddlingthecriticalvalue1c.TheowsofTs,impwithdecreasingtemperaturesclearlyshowaquantumcriticalpointCseparatestheASCandLMphases.Exactlyat1=1c(blueline),Ts,impexhibitsrenormalization-freebehavioratlowtemperaturesandmaintainsacriticalvalue1=6(thevalueofTs,impforthevalence-uctuationxedpoint).When1deviatesslightlyfromitscriticalvalue,Ts,imptracesthecriticalbehavioruntilitcrossesoverbelowacertaintemperaturetoeither1=4fortheLMphase(solidsymbols)orzerofortheASCphase(emptysymbols).Bycontrast,Tc,imphasacriticalvalue1=18and,asthetemperaturedecreases,fallstowardzeroinboththeLMandASCphases,signalingsuppressionofchargeuctuationsattheimpuritysite.Theimpurityentropy 134

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Simphasacriticalvalueln3(thevalueofSimpforthevalence-uctuationxedpoint)andreacheseitherln2fortheLMphaseorzerofortheASCphaseatlowtemperatures. Inthestrongbosoniccouplingregime,onecantunebetweentheASC",#andLCphasesusingalocalmagneticeldhenteringtheHamiltonianEq.( 4 )throughanadditionalterm^H=1 2h(^n1")]TJ /F4 11.955 Tf 13.01 0 Td[(^n1#).Figure 4-14 plotsthedependenceofTs,imp,1 4Tc,imp,andSimpontemperatureatparticle-holesymmetry(1=)]TJ /F7 7.97 Tf 10.49 4.7 Td[(1 2U1)forastrongbosoniccoupling=0.2andvarioushstraddlingthecriticalvaluehc.Here,incontrasttoFig. 4-13 ,Ts,impfallstozeroinboththeLCandASC",#phases,suggestingsuppressionofspinuctuationsattheimpuritysite.However,theowsofTc,impwithdecreasingtemperatureclearlyshowthataquantumcriticalpointC(=",#)separatestheASCandLCphases.Exactlyath=hc(blueline),Tc,impexhibitsrenormalization-freebehavioratlowtemperatures.Whenhdeviatesslightlyfromitscriticalvalue,1 4Tc,imptracesthecriticalbehavioruntilitcrossesoverbelowacertaintemperaturetoeither1=4fortheLCphase(solidsymbols)orzerofortheASC",#phase(emptysymbols).ThebehaviorofSimpwithdecreasingtemperatureisverysimilartothatinFig. 4-13 4.5.3PhaseDiagramsandCriticalCouplings Figure 4-15 (a)showsthephasediagramonthe1-2planeforh=0.For<0,thereisaLMphaseboundedbythecriticalenergies1c,while,for>0,theLCphaseexistsonlyonthelineofparticle-holesymmetry1=)]TJ /F6 11.955 Tf 9.3 0 Td[(U1=2.TherestofthephasespaceislledbytheASCphases.Figure 4-15 (b)showsthephasediagramontheh-2planeatparticle-holesymmetry1=)]TJ /F6 11.955 Tf 9.3 0 Td[(U1=2.For<0,theLMphaseexistsonlyonthelineh=0,while,for>0,theLCphaseisboundedbythecriticallocalmagneticeldhc.TheASC",#phaseslltherestofthephasespace. Wendthatboth1candhcdependlinearlyon2,asshowninFig. 4-15 (a)and(b).Thelinearrelationbetween1cand2canbeunderstoodasfollows.ThepseudogapAndersonmodelcanbemappedviatheSchrieffer-Wolfftransformation[ 10 ]tothe 135

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pseudogapKondomodel[ 54 ]withadimensionlessexchangecoupling 0J=)]TJ /F7 7.97 Tf 19.39 -1.79 Td[(11 U1=2)]TJ /F5 11.955 Tf 11.95 0 Td[(1+1 U1=2+1.(4) ThecriticalcouplingJcsatises0Jc=c[ 54 ],wheretheconstantcdependsonthebathexponentr.CombiningthisconditionwithEq.( 4 ),andassumingthat0J(1=0)=4)]TJ /F7 7.97 Tf 60.64 -1.79 Td[(1=U1c,oneconcludesthatthecriticaldot-1energiesofthepure-fermionicpseudogapAndersonmodelsatisfy 1c'U1 2)]TJ /F4 11.955 Tf 13.15 8.09 Td[()]TJ /F7 7.97 Tf 6.77 -1.79 Td[(1 c)]TJ /F4 11.955 Tf 19.69 8.09 Td[()]TJ /F7 7.97 Tf 6.77 4.34 Td[(21 c2U1,(4) or +1c')]TJ /F4 11.955 Tf 23.11 8.09 Td[()]TJ /F7 7.97 Tf 6.78 -1.79 Td[(1 c)]TJ /F4 11.955 Tf 19.69 8.09 Td[()]TJ /F7 7.97 Tf 6.77 4.34 Td[(21 c2U1and)]TJ /F7 7.97 Tf 0 -7.97 Td[(1c')]TJ /F6 11.955 Tf 21.92 0 Td[(U1+)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(1 c+)]TJ /F7 7.97 Tf 6.77 4.34 Td[(21 c2U1.(4) AswehavediscussedbeforeinSec. 4.2 ,themaineffectofthebosonmodeinthePAHmodelistoreducetheCoulombrepulsion.Therefore,U1inEq.( 4 )mustbereplacedbyitseffectivevalueU1,eff=U1)]TJ /F4 11.955 Tf 12.12 0 Td[(22=!0.ThisimpliesthataslongasU122=!0,thedependencesofthecriticalenergies1conthebosoniccouplingare+1c()')]TJ /F4 11.955 Tf 23.11 8.09 Td[()]TJ /F7 7.97 Tf 6.78 -1.79 Td[(1 c)]TJ /F4 11.955 Tf 19.68 8.09 Td[()]TJ /F7 7.97 Tf 6.78 4.34 Td[(21 c2U1)]TJ /F5 11.955 Tf 13.38 8.09 Td[(2 !01)]TJ /F4 11.955 Tf 17.02 8.09 Td[(2)]TJ /F7 7.97 Tf 13.05 4.34 Td[(21 c2U21, (4))]TJ /F7 7.97 Tf 0 -7.98 Td[(1c()')]TJ /F6 11.955 Tf 21.92 0 Td[(U1+)]TJ /F7 7.97 Tf 6.78 -1.8 Td[(1 c+)]TJ /F7 7.97 Tf 6.78 4.34 Td[(21 c2U1+2 !01)]TJ /F4 11.955 Tf 17.01 8.09 Td[(2)]TJ /F7 7.97 Tf 13.05 4.34 Td[(21 c2U21. (4) Similarreasoningcanbeusedtounderstandthelinearrelationbetweenhcand2.ThelocationsofthecriticallocalmagneticeldhcaredeterminedbyconsideringthemappingoftheAndersonmodelwithnegativeU1,efftoacharge-Kondomodelwithdimensionlesscoupling 0W=)]TJ /F7 7.97 Tf 19.39 -1.79 Td[(11 jU1,eff=2j)]TJ /F6 11.955 Tf 17.93 0 Td[(h=2+1 jU1,eff=2j+h=2.(4) Byanalogywiththespincase,thecriticalcouplingWcisgivenby0Wc=b,wheretheconstantbdependsonthebathexponentr.Assumingthat0W(h=0)=4)]TJ /F7 7.97 Tf 53.41 -1.79 Td[(1=jU1,effj 136

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b,onecanndthecriticaleld hc'2 !0(2)]TJ /F5 11.955 Tf 11.95 0 Td[(20))]TJ /F4 11.955 Tf 13.15 8.09 Td[(2)]TJ /F7 7.97 Tf 13.05 -1.79 Td[(1 b,(4) where0=(!0U1=2)1=2marksthelevelcrossingbetweentheweakandstrongbosoniccouplingregimes. 4.5.4LinearConductance ThelinearconductanceattemperatureTfortheboson-coupleddouble-quantum-dotsystemcanbecalculatedfromtheLandauerformula[ 98 ] g(T)=2e2 hZ1d!()]TJ /F5 11.955 Tf 11.12 8.09 Td[(@f @!)[)]TJ /F1 11.955 Tf 9.3 0 Td[(ImT(!)],(4) wheref(!)=[exp(!=T)+1])]TJ /F7 7.97 Tf 6.59 0 Td[(1istheFermi-DiracdistributionandthetunnelingmatrixisdenedtobeT(!)=0Pi,jViGij(!)Vj.Theterm)]TJ /F1 11.955 Tf 9.3 0 Td[(ImT(!)enteringEq.( 4 )canbeexpressedas[ 57 ])]TJ /F1 11.955 Tf 9.3 0 Td[(ImT(!)=[1)]TJ /F4 11.955 Tf 11.95 0 Td[(2)]TJ /F7 7.97 Tf 6.77 -1.8 Td[(22(!)]~)]TJ /F7 7.97 Tf 6.77 -1.8 Td[(1(!)A11(!)+)]TJ /F7 7.97 Tf 6.77 -1.8 Td[(22(!)+2(!)]TJ /F5 11.955 Tf 11.95 0 Td[(2)~)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(1(!)2(!)G011(!), (4) where~)]TJ /F7 7.97 Tf 6.77 -1.8 Td[(1(!)isasdenedinEq.( 4 ),2()=()]TJ /F7 7.97 Tf 33.46 -1.8 Td[(2=)[()]TJ /F5 11.955 Tf 12.15 0 Td[(2)2+)]TJ /F7 7.97 Tf 18.93 4.34 Td[(22])]TJ /F7 7.97 Tf 6.58 0 Td[(1isaLorentzianofwidth)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(2centeredonenergy2,A11(!)=)]TJ /F5 11.955 Tf 9.3 0 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1ImG11(!)isthedot-1spectralfunction,andG011=ReG11(!).Here,thedot-1localGreen'sfunctionG11(!)takesintoaccountbothCoulombelectron-electroninteractionandbosoniccoupling.TheNRGallowsustocalculatetheimpurityspectralfunctionA11(!)andweobtainG011(!)viatheKramers-Kronigtransformation.GivenA11(!)andG011(!),wethencalculatetheconductancegusingEqs.( 4 )and( 4 ).Thetechnicaldetailsareasfollows. WecalculatetheimpurityspectralfunctionA11(!,T)usingthetechniquesdiscussedinSection 3.4.6 withthechoiceofbroadeningwidthb=0.655ln.FromA11(!,T),wecalculatetherealpartoftheretardedimpurityGreen'sfunctionviathe 137

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Kramers-Kronigtransformation ReG11(!,T)=)]TJ /F18 11.955 Tf 9.3 0 Td[(PZ1A11(!0,T)d!0 !0)]TJ /F5 11.955 Tf 11.96 0 Td[(!.(4) Thereafter,itisstraightforwardtouseEq.( 4 )tocalculatetheimaginarypartofthetunnelingmatrix)]TJ /F24 11.955 Tf 9.3 0 Td[(ImT(!),whichissubstitutedintotheLandauerformulaEq.( 4 )toobtainthelinearconductanceg. AlthoughtheGreen'sfunctionscarryaspinindex,wecanusesymmetrypropertiestoreducethecomputationaleffort.(i)Forthecaseoftheweakbosoniccoupling=0.1,thequantumphasetransitionisaccessedatzeromagneticeld,soG"11(!)=G#11(!).(ii)Forthecaseofthestrongbosoniccoupling=0.2,alocalmagneticeldisappliedtotunethesystemtoitscriticalpoint,butbytime-reversalA"11(!)=A#11()]TJ /F5 11.955 Tf 9.3 0 Td[(!)andhence[byEq.( 4 )]ReG"11(!)=)]TJ /F1 11.955 Tf 9.3 0 Td[(ReG#11()]TJ /F5 11.955 Tf 9.3 0 Td[(!).Asaresult,thetunnelingmatrix)]TJ /F24 11.955 Tf 9.3 0 Td[(ImT(!)givenbyEq.( 4 )isinvariantunder!!)]TJ /F5 11.955 Tf 25.13 0 Td[(!and"!#.Thisproperty,whencombinedwiththeinvarianceoftheweightfunction)]TJ /F5 11.955 Tf 9.3 0 Td[(@f=@!under!!)]TJ /F5 11.955 Tf 24.94 0 Td[(!,leadsviaEq.( 4 )toexactequalityoftheconductancegforthespinupanddownchannels.(Thisisconrmedbyournumericalcalculations.) Figure 4-16 showsthebehaviorsofthelinearconductancegaroundthecriticalpoint1=+1cforaweakbosoniccoupling=0.1anddifferenttemperaturesTexpressedasmultiplesofTK0=7.010)]TJ /F7 7.97 Tf 6.59 0 Td[(4,whichistheKondotemperaturefortheone-impurityAndersonmodelwithU=)]TJ /F4 11.955 Tf 9.29 0 Td[(2d=0.5and)-471(=0.05.InFig. 4-16 (a)showinggvs1(1=1)]TJ /F5 11.955 Tf 12.93 0 Td[(+1c),twofeaturesstandout:(i)AttemperatureT=0.01TK0,thelinearconductancegisstructurelessandreachesitsunitaryvalue2e2=h,signalingperfectelectrontransmissionthroughthesystem;(ii)Athighertemperatures,however,thelinearconductancegexhibitsamaximumat1=+1candminimaoneithersideintheLMandASC)]TJ /F1 11.955 Tf 10.41 1.79 Td[(phases.Thispeak-and-valleystructurebecomesmoreprominentwithincreasingtemperature.Furthermore,weobservea1=Tscalingoftheconductanceginthevicinityofthecriticalpoint,asshowninFig. 4-16 (b). 138

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Figure 4-17 presentsthelinearconductancegaroundthecriticalpointh=hcforastrongbosoniccoupling=0.2.ThelinearconductancegexhibitsverysimilarbehaviorsasitdoesinFig. 4-16 .Inparticular,itexhibitsh=Tscalinginthevicinityofthecriticaleld. 4.6Summary WehaveconductedastudyofthepseudogapAnderson-HolsteinmodelofamagneticimpuritylevelthathybridizeswithafermionichostwhosedensityofstatesvanishesasjjrattheFermienergy(=0)andisalsocoupled,viatheimpuritycharge,toalocal-bosonmode.Wefoundtworegimes,dependingonthestrengthofthebosoniccoupling,thatexhibitdistinctivequantumuctuations.Theweakbosoniccouplingregimeischaracterizedbyspinuctuationswhilechargeuctuationsarepredominantinthestrongbosoniccouplingregime. For0
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Figure4-1. Schematic)]TJ /F1 11.955 Tf 6.77 0 Td[(-dphasediagramsofthepseudogapAndersonmodel[Eqs.( 4 )( 4 )with=0]forbandexponents(a)0
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Figure4-2. (Color)SymmetricPAHmodelatweakbosoniccoupling:(a)NRGenergyENvseveniterationnumberNoftherstexcitedmultiplethavingquantumnumbersS=1,Q=0,calculatedforr=0.4,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.5,!0=0.1,=0.05<0'0.158,andsevenvaluesof)]TJ /F2 11.955 Tf 9.43 0 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c1labeledontheplot.(b)Schematic)]TJ /F1 11.955 Tf 6.78 0 Td[(Tphasediagramfor<0,showingtheT=0transitionbetweentheLM()]TJ /F5 11.955 Tf 10.09 0 Td[(<)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c1)andSSC()]TJ /F5 11.955 Tf 10.1 0 Td[(>)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(c1)phases.DashedlinesmarkthescaleT1ofthecrossoverfromthehigh-temperaturequantum-criticalregimetooneorotherofthestablephases.(c)CrossoverscaleT1vsj)]TJ /F2 11.955 Tf 9.43 0 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c1jinboththeLMphaseandtheSSCphase,showingthepower-lawbehaviordescribedinEq.( 4 ).Here,T1=D)]TJ /F8 7.97 Tf 6.59 0 Td[(N1=2,whereN1istheinterpolatedvalueofNatwhichENin(a)crossesoneorotherofthehorizontaldashedlines. 141

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Figure4-3. (Color)SymmetricPAHmodelatstrongbosoniccoupling:(a)NRGenergyENvseveniterationnumberNoftherstexcitedstatehavingquantumnumbersS=Q=0,calculatedforr=0.4,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.5,!0=0.1,=0.2>0'0.158,and)]TJ /F2 11.955 Tf 9.43 0 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c2valueslabeledontheplot.(b)Schematic)]TJ /F1 11.955 Tf 6.78 0 Td[(Tphasediagramfor>0,showingtheT=0transitionbetweentheLCandSSCphasesandthescaleT2ofthecrossoverfromthequantum-criticalregimetoastablephase.(c)CrossoverscaleT2vsj)]TJ /F2 11.955 Tf 9.44 0 Td[()]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F8 7.97 Tf 6.78 -1.8 Td[(c2jintheLCandSSCphases,showingthepower-lawbehaviordescribedinEq.( 4 ).Here,T2=D)]TJ /F8 7.97 Tf 6.59 0 Td[(N2=2,whereN2istheinterpolatedvalueofNatwhichENin(a)crossesoneorotherofthehorizontaldashedlines. 142

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Figure4-4. Schematicrenormalization-groupowsonthe)]TJ /F1 11.955 Tf 6.77 0 Td[(-UeffplaneforthesymmetricPAHmodelwithabandexponent0
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Figure4-5. PhaseboundariesofthesymmetricPAHmodel:Variationwithbosoniccouplingofthecriticalhybridizationwidths)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c1(emptysymbols,separatingtheLMandSSCphases)and)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(c2(lledsymbols,separatingtheLCandSSCphases).DataareshownforU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2"d=0.5,!0=0.1,andfourbandexponentsrlistedinthelegend. 144

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Figure4-6. (Color)SymmetricPAHmodelatweakbosoniccoupling:Temperaturedependenceoftheimpuritycontributionto(a)thestaticspinsusceptibilityTs,imp,(b)thestaticchargesusceptibilityTc,imp,(c)theentropySimp,and(d)zoominof(c)aroundcriticalvalueforr=0.4,U=)]TJ /F4 11.955 Tf 9.29 0 Td[(2d=0.5,!0=0.1,=0.05<0'0.158,andsevenvaluesof)]TJ /F2 11.955 Tf 9.43 0 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c1labeledinthelegend.Ns=3000stateswereretainedaftereachNRGiteration. 145

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Figure4-7. (Color)SymmetricPAHmodelatstrongbosoniccoupling:Temperaturedependenceoftheimpuritycontributionto(a)thestaticspinsusceptibilityTs,imp,(b)thestaticchargesusceptibilityTc,imp,(c)theentropySimp,and(d)zoominof(c)aroundcriticalvalueforr=0.4,U=)]TJ /F4 11.955 Tf 9.29 0 Td[(2d=0.5,!0=0.1,=0.2>0'0.158,andsevenvaluesof)]TJ /F2 11.955 Tf 9.44 0 Td[()]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c2labeledinthelegend.Ns=3000stateswereretainedaftereachNRGiteration. 146

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Figure4-8. (Color)SymmetricPAHmodelnearthespin-sectorquantumphasetransition:Localspinresponseforr=0.4,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.5,!0=0.1,and=0.05,atornearthecriticalhybridizationwidth)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(c1'0.3166805.CirclesareNRGdataanddashedlinesrepresentpower-lawtting.(a)Staticlocalspinsusceptibilitys,loc(h!0,T=0)vs)]TJ /F2 11.955 Tf 9.28 0 Td[()]TJ /F4 11.955 Tf 11.8 0 Td[()]TJ /F8 7.97 Tf 6.78 -1.79 Td[(c1intheSSCphase.(b)LocalmagnetizationMloc(h!0,T=0)vs)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(c1)]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F1 11.955 Tf 10.1 0 Td[(intheLMphase.Inset:continuousvanishingofMloc(h!0,T=0)as)]TJ /F1 11.955 Tf 10.09 0 Td[(approaches)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(c1frombelow.(c)LocalmagnetizationMloc()-277(=)]TJ /F8 7.97 Tf 34.37 -1.79 Td[(c1,T=0)vslocalmagneticeldh.(d)Staticlocalspinsusceptibilitys,loc(T;h!0,)-278(=)]TJ /F8 7.97 Tf 41.25 -1.8 Td[(c1)vstemperatureT. 147

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Figure4-9. (Color)SymmetricPAHmodelnearthecharge-sectorquantumphasetransition:Localchargeresponseforr=0.4,U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.5,!0=0.1,and=0.2,atornearthecriticalhybridizationwidth)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c2'0.6878956.CirclesareNRGdataanddashedlinesrepresentpower-lawtting.(a)Staticlocalchargesusceptibilityc,loc(!0,T=0)vs)]TJ /F2 11.955 Tf 9.43 0 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c2intheSSCphase.(b)LocalchargeQloc(!0,T=0)vs)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(c2)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F1 11.955 Tf 10.1 0 Td[(intheLCphase.Inset:ContinuousvanishingofQloc(!0,T=0)as)]TJ /F1 11.955 Tf 10.1 0 Td[(approaches)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c2frombelow.(c)LocalchargeQloc()-277(=)]TJ /F8 7.97 Tf 34.37 -1.8 Td[(c2,T=0)vslocalelectricpotential.(d)Staticlocalchargesusceptibilityc,loc(!0,)-278(=)]TJ /F8 7.97 Tf 41.25 -1.79 Td[(c2)vstemperatureT. 148

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Figure4-10. (Color)SymmetricPAHmodel:(a)DependenceofthelocalmagnetizationMloc(h!0,T=0)on)]TJ /F2 11.955 Tf 9.43 0 Td[()]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F8 7.97 Tf 6.78 -1.79 Td[(c1andofthelocalchargeQloc(!0,T=0)on)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(c2)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F1 11.955 Tf 6.77 0 Td[(,forU=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.5,!0=0.1,=0.05(magneticresponse)or0.2(chargeresponse),andbandexponentsr=0.2andr=0.4.(b)Staticlocalspinsusceptibilitys,loc(T;)-278(=)]TJ /F8 7.97 Tf 34.97 -1.79 Td[(c1)andstaticlocalchargesusceptibilityc,loc(T;)-278(=)]TJ /F8 7.97 Tf 34.97 -1.8 Td[(c2)vstemperatureT.Allparametersotherthan)]TJ /F1 11.955 Tf 10.1 0 Td[(takethesamevaluesasin(a). 149

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Figure4-11. (Color)PhaseboundariesofthePAHmodelonthed-)]TJ /F1 11.955 Tf 10.1 0 Td[(planeforweakbosoniccouplings<0=0.15812(1).Dataareshownforr=0.4(left),r=0.6(right),U=0.5,h=0,!0=0.1,andthreeweakbosoniccouplingslistedinthelegend. 150

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Figure4-12. (Color)PhaseboundariesofthesymmetricPAHmodelontheh-)]TJ /F1 11.955 Tf 10.1 0 Td[(planeforstrongbosoniccouplings>0=0.15812(1).Dataareshownforr=0.4(left),r=0.6(right),U=)]TJ /F4 11.955 Tf 9.3 0 Td[(2d=0.5,!0=0.1,andthreestrongbosoniccouplingslistedinthelegend. 151

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Figure4-13. (Color)U2=0double-quantum-dotdeviceatweakbosoniccoupling:Temperaturedependenceoftheimpuritycontributionto(a)thestaticspinsusceptibilityTs,imp,(b)thestaticchargesusceptibilityTc,imp,and(c)theentropySimp,forU1=0.5,)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(1=0.05,2=0,)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(2=0.02,h=0,!0=0.1,=0.1,anddifferentvaluesof1straddlingthelocation+1c')]TJ /F4 11.955 Tf 21.92 0 Td[(0.124985or)]TJ /F7 7.97 Tf 0 -7.98 Td[(1c')]TJ /F4 11.955 Tf 21.92 0 Td[(0.375015ofthespin-sectorquantumphasetransition.Ns=3000stateswereretainedaftereachNRGiteration.Propertiesatthetransition(lineswithoutsymbols)arethoseexpectedatalevelcrossingbetweentheLM(lledsymbols)andASC(emptysymbols)phases. 152

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Figure4-14. (Color)U2=0double-quantum-dotdeviceatstrongbosoniccoupling:Temperaturedependenceoftheimpuritycontributionto(a)thestaticspinsusceptibilityTs,imp,(b)thestaticchargesusceptibilityTc,imp,and(c)theentropySimp,forU1=)]TJ /F4 11.955 Tf 9.3 0 Td[(21=0.5,)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(1=0.05,2=0,)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(2=0.02,!0=0.1,=0.2,anddifferentvaluesofthelocalmagneticeldhstraddlingthelocationhc'0.284959ofthecharge-sectorquantumphasetransition.Ns=3000stateswereretainedaftereachNRGiteration.Propertiesatthetransition(lineswithoutsymbols)arethoseexpectedatalevelcrossingbetweentheLC(lledsymbols)andASC",#(emptysymbols)phases.ComparisonwithFig. 4-13 showsclosesimilaritiesbetweenthebehavioroftheentropynearthespin-andcharge-sectorquantumphasetransitions,butaninterchangeofthespinandchargesusceptibilities. 153

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Figure4-15. (Color)U2=0double-quantum-dotdevice:(a)Phasediagramonthe1-2planeforU1=0.5,)]TJ /F7 7.97 Tf 6.78 -1.8 Td[(1=0.05,2=0,)]TJ /F7 7.97 Tf 6.78 -1.8 Td[(2=0.02,!0=0.1,andh=0.(b)Phasediagramontheh-2planeforU1=)]TJ /F4 11.955 Tf 9.3 0 Td[(21=0.5)]TJ /F7 7.97 Tf 38.76 -1.79 Td[(1=0.05,2=0,)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(2=0.02,and!0=0.1.Notethelinearityofthephaseboundarieswhenplottedagainstthesquareofthebosoniccoupling. 154

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Figure4-16. (Color)U2=0double-quantum-dotdevicenearthespin-sectorquantumphasetransition:(a)Linearconductancegvs1=1)]TJ /F5 11.955 Tf 11.96 0 Td[(+1cforU1=0.5,)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(1=0.05,2=0,)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(2=0.02,!0=0.1,=0.1,anddifferenttemperaturesTspeciedinthelegendasmultiplesofTK0=710)]TJ /F7 7.97 Tf 6.59 0 Td[(4.TheretentionofNs=1000statesaftereachNRGiterationaccountsforthesmallshiftin+c1')]TJ /F4 11.955 Tf 21.92 0 Td[(0.1249871relativetothecaseNs=3000showninFig. 4-13 .(b)Thesamedatascaledas(g)]TJ /F6 11.955 Tf 11.95 0 Td[(gmin)TK0=Tvs1=T,wheregministheminimumontheASC)]TJ /F1 11.955 Tf 10.41 1.79 Td[(sideforeachcurve. 155

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Figure4-17. (Color)U2=0double-quantum-dotdevicenearthecharge-sectorquantumphasetransition:(a)Linearconductancegvsh=h)]TJ /F6 11.955 Tf 11.95 0 Td[(hcforU1=)]TJ /F4 11.955 Tf 9.3 0 Td[(21=0.5,)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(1=0.05,2=0,)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(2=0.02,!0=0.1,=0.2,anddifferenttemperaturesTspeciedinthelegendasmultiplesofTK0=710)]TJ /F7 7.97 Tf 6.58 0 Td[(4.TheretentionofNs=1000statesaftereachNRGiterationaccountsforthesmallshiftinhc'0.2849588relativetothecaseNs=3000showninFig. 4-14 .(b)Thesamedatascaledas(g)]TJ /F6 11.955 Tf 11.96 0 Td[(gmin)TK0=Tvsh=T,wheregministheminimumontheASC#sideforeachcurve. 156

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CHAPTER5CONCLUSIONSANDFUTUREDIRECTIONS 5.1Conclusions Inthiswork,usingthenumericalrenormalization-groupmethod,wehavetheoreticallystudieddissipativequantumimpuritymodelsinwhichthemagneticimpuritynotonlyhybridizeswithafermionichostbutalsoiscoupled,viatheimpuritycharge,tobosonicdegreesoffreedom.Thesemodelsarerelevanttoquantumcriticalityofheavy-fermionmaterialsandelectrontransportinnanoscaledevices.WendthatthebehaviorofamagneticimpurityisstronglymodiedwhenadissipativeenvironmentrepresentedbybosonicdegreeoffreedomisincorporatedintotheconventionalAndersonmodel. Ingeneral,theimpurity-bosoncouplingreducestheCoulombinteractionbetweentwoelectronsintheimpuritylevelfromitsrepulsivebarevalueUtoaneffectivevalueUeff.Intheatomiclimitofzerohybridization,andforweakimpurity-bosoncoupling,Ueffispositiveandthegroundstateoftheimpuritymodelsliesinthesectornd=1wheretheimpurityhasaspinzcomponent1=2.However,forsufcientlylargeimpurity-bosoncoupling,Ueffisdrivennegative,placingthegroundstateinthesectornd=0ornd=2wheretheimpurityisspinlessbuthasachargeof)]TJ /F4 11.955 Tf 9.3 0 Td[(1or+1.Inthemoreinterestingcaseofnonzerohybridization,theboson-inducedrenormalizationofUcanservetotransferthecomplexspincorrelationsoftheconventionalKondoeffectintothechargesector,leadingtoamany-bodyKondoquenchingofalocalizedchargedegreeoffreedom. Thecharge-coupledBose-FermiAndersonmodeldescribesamagneticimpuritythathybridizeswithametallichostandiscoupledtoadispersivebosonicbathwithspectralfunctionproportionalto!s.Atparticle-holesymmetry,furtherincreaseintheimpurity-bosoncouplingmayleadtoaquantumphasetransitionfromtheKondophasetoalocalizedphasewheretheground-stateimpurityoccupancy^ndacquiresanexpectationvalueh^ndi06=1.Forasub-Ohmicbathcharacterizedby0
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theresponseoftheimpurityoccupancytoalocallyappliedelectricpotentialfeaturesacontinuousphasetransition.Inaddition,thehyperscalingofcriticalexponentsand!=Tscalingsuggestaninteractingcriticalpoint.FortheOhmiccases=1,thetransitionisinsteadofKosterlitz-Thoulesstype.Awayfromparticle-holesymmetry,thequantumphasetransitionisreplacedbyasmoothcrossover,butsignaturesofthesymmetricquantumcriticalpointremaininthephysicalpropertiesatelevatedtemperaturesand/orfrequencies. InthepseudogapAnderson-Holsteinmodelofamagneticimpuritythathybridizeswithapseudogappedhostandiscoupledtoalocalbosonmode,thepseudogapinhibitsKondoscreeningandgivesrisetophasesinwhichtheimpurityretainsunquencheddegreesoffreedom.Forweakimpurity-bosoncouplings,theresponseofthesystemtoalocallyappliedmagneticeldrevealstransitionsbetweentwostrong-coupling(Kondo)phasesandalocal-momentphase.Forstrongimpurity-bosoncouplings,bycontrast,theresponseofthesystemtoalocallyappliedelectricalpotentialindicatestransitionsbetweenanothertwostrong-couplingphasesandalocal-chargephase.Particle-holeasymmetryinthemodelwithweakimpurity-bosoncouplingsactsinamanneranalogoustoalocalmagneticeldappliedtothemodelwithstrongimpurity-bosoncouplings.Criticalexponentscharacterizingthefourcriticalpointsarefoundtobenumericallyidentical,suggestingthattheybelongtothesameuniversalityclassasthatofthepseudogapAndersonmodel.OnespeciccaseofthepseudogapAnderson-Holsteinmodeleffectivelydescribesacharge-coupleddouble-quantum-dotdevice,inwhichthephasetransitionsaremanifestedinthenite-temperaturelinearelectricalconductance. 5.2FutureDirections Dissipativequantumimpurityproblemsdescribeinterestingandexcitingphysics.Theyareofintrinsictheoreticalinterestastractabletoymodelsforbulkquantumphasetransitionsandforthenoveluniversalityclassesofquantumphasetransitionsthat 158

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theyexemplify.Theseproblemsalsodescribenanoscalesystemsofrelevancetonanoelectronicsandquantumcomputing,twoareasoftheforefrontofthetechnologicalrevolution.Here,wesummarizeseveralissuesthatmightbeaddressedinthefuture. (1)AswediscussedinSec. 1.4 ,theimpuritymodelwestudiedinChapter 3 isassociatedwiththelatticemodelEq.( 1 )withintheframeworkoftheextendeddynamicalmean-eldtheory.Tofullyunderstandthislatticemodel,self-consistencyoughttobeimposedproperly.Thismightbepursuedinfuturework. (2)Onestraightforwardgeneralizationofourstudyistoreplacethemetallichostinthecharge-coupledBose-FermiAndersonmodelEq.( 3 )withapseudogappedhost.PreliminaryworkonthismodelindicatesthatthereisstillaninteractingquantumcriticalpointseparatingtheKondoandlocalizedphases.Forbandexponentsrandbathexponentsssatisfyings+2r<1,thequantumcriticalpointremainsintheuniversalityclassofthespin-bosonmodel.Howeverfors+2r>1,themodelshowslow-temperaturebehaviorverydifferentfromthatofthespin-bosonmodel[ 99 ].Thisnovelregimerequiresfurtherinvestigation. (3)InChapter 3 ,weonlyconsideredamagneticimpuritycoupledtoonebosonicbath.Apossibleextensionistoinvestigatemodelsinwhichtheimpurityiscoupledtomultiplebosonicbaths.ThesimplestsuchmodelistheXY-anisotropicspin-bosonmodel ^HXYSB=Sz+Xq!q(xyqxq+yyqyq)+g?Xq[Sx(xq+xy)]TJ /F13 7.97 Tf 6.59 0 Td[(q)+Sy(yq+yy)]TJ /F13 7.97 Tf 6.58 0 Td[(q)],(5) wheretheimpurityiscoupled,viaspinxandycomponents,totwoindependentbosonicbathsdescribedbyxqandyq.Thismodelexhibitsthenovelphenomenonofquantumfrustrationofdecoherence[ 100 101 ].AnothermodelofinterestistheXY-anisotropicBose-FermiKondomodel^HXYBFKM=Xk,kcykck+Xq!q(xyqxq+yyqyq) 159

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+JSsc+g?Xq[Sx(xq+xy)]TJ /F13 7.97 Tf 6.58 0 Td[(q)+Sy(yq+yy)]TJ /F13 7.97 Tf 6.58 0 Td[(q)], (5) whichmaydescribethequantumcriticalityofYbRh2Si2)]TJ /F8 7.97 Tf 6.59 0 Td[(xGex[ 102 ]andcertainquantum-dotsystem[ 53 ].TheNRGtreatmentofthesemodelsisverychallengingbecauseofthelargenumberofstatesthatmustberetainedinthecalculation. (4)AlimitationofourresearchinChapter 4 isthatwewereonlyabletocalculatethelinear-responseconductance,describingthebehaviorinthelimitofaninnitesimalbiasbetweentheelectricalleads.However,manyexperimentsshownite-biasKondofeatures[ 60 103 ].Thus,itisofgreatinteresttodevelopnonperturbativemethodstotreatstrongcorrelationsinnon-equilibriumsituations.Onesuchmethodisthetime-dependentNRGmethod[ 104 105 ],whichisrestrictedtocalculatingthetransientresponseofasysteminitiallyatequilibriumtoasuddenchangeinitsHamiltonianfromHitoHf(bothtime-independent).Animportantgoalforfutureworkisthedevelopmentofanonperturbativenumericalapproachfortreatingmoregeneralnon-equilibriumproblems. 160

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BIOGRAPHICALSKETCH MengxingChengwasborninJuly1981,inKaiyuan,YunnanProvince,China.Heattendedlocalpublicschoolstherethroughhighschool.InSeptember1999,heenrolledatthePhysicsDepartmentofFudanUniversityinShanghai.InJune2003,hegraduatedandobtainedaBachelorofScience.InSeptember2003,hejoinedProfessorSunXin'sgroupatthesamedepartmentandinJune2005,hewasawardedaMasterofScience.InAugust2005,heewoverthePacicOceanandtheNorthAmericancontinenttoGainesvilletostartgraduateschoolattheDepartmentofPhysicsattheUniversityofFlorida.Inthesummerof2006,hejoinedDoctorIngersent'sgrouptostudymagneticimpuritymodelsusingthenumericalrenormalizationgroup.HeandhiswifeHuanHuangrstmetinthefallof1995andmarriedinJuly2005.TheirsonJasonChengwasborninMay2009. 166