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Anderson Impurity Models with Bosons as Descriptions of Molecular Devices and Heavy-Fermion Systems

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Title:
Anderson Impurity Models with Bosons as Descriptions of Molecular Devices and Heavy-Fermion Systems
Creator:
Deng, Lili
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (138 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
INGERSENT,J KEVIN
Committee Co-Chair:
MUTTALIB,KHANDKER A
Committee Members:
HERSHFIELD,SELMAN PHILIP
LEE,YOONSEOK
PHILLPOT,SIMON R
Graduation Date:
5/3/2014

Subjects

Subjects / Keywords:
Conduction bands ( jstor )
Crossovers ( jstor )
Electrons ( jstor )
Ground state ( jstor )
Impurities ( jstor )
Kondo effect ( jstor )
Lead ( jstor )
Molecules ( jstor )
Orbitals ( jstor )
Phonons ( jstor )
Physics -- Dissertations, Academic -- UF
bose-fermi-anderson-model -- electron-phonon-coupling -- kondo-effect -- quantum-phase-transition
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
This dissertation presents the results of theoretical study of a pair of localized atomic or molecular levels hybridizing with a fermionic conduction band and also coupled to one or more bosonic degrees of freedom. Two variants of the Anderson impurity model are investigated using the numerical renormalization group technique. One model describes a two-orbital molecular junction with strong coupling to a local bosonic mode, e.g., an optical phonon. The other model, describing two magnetic impurities coupled to a conduction band and also to a dissipative bosonic bath, is used to study superconducting pairing in the presence of lattice magnetism and Kondo correlation. The molecular junction is modeled in terms of a two-orbital molecule connecting a pair of external leads. Strong inter- and intra-orbital Coulomb repulsion gives rise to Kondo correlation between the local spin inside the molecule and delocalized spins in the conduction band. At the same time, the possibility of conformational changes admits strong coupling between a local phonon mode and the molecular charge, and/or phonon-assisted inter-orbital tunneling. Varying the electron-phonon coupling strength causes a crossing of levels in the ground state of the isolated molecule. When the molecule is connected to the leads, a smooth crossover between a Kondo regime and a phonon-dominated regime is found in the vicinity of this level crossing. In a particular side-orbital configuration, this crossover in the general configuration becomes a first-order quantum phase transition due to the emergence of a new symmetry. Furthermore, investigation is carried out of interplay between the phonon-assisted tunneling and a Holstein-type phonon coupling to the molecular charge, and a crossover between spin-Kondo and charge-Kondo regimes is studied. In the two-impurity Bose-Fermi Anderson model, two magnetic impurities hybridize with a (fermionic) conduction band and are coupled via the difference of their spins to a sub-Ohmic bosonic bath. We also consider an exchange interaction between the impurities, modeling the Ruderman-Kittel-Kasuya-Yosida presented in a lattice setting. The phase diagram of this model is found to feature a Kondo phase, an interimpurity-singlet (IS) phase, and a local-moment (LM) phase. Quantum phase transitions between these phases arise due to competition between three energy scales: the Kondo temperature $T_{K}$, the antiferromagnetic interimpurity exchange $I$, and a bosonic coupling $g$ to the staggered impurity spin. The critical behavior along the Kondo-IS and Kondo-LM boundaries is studied in detail. Singlet superconducting pairing shows enhancement along the Kondo-LM phase boundary, particularly close to the triple point where all three phases meet, suggesting a new mechanism for heavy-fermion superconductivity in the vicinity of antiferromagnetic order. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: INGERSENT,J KEVIN.
Local:
Co-adviser: MUTTALIB,KHANDKER A.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-11-30
Statement of Responsibility:
by Lili Deng.

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Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
11/30/2014
Classification:
LD1780 2014 ( lcc )

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KondoDestructionandValenceFluctuationsinanAndersonModelJ.H.Pixley,1StefanKirchner,2,3KevinIngersent,4andQimiaoSi11DepartmentofPhysics&Astronomy,RiceUniversity,Houston,Texas77005,USA2MaxPlanckInstituteforthePhysicsofComplexSystems,01187Dresden,Germany3MaxPlanckInstituteforChemicalPhysicsofSolids,01187Dresden,Germany4DepartmentofPhysics,UniversityofFlorida,Gainesville,Florida32611-8440,USA (Received26August2011;revisedmanuscriptreceived4June2012;published22August2012) Unconventionalquantumcriticalityinheavy-fermionsystemshasbeenextensivelyanalyzedinterms ofacriticaldestructionoftheKondoeffect.Motivatedbyarecentdemonstrationofquantumcriticalityin amixed-valentheavy-fermionsystem, YbAlB4,westudyaparticle-hole-asymmetricAnderson impuritymodelwithapseudogappeddensityofstates.WedemonstrateKondodestructionatamixedvalentquantumcriticalpoint,whereacollapsingKondoenergyscaleisaccompaniedbyasingular charge-uctuationspectrum.Bothspinandchargeresponsesscalewithenergyovertemperature( !=T ) andmagneticeldovertemperature( H=T ).Implicationsforunconventionalquantumcriticalityinmixedvalenceheavyfermionsarediscussed.DOI: 10.1103/PhysRevLett.109.086403 PACSnumbers:71.10.Hf,71.27.+a,75.20.HrCompetinginteractionsinquantumsystemsgiveriseto zero-temperaturephasetransitions.Ifitiscontinuous,such atransitiontakesplaceataquantumcriticalpoint(QCP). Thereismountingevidence,especiallyinheavy-fermion systems,thataQCPcanunderlieunconventionalsuperconductivity[ 1 ];relatedconsiderationshavebeenapplied tohigh-temperaturecuprateandironpnictidesuperconductors[ 2 ].ItisstandardtodescribeaQCPwithinthe Ginzburg-Landau-Wilson(GLW)framework:criticaldestructionofanorderparametercharacterizingaspontaneouslybrokensymmetrygivesrisetocollectivemodes associatedwithorder-parameteructuations[ 3 ].Inthe contextofantiferromagneticmetals,thisisreferredtoas aspin-density-waveQCP[ 4 ]. Recentexperimentsinheavy-fermionmetalshave clearlyestablishedtheexistenceofanovelclassof antiferromagneticQCPs,characterizedbynon-Fermiliquidbehaviorand !=T scalinginthedynamicalspin susceptibility[ 5 ].ThereareindicationsthatsuchunconventionalQCPsalsopromotesuperconductivity[ 6 ]. TheseQCPsdefyadescriptionintermsofaGLW functional[ 7 8 ];theirunderstandingrequirestheintroductionofquantummodesbeyondorder-parameteructuations.Theproposedadditionalmodesareassociated withthecriticaldestructionoftheKondoeffect[ 7 8 ].In theparamagneticphase,KondosingletsformandgenerateKondoresonances,therebyturningthelocalmoments intosingle-electronicexcitationsandenlargingtheFermi surface.ThedestructionoftheKondoeffectacrossthe antiferromagneticQCPsuppressestheKondoresonances, makingtheFermisurfacesmall.CriticalKondodestructionthereforemanifestsitselfinadiscontinuousevolutionoftheFermisurfaceacrossthetransition,ashas beenobservedthroughquantumoscillationandHall effectmeasurements[ 5 9 ]. TheoreticalstudiesofcriticalKondodestructionhave largelybeenconnedtotheKondo-latticelimitofinteger valence.Inrare-earthintermetallics,superconductivityis believedalsotoariseinthevicinityofvalencetransitions [ 10 ],whichhavebeenfoundtoberstorder.Until recently,therehasbeennosignicantevidencefor aQCPassociatedwithvalenceuctuations.Thesituation haschangedwiththeobservationofmixedvalencyin theytterbium-basedheavy-fermionsuperconductor YbAlB4[ 11 ],whichisquantumcriticalunderambient conditions[ 12 ].Inanappliedmagneticeld,themagnetizationobeys H=T scaling[ 13 ],consistentwiththe !=T scalingseenpreviouslyneartheunconventionalQCPsof antiferromagneticheavy-fermioncompounds.Thesepropertiesimplicate YbAlB4asastrongcandidatefora mixed-valentheavy-fermionQCP,andraisetheprospect thatthematerial'sunusualscalingbehaviorcanbeunderstoodintermsofcriticalKondophysics. Atrstglance,criticalKondodestructionatmixed valenceappearsunlikely.KondodestructioninaKondo latticeamountstothelocalizationof f electrons.While unconventional,thisisphysicallytransparent,becauselocalizationcanreadilyariseforacommensuratellingofan electronicorbital(one f electronpersite).Atmixedvalence,thesituationismoresubtlebecausethe f orbitalhas afractional,generallyincommensurate,per-siteoccupancy,andthereisnomechanismknownforelectron localizationatincommensuratellings.Thisleadstoimportantquestionsofprinciple:cancriticalKondodestructionoccurinthepresenceofvalenceuctuationsand,ifso, howdoesthecriticalitycomparetoitslocal-momentcounterpart?Forinstance,arechargeexcitationspartofthe criticaluctuationspectrum? InthisLetter,weaddresstheseissuesinthemixedvalenceregimeofanAndersonimpuritymodelwhose PRL 109, 086403(2012) PHYSICALREVIEWLETTERSweekending 24AUGUST20120031-9007 = 12 = 109(8) = 086403(5)086403-1 2012AmericanPhysicalSociety

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conduction-electrondensityofstatesfeaturesapseudogap centeredontheFermienergy.Wefocusonanimpurity modelbecauseofthelocalnatureoftheKondo-destruction physics;formally,theKondolatticemodelcanbetreated throughaneffectiveimpuritymodelintheextendeddynamicalmeaneldapproach[ 14 15 ].Giventhatthe commensurate-lling(i.e.,local-moment)limitofthe modelexhibitscriticalKondodestructionandassociated dynamicalscalingproperties[ 16 ],weconsiderthepseudogappeddensityofstatestoprovideaprototypesettingto searchforaKondo-destructionQCPatmixedvalence. Ourmodelhastheadvantageofbeingamenabletostudy usingreliablemethods:thecontinuous-timequantum MonteCarlo(CT-QMC)method[ 17 ]andthenumerical renormalizationgroup(NRG)[ 16 18 19 ]. Surprisingly,wedondcriticalKondodestructionin thismixed-valentmodel.Thecriticalpropertiesinthespin sectorreectthecollapseofanenergyscaleastheQCPis approachedfromtheKondo-screenedsidebutnotfromthe Kondo-destroyedside,muchasintheinteger-valent(localmoment)limit.Bycontrast,thechargesectorshowsa collapsingenergyscaleonbothsidesoftheQCP.The criticalpointdisplays H=T (and !=T )scaling.ThisexistenceproofforaKondodestructionQCPatmixedvalence makesitfeasibletointerpretthe H=T andrelatedscaling propertiesof YbAlB4intermsofaninteractingxed point.Wenotethatthesamemodelisalsorelevantto impurityphysicsin d -wavesuperconductorsandgraphene, wherethedensityofbulkfermionicstatesgoestozeroat thechemicalpotential[ 20 ]. TheAndersonimpurityHamiltonianis HA Xk; kcy kck V dy ck H : c : "dnd Und "nd #; (1) where ckannihilatesaconduction-bandelectronof energy k, dannihilatesanelectronofenergy "din theimpuritylevel, U istheelectronelectronrepulsion withintheimpuritylevel, V isthehybridizationtaken tobemomentumindependent, nd dy d,and nd nd " nd #.Thebanddensityofstatesvanishesina power-lawfashionattheFermienergy( F 0 : Xk k 0j =D jr D j j : (2) Theimpurity-bandinteractioniscompletelyspeciedby theimaginarypartofthehybridizationfunction, PkV2 k 0j =D jr,where 0 0V2. Thecriticalpropertiesofthemodelwithparticle-hole ( p h )symmetry( "d U= 2 )anditsKondolimit( U 0,wherelocalchargeuctuationsarenegligible)have beeninvestigatedinanumberofanalyticandnumerical studies[ 16 19 21 23 ].Thebreakingof p h symmetryis irrelevantforpseudogapexponents r intherange 0 r,leadingtoa mixed-valentQCP[ 19 ]; r 1 servesasanuppercritical dimension'',abovewhichthecriticalpropertieshavea mean-eldcharacter[ 16 22 ]. Here,weinvestigatethe p h -asymmetricpseudogap Andersonmodelbyvarying U forxed 0and "dtopassfromaKondo-screenedstrong-couplingphase ( UUc).WeapplytheCT-QMCtechnique,whichwas recentlyshowntobeabletoreachtemperatures T sufcientlylowtoaccessthequantumcriticalregime[ 23 ]. Wemeasurethedynamicallocalspinandchargesusceptibilities, s ; h TSz Sz 0 i and c ; h T: nd :: nd 0 : i ,respectively,where Sz 1 2 nd nd # : nd: ndh ndi ,and 1 =T (taking kB 1 )plays theroleofthesystemsize.Thecorrespondingstatic susceptibilitiesfollowfrom c;s R 0dc;s Measuringpowersofthelocalmagnetization h Mn zi h 1 R 0dSz ni allowsconstructionoftheBindercumulant[ 24 ] B U; h M4 zi = h M2 zi2.Wesupplementour T> 0 (finite)CT-QMCresultswithstaticquantities calculatedarbitrarilycloseto T 0 ( 1 )usingthe NRGmethodasadaptedtotreatpseudogapimpurityproblems[ 16 18 19 ].NRGresultspresentedbelowwereobtainedwithWilsondiscretizationparameter 9 ,with 0corrected[ 19 ]tocompensateforthebanddiscretization andretainingallmany-bodystatesupto50timesthe effectivebandwidthofeachiteration. Wefocusourdiscussionontherepresentativecaseofa pseudogapexponent r 0 : 6 with 0 0 : 1 D and "d 0 : 05 D .Figure 1 plotsthevariationoftheBindercumulantwith U atdifferenttemperatures.Forsmall U ,charge uctuationsarestrong,andtheBindercumulantliesabove therange 1
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characteristicofthelocal-momentphase.Welocatethe phaseboundarybytheintersectionof B U; curvesfor differenttemperatures[ 23 ]at Uc=D 0 : 06313 0 : 0008 TheNRGgives Uc 0 : 06450 D ,asmallshiftthatcan likelybeattributedtoresidualeffectsofNRGdiscretization.Themixed-valentnatureoftheQCPisdemonstrated inFig. 2(a) ,wherethelocaloccupation h ndi isseento differfromunityat U Uc.Notealsothat h ndi displays signicanttemperaturedependenceinthevicinityof theQCP. WearenowinapositiontolookforacriticaldestructionoftheKondoeffectinthismixed-valentQCP,i.e., thecontinuousvanishingofaneffectiveKondoenergy scalesignaledbythedivergenceofthezero-temperature staticlocalspinsusceptibility sas U approaches Ucfrombelow.Suchadivergenceisindeedseeninour zero-temperature svs U data[Fig. 3(a) ]andinthe temperaturedependenceof sat U Uc[Fig. 3(b) ]. Figure 2(b) showsthe U dependenceofthelocalmagnetization Mloc limH 0limT 0h Mzi ,where H isalocal magneticeldenteringaterm HSz(with gB 1 )added toEq.( 1 ).Since Mloc 0 throughoutthestrong-coupling phase,and Mlocrisescontinuouslyfromzeroonentryto thelocal-momentphase,thisquantityservesasanorder parameterforthequantumphasetransition.Ourresultscan besummarizedas s T;U Uc T xs; s T 0 ;UUc us; (3) where u U=Uc 1 .Wend xs 0 : 80 3 fromCTQMCcalculations,inexcellentagreementwiththeNRG value xs 0 : 7908 3 ;theNRGalsoyields s 1 : 42 2 and s 0 : 1874 2 .Thesepower-lawbehaviorsareall deningcharacteristicsofcriticalKondodestruction. ToprobevalenceuctuationsneartheQCP,weturnto thestaticlocalchargesusceptibility c T;U .Asshownin Fig. 4(a) c T 0 ;U increaseswith U inthestrongcouplingphaseanddivergesas U U c,inamanner similarto s T 0 ;U .Inthelocal-momentphase, thespinandchargeresponsesareverydifferent: s T 0 ;U 1 ,but c T 0 ;U remainsnite, althoughitdivergesas U U c.Inotherwords,thevalenceuctuationenergyscaleisnonzeroinbothphases, vanishingonlywhen U approaches Ucfromeitherside.At U Uc, chasasingulartemperaturedependenceas showninFig. 4(b) .Thesebehaviorsareconsistentwith c T;U Uc T xc;c T 0 ;U j u j c: (4) CT-QMCcalculationsyield xc 0 : 36 3 ,whiletheNRG gives xc 0 : 120 1 (extractedattemperaturesmuchlower thancanbeaccessedbyCT-QMC)and c 0 : 21 1 .The differencebetweenthetwo xcvaluesstemsfromavery slowcrossovertothequantumcriticalregime[Fig. 4(b) inset].Themuchwidercrossoverwindowfor ccompared with s[Fig. 3(b) ]likelyarisesbecause xcu(a)D= D=3000 D=1500 D=750 D=350 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4Mlocu(b) Strong Coupling Local Moment FIG.2(coloronline).Valenceandlocalspinpropertiesvs u U=Uc 1 for r 0 : 6 0 0 : 1 D ,and "d 0 : 05 D : (a)Occupancy h ndi atthelabeledtemperatures[ 29 ].TheQCP ( u 0 )occursatmixedvalence,i.e., h ndi 1 .(b)Localmagnetization Mloc,showingquenchingoftheimpurityspinfor u< 0 buttheemergenceofafreelocalmomentfor u> 0 100101102103-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4c(,U)u(a)D= D=3000 D=1500 D=750 D=350 10110210-410-310-210-1c(T,Uc)T/D(b)NRG CT-QMC 10110210310-1210-810-4 FIG.4(coloronline).Localstaticchargesusceptibility c T;U for r 0 : 6 0 0 : 1 D ,and "d 0 : 05 D (a)vs u U=Uc 1 atthelabeledtemperatures[ 29 ]and(b)vs T atthe criticalpoint U Uc,wherethediscrepancybetweenCT-QMC andNRGdataisduemainlytothedifferencein Ucvalues.Inset: c T;Uc overawiderrangeof T ,showingtheslowcrossover behavior. 100101102103-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4s(,U)u (a)D= D=3000 D=1500 D=750 D=350 10010110210310-410-310-210-1s(T,Uc)T/D (b)NRG CT-QMC FIG.3(coloronline).Localstaticspinsusceptibility s T;U for r 0 : 6 0 0 : 1 D ,and "d 0 : 05 D (a)vs u U=Uc 1 atthelabeledtemperatures[ 29 ]and(b)vs T atthecriticalpoint U Uc. PRL 109, 086403(2012) PHYSICALREVIEWLETTERSweekending 24AUGUST2012086403-3

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thatvalenceuctuationsarepartofthecriticalspectrum. Calculationsforlevelenergies "d 0 : 05 D (andhence differentcriticaloccupancies h ndi )indicatethatthecritical exponentsdenedabovedependonthebandexponent r butnotontheimpurityvalence.Thisimpliesthatthe divergenceofthestaticchargesusceptibilityisauniversal property.Atthesametime,wendthatthecriticalbehaviorinthespinsectorcoincideswiththemodelinitsinteger valencelimit,i.e.,the p h -asymmetricpseudogapKondo model[ 16 ]. Wenowdiscussthedynamicalscalingof s ;T and c ;T .InanalogywiththespinresponseattheKondo destructionQCPintheusualKondolimit[ 23 25 ],wend thatat U Ucboth s ;T and c ;T collapseontothe conformalscalingform,showingpower-lawdependences on T= sin T withexponents s 0 : 20 3 and c 0 : 67 3 [seeFigs. 5(a) and 5(b) ].Forthetemperatures considered,thechargesusceptibilityhasnotyetreached itsasymptoticpower-lawbehavior[basedonFig. 4(b) ]. Ourresultsthusimplythatbothleadingandsubleading termsofthecritical c ;T scaleintermsof T= sin T .Thescalingformmeans s !;T and c !;T obey !=T scaling[ 23 ]at U Uc. WenextconsidertheeffectontheQCPofapplyinga nitelocalmagneticeld H .Consistentwiththe !=T scaling,wend H=T scalingforelds j H j
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extendedrangeofparameters.Ourresultsraisetheprospectofunconventionalquantumcriticalityinmixed-valent systemsbeyond YbAlB4. WethankS.Nakatsuji,A.Nevidomskyy,S.Paschen,F. Steglich,H.Q.Yuan,andL.Zhuforusefuldiscussions. ThisworkhasbeensupportedbyNSFGrantsNo.DMR0710540,No.DMR-1006985,andNo.DMR-1107814 andbyRobertA.WelchFoundationGrantNo.C-1411. ThecalculationswereinpartperformedontheRice ComputationalResearchClusterfundedbytheNSFand apartnershipbetweenRiceUniversity,AMD,andCray. WeacknowledgethehospitalityoftheMaxPlanck InstitutesforChemicalPhysicsofSolidsandPhysicsof ComplexSystems(J.H.P.,K.I.andQ.S.),theAspen CenterforPhysics(S.K.andQ.S.),andtheInstituteof PhysicsofChineseAcademyofSciences(Q.S.) [1]N.D.Mathur,F.M.Grosche,S.R.Julian,I.R.Walker, D.M.Freye,R.K.W.Haselwimmer,andG.G.Lonzarich, Nature(London) 394 ,39(1998) [2]D.M.Broun, NaturePhys. 4 ,170(2008) [3]S.Sachdev, QuantumPhaseTransitions (Cambridge UniversityPress,Cambridge,England,1999). [4]J.A.Hertz, Phys.Rev.B 14 ,1165(1976) ;A.J.Millis, ibid. 48 ,7183(1993) [5]H.v.Lo hneysen,A.Rosch,M.Vojta,andP.Wo le, Rev. Mod.Phys. 79 ,1015(2007) ;Q.SiandF.Steglich, Science 329 ,1161(2010) ;M.C.Aronson,R.Osborn,R.A. Robinson,J.W.Lynn,R.Chau,C.L.Seaman,andM.B. Maple, Phys.Rev.Lett. 75 ,725(1995) ;A.Schro der,G. Aeppli,R.Coldea,M.Adams,O.Stockert,H.v. Lo hneysen,E.Bucher,R.Ramazashvili,andP. Coleman, Nature(London) 407 ,351(2000) ;S.Paschen, T.Lu hmann,S.Wirth,P.Gegenwart,O.Trovarelli,C. Geibel,F.Steglich,P.Coleman,andQ.Si, ibid. 432 ,881 (2004) ;S.Friedemann,N.Oeschler,S.Wirth,C.Krellner, C.Geibel,F.Steglich,S.Paschen,S.Kirchner,andQ.Si, Proc.Natl.Acad.Sci.U.S.A. 107 ,14547(2010) [6]T.Park,F.Ronning,H.Q.Yuan,M.B.Salamon,R. Movshovich,J.L.Sarrao,andJ.D.Thompson, Nature (London) 440 ,65(2006) [7]Q.Si,S.Rabello,K.Ingersent,andJ.L.Smith, Nature (London) 413 ,804(2001) ; Phys.Rev.B 68 ,115103 (2003) [8]P.Coleman,C.Pe pin,Q.Si,andR.Ramazashvili, J.Phys. Condens.Matter 13 ,R723(2001) [9]H.Shishido,R.Settai,H.Harima,andY.O nuki, J.Phys. Soc.Jpn. 74 ,1103(2005). [10]H.Q.Yuan,F.M.Grosche,M.Deppe,C.Geibel,G.Sparn, andF.Steglich, Science 302 ,2104(2003) ;A.T.Holmes, D.Jaccard,andK.Miyake, J.Phys.Soc.Jpn. 76 ,051002 (2007) ;J.-P.Rueff,S.Raymond,M.Taguchi,M.Sikora, J.-P.Itie ,F.Baudelet,D.Braithwaite,G.Knebel,andD. Jaccard, Phys.Rev.Lett. 106 ,186405(2011) [11]M.Okawa etal. Phys.Rev.Lett. 104 ,247201(2010) [12]S.Nakatsuji etal. NaturePhys. 4 ,603(2008) [13]Y.Matsumoto,S.Nakatsuji,K.Kuga,Y.Karaki,N.Horie, Y.Shimura,T.Sakakibara,A.H.Nevidomskyy,andP. Coleman, Science 331 ,316(2011) [14]J.L.SmithandQ.Si, Phys.Rev.B 61 ,5184(2000) ;Q.Si andJ.L.Smith, Phys.Rev.Lett. 77 ,3391(1996) [15]R.ChitraandG.Kotliar, Phys.Rev.Lett. 84 ,3678 (2000) [16]K.IngersentandQ.Si, Phys.Rev.Lett. 89 ,076403(2002) [17]E.Gull,A.J.Millis,A.I.Lichtenstein,A.N.Rubtsov,M. Troyer,andP.Werner, Rev.Mod.Phys. 83 ,349(2011) [18]R.Bulla,Th.Pruschke,andA.C.Hewson, J.Phys. Condens.Matter 9 ,10463(1997) [19]C.Gonzalez-BuxtonandK.Ingersent, Phys.Rev.B 57 14254(1998) [20]J.-H.Chen,L.Li,W.G.Cullen,E.D.Williams,andM.S. Fuhrer, NaturePhys. 7 ,535(2011) ;D.JacobandG. Kotliar, Phys.Rev.B 82 ,085423(2010) [21]D.WithoffandE.Fradkin, Phys.Rev.Lett. 64 ,1835 (1990) ;M.VojtaandR.Bulla, Phys.Rev.B 65 ,014511 (2001);M.T.GlossopandD.E.Logan, Europhys.Lett. 61 ,810(2003) ;L.Fritz,S.Florens,andM.Vojta, Phys. Rev.B 74 ,144410(2006) [22]M.VojtaandL.Fritz, Phys.Rev.B 70 ,094502(2004) [23]M.T.Glossop,S.Kirchner,J.H.Pixley,andQ.Si, Phys. Rev.Lett. 107 ,076404(2011) [24]K.Binder, Z.Phys. 43 ,119(1981) [25]S.KirchnerandQ.Si, Phys.Rev.Lett. 100 ,026403 (2008) [26]S.WatanabeandK.Miyake, J.Phys.Soc.Jpn. 79 ,033707 (2010) [27]P.MonthouxandG.G.Lonzarich, Phys.Rev.B 69 064517(2004) [28]Astrongtemperaturedependenceinthequantumcritical regime,similartothatshowninFig. 2(a) ,hasbeen observedinrecentexperimentalmeasurementsofthe 4 f valenceoftheYbionsin YbAlB4atlowtemperatures [S.Nakatsuji(privatecommunication)]. [29]Datafor 1 wereobtainedusingtheNRGmethod. TheremainingdatarepresentCT-QMCresults. PRL 109, 086403(2012) PHYSICALREVIEWLETTERSweekending 24AUGUST2012086403-5



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ANDERSONIMPURITYMODELSWITHBOSONSASDESCRIPTIONSOF MOLECULARDEVICESANDHEAVY-FERMIONSYSTEMS By LILIDENG ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2014

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c r 2014LiliDeng 2

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Tomywife,sonandparents 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmydeepestgratitudetomyadviser,Pr of.KevinIngersent, forhissupport,patience,andencouragementthroughoutmy graduatestudies.During theseyears,Ihavenotonlylearnedknowledgeandskillsfro mhim,butalsobeen inuencedbyhisenthusiasmforthepursuitoftheunknownan dhisrigorousscientic spirit.IamsureIwillbenetfromthesevirtuesinmyfuture careerandlife. Ithankmysupervisorycommitteemembers:Prof.SelmanHers held,Prof. KhandkerMuttalib,Prof.YoonseokLee,andProf.SimonR.Ph illpot,fortheirtime onreviewingmyPh.D.qualifyingproposalanddissertation manuscriptaswellastheir valuablesuggestionsforimprovingthem. IamgratefultoProf.QimiaoSiandJedediahPixleyfromRice University,Prof. EdsonVernekandGiseleIorioLuizfromtheFederalUniversi tyofUberlandia,Brazil, andProf.EnriqueAndafromthePonticalCatholicUniversi tyofRiodeJaneiro,Brazil, fortheircooperationonmyPh.D.research. IthankmyformergroupcolleagueDr.MengxingChengforhisg eneroushelpon myresearchwhenIwasajuniorgraduatestudent,andmycurre ntgroupcolleague TathagataChowdhuryforusefuldiscussionandsharing.Tha nksalsogotoDavid Hansenforhiscomputersupport,andtoKristinNicholaandP amMarlinfortheir administrativeassistance.Finally,Iamgratefultoallth eotherprofessors,friendsand colleagueswhoassisted,advised,andsupportedmyresearc handwritingeffortsover theyears.FinancialsupporthasbeenprovidedbytheDivisi onofMaterialsResearchof theU.S.NationalScienceFoundation. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1Outline ...................................... 13 1.2OverviewoftheKondoProblem ........................ 13 1.3KondoEffectandElectron-PhononInteractioninNanost ructures ..... 16 1.4KondoPhysicswithDispersiveBosonicBath ................ 25 2NUMERICALRENORMALIZATIONGROUPMETHODANDITSAPPLICATI ON TOQUANTUMIMPURITYMODELS ........................ 32 2.1OverviewoftheNumericalRenormalizationGroupMethod ........ 32 2.2ApplicationoftheNRGtotheSingle-ImpurityAndersonM odel ...... 34 2.2.1Single-ImpurityAnderonModel .................... 34 2.2.2LogarithmicDiscretization ....................... 35 2.2.3MappingtoaSemi-InniteChain ................... 37 2.2.4IterativeSolution ............................ 37 2.3ApplicationoftheNRGtotheBose-FermiKondoModel .......... 40 2.3.1Bose-FermiKondoModel ....................... 40 2.3.2NRGTreatment ............................. 41 3RESULTSFORATWO-ORBITALMOLECULARJUNCTIONMODEL ..... 44 3.1Introduction ................................... 44 3.2ModelHamiltonianandLang-FirsovAnalysis ................ 45 3.2.1ModelHamiltonian ........................... 45 3.2.2ThermodynamicsandLinearConductance ............. 48 3.2.3Lang-FirsovAnalysis .......................... 50 3.3GeneralConguration:BasicStudy ..................... 54 3.3.1IsolatedMolecule ............................ 54 3.3.2NumericalResultsfortheGeneralConguration .......... 57 3.3.2.1Propertiesat T =0 andcrossovertemperature T .... 57 3.3.2.2Propertiesat T > 0 ..................... 62 3.4Side-OrbitalConguration:QuantumPhaseTransition ........... 65 3.4.1IsolatedMolecule ............................ 66 3.4.2NumericalResultsfortheQPT .................... 68 5

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3.4.2.1First-orderQPT ....................... 69 3.4.2.2QPTversuscrossover .................... 77 3.5InterplayofTwoPhononEffects:SpinandChargeKondoCr ossover ... 83 3.5.1PreliminaryAnalysis .......................... 83 3.5.2NumericalResultsfortheChargeandSpinKondoEffect s ..... 85 3.6Summary .................................... 90 4RESULTSFORTHETWO-IMPURITYBOSE-FERMIANDERSONMODEL .. 92 4.1Introduction ................................... 92 4.2ModelHamiltonian ............................... 93 4.3Kondo-ISPhaseTransition .......................... 95 4.3.1NRGSpectrum ............................. 96 4.3.2StaticLocalSusceptibility ....................... 100 4.4Kondo-LMPhaseTransition .......................... 103 4.4.1NRGSpectrum ............................. 105 4.4.2StaticLocalSusceptibility ....................... 109 4.5FullPhaseDiagram .............................. 113 4.5.1PhaseDiagramandNRGFLow .................... 113 4.5.2Kondo-ISPhaseBoundary ....................... 116 4.5.3Kondo-LMPhaseBoundary ...................... 118 4.5.4IS-LMPhaseBoundary ........................ 120 4.6PairingSusceptibility .............................. 121 5CONCLUSIONANDFUTUREWORK ....................... 128 5.1Conclusion ................................... 128 5.2FutureWork ................................... 130 REFERENCES ....................................... 132 BIOGRAPHICALSKETCH ................................ 138 6

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LISTOFTABLES Table page 3-1Energiesofempty,singly-anddoubly-occupiedlow-lyi ngstatesofisolated moleculeingeneralconguration. ......................... 56 3-2Energiesandquantumnumbers P =( n + n b ) mod 2 ofsingly-anddoubly-occupied low-lyingstatesofisolatedmoleculewithouttunnelingfr om/toleads(side-orbital conguration). .................................... 66 3-3Empty,singly-anddoubly-occupiedgroundstatesofthe isolatedmolecule, aswellastheirenergiesingeneralconguration. ................. 83 7

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LISTOFFIGURES Figure page 1-1ScanningelectronmicroscopeimageofaGaAs/AlGaAssem iconductorquantum dot. .......................................... 16 1-2Temperatureandmagneticelddependenceofthezero-bi asKondoresonance. 17 1-3Temperature-dependentconductanceofaKondo-regimeq uantumdot. .... 18 1-4Imageofasingle-moleculetransistorincorporatingin dividualdivanadiummolecules. 20 1-5Differentialconductanceforthesingle-moleculardev iceincorporatingindividual divanadiummolecules. ................................ 20 1-6Differentialconductanceofa C 60 single-moleculedevicewithphononsidebands. 21 1-7Conductanceinthephonon-assistedtwo-orbitaltunnel ingmodelvsgatevoltage. 24 1-8Field-inducedquantumphasetransitioninYbRh 2 Si 2 ............... 26 1-9Thetemperature-versus-magnet-eldphasediagramwit htheindicationof KondoenergyscaleofYbRh 2 Si 2 .......................... 27 1-10ChangesofFermisurfacecyclotronmassacrossaQCPinC eRhIn 5 ...... 28 1-11Thepressure-eldphasediagraminCeRhIn 5 .................. 30 2-1SchematicplottingoftheNRGtreatmentofBFKMHamilton ian. ......... 42 3-1Schematicalplotofcongurationoftwo-orbitalmolecu lejunctioningeneral parameterregime. .................................. 47 3-2Physicalquantitiesinvariatioinwith 2 =! 0 ..................... 58 3-3Occupationofindividualmolecularorbitalsvs 2 =! 0 ............... 59 3-4Crossovertemperature T vsscaled e ph coupling ( = x ) 2 ........... 61 3-5Temperaturedependenceofthemolecularentropy,tempe raturetimesthe molecularsusceptibility,andthephononoccupation. ............... 62 3-6Temperaturetimesthemolecularsusceptibilityvsscal edtemperature T = T . 63 3-7Schematicrepresentationofaside-orbitalcongurati onofatwo-orbitalmolecular junctionwithphonon-assistedinterorbitaltunneling. ............... 65 3-8Physicalquantitiesinthefunctionofe-phcouplingstr ength 0 at T =0 .... 70 3-9Temperaturedependenttotalelectronoccupancy h n mol i andlinearconductance G ........................................... 74 8

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3-10Totalelectronoccupancy h n tot i andlinearconductance G inthefunctionof 0 at T =0 forseveralxedvaluesof eV g ..................... 75 3-11Totalelectronoccupancy h n mol i andlinearconductance G inthefunctionof eV g forseveralxedvaluesof 0 ......................... 76 3-12Totalelectronoccupancy h n mol i andlinearconductance G inthefunctionof eV g forseveralgivenvaluesof U 0 rangingfrom0to U (= U = U ) at T =0 . 78 3-13Dependenceon eV g oftheenergiesofthethreeloweststatesoftheisolated moleculearoundthelevelcrossingpoint eV c g .................. 79 3-14Totalelectronoccupancy h n mol i andlinearconductance G versusof eVg in thecrossovercases. ................................. 81 3-15Totalelectronoccupancy h n mol i ,linearconductance G andlow-lyingenergy levelsinthefunctionof eV g at T =0 forseveralgivenvaluesof ...... 86 3-16Temperaturedependentspinandchargesusceptibility aswellasthe dependent spinandchargeKondotemperature. ........................ 88 3-17Totalelectronoccupancy h n mol i andlinearconductance G diagramsinthe functionof 0 and at T =0 ............................ 89 4-1ScaledNRGeigenvaluesversuseveniterationnumber N forvevaluesof theinterimpurityexchange I neartheKondo-ISboundary. ............ 96 4-2Energyscale T versusdistancefromthecriticalpoint I c andSchematicplot ofthe I T diagramnearKondo-ISQCP. ...................... 99 4-3CriticalbehaviorofstaggeredspinsusceptibilityatK ondo-ISQCPwith T =0 101 4-4Temperature-dependencestaticstaggeredlocalspinsu sceptibility s ( T I ) aroundcriticalvalue I c ................................ 102 4-5NRGspectruminthefunctionofeveninterationnumberNf orvegivenvalue ofbosoniccouplingstrength g roundtheKondo-LMQCP. ............ 106 4-6Energyscale T inthefunctionofthedistancetothecriticalpoint g c .Schematical plotofthecrossoverdiagramnearKondo-LMQCP. ................ 108 4-7Temperaturedependentstaticstaggeredspinsusceptib ility s ( T g ) forseven givenvaluesof g aroundcriticalvalue g c ..................... 110 4-8Staggeredmagnetizationatzerotemperature M s ( T =0) asafunctionof g g c .Staticstaggeredspinsusceptibility s ( T =0, g ) asafunctionof g c g .. 111 4-9PhasediagramforthefullHamiltonianconsideringboth Heisenberginterimpurity RKKYinteraction I andbosoniccoupling g .................... 114 9

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4-10SchematicowdiagramofthefullHamiltonianconsider ingbothHeisenberg exchangeinteraction I andbosonicbath g .................... 115 4-11NRGspectruminthefunctionofeveniterationnumberNf orthreepointson theKondo-ISphaseboundary. ........................... 116 4-12Temperature-dependentstaticstaggeredspinsuscept ibility s ( T g I ) forpoints ontheKondo-ISphaseboundary. ......................... 117 4-13NRGspectrumateveniterationnumbers N forthreepointsontheKondo-LM phaseboundary. ................................... 118 4-14Criticalbehaviorsof T andstaggeredspinsusceptibilityaroundtheKondo-LM phaseboundary. ................................... 119 4-15NRGspectrumasafunctionofeveniterationnumber N forthreepointson theIS-LMphaseboundary. ............................. 120 4-16Stacticsingletpairingsusceptibility d ( I g ) at T =0 inthefunctionof I for vexedvalueof g ................................. 122 4-17Staticsingletpairingsusceptibility d ( I g ) at T =0 asafunctionofthedistance j I I c j fromtheQCPontheKondo-ISphaseboundaryforvexedvalue sof g 123 4-18Staticsingletpairingsusceptibility d ( I g ) at T =0 inthefunctionof g for threexedvaluesof I ................................ 124 4-19Staticsingletpairingsusceptibility d ( I g ) at T =0 alongthecuttinglines I = cg with c beingaconstant. ........................... 125 4-20Statictripletpairingsusceptibility d ( I g ) at T =0 alongthecuttinglines I = cg with c beingaconstant. ............................. 126 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ANDERSONIMPURITYMODELSWITHBOSONSASDESCRIPTIONSOF MOLECULARDEVICESANDHEAVY-FERMIONSYSTEMS By LiliDeng May2014 Chair:KevinIngersentMajor:Physics Thisdissertationpresentstheresultsoftheoreticalstud yofapairoflocalized atomicormolecularlevelshybridizingwithafermioniccon ductionbandandalso coupledtooneormorebosonicdegreesoffreedom.Twovarian tsoftheAnderson impuritymodelareinvestigatedusingthenumericalrenorm alizationgrouptechnique. Onemodeldescribesatwo-orbitalmolecularjunctionwiths trongcouplingtoalocal bosonicmode,e.g.,anopticalphonon.Theothermodel,desc ribingtwomagnetic impuritiescoupledtoaconductionbandandalsotoadissipa tivebosonicbath,is usedtostudysuperconductingpairinginthepresenceoflat ticemagnetismandKondo correlation. Themolecularjunctionismodeledintermsofatwo-orbitalm oleculeconnecting apairofexternalleads.Stronginter-andintra-orbitalCo ulombrepulsiongivesriseto Kondocorrelationbetweenthelocalspininsidethemolecul eanddelocalizedspins intheconductionband.Atthesametime,thepossibilityofc onformationalchanges admitsstrongcouplingbetweenalocalphononmodeandthemo lecularcharge,and/or phonon-assistedinter-orbitaltunneling.Varyingtheele ctron-phononcouplingstrength causesacrossingoflevelsinthegroundstateoftheisolate dmolecule.Whenthe moleculeisconnectedtotheleads,asmoothcrossoverbetwe enaKondoregime andaphonon-dominatedregimeisfoundinthevicinityofthi slevelcrossing.Ina particularside-orbitalconguration,thiscrossoverint hegeneralcongurationbecomes 11

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arst-orderquantumphasetransitionduetotheemergenceo fanewsymmetry. Furthermore,investigationiscarriedoutofinterplaybet weenthephonon-assisted tunnelingandaHolstein-typephononcouplingtothemolecu larcharge,andacrossover betweenspin-Kondoandcharge-Kondoregimesisstudied. Inthetwo-impurityBose-FermiAndersonmodel,twomagneti cimpuritieshybridize witha(fermionic)conductionbandandarecoupledviathedi fferenceoftheirspins toasub-Ohmicbosonicbath.Wealsoconsideranexchangeint eractionbetween theimpurities,modelingtheRuderman-Kittel-Kasuya-Yos idapresentedinalattice setting.Thephasediagramofthismodelisfoundtofeaturea Kondophase,an interimpurity-singlet(IS)phase,andalocal-moment(LM) phase.Quantumphase transitionsbetweenthesephasesariseduetocompetitionb etweenthreeenergy scales:theKondotemperature T K ,theantiferromagneticinterimpurityexchange I ,and abosoniccoupling g tothestaggeredimpurityspin.Thecriticalbehavioralong the Kondo-ISandKondo-LMboundariesisstudiedindetail.Sing letsuperconductingpairing showsenhancementalongtheKondo-LMphaseboundary,parti cularlyclosetothetriple pointwhereallthreephasesmeet,suggestinganewmechanis mforheavy-fermion superconductivityinthevicinityofantiferromagneticor der. 12

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CHAPTER1 INTRODUCTION 1.1Outline Thisdissertationisarrangedasfollows.Backgroundandmo tivationfortheresearch aregiveninChapter 1 .ThehistoryoftheKondoproblemisrstoverviewed.Then theKondoeffectanditsinterplaywithelectron-phononint eractioninnanostructures areintroduced.Thischapterconcludeswithabriefreviewo ftheKondoproblemwith bosonicdegreesoffreedomaswellasitsrelationtoheavyfe rmionsystems.InChapter 2 ,theformalismofthenumericalrenormalizationgrouptech niqueisintroduced, andtwoexamplesaregiventoillustrateitsapplicationsin pure-fermionicquantum impurityproblemsandinBose-Fermiproblems.Chapter 3 presentsresultsforamodel describingamoleculewithtwoactiveorbitalsconnectedto apairofelectricalleadsand alsocoupledtoalocalvibrationalmode.Chapter 4 presentsresultsforatwo-impurity Bose-FermiAndersonmodelinwhichtwomagneticimpurities arecoupledviatheir staggeredspintoadispersivebosonicbath.Chapter 5 containsasummayandoutlook forfuturework. 1.2OverviewoftheKondoProblem TheKondoproblemdatesbacktothe1930s[ 1 ]whendeHaas,deBoerandvan denBergrstobservedaminimumintheresistivityofgoldat lowtemperatures(around 30K).Thisphenomenonwassubsequentlyrecognizedtoarise fromtheinteractionof 3 d transitionmetalimpuritieswithdelocalizedelectronsin theconductionbandsofa hostmetal.ObservationofaresistanceminimumforMo-Nbal loyswithFeimpurities[ 2 ] isoneexampleofexperimentalevidencefortheKondoeffect Theoriginoftheresistivityminimumcouldnotbeexplained theoreticallyuntil1964, whenKondoachievedabreakthrough[ 3 ]usingthe s d modelproposedadecadeearlier 13

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byZener[ 4 ].TheHamiltonianforthe s d modelis H sd = X k k c y k c k + 1 N k X k k 0 J kk ( S + c y k # c k 0 + S c y k c k 0 # + S z ( c y k c k 0 c y k # c k 0 # )), (1–1) where S z S = S x iS y arethespinoperatorsforaspin-1/2impurity, c y k isthecreation operatorforanelectronwithwavevector k ,spin andenergy k intheconduction bandofthehostmetal, J kk 0 isthematrixelementforexchangescattering,and N k is thenumberof k points(thenumberofunitcellsinthehostmetal).Kondocal culated theimpuritycontributiontotheresistivitytothirdorder in J (assuming J kk = J is k -independent),andfound R imp = 3 mJ 2 S ( S +1) 2 e 2 ~ F h 1 4 J 0 ( F )ln k B T D i (1–2) Here, m and e arethemassandchargeoftheelectron, 0 ( F ) isthedensityofstates oftheconductionbandattheFermienergy F ,and D isthehalfbandwidthofthe conductionband.ThenotablefeatureofEq.( 1–2 )isthetermcontaining ln T ,which arisesfromspin-ipscatteringprocessesinvolving S .When R imp iscombinedwith othercontributionstotheresistivity(primarilyelectro n-phononscattering),theresistivity minimumissatisfactorilyexplained.However,thediverge nceofthe ln T termas T 0 invalidatesKondo'sperturbativecalculationatlowtempe ratures.Many-bodytechniques weredevelopedtosolvethisdivergenceintheferromagneti ccase( J < 0 )[ 5 ],butdidnot workfortheantiferromagneticcase( J > 0 )becausetheymerelyshiftedthedivergence toanitetemperature T K thatbecameknownastheKondotemperature. Inthelate1960s,Andersonandhisco-workersformulatedap erturbativescaling method[ 6 ],calledpoorman'sscaling,tosolvethe s d model.Thebasicideaofthis approachistoprogressivelyreducethewidthoftheconduct ionband,eliminatingthe high-energyexcitationsandadjustingthecoupling J topreservethesamelow-energy physics.Theresultingscalingtrajectoryleadstoaneffec tiveantiferromagneticcoupling constant ~ J thattendstoinnityastheeffectivebandwidth ~ D 0 ,whichmeansthe 14

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impurityspinisstronglyboundwiththeconductionbandtof ormasingletatvery lowtemperatures.TheanalysisidentiestheKondotempera ture T K asthesingle temperatureorenergyscalecharacterizingthelow-energy physics.Thus,oneexpects thelow-temperaturepropertiestobefunctionsof T = T K alone. K.G.Wilson'scontributionin1975isamilestoneinthestud yoftheKondo problem[ 7 ].Basedonrenormalizationgroupideasfromeldtheory,he deviseda powerfulnon-perturbativenumericalrenormalizationgro up(NRG)approachwhich thoroughlysolvedthe s d modelbydenitelyestablishingthegroundstateandthe low-temperatureproperties.Lateron,theNRGwasextended [ 8 9 ]tosolveamore generalimpuritymodel:theAndersonmodelproposedin1961 todescribelocalized magneticstatesinmetals[ 10 ].TheAndersonHamiltonianis H = d X n d + Un d n d # + X k k c y k c k + 1 p N k X k ( V k d y c k + V k c y k d ), (1–3) where d y isthecreationoperatorforanimpurityelectronwithenerg y d andspin c y k createsaconduction-bandelectronwithwavevector k ,spin andenergy k n d = d y d U istheCoulombrepulsionbetweentwoelectronsintheimpuri tylevel,and V k isthematrixelementforhybridizationbetweentheimpurit yandtheconductionband. The s d modelisaspecialcaseoftheAndersonmodelforthelimit d U V 2 = D whereonlythesubspace n d = n d + n d # =1 isoccupied.Theexchangeinteractionof theKondomodelis: J kk 0 = V k V k 0 1 U + d 0k + 1 k d (1–4) TheNRGsolutionoftheAndersonmodel[ 8 9 ]givesaricherphysicalpicturethat integratestheKondoeffectwithphenomenarelatedtocharg euctuationinthe impurity.MoredetailsabouttheNRGanditsapplicationtoq uantumimpuritymodelsare discussedinChapter 2 15

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Figure1-1.ScanningelectronmicroscopeimageofaGaAs/Al GaAssemiconductor quantumdotdenedbyfourelectrodes(lightregions).Them iddleelectrode ontheleftisthegateelectrodeandtheotherthreeelectrod escontrolthe tunnelbarriersbetweenthequantumdot(centerdarkregion )andtheleads (topandbottomdarkregions).Reprintedwithpermissionfr om D.Goldhaber-Gordon etal .,Nature 391 ,156(1998).Copyright1998bythe NaturePublishingGroup. 1.3KondoEffectandElectron-PhononInteractioninNanost ructures Advancedfabricationtechnologiesallowobservationofst rongelectron-electron ( e e )interactionsinnanoscopicdeviceswithhighlytunablepa rameters.Inthese systems,theKondoeffectismarkedbytheemergenceofazero -biasanomalyin theelectricalconductance,asmeasuredforinstanceinqua ntumdots[ 11 – 14 ]and single-moleculejunctions[ 15 – 19 ].Inthesesystems,theenhancedelectron-electron ( e e )interactionduetostrongspatialconnementisresponsib leforprovidingunpaired localizedelectronsthatareKondoscreenedbythesurround ingconductionelectrons. ThepossibleoccurrenceoftheKondoeffectinquantumdotsw asrstproposedin 1988[ 20 21 ],anditsexperimentalrealizationwasreportedonedecade laterin1998 [ 11 ],wherethequantumdotwasfabricatedonaGaAs/AlGaAshete rostructureandhad discreteenergylevels,thepositionsofwhichwerecontrol ledbythevoltageappliedon 16

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Figure1-2.Temperatureandmagneticelddependenceofthe zero-biasKondo resonancemeasuredinthedifferencialconductanceforthe quantumdot showninFig. 1-1 [ 11 ].ReprintedwithpermissionfromD.Goldhaber-Gordon etal .,Nature 391 ,156(1998).Copyright1998bytheNaturePublishing Group. thegateelectrode.Thescanningelectronmicroscopicimag eofthisdeviceisshown inFig. 1-1 .Duetoitssmallsize,thedothadalargechargingenergy(i. e.,Anderson U parameter)foraddingoneextraelectron.Whentunedtocont ainanoddnumber ofelectrons,suchaquantumdotislikeamagneticimpurityw ithanetspin1/2,anda many-bodyKondosingletisformedbetweentheunpairedelec troninthedotandthe delocalizedelectronsintheleads.(Whentunedtocontaina nevennumberofelectrons, thequantumdotissaidtobeintheCoulombblockaderegime.I nthisregime,levels ofthequantumdotbelowtheFermilevelarefullyoccupied,a ndallemptylevelsare toofarabovetheFermisurfacetobethermallyoccupied.The electricalconductanceis verylowandnoKondoeffecttakesplace.)Fig. 1-2 providestwopiecesofevidencefor 17

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Figure1-3.Temperature-dependentconductanceofaKondoregimequantumdot.The conductance G approachestheunitarylimitvalue 2 e 2 = h atlowtemperature. Theinsetshowsthecollapseofthethreecurvesontoauniver salscaling curvewhen G isplottedasafunctionof T = T K .FromW.G.vanderWiel et al .,Science 289 ,2105(2000).ReprintedwithpermissionfromAAAS. aKondoeffectinthequantumdotstudiedin[ 11 ].Thethreeguresintheleftcolumn showazero-bias V sd =0 peakthatdisappearswithincreasingtemperature.Thepeak resultsfromagreaterdensityofstatesattheFermienergyo ftheleadsinducedbythe formationofaKondosinglet,andincreasingthetemperatur edestroysthesingletand thenattenuatestheconductance.Thethreeguresintherig htcolumnshowthesplitting ofthezero-biasKondopeakinanappliedmagneticeld.This behaviorresultsfromthe splittingofthedensityofstatespeakattheFermienergy. Subsequentexperimentshaveconrmedandfurtherprobedth eexistenceofKondo physicsinquantumdots[ 12 – 14 ].Enhancementofzero-biasconductanceisthemain signatureoftheKondoeffectinquantumdots,andshows[ 13 ]thattheconductancecan reachtheunitarylimitvalue 2 e 2 = h .Fig. 1-3 hastemperature-dependentconductance curvesforthreedifferentxedvaluesofthegatevoltage,a llofwhichplacethedevice intheKondoregime.Inthelow-temperaturelimit,thecondu ctancesaturatesatthe unitarylimitvalue,whichimpliesperfectelectrontransm issionthroughthequantumdot. 18

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Thenormalizedconductance G = (2 e 2 = h ) isexpectedtocollapsetoauniversalcurve thatisafunctionofnormalizedtemperature T = T K ( T K istheKondotemperature,and k B T K isthecharacteristicenergyscaleoftheKondoeffect.).Th einsetinFig. 1-3 shows near-perfectone-parameterscaling(independentofCoulo mbrepulsion U ,levelposition d andhybridization .). ExperimentalinvestigationsoftheKondoeffectindoublequantum-dotsystems havealsobeencarriedout[ 22 – 25 ].Ruderman-Kittel-Kasuya-Yoshida(RKKY) interactionbetweenspinsindifferentquantumdotsplaysa nimportantroleindetermining physicalpropertiesinthesecases.Competitionbetweenth eKondoeffectandthe non-localRKKYinteractionleadstothepossibilityofcont rollingspinandentanglement inquantumdots.Thetwo-channelKondoeffect(proposedin[ 26 ])wasrstobservedin adoublequantum-dotsystem[ 25 ].Alargerquantumdotservedasasecondreservoir ofelectronsbesidestheleads.Thedouble-quantum-dotsys temcouldbetunedtohave aquantumphasetransitionbetweentwodistinctKondogroun dstates,andtheelectrons inthetworeservoirswereentangledthroughtheinteractio nwithanunpairedelectronin thesmallerquantumdot. Useofmoleculesasactivecomponentsisapromisingdirecti oninthedevelopment ofnanometer-scaleelectroniccircuits.Single-molecule devicesarealsopowerful systemstostudyquantumtransport.TheKondoeffectandrel atedphysicalphenomena havebeenwidelyinvestigatedinvariouskindsofsingle-mo leculedevices,including carbonnanotubes[ 15 ],Coionsbondedtoterpyridinyllinkermolecules[ 16 ],divanadium molecules[ 17 ],and C 60 [ 19 ].Inthesedevices,thespinandchargecongurationsof moleculesreectthedetailsoftheirchemicalproperties. Figure 1-4 isanexampleof asingle-moleculedeviceincorporatingindividualdivana diummolecules[ 17 ].Figure 1-5 showsevidencefortheKondoeffectinthisdevice.Theleftp anelshowsthatthe zero-biaspeakdecreasesinheightasthetemperatureincre ases,whichisaclear indicationoftheKondoeffect.Theinsetgivesmoreinforma tion:theKondopeak(high 19

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Figure1-4.Imageofasingle-moleculatransistorincorpor atingindividualdivanadium ( V 2 )moleculesshownschematicallyattop.Reprintedwithperm issionfrom W.Liang etal .,Nature 417 ,725(2002).Copyright2002bytheNature PublishingGroup. Figure1-5.Differentialconductance G forthedeviceshowninFig. 1-4 .(a) G vsbias voltage V atdifferenttemperatures.Theinsetisacolormapof G asa functionofbiasvoltageandgatevoltage.(b)Colormapof G asafunctionof biasvoltage V andexternalmagneticeld B ,forxedgatevoltage V g = 0.1 V .ReprintedwithpermissionfromW.Liang etal .,Nature 417 725(2002).Copyright2002bytheNaturePublishingGroup. 20

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Figure1-6.(a)Bias-voltagedependenceofthedifferentia lconductanceintheKondo regimeofa C 60 single-moleculedevice.AcentralKondopeakand sidebandsduetoavibrationalmodeof C 60 areshown.(b)Colormapof differentialconductanceasafunctionofelectrodespacin gandbiasvoltage. ReprintedwithpermissionfromJ.J.Parks etal .,Phys.Rev.Lett. 99 026601(2007).Copyright2007bytheAmericanPhysicalSoci ety. conductance,brightyellow)appearsat V g < 1 V,whichcorrespondingtotheKondo regimewherethereisanunpairedelectroninthemolecule.I ncreasing V g > 1 V,the moleculeenterstheCoulombblockaderegimewithanevennum berofelectronsinthe molecule,sotheKondopeakdisappears(lowconductance,da rkred).Therightpanel showsthesplittingofthezero-biaspeakduetoanexternalm agneticeld,andthesize ofthissplittingisproportionaltotheeld B .Thebehavioroftheconductancehereis similartothatinquantumdots(Fig. 1-2 ),andtheinterpretationisalsothesame. Single-moleculedevicesaresusceptibletostrongcouplin goftheelectroniccharge toquantizedmechanicalvibrations(phonons)withsignic anteffectsonelectronic transportproperties[ 27 – 31 ].Sucheffectshavebeenclearlyobservedinanacetylene ( C 2 H 2 )molecularjunction[ 28 ],singleC 60 junctions[ 29 ],andcarbonnanotube devices[ 32 ].Electron-phonon( e ph )interactionsalsoleadtoadditionalmany-body effects,signicantlymodifyingthelow-temperatureKond ophysicsofthesystem.The competitionbetween e e and e ph interactionshasproventohaveimportanteffects 21

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onthetransportpropertiesofsingle-moleculejunctionsi ntheKondoregime[ 33 – 35 ], supplementingthezero-biasKondopeakwithvibrationalsi debandsatnitebias. In[ 34 ], C 60 wasusedastheelectroniccomponentofthedevices.Figure 1-6 (a)shows twosidebandsinthedifferentialconductanceinadditiont othezero-biasKondopeak. Thesidebandsresultfromthedevicecouplingtoaphononmod e,whichisshown schematicallyintheinset.Thedistancebetweenthesideba ndandKondopeak correspondstotheenergyofthephononmode.Figure 1-6 (b)showstherelation betweentheelectrodespacingandtheenergyofphononmodes :thefartherapartthe electrodes,thehighertheenergyofthephononmodes.Thisi sanexampleofhowa phononmodeinthemoleculecanbeadjustedbyvaryingextern alconditions,andthen thewholesystemcanbecontrolled. Theinterplayof e ph and e e interactionsinnanodevicescanbeaddressedinthe Anderson-Holsteinmodelanditsvariants.TheAnderson-Ho lsteinmodelsupplements theAnderson[ 10 ]modelforamagneticimpurityinametallichostwithaHolst ein coupling[ 36 ]oftheimpuritychargetoalocalbosonicmode,usuallyassu medto representaopticalphonon.Themodelhasalonghistorydati ngbackto1970's[ 37 – 48 ], andrecentlyhasbeenwidelyappliedtosingle-molecularju nctions[ 49 – 58 ].The Anderson-HolsteinHamiltonianisH = d X n d + Un d n d # + 1 p N k X k V k ( d y c k + c y k d )+ X k k c y k c k ( b y + b )( n d 1)+ 0 b y b (1–5) Therstfourtermsconstitutetheconventionalsingle-imp urityAndersonmodel: d y createsanelectrononthe d leveloftheimpurity( n d = d y d and n d = P n d ); d isthe impurity d levelposition; U istheCoulombrepulsionbetweentwoelectronsonthesame level; c y k createsaconductionelectronwithwavevector k ,spin ,andenergy k ;and V k isthematrixelementofthehybridizationbetweenthemolec uleandtheleads.The lasttwotermsinEq.( 1–5 )arephononrelated: b y createsonephononofenergy 0 (with ~ =1 ),and isthe e ph couplingconstant. 22

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Equation( 1–5 )canberegardedasavariantofthepure-fermionicAnderson problemwithamorecomplicatedimpuritythathasaninnite numberofinternalstates arisingfromcouplingofthe d leveltothephononmode.Forthecasewith V k =0 ,the phononcouplingtermcanbeeliminatedbyapplyingacanonic altransformation[ 49 ] ~ H = e S He S with S =( =! 0 )( n d 1)( b y b ) .ThisproducesamodiedHamiltonian ~ H = X ~ d n d + ~ Un d n d # + 0 b y b 2 =! 0 (1–6) with ~ d = d + 2 =! 0 and ~ U = U 2 2 =! 0 .TheeffectiveCoulombrepulsion ~ U is reducedbelow U bytheHolsteincoupling,and ~ U canevenbenegativeforsufciently strong e ph couplingstrength .VariousanalyticalmethodsandnonperturbativeNRG calculationshaveshownthatinthecaseswith V k 6 =0 ,increasingtheHolsteincoupling strength fromzeroinducesasmoothcrossoverfromtheconventionals pin-Kondo regimetoacharge-analogKondoregime,inwhichitistheimp urity“isospin”ordeviation fromhalf-llingthatisquenchedbytheconductionband.Th iscrossoverhappensinthe vicinityof ~ U =0 ,orthepointwheretheeffective e e interactionchangesfromrepulsion toattraction. Anotherinterestingtheoreticalmodeltreatingboth e e and e ph interactionsina two-orbitalsingle-moleculejunctionwasintroducedin[ 59 60 ].Inthismodel,twoorbitals arehybridizedwiththerstsitesofleftandrightleads,an dinter-orbitaltunnelingis mediatedbyalocalphononmode.ThetermintheHamiltonian H tun = X ( b y d y d + bd y d ) (1–7) describestunnelingbetweethetwoorbitalsaccompaniedby phononabsorptionor emission.Theseprocessesmayplayanimportantroleinthet ransportproperties ofmoleculardevices.Thismodelhaspreviouslybeeninvest igatedintheregimeof temperaturemuchhigherthantheKondotemperature T K viaanequation-of-motion method[ 59 60 ].Figure 1-7 showsthegate-voltagedependenceoftheconductance 23

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Figure1-7.Conductanceinthephonon-assistedtwo-levelt unnelingmodelvsgate voltageforthreedifferenttemperatures: k B T = =0.015,0.01,0.15 .Other parametersare =0.2 t = t =0.1 U =0.4 0 =0.6 ,allinunitsof = .Thedashedcurvecorrespondsto U =0 and 0 =1 .Splitting ofthethirdCoulombblockadepeakisshown.Reprintedwithp ermission fromE.Vernek etal .,Phys.Rev.B 72 ,121405(2005).Copyright2005by theAmericanPhysicalSociety. atdifferenttemperatures.ThersttwoCoulombblockadepe aks( V g =0 and 0.4 )are associatedwithtransitionsintheoccupancyofthe orbital,whilethethirdandfourth peaks( V g =0.1 and 1.4 )arisefromjumpsinthe orbitaloccupancy.Thepeaksplitting resultsfromphonon-mediatedtunnelingbetweenthe and orbitals.Thetemperature dependenceofthesplittingisduetothermalvariationofth eelectronandphonon occupation.Furthermore,[ 59 ]alsoshowsthatthesplittingincreaseslinearlywiththe phononcouplingconstant ,whichisanotherpieceofevidenceforthesourceofthe splitting. InChapter 3 ,wereportessentiallyexactNRGresultsforthemodelstudi edin [ 59 60 ]thatextendintotheKondoregime T T k ofgreatesttheoreticalinterest. 24

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1.4KondoPhysicswithDispersiveBosonicBath Thereisasecondclassofquantumimpurityproblemswithbos onicdegreesof freedom,inwhichanimpuritycouplesnottoadiscreteboson icmode,butinsteadto acontinuousbathofbosons.Suchproblemshavereceivedmuc hattentionassimple modelsforquantumphasetransitions(QPTs):phasetransit ionsatzerotemperature accessedbyvaryinganonthermalparametersuchaspressure ormagneticeld.The archetypalmodelinthisclassisthespin-bosonmodel,whic hhasbeenusedtodescribe avarietyofquantumdissipativesystems[ 61 ],includingfrictionaleffectsonbiological andchemicalreactionrates[ 62 ],coldatomsinaopticaltrap[ 63 ],andtheentanglement betweenaqubitanditsdissipativeenvironment[ 64 65 ].ItsHamiltonianis H = 2 x + 2 z + X i i a y i a i + z 2 X i i ( a i + a y i ), (1–8) where x and z arePaulimatrices, describestheenergyscaleoftunnelingbetween spinupandspindownstates, a y i isthecreationoperatorforaharmonicoscillator withenergy i ,and i describesthecouplingstrengthbetweentheimpurityspina nd oscillator i .Thebathofoscillatorsischaracterizedbyitsspectralfu nction,whichis usuallyassumedtotaketheidealizedpower-lawform B ( )= X i 2i ( ! i )=2 1 s c s ,0 1 .Thespin-bosonHamiltonianfeaturescompetitionbetween tunneling(the rstterm)anddissipation(thelastterm).FortheOhmiccas e s =1 aKosterlitz-Thouless QPTseparateslocalizedanddelocalizedphases[ 61 ].Forthesub-Ohmiccases 0 < s < 1 therehasbeenalong-standingdebateovertheexistenceofQ PTs.However, inrecentyears,theNRGtechnique[ 66 ]aswellasseveralothermethods[ 67 68 ]have providedclearevidencefortheexistenceofacontinuousQP Tinthespin-bosonmodel withasub-Ohmicbosonicbath. 25

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Figure1-8.Field-inducedquantumphasetransitioninYbRh 2 Si 2 ,revealedbyacolor mapof = d ln = d ln T (with beingtheelectricalresistivity)asafunction oftemperature T andmagneticeld H .Intheblueregions,Fermi-liquid behavior( =2 )isobservedwhiletheorangeregionrepresentsnon-Fermi liquidbehaviorwith =1 .ReprintedwithpermissionfromP.Gegenwart et al .,Nat.Phys. 4 ,186(2008).Copyright2008bytheNaturePublishing Group. Anotherinterestingquantumimpurityproblemthatinvolve sbothafermionic conductionbandandabosonicbathistheBose-FermiKondo(B FK)model.TheBFK modelhasreceivedconsiderableattentioninconnectionwi thanomalouspropertiesof certainheavy-fermionmaterials.Oneofthemostinteresti ngtopicsincondensedmatter physicsisthelow-temperatureexcitationsofinteracting fermions.Landau'sFermi-liquid theoryproposesthatinteractingfermionscanbedescribed byweakly-interacting quasiparticleswiththesamechargeandspinasthenoninter actingparticles[ 69 ].This theorysucceedsinexplainingthetemperature-dependentr esistivity,specicheatand magneticsusceptibilityinavarietyofinteractingsystem s,rangingfromliquidhelium-3 toso-called“heavy-fermion”systemslikeCeCu 6 wherestronginteractionbetween 26

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Figure1-9.Thetemperature-versus-magnetic-eldphased iagramforYbRh 2 Si 2 .The broadredlinespeciesthepositionofacrossoverintheiso thermalHall resistivityandmarksanadditionallow-energyscalechara cterizingKondo correlation.ReprintedwithpermissionfromP.Gegenwart etal .,Nat.Phys. 4 ,186(2008).Copyright2008bytheNaturePublishingGroup. localized 4 f or 5 f electronsproducesaquasiparticleeffectivemassintheco nduction bandthatisahundredormoretimesthebareelectronmass.Si ncethe1990s,Laudau Fermi-liquidtheoryhasbeenchallengedbyaseriesofexper imentsclosetocontinuous magneticQPTsinheavy-fermionmaterials[ 70 – 73 ].Theseparticularphasetransitions arepuzzlingbecausetheydonottintotheGinzburg-Landau -Wilsonpictureofcritical long-wavelengthuctuationsofanorderparameterthatdis tinguishesthetwophases [ 74 ].Asanexample,Fig. 1-8 showsthephasediagramofYbRh 2 Si 2 [ 75 ]deduced fromthetemperatureexponentoftheelectricalresistivit y, = d ln = d ln T [ 76 ].The nonthermalcontrolparameterinthisexampleistheapplied magneticeldplottedon thehorizontalaxis.Theblueareasinthisgurecorrespond toanti-ferromegnetic(AF) regimewith T < T N at H < H c andaparamagneticFermi-liquid(FL)regimewith T < T FL at H > H c .AFermi-liquidresistivityexponent =2 isobservedinboth regimes.Theorangeareaisanon-Fermiliquidregimewith =1 thatisanchoredbya quantumcriticalpoint(QCP)at T =0 and H = H c 27

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Figure1-10.ChangesofFermisurfacepropertiesacrossaQC PinCeRhIn 5 .The pressuredependenceofthe(a)dHvAfrequencyand(b)cyclot ronmass. ReprintedwithpermissionfromP.Gegenwart etal .,Nat.Phys. 4 ,186 (2008).Copyright2008bytheNaturePublishingGroup. ThemaincauseofmagneticQPTsinheavy-fermionmaterialsi sacceptedto bethecompetitionbetweenKondocorrelation,whichattemp tstoscreenthelocal momentoneach f site,andtheinter-siteRKKYinteraction,whichtendstoal ign thoselocalmoments[ 78 ].OneclassofinterestingunconventionalQPTsisso-calle d Kondo-destructionQPTs,inwhichKondosingletsexistonly intheparamagneticphase, andtheKondoeffectdissappearspreciselyattheQCPwithth eonsetofmagneticorder. Figure 1-9 showsthattheKondoeffectonlyexistsintheFLphaseofYbRh 2 Si 2 ,and theKondoenergyscale T vanishesattheQCPseparatingtheAFandFLphases[ 75 ]. Figure 1-10 showsexperimentalevidence[ 77 ]foranotherQCP,thisoneinCeRhIn 5 ThedeHaas-vanAlphandatareplotted[ 75 ]inFig. 1-10 (a)indicateapotentialjumpof theFermisurfacevolumeatapressurecoincidingwiththecr iticalpressure p c 2.3 GPa implyingaKondo-destructiontypeofQCP.Thedivergenceof thecyclotronmassatthe criticalpressureshowninFig. 1-10 (b)alsosupportsthisconclusion. 28

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Theoretically,thecompetitionbetweenKondocorrelation andRKKYinteractionis capturedintheKondolatticemodel,whichisdescribedbyHa miltonian H KL = X ij t ij c y i c j + J X i ^ S i ^ s c i + X ij I ij ^ S i ^ S j (1–10) InEq.( 1–10 ), c y i createsanelectronwithspin atsite i ^ S i and ^ s c i arethespin operatorsforthelocalmomentandforconductionelectrons onsite i ,respectively, t ij isthetight-bindinghoppingparameter, J characterizestheKondocouplingand I ij is theRKKYinteractionstrengthbetweensites i and j .Atheoryof“locallycriticalQPTs” [ 79 80 ]hasbeensuccessfulinexplaininganumberoftheexperimen talphenomena contradictingtheconventionaltheoryofQPTsinheavy-fer mionmaterialsand,especially indescribingtheKondo-destructiontypeofQPT.Thebasici deaunderlyinglocal criticalityisthatthelocaldegreesoffreedomrelatedtoK ondocorrelationbecome criticalattheQPTsimultaneouslywiththeconventionallo ng-rangeorder-parameter uctuations.Theconceptoflocalcriticality[ 79 80 ]hasbeendevelopedwithinthe frameworkofanextendeddynamicmean-eldtheory(EDMFT)[ 81 82 ],whichmapsthe KondolatticemodelintoaBFKmodel,theHamiltonianofwhic his H BFKM = X k k c y k c k + X q q ^ yq ^ q + J k ^ S ^ s c + ^ S X q g q ( ^ q + ^ y-q ). (1–11) Here,alocalizedspin ^ S representingasinglesitespinintheKondolatticeiscoupl ed totheon-siteconduction-bandspin ^ s c aswellastoathree-componentbosonicbath describedby ^ q .Thisbosonicbathcapturesmagneticuctuationsgenerate dvia theRKKYinteractionbyalltheotherlatticesites.Theeffe ctoftheconductionband andthebosonicbathontheimpuritysitearefullydescribed bythedensityofstates ( )= P k ( k ) andthespectralfunction B ( )= P q g 2 q ( ! q ) ,respectively. WithinEDMFT, ( ) and B ( ) mustsatisfyself-consistencyequationstoensurethatthe chosensiteisrepresentativeofthelatticeasawhole.Varo usmethodshavebeenused tosolvetheBFKM,including expansion[ 80 ],quantumMonteCarlo[ 83 84 ],andthe 29

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Figure1-11.Thepressure-eldphasediagraminCeRhIn 5 atconstanttemperature T =0.5 K .Thedashedlinebetween P 1 and P 2 indicatesthephase transitionbetweentwosuperconductingphases,thlower-p ressurephase beingantiferromagneticandthehigher-pressurephasebei ngmagnetically disordered.FromQ.Si etal .,Science 329 ,1161(2010).Reprintedwith permissionfromAAAS. numericalrenormalizationgroup[ 85 86 ].MoredetailsoftheNRGtreatmentoftheBFK modelwillbegiveninSection 2.3 Heavy-fermionsuperconductorsareamongthemostinterest ingstronglycorrelated electronsystemsduetotheevidencenon-phononicpairingm echanisms.Oneofthe possiblepairingmechanismsisnearlycriticalvalenceuc tuation,whichisevidenced inbothCeCu 2 Ge 2 [ 87 ]andCeCu 2 Si 2 [ 88 ].Variousexperimentalresultshaveshown therelationshipbetweenquantumcriticalityandsupercon ductivityinheavy-fermion superconductors.Superconductivitymayoccurfromthepro ximitytoaeld-induced QCP,suchasinCeRhIn 5 [ 89 ],CeCoIn 5 [ 90 ]andUBe 13 [ 91 ].Figure 1-11 showsthe pressure-eldphasediagraminCeRhIn 5 (datafrom[ 92 ],replottedin[ 93 ]).Atzeroeld B =0 ,transitionbetweenanantiferromagneticphase(MO)andan on-magneticphase takesplaceat P = P 1 ,withsuperconductivitypresentonbothsidesoftheQCP.Wi th increaseof B ,theQCPshiftstohigherpressureuntil P = P 2 wheresuperconductivity 30

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issuppressedbythestrongmagneticeld.Thisisanexample showingtheexistenceof superconductivityaroundtheQCPinaheavy-fermionsystem Chapter 4 presentsnumericalresultsforatwo-impurityBose-FermiA nderson modelwhichallowsstudyofsuperconductingpairinginthep resenceofKondo correlationandlatticemagnetism. 31

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CHAPTER2 NUMERICALRENORMALIZATIONGROUPMETHODANDITSAPPLICATIO NTO QUANTUMIMPURITYMODELS Inthischapter,thenumericalrenormalizationgroup(NRG) methodisintroduced andtwotypicalquantumimpuritymodelsaretakenasexample stodescribeitsbasic application.Section 2.1 providesageneraloverviewofthemethod.Amoredetailed descriptionoftheNRGtreatmentofthepure-fermionicAnde rsonmodelisprovided inSection 2.2 .Section 2.3 usestheBose-FermiKondo(BFK)modeltoillustratethe extensionoftheNRGtosolveofproblemswithbothaconducti onbandandadispersive bosonicbath. 2.1OverviewoftheNumericalRenormalizationGroupMethod Thenumericalrenormalizationgroupmethodwasintroduced byK.G.Wilson in1975basedonrenormalization-groupideasfromeldtheo ry[ 7 ].Themethodis non-perturbativeinthestrengthofelectron-electronint eractionsandhasprovedtobe anexcellenttechniqueforsolvingquantumimpuritymodels describingasubsystem withonlyanitenumberofdegreesoffreedom(theimpurity) coupledtoacontinuous environment[ 94 ].TheKondoproblem,describingexchangeinteractionbetw eena localizedspinandconductionelectrons,istheprototypic alexampleofsuchamodel. ThedevelopmentoftheNRGmethodwasamilestoneinthestudy oftheKondomodel, inwhichconventionalperturbativemethodsforsolvingman y-bodyproblemsfaildueto theappearanceoflow-energy(infrared)divergences. TheNRGmethodcanbedividedintothreemainsteps:logarith micdiscretizationof theconductionband,mappingontoasemi-innitechain,and iterativesolution[ 94 ]. (1)Discretization:Theconductionbandcoveringenergies spanningarange D << D isdividedintologarithmicintervalssuchthatthe n th positive-or negative-energyintervalcovers D ( n +1) < j j < D n (2–1) 32

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Here, n =0,1,2... ,and ismeasuredfromtheFermienergy F =0 .TheWilson discretizationparameter > 1 introducesanarticialseparationofenergyscales betweenadjacentintervals.Thisscaleseparationiswhata llowsiterativesolutionofthe mappedproblem,asdescribedinstep(3). (2)Mapping:Justonelinearcombinationofalltheconducti on-bandstates couplestotheimpuritysite.Let f 0 destroyanelectronofspin inthiscombination ofstates.Startingwith f 0 ,onecanusetheLanczosmethod[ 95 ]tomapthediscretized conduction-bandpartoftheHamiltonianontoasemi-innit etight-bandingchain: H band = D 1 X n =0 [ n f y n f n + n ( f y n f n 1, + f y n 1, f n )] (2–2) with f y n creatinganelectronofspin onthe n th siteofthischain.Sites n =1,2,3... ofthisctitiouschainrepresentlinearcombinationsofco nduction-bandstatesthatcan bereachedfromthe n =0 combinationonlyby n ormoreapplicationsof H band .The parameters n and n characterizetheon-siteenergyandthehoppingenergybetw een twoconsecutivesitesinthechain,andfor n 1 ,satisfy n 0, n 1 2 (1+ 1 ) n = 2 (2–3) whichfallsoffexponentiallywithdistancealongthechain .ThefullHamiltonianaddsto H band termscouplingsite 0 ofthechaintopurelylocalimpuritydegreesoffreedom. (3)Iteration:ThefullHamiltoniancanbesolvediterative lybydiagonalizingaseries ofscaled,dimensionlessHamiltonians H N describingnitechains 0 n N for N =0,1,2,... .Basedonthesemi-innitechaininEq.( 2–2 ), H N canbewritteniteratively as H N +1 = 1 = 2 H N + N X ( f y N f N +1, + f y N +1, f N ) (2–4) with N 1 forlarge N .Becauseeachsiteofthechainhas4possiblestates,the dimensionoftheFockspaceoftheHamiltonian H N increasesexponentiallywith N .Therefore,itsoonbecomesimpossibletokeepalltheeigen statesaftereach 33

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diagonalization.Onlythelowest N s many-bodyenergylevelsarekeepforthenext iteration.Thecutoffonthenumberofkeptstates N s makesitpossibletosolvea quantumimpurityproblemhavinganinnitenumberofdegree soffreedom.Staticand dynamicpropertiesofthesystemcanbecalculatedusingthe many-bodyeigenstates fromthediagonalization.TheNRGmethodprovidesanaccura teaccountofthe equilibriumpropertiesateveryenergyortemperaturescal e. FollowingtheinitialsuccessoftheNRGmethodinsolvingth eKondomodel[ 7 ], ithasbeenextendedtotreatmanyotherquantumimpuritypro blems.Thetechnique hasbeensuccessfullyappliedtothesingle-impurityAnder sonmodelbyincluding chargeuctuationontheimpuritysite[ 8 9 ];thetwo-channelKondomodelwithimpurity spinscoupledtotwoconductionbands[ 96 ];thetwo-impurityKondomodelfeaturing competitionbetweenKondoscreeningandmagneticcorrelat ion[ 97 – 105 ];quantum impuritiescoupledtoabosonicbath[ 66 106 ];andtheBose-FermiKondomodelwithan impuritycoupledtobothaconductionbandandabosonicbath [ 110 111 ]. 2.2ApplicationoftheNRGtotheSingle-ImpurityAndersonM odel Inthissection,thesingle-impurityAndersonmodelistake nasanexampleto illustratehowtheNRGtechniqueworksforpure-fermionicp roblems.Thenotation followsthepioneeringpapers[ 8 9 ]. 2.2.1Single-ImpurityAnderonModel TheAndersonHamiltonian[ 6 ]canbewrittenas H = H imp + H band + H mix (2–5) with H imp = d X n d + Un d n d # (2–6) H band = X k k c y k c k (2–7) 34

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H mix = 1 p N k X k ( V k d y c k + V k c y k d ), (2–8) where c y k createsaconduction-bandelectronwithspin ,wavevector k ,andenergy k ; d y isthecreationoperatorforanimpurityelectronwithspin andenergy d ,and n d = d y d ; U istheCoulombrepulsionbetweentwoelectronsintheimpuri tylevel;and V k is thematrixelementforhybridizationbetweentheimpuritya ndtheconductionband.To simplifytheAndersonmodel,theconductionbandisassumed toextendsymmetrically from D to D aroundtheFermienergy F =0 andtobespatiallyisotropic,whichmeans that k onlydependson j k j .Moreover,thehybridizationmatrixelement V k istakento beaconstantV.Thedensityofstatesoftheconductionband ( )= 0 =1 = 2 D is assumedtobeaconstantanddenethehybridizationwidth = 0 V 2 .Asaresult,the single-impurityAndersonmodelcanbewritteninone-dimen sionalform: H D = X Z 1 1 ka y k a k dk + d D + U 2 D X d y d + U 2 D X d y d 1 2 + D 1 2 X Z 1 1 dk ( d y a k + a y k d ), (2–9) where k = k = D 2 [ 1,1] and a y k isthecreationoperatorforanelectronwithenergy k andspin .ThismodiedHamiltonianisfullydescribedbythreedimen sionless parameters: d = D U = D and = D 2.2.2LogarithmicDiscretization TherststepintheNRGtreatmentofEq.( 2–9 )istologarithmicallydiscretizethe conductionband.Adiscretizationparameter > 1 isintroducedtodividethe k space [ 1,1] intoaseriesofintervals.The n th intervalspans [ ( n +1) n ] forpositive k and [ n ( n +1) ] fornegative k ( n =0,1,2,... ). Acompletesetoforthonormalfunctionsineachoftheseinte rvalsisdenedby np ( k )= 8><>: n = 2 (1 1 ) 1 = 2 e i n pk if n 1 < k < n 0 otherwise (2–10) 35

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with p = anyintegerandthefundamentalFourierfrequencyinthe n th intervalbeing n = 2 n 1 1 (2–11) Theoperator a k canbeexpandedinthisbasisas a k = X n p [ a np + np ( k )+ b np np ( k )]. (2–12) Theimpuritycouplesonlytothe p =0 modeineachinterval.Wilsonshowed[ 7 ]thatthe couplingofthe p 6 =0 modeswithin H band vanishesinthecontinuumlimit 1 ,andis smallfor > 1 .Thisallowsanimportantsimplicationofdroppingallmod eswith p 6 =0 Theannihilationoperatorsforthe p =0 modesare(simplifyingthenotationfrom a n 0 and b n 0 to a n and b n ) a n = n = 2 (1 1 ) 1 = 2 Z n ( n +1) a k dk (2–13) b n = n = 2 (1 1 ) 1 = 2 Z ( n +1) n b k dk (2–14) TheresultingHamiltonianis H D = 1 2 (1+ 1 ) X 1 X n =0 n ( a y n a n b y n b n )+ d D + U 2 D X d y d + U 2 D X ( d y d 1) 2 + 2 D 1 = 2 X ( f y 0 d + d y f 0 ) (2–15) where f 0 = h 1 2 (1 1 ) i 1 = 2 1 X n =0 n = 2 ( a n + b n ) (2–16) annihilatesanelectronofspin inthelinearcombinationofbandstatesthatcouples totheimpuritylevel.Afterthebanddiscretization,theAn dersonmodelhasbeen transformedfromthecontinuumHamiltonianEq.( 2–9 )toadiscreteHamiltonianEq. ( 2–15 ),whichisreadyforrecastingintosemi-innitechainform .Allpossibleenergies between 1 and 1 arenowreplacedbyasetofdiscretesampleenergies n TheclosertotheFermienergyonegets,thedenserthesample energiesbecome. 36

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Low-temperaturepropertiesofAndersonmodelaredetermin edbythestatesaroundthe Fermienergy,whicharewellsampled.2.2.3MappingtoaSemi-InniteChain InEq.( 2–15 ),theimpuritysiteonlycouplestotheoperator f 0 ,andthestatesof conductionbandarerepresentedbytheoperators a n and b n .Togetaformwhichis convenientforiterativesolution,onecanusetheLanczosp rocedure[ 95 ]tomap H band to atight-bandingchainwithonlynearest-neighbourhopping .TheHamiltonianbecomes H D = 1 2 (1+ 1 ) X 1 X n =0 n = 2 n ( f y n f n +1, + f y n +1, f n )+ d D + U 2 D X d y d + U 2 D X ( d y d 1) 2 + 2 D 1 = 2 X ( f y 0 d + d y f 0 ), (2–17) with n = (1 n 1 ) p (1 2 n 1 )(1 2 n 3 ) n 1 1. (2–18) Thegeneralruletoconstruct f n isthat f n involvesonlycombinationsof a m b m whennisoddwhile f n involvesonlycombinationsof a m + b m whenniseven.Noting thatthestatedestroyedby f n containsequalweightofpositive-energy( a m )and negative-energy( b m )electronoperators,ithaszeromeanenergy,andforthisre ason thereisnodiagonalterm f y n f n inEq.( 2–17 ). 2.2.4IterativeSolution BasedonEq.( 2–17 ),asequenceofHamiltonians H N isdenedas H N = ( N 1) = 2 h X N 1 X n =0 n = 2 n ( f y n f n +1, + f y n +1, f n )+ ~ d X d y d + ~ U X d y d 1 2 + ~ 1 = 2 X ( f y 0 d + d y f 0 ) i (2–19) where ~ d =(2 d + U ) = ( D (1+ 1 )) ~ U = U = ( D (1+ 1 )) and ~ =8 = ( D (1+ 1 ) 2 ) H N onlycontainstherst N +1 sites(indexed0through N )ofthesemi-innitechain. 37

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TheHamiltonian H N satisestherecursiverelation H N +1 = 1 = 2 H N + N X ( f y N f N +1, + f y N +1, f N ), (2–20) withthecoefcient N approaching1forlargevaluesofN.Asaresult,theHamilton ian inEq.( 2–17 )canberecoveredas H D =lim N !1 1 2 (1+ 1 ) ( N 1) = 2 H N (2–21) Eq.( 2–20 )servesasthebasicequationfortheiterativesolutionoft hesingle-impurity Andersonmodel.ThestartingpointistheHamiltonian H 0 describingtheimpurityplus conductionelectronslocalizedattheimpuritysite: H 0 = 1 = 2 h ~ d X d y d + ~ U X d y d 1 2 + ~ 1 = 2 X ( f y 0 d + d y f 0 ) i (2–22) whichcanbediagonalizedstraightforwardly.Thegenerali deaoftheiterationisthat oncethemany-bodyeigenstatesandeigenvaluesof H N areobtained,theFockspace spannedby H N +1 canbeconstructedasthecross-productofFockspacesspann edby H N andbytheoperators f N +1, ( = # ).Formingmatrixelementsof H N +1 inthisbasis allowseigenstatestobefoundviadiagonalization,whicha reusedinthenextiteration N +2 ,andsoon.Theiterationprocedurecanberepresentedas H N +1 = R [ H N ] [ 7 ], whichtransformsonesetofeigenvaluesandeigenstatesofH amiltonian H N intoone updatedsetofeigensolutionsofHamiltonian H N +1 withthesameformbutnewvalues. Theiterationprocedurecontinuesuntilitreachesaninvar iant“xedpoint”.Becauseof theeven-oddalternationpropertiesofnite-lengthfermi onicchains,axedpoint H is denedbythecondition R 2 [ H ]= H Thediagonalizationofmatrix H N canbespeededupbytakingthesymmetriesof Hamiltonianintoconsideration.TheHamiltonianinEq.( 2–19 )conservestotalspinand 38

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charge.Thespinoperatorisdenedas ^ S N = 1 2 X 0 d y ^ 0 d 0 + 1 2 N X n =0, 0 f y n ^ 0 f n 0 (2–23) where ^ 0 isavectorofPaulimatrices,andthechargeoperatorisden edas Q N = X n d 1+ N X n =0 ( X f y n f n 1). (2–24) Itiseasytoverifythat H N commuteswithboth ^ S N and Q N .Inthiscase,( Q S S z )isaset ofgoodquantumnumbersfortheHamiltonian H N .Therefore,thematrixelementsof H N betweenstateswithdifferentquantumnumbersshouldbezer o.Inotherwords,thefull Fockspacespannedby H N canbedividedintoaseriesofsubspacesthatareindexed bythesetofgoodquantumnumbers( Q S S z ).Asaresult,thefullmatrixrepresentation of H N canbedividedintoaseriesofsmallblockswithmuchsmaller dimension.This propertyofthematrixof H N isveryimportantinthecalculation.TheCPUtimeto diagonalizeadensematrixwithdimension M is O ( M 3 ) ,sothediagonalizationof smallblocks,whichrepresentdifferentsubspaces,savesa lotoftimecomparedto diagonalizingthefullmatrixdirectly. IntheAndersonmodel,thedimensionoftheFockspaceis 4 N +2 .Inviewofthis exponentialgrowthwith N ,afterafewiterationsitbecomesimpossibletoretainallt he many-bodyeigenstatesafterdiagonalizationof H N .Instead,oneretainsonlyapreset number N s ofeigenstates,namely,thoseoflowestenergies.Thevalue sof N s andthe Wilsondiscretizationparametermustbechoseninatrade-o ffbetweendiscretization error(comingfrom > 1 )andcutofferror(comingfromusing N s < 1 ).Thecloser is toitscontinuumvalue =1 ,thehigherthevalueof N s thatmustbeusedtoreasonably approximatethephysicalpropertiesbythermalaveragingo verretainedmany-body states.FortheAndersonmodel[Eq.( 2–5 )],thechoices =3 N s =1000 typically provideagoodcompromise. 39

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2.3ApplicationoftheNRGtotheBose-FermiKondoModel Inthissection,Bose-FermiKondo(BFK)modelistakenasane xampletoillustrate theapplicationoftheNRGmethodinproblemswithbothaferm ionicconductionband andabosonicbath.2.3.1Bose-FermiKondoModel TheBFKmodelwasoriginallyproposedinthediscussionofat wo-bandextended Hubbardmodelwithintheframeworkofextendeddynamicalme an-eldtheory(EDMFT) [ 107 ].TheHamiltonianfortheIsing-symmetryversionoftheBFK modeliswrittenas H = H F + H B + H int (2–25) with H F = X k k c y k c k (2–26) and H B = X q q yq q (2–27) where c y k createsaconduction-bandelectronwithwavevector k ,spin ,andenergy k and yq createsabosonwithwavevector q inthebosonicbath.Theinteractingpartof theHamiltonianis H int = 1 2 J ^ S X kk 0 0 c y k ^ 0 c k 0 0 + S z X q g q ( q + y q ), (2–28) Thersttermdescribestheisotropicexchangeinteraction betweentheimpurityspin ^ S andthefermionicbathviaKondocoupling J ;thesecondtermdescribesthecouplingof the z -componentoftheimpurityspintooscillator q inthebosonicbathwithmagnitude g q Forsimplicity,theconductionbanddensityofstatesistak entobea“tophat' function, ( )= 1 N k X k ( k )= 0 ( D j j ) (2–29) 40

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with ( ) beingtheHeavisidestepfunction.Thebosonicbathisassum edtohavea power-lawspectralfunction B ( )= X q j g q j 2 ( ! q )= 8><>: B 0 1 s 0 s for 0
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Figure2-1.SchematicsummaryoftheNRGtreatmentofBFKMod elHamiltonian[ 111 ]. Circles[squares]indicatesitesofthefermionic[bosonic ]chain.Onlytheend siteofeachchainconnectstotheimpuritysite.Thecouplin gbetween consecutivesitesinthefermionic[bosonic]chainis n [ t n ],decayingas n = 2 [ n ].Thedashedboxes(frominnermosttooutermost)indicatet hezeroth, rst,andseconditerationoftheNRG.Reprintedwithpermis sionfromM.T. Glossop,andK.Ingersent,Phys.Rev.B 75 ,104410(2007).Copyright2007 bytheAmericanPhysicalSociety. inthefermionic[bosonic]chain.Forthefermionicchain, n and n decreaseas n = 2 when n islarge.Forthebosonicchain,however, e n and t n decreasemuchfasteras n withtheincreaseofsiteindex n .Thisdifferencecanbetracedbacktothefactthat theconduction-banddensityofstateshasweightonbothsid esof =0 ,whereasthe bosonicbathhasweightonlyfor !> 0 .ThespiritoftheNRGisthatfermionsand bosonsthathavetheenergyscaleshouldbetreatedatthesam eiteration.Asaresult, thebosonicchainshouldbeextendedbyonesiteonlyatevery secondadditionto thefermionicchain.AschematicsummaryoftheNRGtreatmen toftheBFKmodelis showninFig. 2-1 .Fortherenormalizationgrouptransformation H N +1 = R [ H N ] ,both fermionicandbosonicchainsareextendedbyaddingonesite if N +1 isevenwhileonly afermionicsiteisaddedif N +1 isodd. Assumetheeigenstatesof H N are j r N i withenergy E ( r N ) ,where r indexesthe differentretainedstates.Thebasisstatesof H N +1 canbeconstructedas j l r N i = j r N inj l i N +1 (2–33) 42

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with j l i N +1 beingthestatesofaddedsites.Forodd N +1 ,onlythefermionicchainis extended,and j l i N +1 = j i i N +1 with j i i N +1 beingoneofthestates fj 0 i j"i N +1 j#i N +1 j"#i N +1 g ,thepossiblestatesforanaddedfermionicsite.Foreven N +1 ,bothchainsare extended,and j l i N +1 = j i i N +1 nj n i ( N +1) = 2 with j n i ( N +1) = 2 beinganumbereigenstateof b y ( N +1) = 2 b ( N +1) = 2 witheigenvalue n =0,1,2,... .Sincethebasisofbosonicsite ( N +1) = 2 isofinnitedimension,itisnecessarytotruncatethebasi sat n N b .Thematrix elementsof H N +1 h l 0 r 0 N j H N +1 j l r N i ,canbecalculatedintermsoftheeigenstates andeigenvaluesof H N aswellasthematrixelements h i 0 j f N +1 j i i and h n 0 j b ( N +1) = 2 j n i Theneigenstates j r N +1 i andeigenvalues E ( r N +1) of H N +1 canevaluatedand recordedforthenextiteration. ThefactorfortheexpansionoftheFockspacedimensionfrom H N to H N +1 is 4( N b +1) when N +1 iseven,whichiscomputationallymoredemandingthanthe factorof4when N +1 isodd,orforany N inapure-fermionicproblem.Similartothe AndersonmodeldiscussedinSection 2.2.4 ,onecantakeadvantageofHamiltonian symmetriestodividethefullFockspaceintoasetofsubspac esandtherebyaccelerate thediagonalizationof H N ateachiteration. 43

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CHAPTER3 RESULTSFORATWO-ORBITALMOLECULARJUNCTIONMODEL Thischapterisbasedonapublishedpaperandonanunpublish edmanuscript. Allthepublishedcontentsarereprintedwithpermissiongr antedunderthecopyright policyoftheAmericanPhysicalSocietyfromGiseleIorioLu iz,EdsonVernek,LiliDeng, KevinIngersent,andEnriqueAnda,Phys.Rev.B 87 ,075408(2013).Theunpublished manuscriptbyLiliDeng,KevinIngersent,GiseleIorioLuiz ,EdsonVernek,andEnrique AndaiscurrentlyinpreparationforsubmissiontoPhys.Rev .B. 3.1Introduction Inthischapter,atwo-orbitalmolecularjunctionisstudie dusingthenumerical renormalizationgroupmethod,withparticularfocusonthe interplaybetweenelectron-electron ( e e )andelectron-phonon( e ph )interactionswithinthemolecule.Thismodelmayalso beusedtodescribetwo-levelquantumdotsoracoupledpairo fsingle-leveldots. Werstreportabasicinvestigationofthegeneralcongura tionofthismodelwith bothphonon-assistedinterorbitaltunnelingandaHolstei n-typecouplingbetweenthe molecularchargeandthedisplacementofthelocalphononmo de.Whenthemolecule isisolatedfromtheleads,alevel-crossingisfoundfromsi ngleelectronoccupancyto doubleoccupancyunderincreaseofthe e ph coupling.Whentunnelingtotheleads isallowed,thelevelcrossingisassociatedwithasmoothcr ossoverbetweenKondo andphonon-dominatedregimesthathassignaturesinthermo dynamicpropertiesand inchargetrransportthroughthesystem.TheKondoregimeis characterizedbyan enhancementoftheelectricalconductancethroughthejunc tionandalowphonon occupation,whilethephonon-dominatedregimeismarkedby thesuppressionofthe zero-biasconductanceandadramaticincreaseinthephonon occupation. Thenweconsideraspecicregimeofparametersinwhichoneo fthemolecular orbitals(the“sideorbital”)couplestotherestofthesyst emonlythroughphonon emission/absorptionprocesses.Therearetworeasonsforp articularinterestinthis 44

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specialcase.First,inthiscongurationthecrossoverdes cribedabovebecomesa rst-orderquantumphasetransition(QPT)atacritical e ph couplingstrength,wherethe systemundergoesadiscontinuouschangefromaKondophaset oaphonon-dominated phaseinwhichtheKondoeffectisfullysuppressed.Theorig inofthisQPTisthe emergenceofanewsymmetryduetotheabsenceofdirectelect rontunnelingbetween thesideorbitalandtheleads.Thesecondreasonforstudyin gthiscongurationis thatitisquitesimilartoside-coupleddoublequantumdots ,whichhavebeenwidely investigated[ 112 – 115 ].Thetwo-orbitalmolecularjunctionHamiltonianstudied inthis chaptercanthereforegiveintuitionintothepropertiesof suchquantum-dotsystems. Inthenalpartofthischapter,wefocusonacrossoverbetwe enspinandcharge Kondoregimes.Bothphononeffects(phonon-assistedinter orbitaltunnelingand Holstein-typephononcouplingtothemolecularcharge)con tributetotherenormalization oftheCoulombrepulsionbetweenelectronsinthemoleculea ndcanleadtheeffective e e interactiontobecomeattractive.Intheone-levelAnderso n-Holsteinmodelan effective e e attractionhasbeenshowntoproduceachargeanalogoftheKo ndoeffect [ 44 49 ].TheemergenceofachargeKondoeffectinthetwo-orbitalm odelisdiscussed. Thischapterisarrangedasfollows:Section 3.2 introducesthemodelHamiltonian aswellasitspreliminaryanalysisviaaLang-Firsovtransf ormation.Section 3.3 reports analyticalandNRGresultsforafairlygeneralconguratio n.Section 3.4 presents theinvestigationoftheside-orbitalconguration.Secti on 3.5 addressestheinterplay betweenthetwophononeffectsandtheemergenceofachargeK ondoeffect.A summaryisgiveninSection 3.6 3.2ModelHamiltonianandLang-FirsovAnalysis 3.2.1ModelHamiltonian ThegeneralformoftheHamiltoniandescribingourtwo-orbi talmolecularjunctionis H = H mol + H leads + H mol-leads (3–1) 45

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where H mol describestheisolatedtwo-orbitalmolecule, H leads describestheexternal electricalleads,and H mol-leads describescouplingbetweenthemoleculeandtheleads. AschematicrepresentationofthismodelisshowninFig. 3-1 .Thersttermofthe Hamiltonian( 3–1 )canbedecomposedas H mol = H 0 + H ph + H 1 + H 2 ,inwhich H 0 = X j = j n j + X j = U j n j n j # + U 0 n n + t 0 X ( d y d + d y d ) (3–2) describesthepurelyelectronicdegreesoffreedomoftheis olatedmoleculewithorbitals labeled and ,and H ph = 0 b y b (3–3) describesthelocalphononmodeinthemolecule. H 1 = 0 X ( d y d + d y d )( b y + b ) (3–4) and H 2 = ( n + n n c )( b y + b ) (3–5) describe,respectively,aphonon-assistedinter-orbital tunnelingwithcouplingstrength 0 andaHolstein-typephononcouplingtothecombinedchargei nbothorbitalsinthe moleculewithcouplingstrength .InEqs.( 3–2 )-( 3–5 ), n j = P d y j d j ( j = ) is thechargeoperatoroftheorbital j ,with d y j beingthecreationoperatorforanelectron withspin andenergy j relativetothecommonFermileveloftheleads.Withoutloss ofgenerality,wetake U j and U 0 aretheintra-orbitalandinter-orbitalCoulomb repulsion,respectively, t 0 isadirecttunnelingbetweenthetwoorbitals,and n c canbe thoughtofasxingthereferencechargeofthemoleculeandc anbeformallyeliminated byshiftingthebosonicmodeaccordingto ^b = b n c (withresultingshiftsin ,and t 0 ).Here b y isthecreationoperatorforaphononwithenergy ~ 0 ,where ~ = h = 2 isthe reducedPlanckconstant.Tosimplifyournotation,wehence forthset ~ =1 .Thesecond 46

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Figure3-1.Schematicplotofthecongurationofthetwo-or bitalmoleculejunction. Phonon-assistedinterorbitaltunneling( 0 )andaHolstein-typephonon coupling( )arebothconsidered.Bothorbitalsaredirectlyconnected tothe leadswith V = V = V (hence, = = ). termofHamiltonian( 3–1 )canbewrittenas H leads = X ` k ` k c y ` k c ` k (3–6) where c y ` k createsanelectronwithwavevector k ,spin ,andenergy ` k inthelead ` ( ` = R L standingforrightorleftlead).Finally,thelasttermofEq .( 3–1 )reads H mol-leads = 1 p N k X j = X ` k ( V j ` k c y ` k d j + V j ` k d y j c ` k ), (3–7) where V j ` k isthehoppingbetweenthemoleculeandtheleadsand N k isthenumberof latticesitesineachlead.Forsimplicity,wetake V j ` k = V j toberealandassume ` k = k Thenwedenetheevenandoddlinearcombinationofelectron operatorsinthetwo leads, c y e ( o ) k = 1 p 2 ( c y L k c y R k ), (3–8) 47

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sothatEqs.( 3–6 )and( 3–7 )become H leads = X k k ( c y e k c e k + c y o k c o k ) (3–9) and H mol-leads = r 2 N k X j = X k V j ( c y e k d j + d y j c e k ), (3–10) respectively.Molecularorbital j = couplesonlytotheevencombinationoflead stateswithaneffectivecouplingconstant p 2 V j .Thiscouplingallowsastodenethe totalhybridizationwidth j =2 0 V 2 j ofmolecularorbital j withtheleadstates,where 0 istheFermi-levelelectrondensityofstatesinonelead.We willconsidertheleadstobe characterizedbyaatdensityofstates, 0 =1 = 2 D ,where D isthehalf-bandwidth. 3.2.2ThermodynamicsandLinearConductance Thephysicalpropertiesofourmodelarestudiedbycalculat ingtransportand thermodynamicquantities,e.g.,thelinearconductancean dthemolecularelectron occupation.Withinthecanonicalensemble,anythermodyna micalquantitycanbe calculatedas h X i = 1 Z ( T ) X m h m j ^ X j m i e E m = k B T (3–11) where ^ X representstheoperatorassociatedtothephysicalquantit yofinterest,for instance, ^ X = d y j d j ,fortheelectronoccupationoforbital j or ^ X = b y b forthephonon occupation.Forsimplicity,Boltzmann'sconstant k B ishenceforthsettobe1. j m i isan NRGmany-bodyeigenstatewithenergy E m and Z ( T ) isthecanonicalpartitionfunction Z ( T )= X m e E m = T (3–12) Itisusefultodenethemolecularcontributiontoquantity X as X mol = X tot X host (3–13) 48

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where X tot ( X host )isthetotalvalueof X forasystemwith(without)themolecule.Inour problem,thelocalphononmodeistreatedaspartofthehosts ystem.Accordingly,we denethemolecularentropyas S mol ( T )= S tot ( T ) S leads ( T ) S ph ( T ), (3–14) where S tot isthetotalentropyofthesystem, S leads isthecontributionoftheleadswhen isolatedfromthemolecule,and S ph ( T ) istheentropyofthetruncatedlocal-phonon system,givenby S ph ( T )=ln Z ph ( T ) @ ln Z ph @ (1 = T ) (3–15) with Z ph ( T )= N b X n b =0 e n b 0 = T = 1 e 0 ( N b +1) = T 1 e 0 = T (3–16) Otherthermodynamicquantitiesofinterestarethetempera ture-dependentstatic spinsusceptibility mol ( T ) denedvia T mol ( T )= h S 2 z ih S z i 2 (3–17) andthecorrespondingchargesusceptibility c mol ( T ) denedvia T c mol ( T )= h n 2 mol ih n mol i 2 (3–18) where S z = S z + S z representsthe z -componentofthetotalspininthemoleculeand n mol = n + n isthetotalchargeoperatorofthemolecule. WithintheNRG,wealsocalculatethelinearconductance G ,whichinthe wide-bandlimitofthemetalliccontactsacquirestheform G = dI dV V =0 = e 2 h ( + ) Z 1 1 d @ f ( ) @! X A ( ), (3–19) 49

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where f ( ) istheFermi-Diracdistributionfunctionand A ( ) isthespectralfunctionfor thefermionicoperator[ 116 ] d = d cos + d sin (3–20) inwhich tan 2 = = .Anexplicitexpressionfor A ( T ) canbewrittenas A ( T )= 1 Z ( T ) X n m jh n j d y j m ij 2( e E m = T + e E n = T ) T ( ( E m E n )). (3–21) Theaboveequationproducesaseriesofdiscretedeltafunct ionsthatneedtobe broadenedtorecoveracontinuousspectralfunction,andwe employGaussian broadeningofeachdeltafunctiononalogarithmicscale[ 117 118 ]: ( j jj E j ) e b 2 = 4 p b j E j exp[ (ln j j ln j E j ) 2 b 2 ]. (3–22) Inourcasewherethereisnoappliedmagneticeld,thespect ralfunctionisspin-independent soweset A ( T )= A ( T ) .GreatsimplicationofEq.( 3–19 )isobtainedforthe zero-temperaturelimit,inwhich G = 2 e 2 h ( + ) A (0)= G 0 sin 2 h n mol i 2 (3–23) InEq.( 3–23 ), G 0 =2 e 2 = h isthequantumofconductanceandthesecondequality followsfromtheFriedelsumrule[ 116 ],akintothesingle-impurityAndersonmodel. Below,theFriedelsumrulewillnotbeemployedtocalculate theconductance,itwill ratherbeconrmedbyournumericalcalculation.3.2.3Lang-FirsovAnalysis Beforeembarkingonnumericalcalculation,weseektoident ifytheregioninthe high-dimensionalparameterspaceofthemodelwherethe e e and e ph interactions compete.Togetintuitionabouttherelationbetweenthevar iousparameters,itisvery usefultoperformacanonicaltransformationoftheLang-Fi rsovtype[ 119 ]inorder 50

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todiagonalize(atleast)partoftheHamiltoniantoobtainr enormalizedenergies. Althoughwewillrestrictournumericalstudytoparticular termsoftheHamiltonian,in thispreliminaryanalysiswekeepthefullHamiltonianinor dertogainascompleteas possibleaunderstandingoftheproblem. Asapreliminarystep,werewritetheHamiltonianintermsof evenandoddlinear combinationsofthe and orbitals,deningnewcreationoperators d y e ( o ) = 1 p 2 ( d y d y ). (3–24) Wethenapplyacanonicaltransformationfrom H to ^ H = e S He S ,where S ischosen toeliminatetheterms H 1 and H 2 .GeneralizingtheworkofLangandFirsov[ 119 ],an appropriatechoiceof S is S = 0 0 ( n e n o ) + 0 ( n e + n o n c ) ( b y b ), (3–25) where n p = P d y p d p ( p = e o )arethechargesintheevenandoddorbitals respectively.Withthistransformationweobtainanexactr epresentationoftheoriginal modelHamiltonianas ^ H = ^ H mol + ^ H leads + ^ H mol-leads (3–26) where ^ H mol = X p = e o p n p + X p = e o U p n p n p # + U ? X n e n o + U k X n e n o + J 0 ( B y 4 0 I + e I o + B 4 0 I + o I e + S + e S o + S + o S e )+ X [ K 0 ( n e + n o ) 0 ] ( B y 2 0 d y e d o + B 2 0 d y o d e )+ 0 b y b 2 n 2 c 0 (3–27) inwhich e o = + 2 ( 0 ) 2 2 n c ( 0 ) 0 t 0 (3–28) 51

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aretherenormalizedenergiesofevenandoddmolecularorbi tals, U e o = U + U +2 U 0 4 2( 0 ) 2 0 (3–29) U k = U 0 + 2( 0 2 2 ) 0 (3–30) and U ? = U + U +2 U 0 4 + 2( 0 2 2 ) 0 (3–31) arerenormalizedCoulombinteractionsbetweenelectronsi nmolecularorbitalsofthe same( U e o )anddifferent( U k U ? )parity.Theremainingcouplings, J 0 = 2 U 0 U U 4 (3–32) 0 = 2 0, (3–33) and K 0 = U U 4 (3–34) measuredifferencesbetweenmolecularenergiesorCoulomb interactionsintheoriginal model.InEq.( 3–27 ), S + p ( S p ) y = c y p c p # and I + p ( I p ) y = c y p c y p # ( p = e o )arespinandcharge-raisingoperators,respectively.Finally,wed ene B = exp [ 0 ( b y b )] B y (3–35) TheleadHamiltonianisunaffectedbythetransformation,i .e., ^ H leads = H leads while ^ H mol-leads = 2 p N s X p = e o V p X k ( B y 0 d y p c e k + B 0 c y e k d p ) (3–36) with V e o = V V and B ( y ) + 0 [ B ( y ) 0 ]appearingfor p = e [ p = o ].Theeffective hybridizationwidthcanbeestimatedas ~ e o = e o e ( 0 ) 2 =! 2 0 e o h 1 ( 0 ) 2 2 0 i (3–37) 52

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withthesecondequalityvalidintheweakphononcouplingli mit( 0 0 ). TheLang-FirsovtransformationsimpliestheoriginalHam iltonianbyeliminating explicit e ph interactiontermsandreplacingthembyrenormalizationof theorbital energiesaswellastheinter-andintra-orbitalCoulombrep ulsions.However,this transformationalsointroducesothercomplexitiesintoth eHamiltonian.Forinstance, thetermswithcoefcients J 0 0 and K 0 inEq.( 3–27 )describeelectrontunneling processesbetweenevenandoddorbitalsthatwereabsentfro mtheoriginalform. Moreover,thepresenceoftheoperators B 2 0 and B 4 0 inthesetermsindicatesthateach intra-molecularelectrontunnelingeventisaccompaniedb ythecreationandabsorption ofacloudofphononsasthelocalphononmodeadjuststothech angeinthedifference n e n o betweenevenandoddorbitaloccupancies.Theoperator B 0 thatenters ^ H mol-leads describesasimilarphononcloudthataccompaniestunnelin gbetweenthe moleculeandtheleads. Wecangainsomemoreinsightintothetransformationbynoti cingthatapplyingthe Lang-Firsovtransformationtothephononcreationoperato rgives ^ b y = e S b y e S = b y 0 ( n e n o ) 0 + ( n e + n o n c ) 0 (3–38) whichcorrespondstoadisplacementofthephononoperators .FromEq.( 3–38 ),we have ^ b y ^ b = b y b ,soEq.( 3–35 )canalsobewrittenas B = exp [ ( =! 0 )( ^ b y ^ b )] .As aresult, B 2 0 B 4 0 and B 0 inducemixingamongsubspaceswithdifferentoccupation numbers ^ n b ^ b y ^ b forthetransformedphonons. Inthefollowingwewillfocusouralgebraicanalysisonthes ubspacewithzero occupancyofthetransformedphononoperators, ^ n b =0 ,sothatthethermalaverageof theoriginalphononoccupationis h n b i 0 ( n e n o )+ ( n e + n o n c ) 0 2 (3–39) 53

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Thisapproximationisvalidonlywhenthephononenergy 0 islargerthananyother energyscaleinthemodelaswellasthethermalenergy k B T .Inthiscase,allthe low-lyingstatesofthesystemarecharacterizedby h ^ n b i 0 .Thisapproximation becomesexactintheanti-adiabaticlimit 0 !1 [ 36 ].Inthenextsectionswewillcarry outcalculationsintheanti-adiabaticregime,characteri zedbytherelation 0 thatfullltheconditionofvalidityoftheapproximation( 3–39 ). 3.3GeneralConguration:BasicStudy TherichestbehaviorofthemodeldescribedbyEq.( 3–1 )arisesincaseswhere thetwomolecularorbitalslieclosetotheFermienergy F =0 sothattheycanboth contributestronglytothelow-energyphysics.Forsimplic ity,wefocusprimarilyinthis sectiononsituationswithequal e ph coupling 0 = ,equalCoulombinteractions U = U = U 0 = U ,symmetricplacementoftheorbitalswithrespectivetothe chemical potentialoftheleads(i.e., = = ,asmallpositiveenergyscalecomparedwith Coulombrepulsion),andthechoiceofreferencecharge n c =0 .However,wealso presentresultsformoregeneralparameterchoicesatsever alpointsthroughoutthis section.3.3.1IsolatedMolecule Webeginbyexaminingthelow-lyingstatesoftheisolatedmo lecule,usingthe transformedHamiltonian ^ H mol inEq.( 3–27 )tondtheenergies.Forthecase U = U = U consideredthroughoutthissection, K 0 =0 inEq.( 3–34 ).Thentheonlyexplicit e ph couplingremainingin ^ H mol entersthroughtheterms 0 ( B y 2 0 d y e d o + H c .) and J 0 ( B y 4 0 I + e I o + H c .) .Thissectionisconcernedonlywithcaseswhere 0 = issmall. Ifonealsotakes j J 0 j = 1 2 j U 0 U j tobesmall,thenthelow-lyingmolecularstateswill containonlyaweakadmixtureofcomponentshaving h ^ n b i > 0 ,where(asbefore) ^ n b isthenumberoperatorforthetransformedbosonmodedened inEq.( 3–38 ).Under thissimplifyingassumption,itsufcestofocusontheeige nenergiesof P 0 ^ H mol P 0 with P 0 projectingintothe ^ n b =0 Fock-spacesector.Table 3-1 liststhelow-lyingenergiesinthis 54

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sectorforthecase =0 wherethe and molecularorbitalsareexactlydegenerate. Alsolistedaretheenergiesofthesestatesforthespecialc ase 0 = and U 0 = U extendedtoincludetheleadingperturbativecorrectionsf or > 0 .Thesecorrections containamultiplicativefactor jh ^0 j B 2 0 j ^0 ij 2 =exp[ 2( =! 0 ) 2 ] (for 0 = )reectingthe reductionwithincreasing e ph couplingoftheoverlapofthephonongroundstatesfor Fock-spacesectorsofdifferent n mol .Hereandbelow,wedenoteby j ^0 i thestatehaving n mol =^ n b =0 ,whichmustbedistinguishedfromthestate j 0 i inwhich n mol = n b =0 ItcanbeseenfromTable 3-1 thatfor =0 thesinglyoccupiedsectorhastwo states—dependingonthesignof 0 ,either j (1)1 i and j (1)2 i or j (1)3 i and j (1)4 i —with lowestenergyenergy E (1) min = ( + j 0 j ) 2 =! 0 .Incasesofsmall j U 0 U j and/or large j 0 j ,theloweststateinthedoublyoccupiedsectoris j (2)1 i withenergy E (2) min = 1 2 ( U + U 0 ) 4( 2 + 0 2 ) =! 0 ~ ,where ~ = q (8 0 =! 0 ) 2 + ~ J 2 0 (3–40) with ~ J 0 = J 0 h ^0 j B 4 0 j ^0 i 2 = 1 2 ( U 0 U )exp( 8 0 2 =! 2 0 ). (3–41) Onecanuseenergies E (1) min and E (2) min todeneaneffectiveCoulombinteraction, ~ U = E (2) min 2 E (1) min = 1 2 ( U + U 0 ) 2( j 0 j ) 2 0 ~ (3–42) For U 0 = U ,thisvaluesimpliesto ~ U = U 2( + j 0 j ) 2 =! 0 ,whichdecreaseswith increasing e ph couplingatagreaterratethanincaseswithonlyHolstein-t ypephonon coupling ,suchastheone-orbitalmolecularjunctionmodel[ 49 ].Theenhancementof e ph renormalizationoftheCoulombinteractioninmoleculesha vingmultiple,nearly degenerateorbitalsimprovestheprospectsofattainingar egimeofeffective e e attractionandmayhaveinterestingconsequencesintheare aofsuperconductivity. Table 3-1 alsoindicatesthatthegroundstateoftheisolatedmolecul ecrossesfrom singleelectronoccupancy(forweaker e ph couplings)todoubleoccupancy(forstronger 55

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Table3-1.Low-lyingeigenstatesof ^ H mol describingtheisolatedmolecule,asdenedin Eq.( 3–27 ),andeigenenergiesapproximatedbyprojectionintothese ctorof theFockspacehavingoccupancy ^ n b =0 forthetransformedphononmode denedinEq.( 3–38 ).Eigenstates j ( n mol ) i ( =0) i for = = =0 are groupedaccordingtotheirtotalelectronnumber n mol ,andspeciedinterms ofoperators d y p denedinEq.( 3–24 )and B denedinEq.( 3–35 )actingon j 0 i ,thestatehaving n mol = n b =0 ; c 1 and c 2 arerealcoefcientssatisfying c 2 1 + c 2 2 =1 thatreducefor U 0 = U to c 1 =1 c 2 =0 E ( n mol ) i ( =0) isthe energyofstate j ( n mol ) i ( =0) i ,expressedintermsof x = = p 0 x 0 = 0 = p 0 U =( U + U 0 ) = 2 ,and ~ isdenedinEq.( 3–40 ). E ( n mol ) i ( > 0) isthe approximateenergyofthesamestateinthespecialcase U 0 = U and 0 = > 0 ,butincludingtheleadingperturbativecorrectionfor > 0 expressedintermsof y = 0 ( = ) 2 exp[ 4( =! 0 ) 2 ] .For U 0 = U and 0 = > 0 ,thevalues E ( n mol ) i ( > 0) wouldbethesameapartfromthe interchangeoftheenergiesoftheeven-andodd-parity n mol =1 states. n mol i j ( n mol ) i ( =0) i E ( n mol ) i ( =0) E ( n mol ) i ( > 0) 01 j 0 i 00 11 d y e B 0 + j 0 i ( x + x 0 ) 2 4 x 2 1 4 y 2 d y e # B 0 + j 0 i ( x + x 0 ) 2 4 x 2 1 4 y 3 d y o B y 0 j 0 i ( x x 0 ) 2 1 4 y 4 d y o # B y 0 j 0 i ( x x 0 ) 2 1 4 y 21 ( c 1 d y e d y e # B 2( 0 + ) + c 2 d y o d y o # B y 2( 0 ) ) j 0 i U 4( x 2 + x 0 2 ) ~ U 16 x 2 1 6 y 2 1 p 2 ( d y e d y o # + d y e # d y o j 0 i ) U 0 4 x 2 U 4 x 2 1 6 y 3 1 p 2 ( d y e d y o # d y e # d y o j 0 i ) U 4 x 2 U 4 x 2 1 6 y 4 d y e d y o j 0 i U 4 x 2 U 4 x 2 5 d y e # d y o # j 0 i U 4 x 2 U 4 x 2 6 ( c 2 d y e d y e # B 2( 0 + ) c 1 d y o d y o # B y 2( 0 ) ) j 0 i U 4( x 2 + x 0 2 )+ ~ U + 1 2 y e ph couplings)atthepointwhere E (2) min = E (1) min ,whichreducesfor =0 andsmall ~ J 0 to ( + j 0 j ) 2 0 = U + U 0 6 (3–43) Wewillseethatthislevelcrossingintheisolatedmolecule iscloselyconnectedto acrossoverinthefullproblemthatresultsinpronouncedch angesinthesystem's low-temperatureproperties.Thelowestenergyofanymolec ularstatehavingthree electrons(notshowninTable 3-1 )is E (3) min ( =0)= U +2 U 0 (3 + j 0 j ) 2 =! 0 ,while 56

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thesolefour-electronstatehasenergy E (4) 1 ( =0)=2 U +4 U 0 16 2 =! 0 .Forallthe casesconsideredinSec. 3.3.2 below,theseenergiesaresufcientlyhighthatstates with n mol > 2 playnoroleinthelow-energyphysics. 3.3.2NumericalResultsfortheGeneralConguration Thissubsectionpresentsnumericalsolutionsofthefullpr oblemundervariationof the e ph coupling.Weprimarilyfocusonareferencecasewith 0 = U 0 = U =0.5 Theorbital-leadshoppingcoefcientistakentobe V = V = V =0.075 ,andhencethe hybridizationwidth =0.0177 forbothorbitals.Allcalculationswereperformedusing anNRGdiscretizationparameter =2.5 ,allowingupto N b =60 phononsinthelocal mode,andretaining2000-4000many-bodystatesaftereachi teration.Thesechoices aresufcienttoreduceNRGdiscretizationandtruncatione rrorstominimallevels. 3.3.2.1Propertiesat T =0 andcrossovertemperature T Werststudytheevolutionwith 2 =! 0 ofzero-temperaturepropertiesofageneral congurationofthismodel,includingtheorbital(molecul ar)occupancy,phonon occupancy,andlinearconductancethroughthismolecularj unction.Wealsocalculate thecrossovertemperature T characterizingthequenchingofthemolecularspin degreeoffreedom,determinedviathestandardcriterionfo raspin-1/2localizedlevel [ 7 ] T mol ( T )=0.0701 with mol thespinsusceptibilityofthemoleculedenedinEq. ( 3–17 ). Webeginbyconsideringthebehaviorfor =0 .Figure 3-2 (a)showsthe zero-temperaturemolecularcharge h n mol i ,whileFig. 3-3 displaysthecorresponding occupanciesofindividualmolecularorbitals: h n i and h n i inpanel(a),and h n e i and h n o i inpanel(b).For = V 2 = D 0.0177 h n mol i'h n e i' 0.5 ,whichmaybe understoodasaconsequenceofthegroundstatebeingcloset othatfor U = V = 1 and =0 :aproductof(1) 1 2 c y e d y e # c y e # d y e p 2 c y e c y e # j 0 i where c e =(2 N k ) 1 = 2 P k c e k annihilatesanelectroninthelinearcombinationofleft-a ndright-leadstatesthat tunnelsinto/outofthemolecularorbitals,and(2)otherle addegreesoffreedomthat 57

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0 0.5 1 1.5 2total charge 10 -4 10 -3 10 -2 10 -1 10 0T 00.020.04 0.06 0.08 l 2 / w 0 0 2 4 6 8 10 12phonon occupation 0.020.04 0.06 0.08 0 0.2 0.4 0.6 0.8 1G(2e 2 /h) d =0.025 d =0.05 d =0.075 d =0.1 (b) U=0.5 (a) (d) (c) Figure3-2.Variationwith 2 =! 0 of(a)theground-statemolecularcharge h n mol i = h n e + n o i ,(b)theground-statephononoccupation h n b i ,(c)the crossovertemperature T ,and(d)thezero-temperaturelinearconductance G ,allcalculatedfor U 0 = U =0.5 0 = ,andthefourvaluesof = = listedinthelegend.Inthecase =0.05 ,theorbitalenergysplittingisin resonancewiththephononenergy,i.e., =2 = 0 aredecoupledfromthemolecule.Thetotalchargeincreases with andapproaches h n mol i = h n i =1 for ,inwhichlimitthelargeCoulombrepulsion U leads tolocal-momentformationinthe orbital,followedatlowtemperaturesbyKondo screening. Turningon e ph couplings 0 = lowerstheenergyoftheeven-paritymolecular orbitalbelowthatoftheoddorbital,andinitiallydrivest hesystemtoward h n e i =1 h n o i =0 ,andtowardamany-bodysingletgroundstateformedbetween theleadsand alocalmomentintheeven-paritymolecularorbital.Thespi n-screeningscale T inFig. 3-2 (c)showsaninitialdecreasewithincreasing 2 =! 0 thatisverystrongforthesmaller 58

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00.020.04 0.06 0.08 l 2 / w 0 0 0.5 1level occupation 0.020.04 0.06 0.08 0 0.5 1 1.5 2level occupation d =0.025 d =0.05 d =0.075 d =0.1 n a n e (a) (b) n b n o Figure3-3.Occupationofindividualmolecularorbitalsvs 2 =! 0 for U 0 = U =0.5 0 = andthefourvaluesof = = listedinthelegend:(a) h n i (open symbols)and h n i (solidsymbols);(b) h n e i (opensymbols)and h n o i (solid symbols). valuesfor ,wherethe e ph couplingdrivesthesystemfrommixedvalenceintothe Kondoregime.Forlarger ,wherethesystemisintheKondolimitevenat =0 ,there isamuchmilderreductionof T causedbythephonon-inducedrenormalizationof andthephonon-inducedshiftofthelledmolecularorbital furtherbelowthechemical potential. Uponfurtherincreaseinthe e ph coupling, h n mol i and T bothshowrapidbut continuousrisesaroundsomevalue = x .Thecrossovervalue 2x =! 0 0.042 ,which isindependentof for U ,coincidescloselywithits =0 value U = 12 0.0417 for theisolatedmolecule,whereitdescribesthecrossingofth esinglyoccupiedstate j (1)1 i andthedoublyoccupiedstate j (2)1 i (seeTable 3-1 ).For > 0 ,thecrossoverofthe ground-statemolecularchargefrom1to2issmearedoverthe range j U 12 2 =! 0 j 2 59

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suggestingafullwidthforthecrossover ( 2 =! 0 ) 4 = 12=0.006 ,ingoodagreement withFigs. 3-2 (a)and 3-3 Intheregime & x ,thesystemminimizesthe e ph energybyadoptingorbital occupancies h n e i' 2 h n o i' 0 (showninFig. 3-3 toholdforallthe valuesconsidered). Here, T approachesthescale 12 2 =! 0 U atwhichoccupationof n mol =1 molecular statesbecomesfrozenout.Overtheentirerangeof and 2 =! 0 illustratedinFigs. 3-2 and 3-3 ,theground-statephononoccupation h n b i [Fig. 3-2 (b)]closelytracks n b andthe T =0 conductance[Fig. 3-2 (d)]iseverywherewell-describedbyEq.( 3–23 ). WenotethattheequilibriumpropertiesshowninFigs. 3-2 and 3-3 exhibitnospecial featuresintheresonantcase =0.05 inwhichthemolecularorbitalspacing exactlymatchesthephononenergy.Weexpecttheresonancec onditiontoplaya signicantroleonlyindrivensetupswhereanonequilibriu mphonondistributionserves asanetsourceorsinkofenergyfortheelectronsubsystem. Thepropertiespresentedabovearelittlechangedunderrel axationofthe assumptions 0 = and U 0 = U .Forreasonsofspace,weshowdataonlyfor thevariationofthecrossovertemperature T with e ph couplingwithdifferentxed valuesof U 0 = U [Figs. 3-4 (a)and 3-4 (c)]or 0 = [Figs. 3-4 (b)and 3-4 (d)].Ineachcase, T isplottedagainst ( = x ) 2 ,where x isthevalueof thatsatisesthecondition E (2) min ( =0)= E (1) min ( =0) forcrossoverfromsingletodoubleoccupationoftheisolat ed molecule.For U 0 =0.5 U and U 0 =2 U ,itmustberecognizedthat J = 1 2 ( U 0 U ) is notsmall,callingintoquestionthevalidityoftheapproxi mation ^ n b =0 usedtoderive theenergiesinTable 3-1 .Whatismore,thedatashownarefornonzeroorbitalenergy splittings =0.05 (toppanels)and =0.1 (bottompanels).Nonetheless,theplots allexhibitgooddatacollapsealongthehorizontalaxis,sh owingthat x calculatedfor ^ n b =0 and =0 capturesverywellthescalecharacterizingthecrossoverf romthe Kondoregime( x )tothephonon-dominatedregime( & x ). 60

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10 -5 10 -3 10 -1T l¢=0.5l l¢=l l¢=2l 0.5 1 1.5 2 ( l/l x ) 2 10 -5 10 -3 10 -1T 10 -5 10 -3 10 -1T U ¢ =0.5U U ¢ =U U ¢ =2U 0.5 1 1.5 2 ( l/l x ) 2 10 -5 10 -3 10 -1T (c) (a) U ¢ =U l¢ = l d =0.05 d =0.1 l¢ = l U ¢ =U (b) (d) d =0.05 d =0.1 Figure3-4.Crossovertemperature T vsscaled e ph coupling ( = x ) 2 .Theleftpanels showdifferentratios U 0 = U for 0 = whiletherightpanelsshowdifferent 0 = for U 0 = U .Thetoppanels(a),(b)correspondto =0.05 ,andthe bottompanels(c),(d)treat =0.1 .Alldataarefor U =0.5 .Verticaldashed linesat = x [calculatedviathecondition E (2) min ( =0)= E (1) min ( =0) ] separatetheKondoregimefromthephonon-dominatedregime ThedatainFig. 3-4 showgreaterspreadalongtheverticalaxis,particularlyi nthe Kondoregimeundervariationof U 0 = U .However,wendthatineachpanel,thevalueof T inthephonon-dominatedregimecanbereproducedwithgoodq uantitativeaccuracy byapplyingthecondition T mol ( T )=0.0701 tothesusceptibilityoftheisolated molecule,calculatedusingthe11stateslistedinTable 3-1 .Thisprovidesfurther evidencefortheadequacyoftheapproximation ^ n b =0 employedintheconstructionof thetable.Moreimportantly,Fig. 3-4 showsthatthephysicsprobedinFigs. 3-2 and 3-3 forthespecialcase 0 = and U 0 = U isbroadlyrepresentativeofthebehaviorovera wideregionofthemodel'sparameterspace. 61

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0 1 2 3 4S mol /ln 2 l 2 / w 0 =0.025 l 2 / w 0 =0.0391 l 2 / w 0 =0.04389 l 2 / w 0 =0.064 0 0.1 0.2 0.3T c mol 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 T 0 5 10 15phonon occupation (a) (b) U=0.5 d =0.1 (c) Figure3-5.Temperaturedependenceof(a)themolecularent ropy,(b)temperaturetimes themolecularsusceptibility T mol ,and(c)thephononoccupation.Dataare for U 0 = U =0.5 =0.1 0 = ,andthefourvaluesof 2 =! 0 listedinthe legend.In(a),thehorizontaldashedlinesmark S mol =ln2 ln3 ,and ln5 .In (c),thedashedlineshowstheoccupationofafreephononmod eofenergy 0 =0.1 3.3.2.2Propertiesat T > 0 Tothispoint,wehaveconcentratedonground-state( T =0 )propertiesandthe temperaturescale T characterizingthequenchingofthemolecularmagneticmom ent. Wenowillustratethefulltemperaturedependenceofthreet hermodynamicpropertiesin situationswherethemolecularorbitalsarearrangedsymme tricallyaroundthechemical potential.Figure 3-5 plotsthevariationwith T ofthemolecularentropy,molecular 62

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10 -2 10 -1 10 0 10 1 10 2 T/T* 0 0.05 0.1 0.15 0.2 0.25T c mol (T) l 2 / w 0 =0 0.009765 0.01914 0.025 0.034515 0.0391 0.0417 0.04389 0.04726 0.05625 0.064 0.0765 d =0.1 U=0.5 Figure3-6.Temperaturetimesthemolecularsusceptibilit y T mol vsscaledtemperature T = T for U 0 = U =0.5 0 = =0.1 ,andvaluesof 2 =! 0 spanningthe crossoverfromtheKondoregimetothedoublyoccupiedregim e.The collapseovertherange T 10 T ofallcurvescorrespondingto 2 =! 0 0.0391 demonstratestheuniversalphysicsoftheKondoregime.No suchuniversalityispresentintheboson-dominatedlimit. susceptibility,andphononoccupationfor U 0 = U =0.5 =0.1 V =0.075 ( =0.0117 ), andfourdifferentvaluesof 0 = .Aslongasthetemperatureexceedsallmolecular energyscales,theentropyandsusceptibilityareclosetot hevalues S mol =ln16 and T mol =1 = 4 attainedwheneveryoneofthe16molecularcongurationsha sequal occupationprobability,whilethephononoccupationisclo setotheBose-Einsteinresult forafreebosonmodeofenergy 0 [dashedlineinFig. 3-5 (c)].Oncethetemperature dropsbelow U ,mostofthemolecularcongurations(andallwithtotalcha rge n mol > 2 ) becomefrozenout.For x (exempliedby 2 =! 0 =0.025 inFig. 3-5 ),there isaslightshoulderintheentropyaround S mol =ln5 andaminimuminthesquare ofthelocalmomentaround T mol =1 = 5 ,thevaluesexpectedwhentheemptyand 63

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singlyoccupiedmolecularcongurations(therstvestat eslistedinTable 3-1 )are quasidegenerate.Atlowertemperatures,thereisanextend edrangeoflocal-moment behavior( S mol =ln2 T mol 1 = 4 )associatedwithsingleoccupancyoftheeven-parity molecularorbital(states j (1)1 i and j (1)2 i ).Eventually,thepropertiescrossoverbelow thetemperaturescale T denedabovetothoseoftheKondosingletstate: S mol =0 T mol =0 For justbelow x (see 2 =! 0 =0.0391 inFig. 3-5 )thereareweakshouldersnear S mol =ln5 and T mol =1 = 5 ,asinthelimitofsmaller e ph couplings.Inthiscase, however,thesefeaturesreecttheneardegeneracyofthefo ur n mol =1 congurations andthelowest-energy n mol =2 conguration: j (2)1 i inTable 3-1 .Atslightlylower temperatures,thestates j (1)3 i and j (1)4 i becomedepopulatedandthepropertiesdrop through S mol =ln3 and T mol =1 = 6 beforenallyfallingsmoothlytozero.Eventhough thereisnoextendedregimeoflocal-momentbehavior,theas ymptoticapproachof S mol and T mol totheirgroundstatevaluesisessentiallyidenticaltotha tfor x after rescalingofthetemperatureby T .AsshowninFig. 3-6 ,throughouttheregime < x T mol followsthesamefunctionof T = T for T 10 T .Thisisjustonemanifestation oftheuniversalityoftheKondoregime,inwhich T K T istheKondotemperatureand servesasthesolelow-energyscale. Asmallincreasein 2 =! 0 from 0.0391 to 0.04389 ,slightlyabove 2x =! 0 =0.0417 bringsaboutsignicantchangesinthelow-temperaturepro perties.Whiletherearestill weakfeaturesintheentropyat ln5 and ln 3,thenalapproachtothegroundstateis morerapidthanfor < x ,ascanbeseenfromFig. 3-6 .Notealsotheupturnin h n b i as T fallsbelowabout 10 T —afeatureabsentfor < x thatsignalstheintegralrole playedbyphononsinquenchingthemolecularmagneticmomen t. Finally,inthelimit x (exempliedby 2 =! 0 =0.064 inFig. 3-5 ), E (2) 1 isbya considerablemarginthelowesteigenvalueof P 0 ^ H mol P 0 ,sowithdecreasingtemperature, S mol and T mol quicklyapproachzerowithlittlesignofanyintermediater egime.Even 64

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Figure3-7.Schematicrepresentationofaside-orbitalcon gurationofatwo-orbital molecularjunctionwithphonon-assistedinterorbitaltun neling. Phonon-assistedtunnelingischaracterizedbyacouplingc onstant 0 ,andis associatedwiththeemissionorabsorptionofaphononoffre quency 0 .The lower( )orbitalhastunneling V 0 to/fromtheleads,whiletheupper( ) orbitalhasnodirecttunneling( V =0 ). thoughthequenchingofthemoleculardegreesoffreedomari sesfromphonon-induced shiftsinthemolecularorbitalsratherthanfromamany-bod yKondoeffectinvolving strongentanglementwiththeleaddegrees,the !1 groundstateisadiabatically connectedtothatfor =0 3.4Side-OrbitalConguration:QuantumPhaseTransition Inthissection,wewillfocusonaside-orbitalconguratio n(seeFig. 3-7 ),whichcan bemappedtoaside-dotcongurationofadouble-quantum-do tsystem.Weassume thatonlythelowerorbital connectsdirectlytotheleads( 6 =0 ),whiletheupper orbital connectsonlytotheorbital viaphonon-assistedtunneling 0 6 =0 ,soweset = t 0 =0 .Moreover,weonlyconsiderthephonon-assistedinterorbi taltunneling ( 0 0 )andignoretheHolstein e ph interaction( =0 )throughoutthissection.Inthe lastpartofthesection,wewillcomparetheseresultswitho nesforcaseswithnonzero or t 0 65

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Table3-2.Energiesandquantumnumber P =( n + n b ) mod 2 ofsingle-and double-occupiedlow-lyingstates j ( n mol ) i i oftheisolatedmoleculewithout tunnelingfrom/toleads.Wecalculatedalloftheseeigenen ergiesinthe subspacewithzerotransformedphonons.Operator B isdenedinEq. ( 3–35 ),whileotherquantitiesusedinthetableare =( + ) = 2 x = = p 0 x 0 = 0 = p 0 U =( U + U ) = 2 z 0 = e 2 x 0 2 =! 0 = K 0 0 0 =4 x 0 2 +( U + U U 0 ) = 4 A 1 =1 1 2 ( = 0 ) 2 z 0 4 ,and A 2 =( = 0 ) z 0 2 E (4) 2 and E (5) 2 inthistableareapproximatevaluesvalidinthelimit z 0 1 n mol i j ( n mol ) i i E ( n mol ) i P 11 1 p 2 B 0 d y e + B y 0 d y o j 0 i x 0 2 0 z 0 0 2 1 p 2 B 0 d y e B y 0 d y o j 0 i x 0 2 + 0 z 0 1 3 1 p 2 B 0 d y e # + B y 0 d y o # j 0 i x 0 2 0 z 0 0 4 1 p 2 B 0 d y e # B y 0 d y o # j 0 i x 0 2 + 0 z 0 1 21 d y e d y o j 0 i 2 + U 0 1 2 d y e # d y o # j 0 i 2 + U 0 1 3 1 p 2 d y e d y o # + d y e # d y o j 0 i 2 + U 0 1 4 A 1 p 2 d y e d y o # d y e # d y o j 0 i 2 + U +( 2 = 0 ) z 0 2 0 + A 2 p 2 d y e d y e # B 2 0 + d y o d y o # B y 2 0 j 0 i 5 A 2 p 2 d y e d y o # d y e # d y o j 0 i 2 4 x 0 2 +( U + U 0 ) = 2 J 0 z 0 4 0 + A 1 p 2 d y e d y e # B 2 0 + d y o d y o # B y 2 0 j 0 i ( 2 = 0 ) z 0 2 6 1 p 2 d y e d y e # B 2 0 d y o d y o # B y 2 0 j 0 i 2 4 x 0 2 +( U + U 0 ) = 2+ J 0 z 0 4 1 3.4.1IsolatedMolecule BasedontheLang-FirsovtransformationinSec. 3.2.3 ,Table 3-2 liststheenergies ofsingly( n mol =1 )anddoubly( n mol =2 )occupiedlow-lyingstatesoftheisolated moleculewithoutmolecule-leadhybridization( = =0 ).Theyarecalculatedin therestrictedFockspacedescribedbytheHamiltonian P 0 ^ H mol P 0 ,where P 0 projects ontothesubspacewithtransformedphononoccupationnumbe r ^ n b = h ^ b y ^ b i =0 .In manyoftheselow-lyingstateenergies,therearecorrectio nscontainingamultiplicative factor z 0 = h ^0 j B 2 0 j ^0 i = e 2 0 2 =! 2 0 ,where j ^0 i (with ^b j ^0 i =0)representsthevacuumofthe transformedphononoperatorspacewith ^ n b =0 .Clearly,withincreasing e ph coupling 66

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0 ,theoverlapbetweenstateswithdifferentvaluesof n e n o isreducedthroughthe factor z 0 .Notethattheoperators B 2 0 and B 4 0 admixsubspacescontaining ^ n b > 0 Weignorethecontributionsfromcomponentswithnonzerotr ansformedphononsinthe calculationaswedidinSec. 3.3.1 .InTable 3-2 ,thesinglyoccupiedisolatedmolecule hasatwo-folddegenerategroundstate j (1)1 i and j (1)3 i .Intheweak-phonon-coupling limitwith ( 0 =! 0 ) 2 1 ,thesinglyoccupiedgroundstateenergyreducesto E (1) 1 = E (1) 3 = 0 2 0 (1 0 ) ( ) 0 4 4 0 =~ (3–44) indicatingthattheeffectiveenergy ~ ispusheddownwardsaslongas 0 .For thedoublyoccupiedstates, j (2)5 i or j (2)6 i isthegroundstatewheninthestrongphonon couplingregimewith z 0 1 ,andtheeffectiveCoulombinteractioninthemoleculecan beestimatedas ~ U = min f E (2) 5 E (2) 6 g 2 E (1) 1 = U + U +2 U 0 4 2 0 2 0 (3–45) Forlarge e -phcoupling 0 ~ U becomesnegativeandtheeffectiveinteractionbetween twoelectronsinthemoleculeisattractive.Suchanattract iveeffectiveCoulomb interactionissimilartothatobtainedintheAnderson-Hol steinmodel[ 44 49 ]. FromtheenergiesinTable 3-2 ,wenoticethatthereisalevelcrossing,similarto thatinSec. 3.3.1 ,fromasinglyoccupiedtoadoublyoccupiedgroundstateatt hepoint where E (1) 1 = min ( E (2) 5 E (2) 6 ), (3–46) orat 0 = 0x where ( 0x ) 2 0 = 1 12 ( U + U )+ 1 6 U 0 + 1 6 ( + ). (3–47) Wenotethat j n e n o j =2 forboth j (2)5 i and j (2)6 i sincethecoefcient A 2 in j (2)5 i approacheszeroforlarge 0 .Thisfavorscongurations n e =2, n o =0 and n e =0, n o =2 67

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wherethegroundstateisdoublyoccupied.Forthephononocc upancy h n b i ,uponsetting =0 inEq.( 3–39 ),weobtain h n b i = h b y b i = 0 0 2 h ( n e n o ) 2 i (3–48) Byslowlyincreasing 0 ,weexpectanabruptchangeof h n b i aroundthelevelcrossing sincethelevelcrossingisaccompaniedbyanabruptchangeo f ( n e n o ) 2 from 1 to 4 Theaboveanalysiswasperformedneglectingtheeffectofth eleads.Inthegeneral caseconsideredinSec. 3.3.2.1 ,whenmolecule-leadhybridizationsaretakeninto account,thislevelcrossingbecomesasmoothcrossoverfro maKondoregime,inwhich h n mol i =1 ,toaphonon-dominatedregimewhere h n mol i =2 .However,intheside-orbital case = t 0 = =0 consideredinthissection,thecreationorannihilationof one electroninorbital mustbeaccompaniedbytheabsorptionoremissionofonephon on. Hencethequantity P =( n + n b ) mod 2 (3–49) isagoodquantumnumberforthisside-orbitalconguration .Becauseorbital has directtunnelingwithleads,thereisnosimilarconserved P .Table 3-2 liststhevalueof P foreachlow-lyingstate.Wenotethatthesingly-occupiedg roundstate( j (1)1 i and j (1)3 i )has P =0 ,andofthepossibledoubly-occupiedgroundstates, j (2)5 i has P =0 while j (2)6 i has P =1 .Asaresult,if j (2)5 i isthelarge0 groundstate, P =0 oneither sideofthelevelcrossing,andweexpecttogetacrossoverin thefullsystem.However, if j (2)6 i isthegroundstateforlarge 0 ,thelevelcrossingchangesthevalueof P ,and thisabruptchangeisexpectedtoyieldaQPTnear 0 = 0x .OurNRGcalculations reportedinSec. 3.4.2 willconrmthisexpectation. 3.4.2NumericalResultsfortheQPT AllNRGcalculationsreportedbelowwereperformedwithdis cretizationparameter =2.5 ,retaining800many-bodystatesaftereachiteration.Then umberofphonons 68

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wasrestrictedto n b N b =30 .Below,wetakethehalfbandwidthoftheconduction bandtobe D =1 ,andallotherparametersareexpressedasmultiplesof D 3.4.2.1First-orderQPT Werststudyaside-orbitalcongurationwith =0.012 and =0 .Wetakethe energyoforbital tobebelowthechemicalpotentialoftheleads( = 0.05 < 0 )and theenergyorbital tobeabovethechemicalpotential( =0.05 > 0 ).Toensurethat themoleculepossessesonlyoneelectronintheabsenceof e ph interaction,wealso considerstrongintra-orbital e e repulsion( U = U = U =0.25 ).Inthisrstcasestudy, weneglecttheinter-orbital e e repulsion( U 0 =0 )andleavesituationswith U 0 > 0 to beinvestigatedlater.Sinceweareonlylookingattheeffec toftheinter-orbitalphonon assistedhopping,weset = t 0 =0 aswediscussedatthebeginningofthesection. When 0 =0 ,orbital isdisconnectedfromtherestofthesystemandalocalmoment in orbital isKondoscreenedattemperature T =0 if U > > ToinvestigatetheQPTexpectedtoarisefromchangingthe e ph coupling 0 ,in Figs. 3-8 (a)-(c),weshowthemolecularelectronandphononoccupanc iesandthelinear conductanceasfunctionsof 0 at T =0 .Figure 3-8 (a)showselectronoccupancies vs 0 .Trianglesup(blue)andsquares(red)showtheoccupancies oftheorbitals and ,respectively,andtrianglesdown(green)andstars(orang e)representthe occupanciesoftheorbitals e and o .Thetotalelectronoccupationofthemoleculeis plottedusingcircles(black).Weobservethatalltheseocc upanciesexhibitasharpjump at 0c 0.063 ,whichcomparesquitewellwiththelocation 0x =0.064 ofthemolecular levelcrossingaspredictedfromEq.( 3–47 ).Inthefullsystemwith > 0 ,thecritical valueof 0 isthelocationoftheQPTwherethegroundstatevalueof P jumpsfrom0to 1.Quantitatively,theseresultscanbeunderstoodasfollo ws:at 0 =0 ,theorbital is singlyoccupied( h n i' 1 )sinceitliesbelowtheFermilevelandhasastrongCoulomb repulsion( U j j ).Ontheotherhand, h n i' 0 becausetheorbital liesabove theFermilevelandisdisconnectedfromorbital ( 0 =0 )andfromtheleads( =0 ). 69

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0 0.5 1 1.5 2 < n a > < n b > < n mol > < n e > < n o > 0 2.5 5 < n b > by NRG < n b > by Eq. (3-48) 0 0.5 1G ( 2e 2 /h ) 0 0.0250.050.075 0.1 l 10 -4 10 -2 T <( n e -n o ) 2 > l c Kondo phase phonon dominated phaseelectron occupancy phonon occupancy(a) (b)(c) '< P b >(d) Figure3-8.Variationofpropertieswith e ph coupling 0 foraside-orbitalconguration (a) T =0 electronoccupancies h n i h n i h n e i h n o i and h n mol i = h n i + h n i = h n e i + h n o i inthemolecule.(b) T =0 phonon occupancy h n b i calculateddirectlyviatheNRGandcalculatedindirectlyv ia Eq.( 3–48 )usingNRGdatafor h ( n e n o ) 2 i (alsoshown).(c) T =0 linear conductance G throughthemoleculeandexpectationvalueofoperator P denedinEq.( 3–49 ).(d)Characteristictemperaturescale T denedvia thecondition T s ( T )=0.0701 .Alldataarefor = =0.05 U =0.25 U 0 =0 =0.012 =0 =0 t 0 =0 and 0 =0.1 70

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As 0 increases,this e ph couplinginducesinterorbitaltunneling,sooneshouldexp ect h n i todecrease,while h n i increases.Figure 3-8 (a)doesshowthisbehaviorof h n i and h n i for 0 < 0c ,althoughthechangeinoccupanciesisquiteweak.Thisresu ltsin h n mol i = h n i + h n i remainingnearunityoverthesamerangeof 0 .When 0 exceeds 0c ,weobservethatallthemolecularoccupanciesincrease.In particular, h n mol i jumps fromnear 1 tonear 2 at 0 = 0c andremainsclosetothisvaluefortheremainderofthe 0 rangecoveredinFig. 3-8 .Wealsonotethat h n e i = h n o i = h n mol i = 2 forallvaluesof 0 .Thisisbecause h ( n mol ) i j n e j ( n mol ) i i = h ( n mol ) i j n o j ( n mol ) i i forboththesingly-occupied( j (1)1 i and j (1)3 i )andthedoubly-occupied( j (2)5 i or j (2)6 i )groundstatesofTable 3-2 .Figure 3-8 (b)showsthephononoccupancy h n b i vs 0 calculateddirectlyusingtheNRG(blue squares).ThisiscomparedwiththeresultEq.( 3–48 )oftheapproximation h ^ n b i =0 (redtriangles),computedusingtheNRGvalueof h ( n e n o ) 2 i (blackcircles).Justlike theelectronoccupancies, h n b i exhibitsadiscontinuityat = c .Weobservethatthe exactandapproximate h n b i curvesalmostcoincidefor 0 0c andfor 0 0c ,butshow signicantseparationintheintermediateregime, 0 0c .For farawayfrom 0c ,the transformedphononsarealmostdecoupledfromtheelectron s,sotheapproximation madetoobtainEq.( 3–48 )isgood.Ontheotherhand,for 0 0c ,themolecularground statescontainscomponentswithnon-zeronumbersoftransf ormedphonons,sothe approximation h ^ n b i =0 whenderivingEq.( 3–48 )isnotsogood. Aswementionedbefore,theQPTobservedinthephysicalquan titiesabove arisesfromachangeinthequantumnumber P [denedinEq.( 3–49 )]from 0 to 1 .To conrmthispicture,inFig. 3-8 (c)weshowtheexpectationvalueof P (redsquares) versus 0 .Wesee,indeed,thatfor 0 < 0c h P i =0 whileitjumpsto 1 precisely at 0 = 0c ,whichclearlyindicatesarst-orderQPT.ThisQPTisalsoo bservedin thelinearconductance G ofthesystemshownasblackcirclesinFig. 3-8 (c).For 0 < 0c ,themoleculehasanelectronoccupancynear1.Theelectron spininthe moleculeisscreenedbytheconductionelectronsinthelead s,formingamany-body 71

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Kondosinglet.ThroughoutthisKondophasethereisaresona nceattheFermi-level thatisresponsibleforaunitaryconductance G = G 0 =2 e 2 = h .For 0 > 0c ,the systemliesinaphonon-dominatedphase,inwhichstrong e ph couplingbringsabout doubleelectronoccupancyofthemolecule,suppressingthe localmomentandthe Kondostate.BecauseofthesuppressionoftheKondoeffecti nthisphase,thereisno resonancetocarrychargethroughthemoleculeandthecondu ctancevanishes.The conductancecalculatedviatherstequalityinEq.( 3–23 ),showninFig. 3-8 (c),isin excellentagreementwiththevalueobtainedfrom h n mol i usingthesecondequalityin Eq.( 3–23 ). Allthequantitieswehaveanalyzedsofarwerecalculatedat T =0 .Figure 3-8 (d)plotsthecrossovertemperature T denedaspreviouslyviathecondition T mol ( T )=0.0701 .Wenotethereisarapidincreaseof T at 0 0c .For 0 < 0c sincethesystemcrossesovertoaKondogroundstatefor T T ,weassociate T withthecharacteristicKondotemperature T K .For 0 > 0c ,however, T isinstead identiedasacharacteristictemperature T ph forthelocalquenchingofthemagnetic momentmediatedbyphonons.Notethatfortheparametersstu diedinFig. 3-8 T K lies betweenfrom 10 5 and 10 4 ,whereas T ph islargerthan 10 2 Thedecreaseof T K as 0 initiallyincreasesfromzeroresultsfromarenormalizati on ofthemolecule-leadshybridizationwidth ~ denedinEq.( 3–37 )andfromashiftofthe effectiveorbitalposition ~ denedinEq.( 3–44 ).Thistrendin T K canbeunderstand usingHaldane'sformula[ 121 ]for T K inthesingle-impurityAndersonmodel: T K r U 2 e d ( d + U ) = 2 U (3–50) with d < 0 beingthelevelenergy, U beingtheCoulombrepulsionand beingthe impurity-leadhybridizationwidth.Thedependenceof T K on 0 showninFig. 3-8 (d)for 0 < 0 0c tsEq.( 3–50 )verywellifwereplace and d by ~ and ~ respectivelyin 72

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Eq.( 3–50 ).Inthelimit 0 0 0c onends ln T K ln T K ,0 j j 2 0 0 2 + O h 0 0 4 i (3–51) with T K ,0 theKondotemperatureat 0 =0 Sinceinbothphases lim T 0 T mol ( T )=0 ,and T undergoesnojumpat 0 = 0c wedonotexpectanystrikingchangeinthetemperaturedepen denceof T mol atthe criticalcoupling 0 = 0c .Bycontrast,the T =0 occupanciesandthelinearconductance havebeenshowntobediscontinuousattheQPT,suggestingth atthesequantitieshave differenttemperaturedependencesintheKondoandphonondominatedphases.Figure 3-9 showsthetemperaturedependenceof h n mol i and G forseveralvaluesof 0 around 0c 0.063 ,thevalueinferredfromFig. 3-8 .Weseethatasthetemperatureislowered, thesysteminitiallyapproachesaplateauwith h n mol i =1.5 and G =0.5 G 0 .Atstill lowertemperature,thesepropertiescrossovertothosecha racterizingthegroundstate: h n mol i 1 and G G 0 for 0 < 0c (theKondophase),or h n mol i 2 and G 0 for 0 > 0c (thephonon-dominatedphase).Itshouldbenotedthatthepl ateauvaluesof h n mol i and G arejusttheaverageoftheirrespectiveground-statevalue sinthetwophases. Thebehaviorof h n mol i vs T and G vs T near 0 = 0c allowsustodenea secondcrossovertemperature T x ,atwhichthequasi-degeneracybetweenKondo andphonon-dominatedgroundstatesisbrokenandthesystem crossesoverintoits low-temperaturephase.Unlike T T x vanishesas 0 approaches 0c .Numerically,we ndthat T x /j 0 0c j ,abehaviorcharacteristicofarst-orderQPT. Forthelastpartofthissubsection,weinvestigatehowthe T =0 electron occupancyandlinearconductanceareinuencedbytheappli cationofagatevoltage V g thatactsequallyonbothmolecularorbitals,changingthei renergiesas i ( V g )= i (0) eV g ( i = ),with e beingtheelementarycharge.Figure 3-10 shows h n mol i and G vs 0 for (0)= (0)= 0.05 andseveralvaluesof V g .Values 0 < eV g < 0.05 resultsinashiftintheenergyoforbital upwardstowardstheFermienergy F and 73

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1 1.5 2< n mol > -2.2 5 10 -7 -1.7 5 10 -9 -1.6 5 10 -11 8.5 5 10 -11 8.3 5 10 -9 7.9 5 10 -7 10 -12 10 -9 10 -6 10 -3 T 0 0.5 1G ( 2e 2 /h ) -2.2 5 10 -7 -1.7 5 10 -9 -1.6 5 10 -11 8.5 5 10 -11 8.3 5 10 -9 7.9 5 10 -7 dl = l c l Kondo phonon dominated Kondo phonon dominated ' l > l c l > l c l < l c l < l c ' (a) (b) ' ' ' Figure3-9.Temperaturedependent(a)totalelectronoccup ancy h n mol i and(b)linear conductance G ,forseveralvaluesof 0 aroundthecriticalpoint 0c .Dataare for = =0.05 U =0.25 U 0 =0 =0.012 =0 =0 t 0 =0 and 0 =0.1 causeorbital toriseevenfurtherabove F .Thereductionin j ( V g ) j drivesthe systemintothemixed-valenceregimewhen j ( V g ) j < .Asaresult,both h n mol i and G = G 0 decreasefrom 1 at 0 =0 .As 0 increases,duetotherenormalization ofthemolecule-leadshybridizationwidth ~ = e 0 2 =! 2 0 [seeEq.( 3–37 )]andthe downwardsshiftofthemolecularorbital as ~ ( V g )= (0) eV g [Eq.( 3–44 )forthe parameterchoice 0 = usedinthecalculation],thesystemisdrivenbacktowards h n mol i' 1 G G 0 ,andtheKondoregime.Furtherincreaseof 0 drivesthesystem intothephonon-dominatedphaseatarst-orderQPTlocated at 0 = c ( V g ) .The 74

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0.5 1 1.5 2< n mol > 0 0.01 0.02 0.03 0.04 0.05 00.020.04 0.06 0.080.1 l 0 0.5 1G ( 2e 2 /h ) 0 0.01 0.02 0.03 0.04 0.05 0 0.0250.05 -eV g 0.04 0.05 0.06( l c ) 2 / w 0 (a) (b) -eV g' Figure3-10.(a)Totalelectronoccupancy h n tot i and(b)linearconductance G ,as functionsof 0 at T =0 forseveralxedvaluesofthegateenergy eV g Theinsetof(a)showsthelinearrelationbetween 0 2 c =! 0 and eV g .Data arefor = =0.05 U =0.25 U 0 =0 =0.012 =0 =0 t 0 =0 and 0 =0.1 dependenceof 0c ongatevoltagecanbeestimatedfromEq.( 3–47 )uponreplacing i by i ( V g )= i (0) eV g ,whichgives 0 2 c 0 = 1 3 ( eV g )+ 1 12 ( U + U )+ 1 6 U 0 + 1 6 ( (0)+ (0)), (3–52) showingalinearrelationbetween 0 2 c =! 0 and eV g .Thislineardependenceisconrmed intheinsettoFig. 3-10 (a),whereweshow 0 2 c =! 0 vs eV g .Theslopeofthetted straightlineisabout 0.44 ,quiteclosetothevalue 1 = 3 estimatedfromEq.( 3–52 ),based ontheanalysisoftheisolatedmolecule.Wenotethatthejum pof h n mol i and G atthe 75

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0 0.5 1 1.5 2< n mol > 0.05 0.06 0.07 0.075 0.08 0.085 0.09 0 0.05 0.1 0.15 -eV g 0 0.5 1G ( 2e 2 /h ) (a) (b) l -eV g -eV g Kondo plateau c m Figure3-11.(a)Totalelectronoccupancy h n mol i and(b)linearconductance G ,as functionsof eV g at T =0 forseveralxedvaluesof 0 rangingfrom 0.05 to 0.1 .Dataarefor = =0.05 U =0.25 U 0 =0 =0.012 =0 =0 t 0 =0 and 0 =0.1 QPTdecreaseswithincreaseof eV g .Thisbehaviorisexplainedbythedecreaseof theenergydifferencebetween E (2) 5 and E (2) 6 ,moredetailsofwhichwillbegiveninSec. 3.4.2.2 Beforeclosingthissubsection,inFig. 3-11 (a)and 3-11 (b)weshow,respectively, the T =0 electronoccupationandlinearconductancevs eV g forvariousvaluesof 0 Fromanexperimentalpointofview,thisanalysisisinteres tingbecausethegatevoltage V g isadjustableinrealmolecularjunctions[ 16 17 ].For eV g & 0.15 ,themoleculeis emptyofelectronsand,therefore,both h n mol i and G arezero.Withdecreaseof eV g 76

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wenoteasmoothincreaseof h n mol i and G = G 0 fromzeroto 1 aroundacharacteristic gatevoltage V m g suchthat ~ ( V m g )=0 ,whichgives eV m g = ~ (0) .For 0 =0.05 and 0 =0.06 weseeawell-denedplateauoftheconductance G = G 0 precisely when h n mol i =1 ,whichcorrespondstotheKondoplateau.As eV g furtherdecreases for 0 =0.05,0.06 wenoteasharpincreaseof h n mol i from1to2andadropof G from G 0 tozero.Thistransitionisthesameasdiscussedpreviously inFig. 3-10 .The estimatedcriticalgatevoltage V c g atwhichthetransitionoccurscanalsobederivedfrom Eq.( 3–47 )orjustbyinvertingEq.( 3–52 ),fromwhichweobtain eV c g = 3 0 2 0 U + U +2 U 0 4 (0)+ (0) 2 (3–53) ThewidthoftheKondoplateaucanbeobtainedfromthecondit ion ( eV m g ) ( eV c g )= ( ~ (0)) ( ~ (0) ~ U )= ~ U .Since ~ U decreaseswithincreasing 0 [cfEq.( 3–45 )] weexpectanarrowingoftheKondoplateau.Indeed,thisbeha viorisconrmedbydata inFig. 3-11 (b).ThedisappearanceoftheKondoplateaufor 0 0.08 resultsfroma directcrossoverfromanemptystatetoadoubly-occupiedst ate,whichisexpectedto happenaroundthepointwith ~ U =0 .UsingEq.( 3–45 ),wend 0 =0.079 ,consistent withourNRGresults.Forstilllargervaluesof 0 ,theeffective ~ U becomesnegative,so e e interactionsinthemoleculebecomeattractive.Asaresult ,doubleoccupancyofthe moleculeispreferred,leadingtodisappearanceofthemole cularmagneticmomentand oftheKondoeffect.3.4.2.2QPTversuscrossover Tothispoint,wehavefocusedsolelyoncaseswith U 0 = t 0 = =0 and U = U = U > 0 .Inthissubsectionwestudytheconsequencesofrelaxingth ese conditions.Aswediscussedpreviously,thepresenceofaQP Tismadepossiblebythe conservationofquantumnumber P arisingfromsetting t 0 = = =0 .If P isnot conserved,orisconservedbutdoesnotundergoajump,thent heQPTwillreplaced byasmoothcrossoverbetweenKondoandphonon-dominatedre gimes(notdistinct 77

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1 1.2 1.4 1.6 1.8 2< n mol > U'=0 U'=U/5 U'=2U/5 U'=3U/5 U'=4U/5 U'=U -0.2 -0.15 -0.1 -0.05 0 -eV g 0 0.2 0.4 0.6 0.8 1G ( 2e 2 /h ) -0.14-0.1 0 0.5 1 (a) (b) D < n mol > D G Figure3-12.(a)Totalelectronoccupancy h n mol i and(b)linearconductance G as functionsof eV g forseveralvaluesof U 0 rangingfrom0to U (= U = U ) h n mol i and G aredenedtobethechangeof h n mol i and G atQPT.Data arefor = =0.05 U =0.25 =0.012 =0 0 =0.06 =0 t 0 =0 and 0 =0.1 .Insetgurein(b)shows eV g dependent G at U 0 =3 U = 5 (blue)and 4 U = 5 (red)withoneordersmaller =0.0012 ,and allotherparametersthesameasinthemainpanels. phases).Inthefollowingwewilladdressthispointindetai lbyrelaxingtheconditions above. Figure 3-12 and 3-13 illustratetheeffectofintroducinganinter-orbitalinte raction U 0 > 0 .Westartat U 0 =0 ,forwhich E (2) 6 < E (2) 5 [Fig. 3-13 (a)]atthelevelcrossing point eV c g ,sothemoleculardoubly-occupiedgroundstateis j (2)6 i with P =1 ,a valuedifferentfromthat P =0 ofthemolecularsingly-occupiedgroundstatefor 78

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-0.085 -0.07 -0.01 0 0.01 E -0.115 -0.1 -0.14 -0.125 -eV g -0.01 0 0.01 -0.16 -0.145 -0.02 -0.005 -0.01 0 0.01 E (1) E (2) E (2) -0.055 -0.04 D E (2) U'=0 U'=U/5 U'=2U/5 U'=3U/5 U'=4U/5 U'=U -eV g c (a) (b) (c)(d) (e)(f) 1 56 Figure3-13.Dependenceon eV g oftheenergiesofthethreeloweststatesofthe isolatedmoleculearoundthelevelcrossingpoint eV c g fordifferentvalues ofthemolecularinter-orbitalinteraction U 0 .Thereddashedlineisthe energyofsingly-occupiedstates j (1)1 i and j (1)3 i with P =0 ,andthered (blue)solidlineistheenergyofthedoubly-occupiedstate j (2)5 i ( j (2)6 i )with P =0 ( P =1 ).In(a)-(d), E (2) = E (2) 5 E (2) 6 ispositive,whileitchanges signtobenegativein(e)and(f).Dataarefor = =0.05 U =0.25 = =0 0 =0.06 =0 t 0 =0 and 0 =0.1 79

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eV g > eV c g .Asaresult,wehavearst-orderQPTshowninFig. 3-12 .Notethatthe nite hybridizesstateswiththesamequantumnumber P ,therefore,thegroundstate intheKondophasehasnon-negligiblecontributionsfrom j (2)5 i if j E (5) 5 E (1) 1 j issmall comparedwith .Fortheresultsshownpreviouslyfor U 0 =0 ,theenergydifference E (2) = E (2) 5 E (2) 6 islargeenoughsuchthat E (2) > ,hencethegroundstateinthe Kondophasecloseto eV c g isalmostpurelysingly-occupied,andthatiswhywesee thediscontinuities h n mol i 1 and G G 0 atthecriticalpoint,asshownbytheblack circlesofFig. 3-12 For U 0 = U = 5 and 2 U = 5 ,plottedinFig. 3-12 usingorangesquaresandgreen diamonds,respectively,therst-orderQPTisstillappare nt.However, E (2) has decreasedto 0.01 and 0.007 (bothsmallerthat )for U 0 = U = 5 and U 0 =2 U = 5 respectively[seeFigs. 3-13 (b)and 3-13 (c)].Asaresult,thegroundstateoftheKondo phasecloseto eV c g hasanon-negligiblecontributionfrom j (2)5 i ,sothat h n mol i > 1 and G < G 0 Thereductionin E (2) withincreasing U 0 isaccompaniedbyadecreaseof h n mol i and G showninFig. 3-12 .For U 0 =3 U = 5 (bluetrianglesinFig. 3-12 ),weseevery tinydiscontinuities h n mol i and G .Inthiscase, E (2) becomesmuchsmallerthan asshowninFig. 3-13 (d),so j (2)5 i makesalargecontributiontotheKondo-phase groundstatenear eV c g .Uponfurtherincrease U 0 E (2) eventuallybecomesnegative, ascanbeseenfromFigs. 3-13 (e)and 3-13 (f)forfor U 0 =4 U = 5 and U 0 = U .In thesecases,thedoubly-occupiedgroundstatechangesfrom j (2)6 i to j (2)5 i ,which hasthesamevalue P =0 as j (1)1 i .Asaresult,therst-orderQPTisreplacedbya smoothcrossover,andnodiscontinuitiesareseenin h n mol i and G versus eV g (red trianglesandmagentastarsinFig. 3-12 ).WealsonotefromFigs. 3-13 (a)3-13 (f)that thepositionoftheQPT(orcrossover)shiftstomorenegativ evaluesof eV g as U 0 increases.Thisbehaviorcanbeunderstoodfromthedepende nceof eV c g on U 0 given byEq.( 3–53 ).Toemphasizetheimportanceofthecomparisonof E (2) with ,wenow 80

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-0.0100.010.020.03 t =0 t = l /12 t= l /6 t= l /3 1 1.2 1.4 1.6 1.8 2< n mol > G b =0 G b = G a /12 G b = G a /3 G b = G a -0.04-0.0200.02 -eV g 0 0.2 0.4 0.6 0.8 1G ( 2e 2 /h ) (a) (b) (c) (d) ''' ' '' Figure3-14.(a)Totalelectronoccupancy h n mol i and(b)linearconductance G forfour differentvaluesof ;(c) h n mol i and(d) G forfourdifferentvaluesof t 0 .All areshownasfunctionsof eV g at T =0 .Dataarefor = =0.05 U =0.25 U =0 =0.012 0 =0.06 =0 ,and 0 =0.1 ; t 0 =0 for(a) and(b), =0 for(c)and(d). consider =0.0012 (oneordersmaller)andplot G intheinsetofFig. 3-12 (b)fortwo differentvaluesof U 0 .For U 0 =3 U = 5 (bluetriangles), E (2) becomeslargerthan andthemagnitudeofthediscontinuity G [bluetrianglesintheinsettoFig. 3-12 (b)]is muchgreaterthanthatfor =0.012 [samesymbolsinthemainpanelofFig. 3-12 (b)]. However,for U 0 =4 U = 5 (redtriangles), E (2) isnegativeandweseeonlyacrossover behavioreventhough j E (2) j > 81

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Letusnowtoinvestigatetheeffectofbreakingtheconditio nsleadingtothe conservationof P ineachphase.Thiscanbeachievedeitherbyturningonanonzero ,thedirecttunnelingbetweenorbital andtheleads,orbydirectlycouplingthe molecularorbitalsthroughaphonon-independenttunnelin g t 0 6 =0 .First,inFigs. 3-14 (a) and 3-14 (b),weshow h n mol i and G asfunctionsof eV g for U 0 = t 0 =0 =0.012 andfordifferentvaluesof .Weseethatthediscontinuitiesof h n mol i and G seenfor =0 arereplacedfor > 0 byasmoothvariation,indicatingthattherst-order QPTisreplacedbyacrossover.Thiscanbeunderstoodinthef ollowingway:nite valuesof hybridizestateswithdifferentvaluesof P ,so j (2)6 i hybridizeswiththe singly-occupiedgroundstates j (1)1 i and j (2)3 i when j E (2) 6 E (1) 1 j issmallcomparedwith .Asaresult,thegroundstatearoundthelevelcrossingpoin t eV c g isamixtureboth ofsingly-occupied j (1)1 i and j (1)3 i andofdoubly-occupied j (2)6 i .Therefore,thesystem evolvescontinuouslyfromadoubly-toasingly-occupiedgr oundstateas eV g crosses eV c g .Wenallynotethatthewidthofthecrossoverincreasesas increases,andthe widthisroughlyproportionalto Wecanalsodestroytheconservationof P byintroducinganon-zerointerorbital tunneling t 0 ,whichallowsforelectrontransferbetweenthetwomolecul arorbitals withoutphononabsorption/emissionprocesses.Figures 3-14 (c)and 3-14 (d)show h n mol i and G vs eV g forseveralvaluesof t 0 .Wenotethatthecurvesaresmoothed outfornonzero t 0 ,showingthattherst-orderQPTisreplacedbyacrossover. From Eq.( 3–28 ),fornonzero t 0 ,thedegeneracyoftheevenandoddmolecularorbitalsis lifted.ThenEq.( 3–53 )forthepositionofthecrossoverbecomes eV c g = 3 0 2 0 U + U +2 U 0 4 + 2 + t 0 (3–54) WeseeinFigs 3-14 (c)and 3-14 (d)thatthepositionofthecrossovermovestolarger valuesof eV g .Thisisconsistentwithpredictionsbasedonexaminationo fthelevelsof theisolatedmolecule. 82

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Table3-3.Empty,singly-anddoubly-occupiedgroundstate softheisolatedmolecule,as wellastheirenergiesinthepresenceofbothphonon-assist edinterorbital tunneling( 0 )andHolstein-typephononcoupling( ).Energiesarecalculation inthesubspacewithzerotransformedphonons.Theoperator B isdenedin Eq.( 3–35 )andtheotherquantitiesaredenedinthecaptionofTable 3-2 n mol i j ( n mol ) i i E ( n mol ) i 01 B y j 0 i 0 11 1 p 2 d y e B 0 + d y o B y 0 j 0 i + x 2 x 0 2 + tz 0 2 1 p 2 d y e # B 0 + d y o # B y 0 j 0 i + x 2 x 0 2 + tz 0 21 d y e d y e # B 2 0 + j 0 i 2 + U 4 x 0 ( x + x 0 ) 3.5InterplayofTwoPhononEffects:SpinandChargeKondoCr ossover Inpreviousinvestigationsof e ph couplinginasingle-impurityAnderson-Holstein model[ 44 49 ],acharge-analogKondoeffecthasbeenpredictedtoarisef roma negativeeffectiveCoulombrepulsion.Inthissection,wef ocusontherenormalization ofCoulombrepulsioninthetwo-orbitalmolecularjunction modelproducedbya combinationofphonon-assistedinterorbitaltunneling( 0 )andHolstein-typephonon couplingtobothorbitals( ).Wendasmoothcrossoverbetweenspinandcharge Kondoregimesinducedbythephononeffects.3.5.1PreliminaryAnalysis Inthissection,weconsiderafully-connectedconguratio nofthetwo-orbital molecularjunctiondescribedbyEq.( 3–1 )andshowninFig. 3-1 .Inthefollowing analysis,weassumeforsimplicitythatthetwomolecularor bitalssharethesame hybridizationwiththeelectrodes, = = ,ignorethedirectinterorbitaltunnelingby setting t 0 =0 ,andsetthereferencechargeofthemoleculeinEq.( 3–5 )tobe n c =1 Beforepresentingnumericalresultsforcaseswith 6 =0 and 0 6 =0 ,we willrstanalyzethelow-lyingeigenstatesoftheisolated moleculeasgivenbythe Lang-FirsovtransformationdescribedinSec. 3.2 .Thecalculationiscarriedoutin thesubspacewithzerotransformedphononsdenedby P 0 ^ H mol P 0 with ^ H mol beingthe 83

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transformedmolecularHamiltoniangivenbyEq.( 3–27 )and P 0 beingtheprojection operatorintroducedinSec. 3.4.1 .Forsimplicity,wetakealltheinter-andintra-orbital Coulombinteractionstobeequal: U = U = U 0 = U .Inthiscase,exchange termswithcoefcient J 0 and K 0 vanishinthetransformedmolecularHamiltonian ^ H mol andthecalculationisgreatlysimplied.Table 3-3 listsenergiesoftheempty( j (0)1 i ), singly-occupied( j (1)1 i and j (1)2 i )anddoubly-occupied( j (2)1 i )groundstates.The effectiveCoulombrepulsion U e isgivenby U eff = E (2) 1 2 E (1) 1 = U 2( 0 + ) 2 0 (3–55) Wenotethatbyincreasing 0 and/or U e isreducedandcanbecomenegative, whichmeansthatthephonon-assistedtunnelingandHolstei ntermscooperate inthereductionoftheCoulombrepulsionorevenitsreplace mentbyaneffective electron-electronattraction.Intheabsenceofanexterna lmagneticeld,theisolated moleculehasadoubly-degeneratesingly-occupiedgrounds tate j (1)1 i and j (1)2 i solong as E (1) 1 = E (1) 2 < 0 and U eff > 0 .Applyingagatevoltagetothemoleculewillshifteach orbitalenergyto i ( V g )= i (0) eV g .FromTable 3-3 ,theconditions E (1) 1 = E (1) 2 < 0 and U eff > 0 aresatisedfor (0)+ (0) 2 U + 3 0 2 +4 0 + 2 0 < eV g < (0)+ (0) 2 2 0 2 0 (3–56) Whentheleadhybridizationisturnedon,a(spin)Kondoeffe ctisexpectedwithinthe rangeofgatevoltagesgivenbyEq.( 3–56 ).Ontheotherhand,for U e < 0 theempty groundstate j (0)1 i andthedoubly-occupiedgroundstate j (2)1 i oftheisolatedmolecule crossinenergyatthedegeneracypointwhere E (0) 1 = E (2) 1 ,namely eV g = (0)+ (0) 2 U 2 2 0 ( 0 + ) 0 (3–57) 84

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Inthevicinityofthisdegeneracypoint,oneexpectsacharg eanalogoftheKondo effectinthesystem.Inthefollowingwewilltesttheseanal yticexpectationswithNRG calculations.3.5.2NumericalResultsfortheChargeandSpinKondoEffect s Toillustrateouranalyticpredictionsabovewenowperform NRGcalculations usingthesameNRGparametersaspreviously.Forthemodelpa rameter,weset = ==0.01 andconsidertheorbitals and tolie,respectively,belowand abovetheFermilevelfor V g =0 bysetting (0)= (0)= 0.05 .Wealsosetallthe inter-andintra-orbitalCoulombrepulsionstobeequal, U = U = U 0 = U =0.1 .Finally wetake 0 =0.1 ,whichismuchlargerthan ,sothatthesystemisintheanti-adiabatic regime. Letusrststudyacasewithaxedvalue 0 =0.02 andvary tolookfora crossoverbetweenthespin-Kondoandthecharge-Kondoregi mes.Figures 3-15 (a)and 3-15 (b)showthemolecularelectronoccupancy h n mol i andthelinearconductance G vs eV g forseveralvaluesof at T =0 .WenoteinFig. 3-15 (b)thatas increases theconductanceplateauwith G G 0 seenfor =0.04 (blackcircles)narrowsdown toasharpconductancepeakasseenfor =0.08 (magentacircles).Theplateau intheconductancecoincideswitharegime h n mol i' 1 intheelectronoccupancyof themolecule,asseeninFig. 3-15 (a).Thedecreaseoftheplateauwidthindicatesthe crossoverfromaspin-Kondoregimetoacharge-Kondoregime inwhichtheoccupancy ofthemoleculechangesdirectlyfrom 2 to 0 as eV g increases.Inthespin-Kondo regime,theeffectiveCoulombrepulsion U e ispositive,andthegroundstateofthe decoupledmoleculeissinglyoccupiedwithintherangeofga tevoltage V g givenby Eq.( 3–56 ).Figures 3-15 (c), 3-15 (d)and 3-15 (e)showtheenergiesoftheisolated moleculevs eV g for =0.06 =0.065 and =0.07 ,respectively.Theempty, singly-anddoubly-occupiedstatesoflowestenergyaresho wn,respectivelyinsolid black,dashedred,anddot-dashbluelines.For =0.06 [Fig. 3-15 (c)], E (1) 1 isthe 85

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0 0.5 1 1.5 2< n mol > -0.04-0.0200.020.04 -eV g 0 0.5 1G ( 2e 2 /h ) l =0.04 l =0.05 l =0.06 l =0.065 l =0.07 l =0.075 l =0.08 -0.010.01 E (0) E (1) E (2) -0.010.01E 00.010.02 -eV g -0.010.01 (a) (b) l =0.07 l =0.06 l =0.065 (c) (d) (e) 1 11 Figure3-15.(a)Totalelectronoccupancy h n mol i and(b)linearconductance G as functionsof eV g at T =0 forseveralvaluesof .Energiesofthe molecularemptystate( E 0 ),andofthesingly-anddoubly-occupiedground states( E 1 and E 2 respectively)asfunctionsof eV g for(c) =0.06 ,(d) =0.065 ,and(e) =0.07 .Dataarefor (0)= (0)=0.05 U = U = U 0 = U =0.1 0 =0.02 t 0 =0 and 0 =0.1 ; = =0.01 for (a)and(b), = =0 for(c),(d),and(e). 86

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lowestenergyovertherange 0.001 < eV g < 0.017 .Thisimpliesaspin1 = 2 pairof singly-occupiedgroundstates j (1)1 i and j (1)2 i givesrisetotheconventionalspin-Kondo effectandformstheconductanceplateau.Withincreaseof ,therangeof eV g with singly-occupiedmoleculargroundstatesshrinks,andthat iswhytheconductance plateaunarrows.For =0.065 [Fig. 3-15 (d)]weseethatthelowestenergyofthe isolatedmoleculeis E (2) 1 for 0.01 andis E (0) 1 for & 0.01 .Atthiscrossingpoint theconductanceplateaubecomesaconductancepeakasseeni nthepurplestars inFig 3-15 (b).Noticethatthepeakisindeedcenteredatabout eV g =0.01 ,being consistentwiththeresultfortheisolatedmolecule.Furth erincreasing ,weenterthe charge-Kondoregimewith U eff < 0 .Forinstance,for =0.07 showninFig. 3-15 (e), thereisamolecularground-statecrossingbetweenthedoub ly-occupiedandtheempty stateatabout eV g =0.012 ,whichisexpectedfromEq.( 3–57 ).Atthispoint,the doubly-degeneratemoleculargroundstates j (0)1 i and j (2)1 i formachargedoublet, givingrisetothechargeKondoeffect,sothecorresponding conductancepeakreaches G = G 0 .Sincethedegeneracyoccursonlyatthispoint,thechargeK ondoconductance peakisverysharp,asshowninFig. 3-15 (b)for 0.07 TopresenttheuniversalityofthespinandchargeKondoeffe cts,inFigs. 3-16 (a) and 3-16 (b)weshow,respectively, T mol ( T ) vs T = T K and T c mol ( T ) vs T = T c K for severalvaluesof coveringboththespin-andcharge-Kondoregimes.Here, T K and T c K arethespin-andcharge-Kondotemperaturesdenedvia T K mol ( T K )=0.0701 and T c K c mol ( T c K )=4 0.0701 .Inthefollowingcalculations,foreachvalueof thegatevoltageisadjustedsuchthattheconductance G reachesitsmaximumvalue G 0 toguaranteethepresenceofspinorchargeKondointhesyste m.Wenotethat the T mol ( T ) vs T = T K curvesfor 0.06 inFig. 3-16 (a)collapseontoanuniversal spin-Kondocurve,whichisbetterseenintheinsetofFig. 3-16 (a).For 0.07 we noticedeviationsfromtheuniversalcurve.Ontheotherhan d,thecurvesof T c mol ( T ) vs T = T c K showninFig. 3-16 (b)exhibitacharge-Kondouniversalityfor 0.07 87

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10 -2 10 -1 10 0 10 1 T/T K 0 0.05 0.1 0.15 T c mol l =0 0.01 0.02 0.03 0.04 0.05 0.06 0.065 0.07 0.075 0.08 0.085 (a) universal spin Kondo 10 -2 10 -1 10 0 10 1 T/T K 0 0.2 0.4 0.6 T c c,mol c (b) universal charge Kondo 00.020.04 0.06 0.08 l 10 -5 10 -4 10 -3 10 -2 10 -1 T T K T K (c) c Figure3-16.(a)Temperaturetimesspinsusceptibility T mol ( T ) vs T = T K fordifferent valuesof spanningboththespinandchargeKondoregimes.Insetshows thatcurvesfor 0.06 satisfyspin-Kondouniversality.(b)Temperature timeschargesusceptibility T c mol ( T ) vs T = T c K forthesame valuesasin (a).Theinsetshowsthatcurvesfor 0.07 satisfycharge-Kondo universality.(c)Spin-Kondotemperature T K andcharge-Kondo temperature T c K versus .Dataarefor = =0.05 U = U = U 0 = U =0.1 = =0.01 0 =0.02 t 0 =0 0 =0.1 ,and V g isdeterminedby G ( 0 =0.02, eV g )= G 0 88

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0 0.01 0.02 0.03 0.03 0.04 0.05 0.06 0.07 0.08 0.09 l 'l 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 (a) 2 0 1 0 0.01 0.02 0.03 0.03 0.04 0.05 0.06 0.07 0.08 0.09 l 'l 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 phonon dominated regime II phonon dominated regime I charge Kondo regime Spin Kondo regime (b) Figure3-17.Mapsof(a)totalelectronoccupancy h n mol i and(b)linearconductance G on the 0 planeat T =0 .Dataarefor = =0.05 U = U = U 0 = U =0.1 = =0.01 eV g =0.01 t 0 =0 and 0 =0.1 89

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asbetterseenintheinsetofFig. 3-16 (b).Theseresultscorroborateourprevious interpretationthatthesystemisinthespin-Kondoregimef or 0.06 butisinthe charge-Kondoregimefor 0.07 .Thecrossoverhappensabout =0.065 ,consistent withconclusionderivedfromFig. 3-15 .InFig. 3-16 (c)weplot T K and T c K vs .Wenote thatinthespin-Kondoregime T K < T c K ,while T K > T c K inthecharge-Kondoregime. Theintersectionofthetwocurvesoccursaround =0.065 ,thepointofcrossover betweenthetworegimes. Finally,inFigures 3-17 (a)and 3-17 (b)weplotcolormapsof h n mol i and G on the 0 planeforaxedgatevoltage eV g =0.01 .Inthesemapsweobserve threedistinctregimes:aspin-Kondoregime,inwhich h n mol i 1 and G G 0 ;a phonon-dominatedregimeI,inwhich h n mol i 0 and G 0 ;andaphonon-dominated regimeII,where h n mol i 2 and G 0 .Thesethreedistinctregimescorrespond tothreedifferentmoleculargroundstates:thesingly-occ upiedonewithdegeneracy between j (1)1 i and j (1)2 i ,theemptystate j 0 i ,andthedoublyoccupiedstate j (2)1 i ,all showninTable 3-3 .Wenotethatfor & 0.07 ,thephonon-dominatedregimesIand IIareseparatedbyaconductanceridgethatcorrespondstot hecharge-Kondoregime indicatedbythewhitearrowinFig. 3-17 (b). 3.6Summary Wehavestudiedamolecularjunctionwithtwoactiveorbital sconnectingapairof metallicleads.Lang-Firsovcanonicaltransformationand theNRGhavebeenapplied toinvestigatecompeting e e and e ph interactionspresentinthemolecule.Inthe rstpartofthestudy,weconsideredageneralconguration ofthismodelinvolving bothphonon-assistedinter-orbitaltunnelingandaHolste in-typephononcoupling tothemolecularcharge.Alevel-crossingfromasingly-occ upiedgroundstatetoa doubly-occupiedgroundstateoftheisolatedmoleculeisfo undunderincreaseof the e ph couplingcharacterizingbothphononeffects.Asmoothcros soverofthefull 90

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systembetweenKondoandphonon-dominatedregimesorigina tinginthismolecular level-crossinghassignaturesinpropertiesbothatzeroan dnonzerotemperatures. Inthesecondpart,wefocusedonaside-orbitalconguratio nofthismodelwith phonon-assistedinterorbitaltunnelingbutnoHolstein-t ypecoupling.Wefounda rst-orderQPT,occuringatacriticalvalueofthe e ph couplingstrength 0c ,which distinguishestwodistinctphases:aKondophasewiththema ny-bodyKondoeffect characterizedbyazero-biasanomalyintheelectricalcond uctancethroughthejunction; andaphonon-dominatedphaseinwhichKondoeffectiscomple telydestroyed.Variation ofotherparameters,e.g.,gatevoltage V g ,whichisexperimentallyadjustable,canalso accessthisQPT.Weidentiedtheconditionthatgivesriset otherst-orderQPTas opposedtoacrossoverinthelow-temperaturebehaviorthes ystem.Intheside-orbital conguration,theconservationof P =( n + n b ) mod 2 providesanadditionalsymmetry ofthesystemsothateachstateofthemolecularjunctionhas adenitevalue P =0 or 1 ofquantumnumber P .AQPToccurswhentheground-statevalueof P changesat thepointofalevelcrossing;otherwise,thereisjustasmoo thcrossover.Incaseswhere theconservationof P isbroken,therst-orderQPTisalsoreplacedbyacrossover Finally,westudiedtheinterplayoftwophononeffects:Hol stein-typephonon couplingandphonon-assistedinterorbitaltunneling.The twophononeffectswerefound tocooperatewitheachothertorenormalizetheeffectiveCo ulombrepulsionbetween twoelectronsinthemolecule.Forsufcientlystrong e ph interaction,theeffective e e interactioninthemoleculebecomesattractive,andgivesr isetoacharge-analog Kondoeffect.Asmoothcrossoverfromaspin-Kondoregimeto acharge-Kondoregime isobservedinthecalculations.Colormapsoftheelectrono ccupancyandthelinear conductanceshowtwoKondoregimesaswellastwophonon-dom inatedregimes. 91

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CHAPTER4 RESULTSFORTHETWO-IMPURITYBOSE-FERMIANDERSONMODEL ThischapterisbasedonamanuscriptbyJedediahPixley,Lil iDeng,Kevin Ingersent,andQimiaoSithathasbeensumittedtoPhys.Rev. Lettandisavailable atarXiv:1308.0839. 4.1Introduction AsexplainedinSection 1.4 ,experimentsonheavyfermionshaveforcedtheorists toconsiderunconventionalQCPsbeyondtheLandauapproach .Onetheoryof unconventionalquantumcriticalitypostulatesanovelfor moflocallycriticalquantum phasetransitioninvolvingcriticaldestructionofKondoe ffect.Withinthisframework,itis alsointerestingandimportanttoconsidersuperconductiv ity.Itisopenquestionwhether aKondodestructionQCPpromotesunconventionalsupercond uctivity[ 75 ].Sincethe on-siteCoulombrepulsiondisfavorsconventional s -wavepairing,thisproblemcanonly beinvestigatedusingamulti-impuritymodelinwhichsuper conductingpairingbecomes possible. Inthischapter,atwo-impurityBose-FermiAndersonmodeli sstudiedusingthe numericalrenormalizationgroup.Themodelcontainstwoim purities,whichhybridize withafermionicbathandalsocoupleviatheirstaggeredspi ntoabosonicbath.The impuritiesinteractwitheachotherthroughadirectexchan geinteraction,providingthe opportunitytostudysuperconductingpairinginthepresen ceoflatticemagnetismand Kondocorrelation.Thischapterisarrangedasfollows:Sec tion 4.2 introducesthemodel Hamiltonian.Sections 4.3 and 4.4 discusstwodifferentquantumphasetransitions:one thatoccursforzerobosoniccouplingandanotherthatoccur sforzerodirectinterimpurity exchange.ThefullphasediagramispresentedinSection 4.5 .Section 4.6 reports resultsforsuperconductingpairinginthismodel. 92

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4.2ModelHamiltonian TheHamiltoniandescribingthetwo-impurityBose-FermiAn dersonmodelcanbe written H = H imp + H env + H mix (4–1) Theimpuritypart H imp = H d + H 12 describestheisolatedtwoimpuritiesinteractingviaa directinterimpurityexchangeinteraction,where H d = X i =1,2 ( d n di + Un di n di # ), (4–2) H 12 = I S 1 S 2 (4–3) with n di = n di + n di # and n di = d y i d i .Theoperator d i destroysanelectronwithspin inimpurity i =1 or 2 .Weassumethatthetwoimpuritieshavethesamelevelenergy d andintra-impurityCoulombrepulsion U .Theimpurityi spinoperatorisdenedas S i = 1 2 ( d y i d y i # ) ( d i d i # ) T with beingavectorofPaulimatrices. TheenvironmentalpartofthemodelHamiltonianisgivenby H env = H c + H with H c = X k k c y k c k (4–4) H = X q q yq q (4–5) Intheconduction-bandterm H c ,theoperator c k destroysanelectronwithwavevector k ,energy k andspin H describesthebosonicbathwith q destroyingabosonwith wavevector q andenergy(for ~ =1 ) q .Wetakethedensityofstatesfortheconduction bandtobea“top-hatfunction” c ( )= 0 ( D j j ), (4–6) with ( x ) beingtheHeavisidefunction, D beingthehalfbandwidthoftheconduction band,and 0 =1 = 2 D .Thebosonicbathisassumedtohaveasub-ohmicdensityof 93

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states ( )= 1 s c s ( )( c ), (4–7) withtheexponent s beingrestrictedtotherange 1 = 2 < s < 1 ThelastpartofthemodelHamiltonianis H mix = H dc + H d describingtheinteraction betweentheimpuritiesandtheirenvironment(thefermioni cbandandthebosonicbath). Thelocalhybridizationofeachimpuritywiththeconductio nelectronsisdescribedby H dc = V p N k Xi k ( e i k r i d y i c k + H.c. ) (4–8) with N k beingthenumberofunitcellsinthesolid(thenumberof k points).Forsimplicity, V isassumedtobeindependentof k and i .Itisconvenienttodenethehybridization width = 0 V 2 ofeachimpurity.Thestaggered z componentoftheimpurityspins S z 1 S z 2 coupleswithcouplingstrength g tothedisplacementofthebosonicbathinthe nalHamiltonianterm H d = g ( S z 1 S z 2 ) X q ( yq + q ). (4–9) Theimpurity-bandhybridization H dc leadstoanRKKYinterimpurityinteraction t 12 S 1 S 2 .Interactionstrength t 12 isdependenton R = j r 1 r 2 j ,whichistheseparation betweentheimpurities.Thevalueof t 12 isacomplicatedandnonuniversalfunction ofthebanddispersion k Tosimplifyourmodel,wetake R tobeinnite,causing t 12 tovanish.Instead,theexchangeinteractionbetweenthetw oimpuritiesisdescribed bytheHamiltonianterm H 12 inwhichthestrengthoftheinterimpuritycouplingcanbe controlled“byhand”. Wehaveusedthenumericalrenormalizationgroup(NRG)toin vestigatethistwo impurityBose-FermiAndersonmodel.As R istakentobeinnite,thetwoimpurities hybridizeseparatelywithlinearlyindependentcombinati onsofconduction-bandstates creatingtwofermionicchainsintheNRGformulation.Addit ionally,wehaveonebosonic bath,whichwithintheNRGneedstohavecut-off N b ontheoccupancyofeachsiteof 94

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thebosonicchain.Asaresult,ateacheven-numberedNRGite ration(whereboththe fermionicandbosonicchainsareextended),thedimensiono ftheNRGbasisgrowsup tobyafactorof 4 2 ( N b +1) .Inourcalculations,wetook N b =4 andretainedupto 1300many-bodystatesaftereachiteration. Theresultsreportedinthischapterwereallobtainedfor U = 2 d =0.001 and =0.25 (allenergieshenceforthbeingexpressedasmultiplesof D =1 ).These parameters,suchthat U d ,placetheimpuritiesintheirmixedvalenceregime, ensuringaveryhighKondotemperature T k 1.39 for I = g =0 .Thiselevated Kondoscalewasdesirabletoallowdirectcomparison[ 122 ]betweenourNRGresults andthosefromacontinuous-timequamtumMonte-Carloappro achthathasdifculties accessingverylow-temperaturelimitoftheAndersonimpur itymodel.Sincetheground stateinthemixedvalenceregimeiscontinuouslyconnected tothatinthelocal-moment regime( U d ),wecanbecondentthatourNRGresultsarequalitativelys imilar tothosethatwouldbefoundincaseswithmuchsmallerKondot emperatures. 4.3Kondo-ISPhaseTransition BeforegoingintothepropertiesofthefullHamiltoniandes cribedbyEq.( 4–1 ),we rstlookintothespecialcase g =0 inwhichthebosonicbathdecouplesfromtherest ofthesystemandcanbeignored,leavingjustatwo-impurity Andersonmodel.This modelanditscounterpart,thetwo-impurityKondomodel,ha vealonghistoryofstudy [ 97 – 105 ]. Wetaketheinterimpurityexchangeinteractiontobeantife rromagnetic(AFM),i.e., I > 0 .Inthiscase,theHamiltonianterm I S 1 S 2 tendstoalignthetwoimpurityspins antiparalleltoformaninterimpuritysinglet(IS).Atthes ametime,eachimpurityspin experiencesanAFMexchangecouplingtotheon-siteconduct ionbandspin,andis thereforeinclinedtoformamany-bodyKondosingletbelowt heKondoenergyscale T K .Therefore,theregime I > 0 ofthetwo-impurityAndersonembodiescompetition betweentendenciestowardISandKondogroundstates.Under conditionsofstrict 95

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0102030 0 0.5 1 1.5 2 2.5E N 010203040 0102030 0102030 0 0.5 1 1.5 2 2.5E N 010203040 N (even iterations) (a) (b) (c) (d) (e) d = -5 5 10 -6 d = 0 d = 8 5 10 -6 d = -4 5 10 -3 d = 9 5 10 -3 d = I/I c -1 Figure4-1.ScaledNRGeigenvaluesversuseveniterationnu mber N forvevaluesof theinterimpurityexchange I neartheKondo-ISboundary.(a)and(b)show cases I < I c intheKondophase,(c)shows I I c onthephaseboundary,(d) and(c)showcases I > I c intheISphase.Eachpanelislabeledwiththe valueof = I = I c 1 .Reddashedlinesindicatethesecond-lowestenergy levelatthecriticalxedpoint,andblueandgreendashedli nesrepresentthe second-lowestenergylevelofthespectrumcorrespondingt otheKondoand ISxedpointsrespectively.Dataarefor U = 2 d =0.001 =0.25 =9 N b =0 (withoutbosonicbath)and N s =800 particle-holesymmetry( d = U = 2 ),thiscompetitiongivesrisetoaquantumphase transition(QPT)[ 98 ].Awayfromparticle-holesymmetry,thisQPTisreplacedby a smoothcrossover[ 102 103 ]. 4.3.1NRGSpectrum Werstexaminethelow-lyingenergystatesofNRGspectrumf orevidenceofthe competitionbetweentheISformationandtheKondoeffect.F igure 4-1 showsthescaled NRGenergyeigenvaluesversuseveniterationnumber N forvedifferentvaluesofthe interimpurityexchange I .Increasing N meansconsideringtheproblemonasequence 96

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ofexponentiallydecreasingenergyscales D N D N = 2 .TheevolutionoftheNRG spectrumwithincreasing N canbeinterpretedasarisingfromtheowoftheeffective (renormalized)valuesofparametersoftheoriginalmodel. Thisowcontinuesuntilthe systemapproachesastableNRGxedpointwherethelevelsre mainatas N !1 TherearetwostablexedpointsvisibleinFig. 4-1 :aKondoxedpointwherethespin oneachimpurityisKondo-screenedseparatelybytheconduc tionelectrons;andanIS xedpoint,wherethetwoimpurityspinsformasingletthata symptoticallydecouples fromtheconductionband.Thesexedpointscanbedistingui shedbyexamining thesecond-lowestscaledNRGeigenvalue.(Thelowesteigen valueintheseplotsis zerobyconstruction.)AttheKondoxedpoint,thiseigenva lueapproachesavalue E K 0.942 seenforlargeenough N inFigs. 4-1 (a)and 4-1 (b).AttheISxedpoint,the second-lowesteigenvalueapproaches E IS =0 ,asseenforlarge N inFigs. 4-1 (d)and 4-1 (e).BetweenthetwostableNRGxedpointsisanunstableone ,theQCPlyingon theboundarybetweentheKondoandISphases.TheQCPhasitso wncharacteristic spectrum,whichcanbeseeninFig. 4-1 (c).Thesecond-lowestenergyofthecritical spectrum E c 0.628 ismarkedwitharedlineinallthepanelsinFig. 4-1 IntheKondophase,theNRGspectrumowstotheKondoxedpoi ntforlarge iterationnumber N .AttheKondoxedpoint,theeffectiveexchange I goestozero andtheimpurity-bandhybridizationwidth renormalizedtoinnity.Sincetheimpurity separationistakentobe R = 1 ,thesystemseparatesintotwodecoupledparts,each consistingofoneoftheimpuritiesKondoscreenedbyitsown conductionband.Forthe caseshowninFig. 4-1 (a),thesystemapproachestheKondoxedpointatquitesmal l N ( N 10 )orhightemperature T .Figure 4-1 (b)showsavalueof I closertobutstill belowthecriticalvalue I c .NowtheNRGspectrumshowsplateausfor 6 < N < 18 attheenergiescorrespondingtotheQCP.Uponfurtherincre asingin N ,theenergies evolverapidly,thenapproachtheKondospectrumfor N > 28 .Thecrossoverfromthe criticalspectrumtotheKondospectrumtakesplaceatahigh eriterationnumberas 97

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onegetsclosertothephaseboundary.Preciselyatthecriti calpoint I = I c ,theNRG spectrumforthecriticalxedpointsurvivesto N = 1 .Becauseitisunstable,wehave tone-tune I verycloseto I c tomakesurethattheNRGspectrumremainsonthecritical plateausovertheentirerange N 40 showninFig. 4-1 (c).Inpractice,itisnumerically impossibletokeeponthecriticalplateausforarbitrarily large N Forany I > I c ,theNRGspectrumowstotheISxedpointatlargeiteration number N .AttheISxedpoint,theeffectiveexchange I goestoinnitywhilethe impurity-bandhybridizationwidth scalestozero.Inthiscase,theimpuritiesare stronglyboundintoaspinsinglet,whichdecouplesfromthe conductionelectrons.Asa result,theNRGspectrumcanbedecomposedintoaproductofa nisolatedsingletand twofree-electronspectra(oneforeachimpurity).InFig. 4-1 (d),correspondingto I just above I c ,theNRGspectrumstaysclosetothatofthecriticalxedpoi ntover 4 < N < 18 andthenowstotheISxedpointfor N > 26 .Theground-statedegeneracyintheIS spectrumisduetothetwodecoupledfree-electronspectraa ttheISxedpoint.With furtherincreaseof I ,theNRGspectrumowstotheISxedpointatasmaller N ,e.g., around N =14 forthecaseshowninFig. 4-1 (e). Thecrossoveraround N = N fromthecriticalspectral(for N < N )tooneorother stablexedpointspectrum(for N > N )canbeusedtodeneacrossovertemperature scale T D N = 2 .Tomakethismorequantitative,wedene N astheinterpolated fractionaliterationnumberwherethesecond-lowestenerg ycrossesathresholdhalfway between E c (dashedredlinesinFig. 4-1 )andeither E K or E IS Astheinterimpurityexchangeapproachesitscriticalvalu e,weexpect T tovanish as T /j I I c j (4–10) where iscalledthecorrelationlengthexponent.Fig. 4-2 (a)showsthecrossover energyscale T versusthedistancefromthecriticalpoint j I I c j onalog-logscale. ThestraightlinesconrmthevalidityofEq.( 4–10 )andshowthatthesameexponent 98

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10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 | I I c | 10 -20 10 -15 10 -10 10 -5 T I < I c I > I c (a) Figure4-2.(a)Energyscale T ,whichcharacterizesthecrossoverfromthe high-temperaturequantumcriticalregimetothelow-tempe ratureKondoor ISregime,plottedonalog-logscaleversusthedistance j I I c j fromthe criticalpoint.Straightlinesdemonstratethepower-lawv anishingof T as I approaches I c fromeitherside.(b)Schematicplotofthe I vstemperature diagramneartheKondo-ISQCP.Thedottedcurveindicatesth eenergy scale T ,andthehorizontaldottedlineindicatestheKondoenergys cale ( T K )ofthesingle-impurityAndersonmodeltowhichthemodelre ducesfor I = g =0 .Datain(a)arefor U = 2 d =0.001 =0.25 =9 N b =0 (withoutbosonicbath)and N s =800 99

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=2.02(7) isobtainedwhethertheQCPisapproachedfromtheKondophas e[blue pointsinFig. 4-2 (a)]ortheISphase(redpoints).Fig. 4-2 (b)showsaschematicplot of T vs I for I around I c .For T < T ,thesystemisineitheritsKondooritsISregime whilefor T > T ,thesystemisinaquantumcriticalregimegovernedbytheun stable criticalxedpoint. T =0 exactlyat I c andasaresultthequantumcriticalbehavior persistsdownto T =0 onlyatthispoint.Itshouldbenotedthatanyxed T > 0 ,the systemevolvescontinuouslywithincreasing I fromtheKondoregimetothequantum criticalregimetotheISregime.Awell-denedphasetransi tionoccursonlyat T =0 wherethequantumcriticalregimecollapsestoasingleQCP.4.3.2StaticLocalSusceptibility Wenowturntothebehaviorofthestaticstaggeredlocalsusc eptibilitynearthe criticalpointat I = I c .Thestaggeredlocalsusceptibilityisacomplex-valuedsp in-spin correlationfunctiondenedas s ( )= i Z 1 0 dte i t h [ S z 1 ( t ) S z 2 ( t ), S z 1 (0) S z 2 (0)] i (4–11) Inthelimit ! 0 oneobtainsthepure-realstaticresponse s (0)= @ h S z 1 S z 2 i @ h j h =0 = lim h 0 h S z 1 S z 2 i h (4–12) toasmallstaggeredmagneticeld h actingonlyattheimpuritysites.Suchaeldenters themodelthroughanadditionalHamiltonianterm H = h ( S z 1 S z 2 ) .Becauseinthis sectionweonlyconsiderthelocalstaggeredsusceptibilit yat =0 ,weemploythe notation s ( T I ) Figure 4-3 showsNRGresultsfor s ( T =0, I ) aroundthecriticalpoint I c .In Fig. 4-3 (a),oneseesthat s ( T =0, I ) risesasoneapproachesthephaseboundary fromtheKondo( I < I c )orIS( I > I c )side,anddivergespreciselyat I = I c .When farawayfrom I c ,impurityspinsareeitherboundinaspinsingletwiththeco nduction band(Kondophase)orboundinaspinsingletwitheachother( ISphase),sothey 100

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0.60.70.80.9 I 0 5 10 15 20c s (T, I) T=0 I c Kondo IS (a) 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 | I I c | 0 5 10 15 20 25c s (T, I) I < I c I > I c T=0 (b) Figure4-3.(a) I -dependentstaticstaggeredlocalspinsusceptibility s ( T =0, I ) around theKondo-ISQCP.Thesharppeakshowninthegureindicates the divergenceof s ( T =0, I ) at I = I c .(b) s ( T =0, I ) asafunctionof distance j I I c j fromthecriticalpoint. j I I c j isplottedonalogarithmic scale,andthestraightlinesindicatealogarithmicdiverg enceof c ( T =0, I ) onapproachto I c .Dataarefor U = 2 d =0.001 =0.25 =9 N b =0 (withoutbosonicbath)and N s =800 101

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10 -10 10 -8 10 -6 10 -4 10 -2 T 0 5 10 15c s ( T, I ) -4 5 10 -2 -3 5 10 -3 -2 5 10 -4 0 8 5 10 -4 7 5 10 -3 6 5 10 -2 I I c = T T T Figure4-4.Temperature-dependenceofthestaticstaggere dspinsusceptibility s ( T I ) forsevenvaluesof I arounditscriticalvalue I c .The T axisislogarithmic,so thestraightlineindicatesalogarithmicdivergenceof s ( T I ) attheQCPas T goestozero.ForcurvesintheKondophaseorintheISphase, s ( T I ) followsthecriticalbehaviorfor T > T ,butapproachesaconstantfor T < T .Theenergyscale T ,indicatedbytheblackarrowsforthethree cases I < I c ,distinguishesthelow-andhigh-temperatureregimes.Dat aare for U = 2 d =0.001 =0.25 =9 N b =0 (withoutbosonicbath)and N s =800 areinsensitivetoanexternalmagneticeldand s ( T =0, I ) iscomparatively small.However,when I getscloseto I c ,thecompetitionbetweenKondo-singletand ISformationmakestheimpurityspinsmoresensitivetoexte rnaleldsand s ( T =0, I ) showsasignicantincrease.Preciselyat I c ,theimpurityspinsseemtobelikefreespins and s ( T =0, I ) becomesinniteatthispoint.Tohaveaclearerviewofthedi vergence of s ( T =0, I ) at I = I c ,Fig. 4-3 (b)shows s ( T =0, I ) versus log j I I c j .Thetwo 102

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straightlinesindicatethat s ( T =0, I ) / log 1 j I I c j (4–13) as I I c fromeitherphase. Furthermore,Fig. 4-4 presentsthevariationof s ( T I ) with log T forseveralxed valuesof I around I = I c .Thecurvesfor I < I c (bluesolidsymbols)behaveas s ( T I )= constfor T T (4–14) asignatureofthequenchingoftheimpuritymagneticdegree soffreedom.Athigh temperatures( T T ),bycontrast,thecurvesshowadifferentbehavior: s ( T I ) / log 1 T for T T (4–15) Wealsonotethat T inferredfrom s ( T I ) (indicatedbytheblackarrowsinFig. 4-4 ) shiftstolowervaluesas I approaches I c ,whichisconsistentwiththebehaviorof T foundfromtheNRGspectrumandplottedinFig. 4-2 (a).When I ispreciselyat I c T is suppressedallthewaytozero.Thecorrespondingcurve(bla ckstars)followsEq.( 4–15 ) overtheentiretemperaturerange,so s ( T I = I c ) divergesas T 0 .Thecurvesfor I > I c (hollowredsymbols)showsimilarbehaviortotheircounter partsfor I < I c .Athigh temperatures( T T )theyfollowthecriticalbehaviorgiveninEq.( 4–15 ),whileatlow temperatures( T T )theyareagaindescribedbyEq.( 4–14 ).IntheISphase,the constantvalueof s ( T =0, I ) signalsnotaKondoeffect,butratherthequenchingofthe impuritydegreesoffreedomviatheformationofaninterimp urityspinsinglet. 4.4Kondo-LMPhaseTransition Inthissection,weinvestigateanotheraspectofthemodelH amiltonianspeciedin Eq.( 4–1 ).Here,weconsiderthebosonicbathgivenbyEq.( 4–5 )anditscouplingterm tothedifferenceofthetwoimpurityspinsasdescribedinEq .( 4–9 ).Tomakethings simpler,weignoretheinterimpurityexchangeinteraction ,setting I =0 inEq.( 4–3 ). 103

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Inthislimit,thecompetitionbetweentheimpurity-bandhy bridizationandthe impurity-bathcouplinggivesrisetoaQPTbetweenaKondo-s creenedphaseandan unscreened-local-moment(LM)phase.Togainabasicunders tandingofthisQPT, weperformaLong-FirsovtransformationontheoriginalHam iltonian H [Eq.( 4–1 )]to illustratehowcouplingtothebosonicbathchangesthegrou ndstatesoftheimpurities inisolationfromthebandsandalsoleadstorenormalizatio noftheimpurity-band hybridization.Wechooseagenerator S = g ( S z 1 S z 2 ) X q 1 q ( yq -q ) (4–16) with q beingtheenergyofthebosonmodewithwavevector q .Byapplyingthe canonicaltransformation e S He S = H +[ S H ]+ 1 2! [ S ,[ S H ]]+ ,wend e S ( H d + H + H d ) e S = H d E g ( S z 1 S z 2 ) 2 (4–17) and e S ( H c + H dc ) e S = H c + V p N k Xi k ( e i k r i e ( 1) i +1 g = 2 d y i c k + H c .) (4–18) with E g = P q g 2 =! q and = P q ( yq q ) =! q = y .Eq.( 4–17 )indicatesthat thecoupling g betweenimpuritiesandthebosonicbathmakesitenergetica llymore favorableforthetwoimpuritiestoadoptastatewith S z 1 S z 2 = 1 ,andthereforedenes anenergyscale E g thatcanbecomparedwithotherscalessuchas T K .Atthesame time,Eq.( 4–18 )indicatesthattheeffectivehybridization eff isreducedas eff =exp g 2 2 8 X q 1 2 q (4–19) leadingtoadecreasein T K .Theseconsiderationssuggestthatforsufcientlylarge bosoniccouplings g ,theKondogroundstate,whichexhibitsSU(2)spinsymmetry isreplacedbyaboson-dominatedgroundstatewithbrokenSU (2)symmetry.This 104

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symmetrydifferecesuggeststhatthegroundstatemustchan geataQPTratherthan evolvingsmoothlywithincreasing g 4.4.1NRGSpectrum WerstinvestigatetheexistenceofaQPTbetweenKondoandL Mphasesby examiningthelow-lyingenergylevelsintheNRGspectrum.F ig. 4-5 showstheNRG spectrumversusevenNRGiterationnumber N atvevaluesofthebosoniccoupling g .TheseplotsshowthreedifferentxedpointsoftheNRGows forlarge N .Onexed point,reachedinFigs. 4-5 (a)and 4-5 (b),isthefamiliarKondoxedpointwherethelocal momentsofthetwoimpuritiesarescreenedbythefermionicb and.Asecondxedpoint, reachedinFigs. 4-5 (d)and 4-5 (e),isalocal-moment(LM)xedpointwherethetwo impuritiesasymptoticallydecouplefromthefermionicban dandformastaggeredlocal moment.InFig. 4-5 ,blueandgreendashedlinesindicatethesecond-lowestene rgy attheKondoandLMxedpoints,respectively.Thethirdxed pointisacriticalxed pointseeninFig. 4-5 (c),thesecond-lowestlevelofwhichisindicatedbythered dashed line.Thecriticalxedpointisunstable,inthatitisreach edfor N !1 onlywhere g is tunedpreciselytothecriticalvalue g c markingtheboundarybetweentheKondoandLM phases. Figure 4-5 (a)showsveryclearlytheNRGspectrumreachedatlarge N inthe Kondophasewith g < g c .TheNRGowrapidlyapproachestheKondoxedpoint when N > 16 .AttheKondoxedpoint,theeffectivebosoniccoupling g iszeroandthe effectivehybridizationwidth ofeachimpuritylevelincreasestoinnity.Inthiscase, thetwoimpuritiesaredecoupledfromthebosonicbathandto tallyscreenedbythe conductionelectrons.WeexpecttheNRGspectrumattheKond oxedpointtobethe crossproductoftheKondoxedpointspectrumofthetwo-imp urityAndersonmodel withthespectrumofafreebosonicbath.Generally,thedisc retizedbosonicbathhas manydifferentfrequencies,soagenericenergyis P j n j j with n j beingthenumberof bosonswithfrequency j onsite j ofthebosonicchain( j =0,1,2... ).Fortherange 105

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01020 0 0.5 1E N 01020 01020 01020 0 0.5 1E N 01020 N (even iterations) d = 0 d = -4 5 10 -3 d = -8 5 10 -5 d = 5 5 10 -3 d = 10 -5 (a)(b)(c) (d) (e) d =g/g c -1 Figure4-5.NRGspectrumversusofeveninteractionnumber N forvevaluesofthe bosoniccouplingstrength g .(a)and(b)showcasesintheKondophase, while(d)and(e)showcasesintheLMphase.(c)presentsthes pectrum preciselyattheQCPontheKondo-LMphaseboundaryat g = g c .Each panelislabeledwiththevalueof = g = g c 1 .Thereddashedlineindicates thesecond-lowestenergyinthespectrumcorrespondingtot hecriticalxed point.Theblueandgreendashedlinesindicatethesecond-l owestenergyin thespectrumcorrespondingtotheKondoandLMxedpoints,r espectively. Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 ofenergiesshowninFig. 4-5 ,oneseesonlymultiplesofthelowestbosonenergyof thediscretizedbath.Forsimplicity,eachstatecandecomp oseas j i ij n i ,where j i i is the i th stateoftheKondospectrumand j n i isthestateofthebosonicbathwith n free bosonswithfrequency N = 2 .Theenergyofthisstatecanbewrittenas E j i i + n N = 2 .In thecaseshowninFig. 4-5 (a),thethird-lowestenergy(0.734494abovethemany-body groundstate j 1 i )isexactlytwicethesecond-lowestenergy(0.367247),sug gesting thattheselevelsrepresent j 1 ij 1 i and j 1 ij 2 i withenergies E j 1 i + N = 2 = N = 2 and E j 1 i +2 N = 2 =2 N = 2 ( E j 1 i =0 ).Furthermore,theenergy0.942156ofthefourth loweststatealmostexactlyequalsthat(0.942155)ofsecon d-lowestenergy( j 2 i )in 106

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Fig. 4-1 (a)when N > 14 .Sothefourth-loweststatecanbewrittenas j 2 ij 0 i .Amore completeanalysisconrmsthatallthelow-lyingstatesatt heKondoxedpointcanbe decomposedinthisfashionintoaproductofKondostateanda free-bosonicstate. Forvaluesof g closerto,butstillbelow g c [e.g.,Fig. 4-5 (b)],theenergyofthe second-lowestlevelplateausnearitscriticalxedpointv alue(reddashedline)fora rangeofintermediateiterations 8 < N < 14 ,andonlyincreasestowarditsKondo xed-pointvaluefor N > 24 .When g ispreciselyat g c [Fig. 4-5 (c)],thesecond-lowest energymaintainsitscriticalvalue(reddashedline)toarb itrarilylargevaluesof N As g increasestobegreaterthan g c ,asshowninFig. 4-5 (d),thesecond-lowest energyremainsnearitscriticalvalueforintermediateite rations N ,buteventually decreasestowardzero.Theasymptoticvalueofzero,marked withagreendashedline, characterizestheLMxedpoint,atwhichthetwoimpurities arecoupledtothebosonic bathwithaninniteeffective g andthecouplingtothefermionicbandissubdominant. FromEq.( 4–17 ),weknowthatthegroundstateattheLMxedpointshouldbed oubly degeneratewith S z 1 S z 2 = 1 ,andthisisconrmedbythesecond-lowestlevelfalling toapproachtheenergy 0 ofthelowestlevel.Withfurtherincreaseof g [Fig. 4-5 (e)],the NRGspectrumowstotheLMxedpointmorequickly( N > 10 )withoutlingeringnear thecriticalxedpointoveranydiscerniblerangeof N SimilartothesituationattheKondo-IStransition,wecand eneanenergyscale T tocharacterizethecrossoverfromthevicinityofthiscrit icalxedpointtooneofthe twostablexedpoints,KondoorLM.Thevanishingof T at g = g c isgovernedbythe correlation-lengthexponent : T /j g g c j (4–20) Onecanthinkof 1 = T assettingthetimescaleforthedecayofuctuationsofthe staggeredlocal-moment.FromtheNRGspectrainFig. 4-5 ,wecandetermine T throughtherelation T / N = 2 with N herebeingtheiterationnumbermarkingthe 107

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10 -6 10 -5 10 -4 10 -3 10 -2 | g g c | 10 -12 10 -10 10 -8 10 -6 10 -4 T g < g c g > g c (a) Figure4-6.(a)Energyscale T ,characterizingthecrossoverfromthecriticalxedpoint toeithertheKondoortheLMxedpoint,plottedasafunction ofthe distance j g g c j tothecriticalpoint.Thelinearvariationonthislog-logp lot isconsistentwithEq.( 4–20 ).(b)Schematicplotofthe T vs g diagramnear theKondo-LMQCP.Thedottedcurveindicatestheenergyscal e T ,andthe horizontaldottedlineindicatestheKondoenergyscale T K ofthe pure-fermioniccase g =0 .Thequantumcriticalregimespansthe temperaturerange T < T < T K .Datafor(a)arefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 108

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crossoverfromthecriticalxedpointtooneofthestablex edpoints(KondoorLM). Thismethodfordetermining N isexactlythesameasdescribedSec. 4.3.1 FromEq.( 4–20 ),weexpect log T tobelinearin log( j g g c j ) ,andtheslopeto givethecorrelation-lengthexponent .Fig. 4-6 (a)conrmsthisexpectationandyields =2.4(2) ,irrespectiveoffromwhichside( g < g c or g > g c )thecriticalpointis approached.Fig. 4-6 (b)isaschematicplotofthecrossoverenergyscale T versus bosoniccoupling g .Thecurve T ( g ) distinguishesthelow-temperatureKondoorLM regimefromthehigh-temperaturequantumcriticalregime. Atthecriticalpoint g = g c only, T equalszeroandthequantumcriticalregimepersistsdownto T =0 .Itshould alsobenotedthatatanyxedtemperature T > 0 ,increasing g fromzeroproducesa smoothcrossoverfromtheKondoregimetothequantumcritic alregime,andthenfrom thequantumcriticalregimetotheLMregime.Onlyat T =0 isthereaphasetransition atasinglepoint g = g c wherethenondegenerateKondogroundstatechangestoa doublydegenerateLMgroundstate.4.4.2StaticLocalSusceptibility Wenowturntodiscussionofthestaticstaggeredspinsuscep tibility s ( T g ) denedinEq.( 4–12 ).TheKondoandLMphasescanbeunambiguouslydistinguishe d throughthelow-temperaturebehaviorof s ( T g ) .IntheLMphase( g > g c ), s ( T g ) is observedfor T T tobeproportionaltotheinverseoftemperature(hollowsym bolsin Fig. 4-7 ).Morespecically, s ( T g > g c )= M 2 s ( T =0) T for T T (4–21) where M s = h S z 1 S z 2 i istheexpectationvalueofthestaggeredimpuritymoment.F ig. 4-8 (a)plots M s ( g T =0) asafunctionof g g c .SincethisquantityiszerointheKondo phaseandnonzerointheLMphase,itservesastheorderparam eterfortheKondo-LM QPT.Thestraight-linetonalog-logscale(inset)indicat esthat M s ( g > g c T =0) / ( g g c ) (4–22) 109

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10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 T 10 3 10 4 10 5c s ( T,g ) -4 5 10 -3 -1.4 5 10 -3 -4 5 10 -4 0 6 5 10 -4 1.6 5 10 -3 5 5 10 -3 g c g-g c = T T Figure4-7.Temperature-dependentstaticstaggeredspins usceptibility s ( T g ) for sevenvaluesof g aroundthecriticalvalue g c .Onthislog-logplot,thecurve with g = g c isastraightlinewithaslopethatcoincideswiththeexpone nt s =0.6 describingthedensityofstatesofthebosonicbath.Curves forboth g < g c and g > g c showsimilarbehaviortothatfor g = g c inthe high-temperatureregime,whileinthelow-temperaturereg ime s ( T g ) approachesaconstantfor g < g c anddivergeslike 1 = T for g > g c .The crossoverscale T foronecaseoneachsideof g = g c isindicatedbya blackarrow.Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 withorder-parameterexponent =0.45(3) IntheKondophase( g < g c ), s ( T g ) approachesaconstantatlowtemperatures (lledsymbolsinFig. 4-7 )becauseofthequenchedimpuritydegreesoffreedominthe screenedKondogroundstate: s ( T g < g c )= const.for T T (4–23) 110

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-0.005 0 0.005 g g c 0 0.05 0.1 0.15 0.2 0.25M s (T=0) 10 -6 10 -5 10 -4 10 -3 10 -2 10 -2 10 -1 (a) 10 -4 10 -3 10 -2 10 -1 g c g 10 1 10 2 10 3 10 4 10 5c s ( T,g) T = 0 (b) Figure4-8.(a)Staggeredmagnetizationatzerotemperatur e M s ( T =0) asafunctionof g g c M s ( T =0) ,servingastheorderparameter,iszerointheKondo phasewhilenonzerointheLMphase.Theinsetshowsthecriti calbehavior whenthecriticalpoint g c isapproachingfromtheLMphase( g > g c ).(b) Log-logplotofthestaticstaggeredspinsusceptibility s ( T g ) at T =0 asa functionof g c g intheKondophase( g < g c ).Thestraightlineshowsa power-lawdivergenceonapproachto g c .Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 111

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Moreover,the T 0 limitof s isfoundtoincreaseas g g c ,andthedataplottedin Fig. 4-8 (b)areconsistentwith s ( g < g c T =0) / ( g c g ) r (4–24) with r =1.52(8) 1 s ( T =0, g < g c ) measurestheeffectiveKondotemperature, whichdecreasestozeroas g g c .ThestrongrenormalizationoftheeffectiveKondo temperatureisbecauseofthecompetingcouplingtothediss ipativebosonicbath,and thevanishingof 1 s ( g = g c T =0) indicatesthecriticaldestructionoftheKondostate atthecriticalpoint g c Inthecriticalregime( T T ),wendananomalousbehaviorof s ( T g ) inboth theKondophaseandtheLMphase(Fig. 4-7 ): s ( T g ) / T x for T T (4–25) wheretheexponent x satises x = s (4–26) with s beingtheexponententeringthedenitionofthebosonicbat hinEq.( 4–7 ).All ourcalculationswereperformedfor s =0.6 T decreasestozeroatthecriticalpoint g = g c ,and s ( g = g c T ) satisesEq.( 4–25 )downto T =0 (bluestarsinFig. 4-7 ). Thecriticalexponentsdenedaboveobeyhyperscalingrela tionsderivedpreviously [ 111 ]forthelocalspinresponseinthesingle-impurityBose-Fe rmiKondo(BFK)model [Eq.( 1–11 )]: 2 = (1 x ), (4–27) r = x (4–28) Suchhyperscalingrelationsareexpectedtoholdataninter acting(i.e.,non-mean-eld) QCP.Inthiscase,thehyperscalingrelationsprescribeall possiblecriticalexponents intermsoftwoindependentexponents,say, and x .Inthetwo-impurityBose-Fermi 112

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Andersonmodelwith s =0.6 ,wend =2.4(2) and x =0.60(1) .HyperscalingEq. ( 4–27 )yields =0.48(6) ,consistentwithinestimateduncertaintywiththedirectl y computedvalue =0.45(3) .Similarly, r =1.44(15) givenbyEq.( 4–28 )isconsistent withthedirectlycomputedvalue r =1.52(8) Itisinterestingtonotethatthevalue x =0.60(1) isidenticaltothatobtainedforthe one-impurityBFKmodel[Eq.( 1–11 )]withthesamebathexponent s =0.6 .However, thecorrelationlengthexponent =2.4(2) issignicantlydifferentfromthat =1.964(2) oftheone-impurityBFKmodel.Eventhoughthecriticalexpo nentsineachcaseobey hyperscaling(providingevidenceforaninteractingQCP), thedifferencebetweenthe valuessuggeststhattheQCPsofthetwomodelslieindiffere ntuniversalityclasses. 4.5FullPhaseDiagram 4.5.1PhaseDiagramandNRGFLow Inthissection,weconsiderthefullHamiltonian[Eq.( 4–1 )]ofthetwo-impurity Bose-FermiAndersonmodelwithbothanonzerocouplingtoth edissipativebosonic bath[Eqs.( 4–5 )-( 4–9 )]andanantiferromagneticinterimpurityexchangeofHeis enberg symmetry[Eq.( 4–3 )].Asbefore,weassumeparticle-holesymmetricimpurityl evelsand inniteimpurityseparation.Thephasediagramisobtained usingtheNRGisshownin Fig. 4-9 .Itcontainsallthreeofthephasesfoundforthetwospecial casesconsidered above:Kondo,interimpuritysinglet(IS),andlocal-momen t(LM).Forsmall I andsmall g ,themodelisintheKondophase.Increasing I withaxedsmall g ,aQPTfromthe KondophasetotheLMphaseisobserved.AKondo-ISQPTisfoun dasweincrease g withaxedsmall I .Furthermore,thephasediagramalsoshowsanIS-LMphase boundaryinaregionwhere I and g arebothmoderatelylarge.Thethreephasesall meetatatriplepoint. Figure 4-10 showsaschematicrenormalization-group(RG)owdiagramo n the g I plane.Thereisastablexedpoint(lledcircle)correspon dingtoeachofthe threephases.AttheKondoxedpoint,thestrongcouplingbe tweentheimpurities 113

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00.20.40.60.81.01.21.4 g 0 0.2 0.4 0.6 0.8 1 I Kondo LM IS Figure4-9.PhasediagramforthefullHamiltonianconsider ingbothHeisenberg interimpurityRKKYinteraction I andbosoniccoupling g .Threedistinct phasesareshown:Kondo,interimpurity-singlet(IS)andlo cal-moment(LM). Thereexistsonetriplepointwhereallthethreephasesmeet .Dashedlines showthephase-spacecutstakeninFig. 4-19 .Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 andthebanddominatesthebosonicandexchangeenergyscale s,sotheeffective hybridization goestoinnity;theimpurityisstronglycoupledtotheferm ionicband butdecoupledfromthebosonicbath.AttheISxedpoint,the effectiveinterimpurity exchangeinteraction I becomesinnite,dominatingtheothertwoenergyscales T K and E g .Inthiscase,theimpuritiesformsaninterimpuritysingle twhichbecomesdecoupled fromboththefermionicbandandthebosonicbath.AttheLMx edpoint,thebosonic couplingenergyscale E g becomesinnite,dominatingboth T K and I ,meaningthatthe impuritiesarestronglycoupledtothebosonicbath. 114

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Figure4-10.SchematicowdiagramofthefullHamiltoniani nEq.( 4.2 )consideringboth Heisenbergexchangeinteraction I andbosoniccoupling g .Threestable xedpointsareshownwithlledcircles:Kondo(KS),ISandL M.The unstableKIcriticalpointandKDcriticalpointaredenoted bythehollow circlesontheaxes.Thearrowsshowthedirectionofthereno rmalization groupow. Inaddition,Fig. 4-9 showstwounstablexedpoints:KI(correspondingtothe criticalpointseparatingtheKondoandISphases,indicate dbyahollowcircleonthe I axis)andKD(correspondingtotheKondo-destructioncriti calpointseparatingthe KondoandLMphases,indicatedbyahollowcircleonthe g axis).NeartheKIxed point,energyscales T K and I arestronglycompetingwhilethebosoniccouplingenergy scale E g isnegligible,leadingtothedecouplingofthebosonicbath fromtherestofthe system.NeartheKDxedpoint, I isnegligibleandthecompetitionbetweentheKondo effectandthedissipationcausedbythebosonicbathgivesr isetothecriticalproperties. OntheKondo-LMphaseboundary,theRGowisfromthetriplep oint(wherethethree 115

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010 0 0.5 1E N 010 n b =0 n b >0 01020 N (even iteration) (a) (b) (c) g=0.0 g=0.2 g=0.4 Figure4-11.NRGspectrumateveniterationnumbers N forthreepointsonthe Kondo-ISphaseboundary.(a)SpectrumoftheQCPonthe I axiswith g =0 ,includingonlyenergylevelswithzerobosonnumber( n b =0 ).(b),(c) Spectraofpointswithnonzero g ;energylevelswith n b =0 areindicatedby blacksolidlineswhilethosewith n b > 0 areindicatedbyreddashedlines. AllthreepanelsshowthesameNRGspectrumforlarge N .Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 phasesmeet)totheKondo-destruction(KD)xedpoint.Like wise,theRGowon theKondo-ISphaseboundaryisfromthetriple-pointtoward thexedpointKI,which separatestheKondoandISphasesonthe I axis. Thesubsectionsthatfollowpresentthenumericalevidence forthepicture presentedinFig. 4-10 4.5.2Kondo-ISPhaseBoundary WerstlookatthecriticalbehaviorontheKondo-ISphasebo undary.Fig. 4-11 showsthelow-lyingNRGspectrumateveniterationnumber N forthreepointsonthe boundary,correspondingto g =0 g =0.2 ,and g =0.4 .Atthepointwith g =0 weignorethedecoupledbosonicbathinthecalculation,sot heNRGspectruminFig. 4-11 (a)onlyhasthecontributionfromtheKondopartcontaining statesarisingfromthe interationoftheimpuritieswiththefermionicband.Inthe panels(b)and(c)representing nonzerobosoniccoupling g ,thelow-lyingNRGspectrumfor N > 8 canbeinterpreted asthesuperpositionoftwoparts:aKondopartandafreeboso nicbath.Energylevels inFigs. 4-11 (b)and 4-11 (c)denotedbysolidblackcurvescoincidefor N > 8 with 116

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10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 T 0 5 10 15 20c s ( T, g, I=I c ) g = 0.0 g = 0.2 g = 0.4 Figure4-12.Temperature-dependentstaticstaggeredspin susceptibility s ( T g I ) for pointsontheKondo-ISphaseboundary.Thestraightlinesin dicatethe logarithmicdivergenceof s ( T g I ) astemperaturegoestozero.This criticalbehaviorisvalidforallthepointsontheKIphaseb oundary, althoughtheslopeofthestraightlinesvariesalongthebou ndary.Dataare for U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 thelevelsinFig. 4-11 (a)fortheproblemwithoutbosons.TheremaininglevelsinF igs. 4-11 (b)and 4-11 (c),drawnwithdashedredlines,areeitherfree-bosonleve ls(e.g.,the lowesttwosuchlevelsinthesepanels)orcombinationsofKo ndoexcitedstateswith freebosonlevels(e.g.,thethirdlevelshown).Weconclude thatpointsontheKondo-IS phaseboundaryowtotheKIcriticalpointonthe I axiswithadecouplingbetweenthe Kondospectrumandthatofthebosonicbath,consistentwith thedirectionofthearrow ontheKondo-ISboundaryinFig. 4-10 TofurtherdemonstratethesamecriticalbehaviorontheKIp haseboundaryas thatattheKIcriticalpoint( g =0 ),wecalculatethetemperature-dependentlocal 117

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01020 0 0.5 1E N 01020 01020 (a) (b) (c) N (even iteration) I=0.0 I=0.2I=0.4 Figure4-13.NRGspectrumateveniterationnumber N forthreepointsonthe Kondo-LMphaseboundary.(a) I =0 ,(b) I =0.2 ,(c) I =0.4 .Allthree casesexhibitthesamespectrumforlarge N .Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 staggeredsusceptibilitydenedbyEq.( 4–12 ).Fig. 4-12 showsthatboundarypointsfor g =0.2 and g =0.4 (redsquaresandbluetriangles)exhibitalogarithmictemp erature dependenceof s ( T g I = I c ) ,similartothatof s ( T g =0, I = I c ) (blackcircles).This suggeststhatallthepointsontheboundaryfollowthesamec riticalbehaviordescribed byEq.( 4–15 ),withonlytheprefactorof log T inEq.( 4–15 )being g -dependent. 4.5.3Kondo-LMPhaseBoundary WenowturntodiscussionoftheKondo-LMphaseboundary.Fig 4-13 shows theNRGspectrumateveniterationnumbers N forthreepointsontheboundary correspondingto I =0.0,0.2 ,and 0.4 .Forsufcientlylarge N ( N > 14 inFig. 4-13 ),the NRGspectraareallthesame,indicatingthatthedirectiono frenormaliation-groupow onthisboundaryisdownwardsfromthetriplepointtotheKDc riticalpointonthe g axis, asshowninFig. 4-10 .Asaresult,weexpectthecriticalbehavioreverywhereont he Kondo-LMphaseboundarytocoincidewiththatoftheKDcriti calpoint. Fig. 4-14 (a)showsthecrossoverenergyscale T asafunctionof j g g c j at I =0.2 .WenotethatEq.( 4–20 )appliedabovefor I =0 alsoholdswellforthepoint onthephaseboundarywithnonzero I =0.2 ,andgivesacorrelation-lengthexponent 118

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10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 |g-g c | 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 T gg c (a) 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 T 10 2 10 3 10 4 10 5c s (T, g, I=0.2) -7 5 10 -3 -1.5 5 10 -3 -5 5 10 -4 0 5 5 10 -4 2.5 5 10 -3 1.2 5 10 -2 g c g-g c = (b) Figure4-14.(a)Crossoverenergyscale T asafunctionofdistance j g g c j fromthe Kondo-LMphaseboundaryatxed I =0.2 .(b)Temperature-dependent staticstaggeredspinsusceptibility s ( T g I ) at I =0.2 forseveralxed valuesof g straddlingtheKondo-LMphaseboundary.Bothguresshow exactlythesamecriticalbehaviorasthatfoundfor I =0 .Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 119

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01020 0 0.25 0.5E N 01020 01020 N (even iteration) (a) (c) (b) I=0.8 I=0.9I=1.0 Figure4-15.NRGspectrumasafunctionofeveniterationnum ber N forthreepointson theIS-LMphaseboundary.ThethreepanelsshowthesameNRGspectrumforlarge N ,indicatingreormalization-groupowtothesame criticalxedpoint.Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1200 =2.5(2) inagreement(withinestimateduncertainty)withthevalue =2.4(2) found for I =0 .Figure 4-14 (b)showsthetemperature-dependentstaggeredspinsuscep tibility neartheKondo-LMphaseboundarywith I =0.2 .Forcurveswith g < g c (lledsymbols), Eq.( 4–25 )holdsinthehigh-temperaturecriticalregime( T T )whileEq.( 4–23 ) holdsinthelow-temperatureKondoregime( T T ).Forcurveswith g > g c (hollow symbols),Eq.( 4–25 )holdsinthecriticalregime( T T )whileEq.( 4–21 )holdsforin thelow-temperatureLMregime( T T ).Forthecurvewith g = g c (violetstars),Eq. ( 4–25 )holdsovertheentiretemperatureregimebecause T =0 atthecriticalpoint. Thecriticalexponent x =0.61(2) deducedfromFig. 4-14 (b)agreeswithinerrorwiththe one x =0.60(1) for I =0 4.5.4IS-LMPhaseBoundary Last,welookintothecriticalbehaviorontheIS-LMphasebo undary.Fig. 4-15 plots theNRGspectrumateveniterationnumbers N forthreepointsontheIS-LMphase boundarywith I =0.8,0.9 ,and 1.0 .Wenotethatthethreepanelsshownthesame NRGspectrumwhen N > 18 ,whichsuggestrenormalization-groupowontheIS-LM boundaryisupwardawayfromthetriplepoint,asshowninFig 4-10 .Wespeculate 120

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thattheowistowardanunstablecriticalxedpoint(notsh own)characterizedby hybridizationwidth =0 4.6PairingSusceptibility Inthissection,weinvestigatethesuperconductingpairin gbetweenthetwoimpurity spins.Pairingoperators yd = 1 p 2 ( d y 1 d y 2 # d y 1 # d y 2 ) (4–29) and yp = 1 p 2 ( d y 1 d y 2 # + d y 1 # d y 2 ) (4–30) createasingletandatripletbetweenthetwoimpuritiesres pectively.Thezero temperature( T =0 )dynamicalpairingsusceptibilityisdenedas ( )= i Z 1 0 dte i t h [ ( t ), y (0)] i = d p (4–31) Onecanwrite ( )= 0 ( )+ i 00 ( ) with 0 ( ) and 00 ( ) beingtherealand imaginaryparts.WithintheNRG,thestaticpairingsuscept ibility ( =0) isobtained fromtheimaginarypart 00 ( ) viatheHilberttransform ( =0)= 0 ( =0)= 1 P Z 1 1 00 ( 0 ) 0 d 0 (4–32) where P denotestheCauchyprincipalvalue. First,weconsiderthesingletpairingacrossthefullphase diagram,focusing especiallyonitsbehaviornearphaseboundaries.Fig. 4-16 showsthesingletpairing susceptibility d ( g I ) alonglinesofxedcoupling g .For g =0 (blackcircles),a divergenceofthesingetpairingsusceptibilityatthephas eboundarypointstoasinglet pairinginstabilityattheQCP,asfoundpreviouslyin[ 105 ],whichinvestigatedthepairing propertiesbetweentwoimpuritieswithoutanybosonicbath .AswediscussedinSec. 4.5.2 ,thecriticalpropertiesontheKondo-ISphaseboundaryapp eartobethesame asthoseonthe I axis.Datafornonzero g inFig. 4-16 showasimilarpairinginstability onthephaseboundary.Theleftwardshiftofthepeakwithinc reaseof g reectsthe 121

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0.4 0.5 0.60.70.80.9 I 0 2 4 6c d (I,g) g=0.0 g=0.2 g=0.5 g=0.8 g=0.9 Figure4-16.Staticsingletpairingsusceptibility d ( I g ) at T =0 asafunctionof I for vexedvaluesof g .Thedivergenceof d ( I g ) ontheboundaryindicates apairinginstability.Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1300 slightdecreaseofthecriticalvalueof I c alongthephaseboundary,seeFig. 4-9 .Fig. 4-17 showsthebehaviorof d ( g I ) closetothephaseboundary.Similartothecritical behaviorofthestaggeredspinsusceptibility s ( T =0, g I ) describedbyEq.( 4–13 ),the singletpairingsusceptibilityneartheKondo-ISphasebou ndarysatises d ( g I ) / log 1 j I I c j (4–33) Thedifferentslopesofthestraightlinesonthelog-linear plotindicatethattheprefactor oftheloginEq.( 4–33 )is g dependent. Bycontrast, d ( g I ) alonglinesofxed I [Fig. 4-18 ]isalmostaconstantinthe Kondophase,hasaslightenhancementaroundtheQCP,andsho wsstrongdrop-offin 122

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10 -4 10 -3 10 -2 10 -1 10 0 | I-I c | 0 2.5 5 7.5c d (I,g) g=0.0 g=0.2 g=0.5 g=0.8 g=0.9 Figure4-17.Staticsingletpairingsusceptibility d ( I g ) at T =0 asafunctionofthe distancetotheKondo-ISphaseboundaryforvexedvalueso f g .The straightlinesindicatealogarithmicdivergenceof d ( I g ) nearthe Kondo-ISQCP.Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1300 theLMphase.ItiscertainlythecasethattheQCPontheKondo -LMphaseboundaryis notassociatedwithanysuppressionofsingletpairing. Thedatashownabovepointtoasignicantdifferencebetwee ntheeffectsonthe pairingsusceptibilityofadynamic( g )andastatic( I )magneticinteractionbetween thetwoimpurityspins.Itisthereforeinterestingtoconsi derthebehaviorunderthe simultaneousvariationof g and I .Startingfromtheindependent-impuritypointat g =0 and I =0 ,wetuneboth g and I acrossthephasediagramalongthreecutswithxed I = g = c (asmarkedinFig. 4-9 ).AsshowninFig. 4-19 ,thesingletpairingsusceptibility growsas g increasesfromzero,ispeakedfor g slightlybelow g c ,andthenfallsoff withintheLMphasebecauseofthebosonicdissipation.Thel argerthecoefcient c ,the 123

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0 0.5 1 1.5 g 0 1c d (g,I) I=0.0 I=0.2 I=0.4 Figure4-18.Staticsingletpairingsusceptibility d ( I g ) at T =0 asafunctionof g for threexedvaluesof I .Ineachcase,averticalarrowofthesamecolor indicatesthelocation g = g c ( I ) oftheKondo-LMphaseboundary.Dataare for U = 2 d =0.001 =0.25 =9 N b =4 and N s =1300 strongeristheenhancementofthesingletpairingsuscepti bilityaroundtheQCP.Larger c resultsincrossingtheKondo-LMphaseboundaryclosertoth etriplepoint.Therefore, wecanassociatethestrongerenhancementofthesingletpai ringsusceptibilitywith criticaluctuationsthattakeplacewhenthereisnontrivi alcompetitionbetweenKondo screeningandbothstaticanddynamicalinterimpurityexch ange. Withintheextendeddynamicmeaneldtheory(EDMFT),theen hancedpairing susceptibilityintheimpuritymodelcausesapairinginsta bilitynearaFermi-surface-collapsing QCPofaKondolattice[ 79 80 ],whichwouldrepresentanewmechanismforsuperconductiv ity inthevicinityofantiferromagneticorder.Intheconventi onalEDMFT,theAnderson latticemodelismappedtoaoneimpurityBose-FermiAnderso nmodel,inwhichthe 124

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0 0.5 1 g 0 0.5 1 1.5 2 2.5 3c d (g,I) I=0.28g I=0.39g I=0.52g Figure4-19.Staticsingletpairingsusceptibility d ( I g ) at T =0 alongthephase-space cuts I = g = c markedinFig. 4-9 .Ineachcase,averticalarrowofthesame colorindicatesthelocation g = g c ( I ) oftheKondo-LMphaseboundary. Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1300 unitcellcontainingonlyonelatticesitecouplestoaboson icbathcapturingmagnetic uctuationsgeneratedviatheRKKYinteractionbyalltheot herlatticesites.Acluster extendeddynamicalmean-eldtheory(c-EDMFT)method[ 123 ]hasrecentlybeen developedtomaptheAndersonlatticemodeltoatwoimpurity Bose-FermiAnderson model,allowingstudyofthesuperconductingpairingbetwe enthetwoimpuritiesin thepresenceoftheKondoeffectandlatticemagnetism.Inth isapproach,theunitcell containstwoadjacentlatticesites(twoimpurities),andt hebosonicbathrepresentthe effectontheunitcellfromalltheotherlatticesites,simi lartothatintheoneimpurity Bose-FermiAndersonmodel.Inthec-EDMFT,thebosonicbath intheeffective two-impurityBose-FermiAndersonmodelmustbedetermined self-consistentlyto ensurethatthetwochosensitesarerepresentativeofthela tticeasawhole.Inthe 125

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0 0.5 1 g 0 0.5 1c p (g,I) I=0.28g I=0.39g I=0.52g Figure4-20.Statictripletpairingsusceptibility p ( I g ) at T =0 alongthephase-space cuts I = g = c markedinFig. 4-9 .Ineachcase,averticalarrowofthesame colorindicatesthelocation g = g c ( I ) oftheKondo-LMphaseboundary. Dataarefor U = 2 d =0.001 =0.25 =9 N b =4 and N s =1300 calculationsreportedinthischapter,wejustsetthedensi tyofstatesofthebosonicbath tohavetheformgiveninEq.( 4–7 ),sotheresultsrepresentonlyarststeptowards thefullc-EDMFTtreatment.ConsideringthatlargerintersiteRKKYinteraction I in theAndersonlatticemodelgivesrisetoastrongerbosonicc ouplingtotheimpurity spinsinthec-EDMFTmethod,itisveryrelevanttoconsidert heeffectofvaryingboth g and I simultaneouslyalongcutssuchasthelines I = cg consideredinFig. 4-19 Therefore,theenhancementofthesingletpairingsuscepti bilityalongsuchlinesisa signicantndingthatopensthepossibilityforanewmecha nismforsuperconductivity nearantiferromagneticQCPsinheavy-fermionsystems.Thi swillbeatopicforfuture work. 126

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Toconcludethissection,webrieypresentNRGresultsfort hestatictripletpairing susceptibility p ( I g ) denedinEq.( 4–31 ).Fig. 4-20 shows p ( I g ) alongthesame phase-diagramcutsmarkedinFig. 4-9 .Thereisclearlynoenhancementoftriplet pairingneartheKondo-LMphaseboundary.Infact,weobserv ethatanyincreaseof I and/or g suppressesthestatictripletpairingsusceptibility.For largeantiferromegnetic exchangeinteraction I > 0 ,twoimpuritiesareinclinedtoformainterimpuritysingle t, whichsuppressestheformationoftripletbetweenthetwoim purities.Ontheother hand,forlargebosoniccoupling g ,twoimpuritiesfavor S z 1 S z 2 = 1 ,meaning opposingdirectionsofthetwoimpurityspins,whichalsote ndstobreakthetriplet.So theobservednumericalresultsforthestatictripletpairi ngsusceptibilityareentirely consistentwithexpectations. 127

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CHAPTER5 CONCLUSIONANDFUTUREWORK 5.1Conclusion Wehavestudiedtheoreticallyandnumericallyquantumimpu ritymodelsinwhich localizeddegreesoffreedomarecouplednotonlytoafermio nicband,butalsoto bosonicdegreesoffreedom.Generally,problemswithboson icdegreesoffreedomcan bedividedintotwocategories. Intherstcategory,thenumberofbosonicdegreesislimite dandthetreatment viathenumericalrenormalizationgroup(NRG)isthesameas formoreconventional problemswithoutbosons.Modelsinthisrstcategoryareus uallyrelevanttoproblems innanoscaledeviceswhereelectron-phononinteractionsa ffectthetransportproperties. Inthesecondcategory,thebosonicdegreesoffreedomforma continuumandthe presenceofthebosonicbathsignicantlycomplicatestheN RGtreatment.Modelsin thesecondcategoryareofinterestaseffectivedescriptio nsofheavy-fermionsystems exhibitingunconventionalquantumcriticality.Inthisdi ssertation,wehaveinvestigateda modelineachcategory. Generally,electron-bosoninteractionscompetewiththes trongelectron-electron interactions,responsiblefortheKondoeffect.Innanocal edevices,suchasquantum dotsorsingle-moleculejunctions,strongcouplingofaloc alizedchargetoadiscrete vibrationalmode(phonons)leadstothereductionofstrong Coulombrepulsionin thedevicesandcanbeevenleadtoeffectiveelectron-elect ronattraction.Attraction betweentwoimpurityelectronssuppressestheappearanceo ftheKondoeffect. Couplingtoacontinuousbosonicbathcreatesinertia,supp ressesspin-ipscattering andtendstodecoupleamagneticimpurityfromconductionel ectrons.Thisgives arisetoaclassofKondo-destructionquantumcriticalpoin tsseparatingKondoand local-momentphases. 128

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Thetwo-orbitalmolecularjunctionmodelconsideredinCha pter 3 hasonly onebosonicmode,anddescribesasingle-moleculerdevicew ithtwoactiveorbitals connectingapairofleads.Bothinter-andintra-orbitalCo ulombrepulsionareincluded inthemodel,andtwokindsofelectron-phononinteractions ,phonon-assistedinter-orbital tunnelingandHolstein-typephononcoupling,havebeenstu died.Variationofthe electron-phononcouplingstrengthproducesalevelcrossi ngfromsingletodouble occupancyinthemolecule,andresultsinasmoothcrossover fromaKondoregime toaphonon-dominatedregimeinwhichKondoeffectissuppre ssed.Inaside-orbital conguration,thiscrossoverbecomesarst-orderquantum phasetransition(QPT) betweenaKondophaseandaphonon-dominatedphase.ThisQPT arisesfroma particularsymmetrythatispresentforpurelyphonon-assi stedtunnelingbetweenthe twomolecularorbitals.InthefullHamiltonianincorporat ingbothphonon-assisted tunnelingandHolstein-typephononcoupling,thetwophono neffectscooperateinthe renormalizationoftheeffectiveCoulombrepulsionandall owtheeffective e e interaction tobecomeattractiveinthemolecule.AchargeanalogoftheK ondoeffecthasbeen foundintheregimeofnegativeeffectiveCoulombrepulsion Thetwo-impurityBose-FermiAndersonmodeldescribestwol ocalizedlevelsthat hybridizewithconductionelectronsandalsocoupletoabos onicbathviathedifference ofthetwoimpurityspins.Theimpuritiesinteractwitheach otherthroughadirect Heisenbergexchangeinteraction,providingtheopportuni tytostudysuperconducting pairinginthepresenceoflatticemagnetismandKondocorre lation.Forthefullmodel Hamiltonian,competitionbetweenthreeenergyscales(Kon dotemperature T K interimpurityinteraction I andbosoniccouplingenergyscale E g )leadstoaphase diagramwiththreedistinctphases:Kondo,interimpuritysinglet(IS)andlocal-moment (LM).WehavestudiedtheKondo-ISphasetransitionwithout bosoniccouplingand thetheKondo-LMphasetransitionwithoutdirectexchangei nteractioin.Ineachcase, wehaveexploredthecriticalpropertiesbycalculatingthe NRGmany-bodyspectrum 129

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andthestaticstaggeredlocalsusceptibility.Therenorma lizationgroupowofthefull modelhasbeenstudiedandthecriticalbehavioroneachphas eboundarieshasbeen computed.Finally,wehaveinvestigatedsingletsupercond uctingpairingacrossthe phasediagram.ComparedwiththecaseoftwoindependentAnd ersonimpurities,we observeenhancementofthesingletpairingsusceptibility aroundtheKondo-destruction QCP,butnoenhancementoftripletpairing.Theobservedenh ancementcouldform thebasisforanewmechanismforsuperconductivityinthevi cinityofantiferromagnetic order,andconnectstosuperconductivityobservedinvario usheavy-fermionmaterials. 5.2FutureWork Andersonmodelswithbosonicdegreesoffreedomcontainint erestingandexciting physics,especiallyindescribingnano-deveicesandheavy -fermionsystems.Inthis section,wesummaryseveralissuesthatmightbeaddressedi nthefuture: (1)InChapter 3 ,westudiedthemodelHamiltonianforatwo-orbitalsinglemolecule junctiononlyintheanti-adiabaticregime,characterized bytherelation 0 .A crossoverfromtheadiabaticregimetotheanti-adiabaticr egimehasbeeninvestigated intheAnderson-Holsteinmodel[ 124 ].Intheanti-adiabaticregime,thephononcan efcientlyrespondtoeachmolecule-bandelectronhopping eventbyformingapolaron, suppressingtherebytheelectronictunnelingrateacrosst hemolecularjunction.Inthe adiabaticregime,thephononistooslowtorespondtothefre quenttunnelingevents, havinglittleeffectontheirrate.Itwouldbeinterestingt oconsiderthiscrossoverinthe two-orbitalmolecularjunctionmodel,wherephononeffect sonthetransportproperties areexpectedtobedifferentintheadiabaticandanti-adiab aticregimes. (2)InChapter 4 ,wefoundthatthecorrelation-lengthexponentcharacteri zing theKondo-LMphasetransitiondiffersfromthatoftheoneim purityBFKmodel,which suggeststhattheQCPsofthetwomodelslieindifferentuniv ersalityclasses.Future NRGworkwillbecarriedouttoinvestigatethissystematica llyfordifferentvaluesofthe bathexponent s .ItwillalsobeinterestingtocompareourresultsforHeise nberg( S 1 S 2 ) 130

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inter-impurityexchangewithfutureNRGresultsforIsing( S z 1 S z 2 )exchange.Itisnotclear whetherloweringtheexchangesymmetrywillaffecttheuniv ersalityclassoftheKondo destructionQPT. (3)AsinChapter 4 ,wecalculatedtheNRGspectrumofthepointsontheIS-LM phaseboundary,andconcludedthattheRGowonthisboundar yisfromthetriple pointtoanunstablexedpointseparatingtheISandLMphase s.Moreworkisneeded tounderstandthepropertiesofthisunstablequantumcriti calpoint.Wecanstartfrom atwo-impurityspin-bosonmodelcontainingtwospinscoupl edtoeachotherandalso coupledviatheirdifferencetoabosonicbath.Aquantumpha setransitionbetweenIS andLMphasesisexpectedfromthismodel. (4)AswementionedattheendofChapter 4 ,ourstudyofthetwo-impurity Bose-FermiAndersonmodelisonlyarststeptowardsaclust er-EDFMTtreatment oftheAndersonlattice.Wearegoingtocarryoutthefullc-E DMFTmethodtostudy superconductingpairingaroundtheQCPseparatingKondoan dLMphases.Inthis method,thebosonicbathisdeterminediterativelytoensur ethatthetwochosensites arerepresentativeoftheAndersonlatticeasawhole.Ateac hiteration,thedensityof statesofthebosonicbathisdeterminedfromthedynamicals pinsusceptibilityobtained inthepreviousiteration.Theiterativeprocessstopswhen thedensityofstatesofthe bosonicbathconverges.Thesolutionofthisself-consiste ntversionofthetwo-impurity Bose-FermiAndersonmodelshouldprovideasignicantadva nceintheunderstanding ofunconventionalsupersonductivityinheavyfermionsyst ems. 131

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BIOGRAPHICALSKETCH LiliDengwasborninDecember1982inBeibei,Chongqing,Chi na.Heattendedthe HighSchoolAfliatedtoSouthwestChinaNormalUniversity inBeibei,Chongqingfrom 1996to2002.InSeptember2002,heenrolledatthePhysicsDe partmentofTsinghua UniversityinBeijing,China.HegraduatedwithaBacheloro fScienceinJune2006.In August2008,heenrolledingraduateschoolinthePhysicsDe partmentoftheUniversity ofFlorida,Gainesville,Florida.Inthespringof2009,hej oinedProf.KevinIngersent's researchgrouptobeginhisPh.D.studies.HewasawardedaMa sterofSciencein December2009.HeandhiswifeDanFenggotmarriedinApril20 11.TheirsonNeil wasborninOctober2012.HereceivedhisPh.D.fromtheUnive rsityofFloridainMay 2014. 138