Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2013-08-31.

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Title:
Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2013-08-31.
Physical Description:
Book
Language:
english
Creator:
Nakayama,Tomoyuki
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Muttalib, Khandker A
Committee Members:
Hirschfeld, Peter J
Maslov, Dmitrii
Hebard, Arthur F
Rao, Murali

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

Notes

Statement of Responsibility:
by Tomoyuki Nakayama.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Muttalib, Khandker A.
Electronic Access:
INACCESSIBLE UNTIL 2013-08-31

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Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0043370:00001


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First,Iwouldliketothankmyadvisor,Prof.KhandkerMuttalib,whohasspentextendedamountoftimetohelpmeonmyworktowardPh.D.Discussionwithhimhasgivenmeplentifulofphysicalinsights.Iwouldalsoliketothankmycollaborators,Prof.PeterWoleandPavelOstrovskii.Myreserchisbasedontheirrecentstudies,henceitwasimpossibletodothisworkwithoutthem.Theyalsogavemeusefuladviceanddirections.Ireallythankmysupervisorycommitteemembers,Prof.PeterHirschfeld,Prof.DmitriiMaslov,Prof.ArthurHebardandProf.MuraliRaoforspendingtheirvaluabletimeinreviewingmydissertationandhearingmyqualifyingexamandnaldefense.MythankalsogoestoHridisPalandChungweiWangforusefuldiscussiononmanybodytheoryandquantrumtransport.Finally,IwouldliketothankthedepartmentofphysicsatUniverstiyofFloridaandInstituteofCondensedMatterTheoryatKarlsruheInstituteofTechnology,Germanyforprovidingmewithagreatenvironmenttoresearch. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1ANDERSONLOCALIZATION ............................ 12 1.1Introduction ................................... 12 1.2Anderson'sOriginalModel ........................... 13 1.3MinimumConductivity ............................. 14 1.4ScalingTheoryofLocalization ........................ 15 1.4.11-ParameterScalingTheory ...................... 15 1.4.2ScalingforFrequency-DependentConductivity ........... 20 1.5PerturbationTheoryofLocalization ...................... 21 1.6Non-LinearSigmaModel ........................... 25 2UNITARYENSEMBLESIN2DIMENSIONS .................... 27 2.1Motivation .................................... 27 2.2DiffusonPropagator .............................. 27 2.3OverallProcedure ............................... 28 2.4CalculationofUnitarySetofDiagrams .................... 29 2.4.14-VertexHikamiBox .......................... 29 2.4.26-VertexHikamiBox .......................... 31 2.5CorrectiontoConductivity ........................... 35 2.6RestorationofGaugeInvariance ....................... 36 2.6.1GeneratingFunctions ......................... 36 2.6.2DiagramswithSingleDiffuson ..................... 36 2.7SecondOrderCorrectiontoConductivity ................... 39 2.8Discussion ................................... 40 3UNITARYENSEMBLEIN3DIMENSIONS ..................... 41 3.1OverallProcedure ............................... 41 3.2DiffusonPropagator .............................. 41 3.3CalculationofUnitarySetofDiagrams .................... 42 3.3.1CoordinateTransformation ....................... 42 3.3.24-VertexVectorHikamiBox ...................... 44 3.3.36-VertexScalarHikamiBox ...................... 44 3.3.42-VertexHikamiBox .......................... 46 5

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......................... 47 3.5HigherOrderDiagrams ............................ 48 4FREQUENCYDEPENDENTCONDUCTIVITY .................. 54 4.1OverallProcedure ............................... 54 4.2DiffusonPropagator .............................. 55 4.3CalculationofUnitarySetofDiagrams .................... 56 4.3.14-VertexVectorHikamiBox ...................... 56 4.3.26-VertexScalarHikamiBox ...................... 56 4.3.32-VertexHikamiBox .......................... 58 4.4Reevaluationofh6aandh2a 59 4.5CorrectiontotheConductivity ......................... 60 4.6Calculationof-function ............................ 63 4.7Discussion ................................... 64 APPENDIX AORTHOGONALENSEMBLEIN2DIMENSIONS ................. 66 A.1OverallProcedure ............................... 66 A.2DiffusonandCooperonPropagators ..................... 66 A.3CalculaitonofOrthogonalSetofDiagmrams ................ 67 A.3.1HikamiA ................................. 69 A.3.2HikamiB ................................. 70 A.3.3HikamiC ................................. 72 A.3.4HikamiD ................................. 73 A.3.5HikimiE ................................. 75 A.3.6HikamiF ................................. 77 A.4DiagramsRestoringGaugeSymmetry .................... 81 A.4.1AdditionalDiagramG .......................... 81 A.4.2AdditionalDiagramH .......................... 82 A.5SummationofAllDiagrams .......................... 82 A.5.1PowerLawDivergence ......................... 82 A.5.2LogarithmicDivergence ........................ 84 A.6Discussion ................................... 85 BUNITARYENSEMBLEINDIFFUSIVEREGIME .................. 87 B.1OverallProcedure ............................... 87 B.2CalculationofDiagramsatUnitarySymmetry ................ 88 B.2.1HikamiDiagramA ........................... 88 B.2.2HikamiDiagramD ........................... 89 B.2.3HikamiDiagramG ........................... 90 B.3SummationofDiagrams ............................ 91 REFERENCES ....................................... 92 6

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................................ 94 7

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Table page 1-1Threesymmetryclasses. .............................. 25 1-2Valuesofcriticalexponent. ............................. 26 8

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Figure page 1-1Graphof(g)versuslng. ............................. 17 1-2Maximallycrosseddiagrams.Thedottedlinesshowimpurityscatteringandthewavylineisthecooperonpropagator. ..................... 21 1-3Cooperondiagram.Thewavylineisthecooperonpropagator. ......... 22 1-4Coherentbackscatteringprocess. ......................... 23 2-1Diffusonpropagator.Thedottedlinesshowimpurityscatteringandthesolidlineisdiffusonpropagator. .............................. 28 2-23-diffusondiagramand4-vertexvectorHikamibox.Theboxontherightonlycorrespondstothenumberedpartofthediagramontheleft. .......... 29 2-32-diffusondiagramand6-vertexscalarHikamibox. ................ 30 2-4Diagramh4a. ..................................... 30 2-5Diagramh4b. ..................................... 30 2-6Diagramh6a. ..................................... 31 2-7Diagramh6b. ..................................... 32 2-8Diagramh6c. ..................................... 32 2-9Diagramh6d. ..................................... 32 2-10Diagramh6e. ..................................... 33 2-11Diagramh6f. ..................................... 33 2-12Selfenergydiagramandgeneratedthreediagrams,h6,a,h2,aandh2,a0. .... 37 2-13Diagramh2a. ..................................... 37 2-14Diagramh2b. ..................................... 37 2-15Diagramh2c. ..................................... 38 3-1Coordinatetransformation.Wealignz-axiswithQ. ................ 42 3-2Twodiffusondiagram. ................................ 48 3-3Four-diffusondiagram.Onlyonediffusoncarriessmallmomentum.Othersareballsitic. ...................................... 49 4-1Diagramh6a.Thisdiagramneedsmoreaccurateevaluation. .......... 54 9

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........... 55 4-3Diagramh6a,reevaluated. .............................. 59 4-4Diagramh2a,reevaluated. .............................. 60 4-5versuslng.Thesolidlineisthe-functionwithoutadditionaltermA,andthedottedlineisthe-functionwithA.NoteaddingAdoesnotchangetheslopeatg=gc. .................................... 65 A-1HikamiDiagramA.Thisdiagramisthesameash4.Itcontainsthreediffusons. 67 A-2HikamiDiagramB.ThisdiagramconsistsoftwovectorHikamiBox,twocooperonandonediffuson. .................................. 67 A-3HikamiDiagramC.Thisdiagramcontainstwocooperonsandonediffuson. .. 68 A-4HikamiDiagramD.Thisdiagramisthesameash6.Itcontainstwodiffusons. 68 A-5HikamiDiagramE.Thisdiagramcontainstwocooperons. ............ 68 A-6HikamiF.Thisdiagramcontainsonecooperonandonediffuson. ........ 68 A-7DiagramD0instandardnotion.Solidlinesarediffusons. ............ 83 A-8DiagramE0instandardnotaion.Wavylinesexpresscooperons. ........ 83 A-9DiagramF0instandardnotation. .......................... 84 A-10DiagramF00instandardnotation. .......................... 84 A-11DiagramG1instandardnotation. .......................... 84 A-12DiagramH1instandardnotation. ......................... 84 10

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1 ],whichassumesthatelectronsundergoscatteringfromimpuritieswithaconstantprobabilityofdt=duringthetimeintervaldt,whichleadstotheequationofmotion: 1!.(1) 12

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2 ].HismodelassumeshypercubiclatticeinwhicheachofthelatticesitehasrandomenergyEi.TherandomnessofimpurityisintroducedasrandomnessofEioneachsite.TheHamiltonianofthismodelisgivenby Notethatt!1limitcorrespondstos!0limit.Assumingai(0)=0i,thepreviousequationis 13

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sE0+Xi6=jV0jf0(s) Hereweintroduceanewvariable: sE0Vc(0). HencetheproblemhasbeenreducedtoexaminethebehaviorofVcinthelimitofs!0.Tosimplifythecomputation,AndersonassumedconstanttransitionenergyVij=VbetweennearestneighborsitesandintroducedasimpleenergydistributionP(E): Andersonsolvedthisequationandconcludedthatthereexistacriticalvaule(W=V)0andifW=Vexceedsthecriticalvalue,theelectronislocalized. 3 ].DrudeconductivityisslightlymodiedbytheBoltzmannequationas 14

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v=~k v. Inthreedimensions,thedensityofelectronsisgivenbyn=k3F wherel=vFisthemeanfreepath.Mottsuggestedthatincreasingthestrengthofdisorderdecreasesthemeanfreepath,butitcannotbesmallerthanlatticespacea(Ioffe-Regelcondition)andtheconductivityreachesaminimumvalue.CurrentlythisideaisconsideredincorrectandtheAndersontransitionisconsideredtobecontinuous.Thisideaissupportedbythescalingtheoryandalsoconrmedexperimantally[ 4 ]. 1.4.11-ParameterScalingTheoryTheapplicationofAndersonmodeltoanactualphysicalsystemwasnoteasy,forrealsystemsalwayshavenitesizewherewavefunctionsreachfromoneedgetotheother.Oneofthemethodtodeterminethelocalizationofelectronsistoexaminehowthetransportpropertyofanitesystemchangeswhenthesystemsizeischanged.Thismethodiscalledscaling.ScalingtheoryoflocalizationwasrstproposedbyAbraham,Anderson,LicciardelloandRamakrishnananditgreatlypromotedthestudyonthetheoryofAndersonlocalization[ 5 ][ 6 ].Weconsiderd-dimensionalhypercubeoflengthLanditsconductanceG.SinceconductanceisproportionaltothecrosssectionLd1andinverselyproportionaltothelengthL,conductanceandconductivityarerelatedas 15

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Figure1-1. Graphof(g)versuslng. startinganyvalueofconductivity,condutivitydecreasesasthesystemsizeisenlargedandthesystemalwaysbecomesaninsulatorinthethermodynamiclimit.Experimentsonthinlmsandwiresalsosupportthisprediction[ 6 ].Ontheotherhand,forathreedimensionalsystem,the-functioncanbeeitherpositiveornegativedependingontheinitialvalueofconductivity.Thissuggeststheexistenceofphasetransitionatg=gc.Since(g)issmallwhengislarge,wecanexpandthe-functioninpowersofg.Uptothelinearorderwehave g. 17

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L0. Thustheconductivitydecreaseslogarithmicallyasthesystemsizeincreases.Themetal-inslatortransitionatg=gcindimensionsd>2ischaracterizedbythedivergenceofcorrelationlength.Thecorrelationlengthnearthecriticalpointcanbeexpressedintermsofstrengthofdisorder,anditshowspowerlawbehaviorwithsomeuniversalcriticalexponent: c Inthemetalicside,sincethedimensionlessconductanceiswrittenasg(L)/L c(d2). Therefore,theconductivityexponent(d2)isthesameascriticalexponentforthreedimensions[ 7 ].Theconducitivityexponentorcriticalexponentisestimatedbytheslopeofthe-functionatcriticalpoint.Toobtainthebehaviorofconductivityinthethermodynamiclimit,westartfromanitesystemsizeL0whichcorrespondstog0,andintegratethe-function.Weintroducethelinearapproximationandexpressthe-functionasfollows: gcforg
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Startingwithg0
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Thephasebreakingtimedependsontemperature: wherepisapositiveinterger.Hencetheconductanceincreaseswithtemperature. 11 ].Sincetheimportantlengthscaleexceptforthesystemsizeisthecorrelationlengthonly,theoneparameterscalingfunctioncanberewrittenas Fornitefrequencies,therecomesinanothercharacteristiclengthL!,whichisthedistanceanelectrondiffusesduringonecycleofappliedelectriceld.Thescalinghypothesisforfrequencydependentconductanceis 20

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InthesmallfrequencylimitL!>>,wehavea!dependentcorrectiontoDCconductivity.Consideringtheknownresultthatthecorrectionisproportionalto!d2 13 ],ShapiroandAbrahamsconcludedthatthescalingfunctionis 14 ].Ifyouexpandtwo-particleGreen'sfucntionsintheperturbationduetoimpurity,thenitturnsouttheweaklocalizationcorrectiontotheconductivityisobtainedfromthesumofthemaximallycrosseddiagrams,whichiscalledCooperondiagram.DuetotheKuboformula,thecorrection Figure1-2. Maximallycrosseddiagrams.Thedottedlinesshowimpurityscatteringandthewavylineisthecooperonpropagator. 21

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Cooperondiagram.Thewavylineisthecooperonpropagator. totheDCconductivityduetothisdiagramisgivenby whereC(q)isCooperonpropagatorandGareretardedandadvancedGreen'sfunctions: 2N021 2 l(2D) (1) Asaresult,the-functionsare (1) 22

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2N01 andtheresultantcorrectionsin2Dand3Dare (1) 2p TheprocessexpressedbytheCooperondiagramiscoherentbackscattering.Althoughthepotentialduetoimpuritiesisrandominspace,thesystemstillhastime-reversalsymmetry.Thismeansthatthetransitionfromktok0hasthesamematrixelementasthetimereversalprocessk0!k,namely Figure1-4. Coherentbackscatteringprocess. symmetry,thiscorrectiondoesnotexistifthesystemdoesnothavethetime-reversalsymmetry.Forexample,ifanexternalmagneticeldBisappliedtothesystem,thenthetime-reversalsymmetryaswellaslocalizationcorrectiondisappear.Supposeauniform 23

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where(x)isanarbitraryscalarfunction.Underthistransformation,thewavefunction(x)getsanextraphasefactor: Ifyouchoosethescalarfunctionas thenwecanremovethevectorpotentialandtheeffectofthemagneticeldwillbeincorporatedintotheoryasthephasefactor.Sincetheabovepathintegralcorrespondstothemagneticuxpassingthroughthesurfacesurroundedbythepath,ithastheorderofBr2withrbeingtheradiusofthepathandthephasefactoriseBr2=~.Thelocalizationeffectdisappearswhenthephasefactorbecomestheorderof1.ThusinthiscasethephasebreakinglengthLmisapproximately Ifthisphasebreakinglengthissmallerthanthesystemsize,weneedtoreplaceLwithLmintheexpressionfortheconductance.Wehave 22lneBl2 Theconductivityincreaseslogarithmicallyasthemagneticeldincreasesandthisincreasebreaksdownthelocalization. 24

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12 ][ 15 ]. Table1-1. Threesymmetryclasses. SymmetryClass TimeReversal SpinRotation Unitary No Yes/No Orthogonal Yes Yes Symplectic Yes No Fororthogonalsymmetry,the-functionis 44g4+O(1=g5), wheretheresultisexpressedintheunitofe2=handforasinglespin.Forunitarysymmetrythe-functionisgivenby 22g23 84g4+O(1=g6).(1)Hencefortheunitaryensemble,therstcorrectionis1=g2and1=gismissing.Thisleadstothecriticalexponentofconductivityofs=1=2,whichismuchsmallerthanthevalueofs1.4obtainedinnumericalstudies[ 16 ].Thuswewonderifthe1=gtermappearsinthreedimensionalsystems. Forsymplecticsymmetry,-fucntionis 44g4+O(1=g5). 25

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Valuesofcriticalexponent. SymmetryClass Theory Numerical Experiment Unitary 0.5 1.430.04[ Orthogonal 1 1.570.02[ 18 ],0.80.9[ 1 1.370.02[ Unlikethecaseinorthogonalandunitarysymmetry,theleadingcorrectionispositive.Weshouldnotethattheseesultsduetonon-linearsigmamodelwereobtainedby2+expansion,where<<1,andcurrentlycalculationsfor3Darenotavailable.Thismotivatedustocalculate3D-functionatunitarysymmetryinadifferentway. 26

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23 ].However,P.Ostrovskii[ 24 ]showedthatincorporatingthecontributionfromlargermomentum(ballisticregime),thecorrectresultfortwodimensionalsystemsisrestored.Inthischapter,IamgoingtoshowtheballisticregimecalculationoriginallydonebyOstrovskiiinadifferentwayanddiscusstheimportanceoftheballisticregimecontributionandthediagramsthatrestoregaugesymmetry.Inhispaper,Ostrovskiiusedcertaintricksthatcouldbeusedonlyintwodimensions.Sinceourgoalistocalculatethecorrectioninthreedimensions,hereIamgoingtodevelopamorestandarddiagrammmaticcalculation,whichcanthenbereliablyextendedtothreedimensionalsystems. 2N02411 27

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Diffusonpropagator.Thedottedlinesshowimpurityscatteringandthesolidlineisdiffusonpropagator. where 2N0 WeuseQtoexpressmomentuminballisticregimeandqtoexpressmomentumindiffusiveregime.InspiteofthefactthatthediffusonpropagatorsaredivergentwhenthechangeinmomentumQissmallanditdoesnotseemthatthecontributionfromlargerQisimportant,ifyoudonotincludethecontributionfromballisticregime1vFQ,thenwecannotretrievetheresultobtainedbynon-linearsigmamodel.Wekeeptheexpression( 2 )asnecessary. 28

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2Z1=l1=Lqdq(q)1 2Z10QdQFxx(Q), whereFxxis,inprinciple,thesumoftheunitarysetofdiagrams.OrigninallythosediagramscontainbothQandq.However,thesmallqappearsintheGreen'sfunctionsonlyyieldextratermswhichisproportionaltovFq,andthosetermsdonotleadtolnLcontribution.Wesetq=0tocalculatediagrams. 12 ][ 25 ]. Figure2-2. 3-diffusondiagramand4-vertexvectorHikamibox.Theboxontherightonlycorrespondstothenumberedpartofthediagramontheleft. 29

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2-diffusondiagramand6-vertexscalarHikamibox. The4-vertexHikamiboxeswhichyieldtwo-loopordercontributionsareasfollows: Figure2-4. Diagramh4a. Figure2-5. Diagramh4b. (1+l2Q2)3 2# 30

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2 Eachofthetwo-loopdiagramswiththreediffusonsconsistsoftwo4-vertexHikamiboxes.Insteadofcalculatingeachdiagram,weaddtheboxesrst,thentakethedotproductwithitself. (1+l2Q2)3 2# wherea=lQ.Note1(1+a2)1=2factoriscanceled. Figure2-6. Diagramh6a. (l2Q2+1)5 2 31

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Diagramh6b. Figure2-8. Diagramh6c. Figure2-9. Diagramh6d. 2N01 (l2Q2+1)2 32

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Diagramh6e. Figure2-11. Diagramh6f. 2N01 (l2Q2+1)2 2N021 (l2Q2+1)3 2 2N01 (l2Q2+1)3 2l 33

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2N01 (l2Q2+1)3 2l 2N021 (1+l2Q2)1 2"1+1 (1+l2Q2)1 2#2 2N021 (1+l2Q2)1 2"1+1 (1+l2Q2)1 2#2 2N021 (1+l2Q2)1 2"1+1 (1+l2Q2)1 2#2 Wesumupallthesix-vertexHikamiboxesandthenmultiplyitwithadiffusonpropagatorwithmomentumQ. (l2Q2+1)5 2 34

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(l2Q2+1)2 (l2Q2+1)3 2 (l2Q2+1)5 2 (1+a2)2(p whereweusedthefactthat Thefactor1(1+a2)1=2cancelednicelyagain.WiththisF1,thecorrectiontotheconductivityisgivenby 2Z1=l1=Lqdq(q)1 2Z10QdQF1,xx(Q). ForlQ>>1,FQ2k2F41 35

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(lQ)2dependenceatmost.Sinceadivergenceinashortlengthscaleisunphysical,weneedtoxthisproblem. 24 ].Inprinciple,therestorationofgaugesymmetryisequivalenttotherestorationofWardidentity,whichguaranteestheconservationofparticlenumber. 23 ].Therefore,weneedtoadddiagramsthatarederivedfromasetofselfenergydiagramsthatgenerateallthediagramscalculatedabove.TheyalsoproposedasimplerecipetovisualizetheproceduretoproduceasetofdiagramsthatsatisfyWardidentity: 1. Takeanyskeletondiagramoftheself-energy. 2. Startingfromtheright,removeoneGreen's-functionlineandipthepartontherightdowntotheholeline. 3. Continuethesameproceduretotheleft.Thegureshowsanadditionaldiagramgeneratedbyaself-energydiagramwhichalsoproducesthemostdivergentdiagramh6a. 2N01 (l2Q2+1)3 2 36

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Selfenergydiagramandgeneratedthreediagrams,h6,a,h2,aandh2,a0. Figure2-13. Diagramh2a. Figure2-14. Diagramh2b. Diagramh2a0issimilartoh2a.Theonlydifferenceisthepositionoftheimpurityline.h2a0hasanimpuritylineontheholeside.h2a 2N01 (l2Q2+1)3 2 37

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Diagramh2c. (2N0)21 (2N0)21 (l2Q2)"1+1 (l2Q2+1)1 2#2 Flippingthetwoimpuritylinestotheholeside,wegeth2c0. (2N0)21 (l2Q2)"1+1 (l2Q2+1)1 2#2 Summingupthesediagrams,weget (l2Q2+1)3 2 2. 38

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2h2=k2F42+p Thefactor2beforeFxxcomesfromtheinterchangeofQandq.Notonly1 (lQ)2dependenceiscancelledbyaddingthegaugerestoringdiagrams.TheQintegralisconvergent.Thetwo-loopcorrectiontotheconductivityis 2Z1=l1=Lqdq(q)1 2Z10QdQFxx(Q)=e2 l. WeaddittotheDrudeconductivity,thenwecalculatethedimensionlessconductance: l 1 23FlnL l=g01 24g0lnL l, where~=1.The-functionis dlnL=L gdg dL=1 24gg01 24g2 Intheunitofe2=hforasinglespin,thedimensionlessconductanceandthescalingfunctionaregivenby 22FlnL l=g01 22g0lnL l 22g2 Thisexactlymatchestheresultbynon-linearmodel. 39

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22Z1=l1=Lq2(q)dq1 22Z10Q2fxx(Q)dQ. Thesmallqintegralisgivenby Nowwehave (F)21 LZ10a2fxx(a)da. 2N0a aarctana, wherea=Ql=QvFand 2N0 Inthediffusivelimit,thisreducesto 2N01 41

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3.3.1CoordinateTransformationUnlikethetwodimensionalcase,wherethecosineoftheanglebetweentwovectorsissimplyexpressedascos(0),theinnerproductbetweentwovectorstakesmorecomplexforminthreedimensions.Tomakeourcalculationtractable,wealignourcoordinatesystemwiththemomentumcarriedbydiffusons.Inouroriginalcoordinate Figure3-1. Coordinatetransformation.Wealignz-axiswithQ. system,Qandkare 42

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Inthiscoordinatesystem, Totransformkxtothenewcoordinatesystem,rstwexz-axisandrotatexandyaxesby. Thenwexyaxisandrotatexandzaxesby. Combiningthetwotransformations,wehave Hencekxis 43

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a Summingupallthe4-vertexHikamiboxes,weget a+(arctana)2 Sinceh4,x/Qx a+arctana a2#2

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(a2+1)2+tan1a a 2N01 a1 2N021 a3 Fordiagramsfrom(d)to(f),werstcalculatehxx,addhyyandhzz,andthendividetheresultby3. 2N01 a1 (a2+1)tan1a a 2N021 a2tan1a a 2N021 a2tan1a a 45

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Thesumof4-vertexboxesand6-vertexboxeshasultravioletdivergenceasitwasintwodimension. 2N01 a+1 (2N0)21 a2 (2N0)21 a2 46

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atan1a a2#+2k2F41 a2. 3(h24(Q)2+h6a(Q))+1 3h2=2k2F4a22a42atan1a+(1+a2)2(tan1a)2 Notefxx=(1=3)fxx+yy+zz,andalsonotethath4andh6havetobedoubledduetointerchangeofQandq.Sincefxxbehavesas1=a4forthelargea,theintegralconvergesintheultravioletlimit.TheQintegralyields Therefore,thecorrectiontotheconductivityis (F)21 L. Thedimensionlessconductanceis 1631 (F)2L l1 32(F)(kFL)+1 1631 (F)2L l1. Althoughdiagramsaremoredivergentinthreedimensionsthanintwodimensions,addingtheextradiagramstakescareoftheultravioletdivergence.Theproblemhereisthatthisdimensionlessconductancedoesnotscale.Thisproblemstemsfrom 47

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Figure3-2. Twodiffusondiagram. where 2N01 1tan1a a 2N0(Ballistic) (3) 2N01 (3) 48

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Theintegralisproportionalto1=aintheballisticlimit.Since(Q)/1=(N0),F(Q)isproportionalto Weexpect4-diffusondiagramonlyyieldshigherordercontributionthan2-diffusondiagram.Wewillcomparethembothindiffusiveandballisticregimes.4-diffuson Figure3-3. Four-diffusondiagram.Onlyonediffusoncarriessmallmomentum.Othersareballsitic. diagramisgivenby where 49

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2N0p (3) Inthediffusivelimit,thisdiagramyieldsthecorrectiontotheconductivity 4e2 (F)3lnL l4. Inballisticregimeq1=q0,q2=Q1,q3=Q2andq4=Q3,wehave v, where SincewearefocusingontheballisticlimitQvF>>1,mostprobablythelargestcontributioncomesfromthetermswithno=inthenumerator.Collectingthetermn=0,weget 50

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4ZQ21dQ11 4ZQ22dQ2X(Q,Q1,Q2)X(Q,Q1,Q2)=ZdcosZdcos1Zdcos2u0 where Insteadofdoingtheangularintegralfork,wedotheangularintegralforQ1andQ2rstandthentheangularintegralforQ,whichdoesnotchangetheorderinaor.AftertheQ2integral,wepickoutoneofthetermsthatissupposedtoyieldthelargestcontribution.ConsideringthatwearefocusingonballisticlimitvFQ>>1,wechooseatermthatcontainsthegreatestpowerofandtheleastpowerof1=(vFQ=)(Thesumofthesepowersisconstantforalltheterms.).Wechoose IntegratingovertheangleofQ1,weagainsingleoutoneofthelargestterms.Weget 51

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Multiplying3andintegratingwithrespecttoQ1andQ2,weget Therefore,thisdiagramissmallerthanthe2-diffusondiagrambyafactorof1=(vFkF).Asarandomcheck,wepickupatermwhichcontainsonelesspowersofaftertheangularintegralofQ2.Onesuchtermis (vFQ1=)2(vFQ12=)(vFQ1) (vF(QQ1))2(vFQ1=)(vFQ2+vF(QQ1)=). Afterthe1integral,wepick Aftertheintegral,wechoose Multiplyingthistermby3andintegratingwithrespecttoQ1andQ2,theresultisproportionalto Asexpected,thisissmallerthanthelargestcontributionbyafactorofvFkF.Besides,wecanexpectthatthetermsthataredivergentintheultravioletlimitallcancelfor 52

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53

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22Z1=l0q2(q,!)dq1 22Z10Q2fxx(Q,!)dQ, whereweassumed!>0andthesummationoverMatsubarafrequencywastakenonlyintherange!<<0.OtherwisefxxiszerobecausethepolesoftheGreen'sfunctionsareinahalf-planeonthesameside.Aswewillseelater,thiscalculationyieldsthep Figure4-1. Diagramh6a.Thisdiagramneedsmoreaccurateevaluation. 54

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Diagramh2a.Thisdiagramneedsmoreaccurateevaluation. Therefore,wesubtractthesetwodiagramsfromourcalculation,evaluatethemmoreacuuratelyandaddthereevaluatedtwodiagrams.ACconductivityisnowgivenby 22Z1=l0q2(q)dq1 22Z1qQ2fxx(Q,!)dQZ1qQ2fxx(Q,!)dQ+Z1qQ2~fxx(Q,!)dQ, wherefand~farethesumofh6aandh2abeforeandafterreevaluation. 2N0a aarctana b, wherea=Ql=QvF,b=1+!n=1!and 2N0 b. TheGreen'sfunctionsaredenedas 55

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2N03b3 Forsmallfrequencies,wehave 2N03 2N0a2+3! 4.3.14-VertexVectorHikamiBox4-vertexHikamiboxesareapartof3-diffusondiagramsasshowninchapter2.Wecalculatethex-componentofthevectorHikamiboxes. a2+b2 2N01 b1b aarctana b 2N01 whereT13=coscos=Qx=Q.Sinceh4,xisproportionaldirectioncosineofQandthesystemisisotropic,wecalculate(hx)2as(h)2=3. b+b aarctan2a b2 56

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b 2N01 ba a2+b2+2 b 2N01 aarctana bb b 2N01 2N021 bi3 2N01 b1b aarctana b2 57

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2N021 bh1a barctana bi2 b(a2+b2)2(2a2+b2)arctan2a b+a(a2+b2)2arctan3a b. 2N01 a2+b2+1 b (2N0)21 bi2 (2N0)21 aarctana b2 58

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a2+b22(a2+b) b+(2a2+b2) b. 2N0h6a,xx+yy+zz=1 2N01 b 2N01 a2+b2+1 b. Thesediagramswillbesubtractedfromourcalculation.Insteadweevaluatethesediagramsmoreaccurately.Thesereevaluateddiagramscontaintwoindependent Figure4-3. Diagramh6a,reevaluated. momentakandk1carriedbytheGreen'sfunctions.Ifwesetq=0,theneachofthemiszero.However,herewekeepqtogettheaccurateresults.Weobtain 2N01 aarctana b2 59

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Diagramh2a,reevaluated. 2N01 aarctana b2, wherea=qlandT13=qx=q.Thesumofthesediagramsis 22Z1=l0q2(q)dq1 22Z1qQ2fxx(Q,!)dQZ1qQ2fxx(Q,!)dQ. 3(h242(Q,!)+h6(Q,!))+1 3h2=+2k2F4 b(a2+b2)2arctan2a b b2. Notefxx=(1=3)fxx+yy+zz,andalsonotethatdiagramsh4andh6havetobedoubledduetointerchangeofQandq.Sincefxxbehavesas1=a4forthelargea,theintegral 60

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b b1ql=+ b b. Inthelastexpressiona=ql,notQl.fxxiscalculatedas 3h6a+2 3h2a=+4k2F4 (a2+b2)2. Integratingwithrespecttoa=Ql,weget b1ql=k2F4 b. Inthelastexpression,a=ql.Notethattheultravioletlimitcontributionfromfxxandfxxcancel.Nowtheconductivityisexpressedas b b 3ba b. First,weevaluatethefrequencydependenceofI(q,!).Forsmallqand!,Ireducesto 32a3 wherec=!>0.Fortheqintegral,theintegrandis 61

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3c 2(1+9c2)+27c2(1+3c2) 4(1+9c2)2ln1+9c2 Inthelaststep,weexpandedforsmall!andkepttheleading!dependence.Theimaginarypartyields 3c+ln1+9c2 Therefore,thefrequencydependenceoftherealpartislinearin!,whereastheimaginaryparthasastrongerdependence!ln!.NowwewillevaluateI(q,!)Forsmallaandc=!, a2+1 Therealpartandimaginarypartsyield 2N0c2lnc 62

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D2[!ln(!)!A], shereweexpressedthesub-leadingtermfortheimaginarypartasA.Inthescalingregime,wedenethefrequencydependentlengthscaleas Thediffusioncoefcientisrelatedtothecorrelationlengthas Therefore,wehave Thedimensionlessconductanceis AddingDrudeconductance!=,weget 63

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dL!=1 g2lng+2C g21A Atthecriticalpointg=gc,the-functioniszero.Hencewehave Atthispoint,theslopeofthe-functionis g3[2lng3+A](4) g3.(4)Thecriticalexponentis 22C g2c>1 2.(4) 4 ),wehaveaddedbyhandasub-leading(non-singular)correctiontermproportionalto!,whichwewerenotabletoobtain.Suchatermisrequiredinordertoshiftgctolowervalueswhichallowsonetohaveazeroofthebetafunctionbeforeitreachesaminimum.However,ourapproximationsarevalidtoleading!ln!termonlyandweassumethatamoreaccuratecalculationmustyieldtheappropriatelinear!term.Thereisanothertermmissing,namelyai!ln(3),whichisrequiredtoreplacetheln(!)byln(2=L2!).Thisisanothernon-singulartermwhichshouldariseinamoreaccuratecalculation.Essentiallythepresenceofthesetermsarerequiredbythescalingansatz.However,furtherworkisneededinordertoactuallyevaluatethem.OntheotherhandasshowninEq.( 4 ),theconductivityexponentisindependentofanyof 64

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thesecorrectionterms.WecanthereforeclaimthattheconductivityexponentisindeedgivenbyEq.( 4 ),( 4 ),andthatithasacorrectiontermwhichmakesitlargerthan1/2.Notethatthenon-linearsigmamodelresultobtainedin2+dimensionsandextrapolatedto=1is1/2,whilethenumericalsimulationsgiveavaluecloseto1.3.Reevaluationofh6aandh2awascrucial.Itsuccessfullyremovedtheconstanttermfromtheultravioletlimit.Ifwehadthisconstant,thentheconductivitywouldbeproportionalto 2N0"1(1)p 2p Thiscorrectionwouldleadtotheviolationofthescalinghypothesis. 65

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(F) l1 (F)1 4k2F4lnL lZ10af(a)da, where(q)representsadiffusonorcooperonpropagatorinthediffusivelimit.Wecalculatesummationoff(Q)forallthediagramsandseeifwegetacancellation.Notethataccordingtothenon-linearsigmamodel,thecorrectiontothe-functioninthetwo-looporderiszero. 2N0p 2N01 wherea=lQ=vFQ.Sometimesitismoreusefultoaddasinglescatteringlinetothecooperonpropagator,wedeneourextendedcooperonoperatoras 2N01 2N0=1 2N0p 66

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Intheballisticlimit,theybehaveas 2N01+1 2N01 Notetheyhave1=aterms.Therefore,if6-vertexhikamiboxhas1=abehavior,thenithasboth1=aand1=a2termsaftermultiplyingorC. 12 ].The4-vertexHikamiboxesand6-vertexHikamiboxesintheguresaremodiedwithoneimpuritylineortwo. FigureA-1. HikamiDiagramA.Thisdiagramisthesameash4.Itcontainsthreediffusons. FigureA-2. HikamiDiagramB.ThisdiagramconsistsoftwovectorHikamiBox,twocooperonandonediffuson. 67

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HikamiDiagramC.Thisdiagramcontainstwocooperonsandonediffuson. FigureA-4. HikamiDiagramD.Thisdiagramisthesameash6.Itcontainstwodiffusons. FigureA-5. HikamiDiagramE.Thisdiagramcontainstwocooperons. FigureA-6. HikamiF.Thisdiagramcontainsonecooperonandonediffuson. 68

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(1+a2)3 2# 2N01 1+a2# 2N01 SummingupalltheHikamiAdiagrams,weobtain (1+a2)3 2#, wherecos=Qx=Q.Sincethesystemisisotropic,Ayisproportionaltosin=Qy=Q.Wesquare~Aandmultiplywith2toget Weneedtodoublethisresultbecausewehavetoconsiderthecase1(Q),2(Q+q)2(Q)and3(q)also. 69

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(a2+1)3=2 2N01 2N01 1+a2# (a2+1)3=2 2N01 1+a2# 2N01 70

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(1+a2)3=2# (1+a2)3=2#. TakingthedotproductandmultiplyingwithC(Q)2,weobtain Next,weassumeC1(q),C2(q+Q)C2(Q)and(Q). (a2+1)3=2 2N01 1+a2# 2N01 2N01 1+a2# 71

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2N01 1+a2# Summingupthetwo4-vertexboxesseparately,weobtain (1+a2)3=2# Thus~L0~R0iszero.UsingC+asacooperonpropagator,HikamiBisnegativeofHikamiA.NoteweneedtodoubleBalsobecauseweneedtoconsiderdiagramswhichareobtainedbyreplacingthepositionsofandC2. 2N01 2N01 72

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2N01 1+a2 2N01 1+a2 Summinguptheoutsideboxandinsideboxseparately,weget WemultiplythetwoboxesandC+2toobtain (a2+1)5 2 73

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2N01 (a2+1)2 2N01 (a2+1)2# 2N01 2N021 (a2+1)3 2 2N021 2"1+1 (1+a2)1 2#2 2N021 74

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(2N0)21 2N021 2"1+1 (1+a2)1 2#2 2N021 Summingupallthesix-vertexHikamiBoxes,wehave (a2+1)5 2 (a2+1)2+4N0k2F5"p (a2+1)2# (a2+1)3 2. (a2+1)3=2 75

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2N01 (a2+1)2 2N01 2N01 (a2+1)3=211 2N0)21 (1+a2)3=2 2N0)21 2N0)21 2N0)21 76

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2N0)21 Summingthemup,weget (a2+1)3=2 (a2+1)24N0k2F51 (a2+1)3=211 (1+a2)3=2+4N0k2F51 (a2+1)5=2 2N01 (a2+1)2 2N01 (a2+1)2# 77

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2N01 2N0)21 2N0)21 2N0)21 2N0)21 2N0)21 Summingthemup,weget (a2+1)5=2 78

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(a2+1)24N0k2F5"p (a2+1)2# Nowweassume(Q)andC(q). (a2+1)3=2 2N01 (a2+1)2 2N01 2N01 (a2+1)2# 2N0)21 79

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2N0)21 2N0)21 2N0)21 2N0)21 Summingupthediagrams,weobtain (a2+1)3=2 (a2+1)2+4N0k2F5"p (a2+1)2# 80

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2N01 (a2+1)3 2 (2N0)21 (2N0)21 (2N0)21 Summingthemup,weget 2 81

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2N01 (a2+1)3=2 (2N0)21 (2N0)21 Summingthemup,wehave A.5.1PowerLawDivergenceWeclassifydiagramsbythenumberofimpurityscatteringlines,forallthediagramswhichcausespowerlawdivergencealwayscontaintwoimpuritylines,whereasdiagramswhichcontainthreeormoreimpuritiesonlyhavelogarithmicdivergenceatmost.Notethatdiffusonandextendedcooperonpropagatorcontainoneimpurity 82

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Theydonotcancelandthiscausesultravioletdivergence.Ontheotherhand,ifweassumecooperonpropagatorscontainoneimpurityline,thensixdiagramscontaindiagramswithtwoimpuritylines.Thesumofthemis Thepowerlowdivergenceisproperlycancelledfororthogonalensembles.ThissuggestusthatweneedtousetheextendedcooperonpropagatorC+insteadofC.NotethatwedonotneedtoaddextradiagramsGandHtogetthiscancellation. FigureA-7. DiagramD0instandardnotion.Solidlinesarediffusons. FigureA-8. DiagramE0instandardnotaion.Wavylinesexpresscooperons. 83

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DiagramF0instandardnotation. FigureA-10. DiagramF00instandardnotation. FigureA-11. DiagramG1instandardnotation. FigureA-12. DiagramH1instandardnotation. 84

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ThisneglectedQ2termmakesthediagramlessconvergentintheultravioletlimit.Thusthelogarithmicdivergenceisexpectedtodisappear.Accordingtothenon-linearsigmamodel,two-loopcorrectiontothebeta-functionintwodimensionsmustbezero.Fromthisacceptedresult,weexpectthatourtwo-loopcorrectiontotheconductivitytakestheformof 20lnL0!, where andD0isthebarediffusoncoefcient[ 26 ].Uptotwo-looporder,theconductivityiswrittenas 20lnL0!. Tocomparewiththeresultofthenon-linearsigmamodel,wealsoneedtoconsiderthecorrectiontothebarediffusoncoefcientasfollows: 85

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NowwerewritetheconductivityintermsofL!.TherelationbetweenL!andL0!is 2ln11 20lnL0!. Therefore,theconductivityisexpressedas Hencethereisnocorrectionatthetwo-looplevel. 86

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(2l)dZ1l=Lad1(a)da1 (2l)dZ1l=LAd1Fxx(a,A)dA=e2 wherea=qlandA=Ql.Inballisticregimecalculation,thereisacleardistinctionbetweentherolesofqandQ.Thereisnodifferencebetweenthemindiffusiveregime.WeseparateFxx(a)intothreeparts,onefromeachdiagram. whereeachtermcontainsangularintegral: Weonlyconsiderdiffusiveregime.Hencethediffusonpropagatorsare 2N0d a2. 87

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Nowweneedtoconsiderthefollowingangularintegral: Weobtain 88

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Intwodimensions,ifyouintegrateFxxwithrespecttoA,ityieldslogarithmicterm.Sinceintegralwithrespecttoaalsoyieldslog.ThistermleadstolnL l2. l2 89

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Aftertheangularintegrals,weobtain In2D,thecorrectiontotheconductivityduetothesediagramsis l+1 21l2 l. ThersttermcancelwithA,butthesecondtermispositive.Wewilladdextradiagramsandcheckifithelps. 90

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Thisyieldsthecorrectiontotheconductivity 21l2 31l3 l+21l2 l=e2 2+7 3lnL l. 3lnL l+1 2. Thisistheresultfor2dimensions.NotethatwedidnotdoublediagramsAandDbecauseqandQhavethesameroleandwedidnotneedtoexchangethem.OnlyconsideringdiagramsAandG,weobtain l. Eitherway,theydonotmatchtheconductivityobtainedinChapter2,whichyieldsthecorrect-function.Indiffusivecalculation,twomomentumscarriedbydiffusonaresymmetric.Whereasthediagramsaddedtorestoregaugesymmetrydonothavethesymmetry.Wecanalsoseetheimportanceofthegaugerestoringdiagramshere,forityieldstherightsignoflnL l.Still,ifwedonotconsiderthecontributionintheballisticregime,wecannotretrievetheresultobtainedbynon-linearsigmamodel. 91

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D.VollhardtandP.Wole,Phys.Rev.B22,4666(1980). [24] P.Ostrovskii,unpublished. [25] S.Hikami,inAndersonLocalization,EDs.H.NagaokaandH.Fukuyama,Springer,Berlin(1982). [26] P.Ostrovskii,privatecommunications. 93

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TomoyukiNakayamawasborninOita,Japanin1977.Aftergettinghisbachelor'sdegreeinphysicsatKobeUniversity,hemovedtotheUnitedStatesin2005andstartedhisgraduatestudiesinphysicsattheUniversityofFlorida.HebeganworkingincondensedmattertheorywithProf.Muttalibin2006andcompletedhisPh.D.programin2011. 94