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Equilibrium and Time-Dependent Properties of Quantum Impurity Systems

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Title:
Equilibrium and Time-Dependent Properties of Quantum Impurity Systems
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Wagner, Christopher E
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[Gainesville, Fla.]
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University of Florida
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english
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Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
INGERSENT,J KEVIN
Committee Co-Chair:
BISWAS,AMLAN
Committee Members:
HERSHFIELD,SELMAN PHILIP
MUTTALIB,KHANDKER A
HENNIG,RICHARD

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

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Abstract:
Quantum impurity problems provide a simple yet powerful way to study complex strongly correlated systems. The Kondo and Anderson models for an impurity attached to a metallic host (with density of states $\rho(\epsilon) = \rho_0$) or semimetallic host (with pseudogapped density of states $\rho(\epsilon) \propto |\epsilon|^r$ where $\rho(0) = 0$ for the Fermi energy) are natural systems for studying entanglement, as they are both interacting and non-local, yet tractable models to solve. Entanglement is especially difficult to study in the presence of a quantum phase transition (QPT), as few systems that exhibit a QPT are solvable. The pseudogap Kondo and Anderson models both exhibit QPTs with a quantum critical point (QCP) which separates a Kondo phase from a local-moment phase in which the Kondo effect is destroyed. This allows for the study of the many-body correlations in these models through the quantification of the entanglement. Previous studies of entanglement in the spin-boson model near an impurity QCP showed the entanglement between the impurity and its environment to take a cusp peak at the QCP. This dissertation establishes that in the pseudogap Anderson model, there is a cusp peak at the Kondo-destruction QCP only in limited cases, and finds a critical scaling of the entanglement on approach to the QCP. Furthermore, the entanglement calculation is generalized to group the impurity with the electrons within a radius $R$. This calculation of the entanglement with a boundary a distance $R$ from an impurity in the pseudogap Kondo model reveals a length scale $R^*$ for the system, which describes a crossover from the maximal entanglement associated with the Kondo-destruction QCP (occurring for $R \ll R^*$ ) and the lower entanglement associated with one or the other of the stable phases separated by the QCP (occurring for $R \gg R^*$ ). This characteristic length can be understood as the size of the Kondo screening cloud, the length over which there is strong interaction between the impurity and conduction band, and it is found to vary inversely with $T^*$ , the characteristic temperature and energy scale of the problem. On approach towards the QCP from either phase, $T^*$ vanishes in a countinuous fashion. Consequently, the length scale $R^* /propto 1/T^*$ diverges and the entire system becomes entangled with the impurity. Future work will adopt the concepts used in the time-dependent numerical renormalization group to calculate the time-dependence of the entanglement entropy. The mathematical set up for this numerical calculation is laid out near the end of this dissertation. ( en )
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Thesis (Ph.D.)--University of Florida, 2018.
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Adviser: INGERSENT,J KEVIN.
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Co-adviser: BISWAS,AMLAN.
Statement of Responsibility:
by Christopher E Wagner.

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EQUILIBRIUMANDTIME-DEPENDENTPROPERTIESOFQUANTUMIMPURITYSYSTEMSByCHRISTOPHERELIASWAGNERADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2018

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c2018ChristopherEliasWagner

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Idedicatethistomymother,NancyWagner.

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ACKNOWLEDGMENTSIwouldrstandforemostliketoacknowledgeandthankmyadvisor,Dr.KevinIngersent,forhisperpetualaidwiththisresearchanddissertation,aswellashiskindness,understanding,andpatiencethroughoutthisjourney.IthankmycommitteemembersDr.AmlanBiswas,Dr.RichardHennig,Dr.SelmanHersheld,andDr.KhandkerMuttalibfortheirsuggestions.Igreatlyappreciatemyotherresearchcollaborators,Dr.JedediahPixleyandDr.TathagataChowdhury,fortheirinterestinthisresearchandtheirinsightfulconversationsthroughoutthisprocess.IwouldliketothankDavidHansenandClintCollinsfortheirimmensehelpinallissuesassociatedwithcomputing.IalsoappreciatepartialsupportofthisresearchunderNSFGrantNo.DMR-1508122.IwishtoacknowledgeandthankthecommunityatSt.AugustineCatholicChurchfortheirprayersandspiritualsupportthroughoutmytimehereinGainesville,especiallyFr.DavidRuchinski,GailFitsimmons,RaulFernandez,BradleyNartowt,andVincentHerzog.Ithankmyparentsfortheirencouragementinmycontinuededucationandpursuitofknowledge.Lastlyandmostimportantly,Ithankmywife,LydiaWagner,forhercontinualsupportthroughoutthisdegree. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1BACKGROUND ................................... 12 1.1BriefHistoryoftheKondoProblemandImpurityModels ......... 12 1.2PseudogappedModels ............................. 16 1.3PoorMan'sScalingandtheRenormalizationGroup ............. 17 1.4EntanglementEntropy ............................. 21 2NUMERICALRENORMALIZATIONGROUPMETHODS ........... 24 2.1DiscretizationandtheTightBindingChain ................. 24 2.2IterativeDiagonalization ............................ 28 2.3FixedPoints ................................... 31 2.4ThermodynamicProperties .......................... 33 2.5ExtractionoftheCharacteristicTemperatureScales ............. 34 2.6DensityMatrixNumericalRenormalizationGroup .............. 35 2.7TheCompleteBasisSet ............................ 38 2.8EntanglementCalculationsUsingtheReducedDensityMatrix ....... 42 2.9NonequilibriumProblems ........................... 45 3ENTANGLEMENTENTROPYANDTHEQUANTUMCRITICALPOINT .. 48 3.1GeneralConsiderations ............................. 48 3.2AndersonModels ................................ 53 3.2.1Particle-HoleSymmetry:U=)]TJ /F1 11.955 Tf 9.3 0 Td[(2d .................. 56 3.2.2MaximalParticle-HoleAsymmetry:U=1 .............. 59 3.2.3GeneralParticle-HoleAsymmetry ................... 62 3.3WorkonRelatedModels ............................ 65 3.4Discussion .................................... 66 3.5Conclusions ................................... 67 4LONG-RANGEENTANGLEMENTNEARAKONDO-DESTRUCTIONQUANTUMCRITICALPOINT .................................. 69 4.1EntanglementwithintheWilsonChain .................... 72 4.1.1TheWilsonChain ............................ 72 4.1.2SystematicsoftheWilsonChainEntanglementEntropy ....... 73 5

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4.2EntanglementEntropyfortheKondoProblem ................ 79 4.2.1RadialDistributionofEntanglementEntropy ............. 79 4.2.2Fixed-PointEntanglementEntropyvsr ................ 83 4.2.3EntanglementEntropyasaFunctionofKondoCouplingJ ..... 87 4.3Discussion .................................... 89 5TIME-DEPENDENCEINQUANTUMIMPURITYSYSTEMS ......... 91 6CONCLUSIONANDFUTUREWORK ...................... 95 6.1Conclusion .................................... 95 6.2FutureWork ................................... 96 APPENDIX:SINGLEPARTICLECORRELATIONFUNCTIONAPPROACH ... 98 REFERENCES ....................................... 100 BIOGRAPHICALSKETCH ................................ 105 6

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LISTOFTABLES Table page 3-1Propertiesatthequantumcriticalpointoftheparticle-hole-symmetricpseudogapAndersonmodelforr=0:4andthreevaluesofU=D. ............... 58 4-1ValuesofthecoecientscandbdenedinEq.(4{6)fortheSTBandforWilsonchainswithdierentbandexponentsr. ....................... 78 4-2ValuesoftheexponentdenedinEq.(4{8)fordierentbandexponentsr,asdeterminedinthelocal-moment(LM)andKondo(K)phases .......... 84 7

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LISTOFFIGURES Figure page 1-1Resistancevstemperaturecurveshowingtheresistanceminimum. ........ 14 1-2Schematicdiagramforapoorman'sscalingapproach. .............. 17 1-3Flowdiagrams. .................................... 19 1-4SchematicphasediagramforthepseudogapAndersonmodelforxedU .... 20 2-1Representationofdiscretizedconductionband. ................... 25 2-2SchematicdiagramoftheNRGtight-bindingchain. ................ 28 2-3Schematicdiagramoftheiterativediagonalizationprocess. ............ 29 2-4Severelysimpliedschematicdiagramoftheenergyspectrum)]TJ /F4 7.97 Tf 6.59 0 Td[(m=2Hmwithincreasingiterationnumberm. ........................... 30 2-5RenormalizationgroupowdiagramforthesymmetricAndersonmodel. .... 32 2-6ExtractionofthecharacteristictemperaturescaleTinthepseudogapKondomodelfromimp(T),theimpuritycontributiontotheuniformmagneticsusceptibility. 35 2-7Representationofthe\environment". ........................ 38 2-8SchematicdiagramofthecompleteNRGbasis. .................. 42 2-9Schematicdiagramofthegeneralpulsemethod. .................. 47 3-1SchematicrepresentationofthemodelHamiltoniansconsideredinthiswork. .. 49 3-2Particle-hole-symmetricpseudogapAndersonmodelwithbandexponentr=0:4fordierentvaluesofU=D. ........................... 57 3-3U=1pseudogapAndersonmodelforr=0:6. .................. 61 3-4PseudogapAndersonmodelwithbandexponentr=0:6 ............. 62 3-5CriticalbehavioroftheentanglemententropySeforthepseudogapAndersonmodelwithr=0:6. .................................. 64 4-1NRGrepresentationoftheKondomodelasatight-bindingWilsonchainofNsitescoupledatoneendtoanimpurityspin. .................... 70 4-2Tight-bindinghoppingparametersfortheWilsonchain. ............. 74 4-3WilsonchainentanglemententropySavgevspartitionsizeLforametallicdensityofstatesdescribedbyEq.(1{11)withr=0. .................... 75 4-4SavgeforWilsonchainsoflengthN=600andavarietyofbandexponentsr. .. 77 8

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4-5FittedcoecientscandbinEq.(4{6)foraWilsonchainwithdiscretization=1,andbandexponentsr=0,0:2,and0:4. .................. 77 4-6ImpurityentanglemententropySimpevs(a)WilsonchainpartitionsizeLand(b)scaleddistancefromtheimpurityR=RKforametallichost(r=0). ..... 80 4-7ImpurityentanglemententropySimpevsWilsonchainpartitionsizeLforapseudogapKondomodelwithbandexponentr=0:4. ..................... 81 4-8DatafromFig.4-7replottedvsR=R,whereR/L=2=kFandR=1=(kFT)withTbeingacrossovertemperatureextractedfromthemagneticsusceptibility. 82 4-9ValuesoftheexponentdenedinEq.(4{8)fordierentbandexponentsr. .. 86 4-10ImpurityentanglementatthequantumcriticalpointandtheKondoxedpoint. 86 4-11ImpurityentanglemententropySimpevsdimensionlessKondocoupling0Jforbandexponentr=0,discretizationparameter=3,anddierentpartitionsizesL. ........................................ 88 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEQUILIBRIUMANDTIME-DEPENDENTPROPERTIESOFQUANTUMIMPURITYSYSTEMSByChristopherEliasWagnerDecember2018Chair:KevinIngersentMajor:PhysicsQuantumimpurityproblemsprovideasimpleyetpowerfulwaytostudycomplexstronglycorrelatedsystems.TheKondoandAndersonmodelsforanimpurityattachedtoametallichost(withdensityofstates()=0)orsemimetallichost(withpseudogappeddensityofstates()/jjrwhere(0)=0fortheFermienergy)arenaturalsystemsforstudyingentanglement,astheyarebothinteractingandnon-local,yettractablemodelstosolve.Entanglementisespeciallydiculttostudyinthepresenceofaquantumphasetransition(QPT),asfewsystemsthatexhibitaQPTaresolvable.ThepseudogapKondoandAndersonmodelsbothexhibitQPTswithaquantumcriticalpoint(QCP)whichseparatesaKondophasefromalocal-momentphaseinwhichtheKondoeectisdestroyed.Thisallowsforthestudyofthemany-bodycorrelationsinthesemodelsthroughthequanticationoftheentanglement.Previousstudiesofentanglementinthespin-bosonmodelnearanimpurityQCPshowedtheentanglementbetweentheimpurityanditsenvironmenttotakeacusppeakattheQCP.ThisdissertationestablishesthatinthepseudogapAndersonmodel,thereisacusppeakattheKondo-destructionQCPonlyinlimitedcases,andndsacriticalscalingoftheentanglementonapproachtotheQCP.Furthermore,theentanglementcalculationisgeneralizedtogrouptheimpuritywiththeelectronswithinaradiusR.ThiscalculationoftheentanglementwithaboundaryadistanceRfromanimpurityinthepseudogapKondomodelrevealsalengthscaleRforthesystem,whichdescribesacrossoverfrom 10

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themaximalentanglementassociatedwiththeKondo-destructionQCP(occurringforRR)andthelowerentanglementassociatedwithoneortheotherofthestablephasesseparatedbytheQCP(occurringforRR).ThischaracteristiclengthcanbeunderstoodasthesizeoftheKondoscreeningcloud,thelengthoverwhichthereisstronginteractionbetweentheimpurityandconductionband,anditisfoundtovaryinverselywithT,thecharacteristictemperatureandenergyscaleoftheproblem.OnapproachtowardstheQCPfromeitherphase,Tvanishesinacountinuousfashion.Consequently,thelengthscaleR/1=Tdivergesandtheentiresystembecomesentangledwiththeimpurity.Futureworkwilladopttheconceptsusedinthetime-dependentnumericalrenormalizationgrouptocalculatethetime-dependenceoftheentanglemententropy.Themathematicalsetupforthisnumericalcalculationislaidoutneartheendofthisdissertation. 11

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CHAPTER1BACKGROUND 1.1BriefHistoryoftheKondoProblemandImpurityModelsThenontrivialeectsofmagneticimpuritiesinnonmagneticmetalsrstrevealedthemselvesinthe1930swhenexperimentalists[ 1 ]discoveredthatsupposedlypuresilverexhibitedaminimumintheresistivity.Priortothis,metalswereunderstoodtohavearesistivitythatmonotonicallyincreasedwithtemperature.Thisminimumwasnotinitiallyattributedtomagneticimpuritiesinthesystem.Laterwork[ 2 ]showedthatthedepthoftheminimumandthetemperatureattheminimumwerebothdependentontheconcentrationcofthemagneticimpurities(seeFig. 1-1 ),specicallyfromimpuritiesfromthe3dtransitionmetals[ 3 { 5 ].In1964,JunKondousedthes-dmodel[ 5 { 8 ]toexploretheoreticallytheeectsofdilutemagneticimpuritiesontheresistance.ThemodelHamiltonian,nowmorecommonlycalledtheKondomodel,includesaconductionbandtermands-dexchangebetweenthebandandspin-Smagneticimpurities: HK=Xk;kcyk;ck;+1 2NcXnSnXk;k0;;0Jk;k0ei(k)]TJ /F9 7.97 Tf 6.58 0 Td[(k0)Rncyk;;0ck0;0;(1{1)wherekistheenergyofaconductionelectronofwavevectork,cyk;(ck;)isthecreation(annihilation)operatorforaconductionelectronofwavevectorkandspin,Ncisthetotalnumberofatomsinthecrystal,Jk;k0istheexchangeinteractionstrengthtakentobepositive,Rnisthepositionvectorofthenthimpurity,Snistheimpurity'sspinoperator,;0arethePaulispinmatriceswithspin;0=1=2.ThesecondterminEq.( 1{1 )forthes-dexchangecanbeexpandedtoseeamoreexplicitform1 Hs)]TJ /F4 7.97 Tf 6.58 0 Td[(d=1 2NcXn;k;k0Jk;k0ei(k)]TJ /F9 7.97 Tf 6.59 0 Td[(k0)Rnh(cyk0"ck")]TJ /F3 11.955 Tf 11.96 0 Td[(cyk0#ck#)Snz+cyk0"ck#Sn)]TJ /F1 11.955 Tf 9.74 1.8 Td[(+cyk0#ck"Sn+i;(1{2) 1Equation( 1{2 )isequaltotheformwrittenbyKondoifareplacementismadeforJk;k0!)]TJ /F1 11.955 Tf 24.58 0 Td[(2Jk;k0,whereKondotookJk;k0tobenegative.

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whereSn=SnxiSnyaretheraising(Sn+)andlowering(Sn)]TJ /F1 11.955 Tf 7.09 1.79 Td[()operatorsdescribingachangeinspinofthemagneticimpurity.Thes-dexchangeinteractioncanbeassumedtobeshort-rangedinrealspaceand,therefore,canbeapproximatedbyaconstantJk;k0=J.KondoperformedperturbationtheorytothirdorderintheinteractionstrengthJ,toobtaintheimpurityspincontributiontotheelectricalresistivity spin(T)=c3mJ2S(S+1)(V=Nc) 8e2~F1)]TJ /F1 11.955 Tf 13.15 8.09 Td[(3zJ 2FlnkBT F;(1{3)wherecistheconcentrationofimpurities,misthemassoftheelectron,Sisthespinattheimpuritysite,Visthesystemvolume,eistheelementarycharge,FistheFermienergy,andzisthenumberofconductionelectronsperatom.Smalltemperaturevaluesgiveapositiveincreasingresistivitywithdecreasingtemperatureduetothesecondterm.Aphenomenologicalequationcanbewrittenfortheresistivityusingtheelectron-phononinteractionterm,atemperatureindependenttermfrompotentialscattering,andthelogarithmictermfromtheimpurityspincontributionofEq.( 1{3 )[ 3 5 ]: (T)=aT5+cPS)]TJ /F3 11.955 Tf 11.95 0 Td[(c1lnkBT F:(1{4)Takingtheaboveequationandsettingthederivativetozeroshowsthataminimumoccursatthetemperature Tmin=c1 5a1=5:(1{5)Thisexplainstheminimumintheresistance,yetitfailsintheT!0limitbymakingtheresistivitydivergentduetotheln(kBT=F)term.Thelogarithmictermisnotuniquetotheresistivity[ 3 ].FortemperaturesbelowtheKondotemperatureTK'D kBe)]TJ /F6 7.97 Tf 6.58 0 Td[(1=0J,where0/1=FisthedensityofstatesattheFermienergy,thereisanincreaseinthenumberofspinipsbetweentheconductionelectronsandtheimpuritysitewithdecreasingtemperature,causingamuchmorecomplicatedmanybodyproblemandthefailureofperturbationtheoryinthelowtemperaturelimit.ThesearchforabetterunderstandingofthelowtemperaturebehaviorsiscalledtheKondoproblem. 13

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Figure1-1. ResistancevstemperaturecurveshowingtheresistanceminimumformanydierentMo-Nballoys.Thetemperatureanddepthoftheminimumbothgrowwiththeconcentrationofthemagnetic(Mo)ion,consistentwithEqs.( 1{3 )and( 1{5 ).Observationsoftheminimummorecommonlyoccurinmetalswithmuchlowerimpurityconcentrationsthanillustratedhere,typicallyontheorderofpartspermillion.ThisgurehasbeenreprintedwithpermissionfromM.P.Sarachik,E.Corenzwith,andL.D.Longinotti,Phys.Rev.135,A1041(1964)foundathttps://doi.org/10.1103/PhysRev.135.A1041.Copyright1964bytheAmericanPhysicalSociety. Amorephysicallyrealisticmodelthanthes-dexchangemodelistheAndersonmodel.TheAndersonmodelHamiltonian[ 4 ]isasfollows: HA=Hc+Himp+Hmix;(1{6)whereHcdescribestheconductionband,Himpdescribesanisolatedimpuritylevel,andHmixdescribesthehybridizationoftheconductionbandwiththeimpuritysite.These 14

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termscanbewrittenas: Hc=Xk;kcyk;ck;; (1{7a)Himp=Xdnd+Und"nd#; (1{7b)Hmix=1 p NcXk;Vk(cyk;d+dyck;); (1{7c)wherecyk;(ck;)isagainthecreation(annihilation)operatorforanelectronintheconductionbandwithwavevectorkandspin,dy(d)isthecreation(annihilation)operatorforthedlevelwithspin,andnd=dydgivesthenumberofelectronswithspinzcomponent=1=2inthedlevel.Thedlevelhasanenergydwhensinglyoccupiedbyeitheranupordownelectron.Whendoublyoccupied(byanupelectronandadownelectron),theenergyincreasesto2d+UduetoCoulombrepulsionbetweenthelocalizedelectrons.TheCoulombrepulsionUcanbewrittenasthestandardHartreeintegral U=Zd3r1d3r2d(r1)d(r2)e2 jr1)]TJ /F7 11.955 Tf 11.95 0 Td[(r2jd(r1)d(r2);(1{8)wheredisthewavefunctionforthedorbital.Similartothes-dexchangeintheKondomodel,thehybridizationcanbeassumedtobenearlylocalinrealspace,andsocanbeapproximatedbyVk=V.ThehybridizationViscommonlyrewrittenusinganewparameter)-529(=0V2[ 4 9 { 11 ]calledthehybridizationwidthbecauseitgivesthequantum-mechanicalbroadening(inverselifetime)oftheimpuritylevelduetomixingwiththeconductionband.SchrieerandWolshowed[ 12 ]thatundercertainconstraints,theAndersonmodelmapsontotheKondomodel.Itisenergeticallyfavorablefortheretobeasinglyoccupiedmagneticstatewhend<0andd+U>0.Ifthereisonlyasmallcouplingbetweenthemagneticstateandtheconductionband(specically,jdj;d+U\thenatsucientlylowtemperaturestheAndersonHamiltonianmapsontotheKondoHamiltonianusingtheSchrieer-Woltransformation: 15

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Jk;k0=2VkVk01 U+d)]TJ /F3 11.955 Tf 11.95 0 Td[(k0+1 k)]TJ /F3 11.955 Tf 11.95 0 Td[(d:(1{9)alongwithapotentialscatteringtermwhichwasomittedfromEq.( 1{1 )[ 3 ]. 1.2PseudogappedModelsTheconductionelectrondispersion(kvs.k)aectstheimpuritypropertiesonlythroughthedensityofstates.Ingeneralthedensityofstatescanbewrittenas ()=1 NcXk()]TJ /F3 11.955 Tf 11.95 0 Td[(k):(1{10)TheimpuritypropertiesatatemperatureTareonlysensitivetothedensityofstatesforjj.kBT.Ifthedensityofstatesisbothnon-zeroandsmoothattheFermienergy(=0),thenthelow-temperaturepropertiesdependonlyveryweaklyonthedetailedshapeof(),andsothedensityofstatescanbeapproximatedbyaconstant()=(0)0[ 10 ].Morerecentwork[ 9 ]hasshownthatthisassumptionnolongerworksforsocalled\gapless"systems,wherethedensityofstatesiszeropreciselyattheFermienergy,butnonzeroelsewhere.Amoregeneralizedapproachwiththeseproblemsisforthedensityofstatestotakeapower-lawdependenceontheenergy ()=8>><>>:0j=Djrj=Dj1;0otherwise,(1{11)withr>0beingthepowerlawexponent.Adensityofstatesthatvanishesinthispower-lawfashioniscalledpseudogapped.Examplesinclude[ 9 ]:unconventionalsuperconductorshavingaquasiparticledensityofstatesvaryingasjjorjj2;zero-gapsemiconductorshavingvalenceandconductionbandstouchingsuchthat()variesindspatialdimensionsasjjd)]TJ /F6 7.97 Tf 6.59 0 Td[(1forsmalljj;andsometwo-dimensionalelectronsystems,suchasgraphenesheets,whicharepredictedtoexhibitalinearpseudogap. 16

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Figure1-2. Schematicdiagramforapoorman'sscalingapproach.Theapproachslowlydecreasesthehalf-widthoftheconductionbandbyjDjbyremovingthestatesinthepositiveenergyrangeD)-222(jDj<
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actinginthefullband[ 13 ].Theamountofchangeinthecoupling~Jbasedonchangingthehalf-widthisgivenbythescalingequation(tothirdorderin~J): d~J dln(D=~D)=~J2)]TJ /F1 11.955 Tf 13.16 8.09 Td[(1 2~J3+O(~J4)(1{12)Thisscalingequationproducesaowdiagramwithxedpointswhere~Jisstationary.Atrivialxedpointoccursat~J=0.Forsmallpositivevaluesof~J,d~J=dln(D=~D)>0,implyingthat~J=0isanunstablexedpoint.Thereisasecondxedpointinthethirdorderequationat~J=2,whichthenumericalrenormalizationgroup(NRG),discussedindepthinCh. 2 ,revealsisanartifactoftheperturbativemethodandthetruestablexedpointisinsteadlocatedat~J=1,asseeninFig. 1-3 a.Theunstable~J=0xedpointdescribesanimpuritycompletelydecoupledfromtheconductionband,andthe~J!1stablexedpointdescribesanimpuritystronglycoupledwiththeconductionband.Eq.( 1{12 )canbeintegratedfromtheoriginalparameters(D,0J)totheeectiveparameters(~D,~J)tondascaleinvariant,whichistheKondotemperatureforthesystem[ 3 11 ] kBTK'Dp 0Je)]TJ /F6 7.97 Tf 6.59 0 Td[(1=0J+O(0J)=~Dp ~Je)]TJ /F6 7.97 Tf 6.59 0 Td[(1=~J+O(~J)(1{13)JustliketheoriginalKondomodel,apoorman'sscalingapproachcanbeappliedtothepseudogappedKondomodel.ForthepseudogappedKondomodelthepoorman'sscalingequationtothirdorderbecomes[ 14 ]: d~J dln(D=~D)=)]TJ /F3 11.955 Tf 9.29 0 Td[(r~J+~J2)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2~J3+O(~J4);(1{14)whereagain~JistherenormalizedcouplingandristhebandexponentforthepseudogapdensityofstatesdescribedbyEq.( 1{11 ).ComparingthistoEq.( 1{12 ),thereistheadditionallinearterm.Forr<1=2,thereisatrivialstablesolutionof~J=0;thexedpointfor~J=2inthestandardmodelnowsplitsintotwosolutions:~J=~Jc=1)]TJ 12.01 9.39 Td[(p 1)]TJ /F1 11.955 Tf 11.96 0 Td[(2rand~J=1+p 1)]TJ /F1 11.955 Tf 11.95 0 Td[(2r.Aswiththe~J=2xedpointmentionedabove,theexistenceof 18

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Figure1-3. Flowdiagramsfora)thestandardKondomodelandforthepseudogappedKondomodelwithb)01=2.a)Forsmall~J,theright-handsideofEq.( 1{12 )ispositive,producinganunstablexedpointat~J=0andastablexedpointat~J=2.Thenumericalrenormalizationgroup(NRG)discussedinCh. 2 showsthexedpointat~J=2tobeanartifactoftheperturbativemethodandatruestablexedpointtooccurat~J!1.b)ComparedtothestandardKondomodel,thepseudogappedKondomodelwithr<1=2changestheowforcouplingsbelowacriticalvalue~Jc,destroyingtheKondoeectandcreatingalocalizedmoment.c)Forr>1=2,theNRGshowsthatthecriticalpointdisappears. thesolution~J=1+p 1)]TJ /F1 11.955 Tf 11.96 0 Td[(2risanartifactofthemethodanditcanbeshownwiththeNRGthatthetruexedpointis~J=1.Fig. 1-3 bshowsthattheadditionofthe)]TJ /F3 11.955 Tf 9.3 0 Td[(r~Jtermchangesthedirectionoftheowinthediagramfor~J<~JcwhencomparedtoFig. 1-3 a.Increasingthebandexponentrincreasesthevalueofthecriticalcoupling~Jc.Whenr1=2(Fig. 1-3 c),theNRGshowsthattheintermediatexedpointdisappearsandthereisaowfromtheunstablexedpointat~J!1tothestablexedpointat~J=0.Thexedpointat~Jcmarksthelocationofacontinuousquantumphasetransition(QPT).AQPTisazerotemperaturetransitionencounteredthroughthevariationofacouplingparameter.ThecontinuousQPTischaracterizedbyanorderparameterpvanishingcontinuouslyontheapproachtothequantumcriticalpoint(QCP)cfromoneside(sayc).Thesesystemsalsohavea 19

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Figure1-4. SchematicphasediagramforthepseudogapAndersonmodelforxedU.Thephasediagramsareforbandexponents(a)0)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2andd<)]TJ /F3 11.955 Tf 9.29 0 Td[(U=2respectively.In(a)thesymmetricstrongcoupling(SSC)phaseoccursonlyalongtheverticallineatd=)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2. characteristicenergyscalekBTassociatedwiththeproblemkBTj)]TJ /F1 11.955 Tf 12.24 0 Td[(cj,whereisthecorrelationlengthcriticalexponent.InthepseudogapKondomodel,thecouplingparameteris~JandtheorderparameterpishSimpzi(thespinzoftheimpurity)wheninthepresenceofaninnitesimalmagneticeldalongthezaxis.Whenthecouplingparameterislessthanthecriticalvalue(~J<~Jc),theorderparameterisnon-vanishingwhichsignalstheabsenceofKondoscreening,leadingtolabellingthecriticalcoupling~JcastheKondo-destructionQCP.Thecharacteristicenergy(ortemperature)scaleTisassociatedwithtransitioningfromthecriticalbehaviorduetotheQCP(T>T)tothenon-criticalbehavioroftheothertwoxedpoints(T
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Forparticle-holesymmetry(d=)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2,correspondingtotheresultsseenaboveinthepseudogapKondomodel),thereisasymmetricstrong-coupling(SSC)phaseinFig. 1-4 (a)thatemergesforlarge,butnotinFig. 1-4 (b).Awayfromparticle-holesymmetry,twoasymmetricstrongcouplingphases,ASC)]TJ /F1 11.955 Tf 10.99 1.79 Td[(andASC+occurringford>)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2andd<)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2respectively,canbereachedforsucientlylargevaluesof.ThesephasesaredescribedinmoredetailinSec. 2.3 .Asthepoorman'sscalingapproachusedthecouplingJasaperturbationtothefreeconductionelectrons,theproblemsofperturbationtheorydiscussedinSec. 1.1 arestillpresentinthismethod,hencepoorman'sscalingdoesnotsolvetheKondoproblem.Theconceptofrenormalizingparametersintroducedhereisuseful,andisthecoreoftherenormalizationgroup.Thenumericalrenormalizationgroup,thenon-perturbativenumericalmethoddiscussedinthenextchapter,willutilizethisideaofrenormalization. 1.4EntanglementEntropyAbipartitesysteminapurestatejiisconsideredtobeentangledwhenthewavefunctioncannotbewrittenasaproductofpurestatesoftwosubsystemsAandB(i.e.ji6=j Aij Bi),meaningthatthetwosubsystemsdonotthemselveshavepurestates.Astandardexampleofthisisthesingletstateofatwospinsystem,wherethewavefunctioniswritten ji=1 p 2(j"#i)-278(j#"i):(1{15)Onemeasureoftheentanglementinapurestateistheentanglemententropy: Se=)]TJ /F1 11.955 Tf 9.3 0 Td[(TrA(AlnA)=)]TJ /F1 11.955 Tf 9.3 0 Td[(TrB(BlnB);(1{16)whereA;B=TrB;A()iscalledthereduceddensitymatrixofsubsystemAorB,TrB;AisthepartialtraceoverstatesinsubsystemBorArespectively,andisthetotaldensitymatrixforthecombinedsystem,whichcanbeobtainedfromthedensityoperatorofthesystemdenedby^=jihj.Eq.( 1{16 )denesSeasthevonNeumannentropyofeitherreduceddensitymatrix.Quantifyingtheentanglementusingthevon 21

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Neumannentropyprovidessomenotablefeatures:(1)whenthesubsystemsAandBarepurestatesthesystemisdetangledandtheentropytakesaminimalvalueofzero;(2)whenthesubsystemsaremaximallyentangled,theentropytakesthemaximalvalueSe=lndwheredisthesmallerofthedimensionsoftheHilbertspacesofsubsystemAandsubsystemB;and(3)theentropyisinvariantwhenswitchingbetweenbasissets,afactthatwillbeusefulinSec. 2.8 .Theentanglemententropyisonlyoneofmanymeasuresofentanglement.Itisalimitingcaseofsomeothermeasuresofentanglement,suchastheentanglementofformation[ 15 ]andtheRenyientropy[ 16 ].Entanglementhasexperimentallybeenmeasuredinsystemsofultra-coldatomicgases[ 17 ]throughtheRenyientropy.Theentanglemententropyisparticularlyusefulinquantifyingcorrelationswithinpurestatesofquantummany-bodysystems.Entanglemententropyexhibits[ 16 18 19 ]nontrivialscalingnearQCPs[ 20 ].Nontrivialtopologicalordercanbeidentiedusingtheentanglemententropy[ 21 { 25 ],especiallywhenlocalparametersareinadequate.Recentinterestintheentanglemententropyhasexpandedthroughitsrelevancetoquantuminformation[ 26 ]andquantumcomputation[ 27 ].Quantumimpurityproblemsprovideagoodbasisforthestudyoftheentanglement,astheycontaininteractingsystemsaswellasexhibitnonlocalproperties,whichisinherenttotheentanglemententropy.Multiplemethodshavebeenemployedtocalculatetheentanglementinquantumimpuritysystems,suchasconformaleldtheory[ 28 ],thedensity-matrixrenormalzationgroup[ 28 ],andthenumericalrenormalizationgroup[ 29 ](discussedinthenextchapter).Workonthespin-bosonmodel,amodelcouplingabosoniceldtoatwo-levelsystem,studiedthecriticalbehaviornearaQCP[ 29 30 ].Theentanglementbetweenthebosonicbathandthetwo-levelsitewasfound[ 29 ]toexhibitacusppeakattheQCPthatseparatesalocalizedanddelocalizedphase,maximizingtheentanglementinthesystem.WhetherthiscusppeakisauniversalpropertyoftheentanglementofimpurityQCPswasanunansweredquestion.TheworkdiscussedinCh. 3 aimedtodetermineifthe 22

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entanglementneartheKondo-destructionQCPinthepseudogappedAndersonmodelalsoexhibitsthiscusppeak.Variousgroundstatescanbeclassiedbyan\arealaw"Seld)]TJ /F6 7.97 Tf 6.59 0 Td[(1[ 31 ],whereasthermalsystemswithhighlyexcitedstatesareclassiedbya\volumelaw"Seld[ 32 ].Logarithmiccorrectionstothearealaw,Seld)]TJ /F6 7.97 Tf 6.58 0 Td[(1log(l),areintroducedbytheexistenceofaFermisurface[ 33 ].Astheentanglemententropyisespeciallysuitedfordescribingthegroundstatesofquantumimpuritysystems,itisnaturaltoexpectthatvaryingthesizeofthesubsystemscangivesomeinsightintothespatialsizeoftheKondoscreeningcloud.ThishasbeenconrmedintheKondomodel[ 34 ]andsomerelatedmodels[ 28 35 ].InsituationswheretheKondoeectcanbedrivencriticalatacontinuousquantumphasetransition[ 9 36 38 { 49 75 ],thefateoftheKondoscreeningcloudandthespatialstructureofentanglementarebothpoorlyunderstood.WhethertheentanglementislongrangedattheKondo-destructionQCPisrelevanttosomeheavy-fermioncompundssuchasCeCu6)]TJ /F4 7.97 Tf 6.59 0 Td[(xAux[ 50 ],YbRh2Si2[ 51 ],andCeRhIn5[ 52 ].TogeneralizetheworkfromCh. 3 andstudytheKondoscreeningcloudusingtheentanglement,apartitionboundarycanbechosenataradiusRfromtheimpuritytoincludetheelectronsclosesttotheimpurity.TheworkinCh. 4 studiestheentanglementinthepseudogapKondomodelwherethereexistsaKondo-destructionQCPinordertobetterunderstandtheKondoscreeningcloudnearaQCP. 23

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CHAPTER2NUMERICALRENORMALIZATIONGROUPMETHODSAsdiscussedinthelastchapter,onlyanonperturbativetheorycanbesucientfordiscussingthelow-temperaturebehavioroftheKondoandAndersonmodels.AnonperturbativenumericalsolutiontotheKondoproblemutilizingtherenormalizationgroup,calledthenumericalrenormalizationgroup(NRG),wascreatedbyKennethWilson[ 53 ]in1975.TheNRGwasoriginallyappliedtotheKondomodel,howeverin1980H.R.Krishna-murthy,J.W.WilkinsandK.G.WilsonextendedthemethodtotheAndersonmodel.Thismethodhasbeenexpandedtodealwithaddedcomplexitiessuchasanenergydependentdensityofstates[ 9 54 55 75 ],multipleconductionbands[ 56 ],andmultipleimpurities[ 56 { 58 ].TheNRGasformulatedinthischapterismodelindependent,i.e.itcanbeappliedtoboththepseudogappedKondoandAndersonmodels(seeSec. 1.2 formoreinformationonpseudogappedmodels).ImplementationoftheNRGcanbebrokenupintothreemajorsteps: (1) logarithmicallydiscretizetheconductionband, (2) mapthediscretizedconductionbandtoatight-bindingchain, (3) iterativelydiagonalizethetight-bindingHamiltonian.Thesestepsarediscussedinthefollowingtwosections. 2.1DiscretizationandtheTightBindingChainAsdiscussedinSec. 1.1 ,perturbationtheoryfortheKondomodelproducesalogarithmicdivergenceinthermodynamicpropertiesatlowtemperatures,asallmagnitudesofenergyequallycontributetothecalculationofthermodynamicproperties.DuetothelackofaperturbativeenergyscaleintheKondoproblem(refertoSec. 1.1 ),alogarithmicdiscretizationoftheconductionbandisperformedtocreateanarticialseparationofenergyscales[ 53 ].Tobeginthisprocess,theconductionbandHamiltonian

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Figure2-1. Representationofthediscretizedconductionband.Thisgureshowsthedierentsizedbinscreatedfromthediscretizationusingtheparameter. ofEq.( 1{6 )iswrittenintheenergybasis: Hc=Xk;kcyk;ck;!Hc=DXZ1)]TJ /F6 7.97 Tf 6.59 0 Td[(1"cy";c";d";(2{1)whereDisthehalfwidthoftheconductionband,"==Disadimensionlessenergyparameter,andthefermionicoperatorsintheenergybasisobeythestandardanticommutationrelation nc";;cy"0;0o=(")]TJ /F3 11.955 Tf 11.96 0 Td[("0);0:(2{2)FormulatingtheconductionbandHamiltonianintheenergybasisisusefultocreatethearticialseparationofenergyscales.Thelogarithmicdiscretizationoftheconductionbandstartswithsettingupbinsofpositiveandnegativeenergies,theabsoluteenergiesextendingoverdecreasingenergyranges.Adiscretizationparameter>1canbeusedtowriteouttheenergyboundariesseparatingdierentbins: "m=8>><>>:1m=0;1)]TJ /F4 7.97 Tf 6.59 0 Td[(z)]TJ /F4 7.97 Tf 6.58 0 Td[(mm>0(2{3)wherethesuperscriptindicatesthepositive/negativeenergyrangeandmindicatestheboundarybetweenthemthand(m+1)thbins.Inpracticethediscretizationparameterisgenerallychosenintherange2<<9.Theparameterzisusuallysettounity,butcanbevariedinordertoimprovedynamicalandthermodynamicquantities[ 59 60 ].Insidethebinsorthonormalfunctions (q);m(")mustbedened,withq=0;1;2;:::beingaharmonicindexandtheplus(minus)subscriptdenotingafunctionforthebinswithpositive(negative)energy.Thefermionicoperatorc";canbewrittenintermsofthe 25

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binoperatorsa(q)m;andb(q)m;forthepositiveandnegativeenergybins[ 9 ]: c";=1Xm=01Xq= (q)+;m(")a(q)m;+ (q))]TJ /F4 7.97 Tf 6.59 0 Td[(;m(")b(q)m;;(2{4)wherethebinoperatorsobeythestandardanticommutationrelations: na(q)m;;a(q0)ym0;0o=m;m0;0q;q0;nb(q)m;;b(q0)ym0;0o=m;m0;0q;q0;na(q)m;;a(q0)m0;0o=nb(q)m;;b(q0)m0;0o=na(q)m;;b(q0)m0;0o=na(q)m;;b(q0)ym0;0o=0:(2{5)UsingEqs.( 2{1 )and( 2{4 )theconductionbandHamiltoniancanberewrittenintermsofthebinoperators: Hc=DXXm;q;q0Z1)]TJ /F6 7.97 Tf 6.59 0 Td[(1" (q)+;m(") (q0)+;m(")a(q)ym;a(q0)m;+ (q))]TJ /F4 7.97 Tf 6.59 0 Td[(;m(") (q0))]TJ /F4 7.97 Tf 6.59 0 Td[(;m(")b(q)ym;b(q0)m;d";(2{6)where,duetotheorthonormalityofthewavefunctionswithrespecttothesubscripts,thecrosstermsa(q)ymb(q0)m00andb(q)yma(q0)m00disappear,andthenon-crosstermsreducetom0=m.Thedenitionofthebinfunctions (q);mfortheq=0casecanbewrittenoutas[ 9 ]: (0);m(")=8>><>>:w(")=F;mj"m+1j><>>:j"jrj"j1;0otherwise:(2{9)Inthelimit!1,thecouplingbetweenq6=q0inHcdisappears[ 53 ].For>1,itisstillareasonableapproximationtodiscardthesecouplings.Sinceonlytheq=0operatorscoupletotheimpurity,droppingtheq6=0termssimplydropsanear-constanttermcontributingtothekineticenergy.Utilizingonlytheq=0contribution,theconduction 26

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bandHamiltonianbecomes: Hc=DX1Xm=0 F)]TJ /F6 7.97 Tf 6.59 0 Td[(2+;mZ"+m"+m+1d"w2(")a(0)ym;a(0)m;+F2)]TJ /F4 7.97 Tf 6.59 0 Td[(;mZ")]TJ /F11 5.978 Tf 0 -6.11 Td[(m+1")]TJ /F11 5.978 Tf 0 -4.83 Td[(md"w2(")b(0)ym;b(0)m;!:(2{10)Theimpuritycouplestoauniquelinearcombinationoftheconductionbandstates.Thislinearcombinationofstatesthatcouplestotheimpuritycanbewrittenoutas: f0;=F)]TJ /F6 7.97 Tf 6.59 0 Td[(11Xm=0(F+;ma(0)m;+F)]TJ /F4 7.97 Tf 6.58 0 Td[(;mb(0)m;)F)]TJ /F6 7.97 Tf 6.59 0 Td[(1Zp (D")c";d";(2{11)with F2=Zw2(")d"=1Xm=0(F2+;m+F2)]TJ /F4 7.97 Tf 6.58 0 Td[(;m):(2{12)Withauniquewaytowriteouttheconductionbandstatesthatcoupletotheimpurity,itispossibletocreateacompleteorthonormalsetofoperatorsfm;fromtheoperatorsam;andbm;,suchthatitcreatesanearest-neighbortight-bindingHamiltonian.TheexactmappingoftheHamiltoniantoatight-bindingHamiltonianwithasemi-innitechainofsitesisdoneusingtheLanczosprocedure[ 9 61 ].RewritingtheHamiltonianwiththetight-bindingoperatorsproduces: Hc=DX1Xm=0)]TJ /F4 7.97 Tf 6.59 0 Td[(m=2hemfym;fm;+tmfym;f(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1);+fy(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1);fm;i:(2{13)wheref)]TJ /F6 7.97 Tf 6.58 0 Td[(1;=0and =1 2(1+)]TJ /F6 7.97 Tf 6.59 0 Td[(1)3=2)]TJ /F4 7.97 Tf 6.59 0 Td[(z:(2{14)ThescaledhoppingcoecientstminEq.( 2{13 )arealloforderunityandthescaledon-siteenergiesdecreasetowardzerowithincreasingm.Asaresult,theoverallenergyscaleoftermsincludingsitemsioforderD)]TJ /F4 7.97 Tf 6.59 0 Td[(m=2.Thisexponentialdecrease,whichisadirectconsequenceofthelogarithmicdiscretizationoftheconductionband,iscrucialforthesuccessoftheNRGmethod.Arepresentationofthissemi-innitetight-bindingchainofsiteswhereonesitecouplestotheimpurityandtherestcoupletotheirnearestneighborscanbeseeninFig. 2-2 .Thesemi-innitechainofconductionbandsitescreatedfortheNRGiscalledtheWilsonchain. 27

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Figure2-2. SchematicdiagramoftheNRGtight-bindingchain.Eachsitehasitsownoperatorfmandthenearestneighborshaveatight-bindingcouplingtm. 2.2IterativeDiagonalizationTheHamiltonianforanarbitraryimpuritymodelcanbewrittenintheformgiveninEq.( 1{6 )withtheHcapproximatedbyEq.( 2{13 )andallconduction-electronoperatorsenteringHmixexpressedintermsoff0andfy0.Forpracticalnumericalwork,however,theinnitesuminEq.( 2{13 )mustbecutshort.TherststepintheprocessistorewritethefullHamiltonianwithlimitnotation H=limN!1D)]TJ /F4 7.97 Tf 6.59 0 Td[(N=2HN;(2{15)wheretheHamiltonianHNdescribesan(N+1)-siteWilsonchainlabeledsites0throughN: HN=N=2hHc;N+~Himp+~Hmixi;(2{16)withthetruncatedconductionbandHamiltonian, Hc;N=XNXm=0)]TJ /F4 7.97 Tf 6.58 0 Td[(m=2emfymfm+tmfymf(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)+fy(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)fm;(2{17)andrescaledimpurity(Himp)andhybridization(Hmix)Hamiltonians, ~Himp=1 DHimp;~Hmix=1 DHmix:(2{18) 28

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Figure2-3. Schematicdiagramoftheiterativediagonalizationprocess.Theeigenstatesjemicrossedwiththestatesfromthenextsitejm+1igivethebasisstatesforthenextiterationjb(m+1)i.Withthosebasisstates,theHamiltoniancanbediagonalizedgivingtheeigenstatesje(m+1)i. ArecursionrelationcanbecreatedtondthenewHamiltonianafteraddinganadditionalsitetothechain: HN=1=2HN)]TJ /F6 7.97 Tf 6.59 0 Td[(1+XheNfyNfN+tN(fyNf(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+fy(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1)fN)i)]TJ /F3 11.955 Tf 11.95 0 Td[(EG;N;(2{19)whereEG;NisthegroundstateenergyofHN,andtherstHamiltonianisexpressedas: H0=Xe0fy0f0+~Himp+~Hmix)]TJ /F3 11.955 Tf 11.95 0 Td[(EG;0:(2{20)Duetothefactorof)]TJ /F4 7.97 Tf 6.58 0 Td[(m=2tm<1,thelow-lyingeigenstatesofHNcanbeaccuratelyrepresentedbylinearcombinationsofthelow-lyingeigenstatesofHN)]TJ /F6 7.97 Tf 6.59 0 Td[(1timesbasisstatesofsiteN,allowingfortheHamiltoniantobeiterativelydiagonalized.AschematicdiagramoftheprocessisshowninFig. 2-3 .Ateachiteration,theHamiltonianHNisdiagonalizedandtheeigenstatesareobtained.Usingtheseeigenstates,thebasisstatesforthenextiterationarecreated,wherethenumberofbasisstatesforHN+1isdeterminedbythedimensionofthebasisforeachsiteoftheWilsonchain.Foraconductionband 29

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Figure2-4. Severelysimpliedschematicdiagramoftheenergyspectrumof)]TJ /F4 7.97 Tf 6.59 0 Td[(m=2Hmwithincreasingiterationnumberm.Forthisrepresentation,itwasassumedthatthezerothiterationgivestwostatesandthedegeneracyofthesitesontheWilsonchainis2.Ateachiterationamaximumof8statesarekept,andsotruncationoccursforthersttimegoingfromiteration3toiteration4. ofspin-1/2electrons,thereare4dierentstatesateachsite:unoccupied(j0i),singlyoccupiedbyaspin-up(j"i)orspin-down(j#i)electron,anddoublyoccupied(j"#i).Foragiveniterationm,thenumberofstatesisroughly4m+1,resultinginthedimensionoftheHamiltonianmatrixgrowingexponentiallywithiterationnumberandrapidlymakingthediagonalizationnumericallyimpractical.AsolutiontothisproblemistotruncatethebasissothattheHamiltonianmatrixisrestrictedtobewithinamaximumsize.Thiscanbedoneinoneoftwoways:(1)specifyamaximumnumberofstatestokeepaftereachiteration,wherethelowestenergystatesarekeptandallhigherenergystatesarediscarded;(2)specifyacutoenergywherestateswithenergieshigherthanthecutoarediscarded.Fig. 2-4 givesasimpliedexampleofthisforcase(1).SomecareneedstobetakenwhencuttingtheWilsonchainshort:itisnecessarytodetermineifthechainislongenoughtocorrectlydescribethelowtemperaturephysics,whichrequiresthatthesystemhasreacheditsgroundstate.Ifthesystemisinthegroundstate,itisevidencedbycomparingtheenergyspectrumofmultipleiterationsandcheckingforself-similarbehavior.Thescalingfactorof1=2multiplyingtheHamiltonianHN)]TJ /F6 7.97 Tf 6.59 0 Td[(1inEq.( 2{19 )isincludedtoscaletheenergiesofeachiterationtobeinsimilar 30

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energyranges,allowingfortheself-similaritycheck.Thecheckingofself-similaritygoesbacktothenotiondiscussedinSec. 1.3 ofRGxedpoints. 2.3FixedPointsFixedpointsintheNRGarereachedwhentheiterativeHamiltonianisinvariantundertwotransformationsoftheforminEq.( 2{19 ),i.e.HN=HN)]TJ /F6 7.97 Tf 6.59 0 Td[(2[ 53 ].Asmentionedintheprevioussection,invarianceoftheHamiltonianisachievedwhenthescaledenergyspectrumofHNandHN+2isself-similar.Beforereachingaxedpoint,theenergyspectrumoftheHamiltonianvarieswithincreasingN,andsoowdiagramsarecreatedtoexplaintheowoftheHamiltonianbasedonthescaledversionsoftheHamiltoniancouplings.AnexampleofsuchaowdiagramisgiveninFig. 2-5 ,specictothesymmetricAndersonmodelwith2d+U=0.Thehorizontalandverticalaxesrepresenttheeective(renormalized)valuesofthehybridizationwidthandtheHubbardinteraction.Theinitialconditionis)]TJ /F6 7.97 Tf 191.75 -1.8 Td[(e=)-435(=0V2andUe=U.WithincreasingNRGiterationnumberN,theeectiveparametersowalongthetrajectoriesshowninthedirectionindicatedbythearrowsuntilthesystemreachesoneofthepointsrepresentedbycircles.ThefourxedpointsshowninFig. 2-5 areasfollows:(1)FreeOrbitalxedpoint:Thisxedpointoccurswhend=U=)-351(=0.TheHamiltonianisthatofthefree-electronchain(theWilsonchain)plusadlevelofzeroenergyfortheimpurity.Thestatesassociatedwiththisxedpointcanbewrittenbytakingeachstateofthefree-electronHamiltonianandcrossingitwiththefourdegeneratestatesfortheimpuritylevel(j0i,j"i,j#i,andj"#i).TheowlinesinFig. 2-5 allpointawayfromthefreeorbitalxedpoint,indicatingthatthexedpointisunstableagainstanydeviationfromtheconditiond=U=)-278(=0.(2)LocalMoment(LM)xedpoint:Thisoccurswhen)-388(=0,d!,andd+U!1.Takingd!removesthepossibilityoftheimpuritybeingunoccupied,whiled+U!1removesthepossibilityofdoubleoccupancy,thusensuringasingleoccupiedimpuritysite.Thisxedpointhasconductionelectronscompletelydecoupled 31

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Figure2-5. RenormalizationgroupowdiagramforthesymmetricAndersonmodel(2d+U=0),showingtheevolutionandeectivecouplingsUeand)]TJ /F6 7.97 Tf 30.08 -1.79 Td[(ewithincreasingNRGiterationnumberN.Flowlinesstartandendatxedpoints.Thexedpointsinthisdiagramarethefreeorbitalxedpointattheorigin,thelocalmomentxedpointontheUeaxis,thestrong-couplingxedpointonthe)]TJ /F6 7.97 Tf 43.73 -1.8 Td[(eaxis,andtheKondo-destructionQCPonthelineconnectingHLMandHSC. fromthespin1/2impurity.Flowarrowsinthevicinityofthisxedpointaredirectedtowardthexedpoint,meaningthatitisstableagainstsmallperturbations,anddescribesalow-temperaturephaseofthesymmetricAndersonmodel.(3)StrongCoupling(SC)orKondoxedpoint:Thisxedpointoccurswhentaking)]TJ /F2 11.955 Tf 47.21 0 Td[(!1forconstantUandd.Inthisinstancealloftheexcitedstatesareignoredandthegroundstateisallthatisleft.Theimpurityisstronglycoupledtotherstsiteofthechain,leadingtotheimpurityeectivelydecouplingtherstelectronsitefromtherestofthechainofconductionelectrons(t1!0inFig. 2-2 ).Inthiscasethechainactsasthoughitwereonesiteshorter,withtherstsiteremoved.Thisxedointtooisstableagainstperturbationsanddescribesaphaseofthemodel.(4)Kondo-destructionquantumcriticalpoint:ForthepseudogapAndersonmodel(withdensityofstatesbandexponentsr>0)thereisaquantumcriticalpoint 32

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(QCP))-450(=)]TJ /F4 7.97 Tf 73.02 -1.79 Td[(cwhichseparatesaregionofparameterspace)]TJ /F3 11.955 Tf 242.45 0 Td[(>)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(cthatexhibitstheKondoeectfromaregionofparameterspace)]TJ /F3 11.955 Tf 254.2 0 Td[(<)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(cwhichisdescribedbythelocalmomentxedpoint.TheQCPgovernsthephysicsonaphaseboundarywhoselocationdependsontheCoulombrepulsionU,i.e.)]TJ /F4 7.97 Tf 33.33 -1.79 Td[(c=)]TJ /F4 7.97 Tf 20.99 -1.79 Td[(c(U).InFig. 2-5 ,thisphaseboundaryisrepresentedbytheowlinefromthefree-orbitalxedpointtotheKondodestructionxedpoint.Anychoiceof)-326(closetobutnotidenticallyat)]TJ /F4 7.97 Tf 310.59 -1.8 Td[(cwillresultinaowthatdivergesfromthephaseboundaryandheadstowardoneorotherofthetwostablexedpoints.ThesymmetricAndersonmodelhasaKondo-destructionQCPfor0
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eachiteration.ThediscretizationoftheconductionbandperformedtoutilizetheNRGisapoordescriptionofthephysicsofthetrueconductionband.Forthisreason,onlytheimpuritycontributiontothermodynamicpropertiesiscalculated.TheimpuritycontributiontoagivenpropertyXimpiscalculatedbytakingtheresultforthefullmodelXandsubtractingotheresultfortheWilsonchainXWC Ximp=X)]TJ /F3 11.955 Tf 11.96 0 Td[(XWC:(2{23)Onesuchthermodynamicpropertyistheimpuritycontributiontothemagneticsusceptibilityimp(T)whichisusedtoobtainthecharacteristictemperaturescalesTKandTasdiscussedinthenextsection. 2.5ExtractionoftheCharacteristicTemperatureScalesInSec. 1.1 ,perturbationtheorybrokedownbelowtheKondotemperatureTK,thecharacteristictemperaturescalefortheKondomodelwherethesystemcrossesoverfromthevicinityofthelocalmomentxedpoint(T>TK)totheKondoxedpoint(TT)toeitherthelocalmomentorKondoxedpoint(TJc[ 9 ]).ThecloserJistothecriticalcoupling,thelowerthecrossovertemperatureTfromthecriticalvaluetothenon-criticalvalue.TheworkinCh. 4 denes4Ttobe 34

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Figure2-6. ExtractionofthecharacteristictemperaturescaleTinthepseudogapKondomodelfromimp(T),theimpuritycontributiontotheuniformmagneticsusceptibility.ForeachvalueofJ,4TisdenedtobethetemperatureatwhichTimp(T)(thinsolidlines)reachesthemidpoint(horizontaldashedline)betweenitsT!0limitingvalueforthatJ(namely,1=4forJJc)andthecorrespondinglimitingvalueforJ=Jc(thickhorizontalline).Datashownareforbandexponentr=0:4andJ=(110x)Jcwiththevaluesofxshowninthelegend.Thisgurecanbefoundinapaperbytheauthor:C.Wagner,T.Chowdhury,J.H.Pixley,andK.Ingersent,Phys.Rev.Lett.121,147602(2018).(Ref.[ 94 ]) thetemperatureatwhichkBTimpreachesthemidpointbetweenitscriticalvalue(theonethatpersiststoT=0atJ=Jc)anditszero-temperaturelimit[ 9 ].Thistemperatureistakentobe4T(ratherthanT,say)sothatforr!0+where4kBTimp(4T)!0:125,Tsmoothlyapproachesthemetallic(r=0)Kondotemperature,givenbytheempiricaldenitionabove.TheequivalenceoftheAndersonmodeltotheKondomodelinlimitingcasesimpliesthatthesemethodscanbeusedtodetermineTKfortheAndesonmodelandTforthepseudogappedAndersonmodel. 2.6DensityMatrixNumericalRenormalizationGroupTheNRGasdescribeduptothispointallowsforthecalculationoflow-temperaturethermodynamicpropertiesandexcitationsfromthegroundstate.Excitationsofasystemontheorderof~!)]TJ /F4 7.97 Tf 6.59 0 Td[(m=2D)]TJ /F4 7.97 Tf 6.59 0 Td[(N=2DforaWilsonchainoflengthN+1no 35

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longerexistatiterationN,duetotruncatingthelargestenergies.Tocalculatelargeexcitations,itmustthenbeassumedthattheenergydierencesfromthegroundstateatiterationmareagoodapproximationoftheenergydierencesfromthetruegroundstate,i.e.Em)]TJ /F3 11.955 Tf 13.26 0 Td[(EG;mE1)]TJ /F3 11.955 Tf 13.26 0 Td[(EG;1,anapproximationthatworkswellforlargeiterationnumberm.Itisalsoassumedthatthegroundstatemany-bodyeigenfunction,jG.S.imcorrectlyapproximatestheimpuritycontributiontotheexpectationvalueofthermodynamicproperties,anapproximationwhichisaccuratefortheexpectationvaluesoflocaloperators(thoseonlyactingontheimpurityandsitesn
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dynamiccalculationsiscrucialtoobtaininganaccurategroundstateforthehigh-energytransitioncalculations.Asolutiontothisproblemistorstcalculatethegroundstateinformation,afterwhichtheexcitationsarecalculatedusingthetruegroundstate.ThisrequirestwoNRGruns:onetocalculatethegroundstatewithtemperatureTNTKandthesecondtocalculatedowntothetemperaturescaleofinterestTm.Alternatively,itispossibletostorethenecessaryinformationduringtherstruntoavoidthesecondrun.Duringtherstrun,alltransformationsbetweeneigenstatesarestoredinaunitarymatrix,andthegroundstateinformationiscalculatedatiterationN.Usingthegroundstateinformation,thedensityoperatorcanbeconstructed, ^N=1 ZXne)]TJ /F4 7.97 Tf 6.58 0 Td[(NENnjniNNhnj;(2{26)whereZ=Pne)]TJ /F4 7.97 Tf 6.59 0 Td[(NENnisthepartitionfunctionforthenaliteration.Thesecondrunthenrepeatstheprocessdowntotheiterationfortherelevantexcitationenergy~!Tm.Tocalculatepropertiesatiterationm,thedensityoperatorcanbeexpressedintermsoftheproductoftwosubsystems:thechainuptositem,andthechainfromsitem+1tositeN.Thesecondsubsystem,calledthe\environment",isvisualizedinFig. 2-7 .Usingthissubsystemconventionthedensityoperatorbecomes ^m;N=Xn1;n2;l1;l2l1n1;l2n2jl1ienvjn1immhn2jenvhl2j:(2{27)Theenvironmentstatescanbetracedoutoftheaboveformbyusingtheunitarytransformationcalculatedduringtherstrun,andwhatisleftisknownasthereduceddensityoperator ^redm;N=Xn1n2redn1n2jn1immhn2j;(2{28)with redn1n2=Xlln1;ln2:(2{29) 37

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Figure2-7. Representationofthe\environment"byseparatingsitesm0mfromsitesm0>m.TheNRGcalculationiscarriedouttoaniterationnumberN.Ataniterationm
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m0wheretherearetoomanystatestokeep.InthiscaseNm0kofthelowestenergystatesarekeptandtherest(Nm0d=dimpdm0+1)]TJ /F3 11.955 Tf 12.43 0 Td[(Nm0k)arediscarded.Allstatescreated(bothkeptanddiscarded)atm0areorthonormaltoeachother.Atiterationm0+1,Nm0kdorthonormalstatesarecreatedusingtheNm0kstateskeptafteriterationm0,allofwhichareorthonormaltothestatesdiscardedatiterationm0.Bystoringtheeigenstatesdiscardedatm0andmultiplyingthembyadegeneracydassociatedwithnotusingthediscardedstatesduringiterationm0+1andcombiningthemwithallstatescreatedduringiterationm0+1,thebasissetfortheproblemiscomplete.Tomovetoiterationm0+2afteriterationm0+1,onlyNm0+1kofthelowestenergystatesarekeptandtherest(Nm0+1d=Nm0kd)]TJ /F3 11.955 Tf 12.32 0 Td[(Nm0+1k)arediscarded.Atiterationm0+2,Nm0+1kdorthonormalstatesarecreatedusingtheNm0+1kstateskeptafteriterationm0+1,allofwhichareorthonormaltothestatesdiscardedatiterationm0andm0+1.Theeigenstatesdiscardedatiterationm0+1arestoredandmultipliedbyadegeneracyd(justasbefore),andthestoredeigenstatesfromiterationm0aremultipliedbyanotherfactorofd.Combiningthestoredeigenstateswiththeeigenstatescreatedatiterationm0+2ensuresacompletebasisset.ContinuingthrougheachiterationuptothelastiterationN,alleigenstatesdiscardedwiththeappropriatedegeneraciesincludedareorthonormal,andalleigenstatescreatedduringthenaliterationarediscarded(NNd=NN)]TJ /F6 7.97 Tf 6.59 0 Td[(1kd).KnowingthatifthebasissetiscompletethetotalnumberofstatesshouldbeNS=dimpdN+1,itispossibletoshowthatallstatesarepreservedbyaddingupthenumberofdiscardedstateswiththeirdegeneracies: NS=NXm=m0NmddN)]TJ /F4 7.97 Tf 6.59 0 Td[(m;=(dimpdm0+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Nm0k)dN)]TJ /F4 7.97 Tf 6.59 0 Td[(m0+N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xm=m0+1(Nm)]TJ /F6 7.97 Tf 6.59 0 Td[(1kd)]TJ /F3 11.955 Tf 11.96 0 Td[(Nmk)dN)]TJ /F4 7.97 Tf 6.59 0 Td[(m+NN)]TJ /F6 7.97 Tf 6.59 0 Td[(1kd;=dimpdN+1:(2{31) 39

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Sincealltheeigenstatesarecollectedandorthonormal,acompletebasissethasbeencreatedusingthesetofdiscardedstates.Fig. 2-8 givesasignicantlysimpliedexampleoftheinductionabove.Inthegure,eachsitehasdegeneracy2,thenumberofstateskeptaftereachiterationis4,andtherstiterationtodropstatesism0=2.Ateachiteration,thereddiscardedstatesareorthonormaltothereddiscardedstatesofeveryotheriteration,andaslongasthedegeneracyofeachiscorrectlyenumeratedthetotalsetofbasisstatesiscomplete(albeitthesizeofthesetofbasisstatesincreasesbyafactorofd=2aftereachiteration).ForaHamiltoniancalculatedtothemthiteration(withachainoflengthm+1),ifastateisdiscardedafteriterationm)]TJ /F1 11.955 Tf 12.54 0 Td[(1,itwouldhaveproduced2timesasmanystatesifitwereinsteadkeptduetothed=2degeneracyassociatedwithsitem.Ifastateisdiscardedafteriterationm)]TJ /F1 11.955 Tf 12.3 0 Td[(2,itwouldhaveproduced4timesasmanystatesifitwerekeptthroughiterationmduetothed=2degeneracyfromsitem)]TJ /F1 11.955 Tf 11.8 0 Td[(1andd=2degeneracyfromsitem.ThedegeneracymultiplierdN)]TJ /F4 7.97 Tf 6.59 0 Td[(mforagivenstatediscardedatiterationmforaHamiltonianwithNiterations,then,comesfromtheconceptofenvironmentalstatesdiscussedintheprevioussection.ForasystemwithNiterations(orofchainlengthN+1),thecompleteFockspaceFNcanbewrittenusingthenotation FN=fjlemig(2{32)wherellabelsthediscardedstateandeisthesetofenvironmentstatesatiterationm.TheFockspacecontainsstatesfromallvaluesofmintherangem0mN.Thesebasisstatesprovidethestandardouterproductsumtounity Xl;e;mjlemihlemj=1:(2{33)Thediscussionofthekeptstatesforaniterationmusedtoobtainstatesatfutureiterationsleadstoausefulidentitybetweenthediscardedstateslandkeptstatesk.Thekeptstateskatiterationmwillbediscardedatsomefutureiterationm0>m,andsothe 40

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sumofkeptstatesatmisequaltothesumofdiscardedstatesfromallfutureiterations,i.e.: Xk;ejkemihkemj=Xl;e;m0>mjlem0ihlem0j:(2{34)ThedensityoperatorgiveninEq.( 2{26 )canberewrittenusingthecompletesetofbasisstates: ^=1 ZNXm=m0Xl;eexp()]TJ /F3 11.955 Tf 9.3 0 Td[(Eml)jlemihlemj;(2{35)wherethepartitionfunctionZiscreatedfromthecompleteFockspaceFN, Z=NXm=m0Xl;eexp()]TJ /F3 11.955 Tf 9.3 0 Td[(Eml):(2{36)Usingthecompletesetofbasisstatesremovesmostofthetruncationerrorfromdiscardinghighenergystates,howeverthetruncationerrorisnotfullyxedsincethehighenergyeigenstatesandeigenvaluesareonlyapproximate.ItwasmentionedinSec. 2.4 thatthecalculationofthermodynamicpropertiesataniterationNshouldbedonefortemperaturesTontheorderofTN=)]TJ /F4 7.97 Tf 6.58 0 Td[(N=2D=kB.NghiemandCosti[ 65 ]wereabletousethecompletebasissettoallowforthecalculationofthermodynamicpropertiesatiterationNtoworkforalltemperaturesT>TN(thusallinversetemperatures
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Figure2-8. SchematicdiagramofthecompleteNRGbasis,showingNRGeigenvaluesvs.iterationnumbermwithamaximumiterationNandmaximumchainlengthN+1.ThisdiagramcanbecontrastedwiththediagramforthestandardNRGprocedureinFig. 2-4 .Thestatesgeneratedateachiterationareinthegreenandredboxes.Thestatesdiscardedattheendofaniterationareinredandthestateskeptforthenextiterationareingreen.Thewhiteboxesshowthedescendantsofdiscardedstatesatpreviousiterations.Foreachiterationafterastateisdiscardeditaccruesamultiplicativedegeneracyfactordassociatedwiththenumberofstatesitwouldhaveproducedhadthestatebeenkept.Thetotaldegeneracyassociatedwithastatediscardedatiterationmisgivenontheright.ThetotalpartitionfunctionthatgetsgeneratedusingthecompletebasissetisthesumofthepartitionfunctionsZNfromeachiterationN. Employingthedenitionfor^inEq.( 2{35 ),theexpectationvaluecanbewrittenoutas: h^Oi=1 ZNXm=m0e)]TJ /F4 7.97 Tf 6.59 0 Td[(Emlhlemj^Ojlemi:(2{38) 2.8EntanglementCalculationsUsingtheReducedDensityMatrixTheintroductiontoentanglementgiveninSec. 1.4 paintsapictureinbroadstrokes,usingageneraldenitionforentanglemententropyinEq.( 1{16 ).ToapplyEq.( 1{16 )totheNRGformulationofanimpurityproblem,thesystemneedstobepartitioned.Withthisformulation,subsystemAcontainstheimpuritysitealongwithsomenumberLofconductionsitesclosesttotheimpurity.ThereduceddensitymatrixA,takenby 42

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tracingoutthestatesinpartitionB,whendiagonalized,produceseigenvaluesequaltotheprobabilitypithatagivenstatejiiAinsubsystemAappearsinthegroundstateoftheentiresystem.Usingtheseeigenvalues,theentanglemententropycanbere-expressed: Se=)]TJ /F1 11.955 Tf 9.3 0 Td[(TrA(AlnA)=)]TJ /F4 7.97 Tf 14.74 15.22 Td[(dAXi=1pilnpi(2{39)wheredAisthedimensionofacomplete,orthonormalbasisforpartitionA.TheminimumvalueforSeiszero(forpi0=1andpi6=i0=0)anditsmaximumvalueisSe=lndA(forpi=1=dAforalli)whenAisthesmallersubsystemorSe=lndBwhenBisthesmallersubsystem.ThisformcaneasilybeappliednumericallyinthecaseofsmallL(L=0;1),butquicklybecomescumbersomeasitisnoteasilygeneralizablenumerically.ThismethodisusedinCh. 3 .AmoreeectivenumericalgeneralizationofEq.( 1{16 )usesthecompletebasissetdevelopedintheprevioussectiontocreateafulldensitymatrixthatcanthenbepartitioned.Thispartitioneddensitymatrix,whichisreferredtoasthereduceddensitymatrix,wasbrieymentionedinSecs. 1.4 and 2.6 .Tofullyexpandonthenumericalimplementation,afullerdescriptionofthereduceddensitymatrixisrequired.Usingthecompletebasisset,thedensitymatrixcanbewrittenoutas: =0BBBBBBBBBB@m0(l;l0)m0+1(l;l0)0...0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(l;l0)N(l;l0)1CCCCCCCCCCA;(2{40)wherem(l;l0)aretheblocksofthedensitymatrix,mistheNRGiterationnumber,m0istherstiterationwithdiscardedstates,NisthenaliterationnumberforaWilsonchainoflengthN+1,andthestateslandl0arediscardedatiterationm.Theblocksofthe 43

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matrixcanbewrittenoutas m(l;l0)=Xe;e0hlemj^jl0e0m0i=dN)]TJ /F4 7.97 Tf 6.58 0 Td[(ml;l0e)]TJ /F4 7.97 Tf 6.58 0 Td[(Eml Z;(2{41)wherethedensityoperator^comesfromEq.( 2{35 ).AsstatedinSec. 2.7 ,thesetofstatesfjlemigcomposethecompleteFockspace,withelabellingthesetofenvironmentalstatesatiterationm.Summingovertheoverlapofenvironmentalstatesheje0i=e;e0givesthedN)]TJ /F4 7.97 Tf 6.59 0 Td[(mdegeneracy,basedonthedegeneracydofthesitestatesalongthechain.PerformingapartialtraceofthedensitymatrixinEq.( 2{40 )willproducethereduceddensitymatrixAforanumberofsitesLalongtheWilsonchainpartitionedwiththeimpurity.ThepartialtraceoversubsystemBformL)]TJ /F1 11.955 Tf 12.06 0 Td[(1producesthereduceddensitymatrix A=0BBBBBBBBBB@m0(l;l0)m0+1(l;l0)0...0L)]TJ /F6 7.97 Tf 6.58 0 Td[(1(l;l0)RredL)]TJ /F6 7.97 Tf 6.59 0 Td[(1(k;k0)1CCCCCCCCCCA;(2{42)whereRredm(k;k0)isthepartialreduceddensitymatrixatiterationmwithstateskandk0kept(notdiscarded)atiterationm.Thispartialreduceddensitymatrixhasanexplicitformgivenby: RredL)]TJ /F6 7.97 Tf 6.59 0 Td[(1(k;k0)=Xehke(L)]TJ /F1 11.955 Tf 11.95 0 Td[(1)j^jk0e(L)]TJ /F1 11.955 Tf 11.96 0 Td[(1)i;(2{43)usingthedensityoperator^givenbyEq.( 2{35 ).Thiscanbemanipulatedtoobtainareverseiterativerecursionrelation,asgivenbyEq.(30)ofRef.[ 65 ].ObtainingentanglemententropydirectlyfollowsfromthereduceddensitymatrixbydiagonalizingeachblockofEq.( 2{42 )toobtaintheeigenvaluesaofAandthenusingthevonNeumannentropyfromEq.( 1{16 )forthediagonalizedmatrix: Se=)]TJ /F1 11.955 Tf 9.3 0 Td[(TrA(AlnA)=)]TJ /F8 11.955 Tf 11.29 11.35 Td[(Xaalna:(2{44) 44

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2.9NonequilibriumProblemsUnderstandingthetimeevolutionofstrongly-correlatedsystemsoutofequilibriumisstillrelativelynew.Examplesofnonequilibriumproblemsdealingwithquantumimpuritiesaretransportpropertiesthroughquantumdots[ 66 ]andsingle-electrontransistors[ 67 ].Manydierenttechniqueshavebeenproposedtodealwiththesetypesofproblemssuchasthetime-dependentNRG(TDNRG)[ 63 64 ],functionalrenormalizationgroup[ 68 ],perturbativescaling[ 69 ],renormalizedperturbationtheory[ 70 ],and1=N-expansiontechniques[ 71 ].Althoughtheseapproacheshavefoundsomesuccesssolvingnonequilibriumproblems,mosttechniquesareperturbativeinnature,andforreasonsdiscussedearlierwithperturbationtheory,theseapproachespoorlyexplainthetruedynamicsofthesystem.TheTDNRGisoneusefulmethodforcalculatingthetimeevolutionofoperators[ 65 ]duetoitsinherentnonperturbativenaturethatcanaccuratelydescribelowenergyandtemperaturescales.TheTDNRGaimstotackleproblemswhereasystemundergoesaquench,suchasagatevoltagepulse[ 72 ]beingappliedtotheimpurity.TheHamiltonianundergoingthequenchcanbewrittenpiecewise: H(t)=Hi()]TJ /F3 11.955 Tf 9.3 0 Td[(t)+Hf(t);Hf=Hi+H:(2{45)ThistypeofproblemissolvedwiththeTDNRGbycalculatingtheequilibriumpropertiesofeachseparateHamiltonian,HiandHf,calculatingtheoverlapoftheireigenstates(e.g.ihlemjl0e0m0ifinthelanguageofSec. 2.7 )todeterminehowthetransitionfromHitoHfismadeatsomeintermediatetimet.Ithasbeenwellestablishedthatintheshort-timelimitt!0+,thisapproachisexact[ 63 65 73 ].Issuesbegintoariseinthelong-timelimit,however,asthehighenergyscalesoftheproblemareasimportanttothedynamicsofthesystemasthelowenergyscales[ 63 73 ].Forthisreason,utilizingthecompletebasissettoincludethehighenergystatesisanecessarystepforaccuratecalculationsofpropertiesatlargetimescales. 45

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Toapplythecompletebasismethodtotreatasuddenquench,theNRGisruntwice:onerunforHiandoneforHf.DoingthisprovidestheeigenstatesandeigenenergiesofbothHiandHf,aswellastheoverlaps(innerproducts)betweentheeigenstatesofthetwodierentHamiltonians.TakingtheresultsfromtheprevioustwoNRGruns,anewreverseiterativerungivesthetimeevolutionofanoperatorfromthecoreresultoftheAndersandSchillerpaper[ 63 ] h^Oi(t)=NXm=m0trunXr;sei(Emr)]TJ /F4 7.97 Tf 6.59 0 Td[(Ems)tOmr;si!fs;r(m):(2{46)HererandslabelmanybodyeigenstatesofHfthatareeitherkeptordiscardedatiterationm,Omr;s=fhremj^Ojsemifarethematrixelementsofoperator^ObetweenstatesofHf,andi!fs;r(m)isthereduceddensitymatrixofHiexpressedinthebasisoftheeigenstatesofHf[ 65 ]: i!fs;r(m)=Xefhsemj^eqjremif;(2{47)where^eqisthedensitymatrixgivenbyEq.( 2{35 )calculatedfromHi,andthestateswiththesubscriptfcomefromHf.Thelabel\trun"inthesumofEq.( 2{46 )isareminderthat,duetothenatureofthecalculations,atleastoneofrorsmustbeastatediscardedatiterationm,i.e.therearenocontributionsfromtwosetsofkeptstates: trunXr;s=Xl;k+Xk;l+Xl;l0(2{48)wherel;l0arediscardedstatesandkisakeptstate.Thetime-dependentcalculationsareexactfortheshort-timelimitt!0+,butnoisyintheintermediateandlongtimeranges.NghiemandCostifoundthatreducingthesizeofthequenchHreducestheerrorinthelong-timelimit[ 65 ].Theysuggestedthatratherthanusingasinglelargequench,itispossibletousemanysmallquenchesoveraniteamountoftimecreatingwhattheydubbedageneralpulse,seeninFig. 2-9 .NghiemandCosticommentedonthecomputationalcostofusingnquenches:thelargestadditiontothecomputationaltimecomesfromtheoverlapmatricesbetweenquenched 46

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Figure2-9. SchematicdiagramofthegeneralpulsemethodcreatedbyNghiemandCosti.ThisdiagramshowsacomparisonbetweenthestandardquantumquenchandthewayageneralpulsecanbecreatedbyapplyingmultiplesmallquenchestotheHamiltonianoveraniteamountoftime.ThismethodismeanttoutilizesmallstepstoapproximateanearsmoothrisefromHitoHf.Inthisdiagram,thegeneralpulseisappliedwithn+1quenchesoveratotaltimeoftn. Hamiltonians.Sincetheseoverlapmatricesarecalculatedhighlyeciently,theyestimatethatthecomputationaltimeincreaseslinearlywiththenumberofquenches.AfullstudyofthishasbeendoneinRef.[ 74 ]. 47

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CHAPTER3ENTANGLEMENTENTROPYANDTHEQUANTUMCRITICALPOINTThischapterisbasedonapublishedpaper[ 49 ].AllthepublishedcontentsarereprintedwithpermissiongrantedunderthecopyrightpolicyoftheAmericanPhysicalSocietyfromJ.H.Pixley,TathagathaChowdhury,M.T.Miecnikowski,JamieStephens,ChristopherWagner,andKevinIngersent,Phys.Rev.B91,245122(2015).Copyright2015bytheAmericanPhysicalSociety.AsestablishedinSec. 1.4 ,systemswithquantumimpuritiescanprovideinsightintotheentanglemententropypropertiesnearquantumphasetransitions.Here,westudythepseudogapAndersonmodelwhichexhibitsthecriticaldestructionoftheKondoeect,whereKondoscreeningissuppressedatasecond-orderquantumphasetransitionarisingfromthepresenceofthepseudogapintheconduction-banddensityofstatesaroundtheFermienergy[ 9 36 38 { 42 75 76 ].BycombininganalyticandNRGcalculations,weestablishthattheentanglemententropybetweenamagneticimpurityanditsenvironmentcontainsacriticalcomponentinthevicinityoftheseKondo-destructionQCPs.InpseudogapAndersonmodels,weshowthatchargeuctuationsleadSetotakeanonuniversalvalueatthequantumphasetransition.Awayfromparticle-holesymmetry,Secanvarynonmonotonicallywith,andinsomecasesexhibitsacusppeakattheQCP.Theremainderofthischapterisorganizedasfollows.Section 3.1 reviewsgeneralcharacteristicsoftheentanglemententropyandsummarizestheuniversalbehaviorsthatwendatKondo-destructionQCPs.DetailedanalysisofthepseudogappedAndersonmodelsispresentedinSec. 3.2 .ResultsobtainedbycollaboratorsonavarietyofKondomodelsarebrieydiscussedinSec. 3.3 .WediscussourresultsinSec. 3.4 andconcludeinSec. 3.5 3.1GeneralConsiderationsEntanglemententropycapturesthedegreeofquantumnonlocalityintheground-statewavefunction.Specically,itisapropertyassociatedwithapartitionofthesysteminto

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Figure3-1. SchematicrepresentationofthemodelHamiltonianconsideredinthiswork.TheimpuritydegreesoffreedomHimparecoupledviaHhost-imp(wavyline)tothebathdegreeoffreedomHhost.WetraceoutregionB,anddeterminetheentanglemententropyofregionA. tworegionsAandBthateectively\cuts"thegroundstatealongtheboundarybetweentheregions.Upontracingthesystem'sdensityoperator^overregionB,oneobtainsthereduceddensityoperatorinregionA,^A=TrB^.Similarly,onecantraceoverregionAtoobtain^B=TrA^.TheentanglemententropyisthevonNeumannentropyof^Aor^BasgiveninEq.( 1{16 ),whichmeasurestheextenttowhichregionAisentangledwithregionB.Inquantumimpurityproblems,theentanglemententropybetweentheimpurityandtherestofthesystemisdenedbytakingregionAtocontainsolelytheimpuritydegreesoffreedom,whileregionBdescribesthehost(i.e.,therestofthesystem),asshownschematicallyinFig. 3-1 .Upontracingoutthehost,weobtaintheimpurityreduceddensityoperator^impactinginavectorspaceofdimensiondimp.Equation( 1{16 )thengivestheimpurityentanglemententropy[ 29 30 ] Se=)]TJ /F4 7.97 Tf 11.43 16.34 Td[(dimpXi=1pilnpi;(3{1) 49

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wherefpigisthesetofeigenvaluesof^imp.Thecorrespondingeigenstatesfjiigmustrespectthesystem'ssymmetries,aconstraintthatallowstheeigenvaluespitobeexpressedintermsofexpectationvaluesofimpurityoperatorsthatcanreadilybecalculatedusingtheNRG.Sincethehostdegreesoffreedomhavebeencompletelytracedout,theimpurityentanglemententropymeasuresonlytheentanglementbetweentheimpurityandthehostasawhole.Detailsofthehost{suchasthenumber,dispersion,andanyinternalinteractionsfortheconductionband{inuenceSeonlyinsofarastheyaecttheimpuritymatrixelementsthatdeterminetheeigenvaluesof^imp.Foragroundstateofproductformj i=j iimpjihost,onecanchoosep1=1andpi=0foralli>1,implyingthatSe=0.Attheotherextreme,astateofmaximalentanglementbetweentheimpurityanditshostisdescribedbypi=1=dimpforalli,leadingtoSe=lndimp.Acomplicationarisesifthesystemisnotinapurestate,asislikelytobethecasewhenthereisground-statedegeneracy.Forexample,inthelimitwheretheimpurityandthehostaredecoupled,n-folddegeneracyoftheimpuritygroundstateresultsin^imphavingnvaluespi=1=nanddimp)]TJ /F3 11.955 Tf 12.56 0 Td[(nvaluespi=0,implyingthatSe=lnn.Inordertoavoidsuchmisleadingindicationsofentanglement,itisnecessarytobreaktheground-statedegeneracyoftheimpuritytoobtainapurestate.Inthepresentwork,wherewetreatmagneticimpurities,theground-statedegeneracycanbeliftedbytheapplicationofaninnitesimallocalmagneticeldhlocthatcouplessolelytotheimpuritythroughaHamiltoniantermhlocSzimp,whereSimpistheimpurityspinoperatorandtheLandegfactorandtheBohrmagnetonhavebothbeensettounity.Forthisreason,weconsideratwo-parameterfunctionSe(x;hloc),wherexisanonthermal,nonmagneticparameterthattunesthesystemthroughaQCPatx=xc.Inmanycases,weemployareducedvariable=(x)]TJ /F3 11.955 Tf 12.15 0 Td[(xc)=xcsuchthattheQCPislocatedat=0.Italsoprovesconvenienttodenethelocal-eld-dependentpartoftheentanglemententropy 50

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(notethesign) Se(x;hloc)=Se(x;0))]TJ /F3 11.955 Tf 11.96 0 Td[(Se(x;hloc);(3{2)andtointroducetheshorthandnotations S+e(x)=Se(x;hloc=0+); (3{3a)Se(x)=Se(x;hloc=0+); (3{3b)representing,respectively,thedegeneracy-liftedentanglemententropyandthereductioninentanglemententropyduetospontaneoussymmetrybreaking.InthefollowingsectionwereportresultsfortheentanglemententropyinseveralquantumimpurityHamiltoniansofthegeneralform H=Hhost+Himp+Hhost-imp(3{4)whereHhostdescribesafermionicband,Himpdescribestheisolatedimpurity,andHhost-impaccountsforthecouplingbetweenthehostandtheimpurity.Thefermionicbandisassumedtohaveadispersionkgivingrisetoanidealizeddensityofstates c()=N)]TJ /F6 7.97 Tf 6.58 0 Td[(1kXk()]TJ /F3 11.955 Tf 11.95 0 Td[(k)=0j=Djr(D)-222(jj);(3{5)whereNkisnumberofdistinctkvalues,(x)istheHeavisidefunction,Disthehalf-bandwidth,andrisabandexponent.QCPsariseintheAndersonmodelincases00)fromalocal-momentphase(spanning<0),inwhichtheKondoeectisdestroyed.Anappropriateorderparameterforthequantumphasetransitionisthehloc!0+limitofthelocalmagnetizationMloc(;hloc)=)]TJ /F1 11.955 Tf 11.29 0 Td[(limT!0hSzimpi,whichvanishesthroughouttheKondophase,andinthelocal-moment 51

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phaseclosetotheQCPobeys Mloc(;hloc=0+)/()]TJ /F1 11.955 Tf 9.3 0 Td[();(3{6)whereistheorder-parameterexponent.Atthecriticalvalueofthetuningparameter,thelocalmagnetizationsatises Mloc(=0;hloc)/jhlocj1=(3{7)whereisanothercriticalexponent.WeshowinSec. 3.2 thatintheAndersonmodelswithapseudogaphost,andweshowmoregenerallyinRef.[ 49 ]{independentoffeaturessuchasparticle-holesymmetryorasymmetry,thedegreeofimpuritychargeuctuations,whethertheKondophaseinvolvesexact,over-,orunder-screening,andthepresenceorabsenceofcompetitionbetweenfermionicbandsandbosonicbaths{thatuponapproachtotheQCPfromthelocal-momentside,theentanglemententropysatises Se(;hloc)=aM2loc(3{8)whereaisaconstantthatdependsondetailsofthemodel.WhencombinedwithEqs.( 3{6 )and( 3{7 ),Eq.( 3{8 )impliesthat Se(<0;hloc=0+)/()]TJ /F1 11.955 Tf 9.3 0 Td[()e; (3{9a)Se(=0;hloc)/jhlocj1=e; (3{9b)with e=2; (3{10a)1=e=2=; (3{10b) 52

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WesolvethequantumimpurityproblemdiscussedaboveusingtheNRG[ 53 77 ]asadaptedtotreatquantumimpurityproblemsinvolvingapseudogappedfermionicdensityofstates[ 9 75 ].TheimpurityentanglemententropyisfoundviaEq.( 3{1 )usingreduceddensitymatrixeigenvaluespiexpressedintermsofexpectationvalues(convergedforlargeNRGiterationnumberscorrespondingtoasymptoticallylowtemperatures)ofcertainimpurityoperatorsspeciedinthesectionsthatfollow.WeuseaWilsondiscretizationparameter39,arangethathasbeenshownpreviouslytoprovideanaccurateaccountofthecriticalexponents[ 38 47 48 78 ].ClosetotheQCP,wenditnecessarytoemployquadruple-precisionoating-pointcalculationsinordertoaccuratelyresolvetheentanglemententropy,andinparticular,thevalueofSe()denedinEq.( 3{3b ). 3.2AndersonModelsInthissectionweconsidernon-degenerateAndersonimpuritymodelscharacterizedbyahostHamiltonian Hhost=Xk;kcyk;ck;(3{11)whereck;destroysaconductionelectronofwavevectork,spinzcomponent=1 2";#,andenergyksatisfyingEq.( 3{5 ),andanimpurityHamiltonian Himp=dnd+Und"nd#+hloc(nd")]TJ /F3 11.955 Tf 11.95 0 Td[(nd#)=2;(3{12)whereddestroysanelectronofenergydandspinzcomponentelectronattheimpuritysite,nd=dydandnd=nd"+nd#areimpuritynumberoperators,andUistheon-siteCoulombinteraction.Inthepresenceofaninnitesimalsymmetry-breakingeld,theentanglemententropyingeneralneitherexhibitsapeakattheQCPnorattainsitsmaximumpossiblevalueattheQCP.Underconditionsofspin-rotationsymmetryaboutthezaxis,theeigenstatesoftheimpurityreduceddensityoperatorcanbetaken[ 81 ]tobetheconventionalbasisstatesj0i, 53

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ji=dyj0i,andj2i=dy"dy#j0i,witheigenvaluesthatcanbewrittenas p2=hnd"nd#i; (3{13a)p"=f=2+Mloc; (3{13b)p#=f=2)]TJ /F3 11.955 Tf 11.96 0 Td[(Mloc; (3{13c)p0=1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)]TJ /F3 11.955 Tf 11.96 0 Td[(p2; (3{13d)intermsofthelocal-momentfraction(i.e.,single-occupationprobability)f=hndi)]TJ /F1 11.955 Tf 19.74 0 Td[(2p2andthelocalmomentMloc=h1 2(nd")]TJ /F3 11.955 Tf 12.08 0 Td[(nd#)i,suchthat0f1andjMlocjf=2.ThenEq.( 3{1 )canbewritten Se=fS21 2+Mloc f+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)S2p2 1)]TJ /F3 11.955 Tf 11.96 0 Td[(f;(3{14)where S2(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(xlnx)]TJ /F1 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)ln(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)(3{15)isthebinaryentropyfunction.Eq.( 3{14 )canbeinterpretedasthesumofabinaryspinentanglemententropywithweightfandabinarychargeentanglemententropywithweight(1)]TJ /F3 11.955 Tf 12.2 0 Td[(f).TheparallelbetweenthespinandchargepartsofSecanbemadeclearerbydeninga\localcharge"Qloc=p2)]TJ /F3 11.955 Tf 11.96 0 Td[(p0satisfyingjQlocj1)]TJ /F3 11.955 Tf 11.95 0 Td[(f,sothat p2 1)]TJ /F3 11.955 Tf 11.96 0 Td[(f=1 2+Qloc 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f):(3{16)Intheabsenceofasymmetry-breakingeldhloc,wecansetMloc=0inEqs.( 3{13 ),anduseEq.( 3{1 )toobtain Se(hloc=0)=)]TJ /F1 11.955 Tf 9.3 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)]TJ /F3 11.955 Tf 11.95 0 Td[(p2)ln(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)]TJ /F3 11.955 Tf 11.96 0 Td[(p2))]TJ /F3 11.955 Tf 11.96 0 Td[(fln(f=2))]TJ /F3 11.955 Tf 11.95 0 Td[(p2lnp2:(3{17)DierentiatingEq.( 3{17 )withrespecttop2foraxedlocal-momentfractionfyields @Se @p2f=lnp0 p2;(3{18) 54

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whichimpliesthatforagivenvalueoff,Se(hloc=0)isgreatestforequaloccupationoftheimpuritycongurationsj0iandj2i(i.e.Qloc=0),andissmallestwhenoneorotherofthecongurationsisruledout.Inthelocal-momentphaseneartheboundarywiththeKondophase,weexpectaninnitesimaleldhloc=0+toestablishalocalmagnetizationjMlocj1 2withnegligibleshiftoffandp2.Underthesecircumstances,anexpansionofSeinpowersofMlocshowsthespontaneous-symmetry-breakingpartoftheentanglemententropytobe Se()'2M2loc=f/()]TJ /F1 11.955 Tf 9.3 0 Td[()2;(3{19)providingarealizationofEqs.( 3{8 ),( 3{9a ),and( 3{10a ),witha=2=f.Forthepurposesofnumericalstudy,wefocusontheone-channelAndersonimpuritymodeldescribedbytheHamiltonianinEq.( 3{4 )withHhostasgiveninEq.( 3{11 ),HimpasinEq.( 3{12 ),and Hhost-imp=V p NkXk;(cyk;d+H:c:):(3{20)ThehybridizationmatrixelementVbetweentheimpuritysiteandtheconductionbandisconventionallyre-expressedintermsofthehybridizationwidth)-391(=0V2.Wehavecomputedhnd;iandhnd;"nd;#i,thenusedEqs.( 3{1 )and( 3{13 )tondSe.InthefollowingwetakeUanddtobexed,eitheratparticle-holesymmetry(d=)]TJ /F3 11.955 Tf 9.29 0 Td[(U=2)orawayfromit(d6=)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2).WethenndthelocationoftheQCPatacriticalhybridizationwidth)]TJ /F4 7.97 Tf 151.04 -1.79 Td[(c(U;d),andthereafterdenethedistancefromcriticalityas=()]TJ /F2 11.955 Tf 56.19 0 Td[()]TJ /F1 11.955 Tf 12.25 0 Td[()]TJ /F4 7.97 Tf 7.32 -1.79 Td[(c)=)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(c.ThecriticalresponsestoalocalmagneticeldneartheasymmetricQCPsofthepseudogapAndersonmodelbelonginthesameuniversalityclassesastheasymmetricQCPsoftheSimp=1 2Kondomodel[ 9 38 { 40 42 ].Inthesubsectionsthatfollow,werstconsidertwospecialcases(U=)]TJ /F1 11.955 Tf 9.3 0 Td[(2dandU=1)inwhichSe(hloc=0)inEq.( 3{17 )reducestoafunctionofonevariable,thelocal-momentfraction,therebysimplifyinganalysisofthebehavioroftheentanglemententropyinthevicinityoftheQCP.Afterward,wepresentillustrativeexamplesofthe 55

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entanglementpropertiesformoregeneralcases. 3.2.1Particle-HoleSymmetry:U=)]TJ /F1 11.955 Tf 9.3 0 Td[(2dFortheparticle-hole-symmetriccased=)]TJ /F3 11.955 Tf 9.29 0 Td[(U=2,wehavehndi=1andp0=p2=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)=2.Asaresult,Eq.( 3{17 )reducesto Se(hloc=0)=S2(f)+ln2;(3{21)whichincreasesmonotonicallyfromln2toln4asjf)]TJ /F6 7.97 Tf 13.35 4.71 Td[(1 2jdecreasesfrom1 2to0.Equation( 3{18 )tellsusthatforagivenvalueoff,Se(hloc=0)isgreaterforthissymmetriccasethanforanyvalued6=)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2thatleadstop06=p2.Intheconventionalsituationofon-sitecoulombrepulsion(i.e.,U>0),thereisamonotonicevolutionofthelocal-momentfractionfromf=1at)-439(=0tof!1 2for)]TJ /F2 11.955 Tf 12.22 0 Td[(U,reectingtheincreasedadmixtureofthend=0,2excitedcongurationsintothend=1groundstateoftheisolatedimpurity.THisbehavior,whichisexempliedinFig. 3-2 (a)forr=0:4andtwocases,U=D=0:005and0:5,leadstoamonotonicincreaseinSe(hloc=0)fromln2at)-390(=0toln4for)]TJ /F2 11.955 Tf 192.67 0 Td[(U.BothfandSe(hloc=0)increaseinasmooth,featurelessfashionas)-327(risesthrough)]TJ /F4 7.97 Tf 312.68 -1.79 Td[(c.[ThemappingU!)]TJ /F3 11.955 Tf 25.68 0 Td[(U,d!)]TJ /F3 11.955 Tf 26.73 0 Td[(d,)]TJ /F2 11.955 Tf 19.94 0 Td[(!)-326(takesonetoanAndersonmodelwithon-siteattraction,whichforapseudogapdensityofstatesfeaturesquantumphasetransitionsbetweenlocal-chargeandcharge-Kondophases[ 82 ].Sincethemappingtransformsf!1)]TJ /F3 11.955 Tf 12.7 0 Td[(f,itpreservesSe(hloc=0).WewillnotconsidercasesU<0anyfurtherinthispaper.]Wehavealreadyarguedthattheeectofalocalmagneticeldhloc=0+istoreducetheentanglemententropybyanamountthatvariesneartheQCPaccordingtoEq.( 3{19 ).SincetheQCPalwaysoccursatalocalmomentfractionf>1 2,Se(hloc=0)hasapositiveslopeat)-497(=)]TJ /F4 7.97 Tf 167.84 -1.8 Td[(c.Itisthereforethecasethatthedegeneracy-liftedentanglemententropyhasapositiveslopeonbothsidesofthequantumphasetransition.Inotherwords,itisimpossibleforS+etoexhibitapeakattheQCP.Incaseswheretheorder-parameterexponentsatises2<1,weexpectadivergenceofdS+e=d)]TJ ET BT /F1 11.955 Tf 228.15 -687.85 Td[(56

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Figure3-2. Particle-hole-symmetricpseudogapAndersonmodelwithbandexponentr=0:4fordierentvaluesofU=D:(a)local-momentfractionfand(b)symmetry-liftedentanglemententropyS+e,bothvshybridizationwidth)]TJ -214.69 -14.44 Td[(scaledbyitscriticalvalue)]TJ /F4 7.97 Tf 143.69 -1.8 Td[(c.Inthelimitoflarge,S+eapproachesitsmaximumvalueofln41:386.Insetto(b):spontaneous-symmetry-breakingpartoftheentanglemententropySevsjj[where=()]TJ /F2 11.955 Tf 77.1 0 Td[()]TJ /F1 11.955 Tf 11.95 0 Td[()]TJ /F4 7.97 Tf 7.32 -1.8 Td[(c)=)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(c]forthetwocasesshownin(a)and(b).Thelinearvariationsofthepointsinthislog-logplotareconsistentwithEq.( 3{9a )withe=1:8288(1).(c)Local-eld-dependentpartoftheentanglemententropySevshlocatthecriticalhybridizationwidth,showingbehaviorconsistentwithEq.( 3{9b )with1=e=0:3703(1). onapproachtotheQCPfromthelocal-momentside.Bycontrast,for2>1,thespontaneous-symmetry-breakingpartoftheentanglemententropyshouldvanishfasterthanthelinearvariationofSe(hloc=0)andS+eshouldthereforebeessentiallyfeaturelessonpassingthroughthequantumphasetransition.Figure 3-2 (b)plotsS+eoverabroadrange0)]TJ /F3 11.955 Tf 7.31 0 Td[(=)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(c2forr=0:4,acase[ 38 ]where=0:91440(2)>1 2.ThecurvesfordierentvaluesofUarequantitativelydierent,butsharethesameprincipalfeatures:asmoothriseofS+efromzerointhedecoupled-impurity 57

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Table3-1. Propertiesatthequantumcriticalpointoftheparticle-hole-symmetricpseudogapAndersonmodelforr=0:4andthreevaluesofU=D:local-momentfractionfc(fromNRG),andthepredictedandactualvaluesofSe=M2locinaeldhloc=0+,i.e.,2=fcbasedonEq.( 3{19 )andtheratioacomputeddirectlyfromNRGvaluesSeandMloc. U=Dfc2=fca 0.0050.6372203.138633.1386(1)0.050.6469763.091313.0913(2)0.50.6898002.899392.8994(1) limit)]TJ /F3 11.955 Tf 35.28 0 Td[(=U!0,withalinearvariationthrough)-295(=)]TJ /F4 7.97 Tf 206.04 -1.8 Td[(c[notshownindetailinFig. 3-2 (b)]leadingtoasaturationS+e!ln4intheuncorrelatedlimit)]TJ /F3 11.955 Tf 158.24 0 Td[(=U!1.Wendthatforagivenvalueof,S+eintheparticle-hole-symmetricAndersonmodelgenericallyexceedsS+einthecounterpartSimp=1 2Kondomodelwiththesamebandexponentr.ThisisanaturalconsequenceoftheAndersonimpuritybeingentangledwithboththespinandchargedegreesoffreedomofitsenvironment.InthelimitU=D!1,however,theformofS+e()fortheAndersonmodelconvergestothecorrespondingfunctionfortheKondomodel.[theKondofunctionS+e()forseveralvaluesofrcanbeextractedfromFig.3(a)inRef.[ 49 ]byrescalingthehorizontalaxisfrom0Jto=(J)]TJ /F3 11.955 Tf 12.83 0 Td[(Jc)=Jc.]Conversely,asU=Dapproacheszero,thecriticalhybridizationvanishes[ 83 ]as)]TJ /F4 7.97 Tf 28.84 -1.79 Td[(c=D/(U=D)1)]TJ /F4 7.97 Tf 6.59 0 Td[(r,whileS+ebecomeseverclosertoln4throughouttheregion)]TJ /F2 11.955 Tf 130.15 0 Td[()]TJ /F4 7.97 Tf 7.31 -1.79 Td[(c;atrendthatsmoothlymergesintothephysicsofanoninteractingresonantlevel(U=d=0)whereS+e=ln4forany)]TJ /F3 11.955 Tf 84.56 0 Td[(>0.Examinationofthespontaneous-symmetry-breakingpartoftheentanglemententropyallowsquantitativetestingofEq.( 3{19 ).TheinsettoFig. 3-2 (b)illustrateslog-logplotsofSe()thatforU=D=0:5and0:005canbettedtoEq.( 3{9a )withe=0:91440(2).Table 3-1 showsthattheratioSe=M2lociscapturedtohighaccuracybyEq.( 3{19 ).Weconcludeourdiscussionoftheparticle-holesymmetricAndersonmodelbypresentingresultsfortheeectofannitelocalmagneticeld.Figure 3-2 (c)plotsthelocal-eld-dependentpartoftheentanglemententropySe(;hloc)asafunctionofhloc 58

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forxed=0(i.e.,atthecriticalhybridizationwidth).ThelinearvariationofthedataforthreedierentvaluesofU=DfulllsthepredictionofEq.( 3{9b ),whilethettingofquadruple-precisionresultsspanningeldsdowntohloc=10)]TJ /F6 7.97 Tf 6.59 0 Td[(25D(datanotshown)yieldsanexponent1=e=0:37032(2)inexcellentagreementwiththevalue0:37032(4)deducedfromEq.( 3{10b )usingthepreviouslyknownvalue1==0:18516(2). 3.2.2MaximalParticle-HoleAsymmetry:U=1ForthecaseU=1ofmaximalparticle-holeasymmetry,theparametersenteringEq.( 3{17 )reducetop2=0,f=hndi,andp0=1)]TJ /F3 11.955 Tf 11.95 0 Td[(f,sothat Se(hloc=0)=S2(f)+fln2:(3{22)Foragivenf,thisvalueissmallerby(1)]TJ /F3 11.955 Tf 12.3 0 Td[(f)ln2thanitscounterpartforthesymmetricmodel;indeed,Eq.( 3{18 )indicatesthatforxedf,Se(hloc=0)takesitssmallestvaluewhenU=1(andalsowhenU=,leadingtop0=0).Undervariationofthelocal-momentfraction,Se(hloc=0)increasesfrom0atf=0toamaximumvalueofln3atf=2=3,andthendecreasestoreachln2atf=1.Forgivenvaluesofrandd,aquantumphasetransitionbetweenlocal-momentandKondophaseswilloccuratsome)-344(=)]TJ /F4 7.97 Tf 204.16 -1.79 Td[(c.Unlikethesituationatparticle-holesymmetry,whereSe(hloc=0)varieslinearlywithneartheQCP[ 49 ],thevariationforanyd6=)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2isdescribedby Se(;0))]TJ /F3 11.955 Tf 11.96 0 Td[(Se(0;0)'Ajj1)]TJ /F6 7.97 Tf 6.89 0 Td[(~sgn;(3{23)where~isthecharge-susceptibilityexponentattheQCPandAmaybepositiveornegative.ForU=1,thisvariationcanbededucedbywriting dSe(;0)=d=[dSe(hloc=0)=df](df=d)(3{24)andnotingthatwhiledSe(hloc=0)=dfisregularneartheQCP,df=d=@hndi=@/jj)]TJ /F6 7.97 Tf 6.88 0 Td[(~canbenonanalytic[ 42 84 ].For0:55.r<1,thechargesusceptibilityexponent~is 59

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positive[ 84 ],signalingacriticaldivergenceofimpuritychargeuctuationsonapproachtotheQCPfromeitherphase.TheasymmetricQCPsforr'3=8r.0:55insteadshowanondivergentchargeresponse,i.e.,~=0,abehavioralsodisplayedatthesymmetricQCPsthatexistford=)]TJ /F3 11.955 Tf 9.3 0 Td[(U=2and00for=0)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(.PreviousNRGcalculationshaveshownthatcondition(2)issatisedattheparticle-hole-asymmetricQCPsfor0:42.r<1[ 85 ].Inwhatfollows,therefore,wefocusonwhethercondition(1)issatised.ThesignofdSe(hloc=0)=dat=0canbedeterminedusingEq.( 3{24 ).FromEq.( 3{22 )weseethatdSe(hloc=0)=dfispositiveforf<2=3andnegativeforf>2=3.Intheregimed<0thatadmitsinterestingmany-bodyphysics,thelocal-momentfractionhaslimitsf!1for)]TJ /F2 11.955 Tf 38.77 0 Td[(!0andf!1 2for)]TJ /F2 11.955 Tf 29.02 0 Td[(!11.Itmightthereforeappearplausiblethatdf=d)]TJ /F3 11.955 Tf 12.31 0 Td[(<0andhencedf=d<0forallintermediatevaluesof.However,NRGcalculationsshowthisassumptiontobecorrectonlyforlargevaluesofjdj=D.Forsmallerjdj=D,finsteadhasaminimumatanitevalueof,beyondwhichitincreasestoapproach1 2frombelow.Wepresentdataheresolelyfortherepresentativecaser=0:6,buthaveobtainedqualitativelysimilarresultsforotherrvalues.Figure 3-3 (a)plotsfvs)]TJ /F3 11.955 Tf 22.01 0 Td[(=)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(cford=D=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:05and)]TJ /F1 11.955 Tf 9.3 0 Td[(0:5.Ford=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:5D,fdecreasesmonotonicallywithincreasinghybridizationwidth,passingthrough2=3at)]TJ /F2 11.955 Tf 35.8 0 Td[('1:07)]TJ /F4 7.97 Tf 19.02 -1.79 Td[(c,whereSe(hloc=0)risestoasmoothpeakat 1Thelarge-)-326(limitofthelocal-momentfractionisf=1 2,andnotf=2=3correspondingtotheeectivedegeneracyofthreeimpuritycongurationsj0i,j"i,andj#i.Thisisbecausefor)]TJ /F2 11.955 Tf 114.65 0 Td[(jdj,D,themany-bodygroundstatecontainsequaladmixturesof(cy0"j0i+j"i)=p 2and(cy0#j0i+j#i)=p 2. 60

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Figure3-3. U=1pseudogapAndersonmodelwithbandexponentr=0:6for)]TJ /F3 11.955 Tf 9.3 0 Td[(d=D=0:05and0:5:(a)Local-momentfractionfand(b)degeneracy-liftedentanglemententropyS+e,bothvshybridizationwidth)-327(scaledbyitscriticalvalue)]TJ /F4 7.97 Tf 37.55 -1.8 Td[(c.Ford=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:5D,fdecreasesmonotonicallywithincreasing,andpassesthrough2/3at)]TJ /F3 11.955 Tf 122.61 0 Td[(=)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(c'1:07,whereS+epeaksatln3.Ford=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:05D,fdropsthrough2/3(andS+epeaks)at)]TJ /F3 11.955 Tf 58.75 0 Td[(=)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(c'1:00085,andf(andS+e)reachesaminimumat)]TJ /F3 11.955 Tf 124.44 0 Td[(=)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(c'2:05(Inset).Onthewidescaleof)]TJ /F3 11.955 Tf 173.93 0 Td[(=)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(cshowninthisgure,thelocationofthepeakinS+eisvirtuallyindistinguishablefromthepositionofthequantumcriticalpoint. itsmaximalvalueln3.Ford=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:05D,fdropsthrough2=3andSe(hloc=0)peaksat)]TJ /F2 11.955 Tf 11.32 0 Td[('1:0008)]TJ /F4 7.97 Tf 30.72 -1.8 Td[(c,barelyintotheKondophase.However,incontrasttoitsbehaviorford=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:5D,thelocal-momentfractionthenreachesaminimumvaluef'0:834'0:759ln3at)]TJ /F2 11.955 Tf 10.95 0 Td[('2:05)]TJ /F4 7.97 Tf 19.02 -1.79 Td[(cbeforerisingbacktowardf=1 2.Uponfurtherdecreaseofjdj=D(notshown),thepeakinSe(hloc=0)movesevercloserto)-410(=)]TJ /F4 7.97 Tf 168.31 -1.79 Td[(candtheminimumvalueoffandarelatedminimuminSe(hloc=0)=df>0anddf=d>0at=0,meaningthattheconditionsarenevermetfortheoccurrenceofapeakinS+epreciselyattheQCP. 61

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Figure3-4. PseudogapAndersonmodelwithbandexponentr=0:6,impuritylevelenergyd=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:05D,andthreevaluesofU6=)]TJ /F1 11.955 Tf 9.3 0 Td[(2d:(a)Local-momentfractionf,(b)doubleoccupancyp2,and(c)degeneracy-liftedentanglemententropyS+evshybridizationwidth)-327(scaledbyitscriticalvalue)]TJ /F4 7.97 Tf 274.3 -1.8 Td[(c.ApeakinS+eoccurspreciselyatthequantumcriticalpointforU=D=0:075and0:055,butshiftedslightlyintotheKondophase()]TJ /F3 11.955 Tf 208.52 0 Td[(>)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(c)forU=0:051D.Thesecasesareindistinguishableonthewidescaleof)]TJ /F3 11.955 Tf 221.79 0 Td[(=)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(cshowninthisgure. 3.2.3GeneralParticle-HoleAsymmetryFinallyweturntocasesofintermediateparticle-holesymmetry,forwhichnosimplicationofEq.( 3{17 )ispossible.Wefocusonceagainonthecaser=0:6representativeoftherangeofbandexponentsinwhiche=2<1)]TJ /F1 11.955 Tf 13.19 0 Td[(~,potentiallyallowingfortheoccurrenceofapeakinS+eattheQCP.Figure 3-4 showsresultsoverawiderangeof)]TJ /F3 11.955 Tf 185.44 0 Td[(=)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(cvaluesford=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:05DandforU=D=0:051,0:055,and0:075.AsUgetscloserto)]TJ /F3 11.955 Tf 9.3 0 Td[(d,particle-holeasymmetrygrowsstronger,asdochangesnear)-324(=)]TJ /F4 7.97 Tf 180.26 -1.8 Td[(cinthelocal-momentfraction[Fig. 3-4 (a)],thedoubleoccupancy[Fig. 3-4 (b)],andthedegeneracy-liftedentanglemententropy[Fig. 3-4 (c)]. 62

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ForU=D=0:075and0.055,Se(hloc=0)peaksjustinsidethelocal-momentphaseat')]TJ /F1 11.955 Tf 23.42 0 Td[(1:210)]TJ /F6 7.97 Tf 6.59 0 Td[(3and)]TJ /F1 11.955 Tf 9.29 0 Td[(3:210)]TJ /F6 7.97 Tf 6.59 0 Td[(4,respectively.ThenegativeslopeofSe(hloc=0)at)-307(=)]TJ /F4 7.97 Tf 31.09 -1.79 Td[(c,combinedwiththeorder-parameterexponentsatisfying2<1)]TJ /F1 11.955 Tf 12.53 0 Td[(~issucienttocreateacusppeakinS+eattheQCP.Bycontrast,forU=D=0:051,Se(hloc=0)peaksjustinsidetheKondophaseat'510)]TJ /F6 7.97 Tf 6.58 0 Td[(5,therebypreventingtheoccurrenceofanypeakinS+eat=02.However,thetinydisplacementofthemaximuminSe(hloc=0)from=0meansthatinpracticeitwillproveveryhardtodistinguishcaseswheretherereallyisapeakinthedegeneracyliftedentanglemententropyattheQCPfromoneswhereapeakliesclosebyinsidetheKondophase.DeeperintotheKondophase,S+eSe(hloc=0)exhibitsabroadminimum,whichiscenteredat)]TJ /F3 11.955 Tf 68.12 0 Td[(=)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(c'1:65,1:52,and1:13forU=D=0:051,0:055,and0:075,respectively.TheminimuminS+eweakensasUincreasestoward)]TJ /F1 11.955 Tf 9.3 0 Td[(2d,consistentwiththemonotonicriseinentanglemententropythroughouttheKondophasethatisseenatparticle-holesymmetry[ 49 ].ThephysicsneartheQCPisshowninmoredetailinFig. 3-5 ,onceagainforarepresentativecaser=0:6andd=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:05D.Figure 3-5 (a)plotsthecriticalpartoftheentanglemententropyintheabsenceofalocalmagneticeldforU=0:055D.ThedivergentchargesusceptibilityproducesanonanalyticvariationofSe(;0))]TJ /F3 11.955 Tf 12.09 0 Td[(Se(0;0)thatiswellcapturedbyEq.( 3{23 ),eventhoughthepower-lawvariationinthelocal-momentphaseisconnedtoarathernarrowregionofjjvaluesduetotheaforementionedpeak 2Thelocations)]TJ /F3 11.955 Tf 80.54 0 Td[(=)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(cofextremainSe(hloc=0),aswellastheextremalvaluesthemselves,areweaklydependentontheWilsondiscretizationparameteremployedintheNRGcalculations.TheNRGresultsquotedinSection 3.2 werecomputedfor=9.Wehavedeterminedforr=0:6,d=)]TJ /F1 11.955 Tf 9.29 0 Td[(0:05D,andU=0:051D,thatreducingfrom9to3shiftsthepeakinSe(hloc=0)from=5:110)]TJ /F6 7.97 Tf 6.59 0 Td[(5to=6:010)]TJ /F6 7.97 Tf 6.59 0 Td[(5.ThatthisshiftistoaslightlylargerjjvalueprovidesevidencethatthedisplacementofthepeakawayfromthequantumcriticalpointisnotmerelyanartifactoftheNRG,andthatanonzerodisplacementwouldsurviveinthecontinuumlimit!1. 63

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Figure3-5. CriticalbehavioroftheentanglemententropySeforthepseudogapAndersonmodelwithr=0:6,d=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:05D,andU6=)]TJ /F1 11.955 Tf 9.3 0 Td[(2d:(a)Shiftinthezero-eldentanglemententropyforU=0:055Donmovingawayfromthequantumcriticalpointintothelocal-momentphase(<0)andtheKondophase().Power-lawtstoEq.( 3{23 )yield~=0:23(1).(b)Local-eld-dependentpartoftheentanglemententropySevshlocatthecriticalhybridizationwidth,forU=D=0:051,0:055,and0:075.ThedataareconsistentwithEq.( 3{9b )with1=e=0:2340(1). inSe(hloc=0)at=)]TJ /F1 11.955 Tf 9.29 0 Td[(3:210)]TJ /F6 7.97 Tf 6.59 0 Td[(4.Theinferredexponent~=0:23(2)isfullyconsistentwiththedirectlycomputed[ 42 84 ]charge-susceptibilityexponent~=0:210(2).Figure 3-5 (b)showsthelocal-magnetic-eldresponseattheQCP(=0forU=D=0:051,0:055,and0:075).FittingtoEq.( 3{9b )yieldsanexponent1=e=0:2340(1),inexcellentagreementwithEq.( 3{10b )giventhat2==0:23392(8)basedonTableIfromRef.[ 49 ]. 64

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3.3WorkonRelatedModelsReferencedinthischapterwasworkdoneonthesub-Ohmicspin-bosonmodel[ 29 30 86 ].Thismodel,whencouplingthetwo-levelsystemtoabathofharmoicoscillators,hasasecond-orderquantumphasetransition.Theentanglemententropyofthesystem,whenseparatingthetwo-levelsystem(i.e.thequantumimpurity)fromthebosonicbath,showsacusp-likebehaviorwhichisassociatedwiththequantumphasetransition.ThecuspproducesauniversalmaximumintheentanglemententropyattheQCP,whichwasdemonstratedthroughananalysisoftheslopeonbothsidesoftheQCP.CollaboratorsmadefurtherinvestigationsoftheentanglementpropertiesinthepseduogappedKondomodelsandaBose-FermiKondomodel.ThepseudogappedKondomodel,likethepseudogappedAndersonmodel,exhibitsaQCPthatdestroystheKondophaseinpartoftheparameterspace(JJcinthedegeneracy-liftedentanglementwhereitismaximallyentangled,aresultincontrasttothesub-Ohmicspin-bosonmodel[ 29 30 86 ]whichmaximallyentanglesattheQCP.ThespinSimp=1 2Bose-FermiKondomodelshowssimilarresultstothespinSimp=1 2pseudogapKondo 65

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models. 3.4DiscussionOneuniversalfeatureofourresultsisthepresenceofanonzeroentanglementonentrytothelocal-moment(Kondo-destroyed)phase.Sucharesidualentanglementimpliesthatthegroundstateisnotasimpleproductofanimpuritystateandanenvironmentalstate.ThisresulthassignicantimplicationsfortheoreticalandnumericaldescriptionsoftheKondo-destroyedphase.Forexample,withinalargeNmeaneldtheoryforthepseudogapKondomodel[ 36 ],thelocalmomentisrepresentedwithfermionicspinonsfandtheeectiveHamiltonianisaresonant-levelmodelwithahybridizationb=h^biMF=hfyc0iMF(where^bisabosonicoperator).Atthislevel,Kondodestructioncorrespondstob!0,implyingthatthelocalmomentiscompletelyfreeandnolongerentangledwiththeconductionband.Thussuchastaticmean-eldtheorycannotreproducethenonzeroentanglemententropythatwendintheKondo-destroyedphase.Ourresultscanbeunderstood,however,intermsofabosonicoperator^b(!)thathasavanishingstaticcomponentandgiverisestoadynamicalKondoeect.OurresultsalsoimplythattheKondo-destroyedphasecannotbecapturedinvariationalquantumMonteCarostudiesoftheKondolatticethattreatbasastaticvariationalparameter.ItwillbeinterestingtotryandconsidermoregeneralvariationalwavefunctionsthatcantreattheKondo-destroyedphasemoreaccurately.Kondo-destroyedquantumcriticalpointshavebeeninvokedtounderstandtheunconventionalquantumcriticalityobservedinexperimentsonheavy-fermionmetals[ 87 ].AsaresultofthefailureoftheHertz-Millis-Moriyatheory[ 88 { 90 ]ofthespin-density-wavetransitiontodescribetheexperimentaldata[ 91 ],theconceptoflocalquantumcriticality[ 92 ]hasbeenusedtounderstandtheenergy-over-temperaturescalinginthedynamicspinsusceptibility,thepresenceofanadditionalenergyscale,andajumpintheFermi-surfacevolume.Thetheoryoflocalquantumcriticalityisbasedontheextended-dynamicalmean-eldtheoryoftheKondolattice[ 92 ],whichndsthatforsucientlystrong 66

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quantumuctuationstheKondoeectisindeeddestroyedattheantiferromagneticQCP.TheresultsofthepresentstudyimplythatacontinuouslossofentanglementisexpectedattheKondo-breakdownQCPsbelievedtooccurincertainheavy-fermionsystems[ 93 ].TheremarkablesimplicityofthevariationoftheimpurityentanglemententropynearaKondo-destructionQCP,asembodiedinEq.( 3{8 ),canbeattributedtotheapplicationofEq.( 1{16 )withsubsystemAcontainingjusttheimpuritydegreesoffreedomandsubsystemBencompassingjustthehostdegreesoffreedom.IfoneconsideredadierentpartitionofthesysteminwhichhostdegreesoffreedomweresplitbetweensubsystemsAandB,thenSewouldprobeentanglementwithinthehost,whichshouldbemuchmoresensitivethanimpurity-hostentanglementtodetailssuchasthenumberofconductionbands.ThecriticalbehaviornearimpurityQCPsofentanglemententropydenedusingalternativesystempartitionsformsaninterestingquestionwhichisprobedinChapter 4 3.5ConclusionsWehavestudiedthequantummechanicalentanglementbetweenamagneticimpurityanditsenvironmentinseveralmodelsthatfeaturecriticaldestructionoftheKondoeect.IntheKondo-destroyedphaseofeachmodelstudied,wehaveidentiedatermintheentanglemententropyvaryingwithacriticalexponente=2,whereisthecriticalexponentgoverningtheorderparametercharacterizingthequantumphasetransition.Inaddition,wehaveestablishedthattheresponseofSetoalocalmagneticeldgivesrisetoapartoftheentanglemententropythatvarieswithacriticalexponent1=e=2=,whereisthecriticalexponentgoverningtheresponseoftheorderparameterattheQCPtoalocalmagneticeld.WehaveestablishedthatinnondegenerateAndersonmodels,theratioofthecriticalpartoftheentanglemententropytothesquareoftheorderparameterisenhancedoveritsvalueintheSimp=1 2Kondomodelbyafactoroftheinverseoftheimpurity'slocal-momentfraction.OurinvestigationhasshowntheabsenceofanyuniversalbehavioronapproachtoaKondo-destructionquantumcriticalpointfromtheKondo(disordered)phase.InAndersonmodelsthepresenceofchargeucuations 67

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introducestermsthatdependonthevaluesofhnd"+nd#iandhnd"nd#i.WehaveshownthatinthepseudogapAndersonmodelwithabandexponentrontherange0:55.r<1,chargeuctuationsproduceanonanalyticleadingvariationofSeneartheQCPwithacriticalexponentthatdependsonlyonr.Awayfromparticle-holesymmetry,SemayriseonapproachtotheQCPfromtheKondoside,producingacusppeakinSepreciselyatthequantumphasetransition.However,wealsondsituationsinwhichtheentanglemententropydecreasescontinuouslyalbeitnonanalytically,onpassingfromtheKondophasetothelocal-momentphase. 68

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CHAPTER4LONG-RANGEENTANGLEMENTNEARAKONDO-DESTRUCTIONQUANTUMCRITICALPOINTThischapterisbasedonapublishedpaper[ 94 ].AllthepublishedcontentsarereprintedwithpermissiongrantedunderthecopyrightpolicyoftheAmericanPhysicalSocietyfromChristopherWagner,TathagataChowdhury,J.H.Pixley,andKevinIngersent,Phys.Rev.Lett.121,147602(2018).Copyright2018bytheAmericanPhysicalSociety.Inthischapter,weshowthatthenumericalrenormalizationgroup(NRG)canbeusedtoaccuratelycalculatetheentanglementinthegroundstateofaspin-1 2magneticimpurityinametallicorsemimetallichost.InCh. 3 ,weinvestigatedthe\local"entanglementbetweensuchanimpurityanditshost,takingtheimpurityalonetoformsubsystemA[ 49 ].Here,weinsteadcomputeSimpe(R),theimpuritycontributiontotheentanglemententropybetweenaregionofradiusRabouttheimpurityandtherestofthesystem.Forametal,wheretheimpurityspinbecomesfullyscreenedattemperaturesTTK(theKondotemperature),weconrmapreviouslydeduced[ 28 34 35 ]scalingofSimpewithR=RK,whereRK/1=TKisbelievedtobethecharacteristicsizeofthemany-bodyKondocloud.OurmainresultsareforthepseudogapKondomodel[ 36 ],whichfeaturesaKondo-destructionQCPatanimpurity-bandexchangecouplingJ=JcseparatingapartiallyscreenedKondophase(J>Jc)fromalocal-momentphase(J
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Figure4-1. NRGrepresentationoftheKondomodelasatight-bindingWilsonchainofNsitescoupledatoneendtoanimpurityspin.(a)Inrealspace,Wilsonsitencorrespondstoasphericallysymmetricbandstatewitharadialprobabiltydensitypeakedataradius/k)]TJ /F6 7.97 Tf 6.59 0 Td[(1Fn=2fromtheimpurity.(b)TheentanglemententropySe(J;L;N)isfoundbysplittingthemappedsystemintosubsystemsA(theimpurityandtherstLWilsonsites)andB(theremainingN)]TJ /F3 11.955 Tf 11.96 0 Td[(Lsites). forR=R1.OnapproachtotheQCP,theentanglementlengthdivergeslikeRjJ)]TJ /F3 11.955 Tf 12.85 0 Td[(Jcj)]TJ /F4 7.97 Tf 6.59 0 Td[(,leadingtoamaximal,scale-invariantentanglementextendingfromtheimpuritythroughouttheentiresystem.Model.Weconsiderthespin-1 2KondoHamiltonianforasingleimpuritysimilartothatgiveninEq.( 1{1 ) H=Xk;kcykck+J 2NkSimpXk;k0;;0cyk0ck00(4{1)whereckdestroysabandelectronofenergykandspinzcomponent=1 2,Nkisthenumberofkvalues(i.e.,thenumberofhostunitcells),Jisthelocalexchange 70

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couplingbetweenbandelectronsandtheimpurityspinSimp,andisavectorofPaulimatrices.Weconsideradensityofstatesofthe(highlysimplied)formgiveninEq.( 1{11 ).Themodelhasarichphasediagram(asdiscussedinSec. 1.2 and 2.3 )thatcruciallydependsonthebandexponentr[ 9 ].Thecaser=0correspondstotheconventionalKondoprobleminametal[ 3 ].Forsemimetalswith00.AtthisinteractingQCP,thesystemexhibitsacriticalimpurityspinresponsecharacterizedbynontrivial,r-dependentexponents[ 38 ].Weconsidertheimpurity-inducedchangeintheentanglemententropy,denedasSimpe(J;R)Se(J;R))]TJ /F3 11.955 Tf 13.04 0 Td[(S(0)e(R).Here,Se(J;R)istheentanglemententropyofthecombinedimpurity-bandsystemwithsubsystemAconsistingoftheimpurityplusthatpartofthebandwithinradiusRoftheimpuritysite,andS(0)e(R)istheentanglemententropyofthebandalonewhenpartitionedatthesameradiusR[seeFig. 4-1 (a)].SincetheexchangecouplinginEq.( 4{1 )issphericallysymmetric,theimpurityaectsonlythes-wavebanddegreesoffreedom,andforpurposesofcalculatingimpurity-inducedproperties,theproblemreducestoone(radial)dimension.Afterthisreduction,onehas[ 33 95 96 ]S(0)e(R)logRratherthanthefullthree-dimensionalbehaviorS(0)e(R)R2logR.Computationalmethod.WestudytheradialKondomodelusingtheNRG[ 53 77 ]asmodiedtotreatapower-lawdensityofstates[ 9 ].TheHamiltonianismappedontoasemi-innitetight-binding\Wilsonchain"ofsiteslabeledn=0,1,2,:::,coupledtotheimpurityviasite0only.Adiscretizationparameter>1introducesaseparationofenergyscalesthatcausesthenearest-neighborhoppingcoecientstodecayexponentiallyastnD)]TJ /F4 7.97 Tf 6.59 0 Td[(n=2andallowsiterativediagonalizationofKondoHamiltoniansHMhavingniteWilsonchainsoflengthMwithM=1,2,:::,N.Toquantifyentanglement,thesystemdescribedbyHNisdividedintoasubsystemAcomprisingtheimpurityandtherstLchainsites(0nL)]TJ /F1 11.955 Tf 12.39 0 Td[(1)andasubsystem 71

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Bcontainingtheremainingchainsites(LnN)]TJ /F1 11.955 Tf 12.66 0 Td[(1)[seeFig. 4-1 (b)].WeobtaintheentanglemententropySe(J;L;N)=)]TJ /F1 11.955 Tf 9.3 0 Td[(TrA(AlnA)byusingtheNRGsolutionsofHMwithL)]TJ /F1 11.955 Tf 12.62 0 Td[(1MNtocomputethereduceddensityoperatorforsubsystemA:A=TrB()[ 65 97 98 ].Here,/exp()]TJ /F3 11.955 Tf 9.3 0 Td[(HN=kBT)isthedensityoperatoratathermalenergyscalekBTtN,chosentobemuchsmallerthananyotherenergyofphysicalinterestsothattheground-stateentanglementiscalculated.(ForJ
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solutionofnite-chainHamiltoniansHM=Himpf0;fy0+HchainM; (4{2)HchainM=M)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xn=1Xtn)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(fynfn)]TJ /F6 7.97 Tf 6.59 0 Td[(1;+H.c.; (4{3)withM=1,2,:::,N.Here,NischosentobesucientlylargethattN(thelargestenergyscaleofthepartofthesemi-innitechainthatisomittedfromHN)ismuchsmallerthanallenergyscalesofphysicalinterest.TheWilsonchainhoppingcoecientsconvergefor!1tothosefortheexactLanczosmappingofthecontinuum(\=1")Kondomodel.Forexample,inthecaseofametallicdensityofstates[Eq.( 1{11 )withr=0],tndecreasesmonotonicallyfromt10:57Dtowardt1=D=2.Thelog-logplotinFig. 4-2 revealsanexponentialdecayoftn=D)]TJ /F6 7.97 Tf 13.21 4.71 Td[(1 2withincreasingn.Thispatterndistinguishestheexacttight-bindingformulationoftheKondomodelfromastandardtight-binding(STB)chaincorrespondingtoEq.( 4{3 )withtn=D=2.Theeectofthisdierenceontheentanglemententropywillbediscussedbelow.Figure 4-2 alsoplotsjtn=D)]TJ /F6 7.97 Tf 13.05 4.71 Td[(1 2jvsnforthepseudogappedcaser=0:2.Heretnfornodd(even)approachesD=2fromabove(below). 4.1.2SystematicsoftheWilsonChainEntanglementEntropyTheentanglemententropyoftheWilsonchainexhibitseven-oddalternationwithincreasingsizeLofpartitionA.SuchanalternationispresentforaSTBchain,butitbecomesmorepronouncedwithincreasing>1and/orincreasingjrj.Tolteroutthisalternation,whichisanite-sizeeectoflittleinterestforourpurposes,weconsiderathree-pointaverage Savge(L)=1 4Se(L)]TJ /F1 11.955 Tf 11.95 0 Td[(1)+2Se(L)+Se(L+1):(4{4)ThissectionconsidersrstthecaseofametallicbandwithadensityofstatesdescribedbyEq.( 1{11 )withr=0.WeidentifyarangeofLvaluesoverwhichSavgediersnegligiblyfromtheuniversaldependenceexhibitedbyanSTBchain,anddescribe 73

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Figure4-2. Tight-bindinghoppingparametersfortheWilsonchain.Theseareplottedasjtn=D)]TJ /F6 7.97 Tf 13.15 4.7 Td[(1 2jvsnfortheexactLanczosmappingofaconductionbandwithadensityofstatesgivenbyEq.( 1{11 )withr=0(squares)andr=0:2(trianglesandcirclesforoddandevenn,respectively).Onlyforn!1doestnapproachitsuniformvaluetn=1 2forastandardtight-bindingchain. deviationsfoundforsmallandlargevaluesofL.WethenturntotheeectsofvaryingthebandexponentrenteringEq.( 1{11 ).Resultsforr=0:Fig. 4-3 (a)showstheaverageentanglemententropySavgevspartitionlengthLforarepresentativecase=1:04andforvariouschainlengthsNspeciedinthelegend.Savge(L)isalmost(butnotquite)symmetricwithrespecttoreectionaboutL=N=2andpeaksveryclosetoL=N=2.ThevalueSmaxe'Savge(N=2)initiallyincreaseswithincreasingchainlengthN,buteventuallysaturatesasawideplateauformsinSavge(L).Nosuchplateauisobservedinthedatafor=1andN=1200(plottedwithdashedlines).Figure 4-3 (b)plots(solidlines)SavgevsLforaxedchainlengthN=600anddierentvaluesofthediscretizationparameterontherange11:1.Alsoshown(dashedline)arethecorrespondingdataforaSTBchainwithtn=D=2.TheSTB 74

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Figure4-3. WilsonchainentanglemententropySavgevspartitionsizeLforametallicdensityofstatesdescribedbyEq.( 1{11 )withr=0.(a)Datafordiscretizationparameter=1:04withdierentchainlengthsN(solidlines),andfor=1,N=1200(dashedline).(b)DataforN=600withdierentvaluesof(solidlines).Alsoshown(dashedline)isSavgevspartitionsizeLfora600-siteSTBchain.(c)Datafrom(b)replottedvslogL.(d)DataforN=600,=1:04showingthedenitionofapartitionsizeLcharacterizingthecrossoverfromaregimeSavgelogLforLL(reddashedline)toaregimeSavge'SmaxeforLL(bluedashedline).Inset:L(calculatedforN=1200)vs)]TJ /F1 11.955 Tf 11.96 0 Td[(1iswellapproximatedbyL=2=()]TJ /F1 11.955 Tf 11.95 0 Td[(1)(redline). curveisexactlysymmetricaboutL=N=2,whilethoseforWilsonchainsareslightlyasymmetric.Curvesfor>1exhibitaplateausimilartothatseeninFig. 4-3 (a).Asisincreased,theplateauvalueSmaxedecreasesandisreachedatsmallervaluesofL.ForthefermionicSTBchainwithconstanthoppingcoecientsbetweennearestneighbors,theentanglemententropyinthelimitLNisequaltothatofacritical 75

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conformaleldtheory(CFT)[ 99 ].Foranitesystemwithopenboundaryconditions,Savge=c 6ln N sinL N!+b'c 6lnL+bforLN=2; (4{5)wherecisthecentralchargeoftheCFTandbtheboundaryentanglement.Foraspinlesschain,theleft-movingandright-movingfermionseachcarryachargeofc=1=2,sothechainoverallisdescribedbyc=1.Fig. 4-3 (c)replotsthedataforLN=2fromFig. 4-3 (b)asSavgevslogL.TheWilsonchainresults(solidlines)canbewellapproximatedby Savge=8>>><>>>:c 6lnL+bfor10.LL;c 6lnL+bSmaxeforLLN=2:(4{6)Here,candbareindependentofand,whenextrapolatedtotheinnite-sizelimit1=N!0,arenumericallyindistinguishablefromtheirrespectiveSTB-chainvalues:c=1andb'0:478.ForL.10,allWilson-chaindatacoincidebutclearlydierfromthosefortheSTBchain(dashedline),whiletheSTBand=1WilsonchainentanglemententropiesconvergeforL10.Thisisunsurprisinggiventheapproachwithincreasingnofthe=1WilsonchainhoppingcoecientstntotheSTBvaluetn=D=2(seeFig. 4-2 ).ThescaleListhefocusofFig. 4-3 (d).ThemainpanelshowshowLcanbedenedasthehorizontalcoordinateoftheinterceptbetweenthesmall-Landlarge-LasymptotesdenedinEq.( 4{6 ),i.e.,L=exp[(6=c)(Smaxe)]TJ /F3 11.955 Tf 12.27 0 Td[(b)].TheinsetofFig. 4-3 (d)plotsthevariationofLwith(datapoints),demonstratingthatfor.1:1,thescaleiswell-describedbytheempiricalrelationL=2=()]TJ /F1 11.955 Tf 12.99 0 Td[(1)(line).NRGmany-bodycalculationsaretypicallyperformedusingadiscretizationparameterontherange1:53chosentobalancediscretizationerrorsagainsttruncationerrors.Inallsuchcases,L'1,soSavge(L)'SmaxeisalmostindependentofL. 76

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Figure4-4. SavgeforWilsonchainsoflengthN=600andavarietyofbandexponentsr.(a)SavgevsLfordiscretizationparameters=1:0(solidlines)and=1:04(dashedlines).(b)DataforLN=2replottedonalogarithmicLscale. Figure4-5. FittedcoecientscandbinEq.( 4{6 )foraWilsonchainwithdiscretization=1,andbandexponentsr=0,0:2,and0:4.(a)Log-logplotof1)]TJ /F3 11.955 Tf 11.96 0 Td[(cvs1=N,whereNisthechainlength,showingapparentconvergencetoc=1for1=N!0.(b)bvs1=N. Resultsforr6=0:Figure 4-4 plotsSavgevsL(panela)andvslogL(panelb)forxedN=600,for=1(solidlines)and=1:04(dashedlines),andfordierentvaluesofthebandexponentrenteringEq.( 1{11 )describingmetallic(r=0),pseudogapped(r>0),anddivergent(r<0)densitiesofstates.For=1,themaineectofincreasingjrjisaprogressiveincreaseintheasymmetryofSavge(L)aboutL=N=2.Asrincreases(decreases)fromzero,thepeakinSavge(L)movesright(left)fromL'N=2.For 77

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Table4-1. ValuesofthecoecientscandbdenedinEq.( 4{6 )fortheSTBandforWilsonchainswithdierentbandexponentsr.Anumberinparenthesesdenotestheestimatednonsystematicerrorinthelastdigit.chaincb STB1.0000(2)0.4780(7)r=01.0000(1)0.47856(5)r=0:21.0000(1)0.43288(4)r=0:41.0000(1)0.39500(5) 10.LN=2,theentanglemententropyisstilldescribedbyEq.( 4{5 ),ascanbeseenfromFig. 4-4 (b).For=1:04(dashedlinesinFig. 4-4 ),theentanglemententropyforallrvaluesremainsconsistentwithEq.( 4{6 ),whereLisindependentofrandthevalueofSmaxetrackstherdependenceofb,i.e.,Smaxe(r))]TJ /F3 11.955 Tf 11.95 0 Td[(Smaxe(0)'b(r))]TJ /F3 11.955 Tf 11.96 0 Td[(b(0).Figure 4-5 plotsthevariationwithinversechainlength1=Nofthettedvaluesofcandbfor=1andr=0,0:2,and0:4.Table 4-1 liststheresultofextrapolatingcandbtothelong-chainlimit1=N!0,alongwiththecorrespondingvaluesfortheSTBchain.Towithinnumericalaccuracy,thesloperemainsc=1independentofr,asdemonstratedbyalog-logplotof1)]TJ /F3 11.955 Tf 12.89 0 Td[(cvs1=N[Fig. 4-5 (a)],whereastheboundaryentanglementbdecreases(increases)asrisincreased(decreased)fromzero[Figs. 4-4 (b)and 4-5 (b)].AdensityofstatesoftheformofEq.( 1{11 )describesfreefermionsinonespatialdimensionhavingadispersion/jk)]TJ /F3 11.955 Tf 13.05 0 Td[(kFj1=(1+r)sgn(k)]TJ /F3 11.955 Tf 13.05 0 Td[(kF).Itisthereforequitesurprisingthat,apartfromanonuniversalboundarytermb(r),theLdependenceofSavgefor10.LN=2isthesameforr=0(wherethehostsystemexhibitsconformalinvariance)andforr6=0(wherethespaceandtimeaxesaremanifestlyinequivalent).Atpresentwedonotfullyunderstandthephysicaloriginofthisresult.However,itsuggeststhatthepseudogaphostcouldinfactpossessa\hidden"conformalsymmetrywithacentralchargec=1(c=1 2eachforleft-andright-movers). 78

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4.2EntanglementEntropyfortheKondoProblem 4.2.1RadialDistributionofEntanglementEntropyTondSeasafunctionofphysicaldistanceRfromtheimpurity,wenotethatsitenoftheWilsonchainisassociatedwithasingle-electronwavefunction n(r0)thathasitsgreatestradialprobabilitydensityatradiusr0n'n=2=kF,wherekFistheFermiwavevectorandisadimensionlessconstantoforderunity[ 10 ][seeFig. 4-1 (a)].InthephysicallimitN!1and!1, n(r0)approachesaradialdeltafunction.Evenfor>1,weexpectthesmoothedentanglemententropySimpe(J;L)toreasonablyapproximateitscontinuumcounterpartSimpe(J;R=L=2=kF).Wepresentresultsobtainedusingdiscretizationparameter=3,retainingupto600many-bodyeigenstatesaftereachNRGiterationtoreachaWilsonchainofN=161sites.WeemploytheconventionalNRGvalue=21=2=(+1)andworkinunitswhereD=~=kB=gB=1.Resultsforametallichost.FirstweconsidertheconventionalKondomodeldescribedbybandexponentr=0.Figure 4-6 (a)plotstheimpurityentanglemententropySimpevsLforeightvaluesoftheKondocouplingJ.ForallbutthelargestJvalues,SimpestartsforsmallLatthevalueln2indicativeofasingletformedbetween(i)aspin1 2arisingfromanimpuritythatisnegligiblyscreenedbyelectronsoccupyingWilsonsitesn
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Figure4-6. ImpurityentanglemententropySimpevs(a)WilsonchainpartitionsizeLand(b)scaleddistancefromtheimpurityR=RKforametallichost(r=0).TheplotsshowavarietyofKondocouplingsJlabeledinthelegenof(b).Linesareguidestotheeye.In(b)thedatafrom(a)replottedasSimpevsR=RK,whereR=cL=2=kFandRK=1=(kFTK)withTKbeingtheKondotemperatureextractedfromthemagneticsusceptibility.ThecollapseofdatafordierentJvaluespointstoaone-parameterscalingSimpe(J;R)=f0(R=RK).Inset:Datafrommainpanelfor0J=0:05replottedonalog-logscaleshowingan(R=RK))]TJ /F6 7.97 Tf 6.58 0 Td[(1tail(ttedline)forRRK. JandpointingtotheexistenceofauniversalscalingSimpe(J;R)=f0(R=RK).ForR=RK1,Simpedecayslike(R=RK))]TJ /F6 7.97 Tf 6.59 0 Td[(1[seeinsettoFig. 4-6 (b)],consistentwithstudiesofspinchains[ 35 ]andaresonant-levelmodel[ 100 ].Resultsforpseudogappedhosts.OurmaininterestisintheentanglementneartheKondo-destructionQCPsthatoccurforsemimetallicdensitiesofstatesdescribedbyexponents0
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Figure4-7. ImpurityentanglemententropySimpevsWilsonchainpartitionsizeLforapseudogapKondomodelwithbandexponentr=0:4.Symbolsplotdatafor(a)J=(1)]TJ /F1 11.955 Tf 11.96 0 Td[(10x)Jc,and(b)J=(1+10x)Jc,withvaluesofxshowninthelegend.ThicklinesshowthecriticalcaseJ=Jc. WilsonchainpartitionsizeLforvaluesofJclosetoJc.Inthelocal-momentphase[Fig. 4-7 (a)],SimpeinitiallyriseswithincreasingLtoreachaplateaumaximum,onlytofalltowardzeroforlargerpartitionsizes.Thesedatashowthateventhoughtheimpurityspinasymptoticallydecouplesfromtheband,theimpurityinducesadditionalentanglementfornitevaluesofL|orequivalently,atniteenergies'D)]TJ /F4 7.97 Tf 6.59 0 Td[(L=2|manifestingadynamicalKondoeect.IntheKondophase,too,SimpeinitiallyriseswithincreasingLtoreachthesameplateaumaximumasforJ
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Figure4-8. DatafromFig. 4-7 replottedvsR=R,whereR/L=2=kFandR=1=(kFT)withTbeingacrossovertemperatureextractedfromthemagneticsusceptibility.Symbolsplotdatafor(a)J=(1)]TJ /F1 11.955 Tf 11.96 0 Td[(10x)Jc,and(b)J=(1+10x)Jcwithvaluesofxlabeledinthelegend.Linesshowtstodatapoints(notshown)obtainedforothervaluesofr.Insets:Log-logplotsoflarge-RdataforSimpe(J;R))]TJ /F3 11.955 Tf 11.95 0 Td[(Simpe(J;1)vsR=R,calculatedforasingleKondocoupling(a)JJcateachoffourdierentbandexponentsr>0,withpower-lawts(dashedlines). freedomisonlypartiallyscreenedinthepseudogapKondophase[ 9 ].Figure 4-7 alsoshowsthatineitherphase,SimperemainsnearitsinitialplateautolargervaluesofLthecloserJapproachesJc.Wearethusledtooneofourprincipalconclusions:AttheQCP[thicklinesinFigs. 4-7 (a)and 4-7 (b)],theentireconductionbandismaximallyentangledwiththeimpurity,i.e.,thegroundstatehaslong-range,scale-invariantentanglement. 82

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TheprecedingpictureimpliesthattheeventualdecreaseinSimpevsLseenforJ6=JcreectstheRGowawayfromthepseudogapKondoQCP,aowcharacterizedbyacrossovertemperaturesscaleTjJ)]TJ /F3 11.955 Tf 12.1 0 Td[(Jcj,where(r)isthecorrelation-lengthexponent[ 38 ].Followingthesamereasoningaswasappliedforametallic(r=0)host,weexpectTtobeassociatedwithalengthscaleR=1=(kFT).Figure 4-8 replotsther=0:4datafromFig. 4-7 asSimpevsR=RusingvaluesofTextractedfromthemagneticsusceptibility[ 9 ].ThescalingcollapseofdatafordierentJisofasimilarqualitytothatforr=0[seeFig. 4-6 (b)].Thisprovidesstrongevidencefortheexistenceofscalingfunctionsfrsuchthat Simpe(J;R)=fr(R=R)forJ?Jc:(4{7)SignicantdeparturesfromscalingareseenonlyforthesmallestvaluesofR(correspondingtothesmallestLinFig. 4-7 ),andcanbeattributedtotheNRGdiscretization.Figure 4-8 alsoplotsttingcurvesfromsimilardatacollapsesforbandexponentsr=0:2,r=0:3andr=0:451,aswellas[inpanel(b)]themetalliccaser=0. 4.2.2Fixed-PointEntanglementEntropyvsrThissectionprovidesdetailsofthemannerinwhichSimpeapproachesitsvalueateachofthestablexedpoints,aswellastherdependenceoftheimpurityentanglemententropyateachrenormalization-groupxedpoint. 1Forr=0:2and0:3,thecrossoverinSimpebetweenitscriticalandstablexed-pointvaluesiscenteredatsmallerR=RforJJc.ThisisanartifactofanimpuritymagnetizationhSzimpijhj1=inducednearJ=Jcbythesmallmagneticeldhappliedtoliftthelocal-momentground-statedegeneracy.Inourquadruple-precisionruns,thesmallesteldthatwecanuseish=O(10)]TJ /F6 7.97 Tf 6.59 0 Td[(34).Forr=0:2,e.g.,1==0:026givesanon-negligiblehSzimpi'0:13.ForJ!J)]TJ /F4 7.97 Tf -1.11 -7.29 Td[(c,itisthusimpracticaltosimulatespontaneoussymmetrybreaking. 83

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Table4-2. ValuesoftheexponentdenedinEq.( 4{8 )fordierentbandexponentsr,asdeterminedinthelocal-moment(LM)andKondo(K)phases.Anumberinparenthesesdenotestheestimatednonsystematicerrorinthelastdigit.Valueswithouterrorestimatesareassumedratherthancomputed.r(LM)(K) 0.001.000(3)0.20.38(3)0.800(5)0.250.750(4)0.30.58(5)0.693(8)0.330.630(7)0.40.79(4)0.399(5)0.450.89(5)0.199(2)0.510 InsetsinFig. 4-8 demonstratethatSimpehasapower-law-decayingtailinboththelocal-momentandKondophases,namely, Simpe(J;R))]TJ /F3 11.955 Tf 11.95 0 Td[(Simpe(J;1)/(R=R))]TJ /F4 7.97 Tf 6.58 0 Td[(forRR:(4{8)FittedvaluesofarelistedinTable 4-2 andplottedinFig. 4-9 .Towithintheestimatednumericaluncertainty,theextractedexponentsareconsistentwith=2rforJJc.Theseexpressionscoincidewithtwicetheexponentoftheleadingirrelevantoperatoratthelocal-momentandKondoxedpoints,respectively;seeEqs.(4.7)and(4.10)inRef.[ 9 ].Thisisconsistentwiththenaturalinterpretationthatthepower-lawtailsareassociatedwiththerenormalization-groupowtowardthestablexedpointineitherphase,leadingtotheexpectationthattheexponentisacharacteristicofthatxedpoint.ItisprobablethatthedepartureofSimpefromitsvalueonthecriticalplateauisalsodescribedbyapower-lawbehavior,i.e., Simpe(J;R))]TJ /F3 11.955 Tf 11.95 0 Td[(Simpe(Jc;1)/(R=R)0forRR;(4{9)whereonewouldexpect0tobepositiveandacharacteristicpropertyoftheKondodestructioncriticalpoint(and,hence,likelytohaveanontrivialrdependence).However, 84

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numericaluncertaintyinthevalueofthecriticalvalueSimpe(Jc;1)impedesreliabledeterminationof0.TheresultsinSubsec. 4.2.1 showthat,whereasSimpe=0attheweak-couplingxedpoint,theimpurityentanglemententropytakesnontrivial,r-dependentvaluesattheKondo-destructionquantumcriticalpointandattheKondoxedpoint.Thexed-pointvaluesofSimpecanbeobtainedfrommany-bodyNRGcalculationsbytakingthelimitRR(fortheunstablecriticalpoint)orRR(forthestableKondoandlocal-momentxedpoints).TheKondoxed-pointvalueofSimpecanalsobecalculatedusingthesingle-particlemethodoutlinedintheAppendixasthedierenceofSavgeforafreeWilsonchainwithandwithouttherstsitefrozenduetotheformationofalocalspinsingletwiththemagneticimpurity.Thatthemany-bodyandsingle-particleapproachesyieldnumericalvaluesinexcellentagreementprovidesavaluablecheckontheaccuracyofthefullNRGresults.Inordertoremovediscretizationeects,xed-pointvaluesofSimpewerecalculatedforvaluesofbetween1:01and3,thenttedwithapolynomialfunctionofln,allowingextrapolationofSimpetothecontinuumlimit=1,asillustratedinFig. 4-10 (a).ExtrapolatedvaluesofSimpeareshowninFig. 4-10 (b).Asrincreasesfrom0,thecriticalvalueofSimpedecreasesfromln2whiletheKondoxed-pointvalueincreasesalmostlinearlyfrom0.Thetwoxed-pointvaluesmeetatr=1 2,thebandexponentatwhichthequantumcriticalpointmergeswiththeKondoxedpoint.(Noquantumcriticalpointexistsforr>1 2[ 9 ].)AweaksuperlinearvariationcanbeseenwhentheKondoentanglemententropyiscomparedwithaheuristictSimpe=3 2rln2[dashedlineinFig. 4-10 (b)].Thissuperlinearbehaviorissomewhatunexpectedsincethermodynamicpropertiesatstrongcouplinghavebeenshowntoexhibitastrictlylinearvariationwithr[ 9 ]. 85

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Figure4-9. ValuesoftheexponentdenedinEq.( 4{8 )fordierentbandexponentsr,asdeterminedinthelocal-moment(LM,circles)andKondo(K,squares)phases,alongwithlinesshowingthefunctions=2r,1)]TJ /F3 11.955 Tf 11.95 0 Td[(r,and2(1)]TJ /F1 11.955 Tf 11.95 0 Td[(2r). Figure4-10. ImpurityentanglementatthequantumcriticalpointandtheKondoxedpoint:(a)Kondoxed-pointvalueofSimpe(symbols)vs(onalogscale)forbandexponentr=0:1.Apolynomialt(solidline)isusedtoextrapolateSimpetothecontinuumlimit=1.(b)Extrapolated=1valuesofSimpeatthecriticalpoint(squares)andattheKondoxedpoint(circles)vsbandexponentr,alongwithaheuristictSimpe=3 2rln2(dashedline). 86

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4.2.3EntanglementEntropyasaFunctionofKondoCouplingJSubsec. 4.2.1 presentsresultsforSimpe(J;L),thesmoothed(three-point-averaged)impuritycontributiontotheentanglemententropyasafunctionoftheWilsonchainpartitionsizeLfordierentxedKondocouplings0J.Figure 4-11 insteadplotsSimpevs0Jforthemetalliccaser=0witheachdatasetrepresentingadierentxedpartitionsizeL.WithincreasingJ,eachpartitionshowsamonotonicdecreaseofSimpe.ForveryweakKondocouplings0J1,theimpurityspiniscollectivelyscreenedbyessentiallytheentireWilsonchain.TheamountofscreeningthattakesplacewithintherstLsitesoftheWilsonbecomeseversmallerasJ!0+,sotheimpurity'sentanglementwithchainsitesnLapproachesthefullvalueln2foraspinsinglet.Fortheoppositelimit0J1,inthegroundstateofHNgivenbyEqs.( 4{2 )and( 4{3 ),theimpurityisessentiallylockedintoaspinsingletwiththeon-sitecombinationofconductionelectronsannihilatedbythef0operator;chainsites1,2,:::N)]TJ /F1 11.955 Tf 13.21 0 Td[(1behavelikeafreeWilsonchainpartitionedintosegmentsoflengthL)]TJ /F1 11.955 Tf 12.73 0 Td[(1andN)]TJ /F3 11.955 Tf 12.72 0 Td[(L.Asaresult,theimpuritycontributiontotheentanglemententropycanbewrittenSimpe(J;L;N)=Se(J;L;N))]TJ /F3 11.955 Tf 12.46 0 Td[(S(0)e(L;N)'S(0)e(L)]TJ /F1 11.955 Tf 12.45 0 Td[(1;N)]TJ /F1 11.955 Tf 12.46 0 Td[(1))]TJ /F3 11.955 Tf 12.46 0 Td[(S(0)e(L;N),whereS(0)eistheentanglemententropyofachainoflengthNpartitionedintoLandN)]TJ /F3 11.955 Tf 12.55 0 Td[(Lsites.AftermakingNverylargeandperformingathree-pointaverage,thesmoothedimpurityentanglemententropySimpe(J;L)denedatthebeginningofthechapterisnegativeforL.L|overwhichrangeSavge(L;NL=2)growswithincreasingL|andrapidlyapproacheszeroforL&L.Forthevalue=3usedtoproduceFig. 4-11 ,L'1andnegativeSimpevaluesarefoundonlyforL.3.AsLisincreased,thecrossoverinSimpefromln2towardzerotakesplacemoresharplyandcenteredaroundasmallervalueof0J.ThisisanothermanifestationofthenotionpresentedinSubsec. 4.2.1 thatSimpedropsoncetheradiusRofsubsystemAexceedsthecharacteristicsizeRKoftheKondoscreeningcloud.TheinsetofFig. 4-11 replotsthedatafor0J0:3asafunctionofR=RK,whereeachLcurvecorresponds 87

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Figure4-11. ImpurityentanglemententropySimpevsdimensionlessKondocoupling0Jforbandexponentr=0,discretizationparameter=3,anddierentpartitionsizesL.TheinsetshowsthecollapseofcurvesfordierentLvalueswhenthedataspanning0J0:3arereplottedasSimpevsR=RK. toxedvalueofR=cL=2=kF(withkFbeingtheFermiwavevectorsandcaconstantoforderunity)andpointswithinacurvearisefromadecreasewithincreasingJofRK1=(kFTK).WhereasinSubsec. 4.2.1 ,theKondotemperatureTKwasdeducedfromtheimpuritycontributiontothemagneticsusceptibilityviatheconventionaldenitionTimp(TK)=0:0701[ 10 ],inFig. 4-11 weinsteademployedtheperturbativedenition[ 3 ]kBTKDp 0Jexp[)]TJ /F1 11.955 Tf 9.3 0 Td[(1=(0J)+O(0J)]: (4{10)ThecollapseofallcurvesexceptthoseforL=1and2(whichareanomalousforreasonsdiscussedintheprecedingparagraph)isconsistentwiththeexistenceofauniversalscalingfunctionSimpe(J;R)=f0(R=RK),asalsoarguedonthebasisofthedatapresentedinthemainpaper.Similarbehaviorcanbeseeninplots(notshown)ofSimpevs0JatxedLforpseudogappedhosts(i.e.,r>0).Thedataineachphase(Kondoorlocal-moment)canbecollapsedbyplottingSimpeagainstR=R,wherethecrossoverlengthscaleR=1=kFT. 88

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WhileTcanbedeterminedfromimp(T)viatheoperationalprocedurelaidoutinSec. 2.5 ,forvaluesofJsucientlyclosetoJc,goodcollapsecanbeachievedbyinsteadusingtheasymptoticexpressionT/jJ)]TJ /F3 11.955 Tf 11.96 0 Td[(Jcj; (4{11)wherethenumericalvalueofthecorrelationlengthexponenthasanontrivialdependenceonthebandexponentr[ 38 ]. 4.3DiscussionWehavedeterminedthespatialstructureofentanglemententropyintwotypesofquantumimpuritymodels.OurworkdemonstratesthattheimpurityentanglemententropyforasystempartitionedatradiusRaroundaKondoimpuritydependsonlyonRdividedbyR/1=T,whereTisamany-bodyscale.Intheconventionalcaseofametallichost,TistheKondotemperature,whereasTvanisheslikejJ)]TJ /F3 11.955 Tf 12.64 0 Td[(JcjonapproachtotheKondo-destructioncriticalpointinapseudogappedhost.Theimpurityentanglemententropyisbothscaleinvariantandlongrangedatthisinteractingcriticalpoint,whileawayfromcriticalityitfallsolike(R=R))]TJ /F4 7.97 Tf 6.59 0 Td[(forRR.WededucethatthetotalentanglemententropygoeslikeSe(J;R)'blogR+fr(R=R).Ourconclusionshavebeenreachedforamodel[Eqs.( 4{1 )and( 1{11 )]exhibitingstrictparticle-holesymmetry,butweexpectsimilarconclusionstoapplyattheasymmetricinteractingQCPsthatarisefor0:375.r<1upontheadditionofapotential-scatteringtermtoEq.( 4{1 )[ 9 ].Althoughobtainedforasingleimpurity,ourresultsshedlightontheKondolatticemodelandhaveimplicationsforinterpretationofexperimentsonquantum-criticalheavy-fermioncompounds.First,thisworkprovidesinsightintothestructureoftheground-statewavefunctioninaKondo-destroyedphase.Second,ourndingssuggestthattheKondo-breakdownQCPrelevanttoheavy-fermionmetalsisaccompaniedby 89

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long-rangeentanglementbetweenalllocalmomentsandtheentireconductionband.Webelievethatthisscale-invariantentanglementisintimatelyassociatedwiththereconstructionofthecriticalFermisurface. 90

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CHAPTER5TIME-DEPENDENCEINQUANTUMIMPURITYSYSTEMSInSec. 2.9 ,thesetupofthetime-dependentdensitymatrixNRGwasdiscussed,butitwentunutilizedinChapters 3 and 4 .AsegmentofthisdissertationprojecthasbeenspentonwritingthenecessarycodeforthedensitymatrixNRGtoperformthetime-dependentcalculationofexpectationvaluesfollowingaquantumquench.Althoughthecodehasbeennishedandtested,onlypreliminaryresultshavebeenobtainedtodate,andthoseresultsarenotyetfullyunderstood.Inthischapterthefocuswillinsteadbeonthesetupforthecalculationofthetime-dependenceoftheentanglemententropy.UtilizingthecompletebasissettocalculatetheentanglemententropywiththereduceddensitymatrixasdescribedinSec. 2.8 allowscalculationofthereal-timedynamicsoftheentanglemententropybyborrowingsomeoftheconceptsillustratedinSec. 2.9 .Thiscanbeaccomplishedbyreplacingthereduceddensityoperator^Awithatime-dependent^A(t)intheentanglemententropyofEq.( 1{16 ).Thetime-dependentreduceddensityoperatorcanbewrittenout ^A(t)=TrB(^(t))=TrB)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(iHft^eiHft;(5{1)where^isthedensityoperatorfromEq.( 2{35 )inthebasisoftheinitialHamiltonian.Toevaluatetheexponentials,twomultiplicationsbyunityareperformedwiththesumoverthebasisstatesofthenalHamiltonian(Plemjlemiffhlemj=1),resultingin ^A(t)=TrB XlemXl0e0m0jlemiffhlemje)]TJ /F4 7.97 Tf 6.59 0 Td[(iHft^eiHftjl0e0m0iffhl0e0m0j!:(5{2)

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Thiscanalsobeexpressedinmatrixform: A(t)=TrB0BBBBBBBBBB@m0;m0l;l0(t)m0;m0+1l;l0(t):::m0;N)]TJ /F6 7.97 Tf 6.59 0 Td[(1l;l0(t)m0;Nl;l0(t)m0+1;m0l;l0(t)m0+1;m0+1l;l0(t):::m0+1;N)]TJ /F6 7.97 Tf 6.59 0 Td[(1l;l0(t)m0+1;Nl;l0(t)...............N)]TJ /F6 7.97 Tf 6.58 0 Td[(1;m0l;l0(t)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1;m0+1l;l0(t):::N)]TJ /F6 7.97 Tf 6.59 0 Td[(1;N)]TJ /F6 7.97 Tf 6.58 0 Td[(1l;l0(t)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1;Nl;l0(t)N;m0l;l0(t)N;m0+1l;l0(t):::N;N)]TJ /F6 7.97 Tf 6.59 0 Td[(1l;l0(t)N;Nl;l0(t)1CCCCCCCCCCA;(5{3)where m;m0l;l0(t)=Xe;e0fhlemje)]TJ /F4 7.97 Tf 6.58 0 Td[(iHft^eiHftjl0e0m0if:(5{4)Totakethepartialtraceofthematrix,thesystemneedstobepartitionedintosubsystemsAandB.WefollowthesameconventionasinSec. 2.8 ,partitioningtheimpuritywithLsitesoftheWilsonchain.ThenthesumoveriterationsinEq.( 5{2 )canbesplitupasfollows: Xm;m0=Xm;m0
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orinmatrixform: A(t)=0BBBBBBBBBB@~m0;m0l;l0(t)~m0;m0+1l;l0(t):::~m0;L)]TJ /F6 7.97 Tf 6.59 0 Td[(1l;l0(t)~m0;L)]TJ /F6 7.97 Tf 6.58 0 Td[(1l;l0(t)~m0+1;m0l;l0(t)~m0+1;m0+1l;l0(t):::~m0+1;L)]TJ /F6 7.97 Tf 6.59 0 Td[(1l;l0(t)~m0+1;L)]TJ /F6 7.97 Tf 6.58 0 Td[(1l;l0(t)...............~L)]TJ /F6 7.97 Tf 6.59 0 Td[(1;m0l;l0(t)~L)]TJ /F6 7.97 Tf 6.59 0 Td[(1;m0+1l;l0(t):::~L)]TJ /F6 7.97 Tf 6.59 0 Td[(1;L)]TJ /F6 7.97 Tf 6.59 0 Td[(1l;l0(t)~L)]TJ /F6 7.97 Tf 6.59 0 Td[(1;L)]TJ /F6 7.97 Tf 6.58 0 Td[(1l;l0(t)~L)]TJ /F6 7.97 Tf 6.59 0 Td[(1;m0l;l0(t)~L)]TJ /F6 7.97 Tf 6.59 0 Td[(1;m0+1l;l0(t):::~L)]TJ /F6 7.97 Tf 6.59 0 Td[(1;L)]TJ /F6 7.97 Tf 6.59 0 Td[(1l;l0(t)~L)]TJ /F6 7.97 Tf 6.59 0 Td[(1;L)]TJ /F6 7.97 Tf 6.58 0 Td[(1l;l0(t)1CCCCCCCCCCA(5{7)where ~m;m0l;l0=dN)]TJ /F4 7.97 Tf 6.59 0 Td[(LXem;e0m0fhlemmj(t)jl0e0m0m0if~m;L)]TJ /F6 7.97 Tf 6.59 0 Td[(1l;k=~L)]TJ /F6 7.97 Tf 6.59 0 Td[(1;mk;ly=dN)]TJ /F4 7.97 Tf 6.59 0 Td[(LXemfhlemmj(t)jk(L)]TJ /F1 11.955 Tf 11.96 0 Td[(1)if~L)]TJ /F6 7.97 Tf 6.58 0 Td[(1;L)]TJ /F6 7.97 Tf 6.59 0 Td[(1k;k0=dN)]TJ /F4 7.97 Tf 6.59 0 Td[(Lfhk(L)]TJ /F1 11.955 Tf 11.95 0 Td[(1)j(t)jk0(L)]TJ /F1 11.955 Tf 11.95 0 Td[(1)if:(5{8)IncontrasttoEq.( 2{42 ),theaboveformulationisnotdiagonalwithiterationnumberm.Toaccuratelycalculatethetime-dependentproperties,then,itwouldbenecessarytodiagonalizethefullmatrixratherthanperformdiagonalizationsforthemblocksseparately.Apotentialsolutiontothisproblemistomakeapproximationsmotivatedbytheapplicationonehasinmind{here,thecalculationoftheentanglemententropy,whichisazero-temperatureproperty.WhenpartitioningtherstLWilsonsiteswiththeimpurity,thegroundstateinformationwillbealmostentirelycontainedinthereduceddensityoperatorforthelastsiteL)]TJ /F1 11.955 Tf 9.78 0 Td[(1,andthereforeitshouldbesucienttoonlyconsiderthelastsiteL)]TJ /F1 11.955 Tf 12.43 0 Td[(1whencalculatingthereduceddensitymatrixasallsites
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toseeifitisareasonableapproximation,whichthenwouldopenupthetime-dependententanglemententropyasanewresearchpath. 94

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CHAPTER6CONCLUSIONANDFUTUREWORK 6.1ConclusionThisdissertationusedthenumericalrenormalizationgrouptosolveproblemsinvolvingmagneticimpuritiesinmetals(withaatdensityofstates()=0)andsemimetals(withpseudogappeddensityofstates()=0jjr(+D)(D)]TJ /F3 11.955 Tf 12.48 0 Td[()whichdisappearsattheFermienergy=0).Twoimpuritymodels,theKondoandAndersonmodels,wereusedtostudythesesystems.TheworkdiscussedinChapters 3 and 4 focusedontheentanglemententropyinthesesystemsbyutilizingthedensitymatrixNRGinordertocalculatethevonNeumannentropyfortheentanglementSe=)]TJ /F1 11.955 Tf 9.3 0 Td[(TrA(AlnA).Chapter 3 focusesonthedegeneracy-brokenentanglemententropybetweenasinglemagneticimpurityandthebathofelectronsintheasymmetricpseudogappedAndersonmodelfor(1)maximalparticle-hole-asymmetrycreatedbyaninnitelylargeCoulombrepulsionUbetweenelectronslocatedattheimpuritysiteand(2)ageneralparticle-hole-asymmetry(U+2d6=0),wherebothcaseswerecalculatedwithabandexponentr=0:6.Theentanglemententropynearthequantumcriticalpointwasfoundtovarywiththedistancefromthecriticalpoint()]TJ /F2 11.955 Tf 15.62 0 Td[()]TJ /F1 11.955 Tf 13.05 0 Td[()]TJ /F4 7.97 Tf 7.32 -1.8 Td[(c)=)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(casSe(;0))]TJ /F3 11.955 Tf 13.12 0 Td[(Se(0;0)'Ajj1)]TJ /F6 7.97 Tf 6.89 0 Td[(~sgn,where~isthecharge-susceptibilitycriticalexponent.Incontrastwithresultsforthespin-bosonmodel[ 29 30 86 ],theasymmetricpseudogappedAndersonmodeldisplayedacusppeakintheentanglemententropyatthequantumcriticalpointonlyundercertaincircumstances:(1)dSe(hloc=0)=d<0at=0and(2)e=2<1)]TJ /F1 11.955 Tf 12.75 0 Td[(~.Inthecasethatoneoftheseconditionsisnotmet,theentanglemententropydisplaysapeakinsideoftheKondophase)]TJ /F3 11.955 Tf 344.51 0 Td[(>)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(c.Chapter 4 discussestheentanglemententropyinthepseudogappedKondomodelbetweenasinglemagneticimpurityplustheelectronswithinaradiusRfromtheimpurityandtheelectronsoutsideofR.ThisrequiresconsiderationoftheentanglemententropybetweentherstLsitesofaWilsonchainandtherestofthechain.Theentanglement

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entropyintheWilsonchain,whichvarieswithasetofparametersSe(L;N;;r),isequivalenttoacriticalconformaleldtheory[ 99 ](Se/lnL)forsmallpartitionsizesLLandisconstantforlargepartitionsizesLL.TheimpuritycontributionSimpetotheentanglemententropyinthepseudogappedKondomodeltransitionsfromamaximalvaluetoaminimalvalueatadistanceRK/1=(kFTK)inthemetallic(r=0)hostsandatadistanceR/1=(kFT)inthesemimetallic(r>0)hosts,wherethecriticaltemperaturevariesproportionaltothedistancefromthecriticalcouplingT/jJ)]TJ /F3 11.955 Tf 12.46 0 Td[(Jcj.InthecaseofthepseudogappedKondomodelwithsemimetallichosts,theentireconductionbandismaximallyentangledwiththeimpurityatthequantumcriticalpoint,i.e.whenJ=Jc,impliedbythedivergenceofthetransitiondistanceR/jJ)]TJ /F3 11.955 Tf 13.17 0 Td[(Jcj)]TJ /F4 7.97 Tf 6.59 -.01 Td[(.ThexedpointvaluesofSimpeforavarietyofbandexponentsrwerefoundfortheKondoxedpointandthecriticalxedpoint;atthelocal-momentxedpoint,Simpetakesatrivialvalueofln2duetothedecouplingoftheimpurityfromWilsonchain.TheKondoxedpointvaluewasfoundtovaryroughlylinearlywithr,Se'3 2rln2,similartothethermodynamicentropy[ 9 ].Chapter 5 discussesthesetupforthetime-dependentcalculationoftheentanglemententropy.Someissueswiththeformulationwerementionedaswellasapossiblesolutiontotheproblem.Thetime-dependentcodeforthissectioniscurrentlyinthenalstagesofbeingwrittenandinthemiddleoftesting. 6.2FutureWorkAfewavenuestopursueforfutureworkareasfollows:(1)Finishthetestingofthetime-dependententanglemententropyfromCh. 5 andperformtime-dependentcalculationsoftheentanglemententropy.(2)ThecalculationofthevariationofentanglemententropywithdistanceinChapter 4 isanaturalextentionoftheworkonthepseudogapKondomodelsinRef.[ 49 ].ItcouldproveinterestingtoperformthesamecalculationsforthemuchricherpseudogapAndersonmodelasafurtherextentionoftheworkpresentedinChapters 3 and 4 96

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(3)Somepreliminaryworkhasbeenstartedonthetime-dependenceofthelocalmagnetization.InordertotestthecodeforthemethodpresentedinSec. 2.9 ,plotsofthelocalmagnetizationweremadeincomparisontoapaperbyKleineetal.[ 101 ].WehopetoextendtheworkdonetherebylookingmorecloselyattheeectofquenchingacrosstheKondo-destructionQCP.Togoalongwiththiswork,somepreliminaryresultshavebeenobtainedlookingatbreakingthedegeneracyoftheimpurityinthelocalmomentphasebyquenchingthesystemwithaspontaneouslyappliedmagneticeld.Theseresultsforthisarenotyetunderstood,andfurtherresearchmustbedone.(4)Adaptthecodetocalculatethetime-dependenceoftheoverlapofthegroundstatewavefunctionh ij (t)i.(5)Studythetime-dependenteectsoftheimpurityinspecicsystems,suchasquantumdot-superconductorhybrids. 97

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APPENDIX SINGLEPARTICLECORRELATIONFUNCTIONAPPROACHInordertocomprehendtheresultsfortheentanglemententropyinthepseudogappedKondomodelinCh. 4 ,itisimportanttounderstandtheentanglementpropertiesoftheWilsonchain(WC)byitself.InthisAppendixwediscussthemethodofcalculatingtheentanglemententropyintheWC(intheabsenceofaquantumimpurity).WeconsideraWCconsistingofNsites,whereapartitionseparatestherstLsitesofthechain(subsystemAcomprisingofsitesf0,f1,..fL)]TJ /F6 7.97 Tf 6.58 0 Td[(1)fromtherestofthechain(subsystemBcomprisingofsitesfL,fL+1,..,fN)]TJ /F6 7.97 Tf 6.58 0 Td[(1).Weuseapower-lawdensityofstates(")=0j"=Djr(D)-249(j"j),wherethebandexponentrdetermineswhetherthedensityofstatesismetallic(r=0),pseudogapped(r>0),orsingular(r<0)attheFermilevel.Thetight-bindingcoecients(tn)oftheWilsonchainforagivenvalueofrcanbefoundapplyingtheLanczosprocedureandaself-consistentevaluationoftheelectronicwavefunctionoftheWilsonsitesasdescribedin[ 9 ].ThecalculationoftheentanglementintheWCcanbecomputedusingthefull-basisreduceddensitymethodpresentedinSec. 2.8 .ByspecifyingtheKondocouplingJ=0,calculationoftheentanglemententropyusingthefullbasismethodisequaltotheentanglementoftheWilsonchainplusaln2contributionduetothedegeneracyoftheimpuritydegreesoffreedom.However,itisalsopossibletoperformthiscalculationusingtheeigenvaluesofthesingle-particlecorrelationfunctionrestrictedtosubsystemAasdescribedinRef.[ 102 ].ThisformalismstartswiththemostgeneraltightbindingHamiltonianwitharbitraryhoppingtn;mbetweenallsites(n;m)H=Xn;mtn;mcyncm: (A{1)NowletussplitthesitesintoasubsystemAonlycontainingsiteswithindex(i;j).ThesingleparticlecorrelationfunctionforsubsystemAcanbewrittenintermsofthereduced 98

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densitymatrixAasCi;j=hcyicji=TrA(Acyicj): (A{2)TheeigenvaluesofAarethenrelatedtotheeigenvaluesofthesingleparticlecorrelationfunction.Wick'stheoremmustholdsincethesearenon-interactingfermions,thusifthesubsystemHamiltonianhaseigenenergiesk,thenthereduceddensitymatrixcanbewrittenasA=exp()]TJ /F8 11.955 Tf 11.3 8.97 Td[(Pkkaykak) Z;Z=Yk)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(1+e)]TJ /F4 7.97 Tf 6.58 0 Td[(k: (A{3)ThepartitionfunctionZisdeterminedbytheconditionTrA(A)=1.TakingthisformofAandpluggingitbackintothesingleparticlecorrelationfunctionCi;jshowsthattherelationbetweenthesingleparticlecorrelationfunctioneigenvalueskandthesubsystemeigenvalueskis:k=1 1+ek !k=log(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k))]TJ /F1 11.955 Tf 11.96 0 Td[(logk: (A{4)Fromheretheentanglemententropycanbecalculatedusingtheeigenvaluesofthesingleparticlecorrelationfunction:Se=)]TJ /F1 11.955 Tf 9.3 0 Td[(TrA(AlogA)=)]TJ /F8 11.955 Tf 11.29 11.35 Td[(Xk[klogk+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k)log(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k)]: (A{5) 99

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BIOGRAPHICALSKETCHChristopherWagnergrewupinCrete,Illinois,asuburbonthesouthsideofChicago.HeattendedValparaisoUniversityforhisundergraduateworkwhereheearnedaBachelorofScienceinmathematicsandphysics.HeearnedaMasterofSciencedegreeinphysicsatBallStateUniversityimmediatelypriortomarryinghiswifeLydiaandstartingattheUniversityofFloridain2011.HebeganworkingwithKevinIngersentinthesummerof2012,wherehisresearchhasfocusedonthequanticationofentanglementinquantumimpuritysystems,aswellaswritingprogramstoutilizenewcomputationalmethods.Heisalsointerestedinswimming,soccer,gamesandpuzzlesofallkinds,andphilosophy. 105